Unified affine gauge theory of gravity and strong interactions with finite and infinite GL(4, R) spinor fields

Unified affine gauge theory of gravity and strong interactions with finite and infinite GL(4, R) spinor fields

ANNALS OF PHYSICS Unified 120, 292-315 (1979) Affine with Gauge Theory of Gravity and Strong Interactions Finite and Infinite R(4, R) Spinor Fiel...

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ANNALS

OF PHYSICS

Unified

120, 292-315 (1979)

Affine with

Gauge Theory of Gravity and Strong Interactions Finite and Infinite R(4, R) Spinor Fields Y. NE’EMAN* Tel Aviv University, Tel Aviv, Israel AND

DJ. %JAEKI+ Boris Kidric institute of Nuclear Sciences, Belgrade, Yugoslavia Received August 29, 1978

A theory in which the Linear Group in four dimensions a(4, R) and its Affie Extension GA(4, R) bear a direct relationship to the Physics of hadrons, and indirectly to that of the Ieptons is outlined. The Poincare group is embedded in a Global G(4, R) Symmetry. Hadrons correspond to (infinite-dimensional) unitary representations of G(4, R), as defined by their GL(3, R) or X(3, R) bandor content. These E(3, R) bandors are embedded in infinite-component (poly-) fields, representing z(4, R) bandors. All multiplicity free unitary irreducible representations are explicitly constructed, and the formulas for the scalar products of the representations’ Hilbert spaces as well as the matrix elements for all noncompact (shear) operators are given. Leptons are described by nonlinear representations of a(4, R), realized through a Metric tensor field (i.e. Gravity). Nonlinear representations of GL(4, R) are presented in detail. The anholonomic description in terms of tetrad deformations is used. GA(4, R) is then postulated as a local Gauge Theory. It generates an Einstein-like Gravitational Interaction, plus a Confining Strong Interaction which applies only to the linear representations, i.e. to hadrons. Local GA(4, R) symmetry is spontaneously broken to local Poincare invariance for leptons (Einstein-Cartan Gravity). The variations of all the fields in the theory are presented and the Noether currents are derived. The choice of a Lagrangian that could explain hadron confinement is discussed.

1. INTRODUCTION

This article outlines a theory in which the Linear Group in four dimensions

GL(4, R) and its Affine Extension GA(4, R) = T, @ GL(4, R) (i.e. adjoining the translations) bear a direct relationship to the Physics of hadrons-and indirectly, to that of the leptons. In this view, the present favored Strong Interaction Theory of QCB is incomplete, especially in its infrared limit. It constitutes just one contribution * Partially supported by the United States-Israel Binational t Supported in part by R.Z.N.S.

292 00034916/79/080292-24$05.00/O Copyright All rights

0 1979 by Academic Press, Inc. of reproduction in any form reserved.

Science Foundation.

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WITH

GL(4,

293

R)

to Confinement Dynamics, and will in some future development become connected and correlated with the (possibly more important) contributions of GL(4,li> we postulate here. The two might then merge in a way similar to the correlation of Internal Symmetry with the Poincare group via Extended Supersymmetry. We present the following theses: (A)

The Poincare group is embedded in a Global GL(4, R) Symmetry.

(Al) Hadrons correspond to (infinite-dimensional) unitary representations of GA(4, R), as defined by their GL(3, R) or X(3, R) bandor [1,2] content. This is then the Hilbert-space relativistic completion of the algebraic model for generalized Regge trajectories suggested in 1965 [3,4], now including bandspinors as well. As to Relativistic Field Theory, these z(3, R) bandors are embedded in infinite-component (poly-) fields, representing SL(4, R) bandors [l]. We present an explicit construction for such multiplicity-free bandors. (The bars denote the double covering group). (A2) Leptons are described by non-linear representations realized as we show through a Metric tensor field (i.e. Gravity).

of these groups,

(B) GA(4, R) is then postulated as a local Gauge Theory [5,6]. It then generates an Einstein-like Gravitational Interaction, plus a Confining Strong Interaction. One such model was recently exhibited [7]. (Bl) We conjecture the existence of a more sophisticated Lagrangian beyond that of Ref. [7]. In the new dynamics, the Confining contribution would only apply to the linear representations (Al), i.e. to hadrons. Such confined regions (solutions) could then also be reinterpreted as a description of the time-evolution of 3-dimensional lumps [8], generalizing the Dual Model Strings. (B2) For (A2), the Goldstone realization is now replaced by a Higgs-like spontaneous breakdown to local Poincare invariance (Einstein-Cartan Gravity). Space-time outside of the hadronic lumps thus obeys the Principle of Equivalence, possessing a Special-Relativity limit with residual global Poincare invariance. In Part 2 we present an explicit construction for the multiplicity-free unitary irreducible representations of z(4, R). We use a new method in constructing these operators and obtain all multiplicity-free representations. We supply formulae for the scalar products of the representations’ Hilbert spaces and give explicit matrix elements for all non-compact (shear) operators entering covariant derivatives of GA(4, R) gauges; in fact, these also correspond to holonomic covariant derivatives (i.e. for the Einstein Covariance group) for “world-spinors” (bandors) [l]. These representations form a basis for GL(4, R) spinor and tensor fields (infinite; the usual tensor fields correspond to finite non-unitary representations). Another possible application for our results can be found in the description of a “material lump” even if one does not consider the Gravitational unification we mentioned in this context.

294

NE’Ehtm

AND SIJACKI

In Part 3 we introduce the reader to -the notion of Non-Linear representations and present this construction in detail for GL(4, R). We use an anholonomic description in terms of tetrad deformations, including the introduction of the metric field as the representation realizer. The anholonomic definition of GL(4, R) is essential to our next step, i.e. gauging the group. In Part 4 we gauge GA(4, R). We introduce the canonical forms explicitly. The translations are replaced by parallel-transport generators (anholonomized General Coordinate Transformations). We present the variations of all the fields in the theory and derive the Noether currents. In Part 5 we discussthe choice of a Lagrangian that could explain hadron confinement in addition to Newton-Einstein long-range gravitation. It would thus provide the foundations for a natural unification of Gravity and Strong Interactions.

