killing symmetries of generalized minkowski spaces. i. algebraic-infinitesimal structure of...

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0015-9018/04/0400-0617/0 © 2004 Plenum Publishing Corporation

Foundations of Physics, Vol. 34, No. 4, April 2004 (© 2004)

Killing Symmetries of Generalized Minkowski Spaces.I. Algebraic-Infinitesimal Structure of SpacetimeRotation Groups

Fabio Cardone,1 ,2 Alessio Marrani,3 and Roberto Mignani2–5

1Dipartimento di Fisica, Università dell’Aquila, Via Vetoio, 67010 Coppito, L’Aquila, Italy.2 I.N.D.A.M.-G.N.F.M.3Università degli Studi ‘‘Roma Tre,’’ Via della Vasca Navale 84, 00146 Roma, Italy.4 I.N.F.N.-Sezione di Roma III.5 To whom correspondence should be addressed: e-mail: [email protected]

Received August 25, 2003; revised October 14, 2003

In this paper, we introduce the concept of N-dimensional generalized Minkowskispace, i.e., a space endowed with a (in general non-diagonal) metric tensor, whosecoefficients do depend on a set of non-metrical coordinates. This is the first of aseries of papers devoted to the investigation of the Killing symmetries of general-ized Minkowski spaces. In particular, we discuss here the infinitesimal-algebraicstructure of the space-time rotations in such spaces. It is shown that the maximalKilling group of these spaces is the direct product of a generalized Lorentz groupand a generalized translation group. We derive the explicit form of the generatorsof the generalized Lorentz group in the self-representation and their related, gen-eralized Lorentz algebra. The results obtained are specialized to the case of a4-dimensional, ‘‘deformed’’ Minkowski space M46 , i.e., a pseudoeuclidean spacewith metric coefficients depending on energy.

KEY WORDS: generalized Minkowski spaces; Killing equations; infinitesimalgenerators; generalized Poincaré algebra.

1. INTRODUCTION

In the last years, two of the present authors (F.C. and R.M.) proposed ageneralization of SR based on a ‘‘deformation’’ of space-time, assumed tobe endowed with a metric whose coefficients depend on the energy of theprocess considered. (1) Such a formalism (Deformed Special Relativity, DSR)

applies in principle to all four interactions (electromagnetic, weak, strongand gravitational)—at least as far as their nonlocal behavior and non-potential part is concerned—and provides a metric representation of them(at least for the process and in the energy range considered). (1–5) Moreover,it was shown that such a formalism is actually a five-dimensional one,in the sense that the deformed Minkowski space is embedded in a largerRiemannian manifold, with energy as fifth dimension. (6)

In this paper, we introduce the concept of N-dimensional generalizedMinkowski space, i.e., a space endowed with a (in general non-diagonal)metric tensor, whose coefficients do depend on a set of non-metrical coor-dinates. The deformed space-time M4

6 of DSR is just a special case of suchspaces. This is the first of a series of papers devoted to the investigation ofthe Killing symmetries of generalized Minkowski spaces. In particular, weshall discuss here the infinitesimal-algebraic structure of the space-timerotations in such spaces.The organization of the paper is as follows. In Sec. 2 we briefly review

the formalism of DSR and of the deformed Minkowski space M46 . Gener-

alized Minkowski spaces are defined in Sec. 3.1. In Sec. 3.2 we look for thegroup of isometries of such spaces by means of the Killing equations. It isshown that the maximal Killing group of these spaces is the semidirectproduct of a generalized Lorentz group and a generalized translationgroup. The infinitesimal structure of the generalized Lorentz group is dis-cussed in Sec. 4, where we derive the explicit form of its generators inthe self-representation and their related, generalized Lorentz algebra. Thespecial case of the deformed space M4

6 of DSR is considered in Sec. 5.Section 6 concludes the paper.

2. DEFORMED SPECIAL RELATIVITY IN FOUR DIMENSIONS(DSR4)

The generalized (‘‘deformed’’) Minkowski space M46 (DMS4) is

defined as a space with the same local coordinates x of M4 (the four-vectors of the usual Minkowski space), but with metric given by the metrictensor6

6 In the following, we shall employ the notation ‘‘ESCon’’ (‘‘ESCoff ’’) to mean that theEinstein sum convention on repeated indices is (is not) used.

gmn, DSR4(x5) = diag(b20(x5), −b21(x

5), −b22(x5), −b23(x

5))

=ESC offdmn[b20(x

5) dm0−b21(x

5) dm1−b22(x

5) dm2−b23(x

5) dm3], (1)

618 Cardone, Marrani, and Mignani

where the {b2m(x5} are dimensionless, real, positive functions of the inde-

pendent, non-metrical (n.m.) variable x5.7 The generalized interval in M46

7 Such a coordinate is to be interpreted as the energy (see refs. 1–5); moreover, the index 5explicitly refers to the above-mentioned fact that the deformed Minkowski space can be‘‘naturally’’ embedded in a five-dimensional (Riemannian) space. (6)

is therefore given by (xm=(x0, x1, x2, x3)=(ct, x, y, z), with c being theusual light speed in vacuum)

ds2=b20c2 dt2−(b21 dx

2+b22 dy2+b23 dz

2)=gmn, DSR4 dxm dxn=dx f dx. (2)

The last step in (2) defines the scalar product f in the deformed MinkowskispaceM46 . In order to emphasize the dependence of DMS4 on the variable x5,

we shall sometimes use the notation M46 (x5). It follows immediately that it

can be regarded as a particular case of a Riemann space with null curva-ture. From the general condition

gmn, DSR4(x5) gnrDSR4(x

5)=dm r (3)

we get for the contravariant components of the metric tensor

gmnDSR4(x5) = diag(b−20 (x

5), −b−21 (x5), −b−22 (x

5), −b−23 (x5))

=ESC offdmn(b−20 (x

5) dm0−b−21 (x5) dm1−b−22 (x

5) dm2−b−23 (x5) dm3). (4)

Let us stress that metric (1) is supposed to hold at a local (and notglobal) scale. We shall therefore refer to it as a ‘‘topical’’ deformed metric,because it is supposed to be valid not everywhere, but only in a suitable( local) space-time region (characteristic of both the system and the inter-action considered).The two basic postulates of DSR4 (which generalize those of standard

SR) are: (1)

1. Space-Time Properties: Space-time is homogeneous, but space isnot necessarily isotropic; a reference frame in which space-time isendowed with such properties is called a ‘‘topical’’ reference frame(TIRF). Two TIRF’s are in general moving uniformly withrespect to each other (i.e., as in SR, they are connected by a‘‘inertiality’’ relation, which defines an equivalence class of .3

TIRF);

2. Generalized Principle of Relativity (or Principle of Metric Invariance):All physical measurements within each TIRF must be carried outvia the same metric.

