  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.
