vector-spinor space and field equations

Foundations of Physics, VoL 17, No. 1, 1987

Vector-Spinor Space and Field Equations

Nathan Rosen I and Gerald E. Tauber 2

Received June 19, 1986

Generalizing the work o f Einstein and Mayer, it is assumed that at each point o f space-time there exists a vector-spinor space with N~. vector dimensions and N~ spinor dimensions, where N~ = 2k and Ns = 2 ~, k >~ 3. This space is decomposed into a tangent space with 4 vector and 4 spinor dimensions and an internal space with N~ - 4 vector and N~. - 4 spinor dimension. A variational principle leads to

field equations for geometric quantities which can be identified with physical fields such as the eleetromagnetie field, Yang-MiIls gauge fields, and wave functions o f bosons and fermions.

1. I N T R O D U C T I O N

In 1931 Einstein and Mayer (I~ published a paper having as its basic assumption that at each point of space-time there is a five-dimensional linear vector space. By assigning suitable geometric properties to this space and setting up suitable field equations, they were able to obtain the Einstein-Maxwell equations for the electromagnetic and gravitational fields in the absence of matter. The motivation for their work came from the earlier work of Kaluza, (2) He had assumed space-time to be a subspace of a five-dimensional Riemannian space with all the geometric quantities independent of the filth coordinate. By taking as field equations the Einstein equations in five dimensions and interpreting certain components of the five-dimensional metric tensor as the electromagnetic potentials, he had obtained the Einstein Maxwell equations. However, Einstein and Mayer did not like the idea of a five-dimensional space in which it is

1 Department of Physics, Technion-Israel Institute of Technology, Haifa, Israel. 2 School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, Israel.

63

0015-9018/87/0100~0063505.00/0 © 1987 Plenum Publishing Corporation

64 Rosen and Tauber

assumed that all the geometric quantities depend on only four coordinates. On the other hand, with a five-dimensional vector space at each point of space-time, the quantities in this space can depend only on the four coordinates of this point, and they thought that this was a more satisfactory situation.

Recently (3) the present writers generalized the approach of Einstein and Mayer by assuming that the vector space had N dimensions (N~> 5). By making certain assumptions about the geometry of this space and setting up a variational principle, they obtained field equations for quantities that could be interpreted as the electromagnetic field, Yang-Mills gauge fields, (4) and vector mesons.

This work was not entirely satisfactory. For example, the variational principle contained quantities which did not have direct geometric significance. However, the most important objection to the work was that it did not give a complete description of matter: it led to equations for bosons, but not for fermions. Obviously a theory dealing ony with vectors cannot describe fermions--for the latter one needs spinors. The present paper is intended to improve the situation. It contains a generalization of the work of Einstein and Mayer according to which, at each point of space- time, there is a vector-spinor space, i.e., a space of many dimensions con- taining both vectors and spinors and endowed with certain geometric properties. This is in addition to the four-dimensional tangent space at the point which now becomes a subspace of this vector-spinor space. Only geometric quantities enter into the variational principle, from which field equations for bosons, fermions, and gauge fields are derived.

In Section 2 vectors and spinors in the tangent space are briefly discussed. In Section 3 the vector-spinor space is considered, and Section 4 deals with connections and curvatures. Section 5 makes use of a variational principle to derive field equations, and Section 6 investigates some of them from the physical point of view.

2. VECTORS A N D SPINORS IN FOUR D I M E N S I O N S

The points of space-time (the base space) are labeled by coordinates x j (j = 0, 1, 2, 3). At each point there is a tangent space in which there are vectors and tensors (such as V ,j and B~, for example). In particular, there is a Riemannian metric tensor gjk, satisfying the Einstein field equations, and there is the Riemann curvature tev_sor R~.~.

There are also spinors. These are taken to be 4-component spinors, such as Oc~ or ~b' (a, b ' = 1, 2, 3, 4), with a bar and a primed index denoting the complex conjugate. One can think of the components as referring to a

Vector-Spinor Space and Field Equations 65

given tetrad. There are also spinors labeled by several indices (spinorial tensors). For example, there is the spinor metric with components g.,b and g~b, satisfying the relations

~,bc' = ~;c, - " (1) g~'b ~, ~ . ' , g " b ' g b ' ~ . - 6c

This metric is assumed to the Hermit ian, so that

g~'b -~ g~b' = gb'~ (2)

with similar relations for the contravar iant components . The metric enables one to raise and lower indices,

~p , = ga,b~pb, ~, = g~, ~)b' (3)

It should be noted that, whenever an index is raised or lowered, it changes from a primed to an unpr imed one or the reverse.

One can form scalar products of spinors,

(~, t/.,) = g.,b q~")p b = q~,,~p'~ = ¢~"'~., = g'b'¢,,~pb, (4)

In particular, one has

(~, ~') = g . ' , ~ ; " ' ~ = G ' / ' " (5)

There are also the Dirac matrices 7/.,~,, They are Hermit ian

~/.h, = 7~//, (6)

and satisfy the relations

7/~,7k~,. + 7kohy ibm. = 2 g / k 6 ~ (7)

with

F rom (7) one gets

and

7Jr't, = g" '7/4,~, yJ"'~,, = 7/,,; "' (8)

?/"b'/kb~ = 4g~, ?/"b?'kb. = 46~

We can then define the spin matrices SJk.,b by the relation

7./,,bykb_ ),k. b ?.jb = 2s./k~ C

(9)

(10)

(11)

825/17/1-5

66 Rosen and Tauber

There is also the symmetric matrix C "b having as its reciprocal Ca~ so that

C ub = C b~, C ub Ch~. = 6~ (12)

which has the property that

Cb~. 7J~a C a" = -?J"b (13)

The Dirac matrices enable one to assign a vector to a pair of spinors

V i = 7.i"b~,t//h = ?,.i~/b~dOb (14)

or to a spinor and its complex conjugate

V i = ,/S ,h ~TdO~, = 5,i~,,~ @b (15)

In the absence of fields, if one has Lorentzian coordinates so that gj~ = r/j k = diag(+ 1, - 1 , - 1 , - 1 ) , one can take the Dirac matrices in the form

7 o = ( ~ _0I) ' y / = ( O ; / ) , j = 1 , 2 , 3 (16,

where y / = (7./"/,), the o-/are the Pauli matrices, and I is the unit matrix. One then has

~4 = y°y'y2y3 = - i (~ : ) (17)

The spinor metric is taken to have the form

(go,b) = (10 _0I) (18)

This enables one to write down the Hermitian Dirac matrices 7J~,b. One find that

(~/0)2 = /~ (]2]) 2 = --L j = 1, 2, ,3, 4 (19)

and the matrices all anticommute. It should be noted that we also have

7/~ = 0, j = 0 , 1, 2, 3, 4 (20)

With the above choice of Dirac matrices one can take

° ,2,;


In the general case, one can carry out coordinate transformations, with the vectors and tensors in the tangent space transforming in the usual way, One can also transform the tetrads and thus transform the spinors, so that 0 ~---, ~ , ~b~'--, ¢~', etc., according to the relations

~ = S~b~b b, ¢~'= S~'b,~ b' (22)

d~ = (S-~)~,,¢b, ~,,, = (S -~)b'~,,qb, (23)

where S~b may be a function of the coordinates. It is convenient to restrict oneself to transformations under which gdb is invariant. One finds that this imposes the condition

g,,'t, = (S ---1 y,~' (24)

where indices are raised with g"C We see that S "b' (or S.,b) is unitary.

