Gravitational Interaction of Hadrons: Band-Spinor Representations of GL (n,R)Author(s): Yuval Ne'emanSource: Proceedings of the National Academy of Sciences of the United States of America,Vol. 74, No. 10 (Oct., 1977), pp. 4157-4159Published by: National Academy of SciencesStable URL: http://www.jstor.org/stable/67692 .

Accessed: 05/05/2014 18:39

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

.

National Academy of Sciences is collaborating with JSTOR to digitize, preserve and extend access toProceedings of the National Academy of Sciences of the United States of America.

http://www.jstor.org

This content downloaded from 195.78.109.27 on Mon, 5 May 2014 18:39:24 PMAll use subject to JSTOR Terms and Conditions

Proc. Natl. Acad. Sci. USA Vol. 74, No. 10, pp. 4157-4159, October 1977 Physics

Gravitational interaction of hadronsp: Band-spinor representations of GL(n,R)

(gravitation/linear group/double-valued representations/hypermornentum)

YUVAL NE'EMAN

Department of Physics and Astronomy, Tel-Aviv University, Israel; and European Or ganization for Nuclear Research CH 1211 23 Geneva, Switzerland

Contributed by Yuval Ne'eman, July 22, 1977

ABSTRACT We demonstrate the existence of double-valued linear (infinite) spinorial representations of the group of general coordinate transformations. We discuss the topology of the group of general coordinate transformations and its subgroups GA(nR), GL(n,R), SL(nr) for n = 2,3,4, and the existence of a double covering. We present the construction of band-spinor representations of GL(n,R) in terms of Harish-Chandra modules.

It is suggested that hadrons interact with gravitation as band-spinors of that type. In the metric-affine extension of general relativity, the hadron intrinsic hypermomentum is minimally coupled to the connection, in addition to the coupling of the energy momentum tensor to the vierbeins. The relativistic conservation of intrinsic hypermomentum fits the observed regularities of hadrons: SU(6) (- spin independence), scaling, and complex-J trajectories. The latter correspond to volume- preserving deformations (confinement?) exciting rotational bands.

1. Introduction

This note deals with two issues-one mathematical and one other physical. Mathematically, we present a new type of double-valued representation of the group of general coordinate transformations (Einstein's general covariance group) @ and of its linear subgroup : = GL (n,R), the general linear group in n dimensions (n 3 2) over which the representations of @ are built. Alternatively, our new representations can be regarded as representations of another group 9, not included in @, real- izing a global or gauged symmetry of matter fields. Our dou- ble-valued representations are infinite and of discrete type and reduce to sequences of double-valued representations of the Poincare group P. For time-like momenta, they reproduce rotational excitation bands-e.g., with spin

2 2 2 We accordingly have termed them "band-spinors," and more generally (for both single- and double-valued cases) "bandors." Note that it had always been assumed in the folklore of general relativity (and often written in texts) that GL(n,R) has no double-valued or spinorial representations; the existence of band-spinors is thus a nontrivial addition. We provide here the existence proof and a general construction, details being treated elsewhere. Note that one source of the prevalent belief that there are no spinors ("world"-spinors) stems from an unwar- ranted extrapolation of a theorem of E. Cartan (1). As can be seen in the text, Cartan was aware of the restriction of his proof to spinors with a finite number of components.

Physically, we suggest that hadrons interact with gravitation as bandors. Single-valued discrete infinite representations of

The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore be hereby marked "advertisement" in accordance with 18 U. S. C. ?1734 solely to indicate this fact.

SL(3,R) had been suggested (2) as excitations in quasi-orbital angular momentum in the quark model, for a description of the observed "Regge trajectories." Because GL(3,R) is the "little group" for time-like momenta in the general affine group W = GA(4,R), the representations used in ref. 2 can now be reinterpreted as spin-excitations and used for band-tensors. Similar band-tensors had been used for GL(4,R) in ref. 3, in a description of the spinning top. Our new band-spinors will represent nucleons, etc., including their high-spin excita- tions.

