Real sector of the nonminimally coupled scalar field to self-dual gravity

TL;DR

This paper explores the canonical formulation of a nonminimally coupled scalar field in self-dual gravity, focusing on reality conditions, constraint structure, and methods to maintain polynomial form.

Contribution

It introduces a novel approach to implement reality conditions as second class constraints and discusses converting them into first class constraints to preserve polynomial simplicity.

Findings

Reality conditions lead to three real degrees of freedom per space point.

The original complex structure reduces to a real one via Dirac brackets.

An alternative method converts second class constraints into first class constraints.

Abstract

A scalar field nonminimally coupled to gravity is studied in the canonical framework, using self-dual variables. The corresponding constraints are first class and polynomial. To identify the real sector of the theory, reality conditions are implemented as second class constraints, leading to three real configurational degrees of freedom per space point. Nevertheless, this realization makes non-polynomial some of the constraints. The original complex symplectic structure reduces to the expected real one, by using the appropriate Dirac brackets. For the sake of preserving the simplicity of the constraints, an alternative method preventing the use of Dirac brackets, is discussed. It consists of converting all second class constraints into first class by adding extra variables. This strategy is implemented for the pure gravity case.