3. M A N Y - D I M E N S I O N A L VECTOR-SPINOR SPACE

Generalizing the idea of Einstein and Mayer, we assume that at each point of the base space we have a many-dimensional space of vectors and spinors having the four-dimensional tangent space as a subspace. Let us assume that this space consists of an N~,-dimensional vector space and an N~.-dimensional spinor space with

N~ = 2k, N~= 2 ~ (25)

where k is some integer /> 3. This space will be referred to as the VS space. The vectors are labeled by indices such as p, v = 0, 1,..., N v - l, the

spinors by indices such as c~, f i '= 1, 2,..., N~_ For the vectors there is a real metric tensor g~,., for the spinors a Hermitian metric with components g ~ , g~P'. There are also Dirac matrices -i~i~, 7~.t~, the latter being Her- mitian. The matrices are assumed to satisfy the relations

= ~'~ ~ (26) ~ ' ~ 7 ~ + ?~'~y~':l~ 2g 68

One can also define generalized spin matrices sUV~ B by the relation

? ~ Y ~ -- 7~7~';t~ = 2s~'~r~ (27)

The reason for the choice of dimensions as given in (25) is that in the case of special relativity with no fields present one can easily obtain a possible set of Dirac matrices by making use of those considered in the previous section for four dimensions (k = 2, Nv = N, = 4). The idea is to let

68 Rosen and Tauber

k increase in steps of one, and to express the new matrices in terms of the old. Suppose we let k ~ k + 1, N , ~ 2 N , N ~ - - + N ~ + 2 . Let us denote the Dirac matrices ),~q satisfying (26) for N,. dimensions by 7 ¢' and for 2N s dimensions by 7 ~. Then we can write

o) ,= (o ° ,0 °) ( 0 ,,,,)

"/~' = i I , ~ = 2 , 3 ..... N~+~ - y~ ' - 0

(28)

One sees that these satisfy the relations

~,~'l ~ + 3,~, ~ = 2g~'L #, v = O, 1 ..... N~ + 1 (29)

with g~'"= diag( + 1, - l , 1,...). We can also include the matrix

,N 2 ,(01 D (30)

which anticommutes with the above ~,~ and has as its square

(~<+2)2 = - I (31)

We can now take as the spinor metric

g = ( g ~ ' 3 ) = ' / ° = ( £ ? ) I ' .7', /~=1, ..... 2N, (32)

to enable us to go from ~'~n to Y¢'~'/~" In this way, by a series of steps starting from N , = N~, = 4, we can go to

larger N,, N~, satisfying (25). We arrive at a vector space having only one timelike dimension. There are N~ + ! anticommuting Dirac matrices with zero trace,

7~'~ = 0, kt=0, 1,..., N~ (33)

In light of the above we can now consider the general case of the VS space with N~, and N, given by (25). The Dirac matrices, which may be different from those considered above, will satisfy (26) and (33) and can be used to associate vectors with pairs of spinors as, for example,

A t ' = 7~'~,n~U'~0 n = 7~t~b~ ~/~ (34)


One can carry (changes of bases). Thus one has

= ( s

Similarly,

out transformations on the spinors and vectors

~'= S~'.,: :" (35)

~, = (S-~)/~'~,O/J, ( 3 6 )

A/~ = TQA", A~ = (T-~)".A ' (37)

The transformation matrices can be functions of the coordinates x .j. Since the tangent space is a subspace of the VS space, to quantities in

the latter we can assign quantities in the former with the help of projectors. Thus there is the projector h/. such that to the vector A" one assigns the vector A j by the relation

A / = h / . A ", j = 0 , 1,2,3 (38)

It will be assumed that under this projection a timelike vector A ~ (one for which A . A ~ > 0) goes over into a timelike vector AJ(AJA/> 0).

One can also write

g/~ = h~ ha- gt,,, (39)

The indices of the projector can be raised and lowered with the help of the metric tensors, so that (39) gives

h//, h/'k = c~ ~ ( 40 )

We see from (38) that A :~ determines A/, but A / does not determine A ~. In particular, if A i= O, then A" satisfies the equation

h::~A" = 0 (41)

The solutions are vectors orthogonal to the tangent space. In general the number of independent solutions will be n. = N ~ - 4. Let us take a set of orthonormal solutions and denote them by Xp ~ ( P = 1, 2 ..... n~). They satisfy the equations

h/:, Xp" = O, Xp:'XQ¢ ~ = - 3 PQ (42)

the minus sign being taken because vectors orthogonal to the tangent space must be spacelike ( X . X ~ <0).

Let us refer to the subspace spanned by the vectors Xp" as the internal

70 Rosen and Tauber

vector space. It will have G, dimensions. Let us take for the metric tensor of the internal space

gpQ = geQ = --6~ (43)

so that we can write

Xp XQ. = geQ, Xp~'X~2. = 3~ (44)

in analogy with (39) and (40). One can regard Xe ~' as the projector onto the internal vector space. To a vector A ¢~ one can assign the vector

M r' = X P u s [ s' (45)

F r o m the above one finds the relations t3t

hi' hi, , + Xp" XP,, = 3~,~ (46)

o r

g,~. = h;~,h~v gik + Xe~XO~ , gPQ (47)

with #, v = 0, 1,..., N~.- l, j, k = 0, 1, 2, 3, and P, Q = 1, 2,..., n<,. It follows that, if the vector A" has the projections A i, A e, one can write

A" = h y A i + Xe~A e (48)

In analogy to the vector case we can introduce spinor projectors k"~ and /("'~., so that from the spinors ~ , ~b ~' in the VS space we get in the tangent space the spinors

~ = k " ~ ~ , ~"' = k= ~,~ ~' (49)

The spinor metrics are then related,

g.b' = k~ l~b' g~iS (50)

F rom (50) one gets

k ~ k d = o ; (51)

Corresponding to (41), we now consider spinors ~ satisfying the equat ion

k%4,~=0 (52)

They are o r thogona l to the tangent spinor space and span a space of n= =


N , - 4 dimensions, the internal spinor space. One can take a set of orthonormal solutions ~A (A = 1, 2,.., n~) so that

k " ~ G ~ = o (53)

and

( ~ / ~ J g ~ , ~ = - ~ (54)

with gA'B the (Hermitian) metric of the internal spinor space. From (54) one gets

- A ~G =6~ (55)

We can regard ~A ~, ( A as projectors onto the internal spinor space. It is convenient (but not necessary) to take

gA'e = gAB'= ~)A (56)

From the above properties of the spinor projectors one finds the relations

~ . ' + ~ ~J = 6~ (57)

o r

g~'/3 =/U~'kbl~ g~'b + ~A' (B/~ gA'~ (58)

with c~', f l = l , 2 ..... N,., a', b = 1, 2, 3,4, and A', B = 1, 2 ..... n,. A given spinor tp ~ has a projection ~p~' onto the tangent space as given

by (49) and a projection onto the internal space given by

~A = (Atp~ (59)

With the help of (57) one can write

~ , ~ = £ f ~ ; + ~A~ A (60)

In the internat space one can transform the vectors and spinors (i.e., change their bases). For the vectors we have

/IP= TP{yA ~, /tp=(T-1)QpA(2 (61)

It is natural to keep the form of the metric (43) invariant under the transformation. Then the matrix TPQ is a representation of an element of the group O(n,), and we can restrict the latter to SO(n,,).

72 Rosen and Tanber

and

In the case of spinors we write

Let us invariant. We then have

(62)

from which it follows that

gAe' = ( S-~ )8'A (65)

with SA,~= gA,cSC,~. We see that (SA,~) is a unitary matrix and is a representation of an element of the group U(n~). Instead of this matrix we can take one with element e~SA,t~, where now det(SA. B) = 1, and get the group U(1 ) × SU(ns).

