classical tachyons and possible applications docs... · 13"4. classical vacuum instabilities....

RIVISTA I)EL NIJOVO CIMF, I~TO VOL. 9, N. 6 1986

Classical Tachyons and Possible Applications (*).

F~. ]%ECA.MI

Dipar t imenlo di lr dell' Universi t~ Statale - Catania, I t a l i a Istitceto Nazionale di .Fisiea N ~ l e a r e - Sezione di Catania, I t a l i a

(ricevuto il 26 Aprile 1985)

Q uone vide.q cilium de2mre et longius ire ~lfultlplexque loci spati'~m h'anscurrere eodem Tempore quo Solis pervolgant l u m i n a coeIum? ,> (**).

LUCRETIa'S (50 ]~.C., ca . )

. . . . . . . . . . . . . . . . . . . . . . . . . should be thoughts, Wh ic h ten times /aster glide than the S u n ' s beams Driving back shadows over low'ring hills. *

SSAKESPEARE (1597)

4 4 4 5 5 6 6 7 9 10 lI 11 11 13 15 16 16 17 21

21 2'2 22 23

1. Introduction. 1" 1. Foreword. 1"2. Plan of the review. 1"3. Previous reviews. 1"4. Lists of references. Meetings. Books.

PAISa: i . - Part icles and antiparticles in special re la t iv i ty (SR). 2. Special re la t iv i ty with ortho- and anti-chronous Lorcntz tr~msformations.

2"1. The Stiickelberg-Feynman (( switching principle )) in SI2. 2"2. Matter and an t imat te r from SR. 2"3. Fur ther remarks.

l 'A l~ II . - Br~dyons and tachyons in SI~. 3. Historical remarks and preliminaries.

3"1. II istorieal rem~rks. 3"2. Preliminaries t~bout tachyons

4. The postulates of SR revisited. 4"1. Existence of an invariaut speed. 4"2. The problem of Lorentz transformations. 4"3. 0rthogonal and autiorthogonal transformations: digression.

5. A modo l theo ry for tachyons: an <~extended rclativity~) (ER) in two dimensions. 5"1. A duali ty principle. 5"2. Sub- and Super-luminal Lorentz transformations: preliminaries. 5"3. Energy-momentum space. 5"4. Generalized Lorentz transformations (GLT): preliminaries.

(*) Work par t ia l ly supported by C.N.R. and M.P.I. (**) <~ Don ' t you see that they must go ]aster and farther/And travel a larger interval of space in the same amount of/Time than the S u n ' s light as it spreads across the sky? ~>

9. ~E. 1"r CASI 1

24 25

28 28 29 31 31 32 33 34 35 36 37 38 39 39 40 41 42 43 44 45 45 46 47 48 48 49

50 52 54

54 54 55 55 56 59 60 62 64 64 64 65 66 66 66 67 68 68 68 68 69 70 71 71 71 71 72 72 74 76 77

5"5. 5"6.

5"7. 5"8. 5"9. 5"10. 5"i1. 5"12. 5"13. 5"14. 5"15. 5"16. 5"17.

The fundamental theorem of (bidimensional) ER. Explicit form of the Superluminal Lorentz transformations (SLT) in two dimensions. Explicit form of GLTs The GLTs by discrete scale transformations. The GLTs in the (dight-cone co-ordinatcs~. A~tomatic interpretation. An application. Dual frames (or objects). The (~ switching principle )~ for taehyons. Sources and detectors. Causality. Bradyons and tachyons. Particles and antiparticles. Total ly inverted frames. About CPT. Laws and descriptions. Interactions and objects.

5"18. SR with taehyons in two dimensions. 6. Tachyons in four dimensions: results independent of the existence of SLTs.

6"1. Caveats. 6"2. On taehyon kinematics. 6"3. , Intr insic emission )~ of a tachyon. 6"4. Warnings. 6"5. (~ Intr insic absorption ~) of a tachyon. 6"6. Remarks. 6"7. A prel iminary application. 6"8. Tachyon exchange when u" V< c 2. Case of ((intrinsic emission)) at A. 6"9. The ease of (~intrinsic absorption)> at A (when u.V<c2). 6"10. Taehyon exchange when u - ~ ' ~ c 2. Case of ((intrinsic emission)) at A. 6"11. The ease of ((1 " "ntrmsic absorption)) at A (when u.V>~c2). 6"12. Conclusion about the tachyon exchange. 6"13. Applications to elementary-particle physics: examples. Tachyons

as (~ internal lines ~). 6"14. On the variat ional principle: a tentat ive digression. 6"15. On radiat ing tachyons.

7. Four-dimensional results independent of the explicit form of the SLTs: introduction. 7"1. A prel iminary assumption. 7"2. G-vectors and G-tcnsors.

8. On the shape of tachyons. 8" 1. Introduction. 8"2. How would tachyons look like? 8"3. Critical comments on the preliminary assumption. 8"4. On the space extension of taehyons. 8"5. Comments.

9. The causality problem. 9"1. Solution of the Tolman-Rcgge paradox.

9"1.1. The paradox. 9"1.2. The solution. 9"1.3. The moral.

9"2. Solution of the Pirani paradox. 9"2.1. The paradox. 9"2.2. The solution. 9"2.3. Comments. 9"2.4. (( Strong version )> and its solution.

9"3. Solution of the Edmonds paradox. 9"3.1. The paradox. 9"3.2. The solution. 9"3.3. Comment.

9"4. Causality (~ in micro-~ and (( in macro-physics ~. 9"5. The Bell paradox and its solution.

9"5.1. The paradox. 9"5.2. The solution; and comments.

9"6. Signals by modulated taehyon beams: discussion of a paradox. 9"6.1. The paradox. 9"6.2. Discussion. 9"6.3. Further comments.

9"7. On the advanced solutions.

CLASSICAL TACHYONS 3

77 78 79 80 81 82 84 84 84 85 88 89 91 91 91 92 92 92 93 95 95 97 98 99 99 101 102 104 104 105 105

106 106 109 110 111 114

115 116 118

118 120 121 124 127 127 127 128 128 130 137

138

138 141 144 147 148 149 150 151

1O. Taehyon classical physics (results indepcndent of the SLTs' explicit form). 10"1. Tachyon mechanics. 10"2. Gravitat ional interactions of tachyons. 10"3. About (derenkov radiation. 10"4. About Doppler effect. 10"5. Electromagnetism for tachyons: preliminaries.

11. Some ordinary physics in the l ight of ER. 11"1. Introduction. Again about CPT. 1 l"2. Again about the <( switching procedure )>. 11"3. Charge conjugation and internal space-time reflection. 11"4. Crossing relations. 11"5. Fur ther results and remarks.

PART I I I . - General relat ivi ty and ~cachyons. 12. About tachyons in general relat ivi ty (GR).

12"1. Foreword, and some bibliography. 12"2. Black-holes and taehyons.

12"2.1. Foreword. 12"2.2. Connections between BIIs and Ts. 12"2.3. On pscudo-Riemannian geometry. 12"2.4. A reformulation.

12"3. The apparent Superluminal expansions in astrophysics. 12"4. The model with a single (Superlnminal) source.

12"4.1. The model. 12"4.2. Corrections due to the curvature. 12"4.3. Comments.

12"5. The models with more than one radio source. 12"5.1. The case ii). 12"5.2. The cases i) and iii).

12"6. Are <~ superluminal ,) expansions Superluminal? PAwr IV. -- Tachyons in quantum mechanics and elementary-particle physics. 13. The possible role of tachyons in elementary-particle physics and quantum

mechanics. 13"1. Recalls. 13"2. <~ Virtual particles )> and taehyons. The Yukawa potential. 13"3. Prel iminary applications. Neutrinos and tachyons. 13"4. Classical vacuum instabilities. 13"5. A Lorentz-invariant bootstrap. 13"6. May classical microtaehyons appear as slower.than-light quantum

particles? 13"7. About taobyon spin. 13"8. Fur ther remarks.

PAI~T V. - The problem of SLTs in more dimensions. Taehyon electrodynamics.

14. The problem of SLTs in four dimensions. 14"1. On the <~ necessity )) of imaginary quantit ies (or more dimensions).

14"10. 14"11. 14"12. 14"13. 14"14. 14"15. 14"16. 14"17.

14"2. The ]ormal expression of SLTs in four dimensions. 14"3. Preliminary expression ()f GLTs in four dimensions. 14"4. Three al ternative theories.

14"4.1. The four-dimensional approach by Antippa and Everett. 14"4.2. The four-dimensional approach by Goldoni.

14"5. A simple application. 14"6. Answer to the (( Einstein problem )) of subsect. 3"2. 14"7. An auxiliary six-dimensional space-time M(3, 3). 14"8. Formal expression of the Superluminal boosts: the first step in

their interpretat ion. 14"9. The second stop (i.e. prel iminary considerations on the imaginary

transverse components). The case of the generic SLTs. Preliminaries on the velocity composition problem. Tachyon four-velocity. Tachyon four-momentum. Is l incari ty strictly necessary? An attempt. Real, nonlinear SLTs: a tentat ive proposal. Fur ther remarks.

4 ~. R~CAM:

152 153

155 157 158 158 159

15. On tachyon electromagnetism. 15"1. Electromagnetism with taehyonic currents: two alternative, ap-

proaches. 15"2. Tachyons and magnetic monopoles. ]5"3. On the univcrsMity of electromagnetic interactions. 15"4. Further remarks. 15"5. (~ Experimental ~ considerations.

16. Conclusions.

1 . - I n t r o d u c t i o n .

1"1. Eoreword. - The subject of tachyons, even if still speculative, m a y deserve some a t ten t ion for reasons tha t can be divided into a few categories, two of which we want preliminari ly to ment ion right now: i) the larger scheme tha t one tries to build up in order to incorporate spaeclike objects in the rela- t ivistic theories can allow a be t t e r understanding of m an y aspects of the ordinary

relativistic physics, even if tachyons would not exist in our cosmos as (( asymptotically free)) objects; ii) Superluminal classical objects can have a role in e lementary-par t ic le interact ions (and perhaps even in astrophysics); and i t might be t empt ing to ver i fy how far one can go in reproducing the quantum- like behaviour at ~ cl~ssical level just by taking account of the possible existence of faster- than-l ight classical particles.

At the t ime of a previous review (I~ECA~ and MmZ~A~'~, 1974a, hereaf ter called review I) the relevant l i terature was already conspicuous. During the last ten years such l i terature grew up so much t h a t new reviews are cer tainly desirable; bu t for the same reason writ ing down a comprehensive article is already an overwhelming task. We were, therefore, led to make a t ight selection, s trongly depending on our personal tas te and interests. We confined our survey, moreover , to questions related to the classical theory of tachyons, leaving aside for the moment the various approaches to a t achyon quan tum field theory , l~rom the beginning we apologize to all the authors whose work, even if impor tan t , will not find room in the present review; we hope to be able to give more credit to i t on another occasion. In addit ion, we shall adhere to the general rule of skipping here quotat ion of the papers already cited in

review I, unless useful to the self-containedness of the present paper.

1"2. P lan o] the review. - This article is divided in five parts , the first one

having nothing to do with taehyons. In fact , to prepare the ground, in par t I (sect. 2) we shall merely show tha t special relat ivi ty---even without t a c h y o n s - - can be given a form such to describe bo th particles and antiparticle~. P a r t I I is the largest one: initially, af ter some historical remarks and having revisited the postulates of special relat ivi ty, we present a review of the elegant <~ model theory ~> of tachyons in two dimensions; pressing then to four dim(msions, we review the main results of the classical theory of tachyons tha t do not depend

CLASSICAL TACHYONS

on the existence of Supertuminal reference frames (or that are at least independent

of the explicit form of the Superluminal Lorentz ((transformations >>). In particular, we discuss bow tachyons would look like, i.e. their apparent (~ shape ~>. Last but not least, all the common causality problems are thoroughly solved, on the basis of the previously reviewed tachyon kinematics. Par t I I I deals with tachyons in general relativity; in particular the question of the apparent Superluminal expansions in astrophysics is reviewed. Par t IV shows the interesting, possible role of tachyons in elementary-particle physics and in quantum theory. In part V, the last one, we face the (still open) problem of the Super- luminal Lorentz <(transformations ~> in four dimensions, by introducing, for instance, an auxiliary six-dimensional space-time, and finally present the electromagnetic theory of tachyons: a theory that can be relevant also from the <( experimental ,> point of view.

1"3. Previous reviews. - In the past years other works were devoted to review some aspects of our subject. As far as we know, besides review I (ICEOAm and MIG~A.~I, 1974a), the following papers may be mentioned: CALI)IROL), and ]~ECA:~ (1980); ~ECA~xr (:1979a, 1978a); K~C]I (1977); BARASn:E~KOV (1975); Kii~Zl~iTs and SAI~OI~OV (1974); RECA.-~I (1973); BOLOTOVSKu and GI~ZBUICG (1972); CAME~ZI~D (1970); FEINBEI~G (1970), aS well as the short but interesting glimpse given at tachyons by GOL])I~BEI~ and SmTH (1975) in their review of all the hypothetical particles. At a simpler (or more concise) level, let us further list: GI:ASl~ (1983); VOULGAI~IS (1976); KICEISLEI~ (1973, :1969); VELA~DE (1972); GONDI~AND (1971); NEWT0~ (1970); BILANILm: and SU])ARSHA~ (1969a) and relative discussions (BILA~IIJ~ et al. 1969, 1970); and a nice talk by SIIDARSV_A~ (1968). On the experimental side, besides GOLX)~A]3EI~ and Smelt (1975), let us mention: BOICATXV (1980); J0xES (1977); BERLEu et al. (1975); CAlClC0L et al. (1975); I~A~NA MUXT~IY (1972); GIAC0- MELLI (1970).

1"4. Z is ts o] re]erences. Meetings. Books. - As to the existing bibliographies about taehyons, let us quote i) the references at pages 285-290 of review I; at pages 592-597 in RECA~I (1979a); at pages 295-298 in CALDn~0LA and I~EOA~iI (1980); as well as in ICEOA~tI and M_tG~A~I (1972) and in MIGNAM and I~ECA~I (1973); ii) the large bibliographies by P~EPEI,tTS~ (1980a, b); iii) the list by FELD~AN (1974). However, the last one, a librarian's compila- tion, lists some references (e.g, under the numbers 87 9, 13, 14, 18, 21-23) seemingly having not much to do with tachyons; while ref. 38 therein (PEI~.S, 1969), e.g., should be associated with the comments it received from BA~DO and I~ECA:gI (1969). In connection with the experiments only, also the refer- enccs in :BAr~TImTT et al. (1978) ~nd BHA~ et al. (1979) may be consulted.

As to meetings on the subject, to our knowledge: i) a two-day meeting was held in India.; ii) a meeting (First Session of the Interdisciplinary Seminars)

E. R]~CAMI

on ~ Tachyons and Related Topics ~ was held at Eriee (Italy) in September 1976; iii) a (~ Seminar sur le Tachyons ~ exists at the Facultd des Sciences de Tours et de Poitiers (France), which organizes seminars on the subject.

Wi th regards to books, we should ment ion i) the book by TE~LETSKY (1968), devoted in part to tachyons; ii) the book Tachyons, Monopoles, and ttelated Topivs (North-Holland, Amsterdam), with the proceedings of the Erice meeting cited above (see RECA,~, Editor, 1978b).

PART I

Particles and Antiparticles in Special Relativity (SR).

2. - Special relativity with ortho- and anti-chronous Lorentz transformations.

In this par t I we shall ]orget about taehyons. F rom the ordinary postulates of special relat ivi ty (SR) it follows tha t in

such a theory--which refers to the class of mechanical and electromagnetic phenomena- - the class of reference frames equivalent to a given inertial f rame is obtained by means of t ransformations ~ (Lorentz transformations, LT) which satisfy the following sufficient requirements: i) to be linear

(1) x 'v = I2',x',

ii) to preserve space isotropy (with respect to electromagnetic and mechanical phenomena), iii) to form a group, iv) to leave the quadratic form invariant :

(2) ~ , dx ~ dx" ~ ~ dx'" dx '~ .

F rom condition i), if we confine ourselves to subluminal speeds, i t follows

tha t in eq. (2)

(3 ) ~ ' p = d i a g ( + 1, - ] , - ] , - 1) = ~, , , .

Equat ions (1)-(3) imply tha t d e t J 5 2 = l , ( L~ The set of all subluminal (Lorentz) t ransformations satisfying all our conditions consists--as is ~ve]l known- -o f four pieces, which form a noneompaet, nonconneeted group (the full Lorentz group). Wishing to confine ourselves to space-time (~ rotations ~ only, i.e. to the case detJ5 ---- 9- 1, we are left with the two pieces

(da) {Lt+}: Z~ + 1 , detJ5 ~ 9 -1 ,

(db) (L~+}: L00< -- 1, detJ5 = 9- 1,


which give origin to the group of the proper (orthochronous and antichronous) transformations

and to the subgroup of the (ordinary) proper ortochronous transformations

(6) ~ft+ ~ {.Lt+},

both of which being, incidentally, iuvariant subgroups of the full Lorentz group. For reasons to be seen later on, let us rewrite ~f+ as follows:

(5') z(2)-{#/+1}-{+1,-1}.

We shall skip in the following, for simplicity's sake, the subscript ~ in the transformations JSt+, ~+. Given a transformation D , another transformation L'l 'e ,.s always exists such that

(7) /-?' = ( - 1)./2", V~r ,.~+,

and vice versa. Such a one-to-one correspondence allows us to write formally

(7') ff = -

I t follows in particular that the central elements of ~ + are C ~ (+ 1, -- 1). Usually, even the piece (4b) is discarded. Our present aim is to show--on

the contrary-- that a physical meaning can be attributed also to the transformations (4b). Confining ourselves here to the active point of view (cf. I~EOA~ and I~O])I~GVES (1982) and references therein), we wish precisely to show that the theory of SlY, once based on the whole proper Lorentz group (5) and not only on its orthochronous part, will describe a Minkowski space-time populated by both matter and antimatter.

2"1. The St#vkelberg-~eynman (< switching principle >) in SlY. - Besides the usual postulates of SI~ (prineiple of relativity, and light speed invariance), let us assume--us commonly admitted, e.g. for the reasons in GA~vccIo et al. (1980), MIGNA~ and ~EdA~W (1976a)--the following

Assumption: c negative-energy objects travelling forward in time do not exist )).

We shall give this assumption, later on, the status of a fundamental postulate. Let us, therefore, start from a positive-energy particle P travelling forward

in time. As is well known, any orthoehronous LT (4a) transforms it into another particle still endowed with positive energy and motion forward in time, On

8 E. R~CAMI

the contrary, any antichronous (~-nonorthochronous) LT (db) will change s ign--among the o t h e r s ~ t o the t ime components of all the Jour-vectors associated with _P. Any L ~ will t ransform P into a particle P ' endowed in particular with negative energy and motion backwards in t ime (fig. 1).

I t

"x futuPe --" "* X !

, 'dJ~, s 7 ] - - I - - \

/ ' - po.st--",,

s ( I - . "x I

Fig. 1.

In other words, SR together with the natural assumption above implies t ha t a particle going backwards in t ime (G6DEL, 1973) (fig. 1) corresponds in the four-momentum space (fig. 2) to a particle carrying negative energy; and, vice versa, tha t changing the energy sign in one space corresponds to changing the sign of t ime in the dual space. I t is then easy to see t ha t these two paradoxical occurrences (~ negative energy ~ and ~ motion backwards in t ime ~) give rise to a phenomenon tha t any observer will describe in a quite orthodox way, when they are--as they actually are--simultaneous (1-r 1978e, 1979a and references therein).

Fig. 2.

\ / /)" " \ / \

/ / \

rnetr fc (+ - - - ) ; c=1

E = • ~/p; + m~

p,

Notice, namely, tha t i) every observer (a macro-object) explores space- t ime (fig. 1) in the positive t-direction, so t ha t we shall meet B as the first and A as the last event; ii) emission of positive quant i ty is equivalent to absorption of negative quant i ty , as ( - - ) - ( - - ) = ( -F ) ' (+ ) ; and so on.

Let us now suppose (fig. 3) tha t a particle P ' with negative energT (and, e.g., charge -- e) moving backwards in t ime is emit ted by A at t ime tl and

CLASSICAL TACtIYONS

[t<t']

l z , i - -~ ( tq) ;E>O;7;p>O p n = ~ _ J (+, t) ;v > 0 "

(t,x) (t ' , x9

xl x 2 AA ~ (--q);E>O;T;p<O x _ _ CT(ph)= l A I'* . . ^ ~ - ]

(~') ( x " ) ~ ~ (+~);v <u �9 ~ (-~,x) (-GxO

y ",k ,,. h , _ l--:--] (-Cl); E < O ; T ;p .< 0 [ t > t 2 ] "R IP ~p ; - [ ~ _ J (+ , ) ;v>O

(t ,x) (t r, x') /--'-,, ((P_) ; -q ; E< 0 ;_~ LP < 0 I--. .

( t , ,x,) A { ~ 4 " . . . . . . - ~ - - - ~ - ~ B ) ( t 2 , x 2 ) ~ ( p h ) = ~ ( - q ' , i E ~ ? ; v T ~ P o > O , , ~ ' ~ (O);+q;E>O;T;p>O v

a) b) Fig. 3.

absorbed by B at t ime t2 < t~. Then, i t follows tha t at t ime tx the object A <( loses }~ negative energy and charge, i.e. gains positive energy and charge. And tha t at t ime t~ < t~ the object B ~ gains >> negative energy an(t charge, i.e. loses positive energy and ch'~rge. The physical phenomenon here described is nothing but the exchange ]rom B to A of a particle Q with positive energy, charge + e, and going ]orward lit t ime. h~otice tha t Q has, however, charges opposite to P ' ; this means in a sense tha t the present <~ switching procedure }> (in the past ca.lled ~ I~IP ~) effects a <{ charge conjugation }) C, among the others. l~'otice also t ha t (( charge ~, here and in the following, means any additive charge; so t ha t our definitions of charge conjugation, etc., are more general than the ordinary ones (review I, lr~ECAMI~ 1978a). Incidentally, such a switching procedure has been shown to be equivalent to applying the chirality operation ( t ~ E d ~ :rod Zg.~o, 1976). See also~ e.g., ]~EIdHE~]~AC~[ (1971), M~NSKY (1976).

2"2. Matter and antimatter ]rom SlY. - A close inspection shows the application of ally antichronous t ransformation Z 4, together with tile switching procedur% to trnnsform /) into an object

(8) q - P

which is indeed the antiparticle of P. We are saying tha t the concept o] antimatter is a purely relativistic one, and that , on the basis of the double sign in (o = 1)

(9) E - : ~ Vp2 -i m~,

the existence of antiparticles could have been predicted from 1905, exactly with the properties they actually exhibited when later discovered, provided

t ha t recourse to the (~ switching procedure ~) had bee~ made. We, therefore, mainta in t ha t the points of the lower hyperboloid sheet in fig. 2---~ince they correspond not only to negative energy but also to motion backwards in t i m e ~ represent the kinematical states of the antiparticle _P (o] the particle P represented by the upper hyperboloid sheet). Let us explicitly observe tha t the ,switching procedure exchanges the roles of source and detector, so tha t (fig. 1) any observer will describe B to be the source and A the detector of the antiparticle _P.

Let us stress tha t the switching procedure not only can, bu t must be performed, since any observer can do nothing bu t explore space-time along the positive t ime direction. That procedure is merely the translation into a purely relativistic language of the Stfickelberg (1941; see also KLAN, 1929)-Feynman (1949) (~ switching principle ~). Together with our assumption a.bovc, it can take the form of ~ (~third postutate/): (~Ncgative-energy objects travelling forward in t ime do not exist; any negative-energy object P travelling backwards in t ime can and must be described as its anti-object P going the opposite way iu space (but endowed with positive energy and motion forward in time) ~). Cf., e.g., C.~Dn~OLA and I~ECA~ (1980), I ~ E C ~ (1979a) and references therein.

2"3. ~urther remarks. - a) Let us go back to fig. 1. In S]~, when based only on the two ordinary postulates, nothing prevents a priori the event A from influencing the event B. Jus t to forbid such a possibility we introduced our assumption together with the Stiickelberg-Feynman ~ switching procedure ~>. As ~ consequence, not only we eliminate ,~ny particle motions backwards in t ime, bu t we also ~ predict ~) and natural ly explain within SI~ the existence

of ant imat ter .

b) The th i rd postulate, moreover, helps solving the paradoxes connected with the fact t ha t all relativistic equations admit , besides s tandard , retarded ~ solutions, also ~ advanced ~) solutions: The latter will simply represent antiparticles travelling the opposite way (M_m~A~I and I~WCAMT, 1977). For instance~ if Maxwell equations admit solutions in terms of outgoing (polarized) photons of helicity ~ ---- ~t- .1, then they will admi t a.lso solutions in terms of incoming (polarized) photons of helicity )~ ---- -- 1; the actual intervention of one or the other solution in a physical problem depending only on the initial

conditions.

c) Equat ions (7), (8) tell us that~ in the case considered, any L ~ has the same kinematical effect as its (~dual~> t ransformation L ~, just defined through eq. (7), except for the fact tha t it, moreover, transforms P into its antiparticle ~ . Equations (7), (7') then lead (~G~A~I and I~Ec~_~, 1974a, b,

1975a) to write

(10) -- 1 ~= I)T = CPT ,

CLASSICAL TACttYOiWS 1i

where the symmet ry operations P, T are to be understood in the (( strong sense ~): ~or in:stance, T - - reversal of the t ime components oi all four-vectors associated with the considered phenomenon (namely, inversion of the t ime and energy axes). We shall come buck to this point. The discrete operations P, T have the ordinary meaning. When the particle i ~ considered in the beginning can be regarded as an extended object, PAV~I~ and 1%ECAM~ (1982) have shown the <~ strong >> operations P~ T to be equivalent to the space, t ime reflections acting on the space-time both external and internal to the particle world-tube (see subsect. 11"3 in the following).

Once accepted eq. (10), then eq. (7') can be wri t ten

(70 f ,+ = (p~) fit+ ~ (cP~) f t + .

In particular, the tota l inversion Lr ~ - "l t ransforms the process a + b--> --> e + g into the process d + ~ -> b d- a wi thout any change in the velocities.

d) All the ordinary relativistic laws (of mechanics and electromagnetism) are actually already covariant under the whole proper group s eq. (5), since they are CPT symmetr ic besides being covariant under f~+.

e) A few quantities, t ha t happened (cf. subsect. 5"17 in the following) to be Lorentz-invari~nt under the transformations L r ~ f~+, ~re no longer invariant under the transformations J5 e f + . We have already seen this to be true for the sign of the additive charges, e.g. for the sign of the electric charge e of a particle/9. The ordinary derivation of the electric-charge invariance is obtained by evaltmting the integral flux of a current through ~ surface which, under J~+, does move, changing the angle formed with the current. Under L ~ s f~+ the surface (( rotates >> so much with respect to the current (cf. also fig. 6, 12 in the following) t ha t the current enters i t through the opposite face; as a consequence, the integrated flux (i.e. the charge) changes sign.

I~A~ II

Bradyons and Tachyons in $1~.

3. - Historical remarks and preliminaries.

3"1. Historical remarks. - l~et as now take on the issue of tachyons. To our knowledge (Col~BEi~, 1975; I~CAMI, 1978a), the first scientist mentioning objects (~faster t han the Sun's light >> was LVCl~ETIUS (50 B.C., ca.), in his De Rerum Natura. Still remaining in lore-relativistic times, after having

12 ~. ~xcxm

recalled, e.g., Z~eLAOE (1845), let us only melltion the recent progTess represented by the noticeable papers by Tt[o~soN (1889), H~AVISlDE (1892), DES COD-D~ES (1900) and mainly SO~[ERF~mD (1904, 1905).

In 1905, however, together with St~ (EInSTEIn, 1905; POI~CAm~, 1906) the conviction that the light speed c in vacuum was the upper limit of any speed started to spread over the scientific community, the early-century physicists being led by the evidence that ordinary bodies cannot overtake that speed. They beh~ved in a sense like Sudarshan's (1972) imaginary demographer studying the poplflation patterns of the Indian subcontinent: ~ Suppose

a demographer calmly asserts that there are no people North o/ the Himalayas, since none could climb over the mountain ranges/ That would be an absurd conclusion. People o/central As ia are born there and live there: They did not have to be born in India and cross the mountain range. So with /aster-than-light particles ~> (el. fig. 4). Notice that photons are born, live and die just ~( on the top of the mountain )>, i.e. always at the speed of light, without any need to violate S1%, that is to say to accelerate from rest to the light speed.

,\ A\ - c 0 c v c v

a) b)

Fig. 4.

iV[oreover, ToL~_~ (1917) believed to have shown in his antitelephone (( paradox ~> (based on the already well-known fact that the chronological order along a spacelike path is not Lorentz invari,mt) that the existence of Superluminal (v~> c ~) particles allowed information transmission into the past. In recent times that (~ paradox )) has been proposed again and agMn by authors apparently unaware of the existing literature (for instance, some of l~olnick's (1972; see also 1969) arguments had been already (~ answered ~> by CsoN~ (1970) before they appeared). Incidentally, we shall solve it

in subsect. 9"1. Therefore, except for the pioneering paper by SOmG~A~A (1922; recently

rediscovered by CMmI~OLA et al., 1980), after the mathematical considerations by MAJO~A~A (1932) and Wm~E~ (1939) on the spacelike particles one had to wait lmtil the fifties to see our problem tackled again in the works by AZCZEL~S (1955, 1957, 1958), S C ~ D T (1958), TAXGm~LL'~I (1959); and a little later by TANAKA (1960), TERLETSKY (1960) and GICEGO]~Y (1961, 1962). I t started to be fully reconsidered in the sixties: In 1962 the first article

CLASSICAL TACI:[YONS 13

by SUDA/~SttA]N and co-workers ( B ~ A ~ K et al., 1962) appeared, and after that paper a number of physicists took up studying the subject--among whom, for instance, JONES (1963) and F~]3E~G (1967) in the USA and I~ECAlW (1968, 1969a) and colleagues (RECAp, 1968, 1969; OLKH0VSKu and t~ECA~ 1970a, b, 1971) in Europe.

The first experimental searches for SuperluminM particles were carried out by A~vXG]~ et al. (1963, 1965, 1966).

As is well known, SuperluminM particles have been given the name (( taehyons ~) (T) by F v . i ~ a (1967) from the Greek word Tagi~ = fast. (~ Une

particule qui a un nora poss~de ddj~ un ddbut d'existence )) (A particle bearing a name has already taken on some existence)~ was later commented on by A~zELI]~S (1974). We shall call (( luxons ~) (l), following B~LA~K et aI. (1962), the objects travelling exactly at the speed of light, like photons. At last, we shall call ~< bra&yons ~ (B) the ordinary subluminal ( v"< c ~) objects, from the Greek word fi~a~ir ---- slow, as it was independently proposed by C~w~nY (1969), BA~A~D and S ~ z I ~ (1969) and I~ECA~I (1970; see also BALD0 et at.,

1970). Let us recall at this point that, according to DE~oc~I~S of Abdera, every-

thing that was thinkable without meeting contradictions did exist somewhere in the unlimited universe. This point of view~recently adopted also by M. GELL-)/[ANN--WaS later on expressed in the known form (~ A n y t h i n g not

]orbidden is vompulsory ~ ( W ~ E , 1939) and named the (( totalitarian principle ~> (see, e.g., T ~ I ~ , 1970). We may adhere to this philosophy, repeating with SUD~a~SH~ that (( if tachyons exist, they ought to be found. If they do not exist, we ought to be able to say why )).

3"2. Prel iminaries about taehyons. - Tachyons, or spacelike particles, are already known to exist as internal, intermediate states or exchanged objects (see subseets. 6"13 and 13"2). Can they also exist as (( asymptotically free ~) objects?

We shall see that the particular--and unreplaceablewrole in SI~ of the light speed c in vacuum is due to its invarianee (namely~ to the experimental fact that c does not depend on the velocity of the source), and not to its being or not the maximal speed (I~ECA~ and Mo])ICA, 1975; KIaZH~ITS and POLYA- CHENKO, 1964; ARZELI]~S, 1955).

However, one cannot forget that in his starting paper on special relativity EinsTEin--after having introduced the Lorentz transformations--considered a sphere moving with speed u Mong the x-axis and noticed that (due to the relative motion) it appears in the lrame at rest as an ellipsoid with semi-axes

(11) a~ = r VI~-- fl~, a~ ~-- a, = r (~ -- u/e) .

Then EI~S~EI~ (1905) added: (~Ti~r u ~ e svhrump/en nile bewegten Objeete--

veto <truhenden)> System aus betraehtet in ]liieher~hafle Gebilde zusammen. ~i~r Uberlichtgeschwindigkeiten werden unsere ~berlegungen sinnlos; wit werden i~brigens in den ]olgenden Betraehtungen linden, dass die Ziehtgesehwindigkeit in unserer Theorie physikalisch die Rolle tier unendlieh grossen Gesehwindigkeiten spielt ~), which means (SchwARtz, 1977): (~For u ~ e all moving objects-- viewed from the (~stationary~) system--shrink into planelike structures. For supcrlight speeds our considerations become senseless; we shall find, moreover, in the following discussion that the velocity of light plays in our theory the role of an 'infinitely large velocity ~). EI:NS~N referred himself to the following facts: i) for u ~ c, the quantity a~ becomes pure imaginary: ~f a~ ~--a~(u), then

-= 1 - - ~ (U '~ > oZ);

if) in St~ the speed of light v = o plays a role similar to the one played by the infinite speed v = oo in the Galileian relativity (G~II~I , 1632, 1953).

Two of the aims of this review will just be to show how objection i)--which touches a really difficult problem--has been answered, and to illnstrate the meaning of point if). With regard to eq. (12), notice that a priori ~ - ~ - - 1 = = i i ~ 2, since ( - 4 - i ) 2 = - 1. Moreover, we shall alway~ mlderstand that v/1--f i ~ for fl~> 1 represents the upper half-plane solution.

Since a priori we know nothing about Ts, the safest w~y to build up a theory for them is trying to generalize the ordinary theories (starting with the classical relativistic one, only later on passing to the quantum field theory) through (~ minimal exteltsions ~), i.e. by performing modifications as small as possible. Only after possessing a theoretical model we shall be able to start experiments: Let us remember that, not only good experiments are required before getting sensible ideas (GAL~LEI, 1632), but also a good theoretical background is required before sensible experiments can be performed.

The first step consists, therefore, in facing the problem of extending SR to tachyons. In so doing, some ~uthors limited themseIves to considering objects both subluminal and Super]umina], a.ll referred, bowcvcr, to sublumin~l observers ((~ weak approach ~)). Other authors attempted on the contrary to generalize SI~ by introducing both subluminal observers (s) and Superluminal observers (S), and then by extending the principle of relativity ((( strong approach ~)). This second approach is theoretically more worth of consideration (tachyons, e.g., get real proper masses), but it meets, of course, the greatest obstacles. In fact, the extension of the relativity principle to Superluminal inertial frames seems to be straightforward only in the pseudo-Euclidean space-times M(n, n) having the same number n of space axes and of time axes. l~or instance, when facing the problem of generalizing the Zorentz transformations to SuperluminaI frames in four dimensions one meets no-go theorems as Gorini et aL's ( G o ~ I , 1971 and references therein), stating no such extensions exist which satisfy all


the following properties: i) to refer to the four-dimensional Minkowski space- t ime M4 ---- M(1, 3), ii) to be real, iii) to be linear, iv) to preserve the space isotropy, v) to preserve the light speed invarianee, vi) to possess the pre- scribed group-theoretical properties.

We shall, therefore, s tar t by sketching the simple, instructive and very promising (, model theory )~ in two dimensions (n ---- 1).

Let us first revisit, however~ the postulates of the ordinary SlY.

4. - The postulates o f SR revisited.

Let us adhere to the ordinary postulates of S1% A suitable choice of postulates is the following one (review I ; ~Ata~O~ '~E and ICECA~ 1982a, and references therein) :

1) First postulate: principle o] relativity: ~ The physical laws of electromagnet ism and mechanics are covariant (----invariant in form) when going from an inertial frame ] to another frame ]' moving with constant velocity u relative to ] ~).

2) Second postulate: (~Space and t ime are homogeneous and space is isotropic ~. For future col~vcnience, let us give this postulate the form: <~ The space-time accessible to auy inertial observer is four-dimensional. To each inertial observer the 3-dimensional space appears as homogeneous and isotropic, and the 1-dimensional t ime appears as homogeneous ~.

3) Third postulate: principle of retarded causality: <(Positive-energy objects travelling backwards in t ime do not exist; and any negative-energy par t ic le /~ travelling backwards in t ime can and must be described as its antiparticle P , endowed with positive energy and motion forward in t ime (but going the opposite way in space d ~). See subsect. 2"1, 2"2.

The ]irst postulate is i~lspired to the consideration t h a t all inertial frames should be equivalent (for a careful definition of <~ equivalence ~) se% e.g, I C E c ~ , 1979a) ; notice tha t this postulate does not impose any constraint on the relative speed u == lu] of the two inertial observers, so tha t a priori -- c~ < u < ~- oz.

The second postulate is justified by the fact tha t from it the conservation laws of energy, momentum and angular momentum follow, which are well verified by experience (at least in our <~ local )~ space-time region); let us add the following comments: i) The words homogeneous, isotropic refer to space-time properties assumed--as a lways--with respect to the electromagnetic and mechanical phenomena; ii) such properties of space-time are supposed by this postulate to be eovari~nt wi th in the class of the inertial frames; this means t ha t SIC assumes the vacuum (i.e. space) to be (~ at rest )> with respect to each inertial frame. The third postulate is inspired by the requirement t h a t / o r each observer

16 ~ . R ~ C A ~

the (( causes )) chronologically precede thei r own (~ effects ~> (for the definition of causes and effects see, e.g., C~L~II~OL2r and I%ECA-~, .1980). Le t as recall t ha t in sect. 2 the initial s t a t emen t of the th i rd postulate has been shown to be equ iva len t - -as follows from postulates 1) and 2)---to the more na tura l assumption tha t (( negative-energy objects travell ing forward in t ime do not

exist ~.

4"1. Existence oJ an invariant speed. - Let us initially skip the th i rd

postulate. Since 1910 i t has been shown (IG~_TOWSKI, 1910; F R ~ K and I%OT~E,

;1911; HAHn, 1913; PARS, 1921; ~LALA.N, 1937; SEv],:ra, .1955; At~oDI, 1972; DI gOl~O, 1974) tha t the postulate of the light speed invariance is not s tr ict ly necessary, in the sense tha t our postulates 1) and 2) imply the existence of an invar iant speed (not of a maximal speed, however). In fact , f rom the first two postulates i t follows (RLNDLE]~, 1969; BERZI and GoRI~I, 1969; G o r i e r and ZECCA, 1970, and references therein; :LUGIATO and GO~L~I, 1972) tha t one and only one quant i ty w 2 having the physical dimensions of the square of a s p e e d ~ m u s t exist, which has the same value according to all inert ial f rames:

(13) w ~ = inva r i an t .

I f one assumes w---- ~ , as done in Galileian relat ivi ty, t hen one would

get Galflei-Newton physics; in such a case the invariant speed is the infinite one: oo(~)v = c% where we symbolically indicated b y (~ the operat ion of

speed composition. I f one assumes the invariant speed to be finite and real, then one gets

immediate ly Einstein 's re la t iv i ty and physics. Experience has actually shown us the speed c of l ight in vacuum to be the (finite) invar iant speed: c �9 v = c. In this case, of course, the infinite speed is no longer invar iant : co �9 v = V # co. I t means tha t in SI~ the operat ion ~ is not the operat ion + of ari thmetics.

Le t us notice once more tha t the unique role in SI~ of the light speed c

in vacuum rests on its being invar iant and not the maximal one (see, e.g.,

SHA~-ICA~A, 1974; I%ECA~ an4 MODICA, 1975); if t a chyons - - i n par t icular infinite-speed t achyons- -ex i s t , t hey could not take over the role of light in SR (i.e. t h e y could not be used b y different observers to compare the sizes

of thei r space and t ime units, etc.), just in the same way as bradyons cannot replace photons. The speed c tltrns out to be a l imiting speed; bu t any l imit

possesses a priori two sides (fig. 4).

4"2. The problem o] Zorentz trans]ormations. - Of course, one can subst i tute

the light speed invariance postulate for the assumption of space-time homo-

genei ty and space isotropy (see the second postulate). In any case, f rom the first two postulates i t follows t h a t the t rausforma-

CLASSICAL TACI:IYONS 17

tions connecting two generic inertial f rames J, ]', a priori with -- oo < lul <

(l 4) dx~ ---- G'~ dx~

must (cf. sect. 2)

i) t ransform inertial motion into inert ial mot ion,

ii) form a group G,

iii) preserve space i sot ropy,

iv) leave the quadrat ic form invariant , except ]or its sign (I~INDLEI~, 1966, p. 16; LAI~DAU and LIFSt_I/TZ, ]966a, b):

(15) dx~ dx'~ : :]= dx~dx , .

Notice tha t eq. (15) imposes- -among the others--- the light speed to be invariant (Jh~mI~, 1979). Equa t ion (15) holds for any quan t i ty dx~ (position, momentum, velocity, acceleration, current , etc.) tha t is a G-/our-vector, i.e. t ha t behaves as a four-vector raider the t ransformat ions belonging to G. I f

we explicitly confine ourselves to slower-than-light relat ive speeds, u 2 < c 2, then we have to skip in eq. (15) the minus sign, and we are left with eq. (2) of sect. 2. In this case, in fact , one can s ta r t f rom the iden t i ty t ransforma- t ion G = 1, which requir~s the plus sign, and then re ta in such a sign for conti- nui ty reasons.

On the contrary, the minus sign will p lay an impor tan t role when we are ready to go beyond the light-cone discontinuity. In such a persective, let us preliminarily c la r i fy- -on ~ formal g round- -wha t follows (~IACCA~RONE and I~EGA~tT, 1982a, 1984).

4"3. Orthogonal and antiorthogonal trans/ormations: digression.

4"3.1. Le t us consider a space having, in ~ certain initial base, the metric g"5 so t ha t for vectors dx ~ and tensors M:~ it is

d x ~ = g~dx~ , M ~ --- g~'~g~Mv6.

When passing to another base, one writes

Ill the two bases, the scalar products are defined

dx~ dx ~ ~ dx~ g~ dx~, dx~ dx'" ~ dx~ g '~ dx'~,

respectively.

18 ~ . RECAMI

Le t us call A the t r ans fo rma t ion f rom the first to the second base, in the sense t h a t

d~'" ~- A ~ d x Q ,

t h a t is to say

NOW, if we impose t h a t

(16)

we get

(17)

however , if we impose t h a t

(16')

we get that

(17')

4"3.2.

dx ~ __-- (A-1)~ dx'Q.

dx= dx = = ~ dx~ dx '~ (assumpt ion) ,

g ~ = A ~ g , ~ A ~,

dx~ dx = = - - dx~ dx '~ (assumption) ,

g=p - - A ~ ' ~ .

L e t us consider the case (16), (17), i.e.

(16) dx~ dx ~ = ~- dx~ dx ' ' (assumption) ,

and let us look for the proper t ies of t rans format ions A which yield

(18) g~. --~ ~ g~ (assumption) .

I t m u s t be

(17) g~,~ ~-- A ~ g ~ , , A ~ ,

wheref rom

(19) g~'= g~,~ = (gA~)vl ' (gA)~,~ .

At this point , if we impose t h a t in the initial base

(20) g, , ---- ~, , (assumption) ,

t hen eq. (19) yields

~'~ = (vAT) ~" (vA).~,

t h a t is to say

(21) A T ~ A - ~ ~.

C~ASSZCAL ~XCn~O~S

4"3.3. I~OW, in the case (16'), (17'), i .e.

dx~ dx ~ : - - dx~ dx '~ (16')

when

0 7 ' )

19

(assumption) ,

(22a) e i ther

(22b) or

4"3.4.

(16')

when

(17')

i .e. t r ans fo rmat ions A m u s t sti l l be pseudo-or thogonaI :

(21) A T ~ A = 7 .

I n conclus ion , t r ans format ions A when pseudo-or thogonal opera te in such a way t h a t

i) dx~ dz '~ = + dx= dx ~ and

ii) dx~ dx ~' = - - dx~ dx = and

On the cont ra ry , let us now require t h a t

dx= dx= = - - dx~d~ '"

g~, = __ ~ , .

(assumpt ion) ,

a y i j v ~ - fl ,

and s imul taneously let us look for the t r ans format ions A such t h a t

(18) g ~ -= -f- g , , (assmnption) .

In this ease, when in par t icu lar assumpt ion (20) holds, g~,, - - ~,~, we get t h a t t r ans format ions A mus t be pseudo-ant ior thogonal :

(23) A T ~ A = _ ~ .

(20)

i t m u s t be

g ~ = - A ~ g ; , A ~ ,

let us inves t iga te which are the proper t ies of t r ans format ions A t h a t yield

(18') g~ : -- g ~ (assumption) .

I n the par t icu lar case, ~gain, when

g ~ , , - - r /~ , , (assumption) ,

~O E. ~ C A M I

4"3.5. The same result (23) is easily obtained when assumptions (16) and (18') hold, together with condition (20).

In conclusion, t ransformations A when pseudo-antiorthogonal operate in such a way tha t

(24a) either i) dx~ dx '~ ~ -- dx~ dx ~ and g~, ---- ~- U~,

(24b) or if) dx~ dx '~ ~ ~- dx~ dx ~ and g~ ---- -- ~7~,.

4"3.6. For passing from sub- to Super-luminal frames we shall have (see the following) to adopt antiorthogonal (') transformations. Then, our conclusions (22) and (24) show tha t we will have to impose a sign change either in the quadratic form (16'), or in the metric (18'), but not--of course--in both: otherwise one would get, as known, an ordinary and not a Superluminal transformation (cf., e.g., ~_~G~A~ and ~ECA~, 1974c). We expounded here such considerations, even if elementary, since t h a y arose some misunderstandings (e.g., in KOWXLCzY~S~, 1984).

We choose to assume (unless differently s tated in an explicit way)

(25) g~ = + g , , .

Let us add the following comments. One could remember the theorems of :Riemannian geometry (theorems so often used in general relativity) which state the quadratic form to be positive-definite and the g~-signature to be invariant , and, therefore, wonder how it can be possible for our antiorthogonal t ransformations to act in a different way. The fact is t h a t the pseudo-Euclidean (Miz~owski) space-time is not ~ particular l=tiemannian manifold, bu t ra ther

particular Lorentzian (i.e. pseudo-l~iemaunian) manifold. The space-time itself of general relat ivi ty (GI~) is pseudo-Riemannian and not l~iemanniau (only space is l~iemannian in GIr (see, e.g., SAcES and Wu, 1980). I11 other words, the antiorthogonal t ransformations do not belong to the ordinary group of the so-called a arbi t rary ~) co-ordinate transformations usually adopted in GI~, as outlined, e.g., by MOLLEX (1962). However, by introducing suitable scale-invariant co-ordinates (e.g., dflation-eovariant (~ light-cone co-ordinates ~)), both sub- and Super-luminal (~ Lorentz transformations ~ can be formally wri t ten (MACCX~RONE et al., 1983) in such a way as to preserve the quadratic form, its sign included (see subsect. $'8).

Throughout this paper we shall adopt (when convenient) natural units (c ~ 1); and (when in four dimensions) the metric signature ( ~ - - - - - ) , which will be always supposed to be used by both sub- and Super-luminal observers, unless differently s ta ted in an explicit way.

(*) ttcre and in the following, for simplicity's sake, we shall drop the prefix ~ pseudo ~) from the expressions pscudo-orthogonal and pseudo-antiorthogonal.

CLASSICAL TACYIY O~-S 21

5. - A model theory for tachyons: an ~extended re la t iv i ty~ (ER) in two dimensions.

Till now we have not t aken account of tachyons. Le t us finally take them into consideration, s tar t ing from a model theory, i.e. f rom (, ex tended relativ- i ty )~ (Ell) ( M A C c ~ o ~ and I g v . c ~ , 1982a; M)~CCAR]tO~ et al., 1983; BARge et al., 1982; review I) in two dimensions.

5"1. A duality principle. - We got f rom experience t h a t the invar iant

speed is w = c. Once an inert ial f rame so is chosen, the invar iant character of the light speed allows an c~mus t ive par t i t ion of the set {/} of all inertial f rames ] (of. sect. 4) into the two disjoint, complementary subsets {s}, {S} of the frames having speeds ]u] < c and [U[ > c relat ive to so, respectively. In the following, for simplicity, we shall consider oltrselves as ((the observer so*. At the present t ime we neglect Em luminal frames (u--~ U = 0) as (( unphysical )). The first postula.te reqltires frames s and S to be equivalent

(for such an extension of the criterion of (~ equivalence ~> see CALDIIlOLA and IIECA3I:[, 1980; t~ECA~MI, 1979a), and in par t icular observers S- - i f they ex i s t - -

to have at thei r disposal the same physical objects (rods, clocks, nucleons, electrons, mesons, ...) as observers s. Using the language of mnlt idimensionM space-times for fut~tre convenience, we can say the first two postulates to req]~ire t ha t even observers S must be able to fill the i r space (as seen by themselves) with a (( lat t ice work )) of meter sticks and synchronized clocks (TAYLOR and WI-IEELEll, 1966). I t folh)ws tha t objects must exist which are at rest relat ively t~o S and faster th 'm light relat ively to frames s; this, together with the fact t ha t luxons l show the same speed to any observers s or S, implies tha t the objects which are bradyons B(S) with respect to a f rame S must appear as tachyons T(s) with respect to any f rame s, and vice versa:

(26) ]/(S) .... T(s), T(S) = B(s), [(S) = l(s) .

