Quantum Geometrodynamics of Higher Derivative Theories with and without Conformal Symmetry

TL;DR

This thesis develops a canonical quantization framework for higher derivative gravity theories, incorporating quadratic curvature terms and conformal symmetry, providing insights into semiclassical corrections and the role of conformal invariance in quantum gravity.

Contribution

It introduces a novel canonical quantization approach for higher derivative theories with quadratic curvature terms and explores conformal symmetry without a physical length scale.

Findings

Perturbative semiclassical corrections to General Relativity are derived.

A model-independent generator of conformal transformations is constructed.

The approach emphasizes quantizing transformation generators over first class constraints.

Abstract

The main goal of this thesis is to quantize the Einstein-Hilbert action extended by the quadratic curvature terms within the canonical quantization approach, thus formulating quantum geometrodynamics of the higher derivative theories. The motivation is to provide an alternative to the standard canonical quantization based on the Einstein-Hilbert action alone, because the latter does not generate the quadratic curvature terms in the semiclassical limit. A particular formulation of a semiclassical approximation scheme is employed which ensures that the effects of the quadratic curvature terms become perturbative in the semiclassical limit. This leaves the classical General Relativity intact, while naturally giving rise to its first semiclassical corrections. Another topic of interest is a classical theory where the quadratic Ricci scalar and the Einstein-Hilbert term are absent from the action, which then enjoys the symmetry with respect to the conformal transformation of fields (local Weyl rescaling). We pay a special attention to this case, since it provides a natural setting for the absence of the notion of a physical length scale. Certain useful model-independent tools are also constructed in this thesis. Firstly, dimensionless coordinates and the unimodular decomposition of the metric are used to expose the only conformally variant degree of freedom, making the geometrical origin of the physical length scale apparent. With such an approach several earlier results become much more transparent. Secondly, using unimodular-conformal variables a model-independent generator of conformal field transformations is constructed in terms of which a reformulation of the definition of conformal invariance is given. Thirdly, it is argued that a canonical quantization scheme makes more sense to be based on the quantization of generators of relevant transformations, than on first class constraints.