= 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>.
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