The s t a t emen t t h a t the terms B, T, s, S do not have an absolute, bu t only a

relative meaning, and eqs. (26) const i tu te the so-called duality principle (OLKIIOVSKY and ]:~ECA_MI, ] 971 ; 14].:CA~ and ~M_IGNA~I, 1972, 1973a; M[GNANI et al., 1972; /k~TrePA, 1972; MIGNA5I and ~]gCA~MI, 1973a).

This means tha t the relat ive speed of two frames sl, s2 (or $1, S~) will always be smaller t h a n c; an(i the relat ive speed be tween two frames s, S will be always

larger t ha n c. Moreover, the above exhaust ive par t i t ion is invar iant when so is made to va ry inside {s} (or inside {S}), whilst the subsets {s}, {S} get on the cont ra ry interchanged when we pass f rom % e {s} to any f rame So e {S}.

The main problem is finding oat how objects t ha t are subluminM w.r.t. ( - with respect to) observers S apt)e~r to observers s (i.e. to tls). I t is, there-

2 2 ~ . ~ E C A ~

fore, finding out the (Superluminal) Lorentz t rans format ions- - i f t h ey e x i s t ~ connecting the observations made by S with the observations b y s.

5"2. Sub- and Super-~uminal Zoren$z ~rans]ormations: preliminaries. -

We neglect space-t ime translations, i.e. consider only restricted Lorentz t ransformat ions . All frames are supposed to have the same event as thei r

origin. Le t us also recall t ha t in the chronotopical space Bs are character ized b y t imelike, is by lightlike, and Ts by spacelike world-lines.

The ordinary, subluminal Lorentz t ransformat ions (LT) from s~ to s~, or f rom S~ to S~, are known to preserve the four-vector type. After subscct. 5"1, on the cont ra ry , it is clear t ha t the ~ Superluminal Lorentz t ransformations ~> (SLT) f rom s to S, or f rom S to s, must t ransform t imelike into spacelike quantities, and vice versa. With assumption (25) it follows tha t in eq. (15) the plus sign has to hold for LTs and the minus sign for SLTs:

(15) ds ' 2 = ~: ds ~ ( u ~ l ) ;

therefore, in (~ extended re la t iv i ty ~> (ER), with assumption (25), the quadrat ic

fo rm

d s 2 ~-- d x . d x .

is a scalar under LTs, bu t is a pseudoscalar under SLTs. In the present case,

we shall wri te t h a t LTs are such t ha t

(279) dt ' ~ - d x ' 2 = -~ (dr 2 - dx-') (92< 1),

whilc for SLTs it mus t be

(27b) dt '* -- dx '2 = -- (dr ~ -- dx 2) (u: > 1).

5"3. Energy-momentum space. - Since tachyons are just usual particles

w.r.t , the i r own rest f rames ], where the ]'s are Superluminal w.r.t, us, t hey

will possess real rest masses mo ( l~cA~x and I'VlXG~A~I, 1972; L ~ T ~ , 1971a; P A ~ R , 1969). F r o m eq. (27b) applied to the energy-momentum vector p , ,

one derives for free tachyons the relation

(28) E 2 -- p~ = -- ml < 0 (too real),

provided t ha t p~ is so defined to be ~ G-vector (see the following); so t h a t one

has (cf. fig. 5)

(29a)

(29b)

(290)

PUP"=-

2 + m o > O

0

- - m ] < O

for bradyons (timelike case),

for luxons (lightllke case),

for tachyons (spacelike case).

CLASSICAL TACIIYONS

o/I Fig. 5.

Py

E

py ~ o C

c)

23

Equations (27)-(29) tell us that the roles of space and time and of energy and momentum get interchanged when passing from bradyons to tachyons (see subsect. 5"6). Notice that in the present case (eqs. (29)) it is # : 0, 1. Notice also that tachyons slow down when their energy increases and accelerate when their energy decreases. In particular, divergent energies are needed to slow down the tachyons' speed towards its (lower) limit c. On the contrary, when the tachyons' speed tends to infinity, their energy tends to zero; in ER, therefore, energy can be transmitted only at ]inite velocity. From fig. 5a), e) it is apparent that a bradyon may have zero momentum (and minimal energy moe2), and a tachyon may have zero energy (and minimal momentum moV); however, Bs cannot exist at zero energy, and Ts cannot exist at zero momentum (w.r.t. the observers to whom they appear as tachyonsI). Incidentally, since transcendent ( ~ infinite-speed) tachyons do not transport energy but do transport momentum (mov), they allow getting the rigid-body behaviour even in SI~ (BxLA~rU~ and SUDA~SDA~', 1969a; review I; CASTOPJ_NA and I~ECA~, 1978). In particlflar, in elementary-particle physics--see the following~ e.g. subsect. 6"7, 6"13--they might a priori be useful for interpreting in the suitable reference frames diffractive scatterings, elastic scatterings, etc. (MACCARROI~E and I=[ECAMI, 1980b and references therein).

5"4. Generalized Zorentz trans]ormations (GLT): preliminaries. - Equa- tions (27a), (27b), together with requirements i)-iii) of subsect. 4"2, finally imply the LTs to be orthogonal and the SLTs to be antiorthogonat (MAcCARRONE et al., 1983 and references therein):

(30a) GT~IG~----~ (sublnminal case: u ~ l ) [ (-~01 0 ) ]

(30b) G~'U G : -- U (Superluminal case: u ~ ~ 1) U :~ --:l

as anticipated at the end of subsect. 4"3. Both sub- and Super-luminal Lorentz transformations (let us call them ((generalized Lorentz transformations )~,

2 4 E . R E C A M I

GLT) result to be unimodular . In the two-dimensional case, however, the SLTs can a priori be special or not (cf. cqs. (33) and (36') in the following). In any case, in four dimensions (see sect. 12 ; cf. also subscet. 5"5, 5"6), all the GLTs will result to be unimodular and special:

(31) d e t G = ! - 1 , V G e G .

5"5. The fundamental theorem of (bidimensionat) EI~. - We have now to write down the SLTs, satisfying conditions i)-iv) of subsect. 4"2 with the minus

! sign in eq. (15), still, however, with g~ := g~,~ (cf. subsect. 4"3 and MACCA~nO~E and I~ECA~, 1982b), and show tha t the GLTs actual ly form a (new) group G. IJet tls remind explicit ly tha t an essential ingredient of the present procedure is the ~ssumption t ha t the space-time interval dx~ is a (chronotopic~l) vector

even with respect to G (see eq. (14)). Any SLT from ~ sub- to a Super-luminal frame, s --> S', will be identical

with ~ suitable (ordinary) l~T-- le t us call i t the (< dual ~) t r ans fo rma t ion - - except for the fact t h a t i t must change t imelike into spacelike vectors, and vice versa, according to eqs. (27b) and (25).

Alternat ively, one could say tha t a SLT is identical with its dual subluminal LT, provided tha t we impose the pr imed observer S' to use the opposite

! metric signature g,~ = - g~,~, however without changing the sign into the definitions of t imelike and spacelike quanti t ies! (M~G~ANI and I:~ECA,.MI, 1974c;

S~AH, 1977.) I t follows i~hat a generic SLT, corresponding to ~ Superluminal veloci ty U,

will be formally expressed by the product of the dual LT corresponding to the subluminal veloci ty u ~ 1/U, by the matr ix 5 f_~ .S f~- - i l , where here 1 is

the two-dimensional ident i ty :

(32) S L T ( U ) - - - • ( l ) u - - - ~ , u"-< l , U~> I .

(33) 5 z ~- il

The t ransformat ion ~ - ~ 5 f 2 e {~ plays the role of the << transcendent SLT ~) since for u - + 0 one gets SLT(U-> c o ) = -4-5f= • In the present context , the double sign in eq. (32) is required, e.g., by condition ii) of subsect. 4"2; in

fact, given a part icular subluminal Lorentz t ransformation L(u) and the

SLT = + iL(u), one gets

(34a) IlL(u)] [iL-~(u)] :: [iL(u)] [iL(-- u)] ----- -- 1 �9

However ,

(34b) IlL(u)] [--iL-~(u)] ~ [iL(u)J [ - - iL(- - u)]-: ~ - - 1 �9

Equat ions (34) show tha t

[iL(u)J -1 = -- iL-l(u) ~ -- i.L(-- u) .

CLASSICAL TACIIYO_N'S 25

5"6. Explicit form o] the Superluminal Lorentz transformations (SLT) in two dimensions. - In conclusion, the Superluminul Lorentz transformations i iL(u) form a group G together with both the orthochronous and the unti- chronous subluminal LTs of sect. 2 (see fig. 6). Namely, if Z(n) is the discrete

_] ~ v Y ~

L E

/ / / r

Px

Fig. 6.

group of the n-th roots of unity, then the new group G of GIJTs can be ]ormally written down as

(35) G = Z(4) | s

(36) Z(4) ~ (~/]-} -- ( + .1, --:l, --! i, -- i},

where ~ + represents here the bidimensional proper orthochronous Lorentz group. Equation (35) should be compared with eq. (5'). I t is

(37) I G e G . . > - - G e G , V G e G ;

t Go 6 ~ G e G, YGe G.

The appearance of imaginary units into eqs. (33)-(36) in only formal, as

goes into (~ :)througha~(congruenee~transformation(MAccA~O~]~etal., 1983) :

2 6 ~ . ~ c A ~ I

A c t u a l l y , the GLTs given b y eqs. (32), (33), or (35), (36), s imply represent (review I , p. 232, 233) all the space- t ime pseudorota t ions for 0~<~<360 ~

(see fig. 7). To show this, let us write down explicit ly the S L T s in the following ] o r m a l w a y :

(39)

dt - - u dx dt ' ~- -Vi - -

V ' l - u ~

d x - - u d t d x ' ~ :J: i - -

VY-: u~

(Superluminal case; u 2 < 1).

Fig. 7.

It /t I // I / / ' /

/ . /

xr /./

/ / ' /

c ) " b)

The two-dimensions1 space- t ime M(1, 1) ~ (t, x) can be regarded as a complex

plane (*); so t h a t the imag ina ry uni t

(40) i = e x p [~-in]

operates there as a 90 ~ pscudorota t ion . The same can be said, of course, for

the opera t ion 5f~, in accord with eq. (38). Moreover, wi th regard to the a x e s

x ' , t ' , x , t , bo th observers so, S' will agree in the case of a SLT t h a t f---- x,

x ' ~ t. I t follows t h a t eqs. (39) can be immedia t e ly rewri t ten as

(39')

dr' ~ •

d x r ~ :~:

dx - - u dt

dt -- u dx (Superluminal case; u ~ < 1),

(*) Or, rather, as a (( pseudoeomplex ~) plane, since in it one defines, e.g., Izl = Re z 2, and so on. Let us mention, incidentally, that an alternative formalism--which takes advantage of the fact that the Pauli algebra is the natm'al algebra for the 3-dimenslonal Euclidean space, and makes, therefore, recourse to a subalgebra of i ts--has been put forth recently (])): F~IA-Ros-~ et al., ] 985).

CLASSICAl, TACHYON8 27

where the roles of the space and the t ime co-ordinates appear interchanged, bu t the imaginary units disappeared.

Let us now take advan tage of a ve ry i m p o r t a n t s y m m e t r y p rope r ty of the

ordina.ry Loreutz boosts, expressed b y the identities

(4~)

dx -- u dt dt - - U dx

( ~ - l / u ) . dt - - u dx dx - - U dt

Equmtions (39') eventual ly can be wri t ten as

(39")

dt - - U dx

dx - - U dt dx ' : - T v/-U~ - 1

(Superluminal case, U" > 1),

which can be assumed t~s the canonical ]orm of the SLTs in two dimensions. Le t us observe t h a t eqs. (39') or (39") yield for the speed of so w.r.t . S'

dx ' (42) x - - 0 - ~ - - - - ~: -- ===F U ( u ~ < ] ,

where u, U are the speeds of the two dual f lumes s, S'. This confirms t h a t

eqs. (39'), (39") do ac tua l ly refer to Superhtminal re lat ive mot ion. E v e n for eqs. (39) one could have derived t h a t the G-vector ia l veloci ty u~ ~ dx,/d~o (see the following) changes under t r ans fo rma t ion (39) in such a w a y t h a t

/ : : : l I v %,u q' - - u ~ u ~ ; so t h a t f rom us, u~' ~ - 1 i t follows u~,u : - - 1 ( tha t is to say, b radyonic speeds are t r ans fo rmed into tachyonic speeds). We could have derived the (( re in terpre ted fo rm >) (39'), (39") f rom the original express-

ion (39) jus t demanding t h a t the second f l a m e S' move w.r.t, so wi th the Super- luminal speed U =: ]/u, as required by eq. (32).

The group 6 of the (real) GLTs in two dimensions can be finally wri t ten , wi thout a n y recourse to nonreal quant i t ies (*),

a = {+ u {- u ( - u { +

(*) Wishing to get de~(GLT) = -b 1 even in the case of the two-dimensional SLTs,

28 E. XXCAMI

See fig. 6, where the effect of the different GLTs, when s tar t ing from P, is shown: i f , = ( + ~ ) . ~ , ~ = (--~).Lr f f m = (+ ] ) ' ~ r s = ( - - ~ ) - L t

Notice t ha t the t ranscendent SLT, 5 ~, when applied to the motion of a particle, just interchanges the values of energy and impulse, as well as of t ime and space: cf. also subsect. 5"2, 5"3 (review I ; see also Vu 1977a, b).

5"7. Explicit ]orm o] GLTs. - T h e LTs and SLTs together, i.e. the GLTs, can be writ ten, of course, in a form covariant under the whole group G; namely, in (( G-covariant ~ form, they can be wri t ten (fig. 7)

(43)

dt -- u dx

d x - - u dt dx' ---

(generalized case, --c~ < u < + co),

(43a)

with

or ra ther (I%Ec~-~ and MIG~A~I, 1973a), in terms of the continuous parameter 0 e [0, 2:~],

d t ' = -07o(dt- dxtgO) dx' = ~ 'o(dx -- dt tg O) (-- ~ < ~t < + c% 0 < 0 < 2~)

(43b) cos 0 V ] -- tg~ 0

u ~ tg0, ~2 ~= ~2(0) ~ [cos01 ~ , ~ -- + i 1 _ tg~0 ] ,

ro - - + (li - tg~01)-~,

where the form (43a) of the GLTs explicitly shows how the signs in front of t', x' succeed one another as functions of u, or ra ther of 0 (see also fig. 2-4 and 6 in review l).

Apart f rom Somigliana's early paper, only recently rediscovered (C~J~])I- ~OLA et al., 1980), eqs. (39"), (43) first appeared in OL~OVSX-~ and 1%Ec)2~ (1970b, 1971), l~EcA~viI and MIGN~'r (1972), MIG~A~I et al. (1972), and t h e n - - independent ly- - in a number of subsequent papers: see, e.g., A~T~PPA (1972) and I % ~ L ~ u $ ~ and ~A~ASIV).u (1973). Equations (39'), (39") have been shown by I~EC~_MX and M~G~A~Z (1972) to be equivalent to the pioneering-- even if more complicated---equations by PXRF~E~ (1969). Only in MIaNA~I et al. (1972), however, it was first realized tha t eqs. (39)-(43) need their double sign, necessary in order tha t any GLT admits an inverse t ransformat ion (see also ~MIo~A~ and ICECAm, 1973a).

5"8. The GLTs by discrete scale trans]ormations. - I f you want , you can regard eqs. (39'), (39") us entailing a (~ reinterpretat ion ~> of eqs. (39)--such a reinter-

CLASSICAL TACI:IYON S 2 ~

pre ta t ion having nothing to do, of course, with the Stf ickelberg-Feynman (~ switching procedure ~>, also known as (( re in terpre ta t ion principle )~ ((( I~IP ~)). Our in terpre ta t ion procedure, however, not only is s t ra ightforward (cf. eqs. (38), (40)), bu t has been also rendered automatic in terms of new, scale- invar iant ((light-cone co-ordinates )~ (~r et al., 1983).

Le t us first rewrite the GLTs in a more compact form, b y the language of the discrete (real or imaginary) scale t ransformat ions (PAv~6 and I~EcA~, 1977; PAu 1978):

(15') ds '~ = 9~ ds 2 , @~ = • 1 ;

notice tha t , in eq. (36), Z(4) is nothing bu t the discrete group of the dilations !

D: xz----@x, with @2_ • 1. Namely, let us introduce the new (discrete) di lat ion-invariant co-ordinates (KASTlCUP, 1962)

(44) V , - - z x , (z ~-- 4-1 , •

being the intrinsic scale lactor ol the considered object ; and let us observe / ! ! I

tha t , under a dilation D, i t is ~ ~ ~/~ with ~ ~_ ~ x~, while ~' = 9 - ~ . Bradyons (antibradyons) correspond to ~ = + 1 (z = -- 1), whilst tachyons and antitachyons correspond to ~ = =k i. I t is interest ing tha t in the present formalism

the quadrat ic form da2~_ d~i,d~, is invariant , its sign included, under all

the GLTs :

(15 'r) da '2 = + da 2, VG e G .

Moreover, under an orthochronous Lorentz t ransformat ion Z e o~v~+, it holds t ha t ~ , = J 5 ~ , ~ .

I t fo l lows--when going back to eq. (1~), i.e. to the co-ordinates x% ~ - - t ha t the generic GLT ~ G c~a be wri t ten in two dimensions

(45) L e ~e~+ (@2, ~2 = • 1, z' ~-- ~ - l x ) .

5"9. The GLTs i n the (( light-cone co-ordinates )~. Automat ic interpretation. -

I t is known (BJo]~:E~ et al., 1971) tha t the ordinary s u b h m i n a l (proper, orthochronous) boosts along x can be wri t ten in the generic form

~_ ~-1 u, u 2 < 1 ,

in terms of the light-cone co-ordinates (fig. 8)

(46) $ ~ t - - x , $ ~- t ~- x , y , z .

30 n. ~ C A M I

~ t - x

/

t r

)f

Fig. 8.

I t is interest ing t ha t the orthochronous Lorentz boosts along x just correspond to a dilation of the co-ordinates $, $ (by the factors a and a -~, respectively, with a

any positive real number) . In par t icu lar for a -+ -[- o0 we have u -~ c- and for a --> 0 + we have u - + - - (e-). I t is apparen t t ha t a ~ e R, where R is the

(~ rap id i ty ~. The proper antiehronous Lorentz boosts correspond to the negative real

values (which still yield u~<: 1). Recalling definitions (44), let us eventual ly introduce the following scale-

invarian$ (~ l ight-cone co-ordinates ~:

(47) (p ~ ~(0)_ ~1), yj ~ ~](0j "I- ~(I~, ~(2~, ~(~.

In terms of co-ordinates (47), all the two-dimensional GLTs (both sub- and Super-luminal) can be expressed in the synthet ic form (MAcCA~ROm~ et al.,

1983)

{ d~' ~ ~dp , d y / ~ a-1 d~, (4s) ~,= o-1~, ~=-qa, as(o, + ~), o~= i x (lul>~l)'

and all of them preserve the quadrat ic form, its sign included: d~o r d~p' -~ d~ d V.

The point to be emphasized is t ha t eqs. (48) in the Superha-ninal case yield direct ly eq. (39"), i.e. t h ey automatically include the ((reinterpretat ion ~) of

eqs. (39). Moreover, eqs. (48) yield

(49)

- - 6 - -1 U- -~-

= q a , o 2 ~ •

(u2r O < a < + o ~ ) ,

i.e. also in the Superluminal ease they forward the correct (faster-than-light)

relat ive speed without any need el (~ reinterpretation )).

C L A S S I C A L T A C I { Y O ~ 8 ~] -

5"10. A n application. - As an application of eqs. (39"), (43), let ~s consider a. t achyon having (real) proper mass mo and moving with speed V relat ively to us; then we shall observe the relativistic mass

mo - - i m o ~ o

m = ( l l _ #~.1)~ :-: 0 - v~)~ = ( v - ' - 1 ) ~ (V "~ > 1, m0 real),

and, more in general (in G-covariant form),

m o (50) ~ = =L ( ] 1 - v'-I)~ ( - ~ < v < + ~),

just as ant ic ipated in fig. 4a). )~()r o ther applications, see, e.g., review I ; for instance: i) for the generalized

(~ velocity composition law ~ in two dimensions see eq. (33) and table I ia review I , ii) for the generalization of the phenomenon of Lorentz contract ion/di lat ion see fig. 8 of review I.

5"11. Dual ]rames (or objects). - Equat ions (32) and following show tha t a one-to-one correspondence

C ~ (51) v+--+ --

V

can be set between subluminal frames (or objects) with speed v < c and Super- luminal frames (or objects) with speed V - - c 2 / v > c. In such a par t icular conformal mapping (inversion) the speed e is the <( uni ted ,> one, and the speeds zero, infinite correst)ond to each other. This clarifies the meaning of observa- t ion ii), subsect. 3"2, by EInSTEIn. Of. also fig. 9, which illustrates the impor tan t eq. (32). In fact (review I), the relative speed of two (( dual ,) f rames s, S (frames dual one to the other are character ized in fig. 9 by 2~B being orthogonal to the u-~xis) is inf ini te; the figure geometrical ly del)icts, therefore, the circum- sta.nce tha t (so--~ S) = (so -> s). (s -> S), i.e. the fmldamenta l theorem of the (bidimensional) (~ extended rel~t ivi ty ~): (( The SLT: So -* S(U) is the product

P

B 1

C - - - ) t t r

s o

Fig. 9.

32 ~. R~CAM]

of the LT: so-->s(u), where u------1/U, by the transcenden$ SLT ~>: of. subseet. 5"5, eq. (32) (MIGNAm and RECAMX, 1973a).

Even in more dimensions, we shall call <~ dual >> two objects (or frames) moving along the same line with speeds satisfying eq. (51):

(51') v]7 = c = ,

i.e. with infinite relative speed. Let us notice tha t , if p , a n d / ~ , are the energy- momentum vectors of the two objects, then the condition of in/ ini te relative speed can be writ ten in G-invariant way as

(51") p , P , = O.

5"12. The <~ switching prineiplv ~> /or tachyons. - The problem of the double sign in eq. (50) has been already taken care of in sect. 2 for the ease of bradyons (eq. (9)).

Inspection of fig. 50) shows that , in the case of tachyons, it is enough a (suitable) ordinary subluminal orthochronous Lorentz t ransformation JL t to t ransform a positive-energy tachyon T into a negative-energy tachyon T'. For simplicity let us here confine ourselves, therefore, to transformations 15 --/~+ e~'~+, acting on free taehyons (see also, e.g., MARx, 1970).

On the other hand, it is well known in SR tha t the chronological order along a spaeelike pa th is not ~t+-invariant.

However, in the ease of Ts it is even clearer than in the bradyon case tha t the same t ransformation /~ which inverts the energy sign will also reverse the motion direction in t ime (review I ; R~,CxMI, 1973, 1975, 1979a; C~])n~OL~t and R~,CXM~, 1978; see also G ~ u c c I o et al., 1980). In fact, from fig. ]0 we can see t ha t for going from a positive-energy state T~ to a negative-energy state T: i t is necessary to by-pass the <~ t ranscendent ~> state T~ (with V----oo). From fig. 11a) we see moreover tha t , given in the initial frame so a tachyon T travelling, e.g., along the positive x-axis with speed Vo, the <~ critical observer ~ (i.e.

the ordinary sub]uminal observer s o - (to, ~o), seeing T with infinit~ speed)

%'%%% -moC ] �9 ,,/ /++ / /

E

E

Fig. 10.

CLASSICAL TACIIYONS 3~

'tOj / ~r / / l i

/ # / / ]## X c

/ XO ,,,.#

a)

Fig. 11.

t t

'~ l

XO

b)

is simply the one whose space axis x is superposed to the world-line OT; its

speed ur w.r.t. So, along the positive x-axis, is evident ly

(52) Uc - - c"/Vo, uoVo :-: c 2 (~( critical f rame ~)),

dnal to the taehyon speed ]~. Finally, from fig. 10 and ] lb) we conclude tha t any ((trans-critieal~) observer s' ~ ( t ' , x ' ) such tha t u ' V o > c ~ will see the t achyon T not only endowed with negative energy, bu t also travell ing backw'~rds in time. Notice, incidcnta.lly, tha t nothing of this kind happens whert u V o < O, i.e. when the final f rame moves in the direction opposite to the ta chyon's.

Therefore, Ts display negative energies in the same frames lit which they would appear as (~ going backwards in t ime ~, and vice versa. As a consequence, we cau--an( t m u s t ~ a p p l y also to tachyoas the Stf ickelberg-Feynman ~ switching procedure ~) exph)ited lit subsect. 2"1-2"3. As a. result~ point A' (fig. 5c)) or point T I (fig. 10) do not refer to a (~ negative-energy tachyon moving backwards in t ime ~), bu t r a t h e r t o a.n an t i t achyon -T moving the opposite way (in space), forward in tim% ~md wit.h positive energy. Le t ~is repeat tha t the (( switching ~ never comes into the play when the sign of u is opposite to the sign of V0 (review I ; ]~ECA)tI~ 1978c, 1979a; CALDIlCOLA and I~ECAm, 1980).

The <( switching principle ~) has been first applied to taehyons by SUDARSHA~N ,~nd co-workers (BIL~:~'IUK et al. , 1962).

Recently SCnwAxTz (t982) gave the switching procedure an interesting form~/izztion, in which- - in a sense-- i t becomes ~( automat ic ~) (see also KoaF~' and FRI}.:D, 1967).

5"13. Sources and detectors. Causa l i t y . - After the considerations in the previous subsect. 5"t2, i.e. when we apply our third postulate (sect. 4) also to tachyons, we are left with no negative energies (~EC~M~ and Mre~h_~i, 1973b) and with no motions backwards in t ime ()'[ACCAI~RONE and REC~MX, 1980a, b~ and references therein).

34 E. RECAMI

Let us remind, however, tha t a tachyon T can be transformed into an an t i tachyon T (( going the opposite way in space ~ even by (suitable) ordinary subhmina] Lorentz transformations L e ~f~+. I t is always essential, therefore, when dealing with a tachyon T, to take into proper consideration also its source and deteetor, or at least to refer T to an (( interaction region ~. Precisely, when a tachyon overcomes the divergent speed, i t passes from appearing, e.g., as a tachyon T entering (leaving) a certain interaction region to appearing as the ant i tachyon T leaving (entering) tha t interaction region (ARO~S and SUDA~SHA~, 1968; DHAR and SUDA~SEAN, 1968; (}LOCK, 1969; BALDO et al., 1970; CA~m~ZI~D, 1970). More in general, the (( trans-critical ~ transformations L e .~+ (el. fig. 11 and subsect. 5"12) lead from a T emit ted by A and absorbed by B to its Y emit ted by B and absorbed by A (see fig. 1 and 3b), and review I).

The already mentioned fact (subsect. 2"2) tha t the Stfickelberg-Feynman- Sudarshan ~, switching )> exchanges the roles of source and detector (or, if you want , of (~ cause ~) and (~ effect ~)) led to a series of apparent (~ causal paradoxes )) (see, e.g., THOULES, 1969; ROLEICK, 1969, 1972; BENFORD et a l , 1970; S~l~NAD, 1970; STRNAD and KODI~E, 1975a) which--even if easily solvable, a t least in microphysics (CALDIEOLA aud I~ECAMI, 1980, and references therein; MACCAI~- EO~E and REOA_~X, 1980a~ b; see also I~ECAIVIX 1978a, C, 1973 and references therein; TREFIL, 1978; I~ECA_Wrr and 1VIODICA, 1975; CSONKA, 1970; BALDO et al., 1970; SUDAI~SBAE, 1970; B~A~IJJK and SUDAI~SEA~, 1969b; ~EINBERG, 1967; BILANID-K et a l , 1962)--gave rise to much perplexity in the literature.

We shall deal with the causal problem in due t ime (see sect. 9), since various points should rather be discussed about tachyon mechanics, shape and behaviour, before being ready to propose and face the causal (( paradoxes ~). Let us here anticipate t ha t - - even if in E R the judgement about which is the (( cause )) and which is the (~ effect )), and even more about the very existence of a (~ causal connection ~), is relative to the observer--nevertheless at least in microphysics the law of (~ retarded causality ~) (see our third postulate) remains covariant, since any observers will always see the cause to precede its effect.

Actually, a sensible procedure to introduce Ts in relat ivi ty is postulating both a) t achyon existence and b) retarded causality, and then t ry ing to build up an E R in which the validi ty of both postulates is enforced. Till now we have seen tha t such an a t t i tude- -which extends the procedure in sect. 2 to the case of t achyons- -has already produced, among the others, the description within relat ivi ty of both mat te r and an t imat te r (Ts and Ts, and Bs and Bs).

5"14. Bradyons and tachyons. Particles and antiparticles. - Figure 6 shows, in the energy-momentum space, the existence of two different q symmetries ~, which have nothing to do one with the other.

The symmet ry particle/antiparticle is the mirror symmet ry w.r.t, the axis E ~ 0 (or, in more dimensions, to the hyperplane E = 0).

CLASSICAL TACI-IYONS ~5

The symmetry bradyon/tachyon is the mirror symmetry w.r.t, the bisectors, i.e. to the two-dimensional <(light-cone ~.

In particular, when we confine ourselves to the proper orthochronous subluminal transformations Z t e ~f~+, the <( matter ~ or <( antimatter ~ character is invariant for bradyons (but not for taehyons).

We want at this point to put forth explicitly the following simple but important argumentation. Let us consider the two <~ most typical ~ generalized frames: the frame at rest, so ~ (t, x), and its dual Superluminal frame (cf. eq. (51) and fig. 9), i.e. the frame S'~----(t', x') endowed with infinite speed w.r.t, so, The world-line of S'~ will be, of course, superposed to the x-axis. With reference to fig. 7b)~ observer S'~ will consider as time axis t' oar x-axis and as space axis x' our t-axis; and vice versa for so w.r.t. S'~. Due to the <( extended principle of relativity ~) (sect. 4), observers so, S'~ have, moreover, to be equivalent.

In space-time (fig. 1) we shall have bradyons and taehyons going both forward and backward in time (even if for each observer---e.g, for so--the particles travelling into the past have to bear negative energy, as required by our third postulate). The observer so will, of course, interpret all~sub- and Super- luminal~particles moving backwards in his time t as antiparticles; and he will be left only with objects going forward in time.

Just the same will be done, in his own frame, by observer S'~, since to him all--sub- or Super-luminal--particles travelling backwards in his time t' (i.e. moving along the negative x-direction, according to us) will appear endowed with negative energy. To see this, it is enough to remember that the transcendent transformation 5 p does exchange the values of energy and momentum (cf. eq. (38), the final part of subsect. 5"6 and review I). The same set of bradyons and taehyons will be, therefore, described by S'~ in terms of particles and antiparticles all moving along its positive time axis t'.

But, even if axes t' and x coincide, the observer so will see bradyons and tachyons moving (of course) along both the positive and the negative x-axis! In other words, we have seen the following: The fact that S'~ sees only particles and antiparticles moving along its positive t'-axis does not mean at ~11 that so sees only bradyons and tachyons travelling along his positive x-axis! This erroneous belief entered, in connection with tachyons, in the (otherwise interesting) two-dimensional approach by A~TrePA (1972), and later on contributed to lead A ~ P A and EVEl~E~T (1973) to violate sp~ce isotropy by conceiving that even in four dimensions tachyons had to move just along a unique, privileged direction--or (~tachyon corridor ~)--: see subsect. 14'4 in the following.

5"15. Totally inverted jrames. - We have seen that, when a taehyon T appears to overcome the infinite speed (fig. l la ) , b)), we must apply our third postulate, i.e. the <~ switching procedure ~. The same holds, of course, when the considered <( object ~ is a reference frame.

36 E. R~CAMI

More in general, we can regard the GLTs expressed by cqs. (35'), (36') f rom the passive, and no longer from the active, point of view ( I ~ E C ~ and I~OD~I- GUES, 1982). Ins tead of fig. 6, we get then what is depicted in fig. 12. For futm'e

x ? ~ x (-~)

V"

•

Fig. 12.

convenience, let us use the language of mult idimensional space-times. I t is apparent t ha t the four subsets of GLTs in eq. (35') describe the transit ions f rom the initial f rame so (e.g. with right-handed space axes) not only to all f rames fR moving along x wit}l all possible sl~eeds u ~ (-- 0% -{- ~ ) , btl t also to the (~totally inver ted~ frames ] ~ - - ( - - 1 ) i f = (PT)ff~ moving as well along x with all possible speeds u (cf. fig. 2-6 and 11 in review I). Wi th reference to tig. 9, we can say, loosely speaking, tha t , if an ideal f rame ] could undergo a whole tr ip along the axis (circle) of the speeds, t h e n - - a f t e r having over takeu /(oo) - - / ( U = c~)--i t would come back to rest with a le)%handed set of space

axes and with particles t ransformed into antiparticles. For fur ther details,

see I~ECA~I and I~ODRIGImS (1982) and references therein.

5"16. About CPT. - Le t us first remind (subsect. 5"5) tha t the product of two

SLTs (which is always a subluminal LT) can yield a t ransformat ion both ortho-

chronous, L r ~ ~r and antichronous, (-- 1 ) 'Z r ------ (PT)/~ r ---- Z ~ e~r (of. subsect. 2"3). We can then give eq. (10) the following meaning within EIr

Le t us consider in part icular (of. fig. 7a), b)) the ant ichronous GLT(0----- ----]80 ~ ) - - - - - -1 - - P T . In order to reach the value 0----180 ~ star t ing from 0 ~ 0 we must pass the case 0 - - 9 0 ~ (see fig. 12), where the switching

procedure has to be applied (third postulate). Therefore,

(53) GLT(0 = 180 ~ ) ~ -- I ~ ]?T = C P T .

CLASSICAL TACHYO.NS 37

The (( to ta l inversion )) -- 1 - - PT ---- CPT is nothing bu t a par t icular (( rota- t ion ~ in space-time, and we saw the GLTs to consist of all the space-t ime ~( rotat ions ~ (subsect. 5"6). In other words, we can write CPT e G, and the CPT theorem m a y be regarded as a partic~flar, explicit reqlf irement of SI~ (as formula ted in sect. 2), and a ]ortiori of El~ (MIG~A~-I and ICF~OA.~, 1974b, 1975a, and references therein; l~EChm and ZIINO, 1976; ]?AVglO and REOA_M~,

1982). Notice that , in our formalization, the operator CPT is linear and uni tary. F u r the r considerations will be added in connection with the mult idimen-

sional cases (see subsect. 11"1-113).

5"17. Laws and descriptions. Interactions and objeets . - Given a cer ta in phenomenon ph, the principle of re la t iv i ty (first postulate) reqnires two different inertial observers O1, O~ to find t ha t ph is ruled by the same physical laws, bu t it does not require at all O1, O~ to give the same description of ph (cf., e.g., review I ; p. 555 in RECAm, 1979a; p. 715, appendix, in I~ECAMI and 1~ 0])RIGUES, 1982).

We have already seen in EI~ tha t , whilst the (~ re ta rded causali ty ~ is a /aw (corollary of our th i rd postulate), the assignment of the ~r cause a and (( effect ~) labels is relative to the observer (CAM~.~ZrND, 1970) and is to be considered a description detail (so as, for instance, the observed colour of an object).

In EI~ one has to become acquainted with the fact t ha t m a n y descriptioll details, which by chance were Lorentz invariunt in ordinary SR, are no longer invariant under the GLTs. For example, what already said (see subsect. 2"3, point e)) with regard to the possible noninvariance of the sign of the addi t ive charges raider the t ransformat ions L e .~+ holds a ]ortiori under the GLTs, i.e. in EI~. l~Ievertheless, the to ta l charge of an isolated system will have, of course, to be constant during the t ime evolution of the system-- i .e , to be con- served---as seen by any observer (of. also sect. 15).

Le t us refer to the explicit example in fig. 13 (FEI~BF_~G, 1967; BALDO et al., 1970), where the pictures a), b) are the different descriptions of the same interact ion given by two different (ge.uerMized) observers. F o r instance, a) aud b) can be regarded as the descriptions, f rom two ordinary subluminal frames O1, O~, of one and the same process involving the tachyons a, b (c can be a photon,

e.g.). I t is apparen t tha t , before the interact ion, Ox sees one tachyon, while O2

~) ~ * ~ b) t o

Fig. 13.

38 ~. R E C A m

sees two tachyons. Therefore, the very number of particles--e.g, of tacbyons, if we consider only subluminal frames and LTs---observed at a certain time instant is not Lorentz invariant. However, the total number of particles participating in the reaction either in the initial or in the fiual state is Lorentz invariant (due to our initial three postulates). In a sense, EI~ prompts us to deal in physics with interactions rather than with objects (in quantum- mechanical language, with (~ amplitudes ~ rather than with (~ states ~>) (of., e.g., G]~?d~, 1969; BALD0 and I~ECA~, ,1969).

Long ago BALDO et al. (1970) introduced, however, a vector space H - ~ = ~ ( • direct product of two vector spaces ~V' and ~ , in such a way that any Lorentz transformation was unitary in the H-space even in the presence of tachyons The spaces ~ ( ~ ) were defined as the vector spaces spanned by the states representing particles and antiparticles only in the initial (final) state. Another w~,y out, at the classical level, has been recently put forth by SdEWA]~TZ (1982).

5"18. SR with taohyons in two dimensions . - Fm'ther developments of the classical theory for tachyons in two dimensions, after what precedes, can be easily extracted, for example, from review I and references therein, I~ECA~_~ (1978b, 1979a), CO,BEN (1975, 1976, 1978a, b), CALDZROLA and REOAMI (1980), MAd0ARI~ONE and RECA~ (1980b, 1982a), MACdAm~ONE e t a l . (1983).

3: X !

//\ / / / \ /'/

2 / / ./....;lb./., / ,, ,,: . . . . L . . . / , : , " ,,

Fig . 14.

/./'

X f l l

I I

I I I 2 t I

3 . I I

i I I

/ I !

! A /

I I

I

C L A S S I C A L T A C I I Y O ~ S 39

We merely refer here to those pgpers, and references therein. Bu t the m~ny

positive aspects and meaningful results of the two-dimensional EI~--e.g., connected with the deeper comprehension of the ordinary relativistic physics tha t it affords--will be apparen t (besides f rom this sect. 5) also from the future sections dealing with the mult idimensional cases.

In part icular, fur ther subtleties of the so-called (( causali ty problem ~ (a problem already faced in subsect. 5"12-5"14) will be tackled in sect. 9.

Here we shall only make the following (simple, bu t impor tant ) remark. Le t us consider two (bradyonic) bodies A, B t h a t for ins tance- -owing to mutua l attraction--accelerate while approaching each other. The situation is sketched in fig. t4 , where A is chosen as the reference f rame s ~ (t, x) and, for simplicity, only one discrete veloci ty change is depicted. F ro m a Superluminal f rame they will be described ei ther as two tachyons tha t decelerate while approaching each other (in the f rame S " ~ (t", x")), or as two (anti) tachyons that accelerate while receding f rom each other (frame S ' - - (t', x')). Therefore, we expect t ha t

two tachyons Item the kinematical point of view will seem to suffer a repul- sion, if t hey a t t r ac t each other in their own rest frames (and in the other frames in which they are subluminal); we shall, however, see tha t such a behaviour of tachyons m a y be still cons ide red~f rom the energetic~l and dynamical points of v iew--as due to an attraction.

Before going on, let us explicitly r emark tha t the results of the model theory in two dimensions strongly p rompt us go a t t e m p t building up a similar theory (based as far as possible on the same postulates), also in more dimensions.

6. - Tachyons in four dimensions: results independent of the existence of SLTs.

6"1. Caveats. - We have seen tha t a model theory of E t t in two dimensions can be s t ra ightforwardly buil t np (sect. 5).

We have also ant ic ipated (subsect. 3"2) tha t the construct ion of an EI~ is s t ra ightforward as well in the psendo-Euclidean space-times M(n, n), and in subsect. 14"3 we shall approach the case n =- 3 (NIxG•ANI and I~ECA~, ]976b; M~coAI~I~Ol~ and I~CAMI, 1982a, 1984a).

In the 4-dimensional Minkowski space-time M(1, 3), however, if we want

a priori to enforce the principle of re la t iv i ty for bo th s~lb- and Super-luminal (inertial) frames, it comes the following (cf. fig. 7a), b)). Our own world-line

coincides with our t ime axis t; the world-line t' of a t ranscendent (infinite speed) free t achyon moving along *he x-axis wiU coincide, on the contrary, with our

x-axis (in our language). The t ranscendent observer would then call t ime axis (t') what we call x-axis, and analogously would consider our axes t, y, z ~s his three space axes x', y', z'. Conversely, due to our first two postulates (i.e. to the requirements in subsect. 4"2), he would seem to possess one space

axis and three t ime axes! (MAccAI~I~OI~E and I~ECAMI, 1982a, b, and references

4 0 E . R~CAMI

therein ; R~.:(~A~r ] 979a). This point consti tutes the problem of the 4-dimensionM EIr i.e. of the SLTs in four dimensions. We shall deal with it in sect. 14.

In four dimensions, however, we can s ta r t as a first step by s tudying the behaviour of tachyons within the (~ weak approach ~) (subsect. 3"2), i.e. confining preliminari ly the observers to be all subluminal. In this section~ therefore, we shall only assume the existence of sub- and Super-luminM (observed) objects. Tachyons are the spacelike ones, for which in four dimensions it: is ds 2 _~ dt -~ -- dx ~ < 0 and (mo real)

(29c) ~--- - - m o < O .

To go on, therefore, we need only the results in subsect. 5"12, 5"33, which do not imply q.ny SLT. Those results remain, moreover, wdid in four dimensions

t " (see subscc ~. 5 12 and 2"1), provided tha t one takes into account the fact t ha t the relevant speed is now the component V~ of the tachyon velocity V along the (subluminal) boost direction (review I ; MACOA]CRO~E et al.~ 1983~ p. 108; MAC- CAn~O~E and I~EOA~I, 1984a~ sect. 8). ~Namely, if u is the (subluminM) boost velocity, t hen the new observer s' will see instead of the initial t achyon T an an t i t achyon T travell ing the opposite way ((( switching principle ~>) if and only if (MACCA_~RO~E and l~]~c~-~, 1980b)

(52') u" V > c ~ .

l~emember once more that , if u. V is nega.tive, the <~ switching ~ does never

come into play. As an example of results t ha t do not depend on the very existence of SLTs,

let us ct)nsider some tachyon kinematics.

6"2. On tachyon kinematics. - Let us first explore the unusual and un-

expected kinematicM consequences of the mere fact t ha t in the case of tachyons

(see eq. (29c)) it holds

( ' ~ t ' = - - o ) [~{ _{_~/'p2 m~ (too real, V~>:l),

as part ia l ly depicted in fig. 4. To begin with, let us recM1 (F]~I~BEnO, 1967; DIIA~ and SU])AnSt[A~, 1968;

review I) t ha t a b radyon at r e s t - - fo r instance a pro ton p - - , when absorbing a t achyon or an t i t achyon t, may transform into itself: p Jr t -> p. This can bc easily verified (see the following) in the rest f rame of the initial proton. I t can be similarly verified tha t , in the same frame~ the proton cannot decay into itself plus a tachyon. However , if we pass from tha t initiM frame to another

subluminal l rame moving, e.g., along the x-axis with positive speed u ---- u, > > 1/V~ (where V~, assumed to be positive too, is the veloci ty x-component of t), we know from subseet. 5"12-5"1~ tha t in the new frame the t achyon t

C L A S S I C A L TACHYONS 41

entering the above react ion will appear as an outgoing an t i t achyon : p - , p +~ t . I n other words, a p ro ton in flight (but not at res t !) m a y a priori be seen to decay into itself plus a t a chyon (or ant i taehyon) .

Le t us examine the tachyon kinematics with any care, due to its essential

role in the proper discussion of the causality problems and in the e lementary- part icle realm.

6"3. (( In t r ins ic emission ~) o] a tachyon. - Firs t ly , let us describe (MACCAR-

]CONE and I~ECAM~, 1980a, b, and references therein) the phenomenon of (~ intrinsic emission )) of a tachyon, as seen in the rest f r ame of the emit t ing body,

and in generic f rames as well. :Namely, let us first consider in its rest ]rame a

bradyonic body C, with initial rest mass M, which emits towards a second

bradyonic body D a t achyon (or ant i taehyon) T, endowed with (real) rest mass

m ~ m o and 4 - m o m e n t u m p = (ET,p) and travell ing with speed V in the x-direction. Le t M ' be the final rest mass of the body C. The 4 - m o m e n t u m conservat ion requires

(55) M = ~ / p ~ - - m s + ~/p2 + M,~ (rest f rame) ,

t h a t is to say

(55') 2Mlpl = VEmS 4 (M,s_ ~)]~ + 4m=M~,

wheref rom it follows t h a t a body (or particle) C cannot emi t in its rest f r ame

any t achyon T (whatever its rest mass m be), unless the rest mass M of C (( j u m p s ~) classically to a lower value M ' such t h a t (E~ ~ + ~ / ~ )

(56)

so that

(57)

A ~ M ' s - M s = - - m s - 2 M E ~

__ M z < A < - - p ~ < - - m 2

(emission),

(emission).

Equa t ion (55') can read

(55") V = 1 / 1 + 4 m 2 M 2 / ( m ~ ~- A) 2 .

I n par t icular , since infinite-speed Ts car ry zero energy bu t nonzero impulse,

IP] = m o c , then C cannot emi t any t ranscendent t achyon wi thout lowering

its rest mass ; in fact , in the case of infinite-speed T emission, i.e. when E T = 0 (in the rest f r ame of C), eq. (56) yields

(58) A - - m 2 (V---- ~ , ET = 0).

Since emission of t ranscendent t achyons (ant i tachyons) is equivalent to absorp-

4 2 E . ~ c A ~ i

t iou of tr~mscendent ant i tachyous (tachyons), we shall get again eq. (58) also as a l imiting case of tachyon absorptio~l (el. eq. (64)).

I t is ess(mtial to notice tha t A is, of course, an invariant quant i ty ; ill fact , in a generic f rame ], eq. (56) can be read

(59) / I ~ - - m 2 - - 2 p a P l ~ ,

where P~ is now the initiM 4-momentum of body C w.r.t, the generic f rame ]. I t is still apparent t ha t - - M ~ < ~ < - - m 2. I f we recall (ef. eq. (51")) t ha t two objects h:~ving infinite relat ive speed possess orthogonal 4-momenta

(5[") p~.P~ -- 0

we get agMu eq. (58) for the case in which T is t ranscendent w.r.t, body C.

6"4. Warnings . - The word (( emission ~ in eq. (57) aims at indi(~ating-- let us r e p e a t - - a n intrinsic, (~ proper ~ behaviour , in tihe sense tha t it refers to emission (as seen) in the rest f rame of the emit t ing body or particle. In suitable moving frames ], snch an (~ emission ~) can even appear as an absorp- Lion.

Conversely, other (s~litably moving) frames ]' can observe a T emission from C (in flight), which does not satisfy inequalities (57), since it corresponds in the rest f rame of C to an (intrinsic) absorption.

Howcver , i f - - in the moving f rame l - - inequal i t ies (57) appear to be sar tied, this implies tha t in the C rest f rame the process under examinat ion is a t achyon emission, both when ] observes ~n actual emission and when 1 observes on the contrary an absorption. We can state the following theorem:

Theorem 1. (~ ~Nccessary and sufficient condition for a process, observed ei ther as the emission or as the M)sorption of a t achyon T by ~ b radyon C, *o be a Lachyon emission in the C res~ f rame-- / .e , to be, an (~ in tr insie emis-

sion ~>--is t ha t during the process C lowers its rest mass (invariant s ta tement! ) in such a way tha t - - M 2 < A < - m 2,, where M, m, A are defined above.

Le t us ant icipate tha t , in the case of (~ intrinsic absorption )), relat ion (62')

will hold instead of relation (57); and let us observe the following. Since the (inva~iant) quan t i ty A in relation (62 ~) can assume also posit ive values

(coI~trary to the case of eqs. (56), (57)), if an observer ] sees body C to increase

its rest mass in the process, then the (~ proper descr ip t ion , of the process can

be nothing bu t an intrinsic absorption. Le t us stress once agMn tha t the body C, when in flight, can appear to emit

suitable tachyons wi thout lowering (or even cha~ging) its rest mass : in part icular, a part icle in ]light can a priori emit a suitable t achyon t t ransforming

CLASSICAL TACI'KYONS 4~

into itself. But. in such cases, if we pass to the rest, f rame of the initial part.icle, the (~ emi t ted ~) t ' tchyon appears then as an absorbed aut i tachyon t.

At last~ when A in eqs. (56)-(59) can assume only known discrete values

(so as in elcmentary-I3article physics), them--once M is fixed---cq. (56) imposes a link between m and ET, i.e. between m and ]p[.

6"5. (( I n t r i n s i c absorpt ion ~ o] a tachyon. - Secondly, let us consider (MAC- CA~UO~E and I~J~CA_M~, ]980a, b) oltr b radyon C, with rest mass M, to absorb now i n its rest / t a m e a tacbyon (or ant i tachyon) T' which is endowed with (real) rest mass m and 4-momentum p---- ( E T , p ) , in emi t ted by a second bradyon D, and travels with speed V (e.g., along the x-direction).

The 4-momentum col~servation requires t h a t

(60) M -t V/p ~ - m ~ = V/p '-' -~ M '~ (rest frame),

wherefrom it follows t ha t a. body (or particle) (3 at rest can a pr io r i absorb (suitabh~) l, achyons both when increasing or lowering its rest mass ~nd wher~ conserving it. Precisely~ eq. (60) gives

(61.) I P [ : 2M1 V/i.m. ) ,_ A) 2 . 4m. 2 Mo " (rest frame)

which corresponds to

( 6 2 )

so tha t

(62')

zJ = - - m ~- + 2 M E T ,

- - m ~ < A < ~ (absorption).

