Reality conditions for Ashtekar variables as Dirac constraints

TL;DR

This paper demonstrates that reality conditions in Ashtekar variables can be formulated as Dirac second class constraints, affecting the counting of gravitational degrees of freedom and highlighting non-polynomial constraints regardless of their form.

Contribution

It introduces a method to implement reality conditions as Dirac second class constraints, providing a new perspective on the structure of Ashtekar variables in real gravity.

Findings

Reality conditions can be imposed as second class constraints.

Some constraints are inherently non-polynomial.

The approach aligns with the Wick transform perspective.

Abstract

We show that the reality conditions to be imposed on Ashtekar variables to recover real gravity can be implemented as second class constraints a la Dirac. Thus, counting gravitational degrees of freedom follows accordingly. Some constraints of the real theory turn out to be non-polynomial, regardless of the form, polynomial or non-polynomial, taken for the reality conditions. We comment upon the compatibility of our approach with the recently proposed Wick transform point of view, as well as on some alternatives for dealing with such second class constraints.