2. MULTIPLICITY

-

FREE UNIRREPS OF SL(4, R)

The group GA(4, R) is a semidirect product of the group of translations in four dimensions, and of the GL(4, R) group. The latter can be split into the one-parameter group of dilations, and the SL(4, R) group, which is a group of volume preserving transformations in the Minkowski space-time. If the group elements of GL(4, R) are given as 4 x 4 matrices, then the subgroup of dilations consists of constant (diagonal) matrices and they commute with those of SL(4, R). SL(4, R) is a semisimple and noncompact Lie group. The space of the group parameters is not simply connected. The maximal compact subgroup of SL(4, R) is SO(4). The universal covering group of SL(4, R) we denote by E(4, R). This group has the same Lie algebra as SL(4, R). It is a simply connected group and its maximal compact subgroup is SO(4), the covering group of SO(4). SO(4) is isomorphic to SU(2) x SU(2). There is a two element subgroup 2, in the center of E(4, R), such that the factor group of E(4, R) with respect to Z, is isomorphic to SL(4, R), i.e., SL(4, R)/Z, ‘v SL(4, R). Thus, SL(4, R) and SU(4) are the double covering groups of SL(4, R) and SO(4) respectively. The complete center of SL(4, R) is actually Z, , with E(4, R)/Z, N SO(3, 3). E(4, R) is physically relevant becauseit has (infinite dimensional) spinorial unitary irreducible representations (unirreps) which are double-valued unirreps of SL(4, R). The SL(4, R) group has as a subgroup the Lorentz group SO(3, 1), and correspondingly x(4, R) has as a subgroup s0(3, 1) N SL(2, C). The Lorentz group is generated by the angular momentum and the boost operators Ji and Ki , i = 1,2, 3, respectively. We write them as Jnb, a, b = 0, 1, 2, 3, where Jab = -Jbo The remaining nine generators form a symmetric secondrank shear operator T,, , a, b = 0, 1, 2, 3, i.e. Tab = Ton and tr T,, = 0. The commutation relations of the SL(4, R) algebra are given by the following relations:

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GL(4, R)

295

where qab is the Minkowski metric, 7ab = diag(+ 1, - 1, - 1, - 1). These commutation relations can be easily derived in the model where Jnb = x,p,, - xbpa , TcIb

=

xdb

+

XbPa

-

h&

’ Ph

and

[Pa,

xbl

=

hub.

In the three-dimensional spatial subspace of the Minkowski space, the SL(4, R) group reduces to the SL(3, R) group. The latter is an eight parameter group generated by the angular momentum operators Jij and the five shear operators Tii (i,j = 1,2, 3), which transform under the SO(3) subgroup of SL(3, R) as a quadrupole operator. In order to construct explicitly unirreps of E(4, R) it is convenient to make use of the basis of the maximal compact subgroup SU(2) x SU(2) of the E(4, R) group, which is generated byj,“’ = &cijlcJjk + $-T,,iandj,!“’ = $cijk.ljk - $Toi (i,j, k = I, 2,3). These operators are compact, and the remaining (noncompact) generators transform with respect to SU(2) x SU(2) as in the (1, 1) irreducible tensor operator Z. The commutation relations now read:

and [Zij 9 Z&L] = -i(Sj*~Eifinj~) + S&EjmnjF)). (2.3) Physically, the most interesting unirreps of SL(4, R) are the Multiplicity free ones. They contain, in the reduction to the representations of the maximal compact subgroup SU(2) x SU(2), the corresponding representations, labelled by (j, ,jJ, at most once. In what follows, we will explicitly construct all multiplicity free unirreps of SL(4, R). The construction will be carried out in two steps. In the first step we make use of the decontraction formula [4, 91 to evaluate the form of the X(4, R) operators. This formula, by now rather known in Physics, describes a deformation which is the inverse of the Wigner-InGnu group contraction. In the second step, we find all possible irreducible Hilbert spaces with the scalar products corresponding to the unirreps of E(4, R). This part of the work and the completeness of our results is heavily based on the work of Harish-Chandra [IO]. In both steps of the construction we work in the spaces of functions of the parameters of the maximal compact subgroup SU(2) x SU(2). The Wigner-InGnu contraction of the SL(4, R) algebra with respect to that of the SO(4) subgroup consists in defining the new set of (noncompact) operators Yij(e) = EZij , and taking the limit lim,,, Yij(c) = Yij . The corresponding group, obtained in this way from the SL(4, R) group, is T, @ (W(2) x SU(2)), i.e., a semidirect product of the Abelian 9-parameter group T9 and the SU(2) x SU(2) group. The Y
296

NE’EMAN

AND

&JACK1

SU(2) x SU(2). The commutation relations of the contracted group are given by (2.2) where & is replaced by Yij , and the (2.3) now reads: [ Yij , Yk,]

= 0.

(2.4)

The decontraction formula now tells us that the following operators + - i KjT 2(C&2

zg =pY,,

together withjj” The unirreps by the labels (j, of the compact

ji”’

+ (jV,

Yijl,

(2.5)

P E R

andji*’ satisfy (2.2) and (2.3), and thus generate the E(4, R) group. of the SU(2) x SU(2) group are well known. They are characterized ,j,), j, , jZ = 0, 4, l,..., and in the spherical basis the matrix elements generators are

/

~

~)

=

(j2(j2

+

1)

-

m2(m2

k

1)Y2

:l

r?l j> 2

1>.

In this basis (j(l))” --f j,(j, + l), (jc2’)” + j,(j, + 1). The vectors of the orthonormal basis (1 f: 2)) are related to the D-functions by

(t

k,l;

,j)= ((‘& + l%j, + l>>‘/”@&(kJ&,(k2),

(2.7)

where ki is the set of Euler angles (pi , ri), i = 1,2. Let us consider first the multiplicity free Ladder unirreps of %(4, R), i.e. those unirreps of SL(4, R) which when reduced to the unirreps of SU(2) x SU(2) contain (jl , j,) points, of the j,-j, plane, which belong to a straight line. We will prove below that in the latter case one actually has j, = j, = j. Now, since Yij operators commute mutually, (2.4), they do not contain derivatives in the Euler angles and since j, = j, = j, it is sufficient to take only functions of p1 = p2 = & y1 and y2 . The Yii operators are obviously given, in the spherical basis, by y&J

=

H&h43Y2)~

3 E R;

n,p=o,

&I.

