The Space of Connections as the Arena for (Quantum) Gravity

TL;DR

This paper reviews the structure of the space of connections in canonical quantum gravity, highlighting its geometric properties, the emergence of a family of metrics, and implications for the interpretation of quantum theory.

Contribution

It introduces a family of metrics on the space of connections derived from canonical analysis, linking geometric structures to quantum gravity interpretations.

Findings

A 1-parameter family of metrics on the space of connections is identified.

The space of connections can be viewed as a geodesic principle for canonical GR.

Discussion on the potential role of a time variable in quantum gravity.

Abstract

We review some properties of the space of connections as the natural arena for canonical (quantum) gravity, and compare to the case of the superspace of 3-metrics. We detail how a 1-parameter family of metrics on the space of connections arises from the canonical analysis for general relativity which has a natural interpretation in terms of invariant tensors on the algebra of the gauge group. We also review the description of canonical GR as a geodesic principle on the space of connections, and comment on the existence of a time variable which could be used in the interpretation of the quantum theory.