Now let us consider the Dirac matrices satisfying (26). With the help of (57), Eq. (26) can be written

7"~7"~;~ + o;~A ~vA~ + (v,/z) = 2g~'"O~ (66)

where (v,/,) denotes the previous expression with # and v interchanged. We can now carry out all possible projections onto the tangent and internal spaces and in this way get the following twelve equations:

7/",.7~"h + ~)}aD~kD b -}- (k, j ) =

y}A,,Tkc b q- ~fADTkD b AV (k, j ) =

?/",.7~' e + 7/~r~kDB + (k, j ) =

~'/A, 7k" B + 7-moyko~ + (k, j ) =

7JacyPct, q- 7/aD),PDb q- (P, j ) =

7.}Ac~/Pcb -~ 7/AD TPDb "31- ( P, j ) =

7}a,. yPcB "4- 7 /"D TPDB -4- ( P, j ) =

7JAcTe'8 + 7/Ar>TeOB + (P, j ) =

7e",uOcb + 7P"z~FQO~, + (P, Q) =

~PAcTQC b ~- ),PADyQD b -~ (P, Q) =

)"Pac TOC B -t- yPaD TQD B "4- ( P, Q) =

7PAc yQCB ~- )'PAD ?'QD B ~- ( P, Q) =

2g/~:6~ (67a)

0 (67b)

0 (67c)

2g j~ 6 A (67d)

0 (67e)

0 (67f)

0 (67g)

0 (67h)

2gPQc~ (67i)

0 (67j)

0 (67k)

2geO- , ~ (67m)

dA'= gA's,~b~' ' ~A' = (g-1)8'A,OB ~ (63)

consider transformations that maintain the metric gA'B

gA'e = gc'DSC'A'SDB (64)


It is assumed that the bases and the projectors have been chosen so that

~J~B = 7w~, = 0 (68)

Then (67a) goes over into (7), the usual equation for the Dirac matrices in the tangent space. Equations (67b) and (67c) are satisfied, and in place of (67d-h) we get

7JAr)y ~°, + (k, j ) = 2gJk&~ (69a)

~'.J",.TP% + (P, j ) = 0 (69b)

7JADyeV b + (P, j ) = 0 (69c)

y/~'c)'ec8 + (P, j ) = 0 (69d)

7JADyPOB + (P, j ) = 0 (69e)

Equation (69a) is interesting in that it leads to the existence of Dirac matrices ]Y/A B that enables one to assign vectors in the tangent space to pairs of spinors in the internal space.

4. C O N N E C T I O N S A N D CURVATURES

Let us begin with vectors and tensors. We can have, for example, a tensor field with components Q~ke, t~ referring to the VS space, k to the tangent space, and P to the internal space. Let us write for its covariant derivative

~, ~ ~_ ,q~ m, r~, ~"' ~ B R (70) Q k P i l i = Q kP,] '~ . , kP ~ ) , / - - ~ mPlk/J QPkR p/

A comma denotes an ordinary partial derivative, {~} the Christoffel sym- bol formed from ge , F~% the connection in the VS space, and BRm that in the internal space.

As special cases we have

Akll: = Ak:: (71)

where a semicolon denotes a Riemannian covariant derivative, and

A euj = A?:/= A?/+ AQBeo/ (72)

the covariant derivative in the internal space. Under a change of basis in the VS space, so that a vector A ~ trans-

forms according to (37), we require that A~li k transform in the same way,

A~'Uk = T~,,A'II~ (73)

74 Rosen and Tauber

One readily finds that F ~ must transform according to the equation

F~'~.k = T " ~ F ° T k ( T - ~ ) ~ - T ~ , k ( T - ~ ) ~ (74)

In the internal space, if the vector A e transforms according to (61) and we require that A e:k transform in the same way, we find similarly that

BPQk = TeR Bnsk( T-~ )So - T~,k( T - ' )R e (75)

Taking the metric geQ as in (43) and requiring that gpe:k=0, we get

B,OQk = --BQp k (76)

For a vector A~, we have

A.iik = A~,k -- A ; F; .k (77)

In order to determine F~k, let us first consider a simpler connection /~).k giving the covariant derivative

A ~,jk = A~,,k - A ~.F~k ( 7 8 )

The conection /~;¢,k is determined by the conditions

hJ~,lk = O, XPutk = 0 (79)

These given the relations

h J~./~)~,k - hJ " J - , , . , + h , { , .~ } (8o)

X S ~ ' ~ k = Z s k + xO~,Bee~ (81)

so that, using (46), one gets

v p . . . . . . . . (2 s (82) /~"~,k = h/' h./,,,, + X~, X ,,,~ + h,,, h ,, {,,k } + X e X ~B ek

From (47) and (79) we have

gu~lk = 0 (83)

Now let us write

and let us require that

Using (83), we get

F;~k = F;'~k + V ; ~ (84)

g~-'llk = 0 (85)

V ~,,k = -- V,,vk (86)


The general form of V ~ is given by

V..,k = h ' . h n v Wm.k + (X?.h:~ - X?~h: . ) Fsyk + XP~XQ~DI~Q~ (87)

with

Wren k = - - W n m k , DeO/~ -- - D o ~ , k (88)

After having found the connections, let us consider the curvatures. In the tangent space we have the Riemann tensor R~k: and its contractions

R:~ = R~:k~, R = R~k (89)

In the VS space we can define the curvature tensor

P;'~/k = -- F:~.i,k + F:~uk,j -[- F2oj FC:/~k - F ; ~ F ~ i (90)

From (85) it follows that

P .~:k = -- P ,,/~/k (91)

and we can form the contractions

One can write

and this gives

P:~/= h~;.P:~u/k, Po = h/~'Pm (92)

(93)

P0 - R = h:"(hk.luk - hk~,llkfl (94)

The right side can be calculated with the help of (79), (84), and (87). One gets

-- W uk ~:~: ' W'~!/ W,,f Po R = - 2 WX/j: k + ' " ki: t k

-- FOAk Fok / + FQ~Fokk (95)

In order to obtain a positive-definite energy density for the fields, it is found that one must assume that

Fp/k = --Fpkj,

Then we have

I'Vijk = WEUk :j (96)

Po = R + W ak W~k + FeJkFejk (97)

76 Rosen and Tauber

In the internal space there is the curvature tensor

P P BeQjk = - B Oj, k + B ~ko/ + BPe/BRQk - BeRkBeQj (98)

Taking

gPo:k = 0 (99)

we get

B pQ/k = - - B QI,jk

One finds that under the transformation (75) one gets

~"Q/~ = Te~ B%k(T ~)s o

(lOO)

(101)

If we look for a scalar related to the curvature of the internal space, we do not have any corresponding to (89) or (92). We must therefore take a quadratic expression,

Y = B'°ojk B.o Q/~ (102)

Now let us turn to the spinors. The procedure will be similar to that used for the vectors. In the VS space we define covariant derivatives such a s

~llk = t~,k + ~/3/~ (103)

¢~'llk = ¢~',k + Ca'/~'~'k (104)

and require that

g~'tJll* = 0 (105)

In the tangent space we have, for example,

¢.Hk = ¢~;k = ~.,~ -- Cbl~b.k (106)

with

g.'b;k = 0 (107)

We can go father and require that

yi%~ = 0 (108)

The form of F%~ that satisfies this condition is given in the literature. (5~

Vector-Spinor Space and Field Equations

In the internal space we have, for example,

with

so that, if (56) holds,

g A 'B :k ~ 0

G A B ' k ~-" - - G B ' A k

In analogy with (108) we take, with ~ % as in (69a),

7*Aallk=Y e'k+Y'tA/3 j k -+- ~/iDBGADk-- Y'ADGDBk =O

Under the transformation (62) G%k goes over into

dABk = S%GCD/~( S -, )D _ SAD,k( S - , )D

Let us go back to (103), and let us write

with/%/~k giving the covariant derivative

We assume that

One finds then that

a ~-A k ~l.i=g <J=g~'fsl/=O

~'°:flk ~ - a fl ,k "T % A ~ fi, k T a [?, bk ~ ~ A ~ fi B k

From (105) one gets

U~,~k + U,~,k = 0

We can therefore write

U~,~k = ikd~.kb~ Wa'bk + i~"'= " ~A~F.,m~

+ i~A'~,kOt~F.A,k + i~A'~, ~B~DA,e.