(2.8)

The second-order Casimir operator of the Tg @ (SU(2) x SU(2)) group is c, = Y,jYji --f (F),>“.

(2.9

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STRONG

INTERACTIONS

The corresponding generators of E(4, za, = P&9

GL(4, R)

WITH

297

R), in the spherical basis, read

+ f Km2

+ <“i’“3”,

m,

(2.10)

where p2 is a parameter. The unitarity of group representations, i.e. the hermiticity of generators yields p2 E R. The matrix elements of the noncompact Z,, operators can be directly read off in the (1 m,%z,)} basis, i.e.

= (-)j’+

(-)j’-“”

(yk,

’ ’ )( yi,; 1 a ml

i

,C) {j'll

Z/l j),

(2.1 I) (2.11)

(j'll

Z IIj> = ((Zj’ + 1)(2j + 1))“” ( p2 + Yj Li'o" + I) - jo' + 111).

The second-order Casimir operator of the SL(4, R) group now takes the following value C2

z

ZijZji

-

(j(1))z

-

(j("))"

+

(2S2)

~22.

There are thus two classes of the Ladder unirreps of E(4, R), characterized by a real parameter p2 and the min(j) = j, , Wdd(jO , j, ; p2). The matrix elements of the generators are given by (2.6), with j, = j, = j, and by (2.1 l), and the corresponding (j, , j,) content is

~ladd@,0; P2): {jl ,j2) = {CO,01, (1, 11,(2, 2),...$, aladd(+,

(2.13)

iI ; p2): h , j21 = i($ , $3, 63 , 8, (+, %),...I.

These unirreps were previously obtained by different methods [4, 11, 121, and the matrix elements (2.11) agree with those of Ref. - [13]. Let us now turn to the general case of the SL(4, R) multiplicity free unirreps. Since in general j, #.j2 , we will make use of a Hilbert space of functions of Euler angles k, 2 rl> and (A y~2). Th e noncompact operators Yij are given in the spherical basis

L3 =

P2a3lrJ

mP2Y2)>

a,p=O,&l;

The corresponding noncompact generators of the E(4, means of the decontraction formula, i.e. (2.5) yields

f2~R.

(2.14)

R) group are given now by

-G = P~D~JX$ + i Kjc1?2 + (j(2’)2, DiJ&l.

(2.15)

Finally, we write the matrix elements of the Z,, operators, in the basis of orthonormal

298

NE’EMAN

vectors (1 4 ,$>, of the unirreps in the following form

= (+-“;

(q&4

AND

&JACK1

of the maximal compact subgroup W(2)

(4,,

l 1

u

Wi II Z llj&> = -i(--)fi+G ((2ji + 1)(2j,’+

“)( m,

1)(2jl

ji

;

4)

(j;j;

x SU(2),

11z Ilj,j,>,

-mB

+

(2.16)

l)Gj,

+ l)Yi”

x h + ip2 - HjG + 1) - idA + 1) + jXji + 1) - .L(j, -I ]>I) x

A

i

0

1

000

.A

.A

I(

1

.h

00’

1

In this expression, we have allowed p = p1 + ipZ , p1 , pz E R to be an arbitrary complex number, and the subgroup labels can in general be j, ,jZ = 0, 4, 1, $,2,... . The 3 - j symbol, say (j;0 ,, ’ ,, jl ), with the half-integer entries is a compact notation, and one evaluates it by taking the corresponding expression for integer entries and substituting them by the half-integer ones. The matrix elements of the Z,, operators for multiplicity free unirreps agree with those of the general case,which are obtained by making use of Harish-Chandra modules and will be published elsewhere.Owing to the 3 - j symbols, the reduced matrix elements in (2.16) are zero if,j; = j, and/or ji = ,jZ . Thus the representation Hilbert spacecharacterized by all (jr , j,) values splits into eight invariant subspaces. Since only the four possibilities (ji , j.$ = (j, & 1,j, + 1) occur, the (j, ,j,) content of each invariant subspaceis determined by jr f j, (mod 2) j, -j, (mod 2) and the minimal (j, , j,) value. The eight such invariant Hilbert subspaces we label by the minimal (j, , j,) values, i.e. H(min( j,), min( j,)). Explicitly Jw, o>, m,

41, H(0, 1) = H(1, O), H($ ) #) = H(S) &),

f-w, 3,

fm , O), H(B ) l),

H(1, g).

(2.17)

The next - question we want to discussis unitarity of the multiplicity free representations of SL(4, R), or in other words, hermiticity of the corresponding generators. Since E(4, R) is a noncompact group, its unitary representations are necessarily infinite dimensional. Unitarity is a matter which depends on the Hilbert spaceone is working in, i.e., it depends on- the scalar product. Being interested in obtaining all multiplicity free unirreps of SL(4, R), we will start with the most general - scalar product one can can define. In this part we closely follow an analysis of the SL(3, R) unirreps [2]. Since we have defined z(4, R) representations in the spaceof functions of p1 , y1 ,

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& and yz parameters of the maximal compact subgroup SU(2) general scalar product of two functionsfand g is given by

(j-i g) = j dk’ dk f *W dk’, k) g(k),

299

GL(4, R)

x SU(2), the most

k, k’ E W(2)

x SU(2).

