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

374 P. van Nieuwenhuizen, SupergravityHermitian, it is not true that the extra terms C 1MC~+ C2MC2 vanish (their kinetic parts vanish). Notethat we did not have to assume any special properties for MSummarizing, we have the following results1 =Jfldz’~o(F~(z)—a~)sdet(8F~/3z), ~A = (x”, 0”)sdet(3FA/OzB) = fri dC~dC~exp(iC~(DFA/3z”)C~’).(17)H. The super Jacobian is the superdeterminantIn the covariant quantization of supergravity using path integrals, an essential ingredient is the superJacobian. We show here that for a general change of integration variables (x, 0) to 1(x, 0), O(x, 0) thesuper Jacobian is given by the superdeterminant:m(x, 0)/ar). (1)J = det(Ox’(i, ë)/91’) det(80Defining the supermatrices M and M’ byM — (Dx/81 8x/30\ M’ — (31/Ox 31/30 2— \a0/81 89/30)’ — ~09/8x 80/80one need specify only for Ox/0O and 31/00 whether one has left or right directives. One needs rightderivatives, in order that (Ox/31)(31/Ox) + (Ox/00)(00/Ox) be the unit matrix. Hence the super Jacobian isthe determinant of the Bose—Bose part of M (the usual result) times the determinant of theFermi—Fermi part of the inverse of M For infinitesimal changes one has J = ~ Ox’/81’ — Oem/Oem, and aproof that J equals the superdeterminant has been given by DeWitt, based on the principle thatexponentiation of this linearized result gives the superdeterminant. Here we give a direct proof valid forfinite transformations and due to Fung. -We consider the change of variables(x, 0) —* (1, 0) in two steps(i) we change 9~,... ~ to em, ... , 0~,keeping all xfixed;(ii) we change x’ x”~to 1’,. . . , I”, keeping all 0 fixed.The second change of variables has an ordinary Jacobiandet(Ox’ (1, ë)/D1’).Notice that it is essential that this change of variables is performed when the Fermi variables are alreadybarred — otherwise one would find the mixed determinant det(0x’(x, 0)101’). We now consider (i).Consider an integral of the formI=Jdxl...dxkdom...de1P(x,o)— — (3)= J dI’ . . . dl” dotm .. d~1JP(x(1, 9), 0(1, 9))