The current view of hadron dynamics is based on a quark field with a color-gluon mediated super-strong and confining interaction ("QCD"). Bandors represent an intermediate pic- ture between these "fundamental fields" and a phenomeno- logical rendering. They do include some part of the gluon ac- tion, because SL(3,R) is characteristic of excitations induced by volume-preserving stresses-perhaps the confining inter- action itself. They are in fact a somewhat less sophisticated string" or "dual model." In two other articles, we shall present this physical idea in

more detail (4). In particular, we shall show that the band-spinor description fits well into a recent generalization of general relativity, the metric-affine theory (5, 6). This can be regarded as a GA(4,R) gauge, in which the vierbein is coupled minimally to the energy-momentum tensor, and the affine connection is similarly coupled to the intrinsic hypermomentum tensor whose components are the spin, dilation, and shear currents. Note that these three quantities correspond to the observed regularities of the quark model: SU(6) (i.e., spin-independence), scaling, and the Regge trajectories.

The rest of this note is dedicated to the mathematical issue- i.e., the existence of double-valued representations of Y and its subgroups.

2. Topological considerations: The covering group of SL(nR) and GL(nR) We are studying the groups,

@ J D D ( s L 0 [2.1]

D DP DO [2.2] in which N is the unimodular linear SL(4R) and 0 is the special orthogonal SO(4). We do not enter into the further structure induced by the Minkowski metric at this stage. At various stages we shall also deal with the same groups over n = 3 and n = 2; we shall then use the notation 93, &3, etc ....

Because our aim is to find unitary representations of 6', i, 9, and N that reduce to double-valued unitary representations of P and 0, we have a priori two candidate solutions:

(i) e D 0, 93 D 03, 2 D P

(t)@D 7 D g D 7 D (ii)D :DD&D

4157

This content downloaded from 195.78.109.27 on Mon, 5 May 2014 18:39:24 PMAll use subject to JSTOR Terms and Conditions

4158 Physics: Ne'eman Proc. Natl. Acad. Sci. USA 74 (1977)

in which the bars denote double-covering of the relevant groups. In the first case, we would be dealing with single-valued representations of @ and its subgroups, and 0 would be con- tained through its covering 0. In the second case, all groups would display the same bivaluedness as 0, and we would have to go to their respective coverings to find a single-valued rep- resentation containing 0. Because

03 = SU(2)

it is enough that we show that S3 P SU(2) to cancel solution ().

We introduce an Iwasawa decomposition (7) of S. For a noncompact real simple (all invariant subgroups are discrete and in the center) Lie group -B, it is always possible to find

M = ]&A4N [2.3]

in which X is the maximal compact subgroup, A is a maximal Abelian subgroup homeomorphic to that of a vector space, and N is a nilpotent subgroup isomorphic to a group of triangular matrices with the identity in the diagonal and zeros everywhere below it. The decomposition is unique and holds globally

tn A =AQn N= NW n=11} [2.4]

Applying Eq. 2.3 to &3, X is 03. Because this is maximal and unique

S3 t 03

and we are left with solution (ii) only. Applying Eq. 2.3 to

9 = 0A4NJ [2.5]

we also have

7 = 0rA4sdNs [2.6]

Now the groups A and N in an Iwasawa decomposition are simply connected, and AN = AN is contractible to a point. Thus, the topology of & is that of 0. The same result has been shown to hold (8) for @ when the L4 is Euclidean or spherical and holds under some weak conditions for any L4.

By the same token, Y has the topoloy of O(n,R), the double covering of the full orthogonal (which includes the improper orthogonal matrices, with det = -1). Y and Y thus have two connected components.

For n > 3, S is thus completely covered by S, the double covering. However, 0(2) and SL(2R) are infinitely con- nected.

&2 < &2 [2.7]

in which e is the full covering. Topologically, solution (ii) is thus realizable. The single-

valued unitary (and thus infinite-dimensional) irreducible representations of & correspond to double-valued representa- tions of S and reduce to a sum of double-valued representations of 0.

