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 $K$ into the physical Hilbert space $H$. Under this projection, a quantum Cauchy surface isomorphically represents a physical subspace $D\subset H$ with a kinematic subspace $V\subset K$. The isomorphism induces the complete sets of Dirac observables in $D$, which faithfully represent the corresponding complete sets of self-adjoint operators in $V$. 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. (paper)