(2.19)

In this expression K(k), k) is a kernel function and dk and dk’ are invariant measures over SU(2) x SU(2). Obviously, when fc(k’, k) = 6(k’k-l), the Dirac a-function, we recover the usual scalar product. The problem of finding the unitary representations is now reduced to that of finding all possible scalar products, or in other words to the problem of finding all possible kernels. In the Appendix we show that the matrix elements of the noncompact operators in the general case read

where K(& ,j,) are the matrix elements of the kernel. The positive definiteness of the scalar product, i.e., (f,f) > 0 for everyf yields

and the hermiticity

of the scalar product,

i.e., (f, g) = (g,f)*

implies

(2.21) ‘dj, 3j,) = K *(j, , lit>. The noncompact generators of SL(4, R) are hermitian, provided one has in the spherical basis Z$ = (-)“-B Z-,,_, . This condition takes the following form in the Hilbert space

Upon inserting in (2.22) the matrix elements given by (2.16), and making use of the symmetry properties of the 3 - j symbols, as well as (2.21) we arrive at

4XNpl

+ Oh - MA

+ 1) - jl(jl + 1) + jXji + 1) - .idh + 1)l)

= dhjJ(---Pl + ip2- t[jXji + 1) -.il(jl + 1) +.j,‘(ji + 1) -jd.h + l>l). (2.23) This equation provides us with two cases, i.e. p1 = 0, pz E R, and p1 # 0, pz = 0. In the first case, p1 = 0, pz E R, the matrix elements of the kernel are

K(A,.iJ = 1,

for every (jl , j,),

(2.24)

300

NE’EMAN

AND

&JACK1

the irreducible Hilbert spaces are listed in (2.17), and the corresponding unirreps of E(4, R) form the Principal series, which we denote by Il)pr(min(j,), min(j,); p&. In the second case, p1 # 0, pz = 0, the parameter p1 can be either continuous, but of a limited range, or it can be a discrete number. The continuous solution for p1 follows from the condition (2.20). From (2.20) and (2.23) we find that there is a solution for p1 in some of the Hilbert spaces (2.17) if

lP1I < 1- minIA-j2 I,

and

/ p1 I < 2 + min 1j, + j2 I.

(2.25)

An explicit inspection shows that 0 < Ip1I < l,inH(O,O)andH(+,+), 0 < j p1 I < 4, in H(0, &), H(&, 0), H(?, 1) and H(1, +)), no solution in H(0, 1) and H($, $).

(2.26)

-

These unirreps of SL(4, R) form the supplementary series, and we denote them by ID*UPP(min(j,), min(j,);p,). We solve (2.23) in terms of the matrix elements of the kernel for the min(j,) and min(jJ. After some stragihtforward algebra we obtain Wl

K(jl ’ j2) = r(j,

+

p1

+

+ j2 -

+

j2

p1

+ 1) r(min(jl)

JX.&-j21 ’ WjI-j21

1) Win(jd

+p,+2)r(l -Pi+ 2)

+

min(j,)

-

p1

+

1)

+ mW2) + p1 + 1) mW,>

- min(j,)

I -pl + 2)

r(l min(j,) - mW2> I +pl + 2)

x K(min( j,), min( j,)).

(2.27)

The remaining series of unirreps we find from the requirement that ~(j, , j,) # 0 implies that K(j, - 1, j, + 1) = 0 when j, 3 j, , or K(j, + 1, j2 - 1) = 0 when j, > jr . This requirement and (2.23) yield p1

= min lj, -j,

/-

1,

(2.28)

or complementarily, for a givenp, = -4, 0, 3, l,..., one has the irreducible subspaces of the invariant spaces H(min(j,), min( j,)), of those (j, , j,) for which jr - j, > p1 + 1 or j, - j, > p1 + 1. It is easy to see that the condition (2.20) is also satisfied. This class of unirreps form the Discrete series of multiplicity free unirreps of the E(4, R) group, and we characterize them by the minimal allowed (j, , j,) pair, i.e. Ddisc(pl + 1, 0) and W*c(O,pl + 1). For the discrete series (2.23) yields f4.h , j,) =

~(j, + j2 + PI + 1) r(l jl - j2 I + PI + 2, K(min(jl) min(j,)), RA + j2 - p1 + 1) r(l jl - j2 I - P1 + 2) ’

(2 29)

where K(min(j& min(j,)) is either IC(p1 + 1,O) or ~(0, p1 + 1). Whenp, = - 1, K(jl , j,) # 0 implies that K(j, + l,j, - 1) = K(j, - 1, j, -+ 1) = 0

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301

GL(4, R)

WITH

only ifj, = j, , and thus we arrive at the series of Ladder unirreps, which were treated at length above. The irreducibility of the subspaces in which we have defined multiplicity free unirreps of E(4, R) is guaranteed by construction. None of them possess an invariant subspace under the action of the group generators. Irreducible subspaces and the labels of the multiplicity free unirreps obtained in this work, agree with those of Ref. [12]. The second-order Casimir operator for SL(4, R) is c2

=

ZijZji

-

(j(l))2

-

(j(292

+

(442,

P

=I5

+

&.

(2.30)

For the Principal and the Ladder series it is p22, while for the Supplementary and the Discrete series it takes the value -p12. The series of representations obtained above are actually representations of the E(4, R) algebra. However, since we have been working in the space of vectors of the representations of the maximal compact subgroup SU(2) x SU(2) (analytic vectors) and due to a result of Nelson [14], these Lie algebra- representations can be exponentiated to the corresponding continuous unirreps of X(4, R). The necessary and sufficient condition for exponentiability is the hermiticity of Nelson’s operator A, i.e. d = (j(l))2 + zijzji

= 2(j”92 + 2(j’292 + c, .

(2.31)

Since (j(1))2 and (jc2))” and C2 are all hermitian for the above unirreps, the infinite dimensional unirreps of the E(4, R) algebra constructed in this paper can be exponentiated. Let us now briefly consider the reduction of the physically most interesting multiplicity free unirreps of sz(4, R) with respect to the unirreps of the z(3, R) subgroup. The latter describe the spin content - of an ?%(4, R) invariant physical object at rest. There are four series of unirreps of X(3, R) [2, 151. Principal series. We denote them by ap*(J, , K, ;a, S,), cr, 6, E R, and K,, = 0, $, 1. J is the angular momentum label, i.e. label of the SO(3) subgroup of E(3, R). The J content is (J} = (0, 22, 3, 43, 52, 64,... }, J, = 0 for K,, = 0, {J} = (1, 2, 32, 42, 53, 63,... }, J, = I for K, = 0, 1, and {J} = (4, g2, g3, 4” ,... }, J, = Q for K, = 4. The superscript gives the multiplicity in J. Supplementary series. CDsuPP(JO, K, ; CT,S,), o~R,i6,l <+orl,withtheJcontent as for the Principal series. Discrete series. ZWc(JO ; u), u E R. The J content is {J) = (Jo , J, + 1, (Jo + 2)2, (Jo + 3)2, (J, + 4)3, (J, + 5)3,... }, where J, = 2,2, $, 3 ,... . Ladder (Degenerate series. TSadd(J, ; u),aERforJ, = 0, 1,ando = OforJ, = 4. The J content is {J) = (0, 2, 4 ,... > for J, = 0, {Jo} = {I, 3, 5 ,... } for Jo = 1 and {J} = {$, 8, +,...} for J, = $. Reduction of the Ladder unirreps Diadd(O, 0; p2) and ZSadd($, &,p2) of ,SL(4, R) with respect to the unirreps of E(3, R) yields an infinite number of integer angular momentum E(3, R) unirreps [13]. Symbolically we write