Killing Symmetries of Generalized Minkowski Spaces 619

The metric (1) is just a possible realization of the above postulates. Werefer the reader to refs. 1–5 for the explicit expressions of the phenome-nological energy-dependent metrics for the four fundamental interactions.8

8 Since the metric coefficients b2m(x5) are dimensionless, the n.m. coordinate has the form

x5=EE0

where E0 plays the role of a threshold energy, characteristic of the interaction considered (seeRefs. 1–5).

3. MAXIMAL KILLING GROUP OF N-d. GENERALIZEDMINKOWSKI SPACES

3.1. Generalized Minkowski Spaces

We shall call generalized Minkowski space MN6 ({x}n.m.) a N-dimen-

sional Riemann space with a global metric structure determined by the(in general non-diagonal) metric tensor gmn{x}n.m. (m, n=1, 2,..., N), where{x}n.m. denotes a set of Nn.m. non-metrical coordinates (i.e., different fromthe N coordinates related to the dimensions of the space considered). Theinterval inMN

6 ({x}n.m.) therefore reads

ds2=gmn({x}n.m.) dxm dxn. (5)

We shall assume the signature (T, S) (T timelike dimensions and S=N−Tspacelike dimensions). It follows that MN

6 ({x}n.m.) is flat, because all thecomponents of the Riemann–Christoffel tensor vanish.Of course, an example is just provided by the 4-d. deformed

Minkowski spaceM46 (x5).

3.2. Killing Equations in a Generalized Minkowski Space

The Lie groups of isometries of a N-dimensional Riemannian spaceare determined by the N(N+1)/2 Killing equations

tm(x); n+tn(x); m=0 Z t[m(x); n]=0, (6)


where the bracket [..] means symmetrization with respect to the enclosedindices; m denotes, as usual, covariant derivation with respect to the coor-dinate xm, and the contravariant Killing vector tm(x) is the infinitesimalvector of coordinate transformation

xmŒ=xm+dxm(x) (7)

namely

dxm(x)=tm(x). (8)

In the case of a N-d. generalized Minkowski space MN6 ({x}n.m.), being

(as noted above) a special case of a Riemann space with constant (zero)curvature, we can state that it is a maximally symmetric space, i.e., admitsa maximal Killing group with N(N+1)/2 parameters. Moreover, covari-ant derivative reduces to ordinary derivative m=“/“xm, whence the Killingequations (6) become

t[m(x), n]=0Z tm(x), n+tn(x), m=0Z“tm(x)“xn

+“tn(x)“xm

=0. (9)

By virtue of the Poincaré–Birkhoff–Witt (P.B.W.) theorem, and of theLie theorems, any finite element g of a Lie group GL of orderM, acting onMN6 ({x}n.m.), can be written in the exponential form

g=exp 1 CM

i=1aiT i2 , (10)

where {T i}i=1...M is the generator basis of the Lie algebra of GL, and{ai}i=1...M ¥ RM({ai}={ai(g)}).Therefore, by a series development of the exponential:

g=exp 1 CM

i=1aiT i2=C

.

k=0

1k!1 CM

i=1ai(g) T i2

k

, (11)

we get, for an infinitesimal element (gQ dg) (Z {ai(g)}i=1...M ¥ RMQ

{ai(g)}i=1...M ¥ I0 … RM):

dg=1+CM

i=1ai(g) T i+O({a

2i (g)}). (12)


Then, -x ¥MN6 ({x}n.m.), one has, for the action of a finite and an

infinitesimal element of GL, respectively:

gx=5exp 1 CM

i=1ai(g) T i26 x

=5 C.

k=0

1k!1 CM

i=1ai(g) T i2

k6 x=xŒ ¥MN6 ({x}n.m.); (13)

(dg) x=51+CM

i=1ai(g) T i6 x=x+1 C

M

i=1ai(g) T i2 x=xŒ ¥MN

6 ({x}n.m.)

dg: MN6 ({x}n.m.) ¦ xQ xŒ=x+dx(g)(x) ¥MN

6 ({x}n.m.)

ˇS dx(g)(x)=1 C

M

i=1ai(g) T i2 x. (14)

In index notation, Eq. (14) can be written as

dxm(g)(x¯)=51 C

M

i=1ai(g) T i2 x¯

6m, m=1,..., N. (15)

Moreover, from (8) one gets

tm(g)(x)=51 CM

i=1ai(g) T i2 x6

m

. (16)

We can now define the mixed 2-rank N-tensor dwmn(g, {x}n.m.) ofinfinitesimal transformation (associated to dg ¥ GL) as:

dxm(g)(x, {x}n.m.)=51 CM

i=1ai(g) T i({x}n.m.)2 x6

m

— dwmn(g, {x}n.m.) xn. (17)

The number of independent components of tensor dwmn(g, {x}n.m.) is equalto the order M of the Lie group; in general, nothing can be said about itssymmetry properties. From Eqs. (8), (15), (16) it follows

tm(g)(x, {x}n.m.)=dwmn(g, {x}n.m.) xn, (18)

which shows that dwmn(g, {x}n.m.) is the tensor of the rotation parametersinMN6 ({x}n.m.).9

9 In the following, for simplicity of notation, we shall often omit the explicit dependence ofquantities on the non-metrical coordinates {x}n.m..