Equal~ion (6~) tells us t ha t body C in its rest: f rame can absorb T' only when the lachyon speed is

(63) V ----- V l - - l - . i ~ ; ~ M o ' / ( ' m ~ ! A { .

Nol,ice th.~t eq. (62) differs f rom eq. (56), such a difference being in agreement with the fact tha t , if bradyort C varies its veloci ty w.r . t . t . acbyon T', then--i~l

the C rest frame---cq. (60) can t ransform into eq. (55) ((ft. subsect. 5"]2-5"]4). Equatiolls (61), (63) formally coincide, on the contrary, with eqs. (55'), (55"),

respectively; but they refer to different domains of A: in eq. (55") we have

A < - - m ~, whih, in eq.(63) we have A > - - m " . In part icular , eq. (63) yields tha t C can absorb (in its rest frame) infinite-

speed tachyons only when m 2 + A ~ O, i.e.

(64) V ~ oo <=> A ---- -- m 2 (rest frame)

in agreement with eq. (58), as expected.

~ E. ~ECAMI

Quan t i ty A, of course, is again invariant. I n a generic f rame / eq. (62) can be wr i t ten

(65) A -- -- m ~ ~- 2p~/~",

1 ' , being now the init ial C-foatr-momentum in /. Still A ~>-- m ~. Notice also

here t h a t the word absorpt ion in eq. (62') means (~ intrinsic absorpt ion ~, since i t refers to ((absorption (as seen) in the rest f r ame of the absorbing body or part icle >>. This means tha t , if a moving observer / sees relat ion (62) being

satisfied, the (~ intrinsic ~> description of the process, in the C rest f rame, is

a t achyon absorpt ion, both when / observes an ac tual absorpt ion and when J

observes on the cont ra ry an emission. Le t us s ta te the following theo rem:

Theorem 2. (r Necessary and sufficient condition for a process, observed

e i ther as the emission or as the absorpt ion of a t achyon T ' b y a b r adyon C,

to be a t achyon absoI:ption in the C re, st f r amc- - / . e , to be an "intrinsic absorp-

t ion"- - i s t ha t - - m~"<A < @ ~ . ~) I n the par t icular case A - - 0, one s imply

gets

2ME~ = m" ( M ' = M).

When A in eqs. (61)-(65) can assume only known discrete values (just as

in e lementary-par t ic le physics), then---once M is fixed---eqs. (61)-(65) provide

a l ink between m and E T (or IP], or V).

6"6. Remarks. - We shall now describe the tachyon exchange between two bradyonic bodies (or particles) A and B, because of its impor tance not only for causa l i ty bu t possibly also for part icle physics. We have to write down l)he implicat ions of the 4-momentl~m conservat ion at A and at B; in order to de so we need choosing a single f r ame wherefrom to describe the processes

bo th a t A and a t B. Le t us choose the rest f r ame of A. However , before going on, let us explici t ly r e m a r k the i m p o r t ' m r fact tha t ,

when bodies A and B exchange one t a chyon T, the unusual t achyon kinemat ics

is such t h a t the (~ intrinsic descriptions ~ of the processes a t A and at B (in which

the process a t A is described f rom the rest f r ame of A and the process a t B is

now described f rom the res t f r ame of B) can a priori be of the fo l lowing /our

types (MACCAI~I~0NE and ] ~ E c ~ , 1980a, b):

(66)

i) emiss ion-absorpt ion ,

ii) absorpt ion-emiss ion,

iii) emission-emission,

iv) absorp t ion-absorp t ion .

Notice t h a t the possible cases are not only i) and ii). Case iii) can take place


only when the t achyon exchange happens in the receding phase (i.e. while A, B are receding f rom each other) ; case iv) can t ake place only when the t a chyon exchange happens in the approaching phase (i.e. when A~ B are approaching to each other).

Le t us repea t t h a t the descriptions i)-iv) above do not refer to one and

the same observer , b a t on the con t ra ry add together the (~ local )) descriptions

of observers A and B.

6"7. A prel iminary application. - For instance~ let us consider an elastic

scat ter ing be tween two (different) part icles a, b. I n the c.m.s., as is well known,

a and b exchange m o m e n t u m bu t no energy. While no b radyons can be the realistic carriers of such an interact ion, a a infinite-speed tuchyon T can be

on the con t ra ry a sui table in terac t ion carrier (notice t h a t T will appea r as finite-speed t a c hyon in the a, b rest frames). However , if a, b have to re ta in

the i r rest mass during the process, t hen the t achyon exchange can describe t h a t elastic process only when (~ intrinsic absorpt ions ~> t ake place bo th at a

and at b (and this can happen only when a, b are approaching to each other).

6"8. Tachyon exchange when u.V<~ c ~. Case o] (~ intrinsic emission )) at A . - Le t V, u be the velocities of the t achyon T and the br~dyonic b o d y B,

respect ively, in the rest f r ame of A. And let ns consider A, B to exchange a t a chyon (or an t i tachyon) T when u. V < e ~. In the rest f r ame of A we can

have ei ther intrinsic emission or intrinsic absorpt ion f rom the b radyonic body A. Incidental ly , the case u. V < e 2 includes bo th t achyon exchanges in the (~ approaching phase ~) (for intrinsic T emission a t A) and in the (( recession phase )) (for intrinsic T absorpt ion a t A).

Le t us first confine ourselves to the case when one observes in the A rest

f r ame an (intrinsic) t achyon emission f rom A. I n such s case both A and B will see the exchanged t achyon to be emi t t ed b y A and absorbed b y B. I n fact , the observer B would see an an t i t achyon T (travell ing the opposi te way

in space w.r.t, t a c h y o n T, according to the (( switching principle )>) only when u . V > e a, whilst in the present case u . V < c 2.

Impos ing the 4 - m o m e n t u m conservat ion a t A, we get in the A rest f r ame all the eqs. (55)-(59), where for fu ture clar i ty a subscr ipt A should be in t roduced

to ident i fy the quant i t ies (MA, M'A, AA, P~A) per ta in ing to A.

Le t us remain in the res t f r ame of A, and s tudy now the k inemat ica l condi-

t ions under which the t achyon T emi t t ed by A can be absorbed b y the second

body B. Le t M s and P s ~ (Es, Ps) be res t mass and l - m o m e n t u m of body B, respectively. Then

(67)

where M ' n is the B final mass. Le t us define A s ~ M ~ - M~, which rends

46 r . aF~CA~I

arc the relat ivist ic masses of T and B, respect ively, and a ~-- ttV. The invar ian t

quan t i ty A s in a generic f rame ] would be wr i t ten

(68) As = -- m 2 -~- 2puP~

with pz,, P~ the T and B fou r -momen ta in ]. At var iance with the process at A

(intrinsic emission: eq. (56)), now A s can a priori be both negat ive and posi t ive

or null:

(69) - -m2<~As < "- cr (intrinsic absorpt ion) .

lqotiec tha.t, if relat ion (69) is verified, then the process at B will appea r in

the B rest f r ame as an (intrinsic) absorpt ion, wha tever the descript ion of the

process given by ] m a y be. Of course, the k inemat ics associated with eq. (67)

is such t h a t A, can even be smaller t h a n - - mS; bu t such a case ( u V cos ~ > 1)

would correspond to intrinsic emission a t B (and no longer to intrinsic absorp-

tion). In conchlsion, the t achyon exchange here considered is allowed when in

the A rest f r ame the following equat ions are s imul taneously s.~tisfied:

(70)

wi th

(70')

{ /Ix = -- m~ 2M'AET , " - V ) As = - - m ~ + -~ETEn(1 u" ,

- - / ~ < A A < - - m ~ , A B > - - m 2 .

When B is at rest w.r.t. A we recover subseet. 6"5.

Di t Ierent ly f rom A~, quan t i ty A s can even vanish; in this case the second of eqs. (70) simplifies into 2ETE~(1-- u. V ) : m 2. I n the very par t icu lar case when bo th P~ and A~ are null, we get V = V/1 -~ :4M~/mL Fur the r details can be found in MACCAR140~E and ]~ECAMI (1980b), which const i tutes the basis

also of subsect. 6 " 9 - 6 " 1 3 .

6"9. The ease o/ (~ intrinsic absorption )) at A (when it. V < e2). - L e t us consider

t achyon exchanges such t h a t the process a t A appears , in the A rest f rame,

as all (intrinsic) absorpt ion. The condition u. V < e 2 then implies body B to

appea r as emitt ing the t achyon T bo th in the A rest f r ame and in its own rest

f rame. The present ease, there]ore, is just the symmetrical o] the previous one (sub-

sect. 6"8), the only difference being t ha t we are now in the rest f r ame of the

absorbing body A. In conchlsion, this t achyon exchange is allowed when

eqs. (70) are s imultaneously satisfied, bu t with

(71) A A > ~ - - m s, - - M ~ < A ~ < - - m 2 �9


In the par t icular case in which B moves along the same mot ion line. as T

(along the x-axis, let us say), so t h a t P~n( - l -p ) , then

(72) " '~ --- ' .J/~,!p[ E,, x / ( ~ - - , An) ~ t -4m2M~ -T (m2-~ A,)IP~[ (P~V(--':p));

whilst for the analogous s i tuat ion of the case in subsect. 6"8 we WOlfld have obta incd (owing to evident s y m m e t r y reasons) eq. (72) with opposite signs in

its r.h.s. Moreover, when B is a t rest w.r.t, body A, so t ha t P~ = 0, we recover mutat is mutand i s eq. (55'), still wi th - - M~ < Az < - - m ~.

6" 10. Tachyon exchange with u . V>~ c 2. Case oJ ~ intr insic emiss ion ~ at A. -

Still in the A resl, f rame, let us now consider A, B to exchange a t achyon T

wheu u. V>~c '~. Again we can have ei ther <( intrinsic emission ~> or (( intrinsic

absorpt ion ~) a t A. The present cases di]]er f rom the previous ones (subsect. 6"8~ 6"9) in the fact t ha t n o w - - d u e to the (( switching procedure ~) (cf. the th i rd postu-

l a t e ) - - a n y process described by A as a T emission at A and a T "rbsorptiou

a t B is described in the B rest f r ame as a T absorpt ion a t A and a T emission a~, B, respectively.

Le t us analyse the ease of (( intrinsic e m i s s i o n , by body A. Due to the condi-

t ion u . V > c'- (el. eq. (52')) and to the consequent <~ switching ~, in the, rest fl 'ame of B one t.hen observes an an t iaehyon T emi t t ed b y B and absorbed

b y A. Necessary condition for this case to t ake place is t ha t A, B be receding f rom each other (i.e. be in the (~ recession phase ~)).

In any case, for the process a t & (in the A rest f rame) we get the same k inemat ics a l ready e x p m m d e d in subsect. 6"8 and 6"3.

As to the process a t B, i~l the A res t f r ame the body B is observed to absorb

a t achyon T, so t h a t eq. (67) holds. In the B rest f rame, however , one observes an (int.rinsic) T emission, so tha.t theorem 1 is here in order: namely , - - M~ <

< A~ < - - m 2. Notiee tha t , when passing f rom the A to the B rest f r ame (and apply ing the switching procedure), in eq. (67) one has i) t ha t quanti ty E T

changes sign, so t ha t quanti ty ~/p"- - m " appears added to the r.h.s., and no longer

to the 1.h.s.; ii) t h a t the t achyon 3 - m o m e n t u m p changes sign as well (we go in fac t f rom a t achyon T with impulse p to its an t i t achyon T with impulse - - p ) . Cf. a l so KORFF a,lid FRIED (1967).

I n conclusion, the t achyon exchange is k inemat ica l ly allowed when the

two equations (70) are s imul taneously verified, bu t now with

(73) - - M~ < A : < - - . m ~, - - M~ < A ~ < ~ - - m 2.

In the par t icular case when PB and p are eol l inear (we cart have only Psl lP:

recession l)hase), we get

(74) 2M~[R] == En V ' i m 2 '.- A~) "2 [ - 4 m 2 M ~ =- (m2-1 - &)lP l (P~!lp),

with A~ in ~,he range given hy eq. (73).

48 ~. R~CA~I

6"11. The case o] ~ intrinsic absorption ~ at A (when u. V>~ cs). - Due to the present condit ion u. V>~ c s, and to the consequent (~ switching ,, if we observe

the body A in its own res t f r ame to absorb (intrinsically) a t aehyon T, t hen in

the B res t f r ame we shall observe an an t i t achyon T emi t t ed b y A. Necessary condition for this case to t ake place is t h a t A, B be approaching to each o ther

(i.e. be in the (~ approaching phase ~). I n a n y case, for the process a t A in the A res t f r ame we obta in the same

k inemat ics as expounded in subsect. 6"9 and 6"5. As to the process a t B, in

the A res t f r ame the body B is observed to emi t a t achyon T:

(75) V~'~ + M~ = V p ~ - ~ + ViP~ - p ) ~ + ~ ;

m

in the B res t f rame, however , one would observe an (intrinsic) T absorpt ion,

so t h a t i t m u s t be A~>~- -m s. I n conclusion, the present t achyon exchange is k inemat ica l ly allowed when

eqs. (70) are satisfied, bu t now wi th

(76) A A > - - m s, A ~ - - m S .

I n the par t icular case in which P~ and p are collinear, we can have only (-- P~)lip

(approaching phase), and we get

(77) 2M2]pI--~ E~ V ( m s ~- A~) 2 + 4m~M~ - (m ~ Q- AB)]PB[ (PBi[(--P)) ,

wi th A~ ~ - - ms. Final ly, let us recall t h a t in the present case ((( intrinsic a b s o r p t i o n s , a t

B and a t A) bo th quant i t ies AA, A~ can vanish. When A A ---- 0, we s imply get

2MAE v ---- ms. In the par t icular case when A~ ~ 0 one gets 2 E ~ E B ( u . V - 1) ---- = m s, and t hen [p] = (m/2M~) [EB(m s § 4M~) t - mlP~l].

6"12. Conclusion about the tachyon exchange. - With regard to the process

a t B, the k inemat ica l results of subsect. 6"8-6"11 yield wha t follows (M),CCAR-

~o~v, and R E c k , 1980b):

(78a) u. V<cs:

(78b) u- V ) c 2 :

---- - - m s -~ 2p~ P ~ > - - m S ,

AB ~ ~-- _ mS _ 2p~ P~ < -- m s ;

_-- - - m S -~ 2p~ P C < - - mS ,

A s ~ ~ _ mS _ 2 p ~ P C ~ - - mS �9

More specifically, the k inemat ica l conditions for a t achyon to be exchangeable be tween A and B can be summar ized as follows (notice t h a t the case u . V < c s

includes, of course, the case u . V ~ 0):

CLASSICAL TACIIYONS 49

a) in the case of ((intrinsic cnlission ~> a t A :

(79) u . V < c" > An > - - m 2 ~ intr insic abso rp t ion a t B ,

u . V > c ~ > l1 n < - - m 2 ~ intr insic emission a t B ;

b) in the case of ((intrinsic abso rp t ion ~ at A :

(s0) u . V < c ~ ~ / | ~ 3 < - - m "~ ~ intr insic emission a t B ,

u . V > c" =~ -/]B > -- m2 ::~ intr insic abso rp t ion a t B .

6"13. A p p l i c a t i o n s to e lementary-par t ic le p h y s i c s : examples . 1 'aehyons as

(( i n t e rna l l ines ~. - Let us recall t ha t , when e l e m e n t a r y in te rac t ions arc considered

to be m e d i a t e d b y e x c h a n g e d objects , no o rd ina ry (bradyonic) par t ic les can

be the classical, (( realist ic )) carr iers of the t r ans fe r red e n e r g y - m o m e n t u m .

On the con t r a ry , c]assical l a c h y o n s - - i n place of the so-called v ir tua l p a r t i c l e s - -

can a p r io r i ac t as the ac tua l carr iers of the f m t d a m e n t a l snbnuc lca r in terac t ions .

Fo r ins tance, a n y elastic sca t t e r ing can be r ega rded as classically (((real-

is t ical ly )>) med ia t ed b y a sui table t a c h y o n exchange dlu ' ing the a p p r o a c h i n g

phase of the two bodies (cf. subsect . 6"7). I n such a case, eqs. (70), (76)

read, a lways in the A rest f r ame (A A = A R = 0),

(s~) ET ---- m~ , E~ : M A / ( u ' V - - 1) ,

where the a n g u l a r - m o m e n t u m conse rva t ion is no t considered. I n the c.m.s.

we would h a v e Ip~[ = ]Pn] ~ ]P] and

(82) cosO ..... := :1 21P] ~ (elastie sca t te r ing) ,

so t h a t (once !P] is fixed) for each t a c h y o n mass m we ge t a par t ic l l lar 0 .... ;

if m assumes only discrete v a l u e s - - a s expec ted f rom the dua l i ty principle,

subscct . S ' I - - , the~.l 0 reslflts to be classical ly (( quant i zed ~>, a p a r t f rom the cyl indr ica l s y m m e t r y .

More in genera], for cac, h discrete va lue of the Cachyon mass m, the q u a n t i t y

0 .... assumes a. discrete value too, which is me re ly a func t ion of [PI. These

e l e m e n t a r y cons idera t ions neglect the possible mass wid th of the t a c h y o n i c

(( resonances ~) (e.g., of the t a c h y o n mesons) . Le t us recall f rom subsect . 5"3, 6"7

t h a t in the c.m.s, a n y elastic sca t t e r ing appears classically as mediate( t b y an

infini te-speed t a c h y o n h a v i n g p , :~= (0, p ) , wit.h [p] = m. Moreover~ eqs. (81)

impose a l ink be tween m and the d i rec t ion of p , or r a t h e r be tween m and

a ~ p P (where we can choose P = P s ; r e m e m b e r t h a t P~ = -- PA):

m (83) c o s ~ .... - - 2 ] P I ;

5 0 ~ . R ~ C A ~ I

again we find (once IPI is given, and if the in termedia te- tachyon masses are discrete) t ha t also the exchanged 3-momentum results to be (classically) (( quan-

t ized ~ in bo th its magni tude and direction. In part icular , for each discrete value of m, also the exchanged 3-momentum assumes one discrete direction (except, again, for the cylindrical symmetry) , which is a funct ion only of [P].

I t is essential to notice tha t such results cannot be obta ined at the classical level when confining ourselves only to ordinary particles, for the mere fact t ha t bradyons are not allowed by kinematics to be the interact ion carriers.

Of course, also the nonelastic scat ter ing can be regarded us mediated b y suitable t achyon exchanges. We shall come back to this in the following

(subsect. 13"2).

6"14. O n the var ia t iona l p r i n c i p l e : a tentative digress ion. - After having

expounded some tachyon mechanics in subsect. 6"2-6"12, let us tu rn a bi t our b

a t ten t ion to the action S for a free object. In the ordinary case, it is S = ~fds; for a free t achyon let us, ra ther , write

(8~) s = f ldsl �9

By analogy with the bradyonic case, we might assume for a free t achyon the

Lagrangian (c = 1)

(85) L = -~- m0 V ~ -- 1 ( [ - 2 ) 1),

and, therefore, evaluate, in the usual way,

~.L mo V - - _ _ .---~ m V , (86) P -= ~ v + V'I z o : - :1

which suggests eq. (50) to hold in the four-dimensional case too:

m 0 (50') m -

~ / ~ - 1

I f the t achyon is no longer free, we can write as usual

( 8 7 ) F - d p __ d [ moV dt dt W V : - i !

By choosing the reference frame, at the considered t ime ins tant t, in such a

CLASSICAL TACH YOlkS 5 1

way that V is parallel to the x-axis, i.e. IV] : V~, we get then

1 V ~ ] mo (88a) F~ = + +no V ~ - - 1 - - V ( V ~ - ~)3 a, - - ( V ~ _ 1)+ a=

and

(88b) F~ • ~ / m ~ mo = a,, F ~ = §

The sign in eq. (88a) is consistent with the ordinary definition of work :9 ~

(89) dA ~ _= + F- d!

and the fact that the total energy of a tachyon increases when its speed decreases

(el. fig. 4a) and 10). Notice, however, that the proportionality constant between force and acceler-

ation does change sign when passing from the longitudinal to the transverse components.

The tachyon total energy E, moreover, can still be defined as

(90) E - - p ' V - - L m~ - - _ _ : ~ C 2 ,

which, together with eq. (50'), extends to tachyons the relation ~----me ~.

However, the following comments are in order at this point. An ordinary timelike (straight) line can be bent only in a spacelike direction; and it gets shorter. On the contrary, if you take a spacelike line and, keeping two points on it fixed, bend it slightly in between in a spacelike (time) direction, the bent line is longer (shorter) than the original straight line (see, e.g., DOl~LI~G, 1970), For simplicity, let us here skip the generic case when the bending is partly in the timelike and partly in a spacelike direction (even if such a ease looks to be

b

the most interesting). Then, the action integral f Ids] of eq. (84) along the straight a

(spacelike) line is minimal w.r.t, the (( spacelike ~ bendings and maximal w.r.t. the <( timelike )> bendings. A priori, one might then choose for a free tachyon, instead of eq. (85), the Lagrangian

(85')

which yields

L : - - m o V / V ~ - - i ,

8L mo V _ inV . (86') P - ~ v - ~ / v ~ _

52 ~. R~CAMI

Equa t ion (86') would be ra ther interesting, in the light of the previous subsect. 6"13 (el. also subsect. 13"2), i.e. when tachyons are subst i tuted for the (~ vir tual particles ~> as the carriers of the e lementary-par t ic le interactions. In fact, the (classical) exchange of a t achyon endowed with a m o m en tu m antiparallel to its veloci ty would generate an attractive interaction.

For nonfl'ce taehyons, f rom eq. (86') one gets

(87') F - d . _ d t V

and, therefore, when ]V[ : V.,

mo (88a') ~'* -- + ( V ~ - 1)1 a . ,

~n o ~no (88b') F~ -- __. __ a~ F~ -- - - a~.

Due to the sign in eq. (88a'), i t is now necessary to define the work c f as

(89') dfLf ~ -- .F.dl ,

and analogously the to ta l energy ~ as

(90 r) E --= (p V - - L) re~ _ . _ : m e 2 .

V V ~ - I

6"15. On radia$ing tachyons. - Many other results, actual ly independent of the ve ry existence of SLTs, will appear in the following sect. 9-13.

Here , as a fur ther example, let us repor t the fac t t ha t a. t a c h y o n - - w h e n seen by means of its electromagnetic emissions (see the following, and review I ; BALDO et al., 1970)--will appear in general as occupying two posit ions at the same t ime (RECA~rr, 1974, 1977b, 1978a, 1979a; ]3ABUT et al., 1982; see a]so

GRON, 1978). Le t us s tar t by considering a macro-object C emit t ing spherical

electromagnetic waves (fig. 15c)). When we see i t travell ing at constant Super- luminal veloci ty V, because of the distort ion due to the large relat ive speed, ][I > c, we shall observe the electromagnetic waves to be internal ly t angent to an enveloping cone /~ having as its axis the m o t io n line of C (REcAm and M_ZGNA:~I, 1972 ; review I), even ff this cone has nothing to do with ~erenkov's (MIG~A~I and Rv.cA~, ]973b). This is analogous to what happens with an airplane moving at a constant supersonic speed in the air. A first observat ion is the following: as we hear a sonic boom when the sonic contact with the supersonic airplane does s ta r t (BONDI, 1964), So we shall analogously see an (( optic

boom ~ when we first enter in radio contact with the body C, i.e. when we

C L A S S I C A l , T . K C H Y O N $ ~

Fig. 15.

C O V H

�9 !

%

m-

L:

c: c, c o c z c; ( v = = ) - - - - �9 " r - - -

"1" I T" I I I ! ! \ I i

I I I ! !

I I I 1 1 I

0 I j l

. - - . - - ~ 2 . . ~ I l l

l 0 0

a) b)

2 "

tz t~

S

X

/ , / /~2 r C . . . . . . . . . . . c) / " rs ":'-x;~- . . . . . . . . . . . . . . - - 3 ~ " ' . . . . . . , . . , ~ 3

\ SuperLumino~L wor(oL-L/ne

mee t t h e / ' - c o n e surface. I n fact , when C is seen b y us under the angle (fig. 15a))

(91) V c o s a = c ( V ~ IVI),

all the ra4ia t ions emi t t ed b y C in a cer ta in t ime in te rva l a round its posi t ion Co reach us s imultaneously. Soon af ter , we shall receive a t the same t ime the light emi t t ed f rom sui table couples o/points, one on the left and one on the

r ight of Co. We shall thus see the ini t ial body C, a t Co, split in two luminous

objects C1, C2 which will t hen be observed to recede f rom each other with the

Superhuninal ~transverse~> relat ive speed W (~ECAMI et al., 1976; B ~ U ~ et al., 1982):

I + d/bt V . . . . . . . . . . . ( V ~ > I ) , (92) W ---- 2b [1 -5 2d/bt]t' b %/V 2 - 1

where d ---- OH, and t = 0 is jus t the t ime ins tan t when the observer enters in radio contac t with C, or r a the r sees C at Co. I n the s imple case in which C

moves with a lmost infinite speed (fig. 15b)), the appa ren t relative speed of C1

5 4 E. ~XCAMI

and C 2 varies in t, he initial stage as W --~ (2ed/t)t, where now 0]5[ -~ O C o while t = 0 is still the ins tant at which the observer sees C~ ~ C2 ~= Co.

We shall come back to this subject when dealing with astrophysics

(subsect. 12"4); see also the interesting paper by LAKE and Ir (1975). Here let us add the observat ion tha t the radiat ion associated with one

of the images of C (namely, the radiat ion emi t ted by C while approaching us, in the simple case depicted in fig. 15c)) will be received b y us in the reversed

chronological order (cf. :VlIaXANI and ~nch~I , 1973a; I~ECA?4I, 1977b). I t m a y be interest ing to quote tha t the circumstance tha t the image of

a t achyon suddenly appears at a certain position Co and then splits into two images was alreday met by Bac~Y (1972) and BAClCY et al. (1975) while exploit ing a group-theoreticM detinition of the mot ion of a charged particle in a homogeneous field; definition which wan valid for all kinds of particles (bradyons, luxons, tachyons). Analogous solutions, simulating a pair produc- t ion, have been later on found even in the sub lumina l case by BARUT (1978b), when exploring nonl inear evolution equations, and by SALA (1979), b y merely taking account of the finite speed of the light which carries the image of a

moving subluminal object. SALA (1979) did even rediscover--also in subluminal case---that one of the two images can display a t ime-reversed evolution.

At this point, we might deal with the problem of causali ty for taehyons (since the most relevant aspects of t ha t problem do arise w.r.t, the (;lass of the sub lnmina l observers). We shift such a question, however, to sect. 9, because we want preliminarily to touch the problem of t achyon localization.

7. - Four-dimensional results independent of the explicit form of the SLTs."

introduction.

7"1. A pre l iminary assumpt ion . - Let us s ta r t f rom ov_r three postulates (sect. 4). Also in foltr dimensions, when a t t empt ing to generMize SR to Super- luminal frames, the fundamenta l req~irement of such an (~ extended re la t iv i ty )) (cf. subseet. 4"2, 4"3, as well as 5"1, 5"2) is tha t the SLTs change t imelike into spacelike tangent vectors, and vice versa, i.e. inver t the quadrat ic-form sign.

Le t us assume in these sect. 7, 8 tha t such (( t ransformat ions I) exist in four dimensions (even if at the price of giving up possibly one of the propert ies

i)-vi) listed at about the end of subsect. 3"2). Their actuM existence has been

claimed, for instance, by St~A~ (1977, 1978) within the (( quasi-catastrophe ~)

theory.

7"2. G-vectors and G-tensors. - I f we require also tha t the SLTs form a new group G together with the snb]uminM (ortho- and anti-chronous) Lorentz t ransformations, lhe following remarks are then in order. Equat ions (14), (15) introduce the four-position x# an a G-vector; in other words, by definition of

CLASSICAL TACI[XONS 55

G[~Ts, quan t i ty x~, is a four-vector not only w.r.t, the ~roup ~tf!_, bu t Mso

w.r.t , lhe whole ga'OUl) G. As a consequence, tile <~ scalar product)> (lx#dx~ behaves as a pseudoscalar under the SLTs.

Under SLTs it is d s ' ~ - - --de2; it, folh)ws t ha t quan t i ty ~r dx#/ds , 'a.

Lorentz veer;or, is not a. G-vector In order l)o detine the four-veloci ty as a

G-vector , we mus t set

(93a) ul~ -~ dx~/dvo ,

whez'(~ To is the 1)rot)er t ime. &nalogously for the fom'-accelerat ion: a# - dw' /dTo;

and so on. We can expect t ha t also the e lect romagnet ic quant i t ies A~, (Lorentz

vector) and 3'#,' (Lorentz tensor) do not have a pr ior i to be any longer a G-vector and a G-tensor, respect ive ly (cf. sect. 15).

I Iowever , once T~,~ is supposed to be a G-tensor, then under a SLT it is

(93b) T '~ ~ G"~ G ~ T ~ ,

wheref rom it follows t h a t the ord inary invar ian ts

'~re still invar ian t (even under SLTs). This holds, of (.ourse, only for even-rank tensors.

As a l ready ment ioned, if we define u# b y eq. (93a), so to be a G-four-vector , then under a SLT the quan t i ty u 2--- u~,u, becomes ~ ' e = - - u - t Tha t is to

say, a f te r a SLT a b radyonic veloci ty has to be seen as a tachyonic velocity, a.nd vice versa, in agreement with eqs. (26).

Le t us add here, a t this point , t h a t somet imes in the l i t e ra ture it has been avoided the explicit use of a metr ic tensor by mak ing recourse to Einste in 's

notat ions, and b y writ ing the generic chronotopical vec tor as x - = (x0, xx, x2, xa) ~ (ct, ix , iy , iz), so th,t t g~,~ : : ~ , ((( Eucl idean metr ic ~>). Thus one does not have to distinguish be tween covar iant and con t rava r i an t components . h i such a case, since on('. has pract ical ly to deal with a complex manifold, the quadrat ic fo rm which is Lorentz invar ian t is to be defined us the scalar p roduc t of the first vector b y the complex conjugate of the second vector :

(93d) quadratic form -- (dx, d~) = dx~, dy~ ;

in par t icular , the invaria.nt square in terval would be d.~ 2 ~-= (dx, dg') -= d x , dxv.

8. - On the shape o f tachyons.

8"1. In t roduc t ion . - We have a l ready noticed t h a i a t a ( ,hy(m--observed b y means of its l ight s ignals--wil l generMly appeqr '~s occupying two posit ions at ihe same t ime (subscct. 6"]4 and fig. 15).

5 6 ~ . ~ECAM1

Still at ~ prehmimtry level, let us moreover recall tha t free bradyons always admit a particular class of subluminal reference frames (their rest frames) wherefrom they appear- - in Minkowski space-time--as ((points ~) in space extended in t ime along a line. On the contrary, free tachyons always admit a particular class of subluminal (w.r.t. us) reference f rames- - the critical f r ames- - wherefrom they appear with divergent speed V = o% i.e. as <, points ~> in t ime extended in space along a hne (of. fig. 7, 11). Considerations of this kind correspond to the fact tha t the (( locahzation ~> groups (little groups) of the timelike and spacehke representations of the Poincar6 group are S03 and S02,1,

respectively (see, e.g., BA~VT, 1978a), so tha t tachyons are not expected to be locahzable in our ordinary space (of. also 1)I~gEs, 1970; CAWLEY, ] 970; DVFFEY, 1975, ]980; VYw 1977a; SOU~EK, 1981).

I t is, therefore, worthwhile s tudying the shape of tachyons in detail, following B)~uT et al. (1982).

8"2. H o w would tachyons look l ike? - I~et us consider an ordinary bradyon ]r -- t)~ which for simplicity is intrinsically spherical (in partieula.r pointlike), so t ha t when at rest its ~ world-tube ~> in Minkowski space-time is represented by 0 < x ~ -}- y~ + z 2 < r "~. When PB moves with subluminal speed v along the x-axis (fig. 16), its four-dimensional shape (i.e. its world-tube equation) becomes

(94a) 0 < (x - - vt) 2 �9 1 --L- v ~ _}_ y2 _}_ z 2 < r 2

and, in Lorentz-invariant form,

(94b) 0 < ( x~, p~ ) ~ _ x~ x~, < r2 p~p~

where x~ = (t, x, y, z) and p , is the 4-momentum.

(v ~ < 1)

(v ~ < I) ,

I t I

Z / ~ I / ! / / J

/ . i / , ,"

" ' / I

X

Fig. 16.

C L A S S I C A L T A C t t Y O N S 5 7

Let. us now take into examina.tion also the spacelike values of the 4-momen- t l t m p**, still considering, however, only subluminal observers s: We shall regard in these sections the SLTs, as well a.s the ord inary LTs, fl 'om the active

point of view only. B y an act ive SLT let us t r ans fo rm the initial I)n into a tinal t.achyon P ~_~ ]3~ endowed with Superhtminal speed V along x. ])ue to subsect.

7"1, one can expect thai; eq. (94b) will t r ans fo rm for 1)T into

(xz p')'2 (95) 0 <x~,x# <r'-' (V 2 > 1),

where p , has been regarded as a G-four-xTector (for both Bs and Ts it will be

defined p , =~ m o u , ~ modx~/d~o: see subsect. 14"13). :Notice, however , the following: I f a SLT is requested to change the sign of the quadrat ic fo rm d s 2 =

= dx~(tx*,, this means t h a t i t will change t.he t ype of all the <( t angen t vectors ~ (i .e. , for example , the sign of p~p~) but does not mean a t all t h a t i t will

! change sign also t.o x~ x~; this happens only if t.he SLTs dx~ --> dx~ are linear.

[C =--0 for t=O]

Y b)

X

( t=O)

/, ' >Y (

~)

/ . . : , ~o'/If

~:.i L.r ~l ~ ill\\l -

X

Fig. 17.

5 8 E.RECA~I

(Actua.lly, if such a. lilrear S L T has constant coefficients (as required by homogeneity and isotropy)~ then i t is linea.r a.lso the transformatioH J- : x~--~ x'z; cf.~ e.g., I~I~DLE~, ]966.) Therefore, to go s cq. (94b) to eq. (95) it is uecess- a.ry to assume explicitly tha.t SLTs exist which cha.nge, sign both to dx~,dx~

and to x# x~. Equa.tion (95) then yields the four-dimensional sha.pe of ta.chion PT. In the initial fra.me, eq. (95) can be written a.s

( x - Vt)" (96) 0>~ V 2 _ I ~ q - y " q - z 2 ~ - - r '' (V~ I ) .

In conch|sion, if the world-tube of P~ wa.s supposed to be Ulflimited--/.e. if :P~ was supposed to be infinitely extended in t ime-- , then ta.chyon PT appea.rs as occupying the whole space bound by the doubh~, unlimited cone r y2 ~- z ~- = ( x - - Vt)~/(V ~ - 1) and the two-sheeted rotation hyperboloid ~ffo: y~ ~- z ~ = (x - - V t )~ / (V 2 - - 1) - - r2~ where the la.tter is asymptotic to the former; see fig. 17. As t ime elapses, eq. (96) yields the relativistic shape of our tachyon; the whole structllre in tig. 17 (and 18) rigidly moving along x--of course--

v

V

/ /

X

Fig. 18.

with the speed V. l~otice tha t thc colle semi-angle ~ is given by

(97) tg~ = (V '~- 1 ) - ! .

Le t us fix our a t tent ion on the external surfa.ce of P. When it is at rest, the surface is spherical; when sublumina.1, it becomes a.n ellipsoid (fig. 19b));

C L & S S I C A L TACIIYONS 59

a) b) c) d)

Fig. 19.

, ytt

when Superhlminal , such a. surface becomes a two-sheeted hyperboloid (fig. 19d)).

Figure 19c) refers to the l imit ing case when the speed fends to c, "i.e. when

ei ther v -§ 1-- or V - > 1-. Incidenta l ly , let us remind t h a t even in EIr ~he light

st)eed in vacuum goes on being the invar ian t speed, and can be crossed nei ther f rom the left,, nor f rom the right.

Le t us m a k e a comment . Tachyons appeared to be more similar to fields

t h a n t.o particles. I t would be desirable to find out the space- t ime funct ion

yielding the densi ty dis t r ibut ion of a taehyon. For instance, when the t aehyon

shape jus t reduces to the cone ~fo, it would be interest ing to work out the L-funct ion of x and t yielding the t achyon densi ty di t r ibut ion over ~o.

8"3. Critical comments on the pre l iminary assumption. - I n connection with

subsect. 7"1 and 8"2 a critical warning is in order, since we saw at the end of subscct. 3"2 (and shall be t te r see in the following) t ha t real linear SLTs: dx, -+ dxl,

which mee t the requi rements ill-iv) of subsect. 4"2 do not exist in fOUl' dime~tsions. t

We, therefore, expect t h a t real t r ans format ions .~-: x~,-> x~, mapp ing points of

M~ into points of IM~ (in such a way tha t ds~--~-- ds 2) do not exist as well; /

otherwise reM linear SLTs: dx, -> dx~, should exist.

Le t us s ta te it differently. Equal,ion (95) was derived under the hypothes is t ha t SLTs do exist in four dimensions which change the sign bo th of the quadrat ic

t fo rm dx, (tx~, and of the q lmnt i ty x~,x t'. This means t ha t the SLTs: dxt, --~ dx~,

t rans forming dx, dx ~' - § dx~ dx ~' have to be linear. In the case of SLTs linear

and real~ it would exis t as a consequence in M4 a point - to-point t r ans fo rmat ion !

,:V: x ,-+x~,~ and fu r the rmore linear (]~I_~DL~:g, ]966).

The results ill this sect. 8 seem to show, howc~'er, t h a t ill M, we mee t map-

pings tha$ t r ans fo rm manifolds into manifolds (e.g., points into surfaces). !

This seems to 1)redict 1-o us t ha t our SLTs: dxF,-> dx~ in M4 will be linear, bu t not real.

For such nonreal SLTs we shall suggest in subsect. 14"16 an in te rpre ta t ion procedure t h a t will lead us f rom linear nonreal SLTs to real nonlinear SLTs;

of., e.g., fig. 5 in MACCA~rcO~'E and IIE(JA~ (1982a, 1984a). The latLer SLTs,

60 x. RECAMI

!

actuully, cannot be integrated, so tha t no .~ : x~, --> x~ cun be found in this case (SM~z, 1984).

d ' Let us explicitly mention tha t nonlinear SLTs: dx~ -~ x~ can exist which, nevertheless, i) do t ransform inertial motion into inertial motion (e.g., the inertial motion of a point into the inertial motion of a cone), ii) preserve space isotropy and space-time homogeneity, ii) retain the light speed invariance (cf. also subsect. 8"2, 8"4).

8"4. On the spaee ex tens ion o] tachyons. - In the limiting case when it is intrinsically pointlike, tachyon PT reduces to the cone g0 only, and we shall see Pr to be a double cone, infinitely extemled in space (I~]~CA.VlI and MACCA~- ~O~E, 1989; BARUT et al., 1982). But this happens only if the corresponding bradyon I)B exists for -- ~ < t < + ~ . On the contrary, if the lifetime (and extension) of PB ~re ]inite, the space extension (and life) of PT are ]inite

too. Namely if 1)~ in its rest frame is spherical, is born at t ime t~ and is absorbed at t ime i~, then the corresponding ta.chyoa I~ possesses a finite space extension (I~ECAm and MACCAR~O~, 1980, 1983). Under the present hypotheses, in fact, one has to associate with eqs. (93), (94) suitable l imi t ing spacelike hyper- surfaces, which simply become the hyperplanes t ~ i~ and t : t2 when P~ is at rest (fig. 20). The generic Lorentz-invariant equation for a hyperplane is

(98) x# u~ : K ( K ~ const).

X

Fig. 20.

Due to subsect. 7"1 we get tha t eq. (98) keeps its form even under an active SLT, i.e. x , u ' ~ - - - - K ' . The relevant fact is thut we passed from a timelike ua to

, K / a spacelike u~, so tha t the hyperplanes x~u'~ == are now to be referred

CLASSICAL TAeHYO~S 61

to two spatial and one tempora l basis vectors (fig. 21). Such hyperplanes represent ordinary planes (orthogonal to the x-axis, in our case) which move parallely to themselves with the subliminal speed v ' = l / V , as follows f rom

f

their or thogonal i ty to u~.

Fig. 21.

/ /,

In conclusion, in the t achyon case (V ~ > 1), one has to associate with eqs. (95), (96) the addit ional constraints

the shape of a realistic t aehyon PT, obta ined from a finite-lifetime b radyon P~, is got, therefore, by imposing on the s t ructure ~o @ 3r in fig. 17, 18 the following constraints :

,) I t seems to follow tha t our realistic t achyon is const i tu ted not by the

whole s t ructure in fig. 17, 18, bu t only by its port ion confined inside a mobile (( window ~), i.e. bound by the two planes x = xl and x = x~. As we saw, this (( window )) t ravels with the speed v' dual to the t achyon speed V

1 (100) q/ ---- -- ( I r~> 1, v ' 2 < 1)

V

and, if V is constant , its width is constant too (A--t ~ t ~ - tl):

( lOl) Ax ' = A t V ~ - - ~'~ (v' = 1 / V ) .

6 2 n . a r c A ~

Chosen a tixed position x - - 5% such a (~ window ~) l~) cross ~he plane x - - will take a t ime independent of �9 (if V is still constanl) :

(102) At' = At v ~ :-~ AtV/V-"~ 1 .

The problem of the t ime extension of such (~ realistic ~ tachyons does not

seem to have been ye t cla.rified. I f ])~ is not intrinsically spherical, but ellipsoidal, then ])T will be bound

by a double cone W and a two-sheeted hypcrboh)id ~ ' devoid, this t ime, of cylindrical symmet ry (of. BARvT et a t , 1982). Those authors investigated also various limiting eases. Let us ment ion tha t when V - + ~ (while i~, t2 and r remain ]inite) the ~ window ~) becomes ]ixed: Y~x ~ ei~ < x < c[2 ~ ~ .

We ma y conclude tha t , if the lifetime of Ps is very large (as it is usually for macroscopic and even more for cosmic objects), then the corresponding tachyon description is essentially the old one given in subscct. 8"2, and I)T e'~n be associated with actual Superluminal motion. If , on the contrary, the l ifetime of ]?r is small w.r.t, the observation t ime of the corresponding tachyon (as it often happens in the microscopic domain), t.hen P.r would actual ly appear to t ravel with the s u b l u m i n a l (dual) speed v' ~- 1 / V ; even if I)~ is associated with a s t ructnre ~ + .$f travell ing with the Superluminal speed V. In fact, the magni tude of its (( group velocity }~ (i.e. the speed of its ~ f ront ~)) is given by eq. (100). IIowever, within the ~ window )~ confining the real port ion of the taehyon (which possibly carries the tachyon energy and momentum, just as ]?s carried energy-momentum only between t = tx and t----i~), there will be visible a (~ s t ructure ~ evolving a.t Supcrluminal speed, associable, t.here- for% with a taohyonic (~ pha.se velocity )~. W h a t precedes is based on I~ECA_M_X and MACCARr~O~E (1983)~ bu t similar r e s u l t s - ~ v e n if got from quite different s tar t ing po in ts - -were pu t for th by F o x et al. (1969, ]970). See also ALA~AR I~A~A.~USA~ et al. (1983), SOU~:K (1981)~ ](OWALCZYlCISKI (1979), SCn~-~A~

(1971), Cr (1970).

8"5. Comments . - The tachyons ' characteristics exploited in the previous

subsect. 8"4 remind us once more (cf , e.g., subsect. 6".13) of the ordinary q u a n t u m

particles with their (~ de Broglie waves ~: in tha t case too phase veh)city and

g~'oup veloci ty obey eq. (100). To investigate this connection (RECAZ~I and MACCARR0~E, 1983) let uS

recall the ordinary delinitions of Compton wave-length 20 and de Broglie

wave-length )'an (f12 < :1):

4,1 - f it (103a) ~ c ~-- 2dB - - - ----- - ,

oC' fPl �9

CLASSICAL TACHYON8 ~

where we int roduced the new (~ wa.ve-length ~) 2

E/c

sat isfying the rela.tion

1 1 1 (103c) )2 ~t~B

(~ < 1)

Equat ions (103) suggest, of course, the following k inemat ica l in te rpre ta- t ion: Le t 2c represent the intrinsic size of the considered (subluminal, quantum)

p a r t M e ; then ). = 2 c V / 1 - / # is the par t ic le size along its mot ion line in the

fra.me where it t ravels wi th speed v -~ tic; and 2dB/V - )~/V is then the t ime

spent b y the pa.rticle to cross in the sa.me fl 'ame a plaane orthogona.1 to its mot ion line.

Le t us now examine our eqs. (101), (102). In cq. (101) it is na.tural to ident i fy

( c2 ) (10~a) hx'---2' :~oV1 fl '~ ~'=-v'/c, v t ~ p , v '2<c ~

wheref rom

(104b) 2'o --- c a t .

Then, from eq. (102),

10,1c) (fi t ~ v ' /c , v' c'-' ) V ' _ ' 2

By compar ing eqs. (104) with eqs. (103), one recognizes t ha t the chara.cter- istics of a classica.1 t achyon ac tual ly f i t the (( de ]~roglie relat ions ~ v ' = 1 / V and ;t'd~ ;(/ f l ' , with ).' ---- ~ Ax'. However , a. classical (rea.listic) taachyon T obeys all the equa.tions (103) only prov ided that one aattributes to the t achyon

(or, ra ther , to its (( real ~ port ion confined within thc mobile subluminal (~ window ~>) a proper mass mo depending on its in t r ins ic (proper) l ifet ime,

na.mely such t h a t

(105) --;to-- ~ = At -> mo ---- --~. c m0 c 2 c 2 At

Notice tha.t eq. (105) corresponds to the case • o ' A t - - - - E . A x ' / o = h, wi th

Eo - - moo ' , E = moc~/V/ i - (V'fe) ~. Notice, moreover , tha.t the wa.vc-lcngth of the de Broglie waave a.ssocia.ted with a. ta.chyon ha.s an upper l imit (GRo~, 3979) which is essentia.lly equaal to its Compton wa.ve-length: ( dS)ma~

! = h l m o c = 2~.

64 E. I~E (~A~M I

9. - T h e c a u s a l i t y p r o b l e m .

As ment ioned at the end of subsect. 6"15, the discussion tha t will follow in this sect. 9 is independent o I the very existence o I the SLTs, since the most re levant causal problems arise v h e n describing tachyons (and bradyons) f rom the ordinary subluminal frames. We wanted, however, to face the causali ty problem for tachyons in re la t iv i ty only af ter having at least clarified tha t tachyons are not tr ivial ly localizable in the ordinary s p a ~ (cf. sabsect. 8"2-8"5; see also SHAY and Mrr,T.ER, 1977). Actually, a t aehyon T is more similar to a field than

to u particle, as we already noticed at the end of subsect. 8"2. There ~re reasons, however, to believe tha t , in general, most of the tachyon mass be concentra ted near the centre C of T (fig. 17b), 18): At least, we shah refer in the following to the eases in which tachyons can be regarded as (! almost localized )) in space. In what follows, therefore, we shall essentially make recourse only to the results in subscct. 5"12-5"14 (which, incidentally, have been seen to hold also in four dimensions) and to our results about t achyon kinematics (sect. 6). As ment ioned above, we shall confine ourselves only to the subluminM observers (in the presence, of course, of both bradyons and tachyons) and, for s imphcity, to the

orthochronous Lorentz t ransformations only. The results in subsect. 5"]2-5"14, in part icular, showed us tha t each observer

will always see only tachyons (and ant i tachyons) moving with positive energy 1orward in time. As expounded in subsect. 5"13 and 5"17, however, this success is obta ined ~t the price of releasing the old conviction t h a t judgement about what is (, cause ~)and wha t is (( e f fec t , is independent of the observer; in subsect. 5":17 we concluded tha t the assignment of the (( source ~) and (c detector ~ labels is to be regarded as a description detail. As ant ic ipated in subsect. 5"13, this fact led to the proposal of ~ series of seeming (( causal paradoxes ,, t ha t we are going to discuss and (at least , in mierophysics ~)) to solve.

9"]. Solution o I the Tolman-Regge paradox. - The oldest paradox is the

(( ant i te lephone ~ one, originally proposed by TOL~L~_~ (1917; see also B o ~ , 1965) and then reproposed by m a n y authors (cf. subsect. 3"1). Le t us refer

to its most recent formulat ion (I~EGGE, 1981), and spend some care in solving

it since i t is the kernel of m a n y other paradoxes.

9"1.1. T h e p a r a d o x . In fig. 22 the axes t and t' are the world-lines

of two devices A and B, respectively, able to exchange tachyons and moving with constant relat ive speed u (u 2 ~ 1) along the x-axis. According to the terms of the paradox (fig. 22a)), A sends t achyon 1 to B (in other words, t achyon 1 is supposed to move forward in t ime w.r.t. A). The appara tus B is constructed so to send back a t achyon 2 to A as soon as it receives a t achyon 1 from A.

I f B has to emit (in its res t frame) t achyon 2, then 2 must move forward in

CLASSICAL TACH ).'O_N S ~

It 1 ~ t! I t ? . B

... "" .."" /./../

a) b)

Fig. 22.

t ime w.r.t. B, thal; is to say ils world-libra B A , mus t have a slop(: smaller t h a n

the sh)pe B A ' of the x ' -axis (where, BA']]x'); this means t h a t A~ mus t s t ay above, A' . I f the sI)eed of tachy(m 2 is such t ha t A~. falls be tween A ' and A~, it seems tha t 2 reaches back to 5_ (event A.,) be/ore the emission of 1 (event A 0.

This a.ppears to realize an antitelephone.

