Quantum Cauchy Surfaces in Canonical Quantum Gravity

TL;DR

This paper introduces a quantum concept of Cauchy surfaces in canonical quantum gravity, enabling a Schrödinger picture with a physical background time and a clear representation of physical states and observables.

Contribution

It proposes a novel quantum notion of Cauchy surfaces within the refined algebraic quantization framework, linking the physical Hilbert space to a kinematic subspace and defining a background time.

Findings

Defines quantum Cauchy surfaces as isomorphic representations of the physical Hilbert space.

Establishes a correspondence between Dirac observables and self-adjoint operators in a kinematic subspace.

Illustrates the approach with a simple model as an initial example.

Abstract

For a Dirac theory of quantum gravity obtained from the refined algebraic quantization procedure, we propose a quantum notion of Cauchy surfaces. In such a theory, there is a kernel projector for the quantized scalar and momentum constraints, which maps the kinematic Hilbert space $\mathbb K$ into the physical Hilbert space $\mathbb H$. Under this projection, a quantum Cauchy surface isomorphically represents $\mathbb H$ with a kinematic subspace $\mathbb V \subset\mathbb K$. The isomorphism induces the complete sets of Dirac observables in $\mathbb D$, which faithfully represent the corresponding complete sets of self-adjoint operators in $\mathbb V$. Due to the constraints, a specific subset of the observables would be "frozen" as number operators, providing a background physical time for the rest of the observables. Therefore, a proper foliation with the quantum Cauchy surfaces may provide an observer frame describing the physical states of spacetimes in a Schr\"odinger picture, with the evolutions under a specific physical background. A simple model will be supplied as an initiative trial.