Proc. Natl. Acad. Sci. USA Vol. 74, No. 10, pp. 4157-4159, October 1977 Physics Gravitational interaction of hadrons: Band-spinor representations of GL(n,R) (gravitation/linear group/double-valued representations/hypermomentum) YUVAL NE'EMAN Department of Physics and Astronomy, Tel-Aviv University, Israel; and European Organization for Nuclear Research CH 1211 23 Geneva, Switzerland Contributed by Yuval Ne'eman, July 22, 1977

ABSTRACT We demonstrate the existence of double-valued SL(3,R) had been suggested (2) as excitations in quasi-orbital linear (infinite) spinorial representations of the group of general momentum in the for a of the coordinate transformations. We discuss the topology of the angular quark model, description group of general coordinate transformations and its subgroups observed "Regge trajectories." Because GL(3,R) is the "little GA(nR), GL(nR) SL4nr) for n = 2,3,4, and the existence of a group" for time-like momenta in the general affine group double covering. We present the construction of band-spinor J := GA(4,R), the representations used in ref. 2 can now be representations of GLtnR) in terms of Harish-Chandra reinterpreted as spin-excitations and used for band-tensors. modules. Similar band-tensors had been used for GL(4,R) in ref. 3, in a It is suggested that hadrons interact with gravitation as description of the spinning top. Our new band-spinors will band-spinors of that type. In the metric-affine extension of general relativity, the hadron intrinsic hypermomentum is represent nucleons, etc., including their high-spin excita- minimally coupled to the connection, in addition to the coupling tions. of the energy momentum tensor to the vierbeins. The relativistic The current view of hadron dynamics is based on a quark conservation of intrinsic hypermomentum fits the observed field with a color-gluon mediated super-strong and confining regularities of hadrons: SU(6) (- spin independence), scaling, interaction ("QCD"). Bandors represent an intermediate pic- and complex-J trajectories. The latter correspond to volume- ture between these "fundamental fields" and a phenomeno- preserving deformations (confinement?) exciting rotational logical rendering. They do include some part of the gluon ac- ands. tion, because SL(3,R) is characteristic of excitations induced by volume-preserving stresses-perhaps the confining inter- 1. Introduction action itself. They are in fact a somewhat less sophisticated This note deals with two issues-one mathematical and one string" or "dual model." other physical. Mathematically, we present a new type of In two other articles, we shall present this physical idea in double-valued representation of the group of general coordinate more detail (4). In particular, we shall show that the band-spinor transformations (Einstein's general covariance group) @ and description fits well into a recent generalization of general

of its linear subgroup . := GL(n,R), the general linear group relativity, the metric-affine theory (5, 6). This can be regarded in n dimensions (n > 2) over which the representations of @ are as a GA(4,R) gauge, in which the vierbein is coupled minimally built. Alternatively, our new representations can be regarded to the energy-momentum tensor, and the affine connection is as representations of another group ., not included in @, real- similarly coupled to the intrinsic hypermomentum tensor whose izing a global or gauged symmetry of matter fields. Our dou- components are the spin, dilation, and shear currents. Note that ble-valued representations are infinite and of discrete type and these three quantities correspond to the observed regularities reduce to sequences of double-valued representations of the of the quark model: SU(6) (i.e., spin-independence), scaling, Poincare group P. For time-like momenta, they reproduce and the Regge trajectories. rotational excitation bands-e.g., with spin The rest of this note is dedicated to the mathematical issue- 1 +5+9 i.e., the existence of double-valued representations of .9 and its ]2 2 2 subgroups. We accordingly have termed them "band-spinors," and more 2. Topological considerations: The covering group of generally (for both single- and double-valued cases) "bandors." SL(nR) and GL(nR) Note that it had always been assumed in the folklore of general We are studying the groups, relativity (and often written in texts) that GL(n,R) has no D0 double-valued or spinorial representations; the existence of eD D g DeV [2.1] band-spinors is thus a nontrivial addition. We provide here the @D JD PD [2.2] existence proof and a general construction, details being treated in which eV is the unimodular linear SL(4R) and ( is the special elsewhere. Note that one source of the prevalent belief that orthogonal SO(4). We do not enter into the further structure there are no spinors ("world"-spinors) stems from an unwar- induced by the Minkowski metric at this stage. At various stages ranted extrapolation