9"1.2. T h e s o l u t i o n . Firs t of all, since t achyon 2 moves backwards

in t ime w.r.t . &, the event A~ will appear to A as the emission of an an t i t achyon `2.

The observer ((t }~ will see his appara tus A (able to exchange taehyons) emi t suc(~essively towards B the an t i t achyon `2 and the t achyon 1.

At this point , some supportexs of the pa radox (overlooking t a cbyon kinematics, as well as reb~tions (66)) would say tha t , well, the description forwarded b y observer (~ t )) can be or thodox, bu t then the de.vice B is no longer working according to ~.he premises, because B ix no longer emi t th)g a t achyon 2 oll receipt of t achyon 1. Such a s t a t emen t would be wrong, however , sh~ce the fact t ha t (( t ~> sees an (~intrinsic emission ~) at A2 does not mean t ha t (~t'~ will see an ((intrinsic at)sorption }> at B. On |.he contra.ry, we are just in the came of

subsect. 6"10: intrinsic emission b y A, a t A~, with u.V~ > c 2, where u and Vg

are the velocities of B and '2 w.r.l. A, respect ively; so t h a t both A and B suffer an intrinsic emission (of t aehyon 2 or of an t i t achyon `2) in their own res t frames.

Bu t the te rms of the pa radox were cheat ing us even more, and ab initio. In fact , fig. 22a) makes clear tha t , if u.Y~ > c ~, then for t achyon :1 a /ortiori u. V~ :> c ~-, where u and F~ are the velocities of B and 1 w . r . t . A . Due to subsect. 6"10, therefore, the observer (~ t'); will see B emit also t achyon 1 (or,

n~ther, an t i t achyon T). I n conchlsion the proposed cha.iu of events does not include ~my ta.chyon absorp t ion by B.

66 ~. R~CA:MI

For body B to absorb tachyon 1 (in its own rest flame), the world-line of 1 ought to have a slope larger than the x'-axis slope (see fig. 22b)). Moreover, for body B to emit (~ intrinsically ~)) tachyon 2, the slope of 2 should be smaller than the x'-axis'. In other words, when the body B, programmed to emit 2 as soon as it receives 1, does actually do so, the event A, does regularly happen after A~ (of. fig. 22b)).

9"1.3. The mora l . The moral of the story is twofold: i) one should never mix together the descriptions of one phenomenon yielded by different observers, otherwise--even in ordhlary physics--one would immediately meet contradictions: in fig. 22a), e.g., the motion direction of 1 is assigned by A and the motion direction of 2 is assigned by B; this is illegal; ii) when proposing a problem about tachyous, one must comply (C~_LDIRO~ and l~V, OA~, 1980) with the rules of tachyon kinematics (MACO~RO~E and REOAMT, 1980b), just aS when formulating the text of an ordinary problem one must comply with the laws of ordinary physics (otherwise the problem in itself is ~ wrong ~>).

Most of the paradoxes proposed in the literature suffered from the above, shortcomings; for a remarkable, late example, see GIRARD and MAROrrr.-

DON (1984). Igotice that, in the case of fig. 22a), neither A nor B regard event A~ as

the cause of event A~ (or vice versa). In the case of fig. 22b), on the contrary, both A and B consider event A~ to be the cause of event A~: but in this case A~ does chronologically procede A~ according to both observers, in agreement with the relativistic covarianee of the law of retarded causality. We shall come back to such considerations.

9"2. Solution o/the Pirani paradox. - A more sophisticated paradox was proposed, as is well known, by P ~ - ~ I (1970). I t was substantially solved by P~_R~E~TOLA and YEE (1971), on the basis of the ideas initially expressed by SUDARSHA~ (1970a), Brr.ANrUK and 8~-DARSHA~ (1969b), CSONKX (1970), etc.

9"2.1. The p a r a d o x . Let us consider four observers A, B, C, D having given velocities in the plane (x, y) w.r.t, a fifth observer so. Let us imagine that the four observers are given in advance the instruction to emit a tachyon

Y[ x i ~ "3 (c~ ~0) 2 C

A (A2)~4 I,,

Fig. 23.

CLASSICAL TACHYON8 6 7

as soon as they receive a taehyon from another observer, so that the following chain of events (fig. 23) takes place. Observer A initiates the experiment by sending tachyon 1 to B; observer B immediately emits tachyon 2 towards C; observer C sends taehyon 3 to D; and observer D sends tachyon 4 back to A, with the result--according to the paradox---that A receives tachyon 4 (event A1) be/ore having initiated the experiment by emitting tachyon 1 (event As). The sketch of this (( Gedankenexperiment ~) is in fig. 23, where oblique vectors represent the observer velocities w.r.t. So and lines parallel to the Cartesian axes represent the tachyon paths.

9"2.2. The s o l u t i o n . The above paradoxical situation arises once more from mixing together observations by four different observers. In fact, the arrow of each tachyon line simply denotes its motion direction w.r.t, the observer which emitted it. Following the previous subseet. 9"1, it is easy to check that fig. 23 does not represent the actual description of the process by any observer. I t is necessary to investigate, on the contrary, how each observer describes the event chain.

Let us pass, to this end, to the Ylinkowski space-time and study the description given, e.g., by observer A. The other observers can be replaced by objects (nuclei, let us say) able to absorb and emit taehyons. Figure 24 shows that the

c ~

~ 3 ~

'ct ~ B

1 I

X

Fig. 24.

absorption of 4 happens be/ore the emission of 1; it might seem that one can send signals into the past of A. However (el. subsect. 5"12-5"14 and sect. 6, as well as I~ECA~ 1973, 1978V), observer ik will actually see the sequence of events in the following way: The event at D consists in the creation of the pair ~ and 4 by the object D; tachyon 4 is then absorbed at AI~ while g is scattered at C (transforming into tachyon ~); the event As is the emission, by A itself, of tachyon 1 which annihilates at B with taehyon ~. Therefore, according to A, one has an initial pair creation at D, and a final pair annihilation at B, and tachyons 1, 4 (as well as events AI~ As) do not appear causally

68 E. RECA3~I

correlated at all. In other words, according to A, the emission of I does not init iate any chain of events t ha t brings to the absorpt ion of 4, and we are not in the presence of a n y , effect ~ preceding its own ~ cause ~>.

Analogous, or thodox descriptions would be forwarded by the other observers.

For i~stance, the tachyons ' and observers ' velocities chosen by Pn~h.'~ (3970) are such t ha t all tachyons will actual ly appear to observer so as moving in directions opposite to the ones shown in fig. 23.

9"2.3. C o m m e n t s . The comme~tts are the same. as in the previous subsect. 9"1. Notice tha t the * ingredients )) tha t allow us to give the paradox a solution are always the (( switching principle ~ (subsect. 5"12 ; see also ScrrwA_~Tz, 1982) and the t achyon relativistic kinematics (sect. 6).

9"2.4. ( ( S t r o n g v e r s i o n s ) a n d i t s s o l u t i o n . Le t us formulate Pi- rani 's paradox in its strong version. Le t us suppose tha t t achyon 4, when absorbed by A at A~, blows up the whole l~boratory of A, eliminating even the physical possibility tha t taehyon i (believed to be the sequence starter) is subsequently emit ted (~t A~). Following R o o t and TREFIL (1970; see also TnEFIL, 1978), we can see on the cont rary how, e.g., observers so and A will really describe the phenomenon.

Observer so will see the l~boratory of A blow up after emission (at A~) of the ant i tachyon ~ towards D. According to so, therefore, the ant i tachyon I emit ted by B will proceed beyond A (since it is not absorbed at A~) and will eventual ly be absorbed a t some remote sink-point U of the universe. By means of a LT, start ing from the description by So, we can obtain (CALDrROLA and I~ECA~r, 1980) the description given by A.

Observer A, af ter having absorbed at A~ the t achyon 4 (emit ted at D together with .3), will record the explosion of his own laboratory. At A2, however, A will cross the flight of a. tachyonic (( cosmic ray ~ 1 (coming from the remote source g), which will annihilate a t B with the an t i tachyon 3 scat tered at C,

i.e. with the an t i tachyon ~.

9"3. Solution o/ the Edmonds paradox. - The seeming paradoxes arising from the re la t iv i ty of the judgment about (~ cause )> and (~ effect ~> have been

evidenced b y EDMONDS (1977b) in a clear (and amusing) way, with referellce to the simplest t aehyon process: the exchange of tachyons between two ordinary

objects, at rest one w.r.t, the other.

9"3.1. T h e p a r a d o x . We build a long rocket sled with a <(tachyon laser i~ at the lef~ end and a ~ ta rge t flower ~ at the r ight end. A short lever sticks out of the side of the <~ laser ~. I f we t r ip the lever, the t achyon laser emits a ve ry sharp, intense burst of tachyons for which we measure the speed of, let us say, V. These tachyons then hit the (( flower )) and blast i t into pieces.

CLASSICAL TACHYON8 69

The flower a.bsorbs all the tachyons in the pulse as it explodes, so tha t the tachyons disappear.

Now we accelerate the sled (with (( charged ~)tachyon laser and flower ~t taehed to it) up to an incoming speed of -- v -- -- v~ relat ive to our f rame, and then tu rn off its rocket engines. ~oreover , we form a long line of (( astronauts )) floating in space along the x-axis (i.e. along the rocket sled motion line). Each -~stronaut has a (( roulet te wheel ~ in his one hand, and keeps spinning his gambling wheel unt i l he gets, say, the number ]3. When he happens to do so, he quickly puts out a stick in f ront of him which could beat the tr igger on the moving laser. No one in our f rame knows wheu a given as t ronaut will get 13 to come up. Some astronauts may get ]3, bu t too far down the line, or find the tr igger has already passed t h em when they get it. But , finally,

someone gets the r ight number , puts out his stick, finds tha t the lever is almost at his position and he triggers the laser.

Once the laser fires, the observer travell ing with the sled sees- - jus t as b e fo r e - - a burst of tachyons actual ly travell ing from the laser to the flower. I f the sled is moving slowly enough (vV ~ c2), then we also---together with the (( astronauts ~--see the flower blow up at a t ime later t h an the t ime at which the laser fires. However , if the sled is fast enough (vV > c2), we see a pulse of anti- taehyons going from the flower to the laser: Namely, we would see the flower blow up before the laser fires. Therefore, the as t ronaut who triggers the laser sees the laser immediate ly (( swallowing )) a pulse of ant i tacbyons coming from the flower. In other words, the lucky as t ronaut will conclude tha t the flower had to know in advance who was going to get 13 (so tha t it can blow up and create the an t i tachyon pulse just at the right t ime, in order for the beam to arrive at the lucky as t ronaut as he gets the number 13 to come up for him).

9"3.2. T h e s o l u t i o n . Since ~source~ and ~detector ~ are supposed b y EDM0~DS to be at rest one w.r.t, the other, according to bo th laser and f lower- - i.e. in the l abo ra to ry - - the re are no problems about the flight direction of the tachyons. However, if we choose other observers (as the astronauts), they will in real i ty see the laser absorb ant i taehyons T coming from the flower (and not fire taehyons T towards the flower). We have simply to accept it, since we learned (el., e.g., subseet. 5"17) tha t only the ~principle of re ta rded causali ty ~ (third

postulate) is a law, and, therefore, has to be va.lid for each observer; whilst

the assignment of the labels ~ source ~ and ~ detector ~ is a description detail, which does not have to be relativistically invariant .

Then, to answer EDMONDS (I~ECA2dX, 1977a), let us show b y an example

tha t seeming paradoxes as the one above arise also in ordinary special re la t iv i ty (due to the Lorentz noninv~riance of the descriptions). Le t us, therefore, forget about tachyons in the following example.

Le t us suppose we are informed about a cosmic fight taking place between t:wo different kinds of ext ra terres t r ia l beings, each one di'iving his own rocket ,

where the rocke t colours are violet for the first and green for the second species.

Le t us suppose, moreover , t ha t we know the (~ green men ~ to possess an inviolable

na tu ra l inst inct t ha t makes t h e m peaceful; on the con t ra ry the <~ violet men ~

possess an aggressive, warr ior instinct. When we observe the in te rp lane ta ry

ba t t l e b y our telescope, i t can well h a p p e n - - d u e to the Doppler effect, i.e. due to the <~ observa t ion distort ions ~> caused b y the relat ive m o t i o n s - - t h a t , when a <~ violet m a n ~ fires his gun and strikes a green rocket , the violet colour

appears to us as green, and vice versa, because of the rocket motions. Then,

according to thc spirit of E d m o n d s ' paradox , wc should deduce t h a t an inviolable

law of na ture has been bad ly viola ted (the inst inct ive law of those ext ra- terrestr ia l beings). Wi th in SIC, however , we a l ready know how to clarify the

whole s tory : We <~ observe ~) a t first a seeming violat ion of na tu ra l laws; but ,

if we know the re levant physics (i.e. SIC and the rocket velocities), we can

de te rmine the <~ intrinsic (proper) colours ~> of the rockets in thei r own rest

f rames , ~nd solve ~ny doubts. I n other words, any observer is capable of unders tanding the physica l

world in t e rms of his own observat ions only, provided that he is equipped with

a suitable t heo ry (he uses his knowledge of SIC, in this case). Going back to the t achyon <~ pa r adox ~>, we conclude t h a t the lucky astro-

naut , when knowing t achyon mechanics (i.e. the EI~), can calculate the t achyons '

direction in the flower rest f r ame and find out the (~ intrinsic behav iour ~> of

the flower. The as t ronau t will find t h a t in the flower f rame the t achyons are

not emi t ted , bu t absorbed b y the flower; even if the relat ive speed produces

high <( distort ion ~ of the observed phenomenon. I n analogy with our example , i t is not i m p o r t a n t t h a t the flower seem to the as t ronauts to precognize the future , b u t t h a t the flower <( intrinsically ,~ does not.

The discussion of this pa r adox reminded us t h a t i) one can scientifically observe (or observe, tout court) the na tu ra l world only if he is endowed with theoretical ins t ruments , besides exper imenta l and sensorial ins t ruments ; ii) the (~ intrinsic proper t ies ~> (so as the colour) of a body appear to a mov ing observer

dis tor ted b y the relat ive mot ion ; if high relat ive speeds are involved, tha.t dis-

to r t ion can be large as well.

L e t us add a fu r the r comment .

9"3.3. C o m m e n t . I n the case of a b radyon exchange, in which the roles

of q source ~> and q detector ~> are independent of the observer , the emi t t e r

and receiver are well represented b y a male and a female object, respect ively. Such a hab i t is, however , misleading in the case of a t achyon exchange, in which the same object can now appear as the emi t te r , now as the receiver, depending

on the observer. Devices such as ~ guns >> and (( lasers ~> ought to be avoided

in the (( Gcdankenexper imente >> regarding the exchange of tachyons . A round-

shaped device, such as a sphere, sholfld be the r ight one for represent ing objects

able to emi t /absorb tachyons.

C L A S S I C A L TACHYONS 71

9"4. Causality (( in micro- • and (~ in macro-physics ~. - Let us go on investigating the paradoxes arising when two bradyonic objects A, B exchange tachyons T, since there we meet in nuce all the problems than one encounters in the more complicated processes.

Le t us consider, namely, the si tuat ion in which (( laser ~ (A) and (( flower ~ (B) are no longer at rest one w.r.t, the other.

Such a si tuat ion is much more problematic, l~evertheless, no real problems are actually present (cf. sect. 6) as far as the t achyon product ion is supposed to be a (( spontaneous ~, uncontrollable phenomenon, just as part icle produc- t ion in e lementary-par t ic le physics. By convention, let us refer to this as the

case (( of microphysics ~. Problems ntis% however, when the t achyon product ion is a priori regarded

as controllable (we shall refer to this la t ter as the case (( of macrophysics )~). We are going to analyse such problems by means of two paradoxes.

The first one was proposed by ]3ET,L (1979).

9"5. The Bell paradox and its solution.

9"5.1. T h e p a r a d o x . By firing tachyons you can commit a (~perfect murder ~. Suppose tha t A purposes killing B, wi thout risking prosecution. When he happens to see B together with a (~ witness ~ C, he aims his t achyon

pistol at the head of B, unt i l ]3 and O (realizing the danger) s ta r t running away with speed, say, u. Then, A chooses to fire tachyonic projectiles T having a speed V such tha t u V > e 2. In the A rest frame, tachyons T reach ]3 soon and are absorbed by B's head, making him die. Due to the fact t ha t u V > c 2

(and to subsects. 5"12 and sect. 6), however, the witness C- -when questioned by the police--will have to declare tha t actual ly he only saw ant i tachyons come out of B's head and be finally absorbed by A's pistol. The same would be confirmed by ]3 himself, were he still able to give tes t imony.

9"5.2. T h e s o l u t i o n ; a n d c o m m e n t s . Le t us preliminari ly notice tha t B and C (when knowing t achyon mechanics) could at least revenge themselves on A by making A surely liable to prosecution: t hey should simply run towards A ! (cf. subsect. 3"12 and sect. 6).

B u t let us analyse our paradox, as above expounded. I t s main object is emphasizing tha t , when A and B are moving one w.r.t, the other, both A and B

can observe (( intrinsic emissions ~ in their respective rest frames (subsect. 6"10). I t follows tha t i t seems impossible in such cases to decide who is actual ly the beginner of the process, i.e. who is the cause of the t achyon exchange. There are no grounds, in fact , for privileging A or B.

In a picturesque w a y - - a s B~LL put i t - - i t seems tha t , when A aims his pistol at B (which is running away) and decides to fire suitable tachyons T, then B is ~ obliged ~) to emi t ant i tachyons T from his head and die.

7 2 ~ . ~ e A ~

To approach the solution, let us fu'st rephrase the paradox (follo~dng the last lines of subsect. 9"3) by subst i tut ing two spherical objects for A's pistol and B's head. About the propecties of the emit ters/absorbers of tachyons we know a priori only the results got in sect. 6; but , since this paradox simply exploits a par t icular aspect of the two-body interact ions via tachyon exchange, we have just to refer to those results. Their teaching m a y be in terpre ted as follows, if we recall t ha t we are assuming tachyon product ion to be controllable (otherwise the paradox vanishes). The ta.chyon exchange takes place only

when A, B possess suitable ~ tachyonie apti tudes ~), so as an electric discha.rgc takes place between A and B only if A, B possess electrical charges (or, ra ther , ~re a t different potent ia l levels). In a sense, the couph~ of ~ spherical obj~ct~ ~) A, B can be regarded as resembling a Van de Graaff generator. The tachyon spark is exchanged between A and B, therefore, only when observer A gives his sphere (the (~ pistol ))) a suitable (( taehyonic charge ,>, or raises i t to a suitable (( tachyonic potentiM ~). The person responsible for the tuchyon discharge between A and B (which may cause B to die) is, therefore, the one who iutention- Mly prepares or modifies the ~( tachyonic properties ~> of his sphere: i.e., in the case above, i t is A. In the same way, if one raises a conducting sphere A to a positive (electrostatic) potent ia l high enough w.r.t, the ear th to provoke a thunderbol t between A and a pedestr ian B, he shah be the guil ty murderer , even if the thunderbol t electrons actual ly s ta r t from B and end at A.

We have still to stress t ha t the original version of the Bell paradox was, in reMity, a ti t t le different, zkfter the discl~sion of the previous version, we are ready to appreciate it immediately. I t is worthwhile to learn abo~tt it from the words used by BEL~ (3985) himself in his comment : (~ In m y original version victim and witnesses remain at rest , and only the mnrdere r changes speed. Then he can profit also from any prejudice in favour of inert ial observers. The version in which vict im and witness run from the murderer makes a good story. But I th ink I would prefer to commit such a murder while myself on the run past the s ta t ionary victim and witnesses. Then, I would come to rest

to agree with s ta t ionary witnesses, policemen, judges, and jur ies- -point ing

out t ha t in their common frame the death of the vict im came before the pulhng

of the tr igger ~).

~Notice tha t we have been always considering tachyon emissions and a.bsoJ3)- tions, bu t never t achyon s catterings, since--while we know the t achyon mechanics for the former, simple processes--we do not know ye t how tachyons

interact with (ordinary) mat te r .

9"6. Signals by modulated tachyon beams: discussion of a paradox.

9"6.1. T i l e p a r a d o x . Still ((in macrophysics~), let us tackle at last a more sophisticaI~d paradox, proposcd by ourselves (()ALD]:I~0LA~ and I~ECA]~4:I,

CLASSICAL TACHYO~S 73

1980), which can be used to i l lustrate the mos t subt le hints contained in the (( causal i ty >> l i te ra ture (cL, e.g., F o x et al., :[969, 1970).

Le t us consider two ordinary inert ial f rames s--~ (t, x) and s ' z (t'~ x') moving one w.r.t, the other along the x-direction with speed u ~ c, and let us suppose t h a t s sends - - in its own f r a m e - - ~ signal a long the posi t ive x-direc-

t ion to s ' b y means of a modula ted t a c hyon b e a m hav ing speed V ~ c~/u

(fig. 25). Aceordi~)g to s' the t achyon b e a m will ac tual ly appear as an anti-

Fig. 25.

t !

/ ~ B ///

A ~ / // X'

,/ l l l/ ~ X

t a chyon beam emi t t ed b y s ' i tself towards s. We can imagine t h a t observer s, when meet ing s' a t O, hands h im a sealed le t ter and tells h im the following: (( By means of m y " t a c h y o n radio" A and s ta r t ing at t ime t, I will t r ansmi t

to your " t a c h y o n radio" B a mnlt if igured number . The n u m b e r is wr i t t en inside the envelope, to be opened only af ter the t ransmiss ion ~>.

Notice t h a t the (( free will ~ of s ' is not jeopardized nor under question, since s ' can well decide to not :switch on his t achyon radio B. I n such a case

we would be back to the s i tuat ion in subsect. 9"3: I n fact~ s would see his

t achyons T by-pass s ' w i thou t being absorbed and proceed fu r the r into the

space; s', on the cont rary , would see an t i t achyons T coming f rom the space and reaching A. I f s ' knows ex tended re la t iv i ty , he can t r ans fo rm his descrip- t ion of the phenomenon into the (( intrinsic descript ion ~> given b y s, and find

out t ha t s is (( intr insical ly ~) emi t t ing a signal b y tachyons T. He can check

t h a t the signal carried b y those r T corresponds just to the number wr i t ten in advance b y s.

The pa radox is ac tual ly m e t when s' does decide to switch on his t achyon radio B. I n fac t (if t ' is the Loren tz - t r ans fo rmed value of t, and At' --~ A ' O / V ' )

the observer s ' a t t ime t t - At r would see his radio not only b roadcas t the fore-

told mult i f igured n u m b e r (exact ly the one wr i t t en in the sealed le t ter , as s'

can check s t ra ight after) , bu t also emi t s imul taneously an t i t achyons T towards s :

Tha t is to say, t r an s m i t the same n u m b e r to s b y means of an t i tachyons . To m a k e the pa radox more evident , we can imagine s to t r an smi t b y the modu-

74 E. RECA~I

luted tachyon beam one of Beethoven's symphonies (whose number is shut up in advance into the envelope) instead of a plain number.

l?urther related paradoxes were discussed by PArdI6 and I~ECA_W~ (1976).

9"6.2. D i s c u s s i o n . Let us stress that s' would see the antitachyons T emitted by his radio B travel forward in time, endowed with positive energy. The problematic situation above arises only when (the tachyon emission being supposed to be controllable) a well-defined pattern of correlated taehyons is used by s as a signal. In such a case, s' would observe his taehyon radio B behave very strangely and unexpectedly, i.e. to transmit (by antitachyons T) just the signal :specified in advance by s in the sealed letter. He should conclude the intentional design of the tachyon exchange to stay on the side of s; we should not be in the presence of a real causality violation, however, since s' would not conclude that s is sending signals backward in time to him. We would be, on the contrary, in a condition similar to the one studied in subsect. 9"5.2. The paradox has actually to do with the unconventional behaviour of the sources/detectors of tachyons, rather than with causality; namely s', observing his apparatus B, finds himself in a situation analogous to the one (fig. 26) in which we possessed a series of objects b and saw them slip out sucked and (~ aspired ~> by A (or in which we possessed a series of metallic pellets and saw them slip out attracted by a variable, controllable electromagnet A).

From the behaviour of tachyon radios in the above Gedankenexperiment it seems to follow that we are in need of a theory formalization similar to Wheeler and Feynman~s (1945, 1949; see also FLATO and G~YE~CIN, 1977, and GOTT III , 1974a). In particular, no tachyons can be emitted if detectors do not yet exist in the universe that will be able sooner or later to absorb them. This philosophy, as we already saw many times, is a must in ER, since tachyon physics cannot be developed without taking always into account the proper sources and detectors (whose roles can be inverted by a LT); it is not without meaning that the same philosophy was shown (Wm~wL~]~ and F ~ s 1945, 1949) to be adoptable in the limiting case of photons. Let us recall that --according to suitable observers--the two devices A, B are just exchanging infinite-speed tachyons (or antitachyons: an infinite-speed tachyon T going from A to B is exactly equivalent to an infinite-speed antitachyon T travelling from B to A). Any couple of bodies which exchange tachyons are thus reMiz- ing~aeeording to those suitable observers--an instantaneous, mutual, symmet-

rical interaction. Thus tachyons can play an essential role at least as ~ internal lines ~> in bradyonic particle interactions (and vice versa, passing to a Super- luminM frame, bradyons would have a role as internal fines ot tachyonic particle interactions).

This suggests that A and B can exchange that Beethoven's symphony by means of tachyons only if the inner structure of both A, B is (~ already ,~ suited


f ~

b ~

, ,in,~ ~aP"

a~

~ I aaD~

,wl~ o

[ ~ - - , i

41~,,imm'. !

II

n

. :.... :.% . = ~ ' .

Fig. 26.

to such an exchange; this again is similar to what discussed in subsect. 9"5.2, even if the situation is here more sophisticated. I t seems in conclusion that tachyons are not suited for the transmission of really (t new ~> information.

Of course, all problems are automatically (and simply) solved if we adopt the conservative atti tude of assuming the tachyon exchanges between two bradyonic bodies A, B to be spontaneous and uncontrollable, as it happens in microphysies (as far as we know). For simplicity's sake, such a restrictive atti tude might be actually adopted, even if unnecessary. See also, e.g., ]~AVAS (1974) and ~OL~rCK (1974).

What seen in the last three parag~'aphs of this subsection can be of import- ance in connection with the dream of explaining the quantum phenomenology via classical physics (with tachyons). Actually, the fact that tachyons are just the objects sltitablc for transmitting a mutual interaction (in which it is meaningless to say whether A is influencing B, or B is influencing A) meets quite well the apparent requirements of QM and of QFT, that within a quantum system there exist Superlnminal connections (which seem to not imply controllable, Superluminal signals). See, e.g., STAI'I" (1985) and HEGEI~FELD (1985); see %lso \u (1976), Vm:cmr (1980), D'EsPAO~'A'r (1981) and R~a'DIJK and SELLE~ (t985).

9"6.3. F u r t h e r c o m m e n t s . When the signal does not consist of a well-defined pattern of taehyons, but is constituted by a few tachyons only -- typically by ~ single taehyon-- , we saw that no paradoxes stu'vive. If on the contrary claims as the one put forth by NEWT0~ (1.967) were true, then one could send signals into the past even by ordinary antiparticles (which is not true, of cottrse; of. I ~ E c ~ and ~r 1975; IgECA:~, 1970).

5ioreover, to clarify further the terms of the paradox in subsect. 9"6.1, 9"6.2 above, let us explicitly recall that i) the ehrolmlogical order of events can be reserved by an ordinary LT along a spacelike path only; therefore, the order of the events along the A, B world-lines cannot change; ii) also the proper energies (rest masses) of A, B are Lorentz invariaut, together with their ~( jumps )>; iii) while s sees the total energy of A decrease, s' may see it increase (description details...) ; iv) the paradox in subseet. 9"6.1,9"6.2 is connected with the question whether the entropy variatioms and information exchanges are to be associated with the changes in the proper energies: in this case, in fact, they would not necessarily behave as the (~total energies )) (see CAI,DIICOLA and l%ncam, 1980, and PAv~IO and IgEc~-~I, 1976, where the paradoxical situations arising when one deals with macrotaehyons are furthermore discussed).

We mentioned in the previous discussion (subsect. 9"6.2) that the behaviour of tachyon sources/detectors might appear paradoxical to us for the mere fact that we are not accustomed to it. To shed some light on the possible nature of such difficulties, let us report at last the following anecdote (Cso~A, 1970), which does not involve contemporary prejudices: (~For ancient Egyptians,

CLASSICAL TACI-IYON S 77

who knew only the Nile and its t r ibutaries, which all flow South to North, the meaning of the word " sou th" coincided with tile one of "uI)s t ream" , and the meaning of the word " n o r t h " coincided with the one of "downs t r eam" . When Egyt)tians discovered the Euphra tes , which unfor tuna te ly happens to ttow Nor th 1o South, t hey passed through such a crisis t ha t it is ment ioned ill the stele of Tuthmosis I, which tells ItS about that inverted water that goes

downstream (i.e. towards the North) in going upstream ~>. See also, e.g., IIILOE- COOleD (1960).

9"7. On the advanced solutions. - I~elativistic equat ions (both classical and qualital) arc kliowll to admi t ill general advanced, besides re tarded, solutions. For ilistalice, Maxwell equations predict both re tarded aml adwmccd electromagnetic radiations. Naiw,~ly, advanced solutions have been sometimes regarded as actually representil)g motions backwards in t ime. On the contrary, we know from the (( switching principle ~> (subsect. 2"1) alid the very s t ructure of S1r (see I)art I, sect. 2) tha t the ~(advanced ~) waves or objects are nothing

bu t alitiobjects or antiwaves travell ing in the oi)i)osite si)~ce direction. With in ER, actually, when an equatioli admits a solution corresi)onding

to (outgoing) particles or photons, t.hen a class of suitable GLTs tralisform such a solution into another one corresponding to (incoming) antii)articles or (anti)photons. In o ther words, if an equaLioli is G-covariant , it must admi t also of solutions relat ive to incoming antiparticles or I)hotolis, whenever it admits of solutions relative to outgoing particles or photons.

This means tha t all G-covariant relativistic equations must admit both re tarded and advanced solutions. When confining onrselves to sublumilial wqocities u 2, v~< 1, the ordinary relativistic equations already satisfy such a requirement , for the reasons discussed in pa r t I (see, ill partictflar, subsect. 2"3, I)oint d)).

We could, however, ask ourselves why do we usua.lly observe only, e.g.,

the outgoing, ra ther than the incoming radiation. The clue to the question is in taking into account the initial conditions: In ordinary macroi)hysics some initial conditions are by f'u' more probable than others. For insta.nce, the equations of fluid dynamics allow to have on the sea surface both outgoing circula.r concentric waves and incoming circular waves tending to a centre. I t is known, however, t ha t the initial conditi(ms yMding the former are more likely to be met than those yielding the la t ter case.

10. - Tachyon classical physics (results independent of the SLTs' explicit form).

According to subsect. 5"1, the laws of classical physics for iachyolis arc to be der ived jlist by at)plying a SLT to thc ordinary, classical laws of bradyons (this s ta temeli t has been sometimes referred to as the ((rule of ex tended

78 E. RECAMI

relativity )): cf. PA]~:ER, 1969, and I~ECAMT and 1VIIG~ANI, 1974a). To proceed with, we lmcd nothing but the assumption in subsect. 7"1 ; i.e. we need only assuming tha t SLTs exist which carry timelike into spacelike tangent vectors,

and vice versa. I t is noticeable tha t tachyou classical physics can be obtained in terms

of purely real quantities. Subsections 10"1 and 10"2 below do contain important improvements

w.r.t, review I.

10"1. dynamics

Tachyon mechanics. - For example, the fundamenta l law of bradyon

reads

(106) F s ~ c ~ moC ds] dye re~ ( f l ~ < l ) .

~ot ice t ha t eq. (106) in its first form is only Lorentz covariant, while in its second form is G-covariant (cf. subsect. 7"2).

Even for tachyons, then, we. shall have ( I ~ c A ~ and M~.~:v~, 197~a)

d dps (f12 > 1), (107) Fs = + ~ (mous) - + d;:

where me is the t~chyon (real) rest mass and, ~nticipatiag subseet. 14"13, we defined ps := mous also for tachyons. Equat ion (107) is the relativistic Newton law written in G-covariant form: i.e. it is expected to hold for fl~ 51. I t is essential to recall, however, tha t us is to be defined us ~ dx,/d~0 just as in eq. (93a). Quant i ty dye, where 3o is the proper time, is, of course, G-invariant; on the con- t rary , ds ~ 4- c dye for bradyons, bu t ds : • iv d~o for tachyons (cf. subsect. 2"2

and 4"3). Equat ion (107) agrees with eqs. (87) and (87') of subsect. 6"14, where we set

F = dp/dt, and suggests tha t for tachyons dt ~- ~ d~0/V/fl ~ - -1 (see rewiev I), so t ha t in G-covariant form (it---- ~ dv0(]l--fl2]) -~.

)~or the tachyon case, let us notice the following. I f at the considered t ime ins tant t we choose the x-axis so t ha t V -- iV] = V,, then only the force component _F~ will make work. We already mentioned tha t the tota l energy of a tachyon decreases when its speed increases, and vice versa (see fig. 4a) and 10); i t follows tha t F~ when applied to a tachyon will actual ly make a. positive, e lementary work d ~ only if a~ is antiparallel to the elementary displacement dx, i.e. if sign ( a ~ ) ~ - sign (dx). In other words, d ~ in the case of a force F applied to a tachyon must be de]ined (cf. subsect. 6"14) so tha t

m~ a. dx (108) dC~f -- (V i - 1)~ '

C L A S S I C A L TACIIYONS 7 9

where a~ and dx possess, of course, their own signs. Equa t ion (]08) does agree both with the couple of equations (88a), (89) and with the couple of equations (88'a), (89').

I t is evident tha t , with the choice (review I) represented by eqs. (89) and (85) of subscct. 6"14, we shall have (v == v.~, V=-: V~)

(]09a) /f. : : 2 _ mo ( l - - v2) ~- a , for bradyons ,

(109b) F~ ---- -- mo ( V 2 _ 1)_~ a , for tachyons .

On the contrary, still with the choice (89)-(85), we shall have

m 0 (109c) F,,,, . . . . [- (i1_ _ fl~i-)t a~,, (fl2X1)

for both brudyons and t~chyons. Actually, under our hypotheses (v = v . , V = V~), the force t ransverse components ~v , do not make any work; therefore, one h~d no reasons a priori for expect ing any change in eq. (109e) when passing from bradyons to tachyons.

1 0 " 2 . Gravitational interactions of tachyons. - In any gravi ta t ional field a b radyon feels ~h(, (at tractive) gravi ta t ional 4-force

(]10) F l '::: -- moF~o dxq dx ~ ds (is (f12~ 1).

In G-covariant form, then, eq. (1]0) will be wri t ten as (review I ; M_taNA~I and ]~ECA_-Vr~, :1974d; ]r and MIG~A-~I, 1974a; RECAllS, 1977b)

m0 ,~ dxe dxa ( I l l ) F , c -~ I~o dro dro (fl'~Xl),

since the Christoffel symbols behave like (third-rank) tensors nnder any linear transforma.tions of the co-ordinates. Eqnut ions (111) hold in part icular for a

t achyon in "my gravi ta t ional field (both when originatcd b y tachyonic and b y bradyonic sources).

Mlalogously, the equ-d.ion of motion for bosh bradyons and tachyons in a grnvi tat ional field will still read (review I), in G-covariant form,

(~12) a" 4 l~Su~u ~ = 0 (fl2~1),

2 u 2 with a" =~ d x ' / d z o.

8 0 E. RECAMI

Passing to general relativity, this does agree with the equivalence principle: Bradyons, photons and tachyons follow different trajectories in a gravitational field, which depend only on the initial (different) four-velocities and are independent of the masses.

Going back to eqs. (111), we may say that also tachyons are attracted by a gravitational field. However, such an (( attraction >> has to be understood from the energetical and dynamical poi~lt of view only.

In fact, if we consider for simplicity a tachyon moving radially w.r.t, a gravitational source, due to cq. (109b) (i.e. due to the couples of equations either (88a), (89), or (88'a), (89')) it will accelerate when reeedingIrom the source, and decelerate when approaching the source. From the kinematical point of view, therefore, we can say that tachyons seem to be gravitationally repelled. Analogous results were put forth by VA~DYA (:1971), I%AYCHAUDRV~I (1974), HONIG et al. (1974), and so on.

In the case of a bradyonic source, what precedes agrees with the results obtained within general relativity: see, e.g., S~TZmAN and S~TZ~XN (1969), GtmGO~u (1972), HV.TTEL and HELIXWELL (1973), SU~ (1974), ~ L I K ~ and SUD~S~L~N (1976), NA~L~K~ and Dn~R~WD~R (1976), C o ~ ~ad LATH~0P (1978), MALTSEV (1981), C~OR0WSWr (1982), ~INKELSTEIN et al.

(1983), Cxo et al. (198~), etc.

10"3. About Cerenkov radiation. - Let us consider a t iny spherically symmetric electric charge P, in particular pointlike. From subsect. 8"2 (cf. fig. 17 and 18) we know that, when endowed with constant Superluminal speed V (e.g., along x), i) its shape transforms into a double cone %00, ii) its equlpotential surfaces (spherical surfaces in the rest frame) transform into two-sheeted hyperboloids asymptotic to go. Such is the result of the (~ distortion ~> due to the high relative speed V; cf. fig. 27 (see also GLA~DKZ~, 1978a, b; TEl%LETSKY,

~ m n

/

ebsorption k

x

Fig. ~7.

0LASSICAL TACIIYO~'S ~1

1978; CORBES, 1975, 1974; Go~r~: I I I , 1974a; FL)zURu etal., 1973, as well as SHANKAI~A, 1979). _Notice explicit ly tha t we are here dealing with the equi- potent iM snrf~ces of the initial electrostatic field and with thei r t ransforms under ~ SET; complete ly different woldd be the case (el. subsect. 14"]) of the electromagnetic wa.ves actual ly emi t ted by a source (initial spherical waves will have to t ransform a.gain into spherical waves!).

The asymptot ic double con(; ~o in fig. 27 has nothing to do with (3erenkov's, since no actual radiat ion energy is globally emi t t ed by P ~ iJz during its inertial Superlumin~fi mot ion; in fact, one m ay say tha t the seeming emission

associated with the (, re tarded ~ cone is exact ly countcrbManccd by the seeming absorption associated wilh the (( advanced ))cone (BARUT et al., 1982). Moreover, ~ereukov radiut.ion is knowr~ to come not f rom the (~ radiat ing ~> part icle itself, hu t from the charges of ~he (m:~teri~/1) medium; so tha t the expression (~ 0e- read, o r radiat ion in vacuum ~) is itself meaningless: unless one provides a suitable l:heory about the vacuum st ructure (which is not expecte(! to be done within the prc.'~e~,~t classical apI)roach. See MXG~AS"I ~nd tCECA~, 1973b).

Incidentally, it would be nice to know (of. also the (rod of subsect. 8'2) the L-function of the space-time co-ordinates yielding the distr ibution over ~do of

the tachyon charge density. Mtel~vards, on the basis of the Ma, xwell equations for tachyons (;',ee subsect. 15"t in the following) and for a, constant speed V, it would b('~ intercepting to find out solution~ for E(t, x) and H(t, x) corresponding to a null global flux of radiation.

Slime we do not know ye t the ex])licit form of the SLTs in four dimensions, we can resort to th(,~ two-dimensional formulae (subseet. 5"6) to check at least in that~ came the above-seen prediction tha t constant-speed tachyons do not emit ~erenkov radiat ion in the vacuum. ]]el, us consider a free t achyon P~ in the vacuum, i t will appear as a free bradyon :P~ to any Superluminal observer S; accordit~g to S the energy lost by P~ through (~erenkov emission is, therefore, zero: dE/dI . - O. I f we t rans form such a law b y means of a SLT~ e.g. by the t ranscende:~l 2-dime~sional SET, w(~ get again dE' /al l '= O. Provided 1,hat the <( electromagrletie vacuum ~> is invar iant under SETs (apart from tachyons) , we have verified t ha t free taehyons are not expected lo emit ~erenkov radiat ion in vacuum (MIGs"ANI and ~ECA.~I, 1973b; see aJso, e.g., EY and HU~ST, 1977; KIRCH, 1977; BU'LBECK and HUgST, 1984=).

10"4. About Doppler e ] f e c t . - In the two-dimensiona.1 case (subsect. 5'7), the Doppler-effect formula for a sub- or a Super-luminal source, moving along the x-axis, will be (MIG_~-A.~I and RECA.~, 1973a)

~/j~ - ~ ! (:ll3a) v-- v0 . . . . . . . . .

l • ( - - , ~ < u < + ~) ,

where tlle siglt -- ( + ) eorresponds to approa(:h (recession). The eonsequer~ees

82 ~;. I~E capri

are depicted in figures like fig. 23 of review I . l~or Super luminal approach, v happens to be negat ive, so as explained b y our fig. 15c). Le t us, moreover , observe tha t , in the case of recession, the same Doppler shift is associated bo th wi th ~ < e and with U ----- 1/~ > c (MIG~A~ and l%Ec~,'crr, 1974e, ]; R~.CA:~X,

1977b). I n the four-dimensional case, if the observer is still located a t the origin,

eq. (l13a) is expected to generalize (ICEcAMr and M_m~A~ri, 1974) into

(l13b) v = Vo i - - u cos~ ( - c ~ < u < + oo),

where a - - ~ , vec tor s be ing directed f rom the source to the observer . Le t

us notice f rom subsect. 6"15 (eq. (91)), incidentally, t h a t when an observer starts receiving radia t ion f rom ~ Super luminal point l ike source C (at Co, i.e. in the <( op t ic -boom ~ si tuation), the received radia t ion is infinitely blue-shifted.

10"5. l~leetromagnaism ]or tachyons: preliminaries. - The problem of extending e lect romagnet ism to t achyons is not s t ra ight forward, since one does

not know a priori whether the << electromagnet ic tensor ~ ~ has to be s$ili

a tensor under the SLTs; el. subsect. 7"2 (quant i ty ~ , is a tensor under the

t r ans fo rma t ion group .~+, b u t m a y not be have a n y longer as a tensor under a

larger t r ans fo rma t ion group). I f one assumes F~, to be a G-tensor, t hen ord inary Maxwell equat ions keep

the i r fo rm also for t achyons ( R E c A ~ and Mm~A~I, 1974a, p. 277):

(114) ~F~ ' . . . . j~', ~ . ~ " - ~ 0 ,

where j ~ Q o u u is the 4-current of bo th sub- and Super- luminal electric

charges (and where the tilde indicates the dual tensor). Such a choice is

the one adop ted b y C0~BE~ (1974, 1975, 1976, 1978a). I t corresponds to assum-

ing t h a t the electric and magnet ic fields E, H t r ans fo rm under SLTs just

as in eq. (101) of review I , or similar. I f one, on the cont ra ry , does not assume a priori t h a t F ~ is still a tensor

even under G, then one has first of all to de termine or choose the behav iour

e i ther of the components of E, H or of the e lect romagnet ic 4-potent ia l A~ under SLTs. At this stage, let us observe wha t follows. I n subsect. 7"2 we

noticed t h a t two different kinds of (< 4-vectors )) are easily m e t when t ry ing to

ex tend SlY: the ones like u u ~ dxv/dTo tha t are also G-vectors , and the ones like

w~' ~ dx~/ds t h a t are Lorentz vectors , but under a SLT, Su~ (when ds ~ ->--ds~) ,

t r ans fo rm as follows

�9 t

(115) w~ ~ ~ i S S w , -=- • ~wl, .

W h e n we wri te down the ordinary 5Iaxwell equat ions for a pure ly subluminal

CLASSICAL TACHYONS S~

t -current js(s) in terms of the l -potent ia l As

( n 6 a ) [] A~ = is(s)

(116b) asAs = 0 ,

(v~<e2),

where we imposed the Lorentz gauge and, as usual, Fs , = A,.s-- As.~, the left- and r ight-hand sides of eq. (116a) can be both vectors of the same (~ kind ~), or not. In the former case, ]~axwell equations are G-covariant, and we are back to eqs. (114). In the lat ter case, however, eqs. (116) under a SLT, (S-*)~, become (except, possibly, for a sign: see MIGNANI and I~ECA~, 1975b)

(117) [~X~ = -- i j~(S) , asA's = 0 (V'2> e2),

where j ' , ( s )~ j~ (S ) represents the Superluminal l-current . In such a second, al ternative case--which, incidentally, is supported by

our discussion oi tachyon elcctrodynamics to follow in sect. 1 5 ,~ wh e n in presence of both sub- and Super-luminal 4-currents we may, therefore, write

[] (A s q- A'~) = j,(s) - - i js(S ) (v~c~).

I f we introduce the complex l -potent ia l ~ s -- A~ -~ A~ ~ A~,-- iBs , and the complex t -current J s = j#(s) -- i%(S), the previous equations can be writ ten as (MIG~ANI and lgECA~, 1975b)

which extend Maxwell equations, eqs. (116), to the case when both slower- and faster-than-light currents are present. By defining

(119) A s = REX, , B s ~ - - I m X s = § iA~ ~ -~ i (S-~) , ,A ~ ,

we can write the generalized equations for the extended t -potent ia l as follows:

(118') I [] (As -- iBs) = i s ( s ) - ijs(S) (v2~e2),

t ~ ( A s -- iBs) = 0

which can, of course, be split into two real equations.

We shall come back to the problem of the generalized Maxwell equations for bradyons and tachyons in subsect. 15"2, where we shall show tha t Bs can be formally identified with the (( second l -potent ia l >> introduced by C ~ B o and FEI~A~I (1.962) for magnetic monopoles (A]~mDI~ 1968; FE~I~AI~I, 1978; DE FA~IA-t=~OSA et al., 1985, 1986).

84 E. ~ECAMI

11. Some ordinary physics in the light of LR.

l l ' ] . Introduction. A g a i n about CPT. - Looking for the SLTs in the ordi-

nary space-time will pose us a new problem: finding out the t ranscendent t ransformat ion ~f~: 5f4 which generalizes eq. (32) of subsect. S'5 to the 4-dimensional ca.se. However , af ter what we saw in par t I (sect. 2), we are already prepared to accept (cf. subsect. S ' ]6 and 5"6) tha t

(37') G~ G ~ - - G ~ G , VGE G,

even in four dimensions. Actually, f rom iig. 5c) and 6 (now unders tood to hold in four dimcnsioHs),

we see t ha t i) an ordinary LT = Z can carry from Ts to Ts, ii) if a SLT = L s

exists t ha t carries f rom Bs to Ts, then the subluminal t rans tormat ion Ls~]~.Zs will carry from Bs to Bs. Our general results in sect. 2 (e.g., eq. (10)) imply, therefore, t ha t eq. (53) will be valid also in four dimensions (.M_IG~A~I and ICECA:M:[, 1974b):

(53 ~) --I ~-PT----CPT~G;

i~l connection with eq. (53') see all the remarks already expounded in subsect. 2"3. As a consequence, the generalized group G in Minkowski space-time is

expected to be the extension (PArdI5 and I ~ E c ~ , 1.977) of the proper, orthochronous (4-dimensional) Lorentz group ~q~+ by means of the two operations

CPT _ ~ - 1 ~nd ~f:

(120) G = ~(~*+, e l 'T, o~).

In our formalism, the operat ion CPT is a linear (classical) operator in the pseudo-Euclidean space, and will be a un i t a ry (quantum-mechanical) operator when acting on the states space: cf. eq. (53'), and see I~ECA~I (1979a), C0S~A

DE BEAUI~E6A~D (1983). F r om what precedes, and from fig. 5 and 6, we m a y say tha t even in the

4-dimensional energy-momentum space we have two symmetr ies : i) the one w.r.t, the hyperplane E = 0, corresponding to the transi t ion particle ~ anti-

particle, and ii) the one w.r.t, tile light-cone, expected to correspond to the

t ransi t ion b radyon ~- tachyon. In any case, the (( switching procedure ~) (sect. 2 and subscct. S'12) will

surely have to be applied for both bradyons and taehyons also in four dimensions.

Le t us, therefore, reconsider i t in a more formal way.

11"2. A g a i n about the (~ switching procedure ~). - This and the following

subsections do not depend on the existence of tachyons. They depend essen-

tially on our par t I.

CLASSICAL TACHY0)~S ~5

We shall indicate by <( S W P , the switching procedure (previously often called (( I~IP ~>). Let us also call strong conjugation C the discrete operation (*)

(121) 0 - C~o,

where C is the conjugation of all additive charges and Mo the rest mass conjugation (i.e. the reversal of the rest mass sign). /~ECAm and ZI!No (1976) showed that formally (cf. fig. 3b))

<< 8WP ,> = C.

Then, by considering mo as a ]i]th co-ordinate, besides the ordinary four (EINSTEIN and BERG~L~NN, 1938), and shifting to the language of quantum mechanics, they recognized that /)5 ~ C, quantity P~ being the chirality operation, so that

(122) (~ SWP ~> ~- P5 ;

in fact, when dealing as usual with states with definite parity, one may write C - ~ ~ - - P / ~ P s . Notice that in our formalism the strong conjugation C is a unitary operator when acting on the state space (cf. also VILELA-1Vf/-ENDES,

1976). For details and further developments see, e.g., besides the above-quoted

papers, EDMONDS (1974), LAKE and I~OEDER (1975), PAV~I~ and REOA~ (1977), REOA~I (1978a), REOAlViI and RODI~IGUES (1982).

Here we want only to show that, when considering the fundamental par- t ides of matter as extended objects, the (geometrical) operation which reflects the internal space-time of a particle is equivalent to the ordinary operation C which reverses the sign of all its additive charges (PAV~I~ and I~EOAYg, 1982).