Since we are looking for the Killing groups of MN6 ({x}n.m.) (not

necessarily maximal), we have (from Eqs. (15) and (16)) :

t(g) m(x), n+t(g) n(x), m=0 Z dx(g) m(x), n+dx(g) n(x), m=0

Z51 C

M

i=1ai(g) T i2 x6

m, n+51 C

M

i=1ai(g) T i2 x6

n, m=0

Z (dwmr(g) xr), n+(dwnr(g) xr), m=0

Z (dw[mr(g) xr), n]=0. (19)

The last equation implies antisymmetry of dwmn(g):

dwmn(g)+dwnm(g)=0 (20)

which therefore has N(N−1)/2 independent components (such a number,as stressed above, is also equal to the order M of GL, M=N(N−1)/2),i.e., the (rotation) transformation group related to the tensor dwmn(g) is aN(N−1)/2-parameter Killing group.Since a N-d. generalized Minkowski space is maximally symmetric,

we have still to find another N-parameter Killing group of MN6 ({x}n.m.)

(because N+N(N−1)/2=N(N+1)/2).This is easily done by noting that the N(N+1)/2 Killing equations (9)

in such a space are trivially satisfied by constant covariant N-vectorstm ] tm(x), to which there corresponds the infinitesimal transformation

dg: xmQ xmŒ(x, {x}n.m.)=xm+dxm(g)({x}n.m.)=x

m+tm(g)({x}n.m.) (21)

with dxm(g)({x}n.m.), tm(g)({x}n.m.) constant (with respect to x

m).In conclusion, a N-d. generalized Minkowski space MN

6 ({x}n.m.)admits a maximal Killing group which is the (semidirect) product of the Liegroup of N-dimensional space-time rotations (or N-d. generalized, homo-geneous Lorentz group SO(T, S}(N(N−1)/2GEN ) with N(N−1)/2 parameters,and of the Lie group of N-dimensional space-time translations Tr(T, S}NGENwith N parameters:

P(T, S)N(N+1)/2GEN =SO(T, S)N(N−1)/2GEN és Tr(T, S)NGEN (22)

We will refer to it as the generalized Poincaré (or inhomogeneous Lorentz)group P(S, T)N(N+1)/2GEN .

3.2.1. Solving the Killing Equations in a 4-d. Generalized Minkowski Space

We want now to find the explicit solutions of the Killing equationsin a 4-d. generalized Minkowski space M4

6 ({x}n.m.) (S [ 4, T=4−S).A covariant Killing 4-vector tm(x0, x1, x2, x3) must satisfy Eq. (9), namely


˛“t0({x}m.)“x0

=0,

“t0({x}m.)“x1

+“t1({x}m.)“x0

=0,

“t0({x}m.)“x2

+“t2({x}m.)“x0

=0,

“t0({x}m.)“x3

+“t3({x}m.)“x0

=0,

“t1({x}m.)“x1

=0,

“t1({x}m.)“x2

+“t2({x}m.)“x1

=0,

“t1({x}m.)“x3

+“t3({x}m.)“x1

=0,

“t2({x}m.)“x2

=0,

“t2({x}m.)“x3

+“t3({x}m.)“x2

=0,

“t3({x}m.)“x3

=0.

(23)

From the 1st, 5th, 7th, and 10th of these equations, one gets

t0=t0(x1, x2, x3), t1=t1(x0, x2, x3),

t2=t2(x0, x1, x3), t3=t3(x0, x1, x2).(24)

Solving system (23) is cumbersome but straightforward. The finalresult is

˛t0({x}m.)=−z1x1−z2x2−z3x3+T0,

t1({x}m.)=z1x0+h2x3−h3x2−T1,

t2({x}m.)=z2x0−h1x3+h3x1−T2,

t3({x}m.)=z3x0+h1x2−h2x1−T3,

(25)

where z i, h i(i=1, 2, 3) and Tm(m=0, 1, 2, 3) are real coefficients.


We can draw the following conclusions:

10 In fact, although we discussed explicitly the 4-d. case, the extension to the generic N-d. caseis straightforward.

1. In spite of the fact that no assumption was made on the functionalform of the Killing vector, we got a dependence at most linear(linear inhomogeneous) on metric coordinates for all componentsof tm({x}m.). Therefore, in order to determine the (maximal)Killing group of a generalized Minkowski space,10 one can,without loss of generality, consider only groups whose transfor-mation representation is implemented by transformations at mostlinear in the coordinates.

11 Indeed

tm(g)(x¯)=gmn, DSR4({x}n.m.) t

n(g)((x¯

), {x}n.m.)=gmn, DSR4({x}n.m.) dwnr(g, {x}n.m.) xr=dwmr(g) xr.

2. In general, tm ] tm({x}n.m.), i.e., the covariant Killing vector doesnot depend on possible non-metric variables.11 On the contrary,the contravariant Killing 4-vector does indeed, due to the depen-dence of the fully contravariant metric tensor on {x}n.m.:

tm({x}m., {x}n.m.)=gmn({x}n.m.) tn({x}m.). (26)

Such a result is consistent with the fact that dwmn(g), unlikedwmn(g), {x}n.m.), is independent of {x}n.m. (cf. (18) and (19)).

3. Solution (25) does not depend on the metric tensor. This impliesthat all 4-d. generalized Minkowski spaces admit the same covari-ant Killing 4-vector. It corresponds to the covariant 4-vector ofinfinitesimal transformation of the space-time roto-translational groupof M46 ({x}n.m.). Therefore, assuming the signature (+, −, −, −)

(i.e., S=3, T=1), in a basis of ‘‘length-dimensional’’ coordinates,we can state that: (a) z=(z1, z2, z3) is the 3-vector of dimen-sionless parameters (‘‘rapidity’’) of generalized 3-d. boost; (b)h=(h1, h2, h3) is the 3-vector of dimensionless parameters (angles)of generalized 3-d. rotation; (c) Tm=(T0, −T1, −T2, −T3) is thecovariant 4-vector of (‘‘length-dimensional’’) parameters ofgeneralized 4-d. translation.


4. INFINITESIMAL STRUCTURE OF GENERALIZED SPACE-TIMEROTATION GROUPS

4.1. Finite-Dimensional Representation of Infinitesimal Generators andGeneralized Lorentz Algebra

As in the standard special-relativistic case, we can decompose themixed N-tensor of infinitesimal transformation parameters dwmn(g, {x}n.m.)(see (17)) as:12

12 The factor 12 is inserted only for further convenience.

dwmn(g, {x}n.m.)=12 dwab(g)(I

ab)mn ({x}n.m.), (27)

i.e., as a linear combinationofN(N−1)/2matrices (independent of the groupelement g) {(Iab)mn ({x}n.m.)}a, b=1...N 13 with coefficients {dwab(g)}a, b=1...N.

