THE REALITY CONDITIONS FOR THE NEW CANONICAL VARIABLES OF GENERAL-RELATIVITY

We examine the constraints and the reality conditions that have to be imposed in the canonical theory of four-dimensional gravity formulated in terms of Ashtekar variables. We find that the polynomial reality conditions are consistent with the constraints and make the theory equivalent to Einstein's, as long as the densitized inverse 3-metric is positive definite; in a region of space where it is not, the theory is still consistent, although it does not describe (in general) a real local geometry.