302

NE'EMAN AND ~IJACKI

where j, $ j, = J + n, and the angular momentum operator is given by Ji = jj” + ji “) . None of the ladder unirreps of ,.X(4, R) contains in the reduction with respect to SL(3, R) the spinorial ladder unirrep 3 ladd(&;0) of E(3, R). An - analysis shows that the reduction of the YDzZ”,“,(+, 0) and B$F2R(0,+) unirreps of SL(4, R) with respect to the unirreps of SL(3, R) is given by the following symbolic expression

and each SL(3, R) unirrep appears infinitely many times. Summary of the Representations There are four seriesof multiplicity free unirreps of the z(4, R) group, which are characterized by a parameter p = p1 + ipz (see Fig. I), and the minimal (j, ,j,) values appearing in the reduction with respect to the SU(2) x SU(2) subgroup. Supplementary H (O,O)

~HW2,1/2~

H(l,l/Z) HW2,l)

D -

FIG. 1. Allowed

Discrete

values of the parameters p1 and p2

They are defined in the invariant subspacesof (2.17), which are illustrated in Fig. 2. Matrix elementsof the group generators are given in the spherical basisby (2.6) and by (2.19) and (2.16), and those for the Ladder unirreps by (2.11). Principal series. DPr(min(j,), min(j,);p,), p1 = 0, pz E R; defined in irreducible subspaces(2.17), and the matrix elements of the kernel are given by (2.24). Supplementary series. DsuPP(min(jl), min(j,); p,); range of p1 given by (2.26), p2 = 0, and the matrix elementsof the kernel are given by (2.27).

GRAVITY,

STRONG

INTERACTIONS

GL(4, R)

WITH

303

h t

312 I

7I 3

4 8

y

2 5

7 3

4 8

1

IR o-

bI

2 7 5-3-8-l-55--33

4

6

2

7

3R’

2

s/2

I0

I/2

I

3 j,

FIG. 2. Hilbert space identification: point n on the (jI, jJ plane belongs to the subspace H, , where HI = H(0, 0), Hz = H($, &), H, = H(1, 0), H, = H(3/2, l/2), Hj = H(i, 0), He = H(0, $), H, = H(l, $,, H8 = H(&, 1).

Discrete series. IZlidise( p1 + 1, 0) and DdiSc(O,p1 + I),p, = -$,O, 3, I)..., pz = 0, and 1j, - j, 1 > p1 + 1, with the matrix elementsof the kernel given by (2.29). The (j, ,j,) content is illustrated in Fig. 3. Ladder series. Wdd(O, 0; pp) and Wadd($, 4; pz), p1 = -1 and p2 E R, and the (j, , j,) content is given by (2.13). The second order Casimir operator for all these series is given by (2.30): Principal seriesp1 = 0, pz E R; Supplementary series0 < 1p1 ~ < 4, I, pz = 0; Discrete series p1 --_ -2, 0, b, _ I ‘..., ps = 0; Ladder seriesp1 = -I, pz E R.

0

0

0

FIG.

0

3. Lowest (jl, j?) values of DdiSc(p, + I, 0), p1 = -4, 0, :, l,...,

304

NE'EMAN AND GJAEKI -UNITARY REPRESENTATIONS OF GA(4,R)

3. NON-LINEAR

There have been previous attempts to derive Gravity as a (global) GA(4, R) nonlinear realization [16]. In Ref. [16], this was realized for weak (linearized) gravitational fields. In Ref. [17] the GL(4, R) is holonomic, i.e. it is identified with the linear subgroup of the General Coordinate Transformations (GCT) group and is therefore not yet an active symmetry, just a relabelling. In fact, it has since been pointed out [18, 19, 201 that the passive symmetry of the GCT group itself is realized non-linearly over its linear subgroup GL(4, R). To define our m(4, R) we first introduce an arbitrary (non-orthogonal) tetrad frame at a point x of space-time, (Latin indices are anholonomic, Greek are holonomic). There exists a (non-preserved) metric g,, lowering indices. p,“(x) dx” = p”

(pL, a = 0, 1, 293).

(3.1)

p,“(u) are the tetrad fields, p” the one-forms. We can deform the tetrad by applying (only globally at this stage) the group GL(4, R) on the Latin indices. The p” have reciprocal vector fields fa (see (4.3)) D, # fa ,

fa = &z”%3

(3.2)

Pus . Aau = 2Swv.

(3.3)

The GL(4, R) generators obey the commutation

[fab ,fcdl = -hdfcb and together with the “tetrad

translations”

relations

f i71cbfad

(3.4)

fn generate GA(4, R) - T4 0 GL(4, RI,

If0 tfbcl = i77& , K 3.a = 0. It is more instructive

to define the (pseudoorthogonal)

(3.5) (3.6) antisymmetrized

(3.7a)

J~I,= .h~ = Hfaa -fba) and the symmetrized

and traceless generators

4T
of Shear

= WA t- f&d - G%,C.f,,,

together with the translations

rn

P, =

ge nerators

fa obey the commutation

(3.7b) relations

of

GRAVITY,

STRONG

INTERACTIONS

WITH

GL(4, R)

305

S&4, R), (we take the right-invariant tangent vectors for the Hilbert-space operators) given by (2.1) and by the following expressions

[Jab9PC1= i(%ePa- %&J, iTabyPC1= --i(d’b + &‘, - &qabPc), Pa 3Phi = 0. Adjoining

(3.8)

the scale-generator

f=Cfmm m

(3.9)

we generate GA(4, R), where

If, Jnbl= [f, TnJ = 0, [A P,] = -iP,.

