View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 25, 537-555 (1969) Geometry of Spinors* A. Bows Wayne State University, Detroit, Michigan 48202 Submitted by N. Coburn Covariant differentiation of spinors from the point of view of a connexion on a principal fiber bundle of spinor frames has been studied by Guy [3] and Lichnerowicz [7]. The bundle studied has Spin (n) as structural group. In this paper, among other fiber bundles we introduce a principal fiber bundle P[M] of sp inor frames which has a full linear group as structural group and contains the previously mentioned bundle as a subbundle. We then study connections in the rather general setting of P[M]. In addition to the well-known structure equation for a connexion on a principal fiber bundle (see p. 77, [6]), we discuss a second structure equation for a connexion on P[M]. These two equations furnish necessary and sufficient conditions that a form on P[M] be a connection form. (Note that although [3] and [7] deal with IZ = 4, the study generalizes rather easily.) We then study the reduction of connections on P[M]. To this end we will develop a theory for spinors similar to the theory of metric connections on the bundle of (vector) frames L[M]. Actually our study will contain in a natural way this theory of metric connections. An application of this study in terms of the electromagvetic potential vector in general relativity will be given. Latin letters will be used to index tensors and range from 1 to n (n = 2v or 2v + 1). Greek letters will index spinors and range from I. to 2”. The Einstein summation convention is used throughout. 1. SPINORS AND RELATED FIBER BUNDLES Let M be an n-dimensional orientable Riemannian manifold. The follow- ing discussion applies to manifolds with an indefinite metric with only minor modifications (see [2]). * This work was partially supported by the AFOSR grant 20-63 administered by the Office of Research Administration of the University of Michigan. 537 538 BOWN We consider a set of 2v + 1, 2y x 2y complex matrices (where n = 2~ or 2~ + 1) calledJut Dirac matrices where i = 1,2 ,..., 2v + 1; cy, /3 = 1, 2 ,..., 2y. These matrices satisfy where Pi is the Kronecker delta and I is the identity matrix. We define the group S(n) as the set of all 2” x 2 complex matrices a = (clg”) such that there are real numbers hji (i,j = 1,2,..., n) with the property (i = 1, 2,..., n; cq p = 1,2 ,..., 2y) and (ail”) = a-r. In this case we define a map fm on L?(n)by R(a) = h = (hii). It is known (see p. 5-16, [9]) that if 1z is even 2,(&n)) = O(n) (the orthogonal group) and if n is odd fJ.!?(n)) = O+(n) (the proper orthogonal group). We define s(n) as all a E s(n) such that det a = 1. The map xn is the restriction of fn to s(n). Note that the classical group Spin (n) although defined in a manner very similar to s(n) is actually a proper subgroup of 5’(n). We assume that it is possible to construct by extension, using xn and the bundle of orthonormal frames O[M], a principal fiber bundle Y[M] with base M and structure group S(n). For conditions concerning this possibility see [4]. Associated with O[M] we have V = (U,} a covering of M by coordinate neighborhoods. On each U, we have defined a vector field 2 i = 1,2,..., n such that (ei Im) is an orthonormal frame at rnE U, . Hereafter, we deal only with coordi:ate neighborhoods belonging to V. Also we have the projection map n : OIM] + M and a collection of diffeomorphisms a, : U, x O(n)+O[M] given by u (m, c!) = (tji2 I,). Given the coordinate neighborhoods U, , U, E V with U, n UB f 4, then if gk Irn = hkv(m) ,“r Im for m E VA n U, , the transition function of (VA , U,) defined on VA n U, is given by y.4&4 = 44 = V+YmN. GEOMETRY OF SPINORS 539 Associated with the bundle Y[M] we have corresponding quantities which we indicate with the symbol -. For example, if Y&3(4 = h E O(n), then the transition function of (U, , Us) for the