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

The method of solution of the initial value constraints for pure canonical gravity in terms of Ashtekar's new canonical variables due to Capovilla et al. (1991) is further developed in the present paper. There are two new main results. (1) The author extends the method to arbitrary matter coupling again for non-degenerate metrics: the new feature is that the 'CDJ-matrix' adopts a non-trivial 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 second 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 Capovilla et al. It is shown how the Yang-Mills and gravitational Gauss constraints can be solved explicitly 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 also be of interest for Yang-Mills theories on curved backgrounds.