Reduced Feynman propagators for Schrödingerized ill-posed dynamics

Abstract

Schrödingerization maps a non-unitary linear evolution to a unitary dynamics on an enlarged Hilbert space. For ill-posed equations, such as backward diffusion, this does not remove the instability of the recovered physical variable, but it makes the phase-space language of quantum propagation available. We show that the McLachlan projection of a Schrödingerized dynamics has a natural finite-dimensional Feynman interpretation. If V is a full-column-rank reduced basis matrix, its Gram matrix, and the projected Hamiltonian, then the McLachlan equation is the stationary equation of the reduced action After Gram orthonormalization this action generates an ordinary finite-dimensional unitary propagator, hence a quadratic oscillatory path integral. The projection defect measures the leakage of the lifted Hamiltonian flow outside the reduced path sector. For backward diffusion this gives a physical interpretation of McLachlan reduction as a controlled restriction of the Schrödingerized Feynman propagation, analogous to a structured spectral regularization but not to a Feynman-Kac formula.