77

(109)

11o)

111)

112)

(113)

(114)

(115)

(116)

(117)

(118)

(119)

78 Rosen and Tauber

with

I~b'~ = Wb'~, JDAa~k = DB'ak (120)

Let us next consider the curvatures. In the tangent space we have

P"oj~ = --f'abj, k + ['ab&j + ['a~jl'ebk -- ['ac,~ U'bj (121)

Using (107) one finds that

P.b:/k = -- Pb'~jk (122)

Using (108) one gets

a _ _ a c a ~ / c 7i b:ik--7i b :k j - Y,~, b R m # k - )~i bP ~jk + 7~ c, P bjk = 0 (t23)

With the help of (9) and (11) this gives (61

2R"~jk = si"b~, P~hj.k (124)

SO that

s./k~ b pbj~ = 2R ( 125 )

In the internal space we have an analogous situation. There is the curvature

A a D (126) GAB/k = --G Bj, k -~ GABk,j ~- G DjGBk -- G~DkGDBj

Under the transformation (113) it goes over into

GaBjk = SAcGCz~jk( S - I )DB ( t27)

From (110) one gets

G AB'j~ = --G B'Aik (128)

Making use of (112) and the analog of (123), one obtains

tnsRPi j k = siPABGBAj k (129)

where siPAB is given by the definition

y,iaDyko ~ -- 7~aD~°B = 2sJka B (130)

Hence

s>aeGBaa e = ½n, R (131)


One can consider the scalar

Z = G ABjk G ABJk = --G ABjk G BAJk (132)

as analogous to (102) in the vector case. Now let us go over to the VS space. There we have the spinor cur-

vature

P~pjk = - F~/~/,k + F~e~,j + F~j U,~ - F ~ F~/~; ( 133 )

From (105) it is found that

P~'~k = - P /r~ (134)

One can form the scalar

P~ = SJkb k" ~h~p~i k

Since

we have

k a k a

(135)

(136)

SJkb JC ~ ( k a~l E./k --k~ilk~) = P1 - S Jk ba Pab#~

= P 1 - - 2 R (137)

by (125). The left side can be calculated with the help of (116) and (119), and one gets

Pj = 2R - 2s Jkb~( W~*4 WCbk + F~AjPbAk + iW°bj;k) (138)

Another scalar that one can form is

P2 = SJk BA ( A ~ S P~r~J~ (139)

We have, corresponding to (136),

~ s~lljk - ~ s~tikj = ~ c~ GC~jk -- ~ S P~jk (140)

from which we get

sJkBA (A~( ~ Bc~[Ijk -- ~- B~[lkj) = sJkBA GABjk -- P2

= ½ n ~ R - P 2 (141)

by (131). Here again one can calculate the left side using (119) with the result that

P2= ½ n s R - ,ikS ~ -- A C 2S ~ A(F sjF~ ~ + D ~jD ck--iDAsjllh.) (142)

80 Rosen and Tauber

5. VARIATIONAL PRINCIPLE AND FIELD EQUATIONS

Our purpose is to use some of the geometric quantities considered earlier in order to set up a variational principle leading to field equations having a physical significance. In the previous paper dealing with vector spaces (33 the variational principle made use of the scalars of (97) and (102). However, this led to a field equation for a massless vector meson. In order to get a nonvanishing meson mass it was necessary to introduce into the variational principle a term which did not have direct geometric significance. This does not appear to be satisfactory from an aesthetic standpoint. In the present work the approach will therefore be different. The variational principle will be formulated in terms of the projectors as the fundamental quantities. However, as we shall see, this procedure wilt have to be modified in dealing with the gauge fields described by BSQk and G ABk .

Let us take the variational principle in the form

61=0, I= f (R + L)(-g)~/2 dz (143)

where R is the Riemannian curvature scalar as in (89), L is a function of various geometric quantities (it depends also on g;k but not on its derivatives), and dr = dx ° dx I dx 2 dx 3. If we vary gJ*, we get the Einstein field equation

R l jk - ~_ gjk R = --8~T/k (144)

where 8F Tjk = O L / Og ik -- ½ gik L ( t 4 5 )

so that Tjk can be interpreted physically as the energy-momentum density tensor. The Lagrangian density L will be a sum of terms

L = ~ L , (146)

Let us consider first the contributions from the vector spaces. If we go back to (94), we can write

Po "--- R - hJ~llkhkullj + hJ~tljhk~tlk (147)

where - means "equal to within a divergence." Since a divergence does not contribute to the variational integral, we can use (147) in place of (94). Furthermore, one finds that (96) leads to the results

hJ~llk = --hkUlU , hk~ll k = 0 (148)


so that the last term on the right of (147) can be discarded. If we put P0 into the variational integral, this amounts to taking in (146)

Lo = --hJ~l I~ hk~llJ (149)

There is another term of a similar character that one can add,

L 1 = aXP~llkXpUll ~ (150)

where a is a constant, and an underlined index is to be raised. There are other terms that can be added which will be discussed below. The general idea is to have terms depending on the projectors hk~ and Xp ~ that are to be varied in (143). However, in carrying out the variations we must take into account the constraints (39), (42), and (44). For this purpose we make use of Lagrange multipliers by adding a term

L 2 = 2jPX-pUhJ u + ajk(hJUhk _ gjk) + ZpQ(XP~XO _ g p Q ) (151)

Varying 2/p, aj~, and rpQ will impose the proper constraints on the projectors.

Taking into account (148) and some of the constraints, one finds that the most general term (quadratic in the covariant derivatives of the projectors) one can add is

5

L 3 = ~ b,,H,,~ (152) r~1= 1

with H1 = h / by/, t, /~k we v- ~'I~ t,/ " ~"/ "k IIn" ~'FI~' n 2 = a ~ a p n/ ilkn vllk

g v g ~ v Q 8 4 = j v it p H 3 = ~ P v L I p Z~.Q ilkZl vllk , h ~,h/ Xp Ila-X ,,Ilk

H5 = X P ~ h / , , h / ~ l l k X e " l l k - (153)

Additional terms to go into I will be considered later, but they will not involve hJ~ or Xp ~. We can write as field equations

~ { L ( - g ) 1/2} O, N f l ' = _ 3 { L ( - g ) l / 2 } - O (154) MJ ~ ~ ( _g)l/2 (~hJ,, -- ( _g)i/2 c~g,o

In these equations we can express the derivatives with the help of (87). Multiplying each equation by ½ht~ and ½XQ~ and splitting the result into antisymmetric and symmetric parts when appropriate, we get

(1 + bl) W~jk; k = 0 (155)

(1 + 2bl) Wi,.n Wjm. + (1 + b')FeimFpj m + cr~ = 0 (156)

825,'17,'1-6

82 Rosen and Tauber

where

l - - (1 + b 2 - 5bs)Fejgtlk + (1 + 2b~)Fp~_~ Wj~

+ (~ + b')Do,.~_Fo~m + ½,~j~ = 0 (157) ( a 1 + b4 - ~bs)Fejk_tlk + (a + b )Fe,y~ W~

+ (a + 2b3)DQp,~FQjm + ½2jp = 0 ( t58)

(a + b3)DpQkl l k = 0 ( 1 5 9 )

(a+b')Fem,,Fo,,,~+(a+ 2b3)DeR_k_DQek + reQ=O (160)

b ' = b 2 + b 4 - b 5 (161)

The equat ions contain six constants. These can be chosen to give equat ions of various forms.

