On the solution of the initial value constraints for general relativity coupled to matter in terms of Ashtekar's variables

TL;DR

This paper advances methods for solving initial value constraints in general relativity with matter using Ashtekar's variables, extending previous approaches and introducing new independent solution techniques for coupled matter fields.

Contribution

It extends the CDJ method to include arbitrary matter coupling with non-degenerate metrics and introduces an independent approach for solving initial constraints with matter.

Findings

Extended CDJ method to matter coupling with antisymmetric CDJ-matrix parts.

Explicit solutions for Yang-Mills and gravitational Gauss constraints.

New technique for solving initial constraints with scalar or spinor matter fields.

Abstract

The method of solution of the initial value constraints for pure canonical gravity in terms of Ashtekar's new canonical variables due to CDJ is further developed in the present paper. There are 2 new main results : 1) We extend the method of CDJ to arbitrary matter-coupling again for non-degenerate metrics : the new feature is that the 'CDJ-matrix' adopts a nontrivial antisymmetric part when solving the vector constraint and that the Klein-Gordon-field is used, instead of the symmetric part of the CDJ-matrix, in order to satisfy the scalar constraint. 2) The 2nd result is that one can solve the general initial value constraints for arbitrary matter coupling by a method which is completely independent of that of CDJ. It is shown how the Yang-Mills and gravitational Gauss constraints can be solved explicitely for the corresponding electric fields. The rest of the constraints can then be satisfied by using either scalar or spinor field momenta. This new trick might be of interest also for Yang-Mills theories on curved backgrounds.