11"3. Charge conjugation and internal space-time reJlevtion. - Following PAV~I~ and R E C ~ (1982), let us consider in the ordinary space-time i) the extended object (particle) a, such that the interior of its (( world-tube ~> is a finite portion of Md; ii) the two operators space reflection, ~ , and time reversal, 3", that act (w.r.t. the particle world-tube W) both on the external and on the internal space-time:

(123) / ~ = ~ i = ~ i ~ , t 3- = 3-~3-I = 3-~3-~,

(*) In this sect. 11, for ]ormal reasons, we shall indicate the space-time symmetry operations (till now represented by Roman letters) by italics; however, C--= C, P=--P, T=--T, P=--P, T=--T, etc.

8 6 ~ . R E C A M ~

a)

b)

Fig. 28.

X o X ~

~T

1

X o

-'c T

x1

o x x

~ix1 xt

[~ 0

5 ~

x 0

E

x~

5~r

where ~z(3rz) is the i~ternal and ~E(3r~) tile external space reflection (time reversal). The ordinary par i ty _P and t ime reversal T act on the cont ra ry only

on the external space-time:

P - - - - ~ , T - - ~ .

The effects of ~ , '~z and ~ on the world-tube W of a are shown in fig. 28, and the analogmls effects of 3r E , 3r~, 3r in fig. 29.

Le t us now depict W as a sheaf of world-lines w representing---let us s a y - - its const i tuents (fig. 30a)). In fig. 30 we show, besides the c.m. world-line,

a)

b)

Fig. 29.

�9 U u

..qtJ

X 1

cP /a

1

X o

.9"

x o

�9 s I

.go

~=T

<

CLASStCAL T,tCI1Y().N S ] ]7

Fig. 30.

~Z

a ) b) c)

also w~ ~ A alld w2---- B. The operat ion ~ Y will t ransform W inio a second worht- tube ~u consisting of the t ransformed world-lines ~ (see fig. 30b)). :Notice tha t each ~ poinls in the opposite tim(; direction and occupies (w.r.t. the c.m. world-line) the position symmetr ical to the corresponding w.

I f we apply the Stf ickelberg-Feynman <( switching ~> (subsect. 2"1), each world- line ~ t ransforms into ,% new world-line W (cf. fig. 30c)) which points in the positive t ime dh'ection, bu t represents now an anticonstituent. Let us now explicit ly generalize the (( switching principle ~> for extended particlcs as follows: We ident i fy the sheaf W of the world-lines w with the antipart icle g, i.e. W with the world-tube of g. This corresponds to assuming tha t the overall t ime direction of a part icle as a whole coincides with the t ime direction of its (( const i tuents ~>. A prel iminary conchlsion is thai, the antipart icle g of a can be regarded (from the chronotopical, geometrical point of view) us derived from the reflection of the internal space-time of a.

Le t us repeat what precedes in a more rigorous way, following our sect. 2, i.e. recalling t ha t the t ransformat ion L = -- 1 is an actual (even if antichronous) Lorentz t ransformat ion, corresponding to the 180 ~ apace-time ((rotations>:

P T ~ - 1. Now, to apply P T from the active point of view to the worht- tube W of ti t . 30a) means to ro ta te i t (by :180 ~ in four dimensions) into (fig. 30b)); such a ro ta t ion effects also a reflection of the internal 3-space of particle a, t ransforming i t - - a m o n g the o the rs - - in to its mirror image. The same result would be got by applying P T front the passive point of view to the space-time in fig. 30a).

r s we generalize the (~switching principle )> to the c~se of ex tended objects by applying it to the world-tube W of fig. 30b). The world-tube 1'~ docs represe, n~. an (interHally <( mirrored ~>) part icle not only going backwards in

tint(;, bu t also carrying negative elmrgy; lherefore, the (( switching ~) does rigorolLsly t ransform ~" into W (fig. 30c)), the anti-world-tube W representing (i.

In conclusion,

(124) --I --PT : ~ Y ~ g 2 1 Y ~ =: ~ J - = PT~IYl,

8 8 ~ . ~ c ~

wherefrom, since P T : C P T (subseet. 2"3), one derives

(125) C ~ r .

As already anticipated, we have, therefore, shown the operation C, which inverts thc sign of (all) the additive charges of a particle, to be equivalelit to the (geometrical) operatioll of reflecting its internal space-time.

Also the results reported in this subsection support the opinion that in theoretical physics we should advantageously substitute the new operations P - - and T----3" for the ordinary oper~tions P and T, which are merely external

reflections (for instance, onlly the former belong to the fidl Lorentz group). Besides our sect. 2, cf., e.g., review I, ld.ECA_M:I (1978e) and also COSTA DE BEAU- ICEGAI~D (1984).

11"4. Crossing relations. - Besides the C P T theorem, derived from the mere SR, from E:R only it is possible to get also the so-called (~ crossing relations ~. Let us first recall that cross-sections and invariant sc,~ttering amph- tudes can be defined (RECA_~_I and M~G.~A_~, 1974a) even at a classical, purely relativistic level.

We are going to show (MIG~'ANI and REC~M:I, 1974a, 1975a) that,--within El~--the same fr is expected to yield the scattering amplitudes of different processes like

(126a) a ~- b --> c -[- d,

(126b) a -!- ~ ->b -}- d

in correspondence, of course, to the respective, different domains of the kinematical variables.

Let a, b, c, d be bradyonic objects w.r.t, a frame so. The two reactions (126a), (126b) amoug Bs are two different processes Pl, P, as seen by us; but they can be described as the same interaction d s -- d t ~ d~ among Ts by two suitable, different Super]uminal observers S~, S~ (review I; : R E C k , 1979a; C~])n~0LX and/~):cA_~r[, 1980). We can get the scattering amplitude A(pl) of Pl by applying the SLT(S~ -~ s0) -- Z] to the amphtude Al(dl) found by $1 when obser~ng the scattering Pl, i.e. A ( p l ) = LI[AI(dl)]. Conversely, we may get the scattering amplitude A(p2) of p, by applying the SLT(S~-~ so)~ L~ to the amphtude A2(d~) foun4 by S~ whelt observing the scattering p~, i.e. A(p~) = L~[A2(d2)]. But, by hypothesis, Al(dl) =A2(d2)--=A(ds). Then, it follows--roughly speaking--that

(127) A(p~) = A(p~)

for all reactions among bradyons of the kind (126a) and (J26b). Actually, in ordinary QFT the requirement (]27) is satisfied by assuming

CLASSICAL TACHYONS ~9

the ampli tude A to be an analyt ic funct ion tha t can be continued f rom the domain of the invar iant variables relat ive to (126a) to the domain relat ive to (126b). However , our requi rement (127), imposed by EI~ on the processes (126), has a more general nature , besides being purely relativistic in character . For fur ther details see review I.

At last, new <~ crossing-type relations ~> were derived from EI~ ; t hey might serve to check the relativistic covariance of weak and strong interactions (which a priori do not have to be relativist ically covariant) : of. 3/[Ia~A~I and RECAM_I (1974a, 1975a).

11"5. Further results and remarks. - Some results already appeared above; see, e.g., subseet. 9"7 on the in terpre ta t ion of the (~ advanced solutions ~>.

Many fur ther results will appear in pa r t IV (sect. 13), in connection with QM and elementary-par t ic le physics: let us ment ion the ones related with the vacuum decays, v i r tua l particles, a Lorentz- invar iant boots t rap for hadrons, the wave-part icle dualism, etc.

Here, let ns only add the following prel iminary observations. Le t ns consider (fig. 31) two bodies A and B which exchange (w.r.t. a f rame

Fig. 31.

c

_ Q _

r

so) a t ranscendent t aehyon T~ moving along the x-axis. F ro m fig. 3 and sect. 6 we have seen t ha t for t ranscendent particles the mot ion direction along AB is not defined. In such a l imiting case, we can consider Too e i ther as a t aehyon T(v = q- ~ ) going from A to B, or equivalent ly as an an t i t achyon T(v ~ -- c~) going from B to A (el. also fig. 3). In QM language, we could write (Pav~I5 and I~ECAMI, 1976)

( 1 2 8 ) = = - - a s + b 2 = 1 . ITs) -~ alT(v -~ ~ ) ) q- btT(v c~)) ,

Alternatively, it will be immedia te ly realized tha t so can in terpre t his observations also as due to a pair creation of infinite-speed tachyons T and

(travelling along x) at any point C of the x-axis between A and B (MIG~A~I

and I~CA3~, 19760; EDMONDS, 1976; CA];DrROLA and I~ECAHI, 1980): for instance, as the creation of a t ranscendent t aehyon T travell ing towards (and

absorbed by) B and of a t ranscendent an t i tachyon T travell ing towards (and

absorbed by) A. Actually, for each observer the vacuum can become elassi- ta l ly unstable only by emit t ing two (or more) infinite-speed tachyons, in such a way tha t the to ta l 3 -momentum of the emi t t ed set is zero (the to ta l energy emi t ted would be automat ical ly zero: see fig. 4, 5 and 6).

90 E. ~ECMAI

I t is interesting to eheck--cf , subsect. 5"6 and eq. (52) of subsect. S ' 12 - - tha t ~ny (subluminal) observer s~, moving along x w.r.t, so in the direction A to B~ will just see a single (finite-speed) ant i tachyou T~ emit ted by B, passing through point C wi thout any intcract ion, and finally ~bsorbed by A. On the contrary, any observer s2, moving along x w.r.t, so in the direction B to A~ will just see a single (finite-speed) t achyon T~ emi t ted by A, freely travell ing from A to B (without any interact ion at C), and finMly absorved by B.

In what precedes we m a y consider the masses of A and B so large tha t the kinematical constraints, me t in sect. 6, get simplified. In such a case, so, s~ and s2 will '~]1 see an elastic scat ter ing of A and B.

As we have seen above, any observer So can describe the part icular process ph under examinat ion in terms ei ther of '~ vacuum decay or of a suitabh~ tachyon emission by one of the two nearby bodies A, B. One can al ternat ively adopt one of those two languages. ~ o r e generally, the probabi l i ty of such vacmtm decays must be related to the t ranscendent - tachyon emission power (or absorpt ion power) of mat ter .

Fur thermorc , if A and B can exchange tachyons even when they are ve ry far f rom each other, any observer s' (like s~ and s~) moving w.r.t, so will describe ph in terms ei ther of an incoming, suitable tachyonic cosmic ray or of the emission of a slfitable, finite-speed t'~chyon by a mater ia l object. One of the conseql~ences, in brief, is t ha t the tachyon cosmic flux is expected to have for consistency a Lorentz- invar iant 4-momentum distribution, just as depicted in fig. 10 und 5c). The large major i ty of (( cosmic ~ tachyons ought then to appear to any observer as endowed with speed very near to the light speed c (see also V~G~E~, 1979; KA_~OI and KA.~:EFUCHI, 1977). In this respect~ it m ay be iaterest ing to recall t ha t an evaluat ion of the possibh~ cosmic flllx of tachyons yielded even if ve ry rough - - a flux close to the neutrinos ' one (MIG~A~I and I4EOA~I, ] 976a).

As an e lementary i l lustration of other possible considerations, let us at

last add the following. I f so observes the process

(129a) a --> b '-- t ,

where t is an ant i tachyon, t h e n - - a f t e r a suitable L T - - t h e new observer s' can

describe the s%me process as

(129b) a ~ - - t - ~ b .

If , in eq. (129a), the emi t ted t had travelled till absorbcd by a (near or jar) detector U, then in eq. (129b) t must, of course, be regarded as emitMd by a

(near or jar) source U. I f A~ is the mean life of part icle a for the decay (129a), measured by so, it

will bc the Lorentz t ransform of the averagc t ime A~' tha t particle a must spend according to s' before absorbing a (( cosmic ~ tachyon t and t rasforming into b.

CLASSICAL TACI~YONS 91

I )A~ III

General Relativity and Tachyons.

12. - About tachyons in general relativity (GI~).

12"1. Eoreword, and some bibliography. - Spacehke geodesics are (( at home >> in general relativity (GI~), so that tachyons have often been implicit ingredients of this theory.

Some papers dealing with tachyons in GR have been already quoted in subsect. 10"2; other papers are ~ISLLEI~ and Wm~ELE~ (1962), FosTE~ and I~Au (1972), I~Au and F0STE~ (1973), LEI]~0WITZ and I~0SE~ (1973), BA~EgJEE (1973), GOTT III (1974a, b), ARCIDIACONO (1974), GOLD01~I (1975a, b, e, 1978), DAVIES (1975), LAKE and I~OED]~ (1975), I~Au and ZI~IWE~AN (1976, 1977), PArdI6 and l~EcA~g (1977), DE SA]~BATA et al. (1977), BANE~JEE and C~OUDH~I (1977), S~IVASTAVA and PAT}tAK (1977), S~IVASTAVA (1977), G~wVICH and TA~ASEVIC~ (1978), K0WALCZu (1978), I~ECA~ (1978a), CA:~_E~ZIND (1978), MILEWSKI (1978), Jom~I and SRIVASTAVA (1978), DHUI~ANDgA:a (1978), DtIUlCANI)HAR and NA~LI~A~ (1978), CAST0~INA and I~ECA~ (1978), ~AI~LIKAR and D]t~I~ANDHAtr (1978), I~EOAlV~ and S]~A~[ (1979), DADmCH (1979), MA:LLE~ (1979), LXUmCIC et al. (1979), P~ASAD and SI~ItA (1979), RAY (1980), S~ANKS (1980), TALUKDAIr et al. (1981), BA~ERJI and !VIANDAL (1982), MA~N and MOF~AT (1982), S~IVASTAVA (1982, 1984), ISmKAWA and M/YASBITA (1983), ]~ISl~O~ (1983), GU~IN (1983, 1984, 1985).

For instance, S ~ (1974) calculated--see subsect. 10"2--the deflection of n neutral tachyon (coming, e.g., from infinity) in the field of a gravitating body like the Sun. He found the deflection towards the San to decrease monotoni- cally for increasing tachyon speeds, and at infinite speed to be half as much as that for photons. Later on C o ~ and LATI~OP (1978) noticed that the ordinary principle-of-equivMence calculation for the deflection of light by the Sun yields, by construction, only the deflection relative to the trajectories of infinitely fast particles (purely spatial geodesics); the total deflection will thus be the sum of the deflection given by the principle of equivalence and the deflection of the infinite-speed tachyons. This does solve and eliminate the puzzling discrepancy between the deflection of light evaluated by EINSTEIN in 1911, using the principle of equivalence only, and the one calculated four years later using the full theory of GI~.

In the first calculation EINSTEIN (1911) found a deflection of one-half the correct value, since the remaining one-half is exactly forwarded by the deflection of the transcendent tachyons.

We shall here confine ourselves only to two topics: i) tachyons and black- holes, if) the apparent Superluminal expansions in astrophysics.

92 ~. ~ECAMI

Let us recMl tha.t the space-times of SI~ and of G/r are pseudo-l-Ciemannian (subsect. 4"3.5); a priori, one may thus complete the ordinary G~ transformation group (_'~[OLLEIr 1962; SACHS anti Wu, 1980) by adding to it co-ordinate tra.nsformations which invert the geodesic type.

12"2. Black-holes and tachyons.

12"2.1. F o r e w o r d . Black-holes (see, e.g., HAWr~G a.nd ELLIS, 1973) are naturally linked to tachyons, since they are a priori allowed in classical physics to emit only t~chyons. Black-holes (BEI) offer themselves, therefore, as suitable sources and detectors (see subsect. 5"12-5"14) of tachyons; and t~chyonic matter could be either emitted and reabsorbed by a BH, or exchanged between BI~s (see PAu and t~.ECA'~[, 1977; DE SABBATA et al., 1977; KAr~LIK~ and DHUlCA:NDIIAIr ]978; CASTOtCINA and ]~ECA_MI~ 1978; I~.ECA~, 1979a; ~.ECA:~IX and StfArI, 1979; BARUT et al., 1982). This should hold also for hadrons (subsect. 6"13), if they can actually be regarded as t( strong BUs ~ (A_~mhxv et al., 1983; I~EOA~_t, 1982a; CAS!I'OYCINA and ICEOA3~, 1978 ; SALAM~ ] 978 ; SALA3~ and STRATIIDEE~ 1978; C&LDLrCOLA et al., 1978).

12"2.2. C o n n e c t i o n s b e t w e e n B~ts a n d Ts. But the connection between BHs and taehyons is deeper, since the problem of the transition outside/inside the Laplace-Schwarzschild horizon in GR in muthematicMly analogous to the problem of the transition bradyon/tachyon in S/~ (REO~t, 1978a, ]979a). Let Its start by recalling some results in the appendix B of IIawking and Ellis (1973). The vacuum metric in the spherically symmetric case reads

(130) ds -~ --- --/~-'(t, r) dt ~ + X2(t, r) dr 2 4- Y2(t, r) dO

" ~ ~ Y , " 1 ) with d Q ~ d 0 2 + S l n - 0 d ~ . When Y , ,<0 , cq. (130) becomes (G----c :

(131.) as-" ] 2 _dt + l - - r /

which is the known, unique (static) Schwarzschild metric for r > 2m. When Y;~ :Y..~ > 0, eq. (130) yields on the contrary the (spatially homogeneoits) sol-

ution

(131b) ds2 - - - ( 1 - - 2~. )dr~ (1-- 2-~)-~dt~ -'- t2dD ,

which is (part of) the SchwarzschihI solution for r < 2m, since the transform- ~tion t ~-~r carries eq. (131 b) into the form (131a) with r < 2m (see also GOLDO~I,

1975c). In other words, the, solution (131a) holds apriori for r ~ 2m; inside the horizon,

CLASSICAL TAC}IYOI~" S 93

however , it is ((reiltterl)reted ~> into the fo rm (131b), by inver t ing the roles of t and r. In such a way one obtains t h a t the metr ic does not ch~nge signature.

I n the two-dimensional case, however , we haw; seell (subsect. 5"6) t h a t the t rans-

fo rmat ion t ~ -x is just the effect of eqs. (39") when U --+ ~ , i.e. is jus t the t.ranscendent (Superluminal) Lorentz t r ans fo rmat ion (cf. also eq. (39')). And il) four

dimensions the opera t ion t ~-~ r would h~ve the same efteet exl)ected f rom a

(4-dimensional) t ranscendent <, t r ans fo rmat ion ~) (see subsect. 3"2): i t seems to

lead to ~ manifold described t)y three t imel ike co-ordinates and one spaceiike

co-ordin~te .... Such is the problem t h a t one meets to avoid t h a t change of signal:are; a p rob lem t h a t shows up more clearly when eqs. (131) %re wr i t t en down in Cartesian co-ordinates (DE SA_BBATA et at . , 1977). Thai; this is not a t r ivial p rob lem is shown also t)y the difficulties m e t as soon as one el iminates

the privileged role of the radial co-ordinate r b y des t roying the spherical sym-

met ry . Actually, when al~_alysing nonspher i ca] l y symmet r i c per turba t ions , co-ord ina te- independent (( singular surfaces ~) do arise (]V[YSAK and SZE~-EJtES,

1966; ISRA]~I~, 1967; JA~IS et at. , 1968). Clarifying such questions would mean

solving also the m~thelna t ica l problem of the (( SLTs ~> in foyer dimensions.

12"2.3. On p s e l l d o - l ~ i e m , ~ n n i a n g e o m e t r y . In the spherical ly sym-

metr ic case (when it is easy to single out the (( privi leged ~ space co-ordinate r, to be coupled with t), one cnll resort to the Szekeres-Kruska.1 co-ordinates. I f we set

~ := V~-lr/2m -- 11 exp ~ �9 cosh 4~m

(1"~2) (rX2m), v .--: ~ / ; r / 2 m -- 11 exp ~ �9 sinh - - -

4m

defined for r ~ 2m, then the Szekeres -Krusk 'd co-ordinates are chosen as follows :

(133a) ~ t > - - u~ v> ~ v for r > 2m,

outside the horizon, and

(133b) u< -=- v , v< ~ u for r < 2m,

inside the horizon. B u t again, when crossing the horizon, we avoid hav ing to

deal wi th a change of s ignature only a t the price of passing f rom co-ordinates

(133a) to (133b), t h a t is to say, of apply ing to the (everywhere defined) co- ordinates (132) a t r ans fo rma t ion of the k ind (39') with u .... O, i.e. a Super-

lumina l - type ( t ranscendent) t r ans fo rmat ion of the k ind (39") with U - > ~ .

We reached the point where i t becomes agaiu essential the fact t ha t t, he space- t ime of Gt~ is p s e u d o - t ~ i e m a n n i a n (SACIIS and WI:, 1980), and noli

l~iemamiian. ~ a m e l y , if one wishes to m a k e use of the theorems of ]~iemannian

94 ~. ~ECAMI

geometry, one has to limit the grollp of the admissible co-ordinate t ransformations: see M o d e m (1962), p. 234, CA~E~ZI~]) (1970), HALPEZ~ and MALL'~ (1969). This was overlooked, e.g., by KOWALCZY~SK~ (1984).

In a pseudo-I~iemanuian---or Lorentz ian--space- t ime we m a y have co- ordinate t ransformations even changing the ds 2 sign. Therefore, in order to be able to realize whcther we are deMing with a ~ bradyon ~> or a ~ t achyon ~ we mus t - - g iven an initial set of co-ordinates (~, fl, 7, ~) and a space point P - - confine onrselves to the general co-ordinate t ransformations which comply with the following requirement, i f co-ordinates (~, fl, 7, ~) define at P a local observer 0, then a new set of co-ordinates (s fl', ~', 5') is acceptable only if i t detines at the same lP a second local observer 0 ' which (locally) moves slower than light w . r . t . O . To usc Moller's (]962) words, any ((refl~rence frame~) in GI~ can bc regarded as a movirlg fluid; and we must l imit ourselves only

to the general co-ordinate t ransformations leading to a. (( f rame ~ (a', fl', 7', (Y) t ha t can be pictured as a (~ real ~) fluid: This means tha t the velocities of the (( points of rcference ~ - t h e fluid par t i c les - -mus t always be smaller t han e relat ive to the local inertial observer. This has to hold, of course, also for the initial (( f rame ~ (a, fl, 7, 8). For instance, once we introduce everywhere the co-ordinates (132), we cannot pass (inside the horizon) to co-ordinates (]33b).

In terms of the co-ordinates (]31a), or ra ther of the co-ordinates (]32), defined everywhere (for r <> 2m), a falling body which is a b radyon B in the ex- ternM region would seemingly be a t achyon T in the in ternal regioll (see also GOLDO.NI, 1975c). This agrees with the fact that , when adopting suitable co-ordinates bearing a part icular ly direct physical meaning, m a n y authors verified tha t any fMling body does reach the light speed c-- in those co-ordinates-- on any Schwarzsebild surfaces (see, e.g., ZELtDOVIClt and :[~OVI~OV, 1971; MAI~K- LEY, 1973; JAFFE mild SIIABIRO, 1974; CAV_A2,LEI~I -~nd SPI-NELLI, 1973, 1977,

1978; MILEWSKI, 1978). Ill part icular, the co-ordin'~tes r, t of the distant observer have no direct

significance when looking at the speed of a falling body. For instance, DE SA]~BA~rA et al. (1977), following SALTZlW~ and SAI,TZ]~A~ (1969), choose at each space point ]) (r, 0, ? constant) outside the horizon the local i rame 2:(X, T) at rest with respect to the horizon and to the Schwarzschild metr ic

(~g~/~T ---- 0). Of course, frames 27 are not inertial. Then one immediate ly gets

(see, e.g., the book by LI~HT~A~ et al., 1975) tha t the s ta t ionary observer 27 measures the velocity d R / d T = ( 1 - - 2 m / r ) - ~ d r / d t so tha t , independent ly

of the initial velocity, this locally measured speed approaches tha t of light as r approaches 2m. I t should not look strange tha t a fMling body would reach the light spced for r ~ 2m w.r.t, the local s ta t ionary frame X~, since the local inertial frame would also move with the spced of light w.r.t. Z~ . Le t its recall within SR tha t , given a f rame so, if we are in the presence of a body B with speed v = e - - e l - d c and of a second f rame s' with speed u = e - - e ~ - - > e wliere s~ = qsx (for simplicity we refer to t.he case of collinear motions), the

CLASSICAL TACYIYONS ~

speed v' of B w.r.t, s' will be

(134) v ' = c -

which can yield any real values. If @ ~ 0, then v'-+ c; but if @ = 1, then v' -+ 0. And, when v -~ c, the energy of the falling body B does not diverge in X~ ; aetually~ the total energy E of a test particle B is invariant in the local frames ~. For instance~ in the frames ~ where dT is orthogonal to the space

hyperplane, it is E = m o ~/-~oo/V/~----v ~.

12"2A. A r e f o r m u l a t i o n . Obviously, part of what precedes does not agree with the conventional formulation of G~ based on l~iemannian geometry, where space-time is supposed to be a smooth, paracompact, simply connected manifold with metric. ]~ECKMI and SI~_~H (1979) proposed a new formulation, where ((( metric-induced ~>) changes o] topology are allowed when passing from a space-time patch to another (see also Scmv~vTz]~, 1968; IVAN]~NKO, 1979; I~OSE~ 1970; WHEEL]~]r 1968; GSBEL, 1976). Whitin :such a formnlation~ they concluded that an (( external observer ~) will deem a fulling body to be a bradyon for r > 2m and ~ tachyon for r < 2m. Conversely, a body which is a tachyon for r > 2m will be deemed a bradyon for r < 2m; bat such <( bradyons }> will, lof course, be able to come out from a BIt, transforming again into tachyons (cf. also CU~INGHA~, 1975).

Notice that, a priori, the <( external }) observer should be able to get information about the BH interior by means of tachyons. I t should be repeated once more that tachyonic trajectories are perfectly <( at home ~ in GI~.

The motion of a tachyon penetrating the horizon has been studied, e.g., in FuLL]~ and WHEELE~ (1962: see appendix and fig. 6), I%AYCHAUD~raI (1974), NA~LI]O~R and DHUlCAN])~ (1976).

12"3. The apparent Super luminal expansions in astrophysics. - The theoretical possibility of Superluminal motions in astrophysics has been considered since long (G]~EGOlr165 1965, 1972; MIGNANI and I~EOAIV~, 1974e, ]; I~EC~MI, 1974, 1977b, 1978a, d, 1979a).

Experimental investigations, started long ago as well ( S ~ and HOFFEI~, 1963; KNIGHT et al., 1971), led at the beginning of the Seventies to the claim that radio-interferometric observations had revealed--at least in the two quasars 3C279, 3C273 and in the Seyfert type-I galaxy 3C120---expansion of small radio components at velocities apparently a few times greater than that of light (W~rTNEY et al., 1971; COHEN et al., 1971; StIAPFEI~ et al., 1972; SHAPIRO et al., 1973). The first claims were followed by extensive collections of data, all obtained by very-long-baseline-interferometry (VLBI) systems with many radiotelescopes; reviews of the experimental data can be found in C o m ~ et al. (1977), KELLE~HAN~ (1980) and C0n:EN and UxwI~ (1982);

9 6 ~ . RECAM~

see also SffHILLIZZI and DE B~vY~ ~ (1983). The result is, grosso mode, that the nucleus of seven strong radiosources (six quasars, 3C273, 3C279, 3C345, 3C179, NI~AO-140, BL Lac, and one galaxy, 3C120) consists of two components which appear to recede from each other with Superluminal relative speeds

ranging from a few e to a few tens e (PAuLn~-TOTH et al., 1981). A result So puzzling that the journal Nature even devoted one of its covers (April 2, 1981) to the Superluminal expansion exibited by quasar 3C273. Simplifying it, the experimental situation can be summarized as follows:

i) the Superluminal relative motion of the two components is always

a co]linear recession;

ii) such Superluminal (~expansion ~ seems endowed with a roughly constant velocity, which does not depend on the observed wave-length;

iii) the flux density ratio for the two components, E1/J~2, does depend

on the (observed) wave-length and time.

Apparently, those strong radiosources exhibit a compact inverted-spectrum core component (usually variable) and one extended component which separate from the core with Superluminal velocity. But it is not yet clear whether the compact core is indeed stationary or if it too moves. The extended component seems to become weaker with time and more rapidly at high frequencies.

The most recent result, however, seem to show t h a t - - a t least in quasar 3C345 the situation may be more complex (UNwn~ et al., ]983; I~.ADHE~AD et al., 1983; BI~E~A et al., 1983; POROAS, 1983). In the same qnasar an <( extended component )~ does even appear to accelerate away with time (Moo~E et al., 1983; see also I>EARS0N et al., 1981).

Many theoretical models were soon devised to explain the apparent Super- luminal expansions in an orthodox way (l~Fms, 1966; WHrrNEu et al. ]971; CAVALIERE et al., 1971; DENT, 1972; SANDE]~S, 197~; EPSTFA::N and GELLEI~, 1977, and so on). l~eviews of the orthodox models can be found in B~WDFO~D et al. (1977), SCR:EUE~ and I~EADm~,AD (1979), M~SCEE~ and Sco~T (1980),

Ool~ and BR0WEE (1982), PORCAS (1983). The most successful and, therefore, most popular models resulted to be:

a) The relativistic jet model: A relativistically moving stream of plasma is supposed to emanate from the core. The compact core of the (( Superluminal )>

sources is identified with the base of the jet and the (( moving ~) component is a shock or plasmon moving down the jet. If the jet points at a small angle towards the observer, the apparent separation speed becomes Superluminal since the radiation coming from the knot has to travel a shorter distance. Iqamely, if v is the knot speed w.r.t, the core, the apparent recession speed will be (e ---- 1) w ---- v sin a/(1 -- v cos a), with v>w/ (1 ~- w2) ~. The maximal probability for a relativistic jet to have the orientation required for producing the apparent Superluminal speed ~-- independent ly of the jet speed v--is

CLASSICAL TA CII 'gONS 9 7

/)(w) -- (1-~-~2)-1 < 1/~.~ (I~LANDFORD et al., ]977; ~II~,'KELSTELN et al., 1983; CAS~,:LLI~O, 1984). The relativistic jet models, therefore, for the observed <( SuperluminM ~) speeds suffer from statist ical objec, tious, even if selection effects can play in favour of t hem (see, e.g., PO~CAS, 1981; Science ~Vews, 1981; POOLEY, 1981; PI~:ARS0N et al., 1981).

b) The <~screen, models: The ((Superluminal~> emissions are tr iggered by a relativistic signal coming from a. central source and (( i l luminating )) a pre-

existing screen. For instance, for a sphcrica.1 screen of radius R i l luminated by a concentric spherical relativistic signal, the dis tant observer would see a circle expanding with speed w ~ 2 c ( R - - c t ) / ( 2 R c t - - c ~ t 2 ) i ; such a speed will t)e Superlmni~al in the t ime interval 0 < t < � 8 9 V/2)R/c only. When the screen is a ring, the observer would see an expanding double solrrce. The defect of such models is t ha t the apt)arent expansion speed will be w>@ (with @ >> 2c) only for a fraction c2/~ ~- of the t ime during which the radiosource exibits its variations. Of course, one can introduc, e (~ oriented )~ screens--or ad hoe screens-- , bu t t hey are statist ically unfavoured (BLANDFOI~D et al., 1977; CASTELLINO, 1984).

c) Other models: many previous (unsuccessYafl) models have beell aban- d(lncd. The gravi tat ionaMens models did never find any observat ional support , even if a new tyl)e of model (where the magnifying lens is just surrounding tile source) has been recent ly suggested by LIAOFU and CI{O~NG3IING (1984).

In conclusion, the or thodox models are not too much successful, especially if the more complicated Superluminal expansions (e.g. with acceleration) recenl ly observed will be confirmed.

I t ma y be of some interest , thereJore, to explore the possible al ternat ive models in which actual Superluminal motions take place (of., e.g., MmNAsI and RE(JASrI, 1974e).

12"4. The model with a single (Superluminal) source. - The simplest Super- luminal model is t:.he one of a single Superluminal source. In fact, we have seen in subscct. 6" 15 (see fig. ] 5) tha t a single Superluminal source C will appear as the creation of a pair of sources collinearly receding from each other with relative speed W > 2c. This ruodel immedial,ely explains some gross features of the ~( Super]uminM e~pansions ~>; e.g., why converging Superluminal motions are never seen, 'md the high luminosi ty of the <( Super lumina l , component (possibly due to tile optic-boom effect ment ioned in subsect. 6"15 ; see also I~Ec~_~-x, 1977b, 1979a), as well as the oscillations in the received overall intensi ty

(perhaps <~beats ,; el. t~EC.~_~I, 1977b). Since, moreover, the Doppler effec~ will be different for the two images C,, C,2 oE the same source C (subsect. 10"4), a priori the model may even explain why 2'~/2~ does depend on the observed wave-length and on t ime (see subsect. 12"3, point iii)).

98 E. EECA~

Such a model for the (( Superluminal expansions )) was, therefore, proposed long ago (1-r 1974, 1977b, 1978a, d, 1979a; .M_tGNAm and R E c ~ , 19741; t~EC~'~x et al., 1976; G]~o.w, 1978; B ~ u T et al., 1982). W h a t follows is mainly due to l~Ec~viI, MAcc~amo~v., CAS~EL~0 and ~ . Ro~o.w6. Many details

can be found in the MS thesis work by CAS~EmNO (1984), where, e.g., the case of an extended source C is thoroughly exploited, and in R E C ^ ~ et al. (1986).

12"4.1. T h e m o d e l . With reference to fig. 15a) and subsect. 6"]5, let us first consider the case of an expanding universe (homogeneous isotropic cosmol-

ogy). I f we call CoO ---- s = db, with b ~ V [ v / V - ~ - 1, the observed angular ra te of recession of the two images Cx and C~ as a funct ion of t ime will be

2be 1 ~- _4 s (135) O(t) ~ co - - A ~ b2ct , s [1 ~- 2A]�89

provided t ha t s is the (( proper distance ~ between Co and O at the epoch of the radiat ion recept ion by O, and t is the t ime at which O receives those images (t ~-- 0 being the ins tant at which O star ts seeing C, or ra ther receives radiat ion f rom Co). Le t us repeat t ha t eo is the separat ion angular speed of C~ and C,,

observed by O, in the case of a space-time metr ic

ds~ = c2dt ~ - - Its(t) [dr ~ + r ~ d ~ ] ,

where R--~ R(t ) is the (dimensionless) scale factor. Notice tha t (}(t)-~c(,

for t --> 0. I f we call t* and t the emission t ime and the reception t ime, respectively,

then the observed f requency v (see subsect. 10"4 and eq. (l13b)) and the receive4 radiat ion in tens i ty I will be given, of course, by ( R E C k , 1974; I~ECA)n et al.,

1976; CASTELI~O, 1984)

V v , - 1) o (136) ~ = ~ 0 i1 - Veos~l R( t ) ' I = - 4 ~ r 2 ( t - - Vcos ~)2 L ~ J '

where Vo is the intrinsic f requency of emission and ~ o is the emission power of the source in its rest frame. Quant i ty r is again the source-observer (( proper

distance ~) (W~I~B~,RG~ 1972, p. 415) at the reception epoch.

Le t us pass to the case of a nonpo in t l i ke source C. Le t for simplicity C

be one-dimensional with size l w.r.t, the observer O (fig. 15a), and move with speed V in the direction of its own length. Let us call x the co-ordinate of a generic point of the mot ion line, the value x ---- 0 belonging to H. As in subseet. 6"15, be t -~ 0 the ins tant when the observer O enters in radio contact

with C. Once the two (extended) images C1 and C~ get fully separated (i.e. for


t > 1IV), if the intrinsic spectral distribution X(vo) of the source C is known, one can evaluate the differential intensities dI~/dv and dI2/dv observed for the two images (I~EoAm et al., 1976; CASTELL~O, 1984). For the moment let us report only that, due to the extension of the moving images, for each emitted frequency v0 the average observed frequencies (after lengthty calculations) results to be

2Co R(t*) (137) <~"> : : : , ) ' ) ; - -~f (1 - ~,/a,) ) ( t ) ' ~

quantities ~ , ~ being the observed angular sizes of the two images, with ~ > ~ . Moreover, l/d : �89 0~2).

12"4.2. C o r r e c t i o n s due to t h e c u r v a t u r e . Let us consider the corrections due to the curvature of the universe, which can be importaut if the observed expansions are located very far. Let us consider, therefore, a ellrved expanding cosmos (closed Friedmann model), where the length element d2 is given by d2~-- - dr~(1--r~/a2)-x+ # d Q , quantity a = a(t) being the curvature radilts of the cosmos. Again, some details can be found in I~ECAm et al. (1976) and CASTELLINO (1984). For instance, the apparent angular speed of separation 0(t) between the two observed images Ca and Ca (el. eq. (135)) becomes (h = r/a)

1 + 1,h (138) O(t) ---- co ~_ b ~ - [ rt J h2]~,

quantities r and a being the <( radial co-ordinate ~ of Co and the universe radius, respectively, at the present epoch (r ~ - a s i n ( s / a ) , where s is the ((proper distance ~ of Co; moreover, a : e /H ~/2q-- 1 ; H ~_ Hubble constant; q := =_= deceleration parameter; and et<< r). Further evaluations in the above- quoted literature.

12"4.3. C o m m e n t s . Equation(135) yields apparent angular velocities of separation two or three orders of magnitude larger than the experimental ones. I t is then necessary to make recourse to eq. (138), which includes the corrections due to the universe curvature; actually, eq. (138)can yield arbitrari- ly small values of 0(t) provided that h -> 1, i.e. r ~ a. To fit the observation data, however, one has to attribute to the (~ Superluminal expansions ~ values of the radial co-ordinate r very close to a. Such huge distances would explain why the possible blue-shifts--often expected from the local motion of the Super- luminal source C (cf. end of subsect. 10"4)--appear masked by the cosmological red-shift. (Notice, incidentally, that a phenomenon as the one here depicted can catch the observer's attention olfly when the angtflar separation 0 between C1 and Ca is small, i.e. when C1 and C~ are still close to Co .) But those same

100 ~. ~ c ~ i

l~rge distances make also this model <~ improbable }) as a~l explanation of the obscrved (( Superlumi]aal }> expansions, at least in the closed models. One could well resort, then, to open Fr iedmann models. In fac~, the present model with a single (Supcrluminal) source is appealing since i t easily explains a) the appearance of two images with Superluminal relative speed (W > 2c), b) the fact tha t only Superluminal expansions (and not approaches) are observed, c) the fact tha t W is always Superluminal and practically does not depend on ~, d) the relative-motion colline~rity, e) the fact tha t the flux density ratio does depend on v and t, since the observed flux differential intensities for the two images as a function of t ime are given by the formulae (CAsTELLX~'0, 1984)

~]mdt)

dI,dv -- 4~d: V2 --]VL f X(Vo)~0 F,dv~ (i = 1, 2),

(139) ~/~t,(t)

[(+ )]' 2 f , . - 1 R(r,:) T 1 i ,

the integration extrema being

(140a) m~(t) ~ K {VG [VTG']-t :J= 1}, 2

(1.40b) M~(t)--K,~ VT(G' - -2L i -i:~L~-/~-- 2%/V2_ :~)] �89 • 1 ,

where d is the (~ proper distance }> OH at the reception epocb (fig. ]5a)) ; L ~ l/d; T ~--ct/d; K ~V/~-- lR( t*) /R( t ) ; G --%/~----1 ~- VT and G' - -2G-- VT. All equations (139)-(140) become dimensionally correct provided tha t V/e is subst i tu ted for V.

But the present model remains disfavoured since i) the Superluminal expansion seems to regard not the whole quasar or galaxy, but only a <~ nucleus >> of i t ; it) at least in one case (3C273) a.l~ object was visib]e there, even before the expansion s tar ted; iii) i t is incompatible with the acceleration seemingly

observed at least in another case (3C345). ~evertheless we exploited somewhat tha t case, since, A) in general, the

above discussion tells us how it wolfld appear a single Superluminal cosmic source; B) i t might still regard part of the present-type phenomenology(*); C) and, chiefly, i t must be taken into account for each one of the Superluminal objects considered in the following models.

(*) In 1986, e.g., two quasars very similar to each other but with a large angular separation have been observed. It would be interesting to measure their relative recession speed, and so on.

CL&S SICAL TACH~YONS 101

12"5. The models with more than one radio source. - We recalled in subsect. 12"2 that black-holes can classically emit (only) tachyonic matter, so that they are expected to be suitable classical sources--and detectors--of tachyons (PAv~I6 and I~ECAMI, 1977; DE SABBATA et al., 1977; N ~ L ~ and D n ~ A ~ - D~A~, 1978; I~ECA~, 1979a; I~EtA~ and S~_&H, 1979; BA~UT et al., 1982). Notice that, vice versa, tachyons can not only enter the horizon of a black- hole, but also come out from a horizon. As we already said, the motion of a spacelike object penetrating the horizon has been already investigated, within Gl%, in the existing literature (see the end of subsect. 12"2.4).

We Mso saw in subsoet. 5"18 (fig. 14) and in subscct. 10"2 that, in a (( subluminal }> frame, two tachyons may seem--as all the precedent authors claimed-- to repel each other from the kinematical point of view, due to the novel features of tachyon mechanics (subsoet. 10"1, cqs. (109b), (109c)). In reality, they will gravitationally attract ouch other, from the energeticM and dynamicM points of view (subseet. 10"2; see also fig. 4a)).

From subsect. 10"2 a tachyon is expected to behave the same way also in the gravitational field of a bradyonic source. If a central source B (e.g., a

black-hole) emits, e.g., a SuperluminM body T, the object T under the effect of gravity will lose energy and, therefore, accelerate away (subsect. 5"3). If the total energy E = moc~/V/V 2 - 1 of T is larger than the gravitational binding energy E, it will escape to infinity with finite (asymptotically constant) speed. (Since at infinite speed a taehyon possesses zero total energy--see fig. 5e) and subscct. 6"14--, we may regard its total energy as all kinetic).

If on the contrary E < E, then T will reach infinite speed (i.e. the zero- total-energy state) at a finite distance; afterwards the gravitational field will not be able to subtract any more energy to T~ and T will start going back towards the source B, appearing now--possibly--as an antitachyon T (subsect. 5"12 and 11"2). I t should be remembered (subsect. 11"5 and eq. (128)) that at infinite speed the motion direction is undefined, in the sense that the transcendent tachyon can be described either as a tachyon T going back or as an untitachyon going forth, or vice versa.

We shall see, on another occasion (subsect. 13"2), that a tachyon subjected, e.g., to a central attractive elastic force F = -- kx can move periodically back and forth with a motion analogous to the harmonic one, reversing its direction at the points where it has transcendent speed (and maybe alternately appearing--every half an oscillation--now as a tachyon and now as ~n antitachyon). Let us consider, in general, a tachyon T moving in space-time (fig. 32) along the spacelike curved path AP~ so to reach at P the zero-energy state. According to the nature of the force fields acting on T, after P it can proceed along PB (just as expected in the above two cases, with attractive central forces), or along PC, or along PD. In the last case, T wolfld appear to annihilate at t ) with an anti taehyon emitted by D and travelling along the curved world- line DP (subsect. 5"12 and 11"2; see also DAWES, 1975, p. 577).

102 ~. RECAMI

Fig. 32.

-A o

I t is clear t h a t the observed (( Super luminal ~> expansions can be explained

i) e i ther by the spl i t t ing of a centra l body into two (oppositely moving) collinear

t achyons T1 and T2, or b y the emission f rom a centra l source B of ii) a t achyon T,

or iii) of a couple of t achyons T1 and T, (ill the la t te r case, T~ and T~ can for s impl ic i ty ' s sake be considered as emi t t ed in opposite directions wi th the same

speed). I n thi~ respect , i t is interest ing t h a t NE'E.~AN (1974) regarded quasars ---or a t least thei r dense co res - -as possible white holes, i.e. as possible (( lagging

cores ~ of the original expansion. For simplicity, let us confine ourselves to a flat s t a t ionary universe,

12"5.1. T h e c a s e i i) . I n the case ii), be O the observer and a the angle

be tween BO and the mot ion direction of T. :Neglecting for the m o m e n t the

Fig. 33.

g rav i ta t iona l interact ions, the observed apparent relat ive speed be tween T and B

will, of course, be (see fig. 33)

V s in a �9 ( V > 1) . (141) W = ] _ V c o s a

L e t us assume V > 0; then, W > 0 will mean recession of T f rom B, bu t W < 0 will mean approach. Owing to the cylindrical s y m m e t r y of our p rob lem w.r.t.

BO, let us confine ourselves to values 0 < ~ < 380 ~ Le t us ment ion once

more t h a t W -+ oo when cos a -> 1/V ((( opt ic-boom )> situation). I f the emission

angle a of T f rom B w.r.t . BO has the value ~ = ~ , with cos ab ~ 1IV (0 <

CLASSICAL TACI-IYONS 103

< a~ < 90 ~ b ~ (~boom))), t a chyon T ~ppears in the opt ic-boom phase; bu t the recession speed of T f rom B would be too ldgh i~ this ease, as we saw in the, previous subsection.

IncidenLuny, to ~pply the results got iu subsect. 12"4 to the Super luminal object T (or T~ and T~ in the other cases i), iii)), one h~s to t ake account of the

fact t ha t the present t aehyons are born a t a finite t ime, i.e. do not exist before

thei r emission f rom B. I t is theu immedia te to deduce t h a t we shall observe

a) for a > a~, i.e. for a~ < a < 180 ~ , the object T recede f rom B; bu t b) for

0 < ~ < ~ , the object T approach B. More precisely, we shall see T receding

from B with speed W > 2 when

(142) 2(V I 1)--- < t g , ~ <

1 arccos ~ < a < 180 ~ .

V t- VS-V' - - 4 2(V ,~- 1) '

I t should be not iced t h a t eq. (141) can yield values W > 2 whenever V > 2/v/5:

in par t icular , therefore; ]or all possible values V > 1 of V. Due to eqs. (142),

the ~( emission direction ~) c~ of T mus t be, however , contained inside a cer tain

sui table solid angle: a~ < a < a~; such a solid angle always i~cluding, of course,

the opt ic-boom dh'ection ~ . ) 'o r instance, for V - > I we get 0 < t g (o~/2)< ~-, a>c% = 0, wheref rom

([43) 0 < ~ < 53.13 ~ (V --~ 1);

in such ,~ case, we shall never observe T approaching B. On the cont rary , for V-~oo we get �89 - - %/5) < ~g (a/2) < �89 -]- ~/5), ~ = 9 0 ~ < 180 ~ where- f rom -- 63.44 ~ < a < 116.57 ~ a > 9 0 ~ t h a t is to say 9 0 ~ 116.57 ~ . I f we add the requi rement , e.g., W < 50, in order t h a t 2 < W < 50, we have to exclude in eq. (143)--for V - ~ l - - o n l y the t iny angle 0 < ~ < 2.29 ~ so Lh~t in conclusion

2 . 2 9 ~ ~ < 53.13 ~ (V- ~ ~).

Tim same requ i rement 2 < W < 50 will not a f fec t - -on the c o n t r a r y - - t h e above result 90~ < 116.57 ~ for the case V - 7 c~.

Similar calcldations were per formed also b y F ~ . ~ m L S ~ , ~ et al. (1983). The present case ii) suffers f rom some difficulties. First , for ~ > ~2 (for

instance, for 53~162 < 180 ~ in the, case V-> 3) we should observe recession

speeds wi th 1 < W < 2, which is not suppor ted by the da ta ; bu t this can be

unders tood in terms of the ])oppler-shif t selection effects (see subsect. 10"4

and BLAR'DlrOaD et al., 1977). Second, for a < ~ one should observe also

Super luminal approaches ; only for V ~ 1 (V ~ 3) it is at. _~ 0 a.nd, therefore, such Superluminal approaches are not predicted.

104 ~. 2ECAMI

]n cotmhlsiol b this model ii) :~,ppears acceptable only if tile emission mechan- ism of T from 13 is such tha t T has very large kinetic energ~ ~, i.e. speed V ~ 1.

12"5.2. T h e c a s e s i) a n d i i i ) . Let us pass now to analyse the cases i) a:nd iii), still assuming for s imphci ty T 1 and T., to be emit ted with the same speed V in opposite directions. Be a again in the range [0, 180~ In these cases, one would observe faster- than-l ight recessions for c~ > %. When c~ < %, on the contrary, we would observe a single t achyon T ~- T~ reaching the posi-

t ion B, passing it, and continuing its motion (as T~= T~) beyond B with the same velocity bu t with a new, different Doppler shift.

One can perform calculations analogous to the ones in subsect. 12"5.1; see also I~.~KELSTEIN et al. (1983).

In case i), in conclusion, we would never observe Superluminal approaches. For a < ~b we would always see only one body at a t ime (even if T --:- T~ might result as a feeble radiosource, owing to the red-shift effect): the mot ion of T wouhl produce a variability in the quasar. For ~ > ab, as already mentioned, wc would see a Superluminal expansion; again, let us recall t ha t the cases with 1 < W < 2 (expected for large angles e only) could be hidden by the Dop-

pler effect. Case iii) is not very different from case it). I t becomes ((statistically ~)

acceptable only if, for some astrophysical reasons~ the emi t ted tachyonic bodies T~ and T~ carry very high kinetic energy ( V ) 1 ) .