(3.10)

We now introduce a Goldstone field as a non-linear symmetry realizer. We pick a symmetric anholonomic tensor field $aa and project it on the (non-orthogonal) basis defined by the product of two vector fields

hb = &&L~Ab, + &A,,).

(3.11)

We note that g I(” = &bPuaPP so that gab

=

&vda”Ab”

(3.12)

is the anholonomic (non-orthogonal) metric. We see that &, - g&, and our Goldstone field can be identified with the metric field. The construction of non-linear representations was developed in connection with chiral symmetry 121,221. The group 21 : = ?%(4R) will be realized non-linearly, with its ‘Ip (covering) PoincarC subgroup represented linearly. We use the “global procedure” of Ref. [22]. The vacuum state is the “origin” o in an analytical manifold ‘9JI, (points “m”) and ‘$3is then the isotropy subgroup of the origin, with ‘8 the group of transformations: T: (a, m) --f T(u)m is analytic, a E ‘u, T(e)m = m, Qm E 11111 where e is the identity in ‘u,

(3.13)

T(a) T(a’)m = T(aa’)m, Qa, a’ E ‘?I, m E 9X. T(a)m : = t(a, m) is thus a (generally non-linear) analytic function of a and m. T(a) and

306

NE’EMAN

AND

SIJACKI

T’(a) are m-locally equivalent if there exists a (possibly non-linear) operator Q from 6” + @ such that m -+ Q(m) is analytic and has an analytic inverse at D, and (m are the coordinates of m)

QP%)ml = W) Q(m), in a suitable neighborhood

Qa E ‘LI,

of e, and Qm in a neighborhood Q(o)

=

(3.14)

of o. (3.15)

0

We shall thus represent the connected component

of e (=SA(4R)

= : ‘Tr)

is an expansion of a Poincare element p in power series in m. B(p) is a linear representation of the Poincare group and

m' : = Q(m) = jQ %-%Ttp) m>4 (dp the normalized Haar measure on +JJ)establishes an m-local equivalence between B(p) and the restriction of T(a) to ‘p. Denoting the generators of ‘$I asf,@ and those of the complementary invariant subset (the shears Tab) byf,‘, we may write a given element a, in the neighbourhood of e as CI, = (exp i@ . f%)(exp iu . fp),

n, E ‘3,

(3.16)

where (0, U) form a real 19-component vector spanning the adjoint representation. As p * D = o, the orbit 111of o under ‘$i separates the ‘8/p cosets defined by exp ids . f '. The di can thus be used as coordinates for ‘!JI, T(e-i@‘f‘R)(o) = (@).

(3.17)

We now fix for M a dimensionality n = nL. + nN , where nL can accommodate the lepton states including the vacuum, and IZ~ = 9 is the dimensionality of @ = itab) , where icaa) denotes the g,, of (3.12) after the extraction of a trace. As we shall soon see, we can forego this step and instead use g,, .fcab, with nN = 10, thus constructing directly a representation of GA(4, R). The reducible representation we construct here is that of the Hilbert space states, but its one lepton subset (with any number of gravitons) forms an irreducible multiplet which can be used for the lepton field #(x), a Dirac 4-spinor under ‘p. (We do not deal here with the additional features due to the lepton “flavour” group and Iepton-type fermion numbers). Denoting the lepton states by $, we take ‘5X as a lepton-graviton manifold given by the coordinates where

+(aa) = gm I o?.

GRAVITY,

STRONG

INTERACTIONS

The action of $J on ‘331is linear and tensor field, 4(x) as a Dirac 4-spinor unitary representations of the Poincart a graviton by (4, 0). Under the action

WITH

307

GL(4, R)

reducible, since itab)(x) behaves as a spin-two field, and the physical states are products of group. A pure lepton state is given by (0, $), of the a/‘@ cosets,

where the scalar product denotes summation over the (ab) double indices. Using \J3t general covariance (Eq. (3.14)) this can be rewritten in terms of a transformed coordinate I/J, exp(--4,

f‘n)(c

Defining uniquely (c$‘, u’) by the left translation and “a” indices for f%) a-le-im.f"

=

(3.18)

#j = Cb, #) (summations

now carry over [ab]

e-id'.f‘J;e-i":.jl;

(3.19)

we have the action of the whole group, using (3.18), (3.19) a-y@, 4) = a- le&f"((), =

,-iwfy,,

#) qe-wfW)

=

e-im'.f~+4'.f9(@ #)

=

$1 (C’,

q,-iuY’P)

#).

(3.20)

The non-linear representation (3.20) thus acts linearly for u’ of (3.19) through the unitary representation of the Poincart group 3 determined by the action of f@ on ZJ The cosets rotate the lepton states # into lepton @ graviton states. This global %(4, R) treatment can now be completed to ?%(4, R), as SA(4, R) is a normal subgroup of index 2 in G(4, R). The topological features are fixed by the compact subgroups: O(4) versus SO(4). This is equivalent to working throughout with exp(--i Ccna) g(ab),fcnb)), the complementary invariant subset to the PoincarC subgroup in GA(4, R), i.e. adjoining and representing Dilations too. We shall not dwell here upon the other features of this non-linear representation, including its infinitesimal aspect and refer the reader to Ref. 22. However, it is important to note that using the analytic function t(a, m) of (3.13) we can expand the group representation it specifies, as a power series in the m, . We identify each monomial as a distinct coordinate y in a new, linearized manifold, so that t(v) supports a linear representation of VI. Using

we get infinite-dimensional representations of %(4, R), since $ is a representation of v. The necessary condition is that such representations, when reduced to the ‘$ subgroup indeed have a spin-half component which can be identified with (o, +). This is true of the multiplicity-free examples Q!F&($, o) and @l”,(O, &). However, in the simplest case, the expansion will fit into a spin-bandor representation of the Principal series.

308

NE’EMAN AND ~IJAEKI 4. THE a(4,

R) GAUGE AND CONSERVATION

LAWS

The GA[4, R) transformations on the tetrad indices will now be gauged, i.e. they will be allowed local values as in the Weyl or Yang-Mills case. Such theories have been propounded before [23,24] as gauge theories for Metric-Affine Gravity [25], including the action on Hadron Polyfields [5]. The gauging procedure and special gauge choices have been detailed in Ref. [6]. First, we introduce the Yang-Mills potentials for G(4, R) entering in the covariant derivative, D, = %,,-

i-puabfab.

