Path integral for the Hilbert-Palatini and Ashtekar gravity - art. no. 124029

To write down a path integral for the Ashtekar gravity one must solve three fundamental problems. First, one must understand the rules of complex cont our functional integration with holomorphic action. Second, one should find which gauges are compatible with reality conditions. Third, one should eva luate the Faddeev-Popov determinant produced by these conditions. In the pr esent paper we derive the ERST path integral for Hilbert-Palatini gravity. We show that for a certain class of gauge conditions this path integral can be rewritten in terms of the Ashtekar variables. Reality conditions define contours of integration, For our class of gauges all ghost terms coincide with what one could write naively, just ignoring any Jacobian factors arisi ng from the reality conditions, [S0556-2821(98)06824-6].