12"6. Are (~superluminal~) expansions Super luminal? - I f the e m i t ~ d tachyonic bodies T (or T1 and T~) carry away a lot of kinetic energy (V~ 1), all the models i), it), iii) m a y be acceptable f rom the probabilistic point of

view. Contrariwise, only the model i ) - - and the model iii), if ]3 becomes a weak

radiosource aft(;r the emission of T1, T~--remains statistically viable, provided tha t one considers t ha t the Doppler (;fleet can hide the objects emi t ted at large angles (say, e.g., between 60 ~ and 180~ On this point, therefore, we do not

agree with the conclusions in FIS-KEI, STEIN et al. (1983). I n conclusion, the models implying real Superluminal motions invest igated

in subsect. 12"5 seem to be the most viable for explaining the apparent (( S upe r hmina l e• ~); (;specially when taking account of the gravi-

ta t ional interact ions between B and T, or T1 aud T~ (or among T~, T2, B). Actually, if we take the gravi ta t ional a t t rac t ion between B and T (sub-

sect. 10"2) into acc(umV--for simplicity, let us confine ourselves to the case i t )-- , we can easily explain the accelerations, probably observed at least for 3C3~5 and maybe for 3C273 (Sm~NaLIN and Yo~-aZlI~N, 1983)(*).

(*) Notice that, i] very high gravitational attractions are opera~ing, then our models i)-iii) can even prcdic~ a Superluminal approach (following the Supcrlulninal expansion).

C L A S S I C A L TACI tYONS 105

Some calculations in this direction have been recently performed also by SHE~'GHN et al. (1984) and (L~o (1984). But those authors (lid not compare corrcctly their evaluations with the, data, since they ovcrlooked that--because of the finite value of the light speed--the images' apparent velocities do not coincide with the sources' actual velocities. The values W0 calculated by those attthors, therefore, have to be corrected by passing to the values W : : ~-Wo since/(1--cosec); only the values of W arc to be compared with the observation d~ta..

Al l t.he calcul'~tions, moreover, ought to be corrected for the universe cxp~msion, ttowcver, let us re.call (subsect. 12"4) that in the homogeneous isotropic cosmologies---<( conformal ~> expansions--the angular expansion rates are not expected to be modified by the expansion, at least in t.hc ordinary observational conditions. While the corrections due to the universe, curvature would be appreciable only for very distant objects.

Tachyons in Quantum Mechanics and Elementary-Particle Physics.

13. - The possible role of tachyons in elementary-particle physics and quantum mechanics.

In this review we purported (subsect. 1"1) to confine ourselves to the classical theory of tachyons, leaving aside their possible quantnm field theories (el., e.g., BROID0 and TAYLOI~, 1968). We have already met, however, many instances of the possible rolc of tachyons in elementary-particle physics. And we want to develop some more such a.n aspect of tachy()ns in the present sectiom

In subseet. 1"1 we mentioned, moreover, the dream of reproducing the quantum behaviour at a classical level, i.e. wit.hin a classical physics including tachyons (and suitable extended-type models of elementary particles). In the present section we shaft put forth also some hints pointing in such a direction.

l~ight now, let us only recall from sect. 9 and 6 that tachyons are the suitable carriers of mutual, symmetrical interactions between A and ]3, rather than of

~ signals ~>, i.e. of controllable, interpretable messages sent either by A to ]3, or by ]3 to A. If one accepts that fa.ct, then--~s we have seen--one is left within SI~ with no problem at all. But such tachyonic influel:ces, which already

allow ~s to abandon the (~ strong locMity ~> prejudice, would be enouglJ to interpret quantum mechanics in agre<'me, nt with the Bell theorems (of., for instance, H3,]G],~]~]~],H~D, 1985, and STAPP, 1985, p. 3]5, 317).

t e l us tinally mention that we noticed (in subsect. 8"2) tachyons themselves to be more similar to fields than to particles.

106 ~. RECAMI

13"1. Recalls. - We have already seen that EI~ allows a clearer understanding of high-energy physics: in subsect. 11'4 we derived from it, e.g., the so-cMled crossing relations.

Actually, the predicting power of the pure SI~ (even without tachyons) with regard to elementary-particle physics is larger than usually recognized. Once one develops SIC as we did in part I, one succeeds in exphdning--withiu SIC alone--not only the existence of antiparticles (sect. 2 and subsect. 5"14), but aho of the CPT symmetry (subsect. 5"16 and 11"1), as well as of a relation between charge conjugation and internal space-time reflection (subsect. 11"3). For the interpretation of advanced solutions, see subsect. 9"7.

As to tachyons and elementary-particle physics, we recall the results in subsect. 6"2, 6"3 and particularly 6"13, where we mentioned the possible role of tachyons as (~ internal lines ~) in subnuclear interactions, l~or the connections between tachyons and Wheeler-Feynman type theories, see subsect. 9"6.2. In subset . 11"5, at last, we discussed the relevance of tachyons for a classical description of the vacuum decay and fluctuation properties.

13"2. (~ Virtual particles)> and tachyons. The ~ u k a w a p o t e n t i a l . - We already saw in subsect. 6"13 that, tachyons can be substituted for the so-called (( virtual particles )> in subnuch~ar interactions, i.e. that tachyollS can be the ~ realistic )~ classical carriers of elastic and inelastic interactions between elementary particles (SUDA_~SIIAN, 1968; ICECAMI, 1968; CI~AVELI~ et al., 1973; see also all the ref. (s) and (9) in MACeAI~O~E and I~ECAm, 1980b).

Actually, it is known that the ~( virtual particles ~} exchanged between two elementary particles (and, therefore, realizing the interaction)must carry a negative four-momentum square, for simple kinematical reasons (review I):

(144) t :---: p . p " - - B ~ - - p ~ < 0 ,

just as happens for taehyons (cf., e.g., subsect. 6"1, eq. (29e)). Long ago it was checked (ICECAMI, 1969; OLX~OVSI~u and Ir 1969)

whether virtual objects could really be regarded as faster than light, at least within the so-called peripheral models (( with absorption ~) (see, e.g., DAa~, 1964). To evaluate the effect of the absorptive channels in the one-particle exchange models, one has to cut out the low partial waves from the Born amplitude. Namely, an impact parameter (Fol~rier-Bessel) expansion of the Born amplitudes is used, and a cut-off is imposed at a min imal radius R which is varied to fit the experimental data. While considering--for example---different cases of pp interactions via K-meson exchange, values of R were found ranging from 0.9 to 1.1 fm, i.e. much larger than the K-meson Compton wave-length. The same kind of model (at a few GeV/c, with form factors) was also applied to pion-nucleon reactions via o-meson exchange; and also for the p a value (R ~ 0.8 fro) much grea~er tha.n the p-meson Compton wave-length was found.

C~ASSICA~ ~eH~oxs 107

Even if such rough tests are meaningful only within those models, one deduced the v i r tua l K and p mesons of the mtcleon cloud to t ravel fas ter t h an light:

2 for instance, in the first case, for t ~ -- rex, one finds <v> > 1.75 c. According to WIG~ER (1976), (( there is no reason to believe that interaction

cannot be transmitted ]aster than light travels >>. This possibility was considered in detai l by VA~ DA~ and W I ~ E ~ (1965, 1966) already in the Sixties. See also A~uDx~ (1971), CosTA DE BEA~-~E~A~D (1972), MATEEWS and SEETHAI~A- ~A~ (1973), FLATO and GuE~I~ (1977) and Sm:RoKov (1981).

And any action-at-a-distance theory (see, e.g., SUD~SEX~, 1970d; VoLKov, 1971; LEITER, 1971b; ]=[0YLE and NAI~LXKA_R, 1974) implies the existence of spacelike objects, since the infinite speed is not invar ian t (subscct. 4"1).

Moreover, i] hadrons can really be considered as ((strong black-holes >) (subsect. 12"2.1), t hen :strong interact ions cau classically be mediated only by a t achyon exchange, i.e. the strong field (( quanta >> should be Super- luminal (*).

In any case, we van describe at a classical level the v i r tua l cloud of the

hadrons as made of tachyons (see also S U D ~ S t L ~ , 1970b), provided tha t such tachyous, once emit ted, are--(( strongly >>--attached by the source hadron, in analogy with what we discussed for the ordinary gravi ta t ional case (subsect. 12"5). For the description in terms of a ((strong gravi ty >) field, see, e.g., SALA~

(1978), SIVA~A~ and Sz~]tA (1979), I~EOA_M~ (1982a, b) and references therein, and AZvrM~ZAZU et al. (1983). In fact , if the a t t rac t ion is strong enough, the emitted tachyous will soon reach the zero-energy (infinite-speed) s ta te ; and afterwards (cf. fig. 32) they will go back, till reabsorbed by the source hadron. Notice tha t t ranscendent tachyons can only take energy f rom the field. Notice, moreover, tha t classical tachyons subjected to an a t t rac t ive central field can move back and fo r th in a kind of tachyonic harmonic motion (see fig. 34), where the inversion points just correspond to the infinite speed (ef. subsect. 12"5; see also A~rt~o~ov et al., 1969).

Final ly , let us consider a hadron emit t ing and reabsorbing (classical) tachyons. I t will be surrounded by a cloud of outgoing and incoming

T T T

V = o ~ aC6"6"6"r T : V=oo [

Y 7 r

Fig. 34.

(*) Incidentally, in such an approach hadrons should be regarded as pulsating objects (cf. R~cA~, 1982b), with a period of about 10 -~3 s, that pulsation being possibly related with the zitterbewegung (see, e.g., H~sT~]~s, 1985) or, more generally, with the existence of a <~ hidden variable ~>.

108 ~.:. m,:CAm

taci~yons (*). In the (, contimlous ~ approximat ion (and spherically symmetr ic case), tha t clond can be described by the spherical waves:

(1,15) I o:: cxp [ + imo r]p r

We are, of course, confi~fing ourselves to subluminal frames o~fly. We can find out, however, the results forwarded by EI4 formally by put t ing for tachyons mo ~ :~ i# (# reM). I t is noticeable tha t f rom eq. (145) we get, then, the Yuk,~wa potent ia l by set t ing m := ~= i# fl)r tile outgoing and m --= -- i# for the incoming

w a v e s :

(145') Ioc exp [--/~r] I~ ; r I

in other words, at the static limit, the Yukawa. potent ia l can be regarded am the (~ contirmous ~) (classicM) description of a flux of outgoing tachyons and incoming ant i taehyons: see ()ASTOI~INA and I{ECA~ (:1978). Se also I=[ADJIOA:N- _~OU (1966), ~ElC~E~TI and VEI~DE (1966), YAM_AMOTO (1976), EI~ICSE~" 'rod VOYE~LI (1976), FLA~:0 and GLrE]~ (1977) and :FEDERIGHI (1983).

When two h~drons come close to ea.ch other , one of the cloud tachyons - - i n s t e a d of being reabsorbed by the mother h a d r o n - - c a n be absorbed b y the second hadron , or vice versa (thin s ta tement in f rame dependent) . Tha t would be the simplest hadron-hadron interaction. The actual presence of a t achyon exchange wouhl produce a resonance peak in the scattering ampl i tude as a

fm~ction of the momentl~m transfcr t ~ (p~ -- p2) ~ ( S U D ~ S m ~ , 1969a, b, 1970c). Precisely, it would produce a (( negative t enhancement )), fixed when s -~ (p~ + p2) 2 varies, and possibly to be found also in o ther similar processes (D~A~ and SUDARSIIA~, 3968; GLOCK, 1969; BALDO et al., 1970), mlless the tachyons appear to possess a very large width (BuGI~ff et al., 1972; nee also KI~0LIKOWSKI, 1969). A positive theoret ical evide~lce wan pu t for th by GLEESON et al. (1972a). See also VA~ DE~ S P ~ (t973), gun (1973), AK~UA (1976), E~ATSU et al. (1978), review I, p. 266, and ]~ALDO et aI. (1970).

Before closing this section, let us reca.ll tha.t long ago (l~nch:v~, 1968, 1969)

it was suggested tha t the unst'~blc particles ((( re. onanccs )~), bearing masses M * : M + i F formally complex, might be compounds of bradyons and

tachyons. We shall come back to this point in subsect. 13"5 (see also, e.g ,

SU])A~StIAN, 1970d; :EDMONDS, 1974; KESZTH]~]LHYI and ~A6Y, 1974).

(*) Let us remark, by tlle way, that such continuous processes of tachyon cnfission and absorption could be one cause for l,he stochastic motion that in classical physics is to be added to the detcrministic global motion in order to get thc Schr6dinger equation (cf., e.g., CAv~LxRI, 1985).

CLASSICAL TACIIYO_N'S 109

The considerations above can be interest ing also with regard to t teger- feld's (t985) result tha t , in relat ivist ic q u a n t u m theory, a relat ivist ic part icle

or sys tem t h a t at t - - 0 is locMizcd with exponent iMly bounded ta, il at la ter

t imes (( violates causal i ty ~), in the sense t ha t the assumpt ion ol finite propaga- t ion speeds leads to a contradict ion.

More in general, for the possible connections be tween Superhnninal mot ions

and the (( q u a n t u m potent ia l )> (Bmr~ and VIGIE~, 1954, 1958), see, for instance, Vmi~:Ir (1979, 1980). See a.lso STAPP (1977, 1985) and D'EsPA(~.N'AT (1981).

13"3. Pre l iminary applications. Neutr inos and tachyons. - I f subnuclear interact ions are considered as media ted by quanta , no ordinary (bradyonic)

particles can be the carriers of the t ransferred energy-momentum. We have

seen, on the contrary , tha t classical taehyons can a priori act as the carriers

of those interactions.

As pre l iminary examples or applications, ]el us consider the ve r t ex An3-+ - + p + % of a sui table one-part icle exchange diagram, and suppose the

exchanged part icle ( internal line) % to be a tachyonic pion, ins tead of a (( virt l]al ~ pion. Then, f rom subsect. 6"3 and 6"8 we should get 1232 ~ - 938 ~ = 1402 -I-

L 2-1232.~/v~,p[ 2 - 1 4 0 ~, and, therefore (MAccA]C~0~E and ICEc)=~g, 1980b),

(146) [p[~ - 287 MeV/e , E ~ - - 251 MeV,

so tha t , in the c.m. of the As3(1232), the to ta l energy of the t aehyon pion is predic ted to be centred a ronnd 251 MeV.

Again, let us consider the decay ~ -+ ~ -~- ~T raider the hypothesis , now,

t ha t ~T be a tachyon-neut r ino , with m , ~ 0, v ~ c. I t has been shown b y CAWr~EY (1972) t h a t such a hypothes is is not inconsistent with the exper imenta l data.,

and implied for the muon-neut r ino a mass m~ < 1.7 MeV. I n the two limi]~ing cases, from subsecl.. 6"3 and 6"8 in the c.m. of the pion we shmfld get (MACCAR- R o ~ and t%ECA~-UI, 1980b)

(147a) m, =- 0 ~ ]p], ~_ 29.79 MeV/c , vv ~ c,

(147b) m y = ] . 7 ~ IPlv ~ - 2 9 . 8 3 N e V / c , v , _ ~ l . 0 0 1 6 c .

I t is in teres t ing to notice tha t , in the case of tachyon-neut r inos , the magn i tude

of the neutr ino impulse increases w.r.t, the light speed case.

Le t us recall once more f rom subseet. 6"13 t h a t / o r instance any elastic scat-

ter ing can be (, realistically ~> media ted b y a suitable t achyon exchange during

the approaching phase of l~he two bodies. ]in tlle c.m.f. (]Pa[ = [P~[ ---- IPI) we would obta in eq. (82):

m o o ( 8 2 ) c o s 0 o . m . - 1 - - . i p - ~ - ; , , ~ 2

110 ~. ~ c A ~

so that, for each discrete value of the tachyon rest mass ma (subsect. 5'1), the quantity 0 too assumes a discrete value, which is merely a function of ]P[. (Let us recall, with regard to this result, also Landd's (1965, 1975) ideas.) We have always neglected, however, the mass width of the tachyons.

For further considerations about tachyons and virtual fields see, e.g., YA~ DES SPUY (1978) and SOUSEK et al. (1981).

Tachyons can also be the exchanged particles capable of solving the classical-physics paradoxes connected with pair creation in a constant electric field (ZEL'DOVIC~, 1974a, p. 3r and 1972).

For joint probability distributions in phase-space and tachyons see, e.g.,

K~tiGE~ (1978, and references therein), where the ordinary formalism w~s generalized to the relativistic case and shown to yield a unified description of bradyons and tachyons.

13"4. Classical vacuum instabilities. - We saw in subsect. 11"5 that the vacuum can become unstable, at the classical level, by emitting couples of zero- energy (infinite-speed) tachyons T and T. For a discussion of this point (and of the possible connection between the cosmic tachyoa flux and the tachyon emittance of ordinary matter) we refer the reader to subsect. 11"5 (and fig. 31). See also M~G~A~I and I~ECA~ (1976a), as well as fig. 32 in our subsect. 12"5.

Here let us observe that the probability of such a decay of a vacuum ~ bubble ~ into two collinear transcendent tachyons (T and T) is expressible, according to Fermi's golden rule (FE~lv~, 1951), as ~-1oc moc/87ch, where mo is the tachyon rest mass (both tachyons T and T must have the same rest mass, due to the impulse conservation; remember that for transcendent tachyons IPl ---- race); bat we are unable to evaluate the proportionality constant.

More interesting appears considering, in two dimensions (sect. 5), an ordinary particle P ~- P~ harmonically oscillating in a frame ]' around the space origin O'. If the frame ]' moves SuperluminMly w.r.t, another frame ]o(t, x), in the second frame the world-Hue of point O' is a spacelike straight line S; and the world-line of the harmonic oscillator P --= PT (now a tachyon with variable velocity) is depicted in fig. 35. Due to what we saw in subseet. 5"12- 5"14,--as well as in sect. 11, subsect. 12"5 and 13"2~, the ~ sabluminM ~ observer fo will see a vacuum fluctuation propagating in space, with vacuum decays (pair creations of transcendent taehyons) in correspondence with the events 01, 02, Ca..., and with analogous pair annihilations (of transcendent tachyons) in correspondence with the events A1, A~, As... (fig. 35). Cf. also WI~vr~L (1971b) and CATA~A et al. (1982). :Notice that each vacuum instabil- i ty C is just a vacuum decay into a taehyon T and an antit~chyon T, having the same rest mass and oppositely moving with infinite speed; such a process is perfectly allowed by classicM mechanics (see, e.g., subsect. 11"5). Analogously, each event A is nothing but the annihilation (into a ~ vacuum bubble ~>) of transcendent T-T pair.

e~ASSZCAZ Tae.YO~S 111

Fig. 35.

I t A 3

_ _ _ ~ 2 _ _ . . . .

r . . . . X

This is another example of classical description of a typically (( quantM }> phenomenon, i.e. of ~ phenomenon usually regarded ~s belonging to the realm of quantum field theory (QFT). See also, e.g., NAmer (1950), 3lAngm~i~ (1977), FVKUDA (1978), S~Au and MILLEI~ (1978) and SOU6EK (1981).

Let us remark, at this point, that in ordinary theories the possible presence of tachyons is not taken into explicit account. I t follows that the ordinary vacuum is not relativistically invariant, if tachyons on the contrary exist (and, let us repeeot, if account of them is not explicitly taken): cf., e.g., subsect. 3"17 and fig. 13. The fact that in the usual theories the ordinary concept of empty space may not be Lorentz invariant was particularly stressed by NIELSEn (1979), who noticed that, if some large region in space is empty of tachyons as observed from one frame, there is no guarantee that it will be so seen from another frame of reference. NIELSEn et al. (see, e.g., :NIELSEn and NINO~frYA, 1978; ~_~-IELSEN, 1977) als0 developed noninvariant theories, even if independently of the above observations.

13"5. A Loren t z - i nvar ian t bootstrap. - The idea that tachyons may have a role in elementary-particle structure has been taken over by many authors (see, e.g., I~ECAMI, 1968, 1969a; HANA31OTO, 1974; AIC~BA, 1976; I~AFA~ELLI, 1976, 1978; VAn ])E~ SPRY, 1978; CAS~O~InA and I~ECA~, 1978; SzA~0SI and T~EVlSAn, 1978; see alsot~0SEN and SZA3~0SI, 1980, and ref. (s,9) in MACCA~OnE and I~ECAMI, 1980b).

One of the most interesting results is probably the one by CO~BE~, who succeeded in building up a Lorentz-invariant <( bootstrap )> of hadrons or of hadronic ((resonances)) (Co,BEn, 1977a, b, 1978a, b). Let us describe his approach by following initially CASTORIn~ and I~ECAm (1978).

COI~BEN started from the known fact that a free bradyon with rest mass M and a free tachyon T with rest mass m caD_ trap each other in a relativisticMly invariant way; if M > m, the compound particle is always a bradyon B*. If the two particles have infinite relative speed, and P, p are their four-momenta,

112 ~. ~ECA~*

then (subsect. 11"5)

(148a) p ~ P , - - - - 0 ~ p •

In such a case the mass M* of the compound bradyon B* is (subsect. 6"3 and 6"5)

(148b) M* : % / ~ - - m ~ ,

as easily follows from cqs. (58), (59), or from eqs. (64), (65). Le t us now assumc that , inside the (~ composite hadi'on ~, the tachyon T

feels a s trong field similar to the gravitat ional one (see, e.g., l ~ n c ~ , 1982a, and references therein); let us assume moreover tha t the t rapped tachyon has already reached an equilibrium state and is revolving along a circumference around the bradyon B (see also S~P~L~S, 1983). From subscct. 6"1~ and 10"], we then derive tha t any bradyon- tachyon compound-- in its lowest-energy state (~ ground state ~))--is expected to be const i tuted by a tachyoa T having divergent speed w.r.t, the bradyon B, so tha t condition (148a) is satisfied. T reaches in fact its minimal energy when its speed diverges, i.e. the fundamenta l s tate of the system should correspond to a t ranscendent periodic motion of T. One also derives tha t the trapping force, which holds T on a circular orbit, tends to zero when T tends to have infinite speed. In such u case the interaction is negligible, even i / t h e ~ sel /- trapping ~) keeps itsel]. Under condition (148a), therefore, one may consider the B-T compound us a couple of two ]ree particles!

Actual ly C o ~ E ~ (1978a), by using the quantum language, considered two

particles satisfying the equations (M > m)

(149a) Oy~ -~ K2~B (K ~ Melt) ,

(1~9b) [~YJT ---- - - k~ ~PT ( k ~ m e l t ) ,

and such tha t , if v/--~ V2B~PT,

(149c) O~ --~ (K ~ - k 2) k =

Equat ion (149e) comes from postulating the invariant interaction ~, ~B ~ ~T = 0, which is nothing but the quantum-field version of condition (148a); in fact, applied to the eigenstates of energy and momentum, it just implies eq. (148a). (Cf. also eqs. (149c) and (148b).) Plane timelike and spaeelike waves can, therefore, (( lock )) to form a plane wave, tha t is timelike when M ~ m. ~otice t ha t everything still holds when we substi tute D(~)=-~, - ( ie~/~c)A~ for ~ ,

I t would not be possible to combine two timelikc, states in this way, because applying the condition ~ B ~ - - - - 0 (or D(~)y~nD( f~ : 0) to such states leads to imaginary momenta and exponentially increasing (not normalizable)

C L A S S I C A I ~ T A C t I Y O N S 113

wave functions. This corresponds: of course, to the classical fact t ha t condi- t ion (148a) cannot be satisfied by two bradyons.

On the contrary, a bra(ty(m B can combine in a rcl'~tivistically invar iant way with more tha~ one tachyon to yield another b radyon B*. Actually, due to conditions of the type of eq. (]48a), i t can trap no more than three

taehyons, get t ing eventual ly the mass

(t48c) = 2 2 M* ~/'M~--m~ m~--m~,

provided tha t it is real. In such a situation, the three t ranscendent tachyons T1, T.~, T3 c~m be im~gined ~s moving circularly around the axes x, y, z, respec- t ively (tim circle centres always coinci(ling with B). Going back to the quantum- field language (Co~BE~, t977a, 1978b), the extra, conditions ~ u % ~ , ~ J = 0 (i, j : 1, 2, 3, i r require l~he functions ~u~0~ to be orthogonal to each other in space. 15Iore generally, set t ing M : me, the conditious ~ u V ~ u ~ : : 0 (a, fl = 0, l, 2, 3, ~ fl) imply theft no more than three spaeelike states can be

super imposed on one t imel ike state to yield another pa.rticle. (Cf. also 1-)RE:PA -

ICATA, 1976; IIotl , 1976; ~PAGELS~ ]976.) IU QFT a bradyon ~t rest is describe(I, as usual, by a wave funct ion periodic

in t ime and independent of position. A t ranscendent tachyon, on the contrary, corres])onds to a w~ve functio:~ static in t ime and periodic in space: t~ latt ice (el. a~lso sect. 8). In(;ide.qt~tly, the interact ion between a br~dyon ~n(i ~ t ranscendent t acbyon is, therefore, almlogous to the scattering of a wave by a diffraction grat ing (CerumEN, 1978a). The three values of the latt ice spacings in the three directions of space may be regarded as corresponding to the masses of lhe three spacelike states tha t can combiue in the above way with one timelike s ta te ((~(mnE~ ~, 1978b).

By resorting to eqs. (148b), (148c) and to suitable quan tum number considerations, COICBE.~ (1977a, b, 1978a, b) found masses and quan tum numbers of a host of hadrons as composed of other (sub- aud Super-lumina.1) hadrons, thus realizing a rclativisl,ic~.lly invariant bootstrap (CHEw, 11968). There are a. number of ex~mples which appear to verify this, especially in the spectrum of the K-particles ~md thc s ~ 0 mesons l h~t preferential ly decay into K K : we refer the reader to 1:he interesting tables published by CORBE~ in his above-mentioned pqpers, which also contain fur ther details ~nd comments. CO]r " found

also the ma~'.s differences among the members of various isospin multiplets by binding Superluminal leptons to suitable subluminal hadrons.

I t would be interest ing t ry ing to generalize snch an approach even to the qua.rk level.

Actually, many a~tthors suggested tha t qua rks - -more generally, the elementary-pa.rticlc cons t i t ueu t s~migh t be tachyons (see, e.g., HA.~A~OTO, 1972; ?r and ~],~]CAMI, 1975b, p. 539; GuE~'~, 1976; SOU5EK, 1979a, b; see 'also BROW~- and I~go, ]983). RAFA~ELLI (i974~ 1976, 1978) showed tha t

1 1 4 ~:. n ~ c ~

in classical relativistic physics there exists the possibility for a description of an elementary particle which has constituents, if those constituents are tachyons. Free spinning tachyons are then the candidates for elementary- particle constituents. And in the range of Superluminal velocities the theory of a free spinning point particle admits uniquely of a linearly rising trajectory, naturally yielding the constituent confinement (see also subsect. 12"2 and I~ECA~K, 1982a).

~oreover, we shall see--subsect. 15"2--that the duality between electric and magnetic charges is possibly a particular aspect of the bradyons/tachyons duality; and authors as TzE (1974) and BA~UT (1978C) Cmderlined the connections between electromagnetic and dual strings; possibly, a link can thus be found between tachyons and hadron .structure (MI(~-A~I and t~:c~K, 1975b). Let us add that~ more generaily~ quarks have been identified (P~IsI~ 1978) with nonconventional (( monopoles ~)~ i.e. with the monopoles of the field which mediates strong interactions inside hadrons.

Asid% it stands the ((electromagnetic ~) approach by JEtILE (1971, 1972), who noticed that---while the introduction of a gauge variable single-valued in space implies charge conservation but does not change the physical situation-- a (~ pseudogauge )) transformation (with a variable 0 which is single-valued only modulo 2~) is equivalent on the contrary to the introduction of a quantized flux hc/e (see also BERNARDINI~ 1982). I t is~ namely~ a transformation from A' -- O, 3~0' 0 to A~ Ok, where Ak -- (?ic/e) ~kO 0 is invariant; and, because of (e/~v)~ ~ A ~ d r ~ = • one may assume the flux line (singularity of

1~3

3k0) to be a closed loop. A more detailed analysis shows that such singularity loops spinning with velocities larger than c permit a consistent formulation of leptons, as well as of quarks and hadrons, in terms of electromagnetic fields and their probability amplitude distributions. The topological structure of those fields (toroidal field lines) represents the internal quantum numbers in particle physics.

13"6. May classical microtachyons appear as slower-than-light quantum particles? - We have seen in subsect. 8"4 that, if a taehyon P~ has a very large intrinsic (i.e. measured in its own rest frame) li/etime At(P~), just as it is for macroscopic and even more cosmic objects~ then PT can actually be associated with Superluminal motion (subsect. 8"1). On the contrary, we saw that, if the intrinsic lifetime A--t(P~) of PT is small w.r.t, the observation time duration of tachyon PT (as it often happens in microphysics), then PT will actually appear endowed with a slower-than-light (( front velocity )>~ or (~ group velocity )), v . Only its (( phase velocity ~> Vv will be Superluminal: v V~= 1; cf. eq. (100).

In subsect. 8"5 we noticed some formal analogies between these classical tachyons and the (( de Broglie particles )> met in QFT. The analogies become more strict when we analyse the appearance of a tachyonic particle endowed with an additional oscillatory movement~ for example (and for simplicity)

CLASSICAL TACIIYON8 l l5

along the motion line (GAt~IJOClO, 198~t). Let us recall that the <~ shape, of a tachyon (subsect. 8"2) depends also on its speed V; namely, the semi-angle of the cone ~o is given (subsect. 8"2) by tg c~-= (V2--1)-t In such cases, the microphysical tachyon P~ will really appear as a bradyonic object associated with a kind of wave (having Snper]uminal phase velocity). Cf. also TANAKA (1960), Scm~oEI~ (1971), STICEIT and KLAUDF~R (1971), ~lJI~Plrg (1971), ~A_~ANAN (1972), GOTT I I I (1974a, b), STIC~AD and KODRE (1975a, b), THA~ICAPPA~ (1977), and particularly Ir (1978).

At each time instant, the real portion--which does carry energy-momentmn-- of such a wave is the one contained inside a certain moving (( window >~ (see eq. (99), subsect. 8"4) : the whole wave may be possibly regarded, in a sense, as a <~ pilot wave )). Iu this respect, it may become enlightening describing the scattering of two tachyonic particles PT, P~, i.e. of two microphysical bradyons I)~, I)'n observed from a Superluminal frame.

Such considerations can be relevant also to the hoped pnrpose of obtaining quantum mechanics from classical physics with tachyons. In fact, once the de Broglie's hypothesis were derived within classical physics (with tachyons), then a direct path would become available to the basic equations of QI~I, without the use of the standard postulates of QM. See, e.g., KELImR (1982, 1984), where Hestenes' (1966) (<mlfltivector~> algebra is used. IncidentMly, this is one more example of the powerflfl role in physics of Clifford algebras, a role to which it is always advisable to call the reader's attentiom

13"7. About taehyon spin. - I t is known that the little group of a spacelike vector (of., e.g., JOlCDA~, 1978) iS isomorphic to S01.~, the Lorentz group in a pseudo-Euclidean space-time with one timelike and two spacelike dimensions (subsect. 8"1). Since S01,~. is noncompact, its unitary (irreducible) representations '~.re infinite-dimensional, except for the one-dimensional representation. I t was often concluded that, thus, either a tachyon has no spin (i.e. it is a scalar particle), or it has an infinite number of polarization states (CA,'~EI~ZrND, 1970).

However, after the results in sect. 5, 9 and 11 (see, e.g., subsect. 5"17), we are justified in resorting for taehyons to nonunitary representations, which are finite-dimensional (see also CAICEY et al., 1979). ~'or instance, solving the (( relativistic wave equations for any spin >~ in the case of spacclike momentum, the finite-dimensional wave functions form nouunitary representations of the little group SO~.~. Also tachyons can, therefore, be associated with integer and semi-integer spins. This complies better with the philosophy of ER (see, e.g., Col~]~.~, 1978a).

Here we refer, e.g., to SHAY (1978); see also WOLF (1969), ~ x (1970), ~LEU~Y et al. (1973), YACCA]~I~ (1975), PAVw and ~ECA3~I (1976, p. 184), CAM~:.~ZIND (].978) and TANAKA (1979). WOLF (1969) showed moreover that, if a B~rgmam~-Wigner equation1 holds for time-, light- and space-like particles,

116 ~,. RECA-~t I

then W-spin conservation holds for all of them, and not only for timelike

particles. Let us mention, at last, that the ordinary relation between spin and statis-

tics seems to be valid also for tachyons (SuDA~SlrA~ and MUKU~)A, 1970),

but contrary opinions do exist (Fc, LW~EnG, 1967; t t~A~O~O, 1972).

13"8. l~'urther remarks. - In the prescnt sect. 13 we have met some indica- tions not only of the possible role of tachyons in elementary-particle interactions (and perhaps even structure), but also of a possible, eventual reproduc- tion of quantum results within classical physics with tachyons. Let us list

some more hints:

i) Many relativistic wave equat ions~based on perfectly valid representations of the Lorentz group (WIGan]c, 1939)--lead to spacelike solutions: see, e.g., BARU~ and NAC~EL (1977); see also Ko~r~ and F~IE]) (1967). For example,

in a quantum electrodynamics based on the Joos-Weiberg higher-spin wave equations, some solutions for integer-spin particles correspond to tachyons

(EEl, ]973).

ii) In particular, the infinite-component relativistic equations (MaJO~A-~A, 1932) lead also to spacelike solutions (see, e.g., FX0~-SDAL, 1968; G~0])SKu and STI~EATEIr ]968). I t is noteworthy that the timelike and spacelike solutions of the infinite-component Majorana wave equations, taken together, constitute a complete set of solutions (AnE~s et al., 1967; MUKU~DA, 1969). BA~VT and Du]r (1973) recalled that a wave equation with many mass and spin states can be interpreted as describing a composite system in a relativistically invariant way, and then investigated the composite system corresponding to the Majorana equation (by introducing the internal co-ordinates in the c.m.f.). They showed that the internal motion of the two constituents of that composite system can be either oscillatory type or Kepler type. While the timelike solutions of the Majorana equation correspond to bound states of the internM motion, the spa,celike solutions correspond on the contrary to t.he

scattering states of the constituent (~ particles ~). This material was put on a more formal basis by BA~L:~ et al. (1979), thus providing a completely rela-

tivistic quantum theory suitable to describe a composite object; such a result being obtained--let us repeat--only by accepting the spacelike solutions too.

In a further series of papers, B ~ u T and WILS().~ underlined many other circum- stances in which the presence of those solutions in the infinite-component

equations is a good and not an evil (*).

(*) Even in QED, when considering relativistically interacting magnetic dipoles, B~uL'T (1985) found stable p~ =--p~p~< 0 solutions as a result of dynamics. Analogous solutions seem to have been found by that author also in the relativistic Kepler problem.


iii) In general, the existence of spacelike components seems a natural and unavoidable feature of interacting fields (STOYA~OV and TODO~OV, 1968). For instance, it has been proved by DELL'A~To~zo (1961) and GREEI~BERG (1962) that, if the Fourier transform of a local field vanishes in a whatever domain of spaeelike vectors in momentum space, then the field is a generalized ]ree field. But spacelike components seem necessary even to give locality to the fields.

iv) In connection with what we were sayingin subsect. 13"5 about the field- theoretical models of elementary particles (see, e.g., P ~ I s I , 1978), let us recall that the dual-resonance models led to conceive hadrons as nonlocal objects: strings. String models have been widely investigated at both the classical and quantum levels. And they predicted the presence of tachyons in the spectrum of states. To eliminate tachyons, one had to introduce an additional interaction of a particle with the vacuum and spontaneous vacuum transitions (see~ e.g., VonI<ov and Pv, I~VUSHI~r 1977).

3Lore in general, field theories with tachyons are quite popular (TAYLOr, 1976; see also I~IELSE~ and OL~,s~, 1978): but, by assuming the vacuum to be the ground state, an automatic procedure is usually followed to get rid of tachyons~ or rather to turn them into bradyons (see, e.g, I~]:ELSEN~ 1978).

AlSO in the case of the Salam-Weinberg type of models, the gauge symmetry is spontaneously broken by filling the vacuum with tachyons: in this case such tachyons are the Higgs-field particles. However, the vacuum is supposed once more to adjust itself so as to turn the tachyons into bradyons (NIELSEn, 1978).

In conclusion, in the conventional treatment of field theories, tachyons seem to exist only at a formal level. But the procedure itself to get rid of tachyons might have a physical meaning beyond the (~ formal ~) one. In any case, the Higgs particles--yet to be observed experimentally--can be considered at least as tachyons which have been converted into bradyons.

v) The standard theories with positive metric and purely local interaction have not been developed in a convincing way; ]~EISENBERG considered the efforts in that direction to be largely based on (( wishful thinking 7). He was more favourable to Dirac's hypothesis of an indefinite metric in state space (HEIS:ENBERG, 1972).

In quantum theory with an indefinite metric complex-mass states are per- mitred, and cannot be ignored (see, e.g, YAlWAMOTO, 19697 1970a~ b; GLEESON and SUD~SHA~, 1970; JADCZYK, 1970; YOKOYAMA~ 1972; TOYOD• 1973; YA~A~0~0 and KUD0,1975). As we saw towards the end of subsect. 13"2 (REcAlv~ 1968, 1969), complex-mass objects may be related to tachyons: se% e.g, SUI)ARSHAN (1970d, e), VAN DE~ SPUY (1971)~ GLEESOE et al. (1972b), MARQ~s and SW]:EcA (1972); see also DAS (1966) and CORBEN (1975).

118 ~. m~cA,~

vi) Again, W ~ E L (1971a, b) noticed that classical tachyons can appear to ~mdergo a (classical) (( tunnel effect ~>, an effect ordinarily allowed only to quantum objects.

Let us recall that, more in general, the tunnel effect can be described within (( classical physics ~) by extrapolation to imaginary time (ef. subsect. 5"6): see, e.g., McLAUGHI~ (1972), FI~EEI) (1972), JACI~W and I~E]~BI (1976), 'T HOOFT (1976); see also ]3JO~KE_~ and D ~ L L (1964), p. 86.

vii) In a recent paper LEE (1985) has shown that an observer--due to the EP]%type effects present in QM--can get information also about the region beyond a possible horizon by performing quantum measurements in his frame. Such a quantum effect would correspond at the classical level

to the intervention of tachyons.

viii) At last, let us mention that two number fields exist that are asso- ciative and contain <(imaginary units ~ (both properties being apparently necessary in quantltm mechanics (QM)): the complex and the quaternion number field. Starting from the beginning of QM (we mean from the de Broglie wave-particle dualism) and recalling the above Hurwitz theorem, SOU~EK attempted the construction of a quatcrnionic QM, besides the ordinary comph~x QM. ]~e seemingly found that, as the latter describes bradyons, so the former describes tachyons. Namely, in the duality between complex and quaternion QM, there correspond bradyons and tachyons, the electrodynamic Ut gauge field and the Yang-Mills SU2 gauge field; and so on. See SoudEx (1981); see also WEL'~6AR~E~ (1973), ]~I)~O~])S (1977a, 1972), BAR~ET (1978).

PA~T V

The Problem of SLTs in More Dimensions. Taehyon Electrodynamics.

14. - The problem of SLTs in four dimensions.

We have already seen various times (subsect. 3"2, 6"1 and 8"3) that the Lorentz transformations (LT) can be straightforwardly extended to Duper- luminal frames S only in pseudo-Euclidean space-times M(n, n) having the same number of space and time dimensions. In sect. 5 we developed a model theory in two dimensions, i.e. in a M(1, 1) space-time; and those nice results strongly prompted us to at tempt building up a similar theory also in more dimensions, based as far as possible on the same postulates (subsect. 5"18). In four dimensions, M(1, 3), the asymmetry in the numbers of the time and space dimensions carries in very delicate problems (subseet. 5"18). And no four-

dimensional extensions of LTs for the Superluminal ease exist t ha t satisfy all the propert ies i)-vi) listed at the end of subsect. 3"2 (of. also PAI~O~ and STRNAD, 1976).

By trials, i t is easy to write down ~( Snpe rhmi~a l ~) Lorentz t ransformations (SLT) in four dimensions which are real: bu t t hey violate one of the remaining conditions (subseet. 3"2); see, e.g., the interest ing paper by SE~ GuPTA (1966; see also SAAVl~D~A~ 1970). The first proposal of real SLTs in four dimensions is due to O L ~ o v s ~ u and I~ECA3r[ (1970b; see also 1971); such a proposal, soon abandoned by those authors, was independent ly t aken over again by A ~ I e ~ A

and EvEaE~r (1971, 1973), who were inspired in par t by a belief shown by us to be probably erroneous (see the end of subseet. 5"14).

A way out has been already outl ined in subsect. 8"3; we shall come back to it la ter on.

Several promising ways out can be actual ly devised. A priori, one of the simplest could run as follows: a) consider a unique arena M 4 of events, all t angent vectors dx, of which are labelled as dx~ ~ (dt, dx) in a certain f rame so; each observer will have, of coarse, to cover M ~ b y an atlas A of local charts, and endow it with a metr ic g; b) consider, then, two classes of observers: the class C1 of the observers t ha t choose the same atlas A and the same metric g

(for instance, with signature -- 2) as so, and the class C2 of the observers tha t - -whi le still using the atlas A - - a d o p t the metr ic -- g (with signature -4- 2). More specifically, t ake a s C 1 the set of suitable observers t h a t are in straight, uni form motion w.r.t. So, and as C~ the set ((dnal~) to C~, If dx, d x z > 0 in C~, then dxzdx~ < 0 in C~; therefore, C~ and Q are the classes of observers sub- and Super-luminal, respectively, relat ive to so; c) once ordinary Lorentz t ransformat ions are known, the remaining problem is to find out how to pass f rom the description of M ~ yielded by the C~ observers to the one yielded by the C~ observers. Such a problem, being different f rom tha t of looking for SLTs, does not have to face the (( no-go )~ theorem by Go~I~I et al. An interesting point is raised by the fact t ha t the Clifford algebras constructed on M(1, 3) ---- -= {M', A, g} or on 3/[(3, 1) ~_ {M~, A, -- g} lead to completely diMerent (( Clif- ford groups )) (el., e.g., ~IGU:EYI~:EDO, 1985).

An even more promising way out draws inspirat ion from the fact t ha t i) the

natural algebras for IV[ 4 and R 3 are the Dirac and Panli algebra, respectively (see, e.g., I=IESTE2~ES, 1966); if) the imaginary uni t i appearing in Panli algebra corresponds to the pseudoscalar unit, )~5, in Dirac algebra; iii) the space of the Dirae (( biveetors ~> is globally invar iant under the action of ~a (since the multiplication by y5 acts in the Dirac algebra as the Hodge s tar operator). Work in this direction--i .e , utilizing the Clifford algebras const ructed on M~ and on the 3-dimensional Eucl idean space R~--has been pu t for th recently (DE ]~s ROSA et al., 1985, 1986).

B u t we want to warn explicitly the reader tha t many difficulties m ay just

come f rom the ordinary assoeigtion of Minkowski space-time M 4 with the

120 ~. XrCA~

topology of :R~ (a topology tha t does not appear to be the best suited to a par t icular pseudo-Riemannian space like M4). For example, when adopting the so-called Zeeman's space-time ((fine topology~) which contains much more open sets than the ordinary one, bu t allows constructing a light-cone at each po in t - - , many functions tha t are discontinuous in the 1~ 4 topology get continuous on the light-cone within the (( fine topology ~). In such a (( new ~ topology, moreover, the impor tan t relativistic function 1/x/1--fl~ gets ana-

lyt ic everywhere (cf. DE FARIA-I~OSA st at., 1985). In the present sect. 14, however, we shall confine ourselves to cover the

researches t ha t during the last decade aimed at the int roduct ion of four-dimensional SLTs (or, at least, oi the 4-dimen:sional (( shadows ~ of six-dimensional SLTs); and this not only for (( historical ~) reasons, bu t also because the ways a t t e mp te d in the past A) could still be viable (in the light, e.g., of results like the ones in P~EPA~A~A, 1983), B) t hey led, at least, to a physically cogent extension to tachyons of quantit ies as four-velocity and four-momentum, s tar t ing from the analogous b radyonic quanti t ies , and C) the six-dimensional approach implied by some of t h e m - - a t least as an in termediate phase - -migh t

happen to be eventual ly acceptable (cf., e.g., COLE, 1985). Before going on, let us prel iminari ly observe tha t (see fig. 5 and 6) in the

four -momentum space, e.g., the mirror s y m m e t r y w.r.t, the light-cone is a one-to-one m~pping almost everywhere, in the sense t h a t the whole plane E = 0 should be mapped onto the E-axis, and vice versa; b a t one might restore a one-to-one correspondence by associating a direction also with every object at rest (namely, the l imiting direction of its mot ion when coming to rest); or, a l ternat ively, by identifying all the points of the hyperplane E ---- 0, i.e. by adding to the 3-velocity space a single ideal point at infinity.

14"1. On the (( necessity )) o] imaginary quantities (or more dimensions). - Let us s ta r t f rom some e lementary considerations, assuming we want to introduce Superluminal reference frames also (( in four dimensions ~). I f a light

burs t springs out f rom the event 0 (fig. 36), the subluminal ' observer so = (t, x) will observe a spherical light wave expanding with t ime. The Super laminal observer S'~ moving w.r.t, so along the x-axis with divergent speed (having, i.e., as t ime axis t' the x-axis of f rame so) would not observe a spherical wave any longer, bu t a light wave with the shape of a two-sheeted hyperboloid, unless the SLT which connects so with S leads to imaginary quantit ies for the

t ransverse co-ordinates, so to t ransform the hyperboloid back to a spherical surface (:[~]~CA~dI and I~r 1980). Th is shows tha t , if we want to preserve in E/~ the main characters of S]~ (e.g., the equivalence of all inertial frames), we have to release in par t the reali ty condition by introducing also imaginary quantit ies (/~ECA~H and M~G~A~I, 1972, ]973a; CO~BE~, 1974, 1975, 1976; see also ]~A~AOHA~])nA~ et al., 1972~ and ALAGA~ ]~A~A~UJA~ et al., 1973); or which is in a sense equ iva len t - - to increase the number of

CLASSICAL TACtlYONS 121

F ig . 36.

t

-~tr

space-time dimensions. Actually, M A C C ~ o ~ E and ~ECA~r (1982a, 198~) had to introduce an auxil iary six-dimensional space-time M~ ~ M(3, 3) as the abs t rac t background in which the events are a priori allowed to happen. La te r on, t hey went back--for each obse rver - - to a four-dimensional space-time ~(1 , 3) by assuming tha t each observer has access only to a suitable four- dimensional slice o] M6, even if a price has to be paid (in a sense, taehyons should then be described by three t ime co-ordinates and one space co-ordinate), as we shah see in the following.

Le t us ment ion, however, t ha t one could avoid paying t h a t price by mod- i fying Maccarrone and /~eeami's (1982a, 1984) approach into a really six- dimensional theory. In fact , recent work by COLE (1985) and by HAGSr a n d Cox (1985) :showed a six-dimensional space-time to not conflict with every- day experience.

14"2. The formal expression o] SLTs in ]our dimensions. - Wh~t follows is mainly based on MACCARICO2~E et al. (1983) and 3~tCCAI~I~O~E and RECAIVII (1982a, 1984), and references therein. Le t us s ta r t from the postulates of SI~

as pu t for th in sect. 4; let us receJll in par t icular t h a t we gave the second postu-

late the form: (( The space-time accessible to any inert ial observer is foul'- dimensional. To each inert ial observer the 3-dimeD.siona] space appears as

homogeneous and isotropic, and the 1-dimension2A t ime ~s homogeneous ~). Le t us recall also tha t the t ransformations G~ connecting (see eq. (14)) two generic inertial frames ], ]', apriori with -- ~ < lul < ~ 0% must (cf. subsect. 4"2) i) t ransform inertial mot ion into inert ial mot ion, if) form a group G, iii) preserve

122 ~. ~cA~

space isotropy, iv) leave the qnadratic form invariant, except for its sign (see eq. (15)):

(15) ' '~ . dx~ dx = ~: dx~ dx ~ (u ~ X c ~)

Let as recall ut last the whole subsect. 4"3, as well as the two-dimensional theory (expounded in sect. 5) which, whenever possible, has been already expressed in a multidimensional language.

F rom the above 10ostulates (sect. 4), i.e. from the above requirements i)-iv), i t follows tha t in four dimensions the GLTs are expected to be unimodular and special (see subsect. 5"~ ~nd eq. (31)) : det G -~ -k 1, VG E G, and such tha t the LTs are orthogonal, G T ~1(~---- -k U, while the SLTs are untiorthogonul, G T U G - ~ - - ~ (cf. eqs. (30): notice t ha t now, however, U is four-dimensional).

In analogy with subsect. 5"5, the SLTs differ from the LTs only for the fact t ha t they are antiorthogonal, i.e. satisfy eq. (15) with the minus sign, stil]

f however- -wi th g,~ ---- g~ (subsect. 4'3). Alternatively, one might also say that a SLT is identical to its dual sublmninal LT, provided that we require the

!

primed observer S' to use the opposite metric signature g,, = - g~ (without changing, however, the signs into the definition of timelike and spaceHke quantities! See also S ~ z , 1984).

In any case (cf. subsect. 5"5), the fundamenta l theorem of four-dimensional EI~ can be writ ten (u S < 1, U ~ > 1)

(150a) SLT(U) = • 5 P . L T ( u ) [ 150b) 5 P ~ 3 ~ 4 - - i l , SPEGJ (ul[U , u = l / U ) ,

i.e. a generic SLT, corresponding to a velocity U, will be formally expressed by the product of the dual (subluminul) LT, corresponding to the velocity u]l U, with u : e~/U, by the matr ix i l . Due to the imaginary uni t i, the SLTs satisfy eq. (15) with the minus sign. The operation S p ~_ Sf4 -- il plays the role of (( transcendent SLT >>, since when u--~0 we get

(151) SLT(U = ~)---- = k i l .

Incidentally, the transcendent (~ t ransformation ~>~ is simply given by eq. (150b), and does not affect the speed u (nsmely, does not operate any change fl ~ 1/fl, differently from what s tated in some previous p~pers).

