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

242 P. van Nieuwenhuizen, SupergravitvOne can also obtain BRST invariance of I(qu) without S, P, A,~.In addition to eq. (17), one onlyneeds to substitute eq. (15) into the transformation rules. This we will discuss in subsection 2.8.2.4. Path-integral quantization of gauge theoriesThe approach to covariant quantization sketched up to this point used the notion of BRST symmetrywhich can be used to derive Ward identities from which one may prove unitarity and gauge invariance.An alternative approach is path-integral quantization. Since it exhibits some quantization aspects in aclear way, and explains why I(qu) is as it is, we will also consider this approach.One starts with the path-integralZ = J [dqY]exp(iI(cl))= (em~,I/ía S, P, A,,,) (1)and multiplies by unity1 = d~[~ ô(F,,(j’(~))— a~)sdet(Fa 1R’o) (2)where qS(~)=4,’ +R’ar and sdet denotes the superdeterminant (see appendix). Using gauge invarianceand replacing in eq. (1) [d4i’]by [d4,’(~)],one obtains, after interchanging the [d4,’(~)]andd~integrations and dropping a field-independent factor, writing [d4,’]instead of [d4,’(a)]Z = J [d4,’dCC dC*~]exp{i[I(cl)+ I(ghost)]}6(F0 — a0). (3)Again one may multiply by unity1 = J [dba]exp(~ba7”3b0) (sdet ~)I/2 (4)If one can show that eq. (3) is independent of aa and if the matrix y’~’3is independent of quantum fields,one may replace 6(Fa — aa) by 6(F,. — 12,,) and integrating over [dba] one arrives at the same result asbeforeZ = J [d4i’ dC” dCi] (sdet ~)II2exp(iI(qu)). (5)Thus the restriction that y”3 be independent of 4,’ clearly emerges from this formalism.We now prove that (3) is indeed independent of a,,. A necessary and sufficient 3 criterion = 6~theisproduct that under ofaJacobian gauge and transformation the variation64,’ of the = R’,,N”~’A0with Faddeev—Popov (super) arbitrary determinant A0 and (F,,.1R’0)N’3 equals unity. As for example shownin ref. [607],this is equivalent to6(R’,4”)/ôçb’ +f”,, 0 = 0. (6)