Constraints in Quantum Geometrodynamics

TL;DR

This paper compares different methods of handling constraints in quantum gravity, introducing a geometrodynamic quantization approach that avoids traditional problems of time by focusing on true dynamical variables.

Contribution

It develops a general formalism for geometrodynamic quantization in quantum gravity, separating quantization from constraint enforcement, and overcoming issues of time evolution.

Findings

The geometrodynamic quantization approach avoids problems of time.

The formalism is applicable beyond homogeneous cosmologies.

It retains essential features of previous models.

Abstract

We compare different treatments of the constraints in canonical quantum gravity. The standard approach on the superspace of 3--geometries treats the constraints as the sole carriers of the dynamic content of the theory, thus rendering the traditional dynamical equations obsolete. Quantization of the constraints in both the Dirac and ADM square root Hamiltonian approaches leads to the well known problems of time evolution. These problems of time are of both an interpretational and technical nature. In contrast, the geometrodynamic quantization procedure on the superspace of the true dynamical variables separates the issues of quantization from the enforcement of the constraints. The resulting theory takes into account states that are off-shell with respect to the constraints, and thus avoids the problems of time. We develop, for the first time, the geometrodynamic quantization formalism in a general setting and show that it retains all essential features previously illustrated in the context of homogeneous cosmologies.