This being established, it is interesting to check a second source of confusion at the origin of the statements found in the literature of general relativity and denying the existence of such double-valued representations. This is based upon an error in the statement of a theorem of Cartan (9): "The three linear unimodular groups of transformations over 2 variables [SL(2C), SU(2), SL(2R)] admit no linear many-valued representa- tion. "

As can be seem from Cartan's proof of this theorem (9), it holds only for SL(2C) and SU(2). Moreover, Bargmann (10)

has constructed the unitary representations of SL (2R), because this is the double-covering spin(3)(+--) of the 3-Lorentz group (+1, -1, -1) and even though only single-valued representa- tions of SL (2R) are required for this role, he has also constructed (7d) multivalued linear representations of that group. The representations

Ch ,h q= + s2 [2.8] q 4' 4 are bivalued representations of spin(3)(+--) = SL(2R) as can be derived from Bargmann's formula

U(b) = exp(4ilh) U(a) [2.9]

for two elements lying over the same element of SL(2R). We take I = 1.

Note that in reducing SL (4R) to SL (2R), the generators are represented on the coordinates by

1= (x1A1 - X282), -2 = 2 (x182 + x281), 2 2

2= - (x182 - x281) [2.10] 2

with commutation relations

[2;1,Y-2] = -i2;3, [Y-3,Y-1] iy-2, [Y-2,13] = iy-l [2.11]

with 3 generating the compact subalgebra (eigenvalues m in ref. 9). However, when using the same algebra as the double covering (10) of SO(1,2), the identification in terms of the (completely different) (1,-1,-1) space is given by

1 = i(xoO1 + XI(O), 2 = -i(X2(0 + xO(2), -3 = i(x1(2 - x2 (1) [2.12]

with the same commutators and the same role for 1. We stress this correspondence because it clarifies some additional aspects connected with arguments (11) against the existence of bivalued representations of &4.

3. The SL(3R) band-spinors: Existence

The unitary infinite-dimensional multiplicity-free represen- tations of SL(3R) are characterized by jo (the lowest j) and c, a real number,

DA@3; jo, C) [3.1]

the ladder representations (2) corresponding to jo = 0 and jo = 1; -o < c < O.

We provide here a construction, based upon the "sub-quo- tient" theorem for Harish-Chandra modules_{12). We return to the Iwasawa decomposition of Eq. 2.6 for S3

73 = 63A N [3.2]

and taking first S3 define AL3, the centralizer of A in l-i.e., in 03. This is the set of all f E 03 such that

( E 31 4 a-1 = a) [3.3] for any a E A. The elements of A span a 3-vector space, and At3 thus has to be in the diagonal. Because det(A'13) = 1, the elements of 03 belonging to A3 are the inversions in the 3 planes: (+1,-1,-1), (-1,+1,-1), and (-1,-1,+1). Together with the identity element, they form a group of order 4, with a multiplication table ni m2 = m3, m2 m3 = m1, m3 in1 = n2,

m2= 1. It appears Abelian in this representation. _ Returning now to @3 and 03, we look for M3 C 03. The in-

This content downloaded from 195.78.109.27 on Mon, 5 May 2014 18:39:24 PMAll use subject to JSTOR Terms and Conditions

Physics: Ne'eman Proc. Natl. Acad. Sci. USA 74 (1977) 4159

versions are given in SU(2) by exp(irni,/2), which yields the non-Abelian group

JR3 ( ii ?nn i 1 ) ~[3.4]

The subgroup Q3 C &3

can now be used to induce the representations of &3. Note that

S31(Q3 = SU(2)/.*3 [3.5]

The representations p (jo, c) of Q3 are given by jo for a rep- resentation of the Ax3 group of "plane inversions" in SU(2), and c for the characters of A, because N is represented trivially. The representations of Y3 will thus be labeled accordingly; from Eq. 3.5 we see that they will be spin-valued representations of Q3. Because AN = AN, univalence is guaranteed. Note that the only multiplicity-free bands* are 0 (S3; 0, c), O (&3; 1, c) with -c <c < and O(&3; 1/2, 0).

