Deformation Quantization of Schutz Quantum Cosmology: Relational Time, Constraints, and Physical States

Abstract: A controlled correspondence is established between reduced relational dynamics and extended constrained quantization in an FLRW cosmology coupled to a Schutz perfect fluid. A canonical transformation in the matter sector yields a deparametrized constraint of the form $\mathcal C=p_T+H$, with $T$ serving as an internal time. In the extended Weyl--Wigner formulation, the quantum constraint is imposed bilaterally. Its antisymmetric part gives the exact Moyal evolution generated by the reduced Hamiltonian, while its symmetric part imposes an independent quantum shell condition. For a fixed self-adjoint realization of $\hat H$, every reduced density operator supported on its negative spectral subspace admits an explicit distributional extension. A spectral dyad with energies $E$ and $E'$ is mapped to a clock Wigner distribution supported at $p_T=-(E+E')/2$, with relational phase determined by $E-E'$. The map preserves off-diagonal coherences, satisfies both star constraints, and has the reduced Wigner state as its $p_T$ marginal; it is therefore injective on the admissible positive-density sector. A flat dust model with $q=1/2$ and Dirichlet data provides an explicit example: its reduced spectrum is $(-\infty,0]$, so the admissible sector contains all normalizable reduced states. The correspondence applies to a fixed constraint representative and quantum realization.