Friedrichs-Selected Schwarzschild Quantum Dynamics

This paper develops the quantum dynamics generated by the Friedrichs-selected Schwarzschild scalar theory established in the preceding work. The selected nonnegative radial operator determines a positive one-particle frequency, its continuum bosonic second quantization, a self-adjoint nonnegative Fock Hamiltonian, and a strongly continuous generated evolution. The terminal selection is preserved through this chain as actual operator equality rather than as a label attached after quantization.

A second construction derives physical Cauchy-time dynamics directly from the charged scalar field and momentum variables of the same Schwarzschild action. Finite spatial banks yield graph-completed self-adjoint nonnegative Hamiltonians and exact finite-window Schrödinger evolution. Strong regulator removal extends the construction to arbitrary occupation and then to all spatial modes on the owned positive interior interval.

The infinite-spatial problem exposes a genuine polarization constraint. The original unit-amplitude Cauchy polarization has a vacuum obstruction because scalar vacuum subtraction leaves pair coordinates whose norm blocks strong convergence. The full-potential equation instead determines a frequency-normalized polarization with unit Klein–Gordon flux and an initially diagonal charged Hamiltonian. In that polarization the normal-ordered infinite-spatial initial Hamiltonian is self-adjoint and nonnegative, and weighted ultraviolet estimates support removal of both spatial and occupation regulators.

The resulting all-spatial physical Hamiltonian generates a strongly continuous unitary propagator that preserves the free-energy domain and satisfies the physical Schrödinger equation. The same action also supplies the canonical variables and Bogoliubov transformation structure, with frequency-weighted square-summable mixing and a self-adjoint quadratic pair generator.

The result therefore has two deliberately separated dynamical lanes: the Friedrichs spectral/Fock evolution is the canonical quantum continuation of the selected radial operator, while the physical Cauchy construction gives an independently derived interior-time evolution from action-owned field variables. The theorem surface remains fixed-background; local spacetime stress, regulator-independent quantum-matter sources, and gravitational response belong to subsequent work.