Recent Mathematical Developments in Quantum General Relativity

TL;DR

This paper reviews recent mathematical advances in non-perturbative canonical quantum gravity, including explicit constructions and solutions that strengthen the theoretical foundation of quantum general relativity.

Contribution

It presents new mathematical results such as the quantum Wheeler's superspace, solutions to the diffeomorphism constraint, and methods to incorporate reality conditions.

Findings

Explicit construction of quantum Wheeler's superspace

Solution of the diffeomorphism constraint in quantum geometrodynamics

Scheme for incorporating reality conditions in quantum connection dynamics

Abstract

After a brief chronological sketch of developments in non-perturbative canonical quantum gravity, some of the recent mathematical results are reviewed. These include: i) an explicit construction of the quantum counterpart of Wheeler's superspace; ii) a rigorous procedure leading to the general solution of the diffeomorphism constraint in quantum geometrodynamics as well as connection dynamics; and, iii) a scheme to incorporate the reality conditions in quantum connection dynamics. Furthermore, there is a new language to formulate the central questions and techniques to answer them. These developments put the program on a sounder footing and, in particular, address certain concerns and reservations about consistency of the overall scheme.