Quantum Cauchy Surfaces in Canonical Quantum Gravity

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.