As an example, let us take a = b 3 = b 5 = 0 , b 1 = - I/2, b2 = - t, b 4 = I. The above field equat ions now take the simple form

Wgik: ~ = 0 (162)

FgimFpj m + o- 0- = 0 ( t63)

DQpmFQj., + ½2je = 0 (164)

--Fej~llk + ½)~ie = 0 (165)

T p Q = O (166)

In setting up the field equat ions, let us assume that 1;f~, is derivable from a potential , i.e.,

Fpj k = Fpjil~ -- Fpkli i ( 1 6 7 )

and let us express W~j k in terms of a vector W ~ by writing

Wu ~ = (_g )V2 S~jk,~ Wm

with

(168)

so that

F r o m (162) and (168) we get

w , . j - w,., = 0 (17o)

Wi = ~bi ( t71)

8(jkm--8[Ukm], ~0123 = 1 (169)


where ~b is an arbitrary scalar function. We can take for convenience ~b = 0 and hence

Wi/k = 0 (172)

but one can also adopt some other equation for qt. Now let us consider (165). It contains 2je, which is unknown. The

only quantity to which it can be related is the potential Fej appearing in (167). Let us therefore write

"~-/e = --2m2t Fpj, m I = const (173)

Then (165) takes the form

Fpjs.ii k 4- m2t Fpj = 0 (174)

From (163) we get a~, and (164) determines Doe, , although this requires some discussion, as will be seen in the next section.

Before going farther let us look at the form of the variational principle in the present example. Writing

L ~ = L o + L j + L 2 + L 3 (175)

we have in this case

L~,= -L ' / ~*~ _½h.J hjVhk*,l " ~[Ik" II/ I~lhkvHn

-- xPuXjhkVl l~hk , , i fn - l -hJJ ' l j~Xp~l lkXPulFk + L e (176)

Evaluating the derivatives, one gets

L~,= Fe!~Fei k + L2 (176a)

Let us now turn our attention to what can be regarded as a gauge field, Beex,. It is present in derivatives appearing in L~, but is itself undif- ferentiated. Hence we put into (146) the term given in (102) multiplied by a constant, 1/2cl,

L4 = Y / 2 c l = (2cl) -1BeQJkBeQjk (177)

This has a character that is different from that of the other terms in L, which involve projectors, but it seems to be necessary. The situation reflects an essential difference between gauge fields and other fields.

Since the other terms to be added to L will be independent of BeQj, we can vary the latter in (t43) and get the corresponding field equation. One finds it can be written in the form

BeQJ~llk = c i(1 + a + b ' ) J pQj (178)

84 Rosen and Tauber

with jeoJ = Fp Fe~j _ ~vk Fek/ (179)

It should be noted that from the standpoint of the tangent space one can regard the indices P, Q as scalar indices. One can therefore write (178) also in the form

BpQjk ;k = J g ~ Q / + C I ( 1 + a + b ' ) J PQ5 { t 8 0 )

where

J~QJ = BQR]e BPRk -- BPRJk BQRk

With the coefficients in the example (a = b ' = 0), we get

Be~kll k = c ~ JPQ/

(181)

(182)

Let us now consider contributions to the variational principle from the spinor space. We have the scalars P1, defined by (135) and satisfying (137), and P2, defined by (139) and satisfying (141). We can omit the terms with R since the latter has already been put into (143) with the help of P0 given by (147). Hence we take as a term in (146) (with e i=const )

L5 = - 2 e l sJkb.k"~lr j kb~ljk + 2e2s/k~A ~B~ll/~A~ilk (183)

since

L 5 =-- e i ( p ~ - 2 R ) + e 2 ( P 2 - ½ n 2 R ) (184)

In analogy with Lo and L~ it is natural to take also

a - cx 7A ~: x L6=e3 k ~IIJk~ I1~+e4¢ ~l/~--~ Ii (185)

Here again we have to take into account the constraints on the field variables, in this case (51), (53), and (55). For this purpose we take

L7 ,~.a Ak%~A ~ ~o - +.¢ Ak~ (~+crb.(k"~fi%~--5~,) -~ ~ l#.A ~ ~ hA) (186)

so that varying ,~f, 2~A, ~ , and ~z A reproduces the constraint equations. In analogy with (152) and (153) one can add a term involving more

complicated expressions that those in (183) and (185). However, since there are many possibilities for such expressions, this will be relegated to the Appendix.

From the variational principle (143) we have

3 { L ( - g ) t/2} M a - ( ~ { L ( - g ) l / 2 } =0, N A ~ --0 (187)


Taking M~/cb ~, M " ~ A ~, NA~k"~. and NA~(B~. one gets the equat ions

2e l sJk~.( - iWCbj;k - Weaj W q k -- F"A:Fb~k )

+ e 3 ( i W " ~ ; j + W a d W 'b j + FaAjFbA/) q- gab = 0 ( 1 8 8 )

2el SJk~b( -- iFbAillk -- Wb~iF~Ak -- Fb,jDBAk)

+ e3(iF"Ajltj + W~'b~F~ + F~DSA9) + "~A = 0 (I 89)

2ezs:~e~( -- iF~,~ii~ -- wabjFbBk -- F~c.:DC,~ )

+ e4(--iF~A:ll: + W~FbA: + F~t~!D~) + 2 ~ = 0 (190)

2e2sJ~CA( -- iD"6711 ~ + f b c j F b B k -[- Dec:/Dee~)

+e4(- iDe~j l I~+ F~K~F~ + DeqDCAi)+ z ' A = O (191)

Let us consider the case e~ = 0, 3 and let us assume that

and that we can write

where, by (10),

W"b/= K?Tb, ~C = const (192)

F~A: = ?fb ~bhA (193)

O"A I~,/.. ,:.:. (193a) : ~/ b ~ A./

With the help of (108) the equat ions become

O-~b = --e'(4K2cS~ + 7Jac~2:db~Zlc A ~d A) (194)

2"A = --e'(i~:"bObAIl: + 4Ktp"A + ?':abDBAj@bB) (195)

~a A = e 4( iy/a:: ~tbA l lj -- 4t@" A - 7Jab DBA/~/gB ) (196)

rB = e 4( iDeA/IEj _ F.AjF B/ _ D~qDCAj ) (197)

where

with

e' = 6el + e 3

If we eliminate ;:A between (195) and (196), we get

i~ J"b ~ hall./+ m 2 ~9 a A + f l ? ":"b D BAj ~ b B = 0

(198)

(199)

f l = (e' -- ea)/(e ' + e4), m s = 4Kf~ (200)

3 The general case, including e 2 :/:0 and additional terms, wilt be discussed in the Appendix.

86 Rosen and Tauber

In the special case DBA; = 0 we have

i~Jab lpbAllj -~ M21PaA = 0 (201)

With DeAj¢O, let us take the complex conjugate of (197) and change indices in order to obtain the equation for gA ~. Making use of the fact that gA B = Z•A, we get

e 4 DB4/_IIj = 0 (202)

Let us assume that DBA/ is derivable from a potential,

DB~j = DaA:j (203)

Then (202) becomes, for e 4 :~ 0,

D~AHg=0 (204)

Making use of (192) and (193), one can write (183) as

L5 = 24el(4x 2 + ~"A ~A) (205)

and (185) in the form

L 6 = 16e3~c 2 + 4(e 3 + e4)~laA~aA + c4DA~.DBAI (206)

TO deal with the spinor gauge field GABj (109) we can add to the Lagrangian density L a term analogous to L 4 in the vector case,

L8 = Z/2c2 = -(2c2) - 1 G A . k GBAjk, C 2 = const (207)

However, we see from (205) and (206) that varying G%j in (143) will not give any terms involving Oa A in the resulting field equation. In order to get such terms, let us go back to (199) and denote the left side by Qa A. Let us now include in L a term

L9-=C3(~taAQaA+tpaAOaA), c3 =cons t (208)

One sees that

L9/c3 = i)"kab(~aA ~bAilk -- ~aAIl~ IPaA) + 2m2~a A ~P% + 2fl K%jDSAj (209)

where

K% = C ~ / ~ ' ~ (2to)


Varying GA,i gives an equation which can be written conveniently in matrix notation

iGJk.k = I(c ~ + I j (211 )

where

I{c) = i[G j~, Gk]

I j-- c2c3KJ+ ic2c:~f~[D, K j] + ic2eo[D, D s]

and where we have defined

Gj = (G'Aj), Gjk = (GeAj~),

Brackets, as usual, denote commutators.