(4.1)

Geometrically, we follow the method [26] of working in the Group Manifold ‘Qt of GA(4, R). The one-forms pab(x,5,s) and pa(x, 5, E) correspond to “perturbed” left-invariant forms on the group manifold: x represents 4-dimensional space-time, ZJis the 6-dimensional Lorentz-group variable, and 8 is the lo-dimensional variable of the shears and dilations. Following Einstein’s Lagrangian, we assume the generalized gravitational Lagrangian density 23 to be gauge-invariant under the homogeneous ?.%(4, R) subgroup only. In that case, spontaneous fibration will occur, the 4 and B variables factorize and we are left with a Principal Fiber Bundle with space-time as base-space and z(4, R) as fiber. The pu tetrads now become gauge-fields for the fa . We have generalized curvatures (or field-strengths) as 2-forms R* = R$ dx” A dx”, and it is conveniant to separate the pLabIand Rrablof the Lorentz subgroup from pcab) and Rtab) R” = dp” + ip[acl I\ pc _ ip(ac)/\ pc, R[abl = dp[abl + ip[acl I\ plcbl + ip(ac)h p(cb),

(4.2)

R(ab) = dp(ab)+ ip[ac] I\ p(cb)+ ip[bc] h p(~~)e

The Affine Torsion Ra and Curvature Rfabl have now terms in pcQC) which are not present in Einstein-Cartan Gravity, aside from the appearance of the Rcab)Shearstrength. As the gravitational Lagrangian density !I3 is only GL(4, R) gauge invariant, we have two ways of dealing with the translations. If we gauge the P, , 6’iB = 0 only after imposing the equations of motion, i.e. on mass shell. Alternatively, we can replace the P, by the Parallel-transport operators

with parameters thus producing an Anholonomized General Coordinate Transformation with 8% = 0. We follow the latter path and have gauge variations for the gauge l-forms 6pA =

GRAVITY,

STRONG

INTERACTIONS

GL(4, R)

WITH

309

6(puA dx”), where the index A stands for the 20 gauge indices (using forms allows us to omit the irrelevant coordinate relabellings): 6pa = DP - ipC&Rcda =

de” -

6

P labI

+

P

(ab)

=

DE[abl

=

dc[abl

ip(aci

/\

l

C +

ipC

/,

E[uc]

_

ipC

h

c(acl

ip~.$&[abl

-

+

ip[acl

ip(cb)

=

DE(ab)

=

dclab) -

A EC _

ipccdRcda,

6

ip[aC]

/\ _ +

ip(cb)

h E[cbl f

I

-

ip(ac)

A e(cb)

-

ip[cbl

h c[aCl

(4.5)

ipC&Rc,labl,

ipC~dRcd(Ub) ip[aCJ

I\

c(Cb)

,y E[acl

-

ip(ca)

+

ip[bC]

/r

E(c~)

,y E[bCl

- ipc&Rcd’ab).

For a matter tensor field or for a polyfield (spinor or tensor), 677= i(dab$&b] + dab$,b) f eaDa)n,

(4.6)

where theStab, ,ftab) and D, are infinite-dimensional matrices in the latter case. For the lepton Poincart spinor zj in the non-linear representation, we list the lowest terms S$J= i(dab) .habjg# + idab . qb]# + i@ . D,# + ...,

(4.7)

where the fcab) matrices act on the lo-components of gced). Using these variations we find the matter Noether currents (2 is the matter Lagrangian) eQu (Energy-Momentum) and Yabu (Hypermomentum):

obeying the conservation laws D,,(eanu) = RnwbceYbcu + RaubeObU,

D,(e Tub“>=

(4.9a)

a2 v

pub

u e = det pus.

-

Note that Ytabls is the spin-current, current [5, 251.

g

(.fnbn)

-

$

(hnb,g

+

%bd

$9

(4.9b)

Y(ib)u the shear current and tr Yu the dilation

310

NE'EMAN

AND SIJAEKI

The gauge ensuresuniversal couplings to the Energy-momentum tensor 0,~ and to Hypermomentum Yobu:

(4.10) Picking the holonomic gauge pu = 8, and restricting to Special Relativity (i.e. making useof the Equivalence Principle) yields in (4.9b) for the intrinsic P of spin and shear: 2, cm1lL

=

%I

(4.11)

>

(4.12)

8, YG) 4‘ = O(&) + g&$!.

We note that in a Weak field limit we might have written (hc,,, = gcr,*)- nOi,)in the global case, for the Tcnb)currents,
1 y(ibf(o)

i a>’

=

CNL@

i h&b)

1 @>

q”“/q2

+

CL@

1 y(8bf

1 a>‘~.

(4.13)

where q = pa - pB . The graviton pole will appear when cy, /3 contain leptons and will be restricted by (4.12) in analogy to PCAC. This separation is analogous to the seggregation of the pion pole term in the Axial Charge as discussedin Ref. [27]. Here it enables us to separate the Shear Charge

where NL (non-linear field solution) stands for the contributions of the graviton pole between leptons, and L (linear field solution) will contain the hadron contributions. However, once we go over to the Gauged case, the Goldstone solution becomes a Higgs-like solution, and the 7’t;b)NLis annihilated.

5. FIELD EQUATIONS

The Lagrangian density for the gaugefields ‘2)can be chosenin many different ways. Two characteristics are most important in this choice (1) The number of propagating irreducible gauge multiplets NG . (2) The order of the Lagrangian in terms of gauge potentials. NC is determined by the existence of kinetic-energy terms for the gauge potentials in the Lagrangian. For example, the Einstein Lagragnian has the structure emphasized by Trautman, 8

=

Rpb’

A

&,I

>

habl

=

%txctPc

A

P”

(5.1)

so that only dp[abJappears, and NG = I. The same is true of the Metric-Afine

GRAVITY,

STRONG

INTERACTIONS

GL(4, R)

WITH

311

Lagrangian [25], in which only dp tabI appears, although the theory involves GA(4, R) since the Rfab] in it is the one in Eq. (4.2) involving pcab)as well, B = RtabJ A [[ab] + p 1f3,, /‘.