13 The pair of indices (a, b) labels the different infinitesimal group generators, whereas—in the(N( <.)-dimensional) matrix representation of the generators we are considering—thecontrovariant (covariant) index is a row (column) index. This latter remark holds true fordwmn(g, {x}n.m.), too.

Such matrices represent the infinitesimal generators of the space-time rota-tional component of the maximal Killing group of MN

6 ({x}n.m.). Sincein this case dwmn(g) is antisymmetric (see (20)), the basis matrices{(Iab)mn ({x}n.m.)}a, b=1...N, too, are antisymmetric in indices a and b:

{(Iab)mn ({x}n.m.)}a, b=1...N=−{(Iba)mn ({x}n.m.)}a, b=1...N. (28)

For the fully covariant dwmn(g) the analogous decomposition reads

dwmn(g)=gmr({x}n.m.) dwrn(g), {x}n.m.)

=12 dwab(g) gmr({x}n.m.)(I

ab)rn ({x}n.m.)

=12 dwab(g)(I

ab)mn ({x}n.m.). (29)

But, since dwmn(g) is independent of {x}n.m., the same holds for itscomponents dwab(g), and therefore (29) implies

(Iab)mn ] (Iab)mn ({x}n.m.). (30)

In order to find the explicit form of the infinitesimal generators in theN-d. matrix representation, let us exploit the antisymmetry of dwmn(g):

dwmn(g)=−dwnm(g)Z dwmn(g)

=12 (dwmn+dwmn)=

12 (dwmn−dwnm)

=12 gam gbn dwab−

12 gbm gan dwab=

12 dwab(g

am gbn−gbm gan)

=12 dwab(g)(d

amdbn−dbmdan). (31)


Comparing (29) and (31) yields:14

14 Equation (32) shows clearly that the factors with a non-metric dependence in gmr({x}n.m.)e(Iab)rn ({x}n.m.) cancel each other (cf. Eq. (30)). The same does not happen when both mand n are contravariant:

(Iab)mn :=gmr({x}n.m.) gns({x}n.m.)(Iab)rs

=gmr({x}n.m.) gns({x}n.m.)(dardbs −d

brdas)

=gma({x}n.m.) gnb({x}n.m.)−gmb({x}n.m.) gna({x}n.m.).

gmr({x}n.m.)(Iab)rn ({x}n.m.) — (Iab)mn=(damdbn−dbmdan). (32)

We get therefore the following explicit form for the mixed matrix represen-tation of the generators:15

15We have, analogously,

(Iab)m n({x}n.m.)=gnr({x}n.m.)(Iab)mr

=gnr({x}n.m.)(damdbr−dbmdar)

=gnb({x}n.m.) dam−gna({x}n.m.) dbm.

(Iab)m n({x}n.m.)=gnr({x}n.m.)(Iab)rn

=gmr({x}n.m.)(dardbn−dbrdan)

=gma({x}n.m.) dbn−gmb({x}n.m.) dan. (33)

It is easy to see that the generators {(Iab)mn ({x}n.m.)}a, b=1,..., N satisfythe following Lie algebra:

[Iab({x}n.m.), Irs({x}n.m.)]

=gas({x}n.m.) Ibr({x}n.m.)+gbr({x}n.m.) Ias({x}n.m.)

−gar({x}n.m.) Ibs({x}n.m.)−gbs({x}n.m.) Iar({x}n.m.). (34)

Equation (34) defines the generalized Lorentz algebra, associated to thegeneralized, homogeneous Lorentz group SO(T, S)N(N−1)/2GEN of the N-d.generalized Minkowski spaceMN

6 ({x}n.m.).


4.2. The Case of a 4-Dimensional Generalized Minkowski Space

4.2.1. Self-Representation of the Infinitesimal Generators

Let us specialize the results of the previous subsection to a 4-d. gener-alized Minkowski space. Assuming therefore that Greek indices take thevalues {0, 1, 2, 3}, and a signature (S [ 4, T=4−S), we can write explicitlythe generator (ab) of SO(S, T=4−S)GEN as the antisymmetric matrix:

Iab({x}n.m.)

=R0 I01({x}n.m.) I02({x}n.m.) I03({x}n.m.)

−I01({x}n.m.) 0 I12({x}n.m.) I13({x}n.m.)−I02({x}n.m.) −I12({x}n.m.) 0 I23({x}n.m.)−I03({x}n.m.) −I13({x}n.m.) −I23({x}n.m.) 0

S . (35)Like any rank-2, antisymmetric 4-tensor, Iab({x}n.m.) can be expressed

in terms of an axial and a polar 3-vector. By introducing the followinginfinitesimal generators (i, j, k=1, 2, 3, ESC on throughout)

S i({x}n.m.) —12 EijkI jk({x}n.m.),

K i({x}n.m.) — I0i({x}n.m.)(36)

(where Eijk is the rank-3, fully antisymmetric Levi-Civita 3-tensor withE123 — 1), components of the axial 3-vector

S({x}n.m.) — (I23({x}n.m.), I31({x}n.m.), I12({x}n.m.)) (37)

and of the polar one

K({x}n.m.) — (I01({x}n.m.), I02({x}n.m.), I03({x}n.m.)), (38)

Iab({x}n.m.) can be rewritten as

Iab({x}n.m.)

=R0 K1({x}n.m.) K2({x}n.m.) K3({x}n.m.)

−K1({x}n.m.) 0 S3({x}n.m.) −S2({x}n.m.)−K2({x}n.m.) −S3({x}n.m.) 0 S1({x}n.m.)−K3({x}n.m.) S2({x}n.m.) −S1({x}n.m.) 0

S . (39)

The set of generators S({x}n.m.), K({x}n.m.) constitute the self-repre-sentation basis for SO(S, T=4−S)GEN. Unlike the case of standard


SR—where S, K do represent the rotation and boost generators, respecti-vely,—one cannot give them a precise physical meaning, because this latterdepends on both the number S of spacelike dimensions and the assignmentof dimensional labelling (we left here unspecified).