I t is impor tan t to stress tha t the group properties and space isotropy can be preserved only by an operator ~ which is a 4 • 4 matr ix symmetrical w.r.t. all the possible axis permutat ions (MACCA~o~v. and l~v.cAlv~, 1982a, 1984). The expression of 5 P tha t appeared in review I w~s suited only for the simple case of eollinear boosts (and the GLTs as wri t ten in review I formed a group only for eollinear boosts). Misunderstanding this fact ~nd overlooking some recent l i terature (e.g, MACCAI~tCONE and I~ECAlVII~ 1982b), 1VIARCHILDON et al.


(1983) a d o p t e d t he express ion for 5 # g iven in rev iew I also for t he case of generic

(nonco]l inear) SLTs. T h e y were led, of course, to incor rec t conclusions.

The g r o a p G of t he genera l ized L o r e n t z ( ( t rans format ions ~ (GLT), bo~h

sub- and Super - lumina l , will be

(152) { 6 : s 1 7 4 z ( 4 ) ,

z ( ~ ) =- { f f ~ 1) ~ + z , - ~ , + i , - i ;

thi:s is analogo~ls to w h a t seen in subsect . 5"6, b u t now o~fr is t h e ]our-dimen-

sional prope r o r thoc l i ronoas L o r e n t z group. A g a i n we h a v e t h a t , if G ~ 6 , t h e n

(VG ~ G) also - - G E G a nd SfG ~ G; ci. eqs. (37). I n par t icu la r , g iven u ce r ta in

L T ~ JL(u) and t he SLT : + iZ (u) , one has [ iL(u)] [iL-~(u)] ~ [ iL(u)]-

�9 ILL(-- u)] ~ - - 1, while, on the con t r a ry , i t is [iL(u)] [-- iZ-~(u)] ~ [iL(u)].

�9 [ - - i Z ( ~ u)] ~ + 1 ; this shows t h a t

(153) [ iL(u)] -~ = - - iL-~(u) = -- i L ( - - u ) .

The g roup 6 is n o n c o m p a c t , n o n c o n n e c t e d and wi th discont inui t ies on the

l igh t -cone ; its cen t ra l e lements , moreover , are C = ( + 1, - - 1, + i l , - i l ) . L e t

us recall f r o m subsect . 11"1 t h a t - - 1 ~ P T = CPT ~ G, a n d t h a t G = #(~r

CPT, 5P). See also subsect . 11"3. Of course, also de t ~9 ~ = + 1, 5P~5 P = - - 1

and ~= ~ E 6 (cf. eq. (150b)).

I n the pa r t i cu la r case of a boos t a long x, our SLTs , eqs. (150), can be

]ormally w r i t t e n ( U = 1/u) (see MACCAI~O~]~ and l~gCh~I, 1984; MACOAm~O~

et al., 1983, a nd references there in)

(154)

dt -- u dx dx - - U dt

dx - - u dt dt - - U dx - - - : F i - - dx' = :~ i V Y - u ~ V ~ - ~ '

dy ' = :~: i d y , dz ~ ~ ~: i dz

(Super lumina l case, u ~ < 1, U s > 1, u ~ 1 /U) ,

where we took a d v a n t a g e of t he i m p o r t a n t ident i t ies (41): see sub sect. 5"6. Not ice

t h a t unde r (( t r a n s f o r m a t i o n s ~> (154) for the roa r -ve loc i ty (snbsect. 7"2) i t h a p p e n s ! !

t h a t u , u , = - u , u , ; eqs. (151) are, therefore , associa ted wi th Supe r lumina l

mot ions , a:s we shall see b e t t e r below. One should no t confuse in the fol lowing

the boos t speeds u, U wi th the four -ve loc i ty componen t s u , of the cons idered objec t .

L e t Its under l ine t h a t ol~r (~ fo rma l ~ SLTs , eqs. (154), do f o r m a group, G,

t o g e t h e r wi th t h e o rd ina ry (o r thochronons and an t iehronous) LTs. I t should

be no t iced t h a t the general ized L o re n t z <( t r an s fo rma t ions >> in t roduce only

1 2 4 ~ . n ~ c x m

real or purely imaginary quantities, with exclusion of (generic) complex quantities. Le t us moreover stress tha t the t ranscendent t ransformat ion 5 ~ does not depeud at this stage on any spatial direction, analogously to the t ransformation LT(u = 0) = 1. This accords with the known fact (subsect. 3"2) tha t the infinite speed plays for Ts a role similar to the one of the null speed for Bs; more generally, the dual correspondence (subsect. 5"11)

holds also in four dimensions. (See also the beginning of sect. 14.)

14"3. Preliminary expression o] GLTs in ]our dimensions. - Subsections 5"8 and 5"9 can be extended to four dimensions (see MACCA~RO~n et al., 1983). F i rs t of all

(156) 6 = ~ | ~*+,

!

where ~ is the discrete group of the dilations D: x, = 0x, with 0 = • 1, • i. Then, by using the formalism of subsect. 5"8, we can end up with eqs. (45)~ valid now also in /our dimensions.

In terms of the light-cone co-ordinates (46) and of the discrete scale parameter 9, the GLTs in the case of generalized boosts along x can be wri t ten

(~57) { d ~ ' : 9adS, d $ ' = 9a-~d$, d y ' = ~dy , d z ' = 9dz,

9 ~ • 1 7 7 0 < a < - ~ o o , - - o o < u = ~ u ~ < - ~ o o ,

where a is any real, positive number. Equat ions (157) are such tha t d~ 'd~ ' - - -- dy ' ~ - dz '~ = ~ ( d ~ d ~ - - dy ~ - dz~). For m a n y fur ther details see the above- ment ioned MACCA~O~E et al. (1983).

I t is more interest ing to pass to the seale-invariant l ight-cone co-ordi-

nstes (47). Equat ions (157) then become ( a - - a~, ~' ~ 9 - ~ )

~ d~]'(8) = d~(~) , (158) d ~ ' : ~dF, d ~ ' : o~-ld~f, d~ '(2) d~]("), ~_=a~, ~ = • a e ( O , § ~ ) , - ~ < u ~ u ~ < - ? ~ ,

where, as usual, 9 ---- -t- i yields the sublnminal and ~ ---- 4- i the Superluminal x-boosts. Now, all the generalized boosts (158) preserve the quadrat ic form,

its sign included:

(159) d ~ ' d y / - - (d~]'(2)) ~ - (d~]'c3))~ : d ~ o d ~ - (d~(~)) 2 - (d~(~)) ~

Actually, eqs. (158) automatically include in the Superluminal case the interpretation o] the first couple o] equations in (154), just as we obtained in subsect. 5"6.

CLASSlOa5 ~ e E ~ o ~ s 125

I n fact , t hey yield (U = l / u )

(154 bis)

dx -- u dt d t ' = • ~ T

~ / 1 - - u ~

dt -- U dx

dx - - U dt dt - - u dx d x ' = • -- T

d y ' = • d z ' = •

U ~ > 1 U ~ i / u ) , (Super luminal ease, u 2 < 1,

where the imag ina ry nni ts d isappeared f rom the first two eqnat ions (cf. subsect.

5"6). See MIGNA~I and t~ECAm (1973a) a~d Col~m~?q (1975, 1976); see also

MACCARgO~E et al. (1983) and PAV~I5 (1971). Moreover, f rom eqs. (158) one derives for the x-boost speed

~-- O~ -I

(158') u - - - - ~ __ c~_~ ~ ~ pa ( O < a < -~ 0% - - o o < u = _ u ~ < ~- oo);

in par t icular , in the Snper luminal case ( 9 = • i), the boost speed follows to be fas te r t h a n light:

a _ _ ~ - 1

u - - - - > l . a - - a - 1

Actua.lly, in the case of Super luminal boosts and in t e rms of the l ight-cone co-ordinates (46), eqs. (158) can be wr i t ten

(158 bis) " S d d~'=~d~, d~'=~-~dr dy'=--,~] y, d~'=--iNd~,

2 u ~ _ _ = u ~ > l , - - o o < c 7 < @ oo,

which are the t ranscr ip t ion of eqs. (154 bis) in t e rms of the co-ordinates (46); now ~ is jus t real. I n par t icn]ar ,

dt ' ---- � 8 9 ~-1)dt - - �89 + 5 - t )dx , d x ' = � 8 9 c~ -1) d x - �89 + 5-1)dt,

so t h a t for the relat ive boost speed one has u = (dx/dt)]d,,= o = (5 + 5--1)/ / ( 5 - 5-1), u 2 > 1. Le t us observe t h a t our co-ordinates ~, ~ are :so defined

t h a t u is sublnminal (Superluminal) whenever in eqs. (158) the quant i t ies c~ and ~-1 have the same (opposite) :sign.

The more difficult p rob lem of the veloci ty composi t ion law will be considered

below. We shall consider below also the meaning of the above-seen, automatic,

<~ par t ia l re in te rpre ta t ion )> of eqs. (154)--formal , bu t wi th good group-theoret i - cal p rope r t i e s - - i n to eqs. (154 b i s ) - -wh ich lost, on the contrary , thei r group

126 E. ~ c A = i

propert ies : see M~kCCA~O~E and I~ECA_~rr (198:1). Incidental ly , let us explicit ly remind tha t the re in te rpre ta t ion we are (and we shall be) dealing with in

this sect. 14 has nothing to do with the <<switching p r o c e d u r e , (also known

as <( re in te rpre ta t ion principle >>). I n analogy with subsect. 5"7~ the <~ par t ia l ly re in terpre ted ~> eqs. (154 bis)

can be combined with the ordinary (ortho- and anti-chronous) LTs in a compac t

fo rm and in t e rms of a continuous p a r a m e t e r 0 e [0, 2~r] as follows (]~]~cA_.'~

and ~[IGNANI, 1973a):

(160)

with

dr' : - Dyo(dt - - dx t g 0 ) , dx' ---- DTo(dx - - dt t g 0 ) ,

dy ~ . . . . ~5 dy , dz' ---- -- [25 dz ,

u~_tgO, ~ , o ~ - t - ( l l - t g " O l ) - ~ ,

V I-- tg~0

~5 - ~ - j ! ] . - tg 20] '

cos 0 ~2 - .c2(0) ~ [eos0i

(0 < 0 <27r, u 'X 1).

Equa t ions (160) show, among the others, how the f(mr various (( signs ~> (r(;ul

or imaginary , posi t ive or negative) of dy ' and dz' do succeed each other as functions of u, or r a the r of 0 (notice t h a t - - oo < u < -~- oo). In brief, i t is dy ' = . . . . ~2yo~%/1-- tg 20dy. Figure 37 just shows it explicit ly. (We should

r e m e m b e r also fig. 12 in subsect. 5"]5.)

Fig. 37.

(Re ~minus)

(]m ~minus)

(Re ~plus)

(Im ~plus)~; t

C I - - c o - - C I

o o

As to eqs. (154 bis), let us ment ion t ha t recent ly CALDIEOL~_ et al. (1980)

discovered an ear ly derivat.ion due to SO:~GI,IA~A (1922). S O m G L ~ A looked

for the mos t general l inear t ransformat ions leaving invar ian t the elec;tro-

magnet ic -wa~e propaga t ion equat ion, and found---besides the ~mual L T s - - also eqs. (1.54 bis), except for thei r double signs (actually necessary to the exist-

ence of the inverse t rans format ions ; for his procedure, see CALDIROLA e$ al.,

~980).

CLASSICAL TACttYONS 127

14"4. Three alternative theories. - We preliminarily saw from fig. 36 and from eqs. (154 his) that, if we look for SLTs satisfying eq. (15) of subsect. 14"2 with the m i n u s sign, we end up with (( transformations )~ which carry in imaginary numbers for the transverse co-ordinates. As we mentioned many times, this problem disappears in (n, n) dimensions, and typically i n (1, 1) dimensions.

We deemed that such problem (the problem of EI~) has to be faced; and in the following we shall t ry to clarify its perspectives (even if a lot of tachyon physics--as we saw--can be developed without trying to introduce Super- luminal frames). We are mainly following, in other words, the approach by 5 ~ I ~ A ~ and l~cA~I, and subsequent co-workers.

However, other authors preferred to skip that problem, reducing it (even in four dimensions) to an essentially two-dimensional problem. Two alternative approaches have been proposed in such a direction.

14"4.1. The f o u r - d i m e n s i o n a l a p p r o a c h b y A n t i p p a a n d E v e - r e t t . A group of authors, initially inspired by a belief criticized in subsect. 5"14,

just assumed a l l tachyons to move exclusively along a privileged direction, or rather along a unique <( tachyon corridor ~. In this case the problem for tachyons becomes essentially two-dimensional, even in four dimensions. Such an approach does violate, however, not only space isotropy, but also light speed invariance. Those violations do not show up only for collinear boosts along the tachyon corridor. According to us, this approach avoids considering the real problem of SLTs in EI~. I t would then be better to investigate tachyons from the subluminal frames only (i.e. in the (~ weak approach ~ only). For details about this theory--which, of course, does not meet imaginaries--see A~IPPA and EVElCET~ (1971, 1973), AI~IPI.A (1972, 1975), EVElCETT (1976) and MAI~O~ILD01~ et al. (1979) ; see also LE~KE (1976, 1977a, b), and TELI and SuTAI~ (1978).

14"4.2. The f o u r - d i m e n s i o n a l a p p r o a c h by G o l d o n i . The third theory is due to Go~Do~I (1972, 1973), who developed an interesting approach in which a symmetry is postulated between the (( slow )> and <(fast )> worlds, and the tachyon rest mass is real; he succeeded, e.g., in producing the (~ tad- poles ,> dynamically (without supposing a nonzero vacuum expectation value of the fields).

Passing from the slow to the fast worlds, however, means interchanging time with space. And in four dimensions, which space axis has the time axis to be intercharged with? The approach mainly followed by us is equivalent to answer: (~ with all the three space axes )~, so to get transformations preserving the quadratic form, except for its sign (see eq. (15), subsect. 14"2) ; afterwards, one has to tackle the appearance of imaginary transverse components. In order to avoid such difficulty, GOLDONI introduced a different metric signature for each observed tachyon, ending up with the four independent space-time

1 2 8 r . n ~ c A ~

metric signatures ( - - - - - - + ) , ( A - - - - - - - ) , ( - - - [ - - - - - ) , ( - - - - ~ - - - ) . I t follows tha t tachyons are not observable in Goldoni's approach, except for the fact t ha t t he y exchange with bradyous (only) internal quan tum numbers.

Some consequences for QFT may be appealing; bu t we deem tha t this approach t o o - - a t the relativistic level avoids facing the real problem by a (( t r ick )).

~ever theless , ra ther valuable seem the considerations developed by GOLD0.'~I (1975a, b, c) in general relat ivi ty.

14"5. A simple application. - Let us go back to subscct. 14"4 and apply it to find out, e.g., how a four-dimensional (space-time) (( sphere ~) t 2 + x ~ ~- y2 ~_

z ~ : A ~, t ha t is to say

(161) �89 -t- �89 ~_ y~ + z2 = A 2 '

deforms under Lorentz t ransformations. In the ordinary subluminal case (eqs. (157) with @ = + 1), eq. (161) in terms of the new (primed) co-ordinates

can be rewri t ten as (0 < a < ~- cr

(162a) �89 '2 + �89 '2 -4- y,2 _~ z,2 = A ~ (subluminal case),

which in the new frame is a four-dimensional ellipsoid. In the ease of a Superluminal boost (eqs. (158 bis)), eq. (161) becomes,

in terms of the new pr imed co-ordinates (0 < a < + oo),

(162b) �89 ~_ �89 2 ~,~_ y , 2 _ z,2._ A 2 (Superluminal case),

which in the new frame is now a four-dimensional hyperboloid. Notice explicitly, however, t ha t the present operat ion of t ransforming

under GLTs a four-dimensional set of events has nothing to do with what one ordinarily performs (in fact, one usually considers a world-tube and then cuts

it with different three-dimensional hyperplanes).

14"6. Answer to the (( E ins te in problem ~ o] subsect. 3"2. - We have still the task

of in terpre t ing physically the SLTs as given by eqs. (]50), (154). Before going

on, however, we wish to answer preliminari ly the (( Einstein problem ~) ment ioned

in subsect. 3"2 (cf. eq. (12)). We have seen in subsect. 5"6, and later on in connection with cqs. (154 bis) (subscct. 14'3), t ha t eq. (12) is not correct, coming from an uncrit ical extension of LTs to the Superluminal case. Le t us consider an object with its centre a t the space origin 0 of its rest f rame; be it intrinsically spherical or, moro generally, let it have the intrinsic sizes Axo----2xo ~ 2r, Ayo ~- 2yo and AZo -~ 2zo along the three space axes, respectively. Ins tead of eq. (12), for the size along the boost motion line x the theory of E R yields the

CLASSICAL TACII YO_N'S 129

real expression (Xo - - r )

(163a) Ax '~ : Axo % / ~ 2 1 (U-~ 1).

~ o problems arise,, therefore, for the object size along the x-axis.

We meet problems, however, for the t ransverse sizes, which become imagill- aries ~ccording to eqs. (154 his):

(163b) Ay ' --- l A y o , Az ' = iAzo .

But let us go back to subsect. 8"2 and fig. 19. I f the considere4 object P -- P~3 is ellipsoidal in its rest f rame, then, when SuperluminM, I ~ ~_ 1)v will appear to be spread over the whole space confined bctwc(m the dollble indefinite corm ~ : y2/y2 o + z2/z~ ~ ( U t - - xp/x~(U'- ' - - 1), and the two-sheeted hyperboloid ~ f : yO./y~ -~ z2 /z~ ( U t - x )2 /x~(U"- - 1 ) - ~! (cf. lCECAYJI and MACCAI~RONE, 1980).

See fig. 17. The distance 2x' o between the two vertices Vt and V2 of d/f, which yields the linear size of PT along x, is 2x' o ~ 2xo X/~ /2 - 1. For instance, for t ~ 0, the position of the two vertices of J f is given by V~.~ -= ~ xo v /U 2 - 1. This, i~lcidcnt~lly, clarifies the meaning of eq. (163a).

Le t us now tm'n our a t ten t ion to the t ransverse sizes. Tile quantit ies Y0 and zo correspond to the intersections of the initial ellipsoid with the iuitiM axes y and z, respectively (for t ~ 0). We have then to look in the tachyonic case for the intersections of . ~ with the t ransverse axes y and z. Since these intersections are not real, we shall formally get, still for t = 0,

Y'o z iyo , zo --- izo ,

which do explain the meaning of eqs. (163b). In fact (see fig. 38), the real quanti t ies y~/i = Yo and z~/i = z o have still the clear, simple meaning of semi- axes of J r . In o ther words, the quanti t ies lY:[---- Y'o/i and [z:l-~ z:/ i jas t tell

I Ct =0) y:/i

~)

Fig. 38.

b)

130 ~. ~CA~

us the shape of the t achyon relevant surface ( they express the t ransverse size of the (~ fundamenta l rectangles )>, i.e. allow us to find out the fundamenta l asymptotes of PT). See I~ECA_M~ and MACCAI~0~E (1980); see also Com~E~- (1975), GZADKICK (1978a, b), TE~LETSKu (1978), GOTT I I I (1974a, b) and FLE~-~Y et al. (1973).

Eve n if in a particular case only, we have practically shown how to in terpre t also the last two equations in (3.54 bis). We shall come back to this point ; bu t let us add here the following. Equat ions (154 bis) seem to t ransform each ellipsoidal (or spherical) surface 5~ into a two-sheeted hyperboloid ~ . Le t us now consider the intersections of any surface 5~ (see fig. 39a)) and of the corresponding 5f~ (fig. 39b)) with all the possible t ransverse planes x - - 5. In fig. 39,

I I I

R3~ ' R ~ R ~ i 2z I I I I I I I I I I

Z I

' I I I I I

a )

Y

I I r I I T

I I 1 I i I i i

I I I I

( t = O)

x R31 V 1 [

Z

b)

, y

I

J I I

I I

/ I I

J /

Fig. 39.

for simplicity, the case of a Superluminal boost along x with speed V -~ c V/{ and t ---- 0 is considered; so tha t OV 1 ---- O1~ = xo -- r and all quanti t ies ()1~ have the same value both in a) and in b). I t is immedia te to realize tha t , when

the intersections of ~1 with the plane x -- 5 are real, then the corresponding

intersections of 5z~ are imaginary (with the same magnitude), and vice versa. l~amely, in the part icular case considered, the intersections of 5f~ are real for

141 < r and imaginary for 151 > r, while the intersections of 5f~ are on the contrary imaginary for ]51 < r and real for ]xl > r. I t is easy to unders tand tha t eqs. (].54 bis) operate in the planes (x, y) and (x, z) a mapping of ellipses # into hyperboles h, in such a way tha t the real part of # goes into the imaginary part of h, and vice versa (see CALDn~OLA et al., 1980). Cf. also fig. 37.

14"7. A n auxiliary six-dimensional space-time ]5(3, 3). - Equat ions (150), as well as (154), call imaginary quantit ies into p lay and, therefore, seem to require

CLASSICAL TACI~YONS 13i

an 8-dimensional space C A (i.e. a 4-dimensional complex space-time) us the kinematical background. However, an essential teaching of SI~ appears to be that the four-position is given by one real and three imaginary co-ordinates--or vice versa---so that formally (with c--= 1) t ime---- ixspace. As noticed by MINKowsxI (1908) himself, one might formally write l s ---- i • (3 X 10 s) m. As a consequence, to interpret the SLTs it can be enough to assume (temporarily, at least) a 6-dimensional space-time M(3, 3) as background; this was first suggested in MIGNA~I and t~ECAMI (1976b). Ever since, much work has been done on such spaces, with or without direct connection with the SLTs: see, e.g., DATTOLI and XV[IG~ANI (1978), VY~N (1978), PA}'PAS (1978, 1979, 1982), ZK~o (1979, 1983), STlC~AD (1978, 1979a, b, 1980), PAV~I~ (1981a, b), JOH~S0N (1981), FRO~I~G (1981), LEWIS (1981), PArrY (1982), CON]~0~TO (1984) and particularly CoLE (1978, 1979, 1980a, b, c, d, e); see also TO~TI (1976), JA~CEWlCZ (1980) and MACCAI~0NE and RECA~ (1982b). The idea of a possible multidimensional time, of course, was older (see, e.g., Bu~G~, 1959; Do~LI~G, 1970; KALITZ~, 1975; DEME~S, 1975).

Alternative approaches, that can be promising also w.r.t, tachyon theory, may be the ones which start from a complexification of space-time, via the introduction ab init io either of complex numbers (GI~EGOtr165 1961, 1962 ; SUDA~- SHAN, 1963; review I; YACC~INT, 1974; 1V~GNA~r and I~ECA~, 1974v; COLE, 1977; K.~LNAY, 1978; MOSKALENKO ~nd MOSXALENKO, 1978 i see ~lso ~OSEN, 1962; DAS, 1966; S~N, 1966; KXL~AY and TOLEDO, 1967; ~BALDO and ~ECAMI, 1969; t~ECAM~, 1970; OLXHOVSKY and t~ECAY~, 1970c; HA~SEN and NEW~AN, 1975; HEsrENES, 1975; PLEBA~SKI and SCH/_LD, 1976; CHARON, 1977; IMAEDA, 1979, and SAcgs, 1982), or of octonions (see, e.g., CASALBUO~-s 1978), or of twist- ors (see, e.g., PEN~OSE and MCCALLVXVf, 1973; HA~SEN and IqEW~AN, 1975) and quaternions (see, e.g., EDD/[0NDS, 1972, 1977a, 1978; WEI~GAtCTEN, 1973; MXGXNANI, 1975, 1978; IMAEDA, 1979). The most promising <~ ~lternative ~> approaches are probably the last one and those making recourse to more general Clifford algebras: see the end of subsect. 13"8 and the beginning of this sect. 14 (and Sov~E:~, 1981).

Let us mention, incidentally, that transformations in Ca-space ~re related to the group SU~ of (unitary) intrinsic symmetries of elementary particles. I t is not without meaning, possibly, that the M(3, 3) formalism has been used to express the law of (( trichromatism ~) (DE~E~S, 1975).

Let us coufine ourselves to boosts along x. W e are left with the problem of discussing the formal equations (154).

Let us consider ( ~ A c C ~ 0 ~ E and I~ECA~, 1984) the GLTs, eqs. (152), as defined in M~ ~ M(3, 3) -- (x, y, z, t~, t~, t~); any observer s in M~ is free to rotate the triad (t} = (t~, G, t~) provided that {t} 2 {x} - - (x, y, z). In particular, the initial observer so can always choose the axes t~, t~, t~ in such a way that, under a transcendent Zorentz trans/ormation (without rotations: "~DLLEtr 1962, p. 18-22, 45-46) 5 ~ ~Sf~, it is x--->t~, y-->t~, z--->t~, t~-->x, t~--->y,

1 3 2 ~. ~ c x ~ z

L -+ z, in agreement with the fact t ha t the ]ormal expression of ~ = i l (where now 1 is the six-dimensional ident i ty) is f i ldependeut of any space direction.

Moreover, if observer so, when aiming to perform a Superluminal boost along x~, rotates {t} so tha t tj -- t (axis t being his ordinary t ime axis: see sect. 4 and the following), then any t ranscendent boosts can be formally described to operate as in fig. 40b).

E(3) /

f

a)

Fig. 40.

J E(3)

J + ~tz=;z

b)

W h a t above means t ha t the imaginary uni t i can be regarded as a 90 ~ ro ta t ion operator also in Me; f rom the active point of view, e.g., i t carries x - - (x, y, z) ~- (t~, t~, tz) - - t. Here the meaning of i, for one and the same observer, is analogous to its meaning in SlY, where i t is used to distinguish the t ime from the space co-ordinates, which are orthogonal to t ime. There-

fore,

~ t ~- x (two-dimensional case),

(164) i ---- exp [i~/2] t ~- x (six-dimensional case).

~o t ice t ha t in ms the GLTs are actual ly (linear) $rans]ormations, and not

only mappings. W h a t precedes (see r e.g., eq. (164)) implies tha t

(165a) ds~ ~ ---- + ds~ for LT, ,

(165b) ds. '2 . . . . ds~ for SLT6,

with obvious meaning of the symbols. The GLTs, as always, can be considered ei ther f rom the active or f rom the passive point of view (in the la t ter case,


t hey will keep the 6-vector fixed and (( ro ta te ~ on the cont rary the six axes without changing- -no t ice - - the i r names during the (~ rotation O.

The subluminal LTs in Yle, to be reducible in four dimensions to the ordina ry ones in agreement with SR (ds: '~ = ~- ds~), must be confined to those t ha t call into p lay one t ime axis, let i t be t ~ t~, while t~ and t3 remain unchanged (or change in 1~6 only in such a way t ha t at'22 ~- dt'a 2 = dt~ + d 0 . As a consequence, because of eqs. (150), also the SLTs in M8 must comply with some constraints (gee MXCCAI~I~ONE and I~EcAsr[, 1984). For instanc% when the boost speed U tends to infinity, the axis t' ---t ' 1 tends to coincide with the

!

boost axis x~, and the axis x~ with the axis t, - - t.

As to the signature in ~e two al ternat ive conventions are available. The first one is this : we can paint in blue (red) the axes called tj (xj) by the initial observer so, and s ta te t ha t the blue (red) co-ordinate squares mus t always be t aken as positive (negative) for all observers, even when they are (~ ro ta ted )) so as to span the region initially spanned by the opposite-colour axes. Under such a convention, a t ranscendent SLT acts as follows:

(166)

I+!

4- +

~-I

dt~ --> dr,' :-= dz 7

J dt~ -+ dt~ = dy

I dt~ --> dt~' = dx

dx --> d x ' ~ dt~

dy -+ d y ' = d6

dz -~ d z ' - dt~ t-

(under ~ ) .

Notice tha t no imaginary units enter eqs. (166). The previous discussion on the action of o~' in Me was performed with such a metr ic choice.

The second possible convent ion (still wi thout changing the names - - l e t us repeat----of the axes tj, x~- during theh" ~, ro ta t ion ,~) would consist in adopt ing the opposite six-dimensional metric in the r.h.s, of eqs. (166); i t corresponds to changing the (, axis signatures ~ during their rota t ion:

(167)

/§

-i-

d6 -> i dt.' --- i dz 7

dry -~ i dt~ = i dy

[ dt~ ~ i dt~' ---- i dx _

dx --> i dx' ---- i dt~

dy - > i dy' = i dt~

dz --~ i dz' ---- i dtz

§

§

(under 3~).

Such a second convent ion implies the appearance of imaginary units (merely due, however, to the change of metr ic w.r.t, cqs. (166)).

134 ~. ~XCAMI

In any case, the axes called tj by the subluminal observer So, and considered by So as subtending a three- temporal space (t~, t~, tz) L (x, y, z), are regarded by the Superluminal observer S'~, and by any other S', as spatiM axes subtending

a three-spat ia l space; and vice versa. According to our second postulate (sect. 4) we have now to assume th a t so

has access only to a 4-dimcusionM slice M~ of Ms, When so describes bradyous B,

we have to assume M4 --- (tl ---- t; x, y, z), so t h a t the co-ordinates t~, t3 of any B are not observable for so. Wi th regard r SLTs we must , e.g., specify, f rom the passive point of view, which is the (( observabil i ty slice )) MI of M~ accessible to S r when he describes his own bradyons. By checking, e.g., eqs. (166) we reMize tha t only two choices are possible: ei ther i) M~4----(t~; x', y', z'), or ii) (t:, t:, x'). The first choice means ass ing that each while ro ta t ing carries with itself the p roper ty of being observable or unobservable, so tha t the axes observable for S' are the t.ransforms of the axes observable for So. The second choice, on the contrary, means assuming the observabi l i ty (or unobservabilit.y) of each axis to be established b y its position ill M6 (as judged by one and the same observer), so tha t two of the axes

(i.e. t~, t:) observable for S' are the t ransforms of two axes (i.e. t~, t/) unobservable for So. In o ther words, the first choice is M' 4 • Ms, while the second choice is M~ ~- ~a (in 518, when it is referred to one and the same observer). ~Totice tha t , roughly speaking, the above propert ies of the two choices get

reversed when passing to the active point of view. The ]irst choice does not lead automaticMly, f rom eqs. (165) in six dimensions,

to the ds~ invarianee (except for the sign) in four dimensions. I t moreover calls all six co-ordinates into play~ even in the case of sublnminal LTs obta ined through suitable chains of SLTs and LTs. This choice, therefore, could be adopted only when whishing to build np a ~ruly six-dimensional theory. The result- ing theory would predict the existence in M~ of a (( t achyon corridor ~) ~nd would violate the light speed invariance in M:: in such a seuse it would be

similar to Antippa 's (1975). The second choice, once assumed in M6 tha t ds~ ~ ---- -- ds~ for SLTs, does

lead al l tomatical ly also to ds~' ---- -- ds~ in four dimensions (M_acc~a~o:vE and ICEc~[r, 1984). Moreover, i t calls actual ly into play four co-ordinates only, in the sense tha t (of., e.g., eqs. (166)) it is enough to kimw initially the co-ordinates (t; x, y, z) in Ma in order to know finally the co-ordinates (t:, t ' , t',; x') in ~-~I~. We adopt the second choice since we want to t r y to go back from six to four dimensious, and since we like to have the light spced invariance preserved

in four dimensions even under SLTs. The (( square brackets )~ appearing in

eqs. (166), (167) just refer to such a choice. To go on, let us s ta r t by adopt ing also the s ignature--f i rs t c o n v e n t i o n ~

associated with eqs. (166). I f we consider in M~ a (tangent) 6-vector dv lying on the slice Md(t, ~ t; x, y, z), then a SLT- - r ega rded from the active point

of view--wil l ((rotate ~ dv into a vector dr ' lying on the slice M'4(t,, t , , t~; x):

CLASSICAL TAC~YO.~'S 135

see fig. 41. In other words, any SLT--as given by eqs. (150), (154)--leads from a bradyon B with observable co-ordinates in M(1, 3) -- (t; x, y, z) to a final tachyon T with ~( observable )~ co-ordinates in M'(3, 1) - - (t~, t2, t3; w), where the w-axis belongs to E(3) -- (x, y, z) and the t-axis belongs to E'(3) -_--- (t~, t2, t3): see fig. 40a). Formal ly : (1, 3) SLT (3, ] ). F rom t,he p a s s i v e point of view, the initial

/ l;

f

! t~zmZt

/ Fig. 41.

observer so has access, e.g., only *,o the slice (t= ~_ t; x , y , z), while the final observer S' (e.g., S'~) has access only to the slice (t~, t~, t " =, x'), so t ha t the co- ordinates t~, t, (and y', z') are not observable (see also POOLE et a l , 1980, and SoBczYK, ]983). Notice tha t x' comes from the (~rotation~> of the boost axis.

At this point two observations are in order: 1) Our second postulate (sect. 4 and subsect. 14"2) requires observer S' to regard his space-time (t', t~, t~'; x') as related to three space axes and one t ime axis; actually renaming them, e.g., ~i, ~'2, ~ and z', respectively. This consideration is the core of our interpretat ion, i.e. the basis ~or understanding how S' sees the tachyons T in his M't. 2) The principle of re lat ivi ty (sect. 4) requires tha t also so describe his tachyons (in M4) just as S' describes his tachyons (in M~); and vice versa. If we unders tand how S' sees his tachyo~m in M:, we can immediately go back to the initial M(1, 3) and forget about six dimensions.

In connection with M:, the effect of a Snperluminal boosts along x will be

136 ~,:. ~ c A m

the following:

(]68)

l+i

dx -- U dt dx -~ d x ' = ~: ~ /U 2 _ 1

dt - - U dx dt~ -.~- dt" = T - - -

x / U ~ - ]

!

d t , -.~" d t v ~- 4- d y

dt~ --7 dt~ .-~ 4. dz

( U ~ l / u , u 2 < 1 , U " > ] ) .

I n eqs. (]68) no imag inar i e s appear . :Bu~ our signature choice (166) implies

t h a t S ' - - f r o m the metr ic point of view, since he uses the s ignature (-[- + + - - ) -

deals with t'j as if t hey were actual ly t ime components , and with x ' as if i t were

actlla]ly a space component . ,

We migh t say, as expected, t h a t a t achyon T will appea r in M 4 to S' (and,

t&erefore, also to so in M,) as described b y the s~me set of co-ordinutes describ-

ing a b r adyon B, provided t h a t three out of those co-ordina~es are regarded as t ime co-ordinates, and only one as a space co-ordinate. Since we do not

unders tand the meuning of such a s t a t emen t , we m a y seek recourse to some formal procedure so to deal eventual ly (at least formally , (~ apparen t ly ~)) with

one t i m e and three space co-ordinates; we c~n hope to under s t and a poster ior i

such a meaning, via the l a t t e r choice (see, e.g., MXG~A_~I and t~ECAm~ 1974e, and MACCAR~O~E et al. , 1983). One of the possible procedures is the following.

Le t us change the signature choice, b y passing f rom eqs. (166) to eqs. (167),

in such a w a y t h a t both (subseet. 4"3.6) so and S' use the signature ( + -- - - - - ) , as if S' too dealt wi th one t ime and three space co-ordinates. Wi~h the choice

(167), eqs. (168) t r ans form into

(168')

t j

dx -- U dt d x - - T i d x ' - =7i - - , : _ - - _ -

V U 2 - 1

dt - U dx dt~ -~ i dt'~ - - ~ i ~ / U 2 - 1

dt~ - 7 i dr" = q- i dy

d t z - ~ i dt'~ -~. 4 . i d z

'+t

where now (~ imag ina ry units ~) do appear , which correspond to the met r ic

change (166) ~- (]67). Equa t ions (168') are, of course, equivalelt t to eqs. (]68).

Equa t ions (168'), and, therefore, eqs. (]68), coincide wi th our eqs. (].54 bis),

C L A S S I C A L TACI-IYONS 137

provided tha t the second one of eqs. (164) is applied to the vector (itr, itS, it', ix'). See the following.

14"8. ~ormal expression o] the Superluminal boosts: the first step in their interpretation. - We reached the point at which to a t t empt at interpreting eqs. (154). At the end of the last subsect. 14"7, we just saw how to t ransform eqs. (168') into eqs. (154 bis). The result has been the same as got in an (~ automatic )) way in subsect. 14"3.

This is a first step in the interpretat ion of SLTs. But we shall have to deal also with the imaginaries remained in the last two of eqs. (154), or of eqs. (168r).

The first two equations in (168r)--in fac t - -a re true transformations carrying a couple of co-ordinates (t, x) belonging to the initial observability slice into a couple of co-ordinates (t', x') belonging to the final observability slice. In other words, t' and x' come from the (( rotat ion ~> of x and t, such a rotat ion taking always place inside both the observability slices oi so and S'. We can just eliminate the i 's on both sides, get t ing the reinterpreted equations (39'), (39") of subseet. 3"6.

On the contrary, the co-ordinates t~', t : - - t ha t S' must interpret as his transverse space co-ordinates ~:, ~ - - a r e the transforms of the initial co-ordinates t~, t~ (unobservable for So), and not of the initial co-ordinates y, z. Precisely, the axes ~:, ~'s derive by applying to the axes t~, t, a 90~ rotat ion ~) which takes place in Ms o~tside the observability slices of so and S'. As a consequence, in the first two equations in (168') we have to subst i tute dz' for idt : and dy' for i dt: so tha t

( E ) dz' = J= idz (Superluminal x-boost). dy' : 4- i dy

The i 's remain here; in fact, the co-ordinates y', z' (regarded as spatial by S') are considered as temporal by So.

Notice tha t , from the active point of view, M 4 and M' 4 intersect each other in Ms jnst (and only) along the plane (x, t) = (t', x'): see fig. 40, 41.

Equat ions (168') have been thus t ransformed into eqs. (154 bis). While eqs. (150) or (154) for U---> oo (transcendent SLT) yield

(169) =LSf: d t ' = ~=idt, d x ' = J=idx, d y ' = 4 - idy , d z '= ~=idz,

in agreement with the fact tha t the formal expression of 5f = il is direction independent, after the (partial) reinterpretations of eqs. (154) into eqs. (154 bis) we get tha t the t ranscendent 8LT along x acts as follows:

d t ' = 4- dx, d x ' = =L dt, d y ' = 4- idy , d z ' : 4- i d z .

In this case, in fact, the reinterpretat ion follows by regarding i as a 90 ~ rots-

138 ~. R~CA~I

t ion operator in the complex plane (x, t) ~ (t', x'), and not in the planes (y, t) or (z, t). Consequently, even if all t ranscendent SLTs (without rotat ions) 5 c are formal ly identical, they will di]fer f rom one another after the reinterpretation.

More details on this (( in terpre ta t ion first step >> can be found in MACCAR- ~O~E and RECA~X (1984, sect. 7). We want to stress explicit ly tha t the reinter- p re ta t ion is a (~ local >> phenomenon, in the sense tha t it clarifies how each observer S' renames the axes and~ therefore, (~ physically interprets ~> his own observations. The in terpreta t ion procedure, thus, is f rame dependent in E R and breaks the gener~Iized Lorentz inv~ri~nce. Equat ions (154), e.g., do form the group G together with the LTs; bu t the (~ part ial ly in terpre ted >) equations (154 bis) do n o t . Moreover, the reinterpretation (when necessary) has to be applied only at the end o] any possible chain o] GLTs: to act differently would mean

(besides the others) to use diverse s ignatures-- in our sense--dar ing the pro- cedare~ and this is illegal. (~otice once m o r e tha t the re interpreta t ion we are discussing in sect. 14 has nothing to do with the Stfiekelberg-Feynman-Sudarsh~n

(( switching procedure ~), also known as (~ re interpreta t ion principle ~>.)

14"9. The second step (i.e. preliminary considerations on the imaginary transverse components). - In subsect. 14"3 and 14"7, 14"8 we have seen how to in te rpre t the first two equations in (154), so to pass to eqs. (154 bis). We are left with the need for a second step in the in terpre ta t ion of SLTs, to under- s tand the geometrico-physical meaning of the last t w o equat ions in (154) or in (168').

How to perform this second step has been already discussed in subsect. 14"6, when answering the Einstein problem. Namely, when applying a SLT in the chronotopical space, the presence of the (( i 's )) in the transverse components causes the shape of a t achyon (e.g., intrinsically spherical) to appear essentially as in fig. 19d), 18 and 17 (see subsect. 8"2 and 14"6). To be honest , we know how to inr the last two equations in (154) only in some relevant cases (cf. subsect. 14"6). This is a ]~roblem still open in pa r t ; we want a t least to clarify and formalize tha t re in terpre ta t ion procedure at oar best. This wiU be

accomplished in the nex t subsect. 14"10, for a generic SLT. Here let us make a comment . The <( Lorentz )> mappings (154)--af ter their

i n t e rp re t a t i on - -do not seem to carry one any longer outside the initial Minkowski

space-t ime M 4, Only for this . reason we always used the convention of Calling

jus t (( t ransformations >) the SLTs (a use well justified in two, or six, dimensions), even if in four dimensions they seem to t ransform manifolds into manifolds, r a the r t ha n points into points; in this respect, the critical comments in subsect. 8"3 ought to be a t tent ive ly reconsidered (see also S ~ z , 1984).

14"10. The case o/the generic S]~Ts. - Le t us ex tend the whole interpreta-

t ion procedure (of the whole set of four equations const i tu t ing a SZT) to the


case of a generic SLT <(without rotat ions )> (]V[OLLE]~, 1962), i.e. of a Super- luminal boost L(U) along ~ generic mot ion line 1. In terms of the ordinary co-ordinates x~', according to eqs. (150) we shall have (u'i If, u ~ l /U , u 2 < 1, U 2 > 1)

,,69, ) - - u 7 n* (~: - - (7- - : l )n~n , ' Y ~ (1 -- u-)-~,

where L(u) is the dual (sublnminal) boost along the same 1. Quant i ty n is t,he uni t vector individnat ing l: n n ~ = - i = - [np; it points in the (con- ventionally) positive direction along /. N~otice tha t u, U m a y be positive or negative. Equat ions (169) express JL(U) in its formal, (~ original ~) form, still to be interpreted.

Z(U; x,) can be obtained from the corresponding Superluminal boost

along x through suitable rotat ions (Zo(x, U ) = iZo(x,u), Lo(X, V) -B(x) r, s = 1, 2, 3) :

(17o) L(U; x~') = R-~ B ( x ) R ,

A ~ (.1 § n~)- ' ,

nx n~ nz

- - n~ 1 - A n ~ - - A n ~ n~

- - ~ - - A n ~ n . 1 - - A n a l

where B(x) is giv(m by eqs. (154-). Till now we dealt with the in te rpre ta t ion of eqs. (150) only in the case of Sllperluminal boosts along a Cartesian axis. To intert)ret , now, also the J~(U; x~) of eqs. (169), let ~s compare L(U) with ~(U) , where

(171) L(U; x~) = R-XB(x )R

and B(x) is the (])artially) re in terpre ted version of eqs. (151), i.e. is given by eqs. (154 bis).

From eqs. (171) and (15t his) we get (.~IACCA~RONE et a l , 1983)

i U ~ '~.~ " i ~ [ - i (1 -~- ~)n%~,] (U ~ I /u, r, s = 1, 2, 3),

where ~ ~ (U 2 - ])-~ with U :-= 1/u, u 2 < 1 , U 2 > 1 . Eqllat ion (172a) can also be writtelt

(172b) L(U;x~ , )= : t : ( - - u Y - - Y % ) yn" i6~ -[- (i -[- uy) n~n~ '

1 4 0 r . RECAMI

where 7 is defined in eqs. (169), wi th [u[ < 1. Notice explici t ly t h a t the four- dimensional SLTs in thei r original m a t h e m a t i c a l f o rm are a lways purely imaginary; this holds in par t icular for a generic SLT (~ without ro ta t ions >).

I t will seem to contain complex quantities only in its (partially) reinterpreted ]orm. But this is a (( local ~> fac t relat ive to the ]inal f rame, and due to a t r iv ia l effect of the re levant space ro ta t ions: its in te rpre ta t ion is pa r t l y re la ted to fig. 42 (in the following).

Le t us also recall t h a t in the case of a chain of GLTs the in te rpre ta t ion

procedure is to be applied only at the end o] the chain (the re in terpre ta t ion , being f r ame dependent , breaks the Lorentz invarian~.e).

We have jus t to compare the m a t r i x in eq. (172) with the ma t r i x in eq. ([69),

including in it i ts imaginary coeffmient, in order to get the in te rpre ta t ion of

eqs. (169). Such a re in te rpre ta t ion will proceed, as usual, in two steps; the fix'st

consisting now in the in te rpre ta t ion of the t ime co-ordinate and of the space

co-ordinate along 5; the second one consisting in the in te rpre ta t ion of the imagin-

a ry space co-ordinates transverse to 1. For instance, let us compare eq. (169)

with eq. (172b), apa r t f rom their double signs:

[ dt . . . . i yd t 4- iyunsdx*, (169)

l dx '~ := - - iuyn" dt 4- i5~ dx" -- i(y - - 1) n ~ n, dx*;

dt ' = - - u ? d t - - ?n, dx*,

(172b) dx '~ - - - $n'dt 4- i~:dx '4- (u 7 4- i )n 'n~dx ' .

J~irst step. Recipe: You can eliminate the imaginary unit in all the addenda containing ~ as a

multiplier, provided that you substitute t ]or r, and r, /or t (notice t h a t rj, =

= / ' ' n - - - - - - n s X S ) .

gecond step. I n the second equat ions in (169) and (172b) if we p u t r = x (xyz) and r' -- x' -- (x 'y 'z ' ) , we can write r = r , 4- r• where r, -- (r , )n

and r• = r - r~, n---- r - ( r . n )n . Then eq. (172b) can be wr i t ten in integral

r ! = ! / fo rm as r, , r• 9r d- ir• a n d - - a f t e r hav ing applied the !

(~ first step ~> rec ipe - -we are left only with r x = i r . , i.e. only with the imag ina ry

t e rms (not containing ? as a mult ipl ier)

(1.73) d ( r~ ) ' = i d ( r • ---- i(5~ 4- n~n~)dx s ,

which enter only the expression dx 'r. (Of course, rz is a space vector or thogonal to I and, therefore, corresponds to two fu r the r co-ordinates only). Since eqs. (173) refer to the space co-ordinates or thogonal to the boost direction,

the i r imaginary (( signs ~) have to be in te rpre ted so as wc did (fig. 19) in subscct.

14"6 (and 14"9) for the t ransverse co-ordinates y ' and z' in the case of Super-

luminal x-boosts: see fig. 42.

C L A S S I C A L T A C I I Y O N S

Fig. 42.

Y ~')'/

x

141

This means tha t , if the considered SLT is applied to a body P~ initially at rest (e.g., spherical in it,s rest frame), we shall finally obtMn a body P'r moving along the mot,ion line 1 with Superluminal speed V : U, such a body P'r being no longer spherical or ellipsoidal in shape, bu t appearing on the cont rary as confined between a t 'wo-sheeted hyperboloid and a double cone, bo th having as s y m m e t r y axis the boost motion line 1. Figure 42 refers to the case in which P~ is intrfllsicMly sphericM; and the double-cone semi-~ngle ~ is given by tg ~----

--= (V 2 - 1) -�89 More in generM, the axis of the tachyon shape will not coincide wi th 1 (but will depend on the t achyon speed V = U).

/ More precisely, the vector r• apar t from its imaginary (~ sign )),--i.e. the

vector r • be described iby the two co-ordinates ,~-(') ~ Yo, r~(~) ,= Z o just as in subsect. 14"6 and 14"9: see fig. 38 and 42.

We see once more tha t this re in terpre ta t ion ((second step ~) works only in par t icular special cases. To clarify a bi t more the present., situation, MACO~- RO~E et al. (1983) emphasized the following points: i) one is not supposed to consider (and reinterpret) the GLTs when they are applied just to a ((vacuum point ~); actually, we know f~'om St~ t ha t each observer has a r ight to consider the vacuum as at rest w.r.t, himself; ii) one should then a p p l y - - a n d eventual ly reinterprel~--the GLTs, in part,icular the SLTs, only to t ransform the space-

t ime regions associated with physical objects; these are considered as extended

objects (Ks 1978), the pointlike situation being regarded only as a l imiting

case; iii) the ex tended- type object is referred to a f rame with the space origin in its centre of symmet ry .

Many problems remMn still open, therefore, in connection with such a (~ second step ~> of the in terpre ta t ion (cf. subsect. 14"]4-14"16).

14"11. Preliminaries on the velocity composition problem. - Let us apply a SLT in the form (172a) along the generic motion line 1 with SuperluminM

142 ~. s~eA~x

speed U --= 1 /u (U ~ > 1, u ~ < 1) to a b radyon P~ having initial fom'-velocity u , and veloci ty v. Again, one should pay a t ten t ion to not confuse the boost speeds u, U with the four-velocity components u*, of PB- For the purpose of generali ty, v and U should not be parallel. We get

{ u '~ ~ - - f ( u ~ U % ) , (174) l u " ~- - p ( u ~ - U u ~ ~ § i u ~ ,

r ~ U r U r - - where u R - ~ - - u ~ n , , u . + u ~ n ~ u ~ - ~ uj~n ~, and n is still the uni t /

vector along l, while y ---- (U ~ - 1) - i so as in eq. (172a). ~o t iee tha t u o is real, l I$

while the second equat ion in (174) can be rewri t ten as (u N = - u n~)

J u~ = - f ( u , - tTuO).- - ( u , , - ~uO)/Vu ~ - 1 (175) , (US> 1), t

I I I / where u u is real too and only u• is purely imaginary; u , , u,, (u j , u• are the longitudinal (transverse) components w.r.t, the boost direction.

I f we define the 3-velocity V' for tachyons in terms of the 4-velocity u~

( j - - - -1 ,2 ,3 ) :

V 'J 1 (176) u'J--V=--;=---- "~ :l u'~ V'-V '~ . . . . . ~ 1 '

eqs. {175) yield

(177)

, U - - v , 1 - - uv~

v', = i v~ V u ~ - 1 _ iv • V i - u~ Uvn - - I v H - - u

( U ~ 1, u 2 ~ 1, U - ~ I / u ) .

