Friedrichs-Selected Schwarzschild Quantum Dynamics

This paper develops the quantum dynamics generated by the Friedrichs-selected Schwarzschild scalar theory established in the preceding work, Critical Schwarzschild Scalar Dynamics. That analysis showed that the action-derived radial operator in the Schwarzschild interior has deficiency indices (1,1) and a genuine U(1) family of self-adjoint terminal laws. The full Schwarzschild ground-state form selects the no-log realization, and completion of the original compact-core quadratic form identifies that realization with the Friedrichs extension KF. The present paper asks what quantum dynamics is owned by that selection. The first construction follows directly from the selected operator. The nonnegative Friedrichs realization determines the positive one-particle frequency ωF=KF1/2, its continuum bosonic second quantization, the self-adjoint nonnegative Fock Hamiltonian HF=dΓ(ωF), and the corresponding strongly continuous unitary evolution. A quantum-ownership theorem propagates the terminal selection through this complete chain: every Hermitian terminal law satisfying the finite full-potential form criterion yields the same Friedrichs operator, frequency, Fock Hamiltonian, and generated continuum evolution. The boundary-law selection is therefore preserved through quantization as actual operator equality. A second construction develops physical Cauchy-time dynamics from the charged scalar field and momentum variables of the same Schwarzschild action. The physical Hamiltonian is first constructed on finite spatial banks, where its graph completion is self-adjoint and nonnegative. Finite occupation-window propagators satisfy the literal time-dependent Schrödinger equation; strong cutoff removal gives an infinite-occupation unitary evolution with cocycle structure, number-domain propagation, graph continuity, and the closed-Hamiltonian Schrödinger law throughout the owned positive interior interval. The infinite-spatial problem reveals an additional structural constraint. The original unit-amplitude Cauchy polarization has a genuine vacuum obstruction: scalar vacuum subtraction leaves particle–antiparticle pair coordinates whose norm prevents strong convergence of the spatial generator on the vacuum. The same full-potential equation determines a frequency-normalized physical polarization with unit Klein–Gordon flux and an initially diagonal charged Hamiltonian. Its normal-ordered infinite-spatial initial Hamiltonian is self-adjoint and nonnegative. The normalized construction is then extended to all spatial modes and arbitrary occupation. Square-summable pair kernels, weighted ultraviolet control, number-domain estimates, and free-energy propagation support removal of both spatial and occupation regulators. The resulting infinite-spatial physical Hamiltonian is proved self-adjoint, and its unitary propagator preserves the free-energy domain and satisfies the actual full physical Schrödinger equation. Interval coherence removes the auxiliary compact lower bound, producing a single strongly continuous physical evolution on 0