Notes on semiclassical Weyl gravity

TL;DR

This paper reviews the semiclassical limit in quantum gravity, compares it in general relativity and Weyl gravity, and discusses how a semiclassical time can be recovered in these theories.

Contribution

It introduces the canonical quantization and semiclassical approximation of Weyl (conformal) gravity, contrasting it with quantum geometrodynamics.

Findings

Semiclassical time can be recovered in Weyl gravity, but with differences from quantum geometrodynamics.

The canonical quantization of Weyl gravity is developed and analyzed.

Conceptual issues in recovering semiclassical time are discussed.

Abstract

In any quantum theory of gravity, it is of the utmost importance to recover the limit of quantum theory in an external spacetime. In quantum geometrodynamics (quantization of general relativity in the Schr\"odinger picture), this leads in particular to the recovery of a semiclassical (WKB) time which governs the dynamics of non-gravitational fields in spacetime. Here, we first review this procedure with special emphasis on conceptual issues. We then turn to an alternative theory - Weyl (conformal) gravity, which is defined by a Lagrangian that is proportional to the square of the Weyl tensor. We present the canonical quantization of this theory and develop its semiclassical approximation. We discuss in particular the extent to which a semiclassical time can be recovered and contrast it with the situation in quantum geometrodynamics.