(212)

Finally, let us consider the energy-momentum density tensor iCi k as given by (145). We can take L with all the projector derivatives evaluated. Thus, we find

where

9

L = ~ L, (214) i = O

Lo+ L l + L ~ = (1 + b l ) W ijk W,i~+ (1 +a+b')FPikFeik

+ (a + bl)DPQ/Deoj (215)

It will be assumed that W~/~, Fpjk, DpQj, and gi~ are independent. L2 is given by (151). We see that L2 = 0, but

OL2/c?g jk = -%~ (216)

where aj~ is given by (156). The term L4 is given by (177), and we take Be~ k as independent of gik L5 is given by (205) but, assuming that Yj"b is independent of gjk it follows that

(?L S/()gjk = 0 (217 )

L 6 is given by (206), where DAB~ is the quantity independent of g/k. L7 is defined by (186), from which it is seen that

L 7 -~ O, OLv/Og jk = 0

L 8 is given by (207), with GBAjk the independent quantity.

(218)

D = (DeA), etc. (213)

88 Rosen and Tauber

Finally, L 9 defined by (208) and (209)

L9=0 , but c?L9/ag:-¢O (219)

Using (145) we can then write

8gTi/c = Pik 4- Q/k- ½ g:kL (220)

where

Pjk = (4 + 5bi) Wi_~_, Wk,,~ + (3 + 2a + 3b')Fpi,~Fp~ ~

+ (a+b3)De~D.o¢2k+ C 1 IBeQj~BpQ~,,, (221)

• _ _ 1 i ~ a A - - Q/k-- ~ C3(Y/ b(~. ~bAllk ~.Allk~bA)+(k,j)}

+ c3fl (KAB:DBAk + KA~k DB~:)

+ e4DABjDBAk _ C2-t GABj¢, GBAkm (222)

If we take W:m, = 0 and consider the example we had earlier, we get

P/k = 3 Fe:,,, Fp#,,, + c t l B e~m B pQx~ ( 223 )

while (214) reduces to

L = 16e'/c 2 + 4(e' + e4)~uA~9~A + FP""Fp ....

+ (2Cl) - ~ BPQm"BpQmn 4- e4DABmDBAm

__ (2C2) -1GAt3m,, GBAmn (224)

In the following section some of the field equations derived above will be investigated from the physical point of view.

6. FIELD EQUATIONS A N D PHYSICS

Let us consider the physical interpretation of some of the field equations of the previous section.

For the vector fields the variational principle involved a number of terms. An example was given in which only a few terms were chosen, leading to the field equations (162)-(166). There are other possibilities leading to different equations, but this example is enough to bring out some of the features of the present approach. For the spinor fields one can also have many terms, but it is found that one can work with a relatively simple expression and obtain interesting results.


Let us begin with the vector fields. The internal vector space may have many dimensions (for k - - 6 in (25) we have nv= 8). One can imagine that this space is made up of subspaces having different properties. For example, the constant m 1 in (173) and (174) may be different for different subspaces.

Let us suppose, for example, that the connection, or gauge field, BeQ k differs from zero only in the subspace with P = 1, 2. Then it has only one independent component

Bl2k = --B12 k = B21 k -= B~ (225)

If we carry out an orthogonal transformation in this subspace so as to retain the form of the metric (43), the transformation matrix T = ( T e R ) in (61) is given by

(cos 0 - s i n 0 ~ (226) T = \sin 0 cos OJ

According to (75) we get B~ ~/}~ with

/lk = B~ + 0k (227)

The curvature tensor is characterized by Bjk with

Bjk = Bt2jk -- B21jk

and according to (98)

Bjk = Bj,~ - Bk,/

(228)

(229)

The field equation (182) gives

with

BJk:k = 4rcJJ (230)

4 n J j = c 1 j12j (231 )

We see that one can interpret Bj~ as the electromagnetic field, Bj as the electromagnetic potential, JJ as the electric charge-current density vector, and (230) as the Maxwell equations.

Now let us look at the field equations (162)-(174). Equations (t67) and (174) are the equations for a vector boson labeled by the index P and having a mass rn~ in appropriate units. Let us suppose that F e j differs from

90 Rosen and Tauber

zero only in the one-dimensional subspace with P = 1. From (88) we see that DH~=0, so that (164) gives )vl =0 , and (174) reads

Flj~llk = 0 (232)

Hence in this case we get only a massless boson. Let us take the case of two dimensions ( P = 1, 2). We now have

D~zk = - O 2 1 k = D k which may be different from zero. Equation (164) gives

DkF2jk + m12F11 = 0

Dk Ft/k - m~2 F2] = 0 (233)

We have here 8 equations to determine the 4 quantities D ~. In general, there will be solutions only if m~ = 0 and Ftj k is proportional to F2jk (or one of them vanishes). Again we get massless particles.

Now let us consider a subspace of three dimensions ( P = 1, 2, 3) in which Fpj differs from zero. Let us write

D12k = D3k, D23k = DI~, D31k = D2k (234)

Then (164)gives, with a slight generalization,

D2K F3j k -- D3k F2j k - I n 1 2 F u = 0

D 3k Flj ~ - D ~k F3/j~ - m12 F2i = 0

D Ik F2j k - - D 2k F I j k - m°2 F~j = 0

(235)

(236)

In (236) we have assumed that for P = 3 the mass is mo, which may be different from ms. We now have the right number of equations to determine (in general) the components D e o k. In the case of a three-dimensional internal space, we get a triplet of massive particles.

Suppose we also have the electromagnetic field discussed above. Then one gets from (167)

Flj~ = Flj.k -- F l k j + F2jBk - F2k Bj

F 2 j k =- F2;,k - F 2 k , j + FljB~ - F lkB / (237)

F3jk = F3j, k -- F3kj

Let us define

~p+j=Fl j+ iF2 j , ~ g j = F v - i F z j , ~oj = F3j (238)


with corresponding expressions for t)+jk, etc. One finds

t) +jk = t) +j,k - t) +~a - it) +jBk + it) +kBj

t ) - - j k = t ) - j , k - - t ) -- k , j "4- it) _ y B k - it) _ k B~

t) oj, = t) o:,, - O o , , j

From (174) one gets

(239)

t) +jk~k -- iBkt) + j k -~- m12[//+j = 0 (240)

t)--j*;k + iBk t ) - j k + m12t) - j = 0

t)Oik;k + mo 2 t)qj = 0 (241)

We recognize (240) as describing positively and negatively charged vector mesons (in appropriate units) and (241) a neutral vector meson. Equations (240) are invariant under the gauge transformation (227) for Bk provided we also have the transformations

+J = t) +y0 , ~ _j = t) _je i0 (242)

In the Maxwell equations (230), if one makes use of (231) and (179), it is found that one can write

j j = (Cl/87ri)( t) ,k t) +jk __ t ) , k t ) _jk) (243)

SO that the contributions of the particle fields to JJ correspond to the above interpretation.

We see that in a three-dimensional subspace we can have a family con- sisting of two charged and one neutral particle, Their masses may be equal, but one can also have the neutral particle mass different from the others. The existence of various components in the internal subspace can be discussed with the help of the concept of "isospin," which in this case has the value unity. If m o = rn x and B j = O, the field equations are covariant under the transformations representing the group S 0 ( 3 ) .

