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

246 P. van Nieuwenhuizen, Supergravity= ÷ 01110Fig. 1. Unitarity in Yang—Mills theory.The proof of unitarity is based on the following Ward identities, which one derives by making a BRSTtransformation of the integration variables in the path-integral6BRsT(C*c~(x)c9.W”(v))=0=(3~W°(x)a. Wb(y)) (2)SBRS.,-(C(x) W~(y))= 0 = (i9~W~(x)W~(y)—C*~(x)e~C’~’(y)). (3)Graphically they are represented in fig.momentum-shell.2. In (2) we have omitted a term fl~since we are on~ ~= (a:::::‘: )q,.(~:‘.::: )~~~voFig. 2.Ward identities for unitarity.The second identity describes a conservation law, but note that one must hit all non-physical (but withphysical momenta) lines by momenta. The first identity shows what happens if one only hits one gaugefield: the gauge field becomes a ghost field which emits a momentum at the other line.Let us now return to fig. 1. Using= 6,.,~— (k~1e~+1çk~)(k ~y’ (4)we omit the k,., terms in one of the two propagators, since we can use yet another Ward identity(3• W”(x) W~’(y 1)...W~’,,(y,,))= 0 (conservation) (5)where all W~,(y,)fields are on-shell with physical polarizations. Thus we consider fig. 1 with 6,.,,. upstairsbut 6,.,,. downstairs in the first diagram. Using the first identity in fig. 2 once to cycle clockwise on theleft, and_once more to cycle clockwise on the right, one makes a full circle and ends up with a factork)’ times the final ghost momentum k,. This is exactly the Faddeev—Popov ghost plus minussign with counter clockwise orientations. Similarly, the term with —,k’~k,.(k R)’ can be used to make afull circle counter clockwise and yields the other orientation of the Faddeev—Popov ghost. Thus, asexpected, although after fixing the gauge all four modes of W”,~propagate for a given a, the complexworld-scalar ghost subtracts two modes, and one ends up with two physical modes in the quantumtheory as well.