4. GA(4R) and GL(4R)

Because the representations of @ are those of iY, the physical states can be described by induced representations of Y over its stability subgroup and the translations. The stability subgroup is GL(3R), and we can thus use the product of our representa- tions of &3 by the 2-element factor group 0(3)/SO(3), because 93 will have the topology of 0(3).

Further complications will arise as a result of the local Min- kowskian metric %ab. The representations we developed fit the case of time-like momenta.

For construction of fields, we should use 94. Our analysis in the previous section can be repeated for this grou; the At4 will correspond to a product of two sets i (an, 1) and 94/Q54 is SU(2) X SU(2)/t4. For band-spinors, we shall need Yi(jo(l) = 1/2, jo 2)

= 0) (1 DU(jo(1) = 0, jo(2) = 1/2) with (Aj(1) = 1, Aj(2) = 1) non- compact action.

Each (j(l), j(2)) level of a band-spinor field satisfies a Barg- mann-Wigner equation (16) for j = I j(') I + I j(2) 1.

* Following our original search (1969-1970) with D. W. Joseph for spinor representations of SL (3,R) (and while we were still unaware of the general-relativity taboo), this result was found by Joseph and proved (13). It has recently been reconfirmed (14) after having been questioned (15).

The covariant derivative of a band-spinor field 4J'a will be griven by

D. 4Ja = j j,a + FT (GvT) f [4.1]

in which ii, A runs over the sets a, aXg, aXgpo-, --i.e., spins

1 5 9 i=_

159

2) 2' 2)'2 GV is an infinite dimensional representation of 9, and r,,X is the usual affine connection.

We would like to thank Professors B. Kostant and L. Michel for their aLdvice. This work was supported in part by the United States-Israel Binational Science Foundation.

1. Cartan, E. (1938) Leqons sur la Theorie des Spineurs II (Her- mann Editeurs, Paris), Article 177, pp. 89-91.

2. Dothan, Y., Gell-Mann, M. & Ne'eman, Y. (1965) Phys. Lett. 17, 148-151.

3. Dothan, Y. & Ne'eman, Y. (1965) in Symmetry Groups in Nu- clear and Particle Physics, ed. Dyson, F. J. (W. A. Benjamin, New York), pp. 287-310.

4. Hehl, F. W., Lord, E. A. & Ne'eman, Y. (1977) Phys. Lett., in press.

5. Hehl, F. W., Kerlick, G. D., & von der Heyde, P. (1976) Zeit. Naturforsch. Teil A 31, 111-114, 524-527, 823-827.

6. Hehl, F. W., Kerlick, G. D. & von der Heyde, P. (1976) Phys. Lett. B 63, 446-448.

7. Iwasawa, K. (1949) Ann. Math. 50,507-558. 8. Stewart, T. E. (1960) Proc. Am. Math. Soc. 11, 559-563. 9. Cartan, E. (1938) Leqons sur la Theorie des Spineurs I (Hermann

Editeurs, Paris), Articles 85-86, pp. 87-91. 0. Bargmann, V. (1947) Ann. Math. 48, 568-640.

11. Deser, S. & van Nieuwenhuizen, P. (1974) Phys. Rev. D 10, 411-420, Appendix A. ̀

12. Harish-Chandra (1954) Trans. Am. Math. Soc. 76,26-65. 13. Joseph, D. W. (1970) "Representations of the Algebra of SL(3R)

with Aj = 2" (University of Nebraska preprint, unpublished, referred to in ref. 15, 16p.

14. Ogievetsky, V. I. & Sokachev, E. (1975) Teor. Mat. Fiz 23, 214-220; English translation pp. 462-466.

15. Biedenharn, L. C., Cusson, R. Y., Han, M. Y. & Weaver, 0. L. (1972) Phys. Lett. B 42, 257-260.

16. Bargmann, V. & Wigner, E. P. (1946) Proc. Natl. Acad. Sci. USA 34,211-223.

This content downloaded from 195.78.109.27 on Mon, 5 May 2014 18:39:24 PMAll use subject to JSTOR Terms and Conditions