Non-Minimally Coupled Scalar Field and Ashtekar Variables

TL;DR

This paper explores the incorporation of a non-minimally coupled scalar field into Ashtekar's variables framework, deriving a first order action and analyzing the resulting non-polynomial equations in the canonical formulation.

Contribution

It introduces a new first order action functional for non-minimally coupled scalar fields within Ashtekar's variables, extending the canonical analysis of gravity.

Findings

The scalar field coupling leads to non-polynomial equations in Ashtekar's formulation.

A conformal transformation relates tetrad and connection variables to Ashtekar variables.

The derived action provides a basis for further quantum gravity studies with scalar fields.

Abstract

The non-minimal coupling of a scalar field is considered in the framework of Ashtekar's new variables formulation of gravity. A first order action functional for this system is derived in which the field variables are a tetrad field, and an SL(2,C) connection, together with the scalar field. The tetrad field and the SL(2,C) connection are related to the Ashtekar variables for the vacuum case by a conformal transformation. A canonical analysis shows that for this coupling the equations of Ashtekar's formulation of canonical gravity are non-polynomial in the scalar field. (to be published in Phys. Rev. D)