of a theorem of E. Cartan (1). As can be we shall also deal with the same groups over n = 3 and n = 2; seen in the text, Cartan was aware of the restriction of his proof we shall then use the notation -93, QV3, etc .... to spinors with a finite number of components. Because our aim is to find unitary representations of @, AT, Physically, we suggest that hadrons interact with gravitation 9, and eV that reduce to double-valued unitary representations as bandors. Single-valued discrete infinite representations of of P and 0, we have a priori two candidate solutions: The costs of publication of this article were defrayed in part by the (i) D A,93DD 37, D P payment of page charges. This article must therefore be hereby marked "advertisement" in accordance with 18 U. S. C. §1734 solely to indicate (ii) DD DeVD this fact. D p 4157 Downloaded by guest on October 2, 2021 4158 Physics: Ne'eman Proc. Natl. Acad. Sci. USA 74 (1977) in which the bars denote double-covering of the relevant has constructed the unitary representations of SL(2R), because groups. In the first case, we would be dealing with single-valued this is the double-covering spin(3)(+--) of the 3-Lorentz group representations of @ and its subgroups, and 0 would be con- (+1, -1, -1) and even though only single-valued representa- tained through its covering 0. In the second case, all groups tions of SL(2R) are required for this role, he has also constructed would display the same bivaluedness as 0, and we would have (7d) multivalued linear representations of that group. The to go to their respective coverings to find a single-valued rep- representations resentation containing 0. Because chh=! q=1+s2 [2.8] 03 = SU(2) q' 4 ~1 it is enough that we show that 83 1 SU(2) to cancel solution are bivalued representations of spin(3)(+--) = SL(2R) as can (i). be derived from Bargmann's formula We introduce an Iwasawa decomposition (7) of W. For a noncompact real simple (all invariant subgroups are discrete U(b) = exp(4ilh) U(a) [2.9] and in the center) Lie group !, it is always possible to find for two elements lying over the same element of SL(2R). We takel = 1. S = N-A4-AN [2.3] Note that in reducing SL(4R) to SL(2R), the generators are in which X is the maximal compact subgroup, A is a maximal represented on the coordinates by Abelian subgroup homeomorphic to that of a vector space, and NV is a nilpotent subgroup isomorphic to a group of triangular 21= 2 (x11 - x282), :2 =2 (x182 + x281), matrices with the identity in the diagonal and zeros everywhere 2 2 below it. The decomposition is unique and holds globally = -2 (xV32 - x281) [2.10] kf -A = Afn =J X = {l} [2.4] 2 with commutation relations Applying Eq. 2.3 to V3, . is 03. Because this is maximal and unique [11,12] = -0;3, [13,z11= i12, [12,131 = il1 [2.111 83 Zb @3 with 3 generating the compact subalgebra (eigenvalues m in ref. 9). However, when using the same algebra as the double and we are left with solution (ii) only. Applying Eq. 2.3 to covering (10) of SO(1,2), the identification in terms of the (completely different) (1,-1,-1) space is given by &=0A'N8 [2.5] have 1 = i(XOO1 + X160), -2 = we also -i(X26o + XO(2), 23 = i(x162 - x2 81) [2.12] P=0.A8N's8 [2.6] Now the groups A and N in an Iwasawa decomposition are with the same commutators and the same role for 2. We stress simply connected, and_A = AN is contractible to a point. this correspondence because it clarifies some additional aspects Thus, the topology of is that of 0. The same result has been connected with arguments (11) against the existence of bivalued shown to hold (8) for @ when the L4 is Euclidean or spherical representations of eV4. and holds under some weak conditions for any L4. By the same token, T has the topoloy of O(n,R), the double 3. The SL(3R) band-spinors: Existence covering of the full orthogonal (which includes the improper orthogonal matrices, with det = -1). 9 and T thus have two The unitary infinite-dimensional multiplicity-free represen- connected components. tations of SL(3R) are characterized by jo (the lowest j) and c, For n > 3, eV is thus completely covered by eP, the double a real number, covering. However, 0(2) and SL(2R) are infinitely con- nected. D(&3; jo, C) [3.1] the ladder representations (2) corresponding to jo = 0 and jo 72 < &2 [2.7] = 1; -o < C < O. We provide here a construction, based upon the "sub-quo- in which e is the full covering. tient" theorem for Harish-Chandra modules_(12). We return Topologically, solution (ii) is thus realizable. The single- to the Iwasawa decomposition of Eq. 2.6 for S3 valued unitary (and thus infinite-dimensional) irreducible representations of S correspond to double-valued representa- 3= 03A N [3.2] tions of V and reduce to a sum of double-valued representations and taking first S3 define s03, the centralizer of A in W-i.e., of 0. in This is the set of all a such that This being established, it is interesting to check a second 3. E 03 source