bundle Y[M] is denoted by yAB and jj,.,,(m) is required to be one of the elements of the set x;‘(h). Note that in order to be able to construct Y[M] we must be able to choose the transition functions in such a way that if U, , U, , UC are in V, then for m E VA n U, n Uc . If this can be done then there is a standard proce- dure for constructing Y[M] (see p. 52, [6]). DEFINITION. A contravariant spinor z,Lat m E M is a map p + #(p) of &l(m) (which is a subset of Y[M]) to the vector space of 2y :< 1 complex matrices, such that #(P - a-7 = a - VW, (1.1) where a E s(n) and p * a-l means right translation of p by a-l. Iffi(p) = m E U, we can define a set of basis spinors at m assoaiated with VA as follows. DEFINITION. If &‘(~a) = (m, I) E U, x S(n) then 0. 0 for (Y = 1, 2,..., 2’ where the 1 is in the oath row. Note that the effect of ,eEIrn on any p E@(m) can be seen using (1.1). Now each contravariant spinor $ at m can be expressed as a linear combina- zet of the 5 Irn . If ii<p) = m E U, and z,h(pJ = (@) where pa = 5Jrn I) 9(P) = e$l Ins(P) Further the set (z~ 1,) is linearly independent and thus forms a basis for the vector space of contravariant spinors at m E U, . Such an ordered set is 540 BOWN called a contra-spinor frame at m. The frame (e 1 ) will be referred to as the natural contra-spinor frame at m associatgi gith U, . We can consider the bundle Y[M] as a collection of contra-spinor frames as follows. Suppose p E Y[M] and eA(rn, a) = p. Then we introduce the correspondence It can be shown that for m E U, n U, , (za I,) = T&9 . (Au I,). DEFINITION. A covayiunt spinor C$at m is a map p -+ 4(p) offi-l(m) to the vector space of 1 x 2” complex matrices such that if a E s(n) $qp * a-l) = c+(p) ’ a-l. We define the natural co-spinor frame associated with lJ, at m E U, , (seIm> by DEFINITION. Let 6i1(pO) = (m,I), then 4;* lm (PO) = (0, 0 ,..., 0, 1, 0 ,... , 0) for 01= 1, 2,..., 2y where the 1 is in the ath place. As in the contravariant case if C$is a covariant spinor at m E U, and n(p) = m, and if q@+,) = (+J where p, = eA(m, I), then Note that it can easily be shown that the co-spinor frame (;a I,) transforms contragradiently to the contra-spinor frame (z= I,). We pointed out that Y[M] can be thought of as a bundle of contra-spinor frames (e, I,> = (c+jAI,>, where a = (a,“) E 5’(n). We can enlarge this bundle to obtain two additional bundles. DEFINITION. The bundle ,?[M] consists of all contra-spinor frames (e, Im) m E M such that if m E U, E V, then where a = (a,3 E g(n). GEOMETRY OF SPINORS 541 DEFINITION. The bundle P[M] consists of all contra-spinor frames (e, I,) m EM such that if m E VA E V, then e, Im = d;A Im where a = (aa”) E GL(2’, C). Thus corresponding to the groups S(n) C S(n) C GL(2’, C) we have the bundles LqM] c &lq c P[M] with the above groups as their corresponding structure groups. It is not difficult to show that 5’(n) is closed in g(n) and S(R) is closed in GL(2’, C). (The manifold structure on GL(2’, C) arises from the fact that GL(2”, C) can be identified with an open set in C?“Xs” = RsXZvXav).We know therefore that there is a unique analytic’ structure for S(n) which makes it a submanifold of GL(2’, C) with the relative topology. Similarly, s(n) is a submanifold of 8(n) with the relative topology (see p. 105, [5]). Thus the same statements apply to the corresponding bundles Y[M], ~[M],P[MJ. 2. COORDINATESANDCOMPLEXTANGENTVECTORSTO P[M] We assign coordinates on P[M] as follows. Suppose p E P[M], QP) = m E U,4 , the coordinates of m in VA are (;f) i = 1, 2,..., n, and P = cx+x lm> for some X = (X,,a) E GL(2’, C). If x; = vya + c--T wya (where 1/--T denotes the complex number (0, 1)) then the real coordinates of p associated with the coordinate neighborhood n-l( VA) are (21, 2” ,...) iy, V& v;,..., v,“: , WC)..., w,z:> = (f, v,“, W;). The complex coordinates of p in +-l( VA) are ($, XYa). We will find these more convenient to work with.