4.2.2. Decomposition of the Parametric 4-Tensor dwmn(g)

We can now exploit the self-representation form of the infinitesimalgenerators of SO(S, T=4−S)GEN (S [ 4) to decompose the infinitesimalparametric 4-tensor dwmn(g).Equation (17), on account of (27), can be written as

dxm(g)({x}m., {x}n.m.)=dwmn(g, {x}n.m.) xn

=12 dwab(g)(I

ab)mn ({x}n.m.) xn, (40)

which is valid in the general case of SO(S, T=N−S)GEN.In the case N=4, Eq. (40) reads, on account of (36) and (37)):

dxm(g)({x}m., {x}n.m.)

=dwmn(g, {x}n.m.) xn=12 dwab(g)(I

ab)mn ({x}n.m.) xn

=12 dwij(g)(I

ij)mn ({x}n.m.) xn+dw0i(g)(I0i)mn ({x}n.m.) xn

=12 dwij(g)(E

ijl Sl({x}n.m.))mn xn+dw0i(g)(K i)mn ({x}n.m.) xn

=12 Eijl dwij(g)(S

l)mn ({x}n.m.) xn+dw0i(g)(K i)mn ({x}n.m.) xn. (41)

We can define an axial and a polar parametric 3-vector by

hi(g) — −12 Ei

jk dwjk(g), (42)

zi(g) — −dw0i(g), (43)

namely

h(g) — (−dw23(g), −dw31(g), −dw12(g)), (44)

z(g) — (−dw01(g), −dw02(g), −dw03(g)). (45)

Therefore, dwab(g) can be written in matrix form as

dwab(g)=R0 −z1(g) −z2(g) −z3(g)z1(g) 0 −h3(g) h2(g)z2(g) h3(g) 0 −h1(g)z3(g) −h2(g) h1(g) 0

S . (46)


Having left the number S of spacelike dimensions and the dimensionallabelling unspecified, we cannot attribute a physical meaning to the para-metric 3-vectors (44), (45) (unlike the case of standard SR, where h(g) andz(g) are the space rotation and boost parameters, respectively).Equation (41) can be rewritten in terms of the 3-d. Euclidean scalar

product · as:

dxm(g)({x}m., {x}n.m.)=−hl(g)(Sl)mn ({x}n.m.) xn−zi(g)(K i)mn ({x}n.m.) xn

=[−h(g) ·S({x}n.m.)− z(g) ·K({x}n.m.)]mn xn. (47)

5. SPACE-TIME ROTATION COMPONENT OF THE KILLINGGROUP IN A 4-D. DEFORMED MINKOWSKI SPACE

5.1. Deformed Homogeneous Lorentz group SO(3, 1)DEF andSelf-Representation Basis of Infinitesimal Generators

We want now to specialize the results obtained to the case of DSR4,i.e., considering a 4-d. deformed Minkowski spaceM4

6 (x5).Let us recall that the N-dimensional representation of the infinitesimal

generators of the Killing group in a N-d. generalized Minkowski space isdetermined (by means of Eq. (33)) by the mere knowledge of its metrictensor.In the DSR4 case we have therefore

(Iab)mn, DSR4 (x5)

=gmrDSR4(x5)(Iab)rn, DSR4=g

mrDSR4(x

5)(dardbn−dbrdan)

=gmaDSR4(x5) dbn−g

mbDSR4(x

5) dan

=ESC off dma(b−20 (x5) dm0−b−21 (x

5) dm1−b−22 (x5) dm2−b−23 (x

5) dm3) dbn

−dmb(b−20 (x5) dm0−b−21 (x

5) dm1−b−22 (x5) dm2−b−23 (x

5) dm3) dan. (48)

From Eq. (48) we get the following 4×4 matrix representation of theinfinitesimal generators of the deformed homogeneous Lorentz groupSO(3, 1)6DEF (space-time rotation component of the deformed Poincarégroup P(3, 1)10DEF):


I10DSR4(x5)=R

0 −b−20 (x5) 0 0

−b−21 (x5) 0 0 0

0 0 0 00 0 0 0

S , (49)

I20DSR4(x5)=R

0 0 −b−20 (x5) 0

0 0 0 0−b−22 (x

5) 0 0 00 0 0 0

S , (50)

I30DSR4(x5)=R

0 0 0 −b−20 (x5)

0 0 0 0

0 0 0 0

−b−23 (x5) 0 0 0

S , (51)

I12DSR4(x5)=R

0 0 0 00 0 −b−21 (x

5) 00 b−22 (x

5) 0 00 0 0 0

S , (52)

I23DSR4(x5)=R

0 0 0 00 0 0 00 0 0 −b−22 (x

5)0 0 b−23 (x

5) 0

S , (53)

I31DSR4(x5)=R

0 0 0 00 0 0 b−21 (x

5)0 0 0 00 −b−23 (x

5) 0 0

S . (54)

Comparing Eqs. (49)–(54) with the 4-d. matrix representation of theinfinitesimal generators of the standard homogeneous Lorentz groupSO(3, 1) shows that deforming the metric structure implies the loss ofsymmetry of the boost generators and of antisymmetry of space-rotationgenerators.


The antisymmetry of the generators in the labelling indices (ab) stillholds:

{(Iab)mn, DSR4 (x5)}a, b=0, 1, 2, 3=−{(Iba)mn, DSR4 (x5)}a, b=0, 1, 2, 3 (55)

or

IabDSR4(x5)=−IbaDSR4(x

5) (a, b=0, 1, 2, 3). (56)

Therefore, there are only 6 independent generators. In matrix form

IabDSR4(x5)=R

0 I01DSR4(x5) I02DSR4(x

5) I03DSR4(x5)

−I01DSR4(x5) 0 I12DSR4(x

5) I13DSR4(x5)

−I02DSR4(x5) −I12DSR4(x

5) 0 I23DSR4(x5)

−I03DSR4(x5) −I13DSR4(x

5) −I23DSR4(x5) 0

S . (57)We can now pass to the self-representation basis of the generators of

SO(3, 1)DEF by introducing the following axial and polar 3-vectors bymeans of the Levi-Civita tensor

S iDSR4(x5) — 1

2 EijkIjkDSR4(x

5), (58)

K iDSR4(x5) — I0iDSR4(x

5), (59)

or

SDSR4(x5) — (I23DSR4(x

5), I31DSR4(x5), I12DSR4(x

5)), (60)

KDSR4(x5) — (I01DSR4(x

5), I02DSR4(x5), I03DSR4(x

5)). (61)

Then, IabDSR4(x5) can be written as:

IabDSR4(x5)=R

0 K1DSR4(x5) K2DSR4(x

5) K3DSR4(x5)

−K1DSR4(x5) 0 S3DSR4(x

5) −S2DSR4(x5)

−K2DSR4(x5) −S3DSR4(x

5) 0 S1DSR4(x5)

−K3DSR4(x5) S2DSR4(x

5) −S1DSR4(x5) 0

S . (62)

In DSR4, like in the SR case, we can identify (apart from a sign)S iDSR4(x

5) with the infinitesimal generator of the deformed 3-d. space


rotation around x i5, and K iDSR4(x5) with the infinitesimal generator of the

deformed Lorentz boost with motion direction along x i5.

