Quantization of diffeomorphism invariant theories of connections with local degrees of freedom

TL;DR

This paper develops a method for quantizing diffeomorphism invariant connection theories, successfully solving constraints and establishing an inner product, with implications for models like Husain-Kuchař and potentially for general relativity.

Contribution

It introduces a new quantization approach for diffeomorphism invariant theories of connections, including solutions to constraints and a suitable inner product, advancing the quantization of models like Husain-Kuchař.

Findings

Solved diffeomorphism constraints for connection theories

Established an inner product satisfying reality conditions

Provided a framework applicable to general relativity

Abstract

Quantization of diffeomorphism invariant theories of connections is studied. A solutions of the diffeomorphism constraints is found. The space of solutions is equipped with an inner product that is shown to satisfy the physical reality conditions. This provides, in particular, a quantization of the Husain-Kucha\v{r} model. The main results also pave way to quantization of other diffeomorphism invariant theories such as general relativity. In the Riemannian case (i.e., signature ++++), the approach appears to contain all the necessary ingredients already. In the Lorentzian case, it will have to combined in an appropriate fashion with a coherent state transform to incorporate complex connections.