If the subspace under consideration has more than three dimensions, then the number of independent components of DpQk is larger than the number of equations (164) determining them, so that there is some indeter- minacy in the form of this tensor. However, this is not important since DpQ k does not represent a physical field, as can be seen from the fact that it does not enter into the energy-momentum tensor Tjk as given by (220), (222), (223), and (224). What is important is that for three or more dimensions one can get a family of massive vector bosons. The masses may be so different that it may not be obvious that the particles belong to the same family.

92 Rosen and Tauber

Above, we have taken as an example the case in which the gauge field is the electromagnetic field. However, one can have other gauge fields BeQ~ associated with groups represented by the transformation matrices Tee and transforming according to (75). Terms involving such fields will appear in (167) and (174) describing the bosons.

Now, let us turn to the internal spinor space. This has generally many more dimensions than the vector space. For k = 6, so that n~ = 8, we have ns=60; for k = 7 , n~= 10, n~= 124. Here, too, we can assume that the spinor space is composed of subspaces having different properties.

Let us suppose we have a subspace of n, dimensions in which there is a gauge field GABk, and let us write

where

GACk = --ie3A.Bk + LACK, e = const (244)

where

GACjk = iecSA B~k + GAcjk

Bjk = Bj, k -- B~,s (249)

and GAcj k is given by an expression analogous to (126) with GABS replaced by (~As and

(~AAj k = 0 (250)

Let us go back to (211 ). If we note that the trace of the commutator of two matrices vanishes, we get

BJk;~ = 4xJ j (251)

with JJ = -- (C2 c3/4Tcen u) ~a A 7Jab ~ba (252)

(248)

(~AAk = 0 (245)

If we carry out a transformation satisfying (65), but with SA8 replaced by ei~°SA s and det(SA~)= 1, we get after the transformation

/~k = Bk + 0k (246)

and, corresponding to (113),

A ~ C 1 D A --1 D GA~k=S cG D~(S ) B--S D,~(S ) B (247)

From (126) one finds


Let us choose the constants so that

c2c 3 = - 4 zcn,e ~ (253)

giving

JJ = e~: A 7/:b ¢bA (254)

Comparing Eqs. (246), (249), and (251) with (227), (229), and (230), we see that in both cases we are dealing with an electromagnetic field, but with different sources. If both sources are present, the field is given by the sum of the solutions of (230) and (251).

Now let us consider Eq. (201). If ~ABj=0 , it takes the form

i7'iab(t)bA; ) + ieB/~bA) + m2 ~aA -= 0 (255)

which is the Dirac equation for a fermion of mass m2 and charge e (which could be _ e, with e the electronic charge or some fraction or multiple of it) in appropriate units. The charge-current density vector in (254) has been chosen so as to be consistent with this charge (if ~"A is normalized). There may be subspaces with different values of e. There may also be subspaces with different values of m 2 as given by (200). In particular, there may be subspaces with m 2 = 0 and e.=0, so that (255) is the equation for a neutrino. In the case of the transformation considered above, ~a A goes over into ~"A given by

~"A = e '~° (J~B( S : ~ )B A (256)

which, for SAB= 6~, becomes ~"A = e ¢~°t)~ A. If in a certain subspace we have (255) holding for particles of mass m2

and charge e, we can have another subspace of the same dimensionality in which there is a corresponding equation for the antiparticles with the same mass but charge -e . The equation can be obtained from (255) by "charge conjugation." Let us take the complex conjugate of (255) and write it in the form

-iTJb~(~bA;j-- ieBj~b A) + m2 ~ A = 0 (257)

NOW let us write

~ A = Cab~bA (258)

with Cab (and C ~'b) as defined by (12) and (13). Multiplying (257) by C da and using (13), one can write the resulting equation in the form

iTJab( ~bA;/ - ieBj~ gA ) + m2q} aA = 0 (259)

94 Rosen and Tauber

as describing the antiparticle corresponding to the particle described by (255). The antiparticle current vector corresponding to (254), but with - e , is given by

JJ -~ -,7.~a A ~Jab ~)bA (260)

Besides the difference in the sign of the charge, there may be other differen- ces between particle and antiparticle because of the difference in the internal-space index.

If GABj ~¢ 0, then (201) includes the interaction of the fermion with this gauge field. The fermion may be in a state with a particular value of A, or it may have components with various values. One can imagine a large subspace which is a product of a number of smaller subspaces, so that the index A represents a number of indices associated with the latter, A = (A1, A2,... ), These indices may be quantum numbers related to various properties of the particles. We do not know at present what are the fundamental particles of which all the others are built. If one makes the assumption (which is very doubtful) that the fundamental particles are the quarks and the leptons, then one can associate some of the quantum numbers with particle properties such as "flavor" and "color." One can then go farther and form linear combinations of products of wave functions of different subspaces the transformations of which are irreducible group representations in order to get approximations to the states of mesons and hadrons. However, the present standpoint is that the geometric approach developed here should lead to particles more fundamental than those known at present, so that one must wait tbr thrther experimental progress in order to known the proper form of the equations.

In addition to the gauge fields, there is the field DBA satisfying (204). It can be regarded as describing a family of scalar massless bosons. The Dirac equation (199) contains an interaction term involving DAsj. However, the situation is not satisfactory since the equation for DBA (204) does not contain any interaction with O~A" It appears, therefore, that one should either modify the variational principle o rowhich is simpler--set DAg-~ O, which shall be done.

Now let us turn to the energy-momentum density tensor Tjk as given by (220) and the following equations. The first point to note is that on the right of (224) there is a constant term. This gives a term on the right side of (144) which can be written -Agj~, with A a constant, so that (144) gives the Einstein field equations with a cosmological term. Since the constants in our equations are associated with elementary particles, the curvature of space-time which this term brings about (as one can readily verify) is very large. One can avoid this by imposing the condition

e' = 6el + e3 = 0 (261)

Veetor-Spinor Space and Field Equations 95

so as to make A = 0. We then get from (200)

j~ = -1 , m2 = -4~c (262)

so that we have to take ~c ~< 0 in (92). We now have (with DAe]=0)

Tjk = Tjk + T ;k + Tjk + (T/k (263) (F) (B) j (4')

where the notation is obvious. One can now ask the question: in the limiting case of special relativity, under what conditions will one have Too>0? It will be assumed that, in addition to the Lorentzian metric gik = r/jk, we have the internal metrics (43) and (56) and the Dirac matrices (6) and (8). Using (222), (223), and (224), we find the following contributions from terms in (263):

We see that we

8~z Too =

1 8~ Too= ~ (Feom)2 4-i ~ (Fpmn) 2 (264a)

( F) P,m ~- 0 P,m,n

8~ Too = - ( 4 c l ) 1 ~ (BpQm,,)2 (264b) (B} p,Q

m , n

8TO T o o = - - ( 4 c 2 ) 1 2 GA'Bmn 2 (264c) (G) A',B

m,n

must take cl < 0, c2 < 0. There remains the term

ic370a (@aA b 0 AiIO--~,AIIOObA)--2e4~AO"A (264d)

Let us assume that in the Dirac equation (201) [GAso] ~ m 2 and that Oa A is a solution for a stationary state, so that O"A,0 = --ico~aA (CO = const > 0), with CO nearly equal to m 2. One sees that there will then be two large components ~,3A, ~fi4A, and two small components tplA, ~2 A, i.e., ]@3A], t@4A] >> I~,~AI, lq)2~I. One can then write to a good approximation

8~z Too = 2(c3co 4- c4) 2 { It~/3A[ 2 -l- [I//4A[ 2 } (265) (~) A

Hence, we must take c3 > 0, e4 > 0. However, it should be noted that the right side of (264d) may become negative if some of the assumed conditions are violated, as may happen in the Dirac theory of the electron (negative-energy states, Klein paradox).

