Relate the refined quantum state to the Fock-space dynamics developed here

Determine a precise mathematical correspondence between the refined quantum state ω^t_V, defined via group-averaged surface layer integrals, and the unitary Fock-space time evolution constructed in this paper. Construct an explicit mapping or equivalence showing how the density operator arising from the refined state relates to the Fock-space state vector and verify consistency of observables and dynamics across both formulations.

Background

Previous work by the authors introduced a refined quantum state for causal fermion systems via unitary group averaging and surface layer integrals, yielding a state at each time but without an explicit dynamical law. In contrast, the present paper derives a unitary time evolution on fermionic and bosonic Fock spaces.

The authors explicitly state that, on a technical level, it is presently unclear how to relate these two constructions. Establishing this relation would integrate the algebraic state-based framework with the dynamical Fock-space evolution, providing a unified picture.