Operator-path-integral equivalence

With a time-slice regulator, inserting field-coordinate resolutions of identity in evolution kernels produces a bosonic phase-space path integral; integrating the canonical momentum gives the configuration-space action. For fermions, resolutions in fermionic coherent states instead produce a Berezin integral and the first-order kinetic term. Vacuum endpoint projection supplies the Feynman i-epsilon prescription. Both constructions reproduce the same free oscillator correlators and their Wick theorem expansions, and thus agree order by order in a regulated perturbation series.