Quantum Cauchy Surfaces in Canonical Quantum Gravity

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 Schrödinger picture, with the evolutions under a specific physical background. A simple model will be supplied as an initiative trial.

27 pages

References in corpus (6)

• Quantum Nature of the Big Bang: An Analytical and Numerical Investigation
• Algebraic Quantum Gravity (AQG) I. Conceptual Setup
• Quantum field theory on a cosmological, quantum space-time
• New scalar constraint operator for loop quantum gravity
• The Wheeler-DeWitt Quantization Can Solve the Singularity Problem
• Rainbow metric from quantum gravity

Cited by in corpus (2)

• Conformally Invariant Brans-Dicke Loop Quantum Cosmology: A Quantum Geometric Model of Linking Theory
• Quantum Reference Frames via Transition Amplitudes in Timeless Quantum Gravity