Of course we have a notion of qft in curved spacetime, though I'm not sure how one can represent a particle state on curved spacetime without a timelike Killing vector field (i.e. a particle should either have an energy momentum tensor associated to some conserved current, else the particle state should be define along each spacelike hypersurface).

I wonder if there is literature on constructing a one particle state density matrix $\rho$ in the context of GR, and if so, in the 3+1 space-time decomposition. Perhaps this defines unambiguously the notion of a state vector. Thank you for suggestions.

Hence how one can define the Schrodinger equation, or Quantum Liouville/Kinetic equations in this context?