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.