We thus see that by setting up a variational principle involving geometrical quantities only, one arrives at field equations for bosons, fermions, and gauge fields.

96 Rosen and Tauber

A P P E N D I X

In addition to the basic terms appearing in the Lagrangian Ls (183) and L 6 (185), one can add more complicated expressions involving the fac- tors k~, k . , ~A ~, or (A . Consider the following combinations:

a a - c~ - c~ b

X bi = k cdlikb , Xbai = k . Ilik ~,

a a ct y.A X Ai = k c~[li~A , xAai : ]£o:~lli % c~

yaAi = ~A=[lika , ~:A 7A ~ c< (AI)

from which we can form (with i, i denoting i, j)

XabiXbai , ~baixabi , xabiXbai , XaAi~Aai , XaA+ YAai

XAai yaAi , yaAi yAai , y B i i yABi , yABi ~:BAi , yBAi yABi (A2)

In constructing a Lagrangian from (A2) the coefficients of terms like X~biXb.i and :?b~iJ?~bi (or yBAi yABi and ~VAs/Y~Ai) must be equal in order to preserve the Hermitian character.

Adding appropriate coefficients ds (i = 1 ..... 7) to these terms, one then obtains

L6' = dl (k"~l Igkb/~l I~]¢b ~/c,f +/c,,/¢11~ kb~l I~k"~ kb/~)

+ d a k ~ l l f J i l i k b ~ k ~, + dsk"~ll f .~l l~A~(Ai3

@ d6(A~lli~ABilif2a~ka B @ dT~A=l[i~A[~lli~B~ B'

Let us now write

(A3)

Ls = L5 + L6 + L6' + L7 (A4)

and carry out the variation indicated in (187). Forming the combinations

a - a a ~ ~ a

M ~k b , M b~A , N A k ~, NA~B~

one gets

2el sJkac( - i W~'bj;k -- WCaj Wabk - fCAj ff bAk )

+ uiW"bj; i + yW"~j W~bj + zF~AjFhA: + ~r~b = O (A5)


where

2e 1 sJe~o ( _ ife'a/l l~ - wb j u ak -- Fbej D Ba~

+ xiFaAjI{j + jWabjFbAj + ZFaBjDBa~ + J.~a = 0 (A6)

2e2sJkeA( -- iF~e~lle - W~F~ee - FaoDCB, )

-x*iF~agl~+ y*W"aj.F~a/+ z*F°~Dea~+ 2~a=O (A7)

2e2sJ~CA( -- iD~cyll~ + F~QFb~k + DEQDBE~ )

+ u*iD~AjlI~+ v*F~A~F.~.+ z*DI~cjDCA~+ Z ~ = O (A8)

U = e 3 + d 4 - 2di,

x = e 3 + d s - d 2 ,

y = e3 + 2d4 - 2dl,

z = e 3 + ds - 2d2 + d6,

Assuming again tha t [see (192), (193)]

W"b/ = ~:7 /'b and

these equat ions become

U * = --e 4 + 2 d 3 - d 7

x * = e 4 + d6 - -d 2

y* = e4 + d5 - 2d2 + d6

z* = e4 - 2d3 + 2d7

f a A i ~ ~/.iab l~tbA

(A9)

a~b = - 4 ( 6 e , + y)K23; - (6e~ + z)y/ac.'//%t)c A ~d A (A10)

2~A = --(6ei + x ) i g J a b ~ b a l l j - - 4(6el + y)~cOaa - (6el + z)yJ"bDeAjObe (A11 ) ca j k B • a b a b Z A = 2e2s A(zTi ht~ ~llk + KS)k bit s B + 7jabDCB/<~ec)

+ ix*7J~'b ~seatl /- 4Ky*~s"A - Z*~//~bOeA/l l lb B (A12)

ZBA = 2e2sJkCa( iDeotle + Sje~b ~Bt~b c + DBeDUce )

+ iu*DBagli/- 4 y * ~ u ~ A -- z*D~c/DCai (At3)

Eliminating '~"A between (A11) and (A12) gives

.~- F s J k B ~, a "l,ltb i(7k~bg)BA ~2 a~j bt'r BIIk +m3$~A + f~TJabDB.~bB

+ f 2( ~CS~kBA Sjk~b + sJkCA 7J~b DBck ) O e e = 0 (A t4)

fz = 2e2/(6el + x + x*)

f3 = (@1 +z+z*) / (6e~ + x + x * )

m 3 = 4~c(6e~ + y + y*)/(6el + x + x * )

with

825/17/1.7

98 Rosen and Tauber

Comparing (A13) with the corresponding equation for "~A ~ and noting that rBA = "?A e, we get an equation which can be written in matrix notation as

2iu*D~llj - 2ie2[s :k, D:II~ ] - 2e2[s j~, F:- F~ + DjD~] = 0 (A15)

where the brackets denote commutators and

Making use of (192) and (193), one can write L~ (183) as

L~ = 9 6 e ~ ~ + 24e~ ~A ~ - 2e~s:~ A s j ~ ~ A tY~B

- 2e2 s / ~ D~c;DCe~ (A 17)

Noting also that k ~l/kb~ = - k b l l / k . ~ = iWba /

- '" = -A - ~ = i lLA j

(AIS) ~ A~!l./k"~ = -k"~H: ~ A ~ = iF"A:

we can write L6, (A3) in the form

L6 '= 16~c2(d4 - 2d~) + 4(d 5 - 2d2 + d6)tp~A ~ A + (d7 - 2d3)DA~jD~Aj (A19)

so that L 6 q- L 6' becomes

L 6 + L 6 ' = 1 6 t c Z U * + 4 ( X + X * ) ~ A ~ ] S - - u * D A ~ D B A j (A20)

With Q"A given by the left-hand side of (A14) the Lagrangian L 9 (208) can be written as

L 9 / c 3 = t 'Th'"h(~S IphAii/,: - - ~aAtlk(pbA ) -}- 2m 3 ~a A ~ a A

+ 2f3 KA~DSAj + if2sJkBA KASjlIk + 2J'2SJkBA Sjk"b tP. A tPb~

+ f 2 K A B j ( D B c k s J k C A - - SJkBcDCAk ) (A21)

where

Varying GABj

notation (211 )

KA~j = ~fb ~ A ~b e (A22)

then gives an equation which can be written in matrix

iGJk;k = I-(a) + I j


where L~at is given by (212) as before, but

i j = c2c3 Kj + l c2c3 f2[Kk , s jk ] -4- ic~u[D, IY]

+ icec3f3[D, K j] + ic2e2[D, [s ~k, l ) k ] ]

+ ½ic2c3f2[D, Kk, s i~]] (A23)

It the energy m o m e n t u m tensor (220), pjx (221) remains unchanged, but in Qjk (222) f l is replaced by f3 and e 4 by u = e4 + dv - 2d6. It is now possible to choose the constants so that some of the terms vanish. For example, if u = 0 the term involving DABjDBAk in the energy m o m e n t u m tensor wilt d rop out. Of course, the greatest simplification is achieved if f2 = 0 (which implies e 2 = 0, as has been assumed in the text).

R E F E R E N C E S

1. A. Einstein and W. Mayer, Sitzungsber. Preuss. Akad. Wiss., 541 (1931). 2. Th. Kaluza, Sitzungsber. Preuss. Akad. Wiss., 966 (1921). 3. N. Rosen and G, E, Tauber, Found. Phys. 14, 171 (1984). 4. C. N. Yang and R. L, Mills, Phys. Rev. 96, 191 (1954). 5. B. F. Laurent, Ark. ±~vs. 16, 263 (1959). 6. O. Klein, Ark. t:[vs. 17, 517 (1960).

vector-spinor space and field equations

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