5.2. Decomposition of the Parametric 4-Tensor dwmn(g) in DSR4

We can now specialize Eq. (41) (expressing the infinitesimal variationof the contravariant 4-vector xm in the self-representation) to the DSR4case, getting (ESC on):

dxm(g), DSR4({x}m., x5)=dwmn, DSR4(g, x5) xn

=12 dwab, DSR4(g)(I

ab)mn, DSR4 (x5) xn

=12 Eijl dwij, DSR4(g)(S

l)mn, DSR4 (x5) xn

+dw0i, (DSR4)(g)(K i)mn, DSR4 (x5) xn. (63)

Therefore, the parametric 4-tensor dwmn(g) can be written as

dwab(g)=R0 −z1(g) −z2(g) −z3(g)z1(g) 0 −h3(g) h2(g)z2(g) h3(g) 0 −h1(g)z3(g) −h2(g) h1(g) 0

S , (64)

where the axial 3-vector h(g) and the polar 3-vector z(g) are defined by:

h(g)=(hi(g)) — (−12 Ei

jk dwij(g))=(−dw23(g), −dw31(g), −dw12(g)),

z(g)=zi(g) — (−dw0i(g))=(−dw01(g), −dw02(g), −dw03(g)) (65)

and correspond to a true deformed rotation and to a deformed boost,respectively.

5.3. The Infinitesimal Transformations of the 4-d., Deformed HomogeneousLorentz Group SO(3, 1)DEF

We can utilize the results of the previous two subsections to writeEq. (63) as:

dxm(g), DSR4({x}m., x5)=−hl(g)(S l)mn, DSR4 (x5) xn−zi(g)(K i)mn, DSR4 (x5) xn

=(−h(g) ·SDSR4(x5)− z(g) ·KDSR4(x5))mn xn. (66)


Therefore, the infinitesimal space-time rotation transformation in thedeformed Minkowski space M4

6 (x5), corresponding to the element g ofSO(3, 1)DEF, can be expressed as:

dg: xmQ xm Œ(g), DSR4(x5)

=xm+dxm(g), DSR4({x}m., x5)

=(1−h1(g) S1DSR4(x

5)−h2(g) S2DSR4(x

5)−h3(g) S3DSR4(x

5)

−z1(g) K1DSR4(x

5)−z2(g) K2DSR4(x

5)−z3(g) K3DSR4(x

5))mn xn, (67)

where 1 is the identity of SO(3, 1)DEF.Then, on account of the physical meaning of the 3-d. parameter and

generator vectors, h(g), z(g), and SDSR4(x5), KDSR4(x5), we can get, fromthe matrix representation of the SO(3, 1)DEF generators, the explicitexpressions of all the different kinds of infinitesimal transformations of thedeformed Lorentz group, namely:

1. 3-d. deformed space (true) rotations (parameters h(g) and genera-tors SDSR4(x5)), which constitute the group SO(3)DEF of rotationsin a deformed 3-d. space, non-abelian, non-invariant proper sub-group of SO(3, 1)DEF:

– (Clockwise) infinitesimal rotation by an angle h1(g) around x15 :

xm Œ(g), DSR4(x5)=(1−h1(g) S

1DSR4)

mn (x5) xnZ R

x0−(g),DSR4(x5)

x1−(g),DSR4(x5)

x2−(g),DSR4(x5)

x3−(g),DSR4(x5)

S

=R1 0 0 00 1 0 00 0 1 h1(g) b

−22 (x

5)0 0 −h1(g) b

−23 (x

5) 1

S Rx0

x1

x2

x3

S

=Rx0

x1

x2+h1(g) b−22 (x

5) x3

−h1(g) b−23 (x

5) x2+x3

S ; (68)



xm Œ(g), DSR4(x5)=(1−h2(g) S

2DSR4)

mn (x5) xnZ R

x0−(g), DSR4(x5)

x1−(g), DSR4(x5)

x2−(g), DSR4(x5)

x3−(g), DSR4(x5)

S

=R1 0 0 00 1 0 −h2(g) b

−21 (x

5)0 0 1 00 h2(g) b

−23 (x

5) 0 1

S Rx0

x1

x2

x3

S

=Rx0

x1−h2(g) b−21 (x

5) x3

x2

h2(g) b−23 (x

5) x1+x3

S; (69)


xm Œ(g), DSR4(x5)=(1−h3(g) S

3DSR4)

mn (x5) xnZ R

x0−(g),DSR4(x5)

x1−(g),DSR4(x5)

x2−(g),DSR4(x5)

x3−(g),DSR4(x5)

S

=R1 0 0 00 1 h3(g) b

−21 (x

5) 00 −h3(g) b

−22 (x

5) 1 00 0 0 1

S Rx0

x1

x2

x3

S

=Rx0

x1+h3(g) b−21 (x

5) x2

−h3(g) b−22 (x

5) x1+x2

x3

S ; (70)

2. 3-d. deformed spacetime (pseudo) rotations, or deformed Lorentzboosts (parameters z(g) and generators KDSR4(x5); they do notform a group (cf. Eq. (78) below):


– Infinitesimal boost with rapidity z1(g) along x15 :

xm Œ(g), DSR4(x5)=(1−z1(g) K

1DSR4)

mn (x5) xnZ R

x0−(g), DSR4(x5)

x1−(g), DSR4(x5)

x2−(g), DSR4(x5)

x3−(g), DSR4(x5)

