SUPERGRAVITY P. van NIEUWENHUIZEN To Joel Scherk 0370 ...

P. van Nieuwenhuizen, Supergravity 363A,. = A~C 1,0b, b arbitrary. (7)Hence AT= AtC*yb* = ATCY4C*7(b*b) from which it follows that CY4C*Y must be a positivenumber times the unit matrix. This is indeed the case in four dimensions (see below), but, for example,in five dimensions this is not the case and there one needs an internal index in order to define Majoranaspinors. We will always scale C such thatTC. (8)A=AIn order that the relation A = ATC is maintained in time, one has for a spinor whose field equation isthe Dirac equation (y”ö,. + M)A = 0C C-1—— T 9X(C-’7~TCä,.+M)OJ )‘L — ‘/~~ ()For massless spinors, one finds the weaker conditionC~y,.Cl=a~y,.T, a2+1. (10)(One easily proves that a2 = +1 from (5).) In four dimensions one finds solutions with a = +1 and —1, but,for example, in five dimensions a = +1. However, although in five dimensions a matrix C exists, one needsmore than one spinor in order to define a Majorana spinor, as we have seen above.Since the dotted and undotted components of A are mixed in the Dirac equation, one now gets thestronger condition Cy,.C-1 = ±y,,Tfor C, and it is clear that if a C exists, it is unique up to a constant.This matrix C is the charge conjugation matrix. That a C exists follows from the fact that the matrices(—y4,,”) (as well as +y4,,T) satisfy the same group multiplication table as y,,. Hence, since according toSchur’s lemma there is only one inequivalent four-dimensional representation of the finite group with 32elements spanned by y-matrices, y,. and ~,.T are indeed equivalent, as are y,. and +y,.T.For spin 3/2 fields one can essentially repeat their steps, and the same results are obtained.In a general representation of {y,,, y,.} = 20,,,. with, however, hermitian y,,, the charge conjugationmatrix C is antisymmetricCr=_C. (11)However, only in special representations is it true that (C- 1)as = — C,.1, (as should already be anticipatedfrom the position of the indices). To see this, note that using Cy,.C-’ = ay,,T twice, one findsTCy,.C-lCr= y,,. (12)(Cl)Hence C-1C~commutes with y,,. Thus C-ICr= kI, or CT= kC so that (cl)T C= k2C and k = ±1.Thus (Cy,)T= ak(Cy,.), (Q,.)T k(Cu) (Cy5y,.)T —akCy5y,. and (Cy5)T= kCy5. Clearly a = +1and a = —1 are both allowed (C = y~y~and C = y~in a Majorana representation). Since the 16 Diracmatrices are linearly independent and there can be at most 6 antisymmetric matrices, k = —1 and C isantisymmetric.Taking the hermitian conjugate of Cy,.C-’ = ay,,T one finds that CrC= 11 and clearly 1>0. Thus.