(5.2)

In a recently suggestedConfining model, No = 2 as dp”, dp[ablboth appear, B = &

(R” A ,o”) A *(Rb I, ,LJ”) + .&

R[flb] ,., *R[a”],

(5.3)

As we would like to involve the Equivalence Principle for Macroscopic Space-Time, the NG = 1 cases represent ordinary gravitons. It is then best to go over to the holonomic gauge pu = 6, and expressp,,abin terms of g,, and its derivatives. The equations of motion can be written as (5.4) (5.5) Using the canonical momenta

the equations become

where Eau and Eabu represents the momentum and hypermomentum of the GA(4, R) gauge fields themselves. The NG = 1 casesgenerally have TV@”= 0 and no contribution to E abufrom the GL(4, R) connections, which makes(5.8) algebraic and solvabe for p,ab. Using the holonomic gauge for the metric affine theory (a non-linear realization, with a g&r Eqs. (5.4) (5.5) are replaced by (r is the holonomic form of puQb) G(uv)(~,

r)

=

&Id,

Pupp(g,r) = KYUY”,

(5.9) (5.10)

where G(““) is the Einstein tensor and P liyp is the Palatini tensor. Equation (5.10) relates the Goldstone field g or its covariant derivatives to the hypermomentum current, and has some relationship with (4.12). Note that the derivation of Noether’s theorem (and identities for a local gauge) involves the application of the equations of motion.

312

NE’EMAN

AND

ibJAEK1

On the other hand, this equation in fact describes torsion (for [ab]) in terms of spin lil~,~l and Non-metricity,

Q,a)

=

&&b)

=

‘%k,,k%)

-

(5.11)

2hdab)

in terms of r&b) . It can be regarded as a Higgs-like phenomenon, in which the momenta of g provide the components of the shear potentials. In fact, the Palatini tensor is used to go over to second order formalism, in which the QUfab)terms in G(p”) are replaced by Shears from (5.10) and the Noether shear current looses its universal (gauge) role, while &’ acquires a contact term. This is another aspect of a Higgs-like spontaneous breakdown: the disappearance of the physical gauge currents for the broken symmetry. If we now turn to the Lagrangian in (5.3) we find that since NG > 1, both equations of motion are differential equations and both pz and p!“] propagate. In that case we have an “outside” where pz carries Newton-Einstein like gravitation and where we can use the Equivalence Principle, and an “inside” where ppbJ also propagates and generates Confinement. We now conjecture that the physical B might have both aspects: behave as in the Metric-Affine for leptons, with a metric g,, as a Goldstone-Higgs field, and as in (5.3) for hadrons which are thereby confined. One such possible model would be

(K is the Planck length, i.e. Newton’s constant; (Y is dimensionless). momenta are (we dispense with the holonomic indices)

The canonical

1 r a- -

0;

with equations (including --

%b]

=

A pbEabed

2

‘abcdp

the material

,‘zR[cdl ;;z

2

contributions) 1

+

401

1 k+eczRC

A pd +

n(ab)

7;

R

=

4n ’

(5.13)

hab)

using (5.7), (5.8)

(cd) A 65(,,~/6p” = e@, ,

R (cd) h 6[(cd)/6p[“b1

&

Dikb,

(the currents are 3-forms). The last equation is the Yang-Mills

=

eYLab]

=

eY(ab)

,

(5.16)

like confining equation,

acting only on hadrons, as the Higgs solution ensures that outside of the confined region where dpab # 0 y(abh’L

and space-time is Metric-affine.

=

0

(5.17)

313

GRAVITY, STRONG INTERACTIONS WITH GL(4, R) APPENDIX

Let us consider the most general scalar product one can define in the Hilbert space of functions of the parameters /3, , y1 , flz and yz of the maximal compact subgroup SU(2) x SU(2) of $%(4, R). For any two functionsfand g, the scalar product is given by tf, g> = j dk’ dk f *WI W,

4 g(k),

k, k’ E W(2)

x W(2),

(A-1)

where K(kl , k,) is a kernel function and dk and dk’ are invariant measures over SU(2) x SU(2). The Hilbert space with this scalar product we denote by H = L2(SU(2) x SU(2), K). A representation of the maximal compact subgroup in the space of functions over itself is given by t VOf)tk)

= ftkk”),

(A.3

for an arbitrary element k” E SU(2) x SU(2), and the unitarity means that

of the representation

( W”) f, Uk”) d = tf, d.

(A.3)

Since dk is an invariant measure we obtain from (A.l), (A.2) and (A.3) the following constraint on the form of the kernel K(k’, k) = K(k’k-l).

(A-4)

In terms of the complete set of functions {((2j1 + 1)(2j2 + 1)l) D&#&) over SU(2) x SU(2) we can write f(k)

=

c

KG

+

WG2

+

1))1’2

.fz:

. ^D:;.tk,)

D&tk,),

D&$32y2)

k = k,k, ,

(A.5)

(‘4.6)

(A.7)

314

NE'EMAN AND SIJA~KI

we find that

In general an arbitrary generator X of E(4, R) has matrix elements given by

x

s

dk’ dk D$k;)

D$(k;)

K(k’k-‘) X(k) D&(kl)

D:mz(k,).

(A-9)

Making use of the additivity relations (A.7) for the D-functions and the fact that dk is an invariant measure we obtain for the multiplicity free representations (A.lO)

The hermiticity condition (i.e., the unitarity of the representations) (A.11)

(L XT) = (g, WI* now reads

(A.12) From the requirement that the scalar product is hermitian, i.e., (L g) from (A.12) for X beingj:’ or j$, immediately follows the condition 4A) Forf

( g, f)*, or

(A.13)

= K*(MJ.

= g, expression (A.8) becomes (A.14)

CL.f> = C i j21:,,,I2K(iA j,j, ml% and the positive definitenessof the scalar product, i.e. (f,f) djl ,h) 2 0, for every j, and jz .

3 0, yields (A.15)

GRAVITY,

STRONG

INTERACTIONS

WITH

GL(4,

&)

315

ACKNOWLEDGMENT We are undebted to Dr. H. Rumpf for the anholonomic version of the Lagrangian of [7]. One of the authors (Dj. S) wishes to thank the Wolfson Chair Extraordinary of Theoretical Physics of Tel Aviv University for its hospitality and financial support.

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