S

=R1 −z1(g) b

−20 (x

5) 0 0−z1(g) b

−21 (x

5) 1 0 00 0 1 00 0 0 1

S Rx0

x1

x2

x3

S

=Rx0−z1(g) b

−20 (x

5) x1

−z1(g) b−21 (x

5) x0+x1

x2

x3

S. (71)


xm−(g), DSR4(x5)=(1−z2(g) K

2DSR4)

mn (x5) xnZ R

x0−(g), DSR4(x5)

x1−(g), DSR4(x5)

x2−(g), DSR4(x5)

x3−(g), DSR4(x5)

S

=R1 0 −z2(g) b

−20 (x

5) 00 1 0 0

−z2(g) b−22 (x

5) 0 1 00 0 0 1

S Rx0

x1

x2

x3

S

=Rx0−z2(g) b

−20 (x

5) x2

x1

−z2(g) b−22 (x

5) x0+x2

x3

S. (72)



xm−(g), DSR4(x5)=(1−z3(g) K

3DSR4)

mn (x5) xnZ R

x0−(g), DSR4(x5)

x1−(g), DSR4(x5)

x2−(g), DSR4(x5)

x3−(g), DSR4(x5)

S

=R1 0 0 −z3(g) b

−20 (x

5)0 1 0 00 0 1 0

−z3(g) b−23 (x

5) 0 0 1

S Rx0

x1

x2

x3

S

=Rx0−z3(g) b

−20 (x

5) x3

x1

x2

−z3(g) b−23 (x

5) x0+x3

S. (73)

The explicit form of the infinitesimal contravariant 4-vectordxm(g), DSR4({x}m., x

5) corresponding to an element g ¥ SO(3, 1}DEF, is therefore

˛dx0(g), DSR4({x}m., x

5)=−z1(g) b

−20 (x

5) x1−z2(g) b−20 (x

5) x2−z3(g) b−20 (x

5) x3

=b−20 (x5)(−z1(g) x1−z2(g) x2−z3(g) x3),

dx1(g), DSR4({x}m., x5)

=−z1(g) b−21 (x

5) x0−h3(g) b−21 (x

5) x2−h2(g) b−21 (x

5) x3

=−b−21 (x5)(z1(g) x0−h3(g) x2+h2(g) x3),

dx2(g), DSR4({x}m., x5)

=−z2(g) b−22 (x

5) x0−h3(g) b−22 (x

5) x1+h1(g) b−22 (x

5) x3

=−b−22 (x5)(z2(g) x0−h3(g) x1−h1(g) x3),

dx3(g), DSR4({x}m., x5)

=−z3(g) b−23 (x

5) x0+h2(g) b−23 (x

5) x1−h1(g) b−23 (x

5) x2

=−b−23 (x5)(z3(g) x0−h2(g) x1+h1(g) x2).

(74)

The covariant components of such a 4-vector are

˛dx0(g), DSR4({x}m.)=−z1(g) x

1−z2(g) x2−z3(g) x3,dx1(g), DSR4({x}m.)=z1(g) x

0−h3(g) x2+h2(g) x3,dx2(g), DSR4({x}m.)=z2(g) x

0−h3(g) x1−h1(g) x3,dx3(g), DSR4({x}m.)=z3(g) x

0−h2(g) x1+h1(g) x2.

(75)


Comparing Eq. (75) with the expression (25) of the covariant Killingvector, we see the perfect correspondence between the space-time rotationalcomponent of tm({x}m.) (unique for all the 4-d. generalized Minkowskispaces) and the covariant 4-vector dxm(g), DSR4({x}m.) related to SO(3, 1)DEF(see point 2 of Sec. 3.2.1).

5.4. The 4-d. Deformed Lorentz Algebra, i.e., the Lie Algebra of the 4-d.Deformed, Homogeneous Lorentz Group SO(3, 1)DEF

Let us specialize Eq. (34) to the DSR4 case, in order to derive the 4-d.deformed Lorentz algebra, i.e., the Lie algebra of the 4-d. deformed,homogeneous Lorentz group SO(3, 1)6DEF. We get

[IabDSR4(x5), IrsDSR4(x

5)]

=gasDSR4(x5) IbrDSR4(x

5)+gbrDSR4(x5) IasDSR4(x

5)

−garDSR4(x5) IbsDSR4(x

5)+gbsDSR4(x5) IarDSR4(x

5)

=das(b−20 (x5) da0−b−21 (x

5) da1−b−22 (x5) da2−b−23 (x

5) da3) IbrDSR4(x5)

+dbr(db0b−20 (x5)−db1b−21 (x

5)−db2b−22 (x5)−db3b−23 (x

5)) IasDSR4(x5)

−dar(da0b−20 (x5)−da1b−21 (x

5)−da2b−22 (x5)−da3b−23 (x

5)) IbsDSR4(x5)

−dbs(db0b−20 (x5)−db1b−21 (x

5)−db2b−22 (x5)−db3b−23 (x

5)) IarDSR4(x5).(76)

On account of the physical interpretation of the infinitesimal genera-tors, one has therefore the following kinds of commutators:

1. Commutator of generators of 3-d. deformed space rotations:

[I ijDSR4(x5), I lmDSR4(x

5)]

=d im(d i0b−20 (x5)−d i1b−21 (x

5)−d i2b−22 (x5)−d i3b−23 (x

5)) IjlDSR4(x5)

+djl(dj0b−20 (x5)−dj1b−21 (x

5)−dj2b−22 (x5)−dj3b−23 (x

5)) I imDSR4(x5)

−d il(d i0b−20 (x5)−d i1b−21 (x

5)−d i2b−22 (x5)−d i3b−23 (x

5)) IjmDSR4(x5)

−djm(dj0b−20 (x5)−dj1b−21 (x

5)−dj2b−22 (x5)−dj3b−23 (x

5)) I ilDSR4(x5)

=ESC off on i, j−d imb−2i (x5) IjlDSR4(x

5)−djlb−2j (x5) I imDSR4(x

5)

+d ilb−2i (x5) IjmDSR4(x

5)−djmb−2j (x5) I ilDSR4(x

5). (77)


2. Commutator of generators of 3-d. deformed boosts:

[Ii0DSR4(x5), Ij0DSR4(x

5)]

=di0(di0b−20 (x5)−di1b−21 (x

5)−di2b−22 (x5)−di3b−23 (x

5)) I0jDSR4(x5)