' = 1/~ u ' where ~ is tile t ransform I t m a y be noticed tha t V, , V• = i~l[~,~,

of v under the dua l (subluminal) Lorentz t ransformat ion L ( u ) with u = 1 ] U , / I u I] U. Again V~ is real and V~ pure imaginary. However , V '~ is a l w a y s positive

so t h a t IV'l is real and even more Superluminal ; in fact,

(17s) z / 2 v '~ v ' ,~+ v ? = v,~ - I v l t ~ > 1 .

More in general, eqs. (177) yield for the magni tudes

(179) 1 - V '~ - - - ( 1 - - v ~ U 2) ( 1 - - U . v ) ~

( v 2 ~ l , U 2 , V ' 2 ~ 1 ) ,

which, incidentally, is a G-eovariant relation. Le t us recall t h a t eqs. (174), (175) and (177) have been derived from the (partially) reinterpreted form of

CLaSSiCaL TAC~u 143

the SLTs ; therefore, they do not possess group-theoretical properties any longer. For instance, eqs. (177) cannot be applied when transforming (under a SLT) a speed initially Superlnminal.

Eqnat ion (179) shows tha t under a SLT a bradyonic speed v goes into a tachyonic speed V'. But we have still to discuss the fact t ha t the tachyon 3-velocity components transverse to the SLT motion line are imaginary (see the second equation in (177)).

We shall proceed in analogy with subsect. 14"6 and 14"10. Le t us initially consider, in its c.m. frame, a spherical object with centre at O, whose external surface expands in t ime for t > 0 (<( symmetrical ly exploding spherical bomb ~)):

(180) 0 < x ~ ~- y~ q~ z ~ < ( R Jr vt) ~ ( t>0) ,

where R and v are fixed quantities. In Lorentz-invariant form (ior the subluminal observers), the equation of the (~ bomb ~) world-cone is (MACC~I~I~O~E et al., 1983)

(180') ! (x~ ~- bz)(x" -~ b') < [(x~ -~u~ ub')~ uz] 2

I x~>0,

<(1- -v2) -~ (x~ ~- b~)(x" -~ b~),

where x, ~ (t, x, y, z) is the generic event inside the (truncated) world-cone, vector ug is the ((bomb)~ centre-of-mass four-velocity and b~ ~ u , R / v . One

can pass to Superlnminal observers S' just recalling tha t (subsect. 8"2) the SLTs invert the quadratic-form sign (eL, however, also subsect. 8"3). I f S' just moves ~long the x-axis with Snperlnminal speed -- U, the first l imiting equali ty in eq. (180') transforms, as usual, i'nto the equation of a double cone symmetrical w.r.t, the x-sxis and travelling with speed V ~- U along the axis x ~ x'. The second inequali ty in eq. (180') transforms on the contrary into the equation

(181) (1-- v ~ V~)x ' 2 - ( V ~ - l ) ( y '~ ~-z ' 2 ) - 2 t ' V ( 1 - v~)x ' - 2 R v V % / ~ - ] x ' <

< ( V ~ - - I ) R ~ - - 2 t ' R v V / V 2 - - 1 -- ( V ~ - - v 2 ) t '2 ( x ' > t ' / V ) .

When it is v V < 1, the equali ty sign in eq. (181) corresponds to a two-sheeted hyperboloid whose position relative to the double cone does change with t ime (fig. 43). The distance between the two hyperboloid vertices, e.g., reads V ~ - -- V1 ---- 2(1-- v 2 V~) -~ [ t ' v ( V 2 - 1) Jr R v / V ~ / ] ~ / . When in eq. (181) it is v V > 1,

the geometrical situation gets more complicated. But, in any case, the (( bomb ~ is seen by the Snperlaminal observers to

explode remaining always confined within the double cone. This means tha t i) as seen by the snbhmina l observers s a (bradyonic)

bomb explodes in all space directions, sending its (( fragments ~> also--e.g.-- along the y and z axes with speeds v~ and v~, respectively; ii) as seen by the Super-

144

Fig. 43.

1~. RECAMI

yt

luminal observers S', however, the (tachyonic) bomb looks to explode in two

~ jets ~ which remain con]ined wi th in the double cone, in such a way tha t no fragments ~) move along the y' or z' axis. In other words, the speeds V~, V:

of the tachyonic-bomb fragments (~ moving )~ along the y', z' ~xes, respectively, would result to be imaginary (_MAcc~J~o~. et at., 1983; see also Co~]~E_~, 197~, 1975).

14"12. Tachyon lout-velocity. - Let us refer, for the particular case of Super- luminal x-boosts in four dimensions, to eqs. (154.) ~nd (154. his). Let us recM1 tha t in this particular case the SLTs--a.fter theft' part ial in te rpre ta t ion-- coincide with the ones proposed by MIG~A~I and REC~MX (review I).

We want to reconsider ab ini t io the problem of introducing the 3- and 4.- velocity vectors for tuchyous.

In agreement with cqs. (150), we have seen tha t , if a subluminal LT carries from the (~ rest frame )) so to a frame s endowed with velocity u relative to so, then the dual SLT must carry from so to the frame S' endowed with velocity U~ -~ qd/u ~, U~ : u , /u ~, U~ : u~/u ~, such tha t U "2 : 1 /u 2. By referring to the auxiliary space-time M6 and ~o the names attributed to the axes by the ini t ial

observer s, the second observer S' is expected to define the 3-velocity of the observed object as follows (subsect. 14"6):

V~ - d ~ ' F~' -- d~ ' V: ~ d~ (Superluminul boost),

where the tilde indicates the tr-~nsformation accomplished by the dual, subluminal LT (actually, d~ = dt~ and d~ = dt~); the tilde disappears when

CLASSICAL TACttYO:NS 14~

the considered SLT is a t ranscendent Lorentz boost : V: d t J d w , V'~ ~ dt~/d~, V : ~ d t J d x . However , due to our postula tes (sect. 4), S' in his te rminology will, of course, define the 3-velocity of the observed t aehyon in the ordinary w a y

dx' dy ~ , dz' (183) V ' ~ dt ' V s dt ' g , ~ d-7'

where dx'~ dy' , d z ' are a p r i o r i given b y eqs. (]54).

Iden t i fy ing eqs. (183) wi th (182), on the basis of eqs. (154 bis) we get (see fig. 44)

dx' dt dy ' d?~ dz' d~ (184) F: = dr' - - dg-' V~, = dt ' - - d 2 ' V; = dt ' - - d 2 '

Fig. 44.

! /

)\ \

e(3) I._ ,

[E; (3) /

where, in the present case, d?~ z d y , d5 = dz. Namely , apa r t from the signs, the SLTs yield the final relat ions (dG ~ dt)

( 1 8 4 ' )

d x ' d t - - u d x d y ' _ i d y V / 1 - u 2

V : ---- d t ' d x - - u d t ' V'J = d t ' d x - - u d t '

dz' . dz ~ 1 - - u ~ V ; = - -

d t ' ~ d x - - u d t '

relat ing the observat ions made by s on Ps with the observat ions m~de b y S'

on PT ( t ransform of PB under the considered Superlumina] boost). The imagin-

a ry units in the t ransverse components mean a p r i o r i t h a t the t achyon P~

146 ~. ~xcAm

moves, w.r.t. S', with velocity V' in the ~4 space-time (see the following). l~rom eqs. (184') one immediately sees that

( ] s 5 ) v ' , ~ , = 1

and, in particular, V'~v~ = 1 when SLT = 5 p. Notice, therefore, that the dual correspondence V ' ~ - c~/v holds only for the velocity components along the SLT direction; that correspondence does not hold for the transverse components (even if V~ ~ v~ and V] =/= %), nor for the magnitudes V' and g. In fact (v ~- ]v], v 2, u 2 < 1),

(186)

1 - - u v ~ 1 i % ~ / ~ - - u 2 .~ V: ~ - - ~ v , ' V~ ~m - ~

v~ iv~ V ~ - - u s . ~o

cf. also subsect. 14"11. That is to say, the transverse components V:, V: are connected with the longitudinal compouents V'~ in the same way as in the ordinary subMminal SI~ (M~ccA~o~E and l~xc~!v~, 1984). Equations (186), as well as eqs. (158'), confirm that eqs. (154) are actually associated with Super- luminal motion, notwithstanding their appearance. Equations (186) can be written, in terms of the Superluminal-boost speed, as (US> 1)

U - v~ i v , v ' U 2 - ] iv~ V ' U ~ - 1 , (186') V'~ - - Uv------~ , V~ - - Uv , - - 1 ' V'~ = U~v~--

which express the velocity composition law in the case of Superluminal boosts. Let us stress, again (see eqs. (178) and subsect. 14"11), that from eqs. (186),

(186') one can verify that always

(186 rl) V r2 > 1

even if V ~ < 0 and V',2<0, so that 1 < V ' 2 < V '~ . This means that V ' - - ]V ' [ is always real and Superluminal. See also eq. (179).

In terms of ]our-velocities, the composition of a subluminal generic four- velocity v~ with a Superlumina] x-boost four-velocity U~ will yield

(187)

V'o = V~Uo § vou~ - - - (v~ U~ + % Up)

V~,8 = iv2,~

(SuperluminM boost, v ~ v ~ = +I, U ~ , U ~ : - I ) ,

which do coincide with eqs. (186'). The x-boost Superluminal speed is --U,

CLASSICAL TACKYOI~8 147

with U ~ - l / u . Let us repeat tha t eqs. (186), (187) should not be applied, when start ing from a Superluminal speed Iv I > 1, since applying the (partial) interpretat ion breaks the group properties.

We shall come back to the problem of the imuginaries in the transverse components of eqs. (186), (186') in subseet. 14"15.

14"13. Tachyon ]our-momentum. - Let us apply the SLTs to the four- momentum, defined in a G-covariant w~y as follows:

(188) p , ~- m o v , , v , =-- dx~/dro (v~X1).

Then p~ is a G-vector, ~nd we can apply to it eqs. (154), oi' (154 bis). The lat ter yield, for the tachyon four-momentum (obtained by applying a Snperluminal boost along x to a bradyon B with 3-velocity v, Ivl--~ v < 1),

- - P o - Up1 p~ = • p~ Up o = T V" l - u~ V ~ : - - 1

_ _ p ~ - Ypo Po -- up~ T (189) P~ = • V ~ - u ~ ~ / ~ i

wherefrom, among the others, p'~,a = • imoV2, a = • i m o v ~ , j V / ~ = • imv~,,.

Do not confuse the four-velocity components v~,3 with the three-velocity components v~.~ ; and so on. At tent ion must be paid, moreover, to the fact t ha t v, v~ refer to the initial bradyon (in the initial frame), while U and its dual velocity u refer to the SLT.

Equations (189) can be rewrit ten (MAOOA~O~E and I~EOA~, 1984) as

V~ - - u 1 - U v ~ p~o = :J: m - - T i n - -

1 - uv~ p~' = • m _ _ v 1 U 2 x / Y~ -- l '

p~ ~ m o V~ = • i m v v = • imov2,

P'3 -~ mo V~ : • imv~ = :J: imp vs .

~otice tha t , even if these equations express the four-momentum of the final taehyon T ~ PT, nevertheless m and v~, G, v, represent the relativistic mass and the 3-velocity components of the initial bradyon B ~ Pr (in the initial frame), respectively; in particular,

m o m - - - - (v 2 ~ v ~ < 1).

148 ].:. RECAMI

By comparing eqs. (189) with the velocity composition law (186)-(186'), i t follows even ]or tachyons t ha t

(190)

mo _ mo V', p o - - V ~ l , P; VV2=-i

p , _ moV; p~__ ~oY~' ( V '2 "= V ' ~ > 1 ) .

Since V' ~nd V' ~ are imaginary, V'2 and V 3 axe imaginary a.s well, in a.gTeement ! /

with the relations V~ =- -J: ivo., V~ - - • ivy.

I~'inally, comparing eq s. (190) with (] 88), one derives tha t even in the tachyon

case the 4-velocity and the 3-velocity are connected as follows:

(1.91)

1 v', Vo = VTa__ i ' v ; - - ~ / ~ ,

G , V'. v'~ = ~ / V ~ - - i ' G - V ~ - - ~ '

where V ' ~ V'L In conclusion, equations (188)-(1.91) tha t we derived in the taehyonic case from eqs. (154 bis) are self-consistent and const i tute ~ natura.1 extension of the corresponding subluminM formulae. For instance, it holds in G-covariant form

Since %, like x , and p#, is a G-vector, wc m a y apply the SLTs directly to v~. By applying a Superluminal boost, one gets

(:t92)

Vo-- Uvl v l - Uvo v~ = ~: V V ~ I ' v~ = ~: V ~ 2 . ~

' ' ivy ,3 • i %'~ V 2 ) 3 = --:-

V1 -- v -~

(U~> 1~ v : < 1).

14"14. I s l inearity strictly necessary ? - We might have expected tha t tra.ns- /

formations Y : x , -> x~ mapping points of M 4 into points of M4 (in such a

way tha t ds ~ - + - ds ~) did not exist. Otherwise real linear S~Ts: dx, -+ dx~ of the t angen t vector space associated with the original ma.nifold map f should h.~ve existed (I~Im)~,m, 1966; S_m~z, 1984). But we saw, already at the end os subseet. 3"2, t ha t real linear SLTs (meeting the requirements ii)-iv) of

subsect. 4"2) do not exist in four dimensions. Oil the contrary, the results in subsect. 8"2, as well as in subsect. 14"6 and

14"11, seem to show tha t in the Supcrluminal case in M 4 we have to deal with

CLASSmAL TAC~YONS 149

mappings tha t t ransform manifolds into manifolds (e.g., points into cones). In d ' subscct. 8"3 we inferred the SLTs: dx, --, x, to be linear b~tt not real; just

as we found in the present sect. 14. We may, h o w e v e r ~ a n d perha.ps more soundly-- , ma.ke recourse to non-

linear (but real) SLTs. d ' real but not linear~ then Supcrhminal I f we consider SLTs: d x - ~ x~,

maps 3-': M~ --~ .'~I~ (carrying points into points) do not exists. We '~h'eady realized this. The impor tant point, in this case, is tha t the (( Supcrluminal mappings >)J- (transforming then manifolds idle manifolds) be compatible with the postulates of SI~ ; in particular (subsect. 4"2), i) trans]orm inertial motion into inertial motion, ii) preserve space isotropy (and homogeneity) , iii) preserve the light speed invariance.

To meet the group-theoretical requirements, we have to stick to eqs. (154) and to their integral form. But their reinterpretat ion--accomplished in this sect. 14 and anticip~tcd in sect. 8---does comply with conditions i)-iii) above. For example., i t leads from a pointlike bradyon moving with constant velocity to a tachyon spatially extended, bu t still travelling with constant velocity. The problem is now to look for real, nonlinear SLTs (i.e. mappings of the tangent vevtor space), and substitute them for the linear nonreal eqs. (154 bis); with the hope tha t the new (nonlinear) SLTs can yield more rigorously the same results met before, thus solving the problems lef~ open by the previously << second step )> reinterpretation. For a discussion of such topics see also Sm~z (1984).

14"15. A n at tempt . - A tentat ive approach to real~ nonlinear SLTs can be suggested by investigating the diffimflty mentioned at the end of subsect. 14"12 (i.e. the still present difficulty of the imaginaries in the transverse components of eqs. (186), (186')).

The 3-velocity W' of the tachyon centre of mass, i.e. of the vertex of the (< enveloping con(; )) ~ (fig. 18 and 42), must be real, in any case. For examplc (see subsect. 14"12), in the trivial case in which v~ = v, = 0, i t is simply W' = W ' =

- - gO

:= V:----V'. ~o re generally, when concerned with the overall velocity W' of the considered tachyon T, the imaginaries in the transverse compon(mts essentially record the already mentioned fact that , by composing U with v, one gets a velocity V' whose magnitude V' is smaller t han V'gO (subsect. 14"12). In the particular case when U and v are directed along x and y, respectively, and lvl << 1, one may conclude tha t (fig. 45)

-gty ' - w ; g --1 (19a) rw' ] = v i e ' - v , - v 2, = w-% = '

which yield also the direction ~ of W' (]~IACC~O~E and I ~ E c ~ , 1984). Notice tha t W " I W ' I c o s t and W~--= ltV'} s ina , bu t W',=# V', and W ~ # g~.

1 5 0 ~ . R~CAMI

Fig. 45.

B

The second equation in (193) can be obtained from the following intui t ive ~nalysis. Le t as recall what seen in subsect. 8"2 and 14"6 for an (intrinsically) spherical object P, initially at rest w.r.t, a certain frame so and with its centre C at the space origin 0 of so. When travelling along x with Superluminal speed

I W ' [ = w ' ' --~ W= it will appear to so as in fig. 19d) (where for simplicity only the plane (x, y) is shown). I t is trivial to extend the previous picture by requiring that , when C • 0, for instance C -- (0, 9), the shape of P will be obtained by shifting the shape in fig. 19d) along y by the quant i ty Ay ---- 9 (if the laboratory containing P travels again with speed W' parallel to the x-axis).

I f P is now supposed to move slowly along y in the laboratory~ and the laboratory travels parallel to x with speed W' w.r.t, so, it is sensible to expect tha t P will appear to so with a shape still similar to fig. 19d), bu t travelling along a (real) line inclined w.r.t, the x-axis by an anglo ~. I t is what we showed in subsect. 14"10: see fig. 42.

The <~ reinterpretat ion )) of the cone vertex velocity (i.e. of the overall tachyon velocity) suggested by the previous intnit ive remarks is, then, the one shown in fig. 45, where we consider for simplicity W: ---- V: ---- 0. Recall t ha t the magnitude of the tachyon overall wdocity is W' -- I W'] : W'~ "~ + W~ 2 = = V~ ' 2 - ]V~] 2, since V: = i%v/T f i - -1 / (Uv= - 1) is imaginary. According to the interpretat ion here proposed for the velocity transverse components, the direction of W' is given by tg a -- W~/W'= = (V~/i)/V'~ (see eq. (188)).

14"16. Real, nonlinear SLTs: a tentative proposal. - Tile interpretat ion proposed in the previous subsect. 14"15 has been shown by M_ACC~RONE to correspond to the real, nonlinear transformations (]IV' I ~--IV'I)

~ 2

A = [ ( 1 - - - + +

C L A S S I C A L TACI~YONS 151

where (subsect. 14"12) ~ is given by the dual~ subluminM Lorentz tra.nsformation (v ~, u ~ < 1)

V~ -- U Vu V 1 - - u 2 V~ ~/1 -- U 2 - ~ - , ~ - ( 1 ~ 1 < 1 ) -

In terms of the 4-velocity, eqs. (194) can be writ ten as (cf. cqs. (191))

(195)

{ w~- ~ A ~ , w ; = ~ I _ ~ = B ,

w~ = ~ B , W'~ = ~ B

17 2 ( w ~ w ' , = - 1 , _= ~ ' ) .

Equations (195) led to the real

d t t =

(196) dy' :

should then hold for all tangent vectors. We are, therefore, l

SLTs : dxg -+ dxz :

~ . ~ ( 1 - - v2) / (1-- ~ ) d t , dx'--~ B V / 1 - - v ~ d t =- C d t ,

~ C dt , dz' ~ ~,C dt ,

which are nonlinear, bu t carry ds 2 - + - - d s ~, t ransform inertiM motion into inertial motion, and preserve space isotropy (and homogeneity) since they do not explicitly depend on the space-time position nor on any p~rticular sp~ee direction. Notice, moreover, tha t alto ---- dt' V ' ~ ---- dt' ~/(1 -- ~ ) / ~ .

Since any kind of real, nonlinear SLTs, so as eqs. (196), consti tute a reinterpretation of eqs. (154), we do not expect them to possess group-theoretical properties (which still seem to be possessed only by SLTs in their mathematical, formal expression (154)).

14"17. Further remarks. - Let us recall here the following further points:

i) At the beginning of subsect. 14"7 we mentioned the possibility of introducing ab ini t io a complex space-time.

if) At the end of snbsect. 13"8 we stressed the possible role of qnaternions in the description of tachyons (see also SOIISEK, 1981 ; MIGNA~I, 1978; EDMONDS,

1978), aS well as of more general Clifford algebras (see also the beginning of sect. 14).

iii) KXL~AY (1978, 1980; KIL~Au and TOLEDO, 1967) showed in par- ticulur how to describe the four-position of extended-type objects (cf , e.g.,

SANTILLI, 1983) by complex numbers (see also OL~gOVSKY and I%ECA~, 1970c). According to tha t author, genuine physical information goes lost when physics is exclusively constrained to reM variables.

iv) Fur ther considerations on the issues of this sect. 14 can be found (besides in the quoted li terature MACCARI%ONE et al., 1983; MACCAI%I%01~E and I{ECAlVII, 1984) in S~az (1984) and in DE FA~IA-I~osA et al. (1985, 1986).

152 ~. ~ : c ~ i

15. - On t a c h y o n e l e c t r o m a g n e t i s m .

We preliminarily introduced the (~generalized Maxwell equations ~ (in terms o[ the four-potential) already in subsect. ]0"5. The method followed there is noticeable since i t does n o t depend on the explicit form of the SLTs.

If we now make recourse, however, to SLTs in their form (154 b i s ) , w e

can generalize Maxwell equations in a more convincing way for the case in which both sub- and Super-luminal charges are present. I t is noteworthy tha t , even if imaginary quantities enter the last two equations in (154 b i s ) , nevertheless the generalized Maxwell equations can be expressed in pro'ely real terms (see, e . g . , R~3OA~ and MIG_~A_~I, 1974a; C0~BE~, 1978a, b); we already mentioned, actually, tha t this seems to happen for all the f l lndamental classical equations for tachyons (review I). Therefore, it is not s t r i c t l y necessary to pass to a multidimensional space-time for exploiting tachyon electromagnetism; but interesting work has been done, for example, in six dimensions (see, e . g . ,

D~,TTOI~ and M3CG~A~I, 1978; COLE, :1980e; PATTY, 1982). Before going on, let us recall that the o r d i n a r y '_~[axwel] equations read

(g,~ : : 0, 1, 2, 3) (*)

(197) ~F~*~ = ]~, ~_P~ == 0 ,

where j , - (0 , , j ) and _~ is the tensor d u a l to the electromagnetic tensor

.F~,~ (~, fl, 7, ~ = O, 1, 2, 3):

(197') F ~ e = ~e,~a~,oF'ee .

Notice tha t / ~ = F v ~ . Typically, the present dual i ty effects the exchanges

(198) E -> i l l , H -> - - i E .

In t~rms of the a u t o d u a l electromagnetic tensor (review I)

(199) T,,, = F , , , . ~ .~,,~, '2,,~ = T, , , ,

which is invariant under the (~ d~mlity exchanges ~> (198), eqs. (197) o~n be

wri t ten as

When in the presence also of ordinary magnetic monopoles (DI~AC, 1931),

(*) While going on using the signature --2, we set here (for simplieity's sake) x ~ ~ - c t , x ~ ~ i x , z"- --- i y , x 3 ~ i z , so ~hat formally '7~ =~,,.. In this way, eqs. (197')- (201) get more elegant.

C L A S S I C A L T A C H Y O N S 153

i.e. also of a magnetic current gu ~ ( ~ , g), eqs. (197) and (200) get << symmetrized )>:

(20in) ~Fz~ ~ j , , ~ _ ~ = igu,

Equations (200), (201) are covariant, besides under the Lorentz group, ulso (umong the others) under the dualiCy transformations: tha t is to say, under eqs. (198) and under more general rotations in the space ~ = E + i H

(see, e.g., AMALDI, 1968; AMALDI and CABIBBO, 1972; FEI~RAI~I, 1978). At tast, let us rec~II tha t under subluminaI x-boosts the electric and magnetic

field components t ransform as follows (u~< 1):

(202)

~ : = E : , E : - - E ; + u ~ : _ E:- - u~; V V ~ ' E: V1 - - uS'

l

15"1. ~leetromagnet ism wi th tavhyonie currents: two alternative approaches. -

Let us suppose the existence of slower- and luster-than-light electric charges, corresponding to the two four-currents ju(s) ~ [~(s), j(s)] ~nd ju(S) = [~(S), j(S)].

In analogy with what we mentioned in subseet. 10"5, the electromagnetic tensor Fu ~ may not be any longer a tensor under the SI~Ts; i.e. it cunnot be expected a pr ior i to be a G-tensor (subsect. 7"2). According to the way one solves this problem, different theories follow (see RV.CA~ and Mx~A~I , 1974a).

I t is then sound to ])ass ~nd investigate how the E and H components are expected to t ransform under SLTs. Le t us confine to Superluminal x-boosts.

i) I f one wishes ordinury Muxwell equations (197) to be G-covariunt, one has to postulate (with a unique choice for the signs, for simplicity's suke) thu t (U 2 > 1)

(203) / E ~ = - - E = , [

E" : i ~ ( E ~ - - U H , ) ,

H; = ir(~, + u2L),

E', = ir(E~ + Y ~ ) ,

~'o = i y ( R ~ - UF~),

with y ~ 1 /~ /U s - 1. Notice tha t eqs. (203) leave G-covariant also eqs. (201a), (201b); see I~cA~l-~ and MIG~A~I (1974a).

This choice was adopted by COnBE~. In his approach, let us I'epeut, Muxwell equations hold in their ordinury form also when in the presence of both sub- and Super-luminal currents (i.e. when u~u~ = • 1 in eqs. (203)):

{~ F ' ~ = j ' , ~ P ~ = 0 , (203 bis) jz ~ ~ uu , uuuu ~- • 1;

154 ~. R~CAMI

for details on such an interest ing theo ry - -wh ich corresponds to a.ssuming l~z ~ to be a G-tensor--,see Colr (1975, 1976, :1978a).

ii) On the contrary, one can t r y to generalize the subluminal t ransformations (202) for the Superlumin~l case, and only a posteriori deduce if 2'u~ is a G-tensor or not, and finally derive how Maxwell equations get generalized. In eqs. (202) each couple of components E~, H~ and E~, H~ transforms just as

the couple of co-ordinates x, t (cf. fig. 7a)); and the components E~, H~ both t ransform just as the co-ordinate y or z.

Subst i tut ing the plane (E,, H~), or the plane (E~, H~), for the plane (x, t), it is then n~tural (of. fig. 7b)) to extend the subluminal t ransformations by

�9 ' ' H:) to (~ ro ta te ~)beyond 45 ~ unti l when E~ allowing the axes E: , H'~ (or E~, coincides with H~ and H: with E , for U--> ~ : see fig. 46. This corresponds to ex tend the two-dimensional Lorentz t ransformations so as in subsect. 5"6, eq . (39").

Then, we m a y ex tend the t ransformat ions for E~ (and H~) by analogy

with the last two equations in (354 bis) or in (160); t ha t is to say: E , - - iE'~, H, : ill',, where for simplicity we confined ourselves to -- ~/2 < 0 < + z~/2. In such an approach, the quantit ies T,~, Fu~, Au., are not G-tensors, since under SLTs they t ransform as tensor except for an extra i (see, e.g., review I and I%ECA_~Z and ~MI@iNA1NI, 3976, 1977). Notice tha t , due to the invariance of Tu~

under the (( dual i ty ~ t ransformations, we may identify i E ' ----- -- H , , i// ', ---- E , in t teavis ide-Lorentz units (i.e. in rationalized Gaussian units).

In review I it has been shown tha t the assumption of the previous Super- luminal t ransformat ions for the components of E and H leads to generalize eqs. (200) in the following (G-covariant) form:

{ ~ Tu~ = ]u(s) - i~u(S) (v '~ l ) , (204) ~uv- - Tu~

which const i tute the (( extended Maxwell equations ~--valid in the presence of bo th sub- and Super-luminal (( electric ~ currents- -according to MIG~A~'~TX

and ~EC).~X (1975b, c, 1974d) and I~ECA~rr and M~G~A~I (1976, 1974a, b). I f we confine ourselves to subluminal observers, eqs. (204) can easily be

writen as (I~ECA~r and MIG~Alh~, 1974a)

(205)

d i v D .... _ o(s)

div B --- -- ~(S)

ro t E = -- 8B/St ~- j (S)

rot H = -i- 8D/~t -]- j(s)

(v~.~l, s( ~ v 2 < ] , S ~ v ~ > l).

Therefore, according to the present theory, if both sub- and Super-lunfinal

CLASSICAL TJLCI-Iu 155

<~ electric ~> charges exist, Maxwell equations get fully symmetrized, even if (ordinary) magnetic monopoles do not exist.

Actually, the generalization o2 eq. (202) depicted in fig. 46, as well as the extended Maxwell equations (204)-(205) seem to comply with the very spirit of SR and to (( complete ~) it.

(Ho)z,y'

0

/e;,y / / / (H0),,/

_ / ~ Hz(q~"), �9 / .~-ry, z (H~ =- iEy'

B, .

(Eo)y Ey((P I) (Eo)y,z

E" z (Hol

z,,(~,,~ (e4, too),,, o b) G (E0)~ c)

l~ig. 46.

15"2. Tachyons and magnetic monopoles. - The <( subluminal ~> equations (102b) seem to suggest tha t a multiplication by i carries electric into magnetic current, and vice versa. Comparison of eqs. (201b) with the generalized equations (204) suggests tha t :

i) the covariance of eqs. (201b) under the dual i ty transformations, e.g. under eqs. (198), besides under LTs, corresponds to the covariance of eqs. (204) under the operation 5 P = 5P4 (subsect. 14"2), i.e. under SLTs. In other words, the covarianee of eq. (201b) under the transit ion charges ~ monopoles corresponds to the covariance oi eqs. (204) under the transit ion bradyons ~ tachyons;

ii) when transforming eqs. (201b) under SLTs (in particular, under the Superhminal transformations previously defined for the electric and magnetic field components) electric and magnetic currents go one into the other. Equa- tions (205) show, more precisely, tha t a Superluminal (( electric ~ positive charge will contribute to the field equations in ~ way similar to the one expected to come from a magnetic south pole; and analogously for the currents. This does not mean, of course, tha t a Superluminal charge is expected to behave just as an ordinary monopole, due to the difference in the speeds (one sub-, the other Super-luminal !). Since eqs. (205) are symmetr ic even i~ ordinary monopoles would not exist, EI~ seems to suggest - -a t least in its most economical vers ion-- tha t only a unique type of charge exists (let us call it the electromagnetic charge), which, if you like, m a y be called <( electric )> when sllbluminal, and <( magnetic ~ when Superlnminal (MIe~A~I and I~EOA2~I, 1975b; I~ECAMI and MIeI~A~I, 1976, 1977). The universali ty of electromagnetic interactions seems, therefore, recovered even at the classical level (i.e. in SLY).

1 5 6 r . a ~ c A ~ i

Let us exploit ~)oint if), by finding out the conditions under which the generalized equations (118), (118') of subsect. 10"5, written there in terms of four-potentials, are equivalent to the present extended Maxwell equations written in the form (204):

(zo6) [ ~ , .~ . = o ] ~ L ~ " .... T,,, j (v,~c,) ,

where ]~ ~ j ~ , ( s ) i j v (S ) . From the identity C ] A ~ = - ~2 - -}- envea v v 21 ) we can derive that eq. (206) holds provided that we set (v ~ ~ c ~)

(207 )

I t is remarkable that cq. (207) c~n be explieited into one of the two following

conditions:

(208b) F*~ - - B~,~ - - B~,~ + iet,,ooAe'~ ,

where 1 , ' * , - - - ( i / 2 ) s ~ , , ~ F e ~ - i l ~ , (so that I '~ - -1~ , - i~*~, in agreement with eqs. (118')). Equation (208b) is a consequence of the identity (FINZI and PA-

STO]{I, ]961 )

i ~,r , . : - - - . - - F [ L y �9 B,,,~, - - B~,~ ~e~,,~ A~ ,~ 2 ~ ( A ~ , ~ - - A~,~ - - i ~ p y ~ B ) - - '*

Equation (208a) is nothing but the Cabibbo and Ferrari (1962; see also FEI{n~a~I, ]978) relation. In fact, those ,~nthors showed that the electromagnetism with ordinary charges and mollopoles can be rephrased in terms of two

four-potentials A~ and B,; and, in particular, gave the <( Dirac term ~> the form of the last addendum on the r.h.s, of eq. (208a).

We gave a n e w physical inteI~pretation to the Cabibho-~errari relation. Moreover, while t],c ordinary approach with the two four-potentials A,, B~ meets difficulties when confronting the gauge symmetry requirements, such difficulties disappear in our theory since B~ is essentially the transform of an A~-type potential under a suitable SLT. Actually, a satisfactory Lagran- gian ~nd H~miltonian formalism has been recently worked out (DE FA-RIA- ROSA et al. , ]985, 1986) within the framework of Clifford algebras for electromagnetism, in which charged tachyons ~re formally shown to be able to interact with ordinary charged matter, and vice versa. I t is noticeable the fact that a quite similar formalism applies--of course--to electromagnetism with monopoles, thus allowing one to solve some long-standing difficulties. By using the magnetic-monopole language, it has been satisfactorily shown, in particular, that both electric and magnetic charges can interact with both

the potentials A, and B~.

CLASSICAl, TACIIYONS 1~7

15"3. On the universal i ty oJ electromagnetic interactions. - Equat ions (205) say tha t , grosso modo, a (( tachyon electron ~) (electric charge -- e) will behave as a (SuperlumillM!) north magnetic charge (-[-g)~ and so on; in the se, nse that: the t~achyonic electron will bring into the field equations a contr ibut ion exact ly at the place where contr ibut ion was on the cont ra ry expected from a magnetic charge.

Since when 1)assing, in the four -momentum space, on the other side of the light-cone the topology does change (see, e,g., SHAII, 1977), it is not easy to find out the relat ion between -t- g and -- e. MIG~ANI and ICECA~ put forth the most naive proposal

(209) g = - - e ;

in such a case (when quantizing) we expect to have

(209') eg - - •

where ~ is the fine-structure constant , instead of the Dirac-Schwinger relat ion

eg = ~ ~c. But this point needs fur ther investigation (on the basis, e.g., of Singe'~ work). In any case, in the present apI)roaeh, SR itself is expected to yield a relation between g and e, so to provide a theory with a unique independent coupling constant . In ordinary classical electromagnetism with monopoles two coupling constants, on the contrary, do at)pear: and this violates at a classical level the universal i ty of electromagnet.Jr interactions, at variance with what one expects in SIC (only at the quantum level the universMity gets recovered, in the ordinary theory wi thout ta.chyons).

As a working hypothesis , let its assume eqs. (209), (209') to be valid in our ~ ta.chyonic )~ theory ; t ha t is to say, in general, ge = na~e.

We know tha% quantizing the ordinary theory with subluminal monopoles, we end up on the cont rary with the different relation eg = �89 (Dn~Ac, 1931), or eg ~ n]~c (Sc~w]NGEg, 1966). To a.void contradict ion, we have a.t least to show tha t , when quantizing the present approach (with (( ta ~hyon monopoles )~), we end up nei ther ~ i th Dirac's, nor with Schwinger's relation.

In fact (ICECA]Wr and M~GNA~I, 1977), let us quantize this theory by using 2~andelstam's method, i.e. following CAm~BO and FFA~RAI~I (1962). In tha t

approach the fieht quant,ities describing the charges (in interact ion with the electromagnetic field) arc defined so tha t

(210) ie ' ]

(p(x, P ' ) = ~(x, P) exp -- ~ .F,,da,~

X

where 27 is a surface delimited by the two considered spacelike paths P and P ' , ending at ])oiat x. ill o ther words, t,he field quantit ies ~ are independent of l~he gauge chosen for the four-potent ial A~' bn t are path dependent . When only

158 ~. RECAMI

subluminal electric charges are present, then _~'.~ = A . , . - A.,. and eq. (210) does not depend on the selected surface Z (it depends only on its boundary P - - P'). I f also subluminal magnetic monopolcs arc present, then _~.~ = A~,~, -- -- A , . ~ - e~,o~B ~'~ where B , is a second four-potential, a.nd the following condition must be explicitly imposed:

(211) [----ie ~ Fa~da~] " :l , exp 2

X--X'

where]tom Dirae relation eg = n~e/2 ]ollows. However, if (( magnetic monopoles ~> cannot be put at rest, as in thc case

of (~ taehyon monopoles ~>, then eq. (210) is again automatically satisfied, without any recourse to the Dirac condition.

15"~. ~urther remarks.

i) I t may be interesting to quote tha t the possible connection between taehyons and (~ monopoles ~ in the sense outlined above (RECAm and :'VIIG~A~-I, 1974a) was first heuristically guessed by Anz~I~S (1958)--who predicted tha t E ~- H for U > c- -and later on by PAICK_E]~ (1969), in its impor tant and pioneering two-dimensional theory (see also WE~GAICTE~', 1973).

ii) As to the first considerations about the motion of a charged tachyon in an externM field, see ]3ACICy (1972) and BAclr165 et el. (197~). Notice, incidentally, tha t even a zero-energy cha.rged tachyon may r~diate (I~EE, 1969) subtracting energy to the field.

iii) The interactions of tachyon soliton charges have been studied, e.g., by vA~" ])EI~ MEI~WE (1978), by means of Bi~cklund transformations.

iv) If we consider the quanta inside the Cauchy-Fresnel (~ evanescent waxes )>, since the momentum component normal to the reflecting plane is imaginary, the one p~rallel to t ha t plane is larger t han the energy. Such part ial (~ tachyon properties }> of those quanta have been studied p,~rticlflarly by Cos ta DE BEAUI~GARD (1973; see also C0STA DE BEAUICEGAI~D et al., ]971), whose resea.rch group ewm performed an experimental investigation (HIIAxD and IMBEI~T~ 1978). Fur the r expcrimenta.1 work is presently being performed, for

example, by ALZET~A at Pisa.

v) AGu~)I~ and P~ATZECK (1982b) observed tha t the motion of an accel- erated charge can be expanded into uniform-motion components, via Fourier analysis. They claimed that , when the acceleration is not zero, bolh (( tachyonie ~ and (( bradyonic ~) components must be present.

15"5. (( Experimental }~ considerations. - The very first experiments looking for taehyons, by ALV:~GEI~ et el. (1963, 1965, 1966), have been already mentioned

CLASSICAL ~AeHYO~S 159

in subsect. 3"1. Let us add that a major research for charged taehyons was first carried on by AJ~VXGE~ and KI~EISL]EI~ (1968).

Most experiments (see Hs and ItUGE?qmOBL]m~, 1978; see also, e.g., PEREPELITSA~ 1977a) looked for the Cerenkov radiation supposedly emitted by charged tachyons in vacuum. In subsect. 10"3 we have, however, seen that we should not expect such a radiation to be emitted.

Searches for taehyons were periormed in the cosmic radiation (see, e.g., t:~AlVfANA MUI~TltY, 1971) and in elementary-particle reactions (see, e.g., BALmAu

et at., 1970; DANBURa et al., 1971; l~tlV~tNA MURT~-Y, 1973; and ~EREPELITSA, 1976). Also taehyonic monopoles were looked for (see, e.g., ]3a~TLE~m and LAHANA, 1972; PEI~EPELITSA, 1977b; and BAI~TLETT et al., 1978).

We indirectly discussed many experimental topics in sect. 13, where the possible role was shown of taehyons in elementary-particle physics and quantum mechanics; and we refer the reader to that section. As to the possible links between neutrinos and taehyons see subsect. 11"5 and 13"3.

Let us add here tha t - -even if one does not stick to the conservative atti- tude of considering tachyons only as ~ internal lines ~> in interaction processes-- any experimental project going beyond simple kinematical evaluations ought to take account (Co,BEN, 1975) also Of the drastic ~ deformations> caused by the huge velocity of the observed objects w.r.t, us: see, e.g., the results on the tachyon shape presented in subseet. 8"2 and 14"6. As noticed by BA~um (1978), one may wonder if we have really correctly looked for taehyons so far.

Within the classical theory of taehyons, it would be important to evaluate how charged taehyons would eleetromagnetieally interact with ordinary matter : for instance, with an electron. The ealcul~tions can be made on the basis of the generalized Maxwell equations, either in Corben's form or in Mignani and l~ecami's (subsec~. 15"1). I t we take seriously, however, sect. 8 on the shape of taehyons, we have to remember that a pointlike charge will appear--when Superluminal--to be spread over a double cone ~; it would be nice (see subsect. 10"3) first to know the L-function of the space-time co-ordinates yielding the distribution of the taehyon charge density over c~. Let us recall, moreover, that y-y interactions should a priori produce both couples of bradyons and couples of taehyons.

At last, for the possible role of taehyons in astrophysics, see sect. 12.

1 6 . - C o n c l u s i o n s .

In this paper, after having shown that BE can be adjusted to describe

both particles and antiparticles, we have presented a review of taehyons with particular attention to their classical theory.

We first exploited the extension of Sl~ to taehyons in two dimensions:

160 ~. R~CA~I

an elegant model theory which allows a better lmderst~nding also of ordinary physics. Then we passed to the four-dimensional results (particularly on tachyon mechanics) which can be derived without assuming the existence of Super- luminal reference frames. We discussed, moreover, the localizability and unexpected apparent shape of the t~chyonic objects, and carefully showed---on the basis just of taehyon mechanics--how to solve the common causal paradoxes.

In connection with GE, particularly the problem of the apparent Super- luminal expansions observed in astrophysics has been reviewed. Later on we examined the important issue of the possible role of tachyons in elementary- particle physics and in quantum mechanics.

At least we tackled the (still open) problem of the extension of the relativistic theories to tachyons in four dimensions, and reviewed the electromagnetic theory of tachyons: a topic that can be relevant also to the experimental side.

A few conclusions may be the following. Most tachyon classical physics can be obtained without resorting to Super-

luminal observers; and in such a classical physics extended to tachyons the ordinary causal problems can be solved.

The elegant reslflts of EIr in two dimensions, however, prompt us to look for its multidimensional extensions (i.e. to try understanding the meaning and the possible physical relevance of all the related problems: sect. 14).

Tachyons m~y, for instance, have a role as objects exchanged between elementary particles, or betwecn black-holes (if the latter exist). They can also be classically emitted by a black-hole and have, therefore, a possible role in astrophysics.

For future research, it looks~ however, even more interesting to exploit the possibility of reproducing quantum mcehalfics at the classical level by means of tachyons. In this respect even the appearance of imaginary quantities in the theories of tachyons can be a releva.nt fact, to be furthcr studied.

***

The author thanks, for encouragement, A. BA~UT, P. CALDIROLA, A. GIGLI~ M. JAI~KEE, P.-O. L~JWDIN~ N. I~0SEiN', D. SCIA_MA~ G. SUDARSIIA~N, A. VAN DEE MEICWE, C. VILLI, and particularly 1~. A. ICiccI and D. WmKr~'S0~. He thanks moreover, for discussions~ A. AGODI~ M. ANILE, J. S. BELL, l~. BROWN~ A. CA- STELLLN0, E. GIAN]NETT0~ _A_. IiNSOLIA, A. ITALIAI~0, A . J . ]~[~LNAY, G.D. ]~[ACCARIt0NE~ A. MAIA~ Ir M/GNAI~I~ M. PAV~I~, A. I~IGAS, M. ROD0~), W.A. I~ODRIGUES, M.A. :F. I~0SA, F. SELLERI, M.V. TENOI~IO, and particularly P. S~mz.

He is grateful to P. PAPM~I and the Redazione of this journal for the generous co-operation. At last, the author exprcsses his thanks to Mr. F. AgEIVCr for his help in the numerous drawings, and to Dr. L. 1-r BALDI_NI for the kind col- laboration over the years.


The first s t imu l i to wr i t e th is rev iew c~me f rom two kind~ p r e l i m i n a r y

i n v i t a t i o n s : one in 1976 b y K. PAULUS (on beha l f on the I n s t i t u t e of Physics) ,

a n d one in 1980 b y Sir D. WILK~SO~ (editor of two series of books for Oxford

U n i v e r s i t y Press) . The p r e sen t work appea red in p r e p r i n t f o rm as r epor t

I N F N / A E - 8 ~ / 8 (Fr~sc~t i , 21 A u g u s t 1984).

Note added in p~oo]s.

Two recent papers (~) called a t tent ion to a series of experimental works (2)--dated 1971 to 1984--which suggest tha t the muon-neutrino coming from pion decay (7:+-+ -~ ~++v) might be tachyonic, in the sense tha t its four-momentum square results to be negative (even if by two standard deviations only: (~world average)> based on ref. (~)). More recent data (s), adopting a new precision measurement of the r~ mass, yielded the value

(N1) p 2 p~p~ = (_ 0.097 ~ 0.072) (MeV)2/c 2 .

Actually, the idea that neutrinos could be tachyons, at least in some cases, has a long story: see subsect. 11"5 and 13"3 of this review and references therein. For instance, it is immediate to show (4) tha t in the pion centre of mass (in which Ipl~ = Ipl~) it is

(N2a) lp]v~29.7901 ~eV/c ~ llalo, if m v = 0, vv=c ;

(N2b) [Ply< IPlo, if ~ 0, v ~ < e ;

(N2e) IPI~ > lP[o, if ~n. # 0; v v > c.

The latest experimental data (N1) are compatible with a tachyon-neutrino with mv_~0.31 MeV/e; from eqs. (55'), (55") it follows that in the pion e.m. V/e~v~ /e ~_

0.000055. Let us mention some consequences, tha t could be experimentally tested. Due to the (( switching procedure )~, when the pion speed in the laboratory is v= <

< e~/V ~ 0.999 945c, we shall s tart seeing some pion ((decays ~ as processes : : - ~ - + -+ ~, the condition being that v=- V < - - c 2 (where v= is measured in the laboratory and V in the pion c.m.). E.g., for pions with a laboratory energy E ~ 31.2 GeV, one event every ~45000 decays will appear in the laboratory as a tachyon absorption in flight, ~ + ~ ~ ~, corresponding to a <~partial-mode ~> mean life in flight of A~'

0.26083 s. Since the electron-neutrino seems to be bradyonic, if the muon-neutrino is on the

contrary faster than light, interesting consequences then follow w.r.t, neutrino oscillations. For simplicity's sake, let us consider Majorana neutrinos with a finite mass.

(1) A. C~ODOS, A. I. HAUSEI~ and V. A. KOST:ELECKY: Phys. Lett. B, 150, 431 (1985); H. VAN DA~, u J. NG and L. C. BI~D~N~IA~: Phys. Lett. B, 158, 227 (1985). (2) E .V. SI~RV~ and K. 0. H. Zloe]~: Phys. Lett. J~, 37, 114 (1971); G. BACKE~- STOSS et al.: Phys. Lett. B, 43, 539 (1973); D. C. L v et at.: Phys. Rev. Lett., 45, 1066 (1980); H. B. AND~n~IZB et al.: Phys. Lett. B, 114, 76 (1982); R. AreOLA et al.: Phys. Lett. B, 146, 431 (1984). (3) B. J]~CKI~L~ANN et al.: Phys. t~ev. Lett., 56, 1444 (1986). (4) G.D. MACCAI~RONE and E. RECAMI: Nuovo Cimento A, 57, 85 (1980), p. 99.

1 6 2 ~ . ~ n c ~

Iu the s tandard formulae (~), the relevant quant i ty ~ve can ordinarily reach only valucs of the order of ~ 10 ~ (eV) ~, whilst under the present hypothesis it could be of the order of 2.5- 10 ~ (eV) 'z. This means tha t , if the muon-neutrino is Supcrluminal, the posit ion of the first oscillation maximum corresponds to values L/I~ various orders of magni tude smal ler than ordinar i ly expected (whcl'e E, L are the neutrino energy and the distance from the source, respccti,cely). Even more, the coherence between the two ~mass eigenstates ~> of the muon-neutrino (a condit ion necessary to interference) would be lost both in the solar neutr inos and in the reactor experiments. I t would be expected to remain st i l l satisfied only for the cosmic radiat ion.

Notice tha t for a tachyon-neutr ino the mass upper l imit is about (0.44-: 0.49) MeV/c ~ (95% confidence level). The ordinary as t rophysical arguments, sett ing stringent l imits on the masses of bradyonic neutrinos, do not hold good in our case.

Fur the r detai ls can be found in ref. (s). Here let us only recall t ha t : i) if the existence of Supcrluminal neutrinos will be confirmed, then there should exist both sub- and Super- luminal neutrinos, even if behaving differently; ii) also tachyons can be associated with semi-integer or integer spins, since in the spacelike case we are allowed to make recourse to nonuni tary representat ions of the Poincard group; iii) fas ter- than-l ight electric charges (dipoles) ought to behave as (Superluminal) magnetic poles (dipoles); iv) if neutrinos possess a finite mass but arc tachyonic, then the rc la t iv is t ica l ly inva~an t dist inction between (left-handed) neutrinos and (right-handed) antineutrinos sti l l holds, since in E R the hel ic i ty gets reversed together with the par t ic le /ant ipar t ic le character.

(s) See, e.g., V. F~,~a~_~-~o and B. SXITTA: Neutrino oscillation experiments, report INFN/AE-85/6 (Frascati , 1985). Cf. also D. H. P~]~];~Ns: Introduction to High Energy _Physics (Addison-Wesley, London, 1982). (s) E. GIANN~TTO, G .D. MACC),RI',ON:E, R. M~A_~-~ and E. I { ~ c 2 ~ : -possible consequences /or neutrino oscillations o] a tachyonic muon-neutrino, prepr in t I)P/777 (Physics Depar tment , Catania Universi ty, 1986), to bc published. Sec also E. RI~CAMI, R. M~G_~-A~I and G . D . M~CC~R~ON~: Are ranon neutrinos ]aster-than-light particles? (Comments on a recent paper by Chodos et al.), repor t INFN/AE-86/1 (Fraseati , 1986).

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