of confusion at the origin of the statements found in the (a 4J a31aa-' = a) [3.3] literature of general relativity and denying the existence of such for any a E A. The elements of A span a 3-vector space, and double-valued representations. This is based upon an error in .M3 thus has to be in the diagonal. Because det(.M3) = 1, the the statement of a theorem of Cartan (9): "The three linear elements of 03 belonging to M3 are the inversions in the 3 unimodular groups of transformations over 2 variables [SL(2C), planes: (+1,-1,-1), (-1,+1,-1), and (-1,-1,+1). Together SU(2), SL(2R)] admit no linear many-valued representa- with the identity element, they form a group of order 4, with tion. a multiplication table ml m2 = m3, m2 m3 = ml, m3 Ml = MZ As can be seem from Cartan's proof of this theorem (9), it = 1. It appears Abelian in this representation. holds only for SL(2C) and SU(2). Moreover, Bargmann (10) Returning now to OV3 and 3, we look for A3 C 03. The in- Downloaded by guest on October 2, 2021 Physics: Ne'eman Proc. Natl. Acad. Sci. USA 74(1977) 4159 versions are given in SU(2) by exp(i7rua/2), which yields the The covariant derivative of a band-spinor field '' will be non-Abelian group given by -*3:(±ign, ±1) [3.4] DM, Vi = Jaz 'Ia + rmv- (Gv,)2 byi [4.1] The subgroup @3 C Y3 in which ii, ,B runs over the sets a, aXli, aXlupp, -**i.e., Q3:=t3A spins 1 5 9 can now be used to induce the representations of &3. Note that 22' 2'2' GIv is an infinite dimensional representation of 9, and rFr; is V3/(Q3= SU(2)/4t3 [3.5] the usual affine connection. The representations p (Jo, c) of Qa are given by jo for a rep- resentation of the .M3 group of "plane inversions" in SU(2), and We would like to thank Professors B. Kostant and L. Michel for their c for the characters of_A, because WV is represented trivially. advice. This work was supported in part by the United States-Israel The representations of 7s will thus be labeled accordingly; from Binational Science Foundation. Eq. 3.5 we see that they will be spin-valued representations of (Q3. Because AN = AN, univalence is guaranteed. Note that 1. Cartan, E. (1938) Legons sur la Theorie des Spineurs II (Her- the only multiplicity-free bands* are D(S3; 0, c), O (eS3; 1, c) mann Editeurs, Paris), Article 177, pp. 89-91. with -Xo < c < X and 30(S; '/2, 0). 2. Dothan, Y., Gell-Mann, M. & Ne'eman, Y. (1965) Phys. Lett. 17, 148-151. 4. GA(4R) and GL(4R) 3. Dothan, Y. & Ne'eman, Y. (1965) in Symmetry Groups in Nu- Because the representations of @ are those of the physical clear and Particle Physics, ed. Dyson, F. J. (W. A. Benjamin, New Y, York), pp. 287-310. states can be described by induced representations of J over 4. Hehl, F. W., Lord, E. A. & Ne'eman, Y. (1977) Phys. Lett., in its stability subgroup and the translations. The stability subgroup press. is GL(3R), and we can thus use the product of our representa- 5. Hehl, F. W., Kerlick, G. D., & von der Heyde, P. (1976) Zeit. tions of e3 by the 2-element factor group 0(3)/SO(3), because Naturforsch. TeilA 31, 111-114, 524-527, 823-827. 93 will have the topology of 0(3). 6. Hehl, F. W., Kerlick,,G. D. & von der Heyde, P. (1976) Phys. Further complications will arise as a result of the local Min- Lett. B 63, 446-448. kowskian metric flab. The representations we developed fit the 7. Iwasawa, K. (1949) Ann. Math. 50,507-558. case of time-like momenta. 8. Stewart, T. E. (1960) Proc. Am. Math. Soc. 11, 559-563. For construction of fields, we should use g4. Our analysis in 9. Cartan, E. (1938)Legons surla Thgorie des SpineursI (Hermann the section Editeurs, Paris), Articles 85-86, pp. 87-91. previous can be repeated for this group; the 4f4 will 10. Bargmann, V. (1947) Ann. Math. 48,568-640. correspond to a product of two sets ± (a,,, 1) and &4/( is SU(2) 11. Deser, S. & van Nieuwenhuizen, P. (1974) Phys. Rev. D 10, X SU(2)/4t4. For band-spinors, we shall need Z(jo(l) = 1/2, jo2 411-420, Appendix A.' = 0) 0 D(jo(1) = 0, jo(2) = '/2) with (Aj(') = 1, Aj(2) = 1) non- 12. Harish-Chandra (1954) Trans. Am. Math. Soc. 76,26-65. compact action. 13. Joseph, D. W. (1970) "Representations ofthe Algebra ofSL(3R) Each (j(l), j(2)) level of a band-spinor field satisfies a Barg- with AJ = 2" (University of Nebraska preprint, unpublished, mann-Wigner equation (16) for j = Ii~'~l+ 1j(2)1. referred to in ref. 15, 16p. 14. Ogievetsky, V. I. & Sokachev, E. (1975) Teor. Mat. Fiz 23, * Following our original search (1969-1970) with D. W. Joseph for 214-220; English translation pp. 462-466. spinor representations of SL(3,R) (and while we were still unaware 15. Biedenharn, L. C., Cusson, R. Y., Han, M. Y. & Weaver, 0. L. of the general-relativity taboo), this result was found by Joseph and (1972) Phys. Lett. B 42,257-260. proved (13). It has recently been reconfirmed (14) after having been 16. Bargmann, V. & Wigner, E. P. (1946) Proc. Natl. Acad. Scd. USA questioned (15). 34,211-223. Downloaded by guest on October 2, 2021