+d0j(dj0b−20 (x5)−dj1b−21 (x

5)−dj2b−22 (x5)−dj3b−23 (x

5)) Ii0DSR4(x5)

−dij(di0b−20 (x5)−di1b−21 (x

5)−di2b−22 (x5)−di3b−23 (x

5)) I00DSR4(x5)

−d00(d00b−20 (x5)−d01b−21 (x

5)−d02b−22 (x5)−d03b−23 (x

5)) IijDSR4(x5)

=−b−20 (x5) IijDSR4(x

5). (78)

3. ‘‘Mixed’’ commutator of generators of 3-d. deformed space andboosts generators:

[I ijDSR4(x5), Ik0DSR4(x

5)]

=d i0(d i0b−20 (x5)−d i1b−21 (x

5)−d i2b−22 (x5)−d i3b−23 (x

5)) IjkDSR4(x5)

+djk(dj0b−20 (x5)−dj1b−21 (x

5)−dj2b−22 (x5)−dj3b−23 (x

5)) I i0DSR4(x5)

−d ik(d i0b−20 (x5)−d i1b−21 (x

5)−d i2b−22 (x5)−d i3b−23 (x

5)) Ij0DSR4(x5)

−dj0(dj0b−20 (x5)−dj1b−21 (x

5)−dj2b−22 (x5)−dj3b−23 (x

5)) I ikDSR4(x5)

=ESC off su i, j−djkb−2j (x5) I i0DSR4(x

5)+d ikb−2i (x5) Ij0DSR4(x

5). (79)

In the ‘‘self-representation’’ basis of SO(3, 1)6DEF, it is easy to showthat commutation relations (76)–(79) read:16

16Use has been made of the relation

EimsEjrsb−2s (x

5)=(dijdmr−dirdmj) 1 C3

k=1(1−dik)(1−dmk) b

−2k (x

5)2 ,

which generalizes to the DSR4 case the well-known formula EimsEjrs=dijdmr−dirdjm.

˛[S iDSR4(x

5), S jDSR4(x5)]

=ESC on 1 C3

s=1(1−dis)((1−djs) b

−2s (x

5)) EijkSkDSR4(x

5)2

=Eijkb−2k (x

5) SkDSR4(x5);

[K iDSR4(x5), K jDSR4(x

5)] =ESC on −b−20 (x5) EijkS

kDSR4(x

5),[S iDSR4(x

5), K jDSR4(x5)]

=ESC on on l, ESC off on jEijlK

lDSR4(x

5) 1 C3

s=1djsb

−2s (x

5)2

=Eijlb−2j (x

5) K lDSR4(x5),

(80)

which define the deformed Lorentz algebra of generators of SO(3, 1)6DEF.


Such relations generalize to the DSR4 case the infinitesimal algebraicstructure of the standard homogeneous Lorentz group SO(3, 1). Theyadmit interpretations wholly analogous to those of the usual Lorentzalgebra. First Eq. (80) expresses the closed nature of the algebra of thedeformed rotation generators; consequently the 3-d. deformed space rota-tions form a 3-parameter subgroup of SO(3, 1)6DEF, SO(3)DEF. On the con-trary, the deformed boost generator algebra is not closed (according tosecond Eq. (80)), and then the deformed boosts do not form a subgroup ofthe deformed Lorentz group. This implies that SO(3, 1)6DEF cannot be con-sidered the product of two its subgroups. This is further confirmed by thenon-commutativity of deformed space rotations and boosts, expressed bythird Eq. (80). Moreover, first and third Eqs. (80) show that both SDSR4(x5)and KDSR4(x5) behave as 3-vectors under deformed spatial rotations.

6. CONCLUSIONS

We want first to stress that, in the 4-dimensional case, with a signatureS=3, T=1, in the limit

gmn, DSR4(x5) Q gmn, SSR4 Z dmn(dm0b20(x

5)−dm1b21(x

5)−dm2b22(x

5)−dm3b23(x

5))

Q dmn(dm0−dm1−dm2−dm3)

Z b2m(x5)Q 1, -m=0, 1, 2, 3 (81)

all results valid at group-transformation level in DSR4 reduce to thestandard ones in SR.Moreover, let us recall that the (parametric) dependence of the metric

of a generalized Minkowski space on the set {x}n.m. of non-metrical coor-dinates reflects itself also at the group level. In particular, such a depen-dence shows up in:

(1) The N×N matrix representation of the infinitesimal generators;

(2) The infinitesimal group transformations;

(3) The structure constants of the Lie algebra of generators.

Let us also notice that, since to any fixed value {x̄}n.m. of {x}n.m. therecorresponds a generalized Minkowski space MN

6 ({x̄}n.m.), we have a familyof N-d. generalized Minkowski spaces

{MN6 ({x}n.m.)}{x}n.m. ¥ R{x}n.m. (82)


where R{x}n.m. is the range of the set {x}n.m.; if the cardinality of the range ofeach element of the set {x}n.m. is infinite, the cardinality of R{x}n.m. (and ofthe family (82)) is .Nn.m.. In correspondence, one gets a family of general-ized Poincaré groups

{P(S, T)N(N+1)/2GEN ({x}n.m.)}{x}n.m. ¥ R{x}n.m. (83)

with the same cardinality structure as (82).This can be summarized as

(Iper) spatial level of N-d. generalized Minkowski spaces:

(1) {MN6 ({x}n.m.)}{x}n.m. ¥R{x}n.m. ,

(2) MN6 ({x}n.m.)—MN

6 ({x̄}n.m.).

ˇ

Z˛Group level of related maximal Killing groups:(1) {P(S, T)N(N+1)/2GEN ({x}n.m.)}{x}n.m. ¥R{x}n.m.

={SO(T, S)N(N−1)/2GEN ({x}n.m.)és Tr(T, S)NGEN ({x}n.m.)}{x}n.m. ¥R{x}n.m.(2) P(S, T)N(N+1)/2GEN ({x}n.m.)— P(S, T)

N(N+1)/2GEN ({x̄}n.m.).

(84)

In the forthcoming papers, we will discuss the finite structure of thespace-time rotation groups and the translation component of the maximalKilling group of generalized Minkowski spaces.

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killing symmetries of generalized minkowski spaces. i. algebraic-infinitesimal structure of...

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