The response field and the saddle points of quantum mechanical path integrals

In quantum statistical mechanics, Moyal's equation governs the time evolution of Wigner functions and more general Weyl symbols that represent density matrix arbitrary mixed states. A formal solution to is given by Marinov's path integral. this paper we demonstrate integral can be regarded as natural link between several conceptual, geometric, dynamical issues in mechanics. unifying perspective achieved highlighting pivotal role which response field, one integration variables integral, plays for pure states even. The discussion focuses on how integral's semiclassical approximation relates its strictly classical limit; unlike Feynman type integrals, latter well defined Marinov case. topics covered include a random force representation based upon concept "Airy averaging", related positivity-violating describing tunneling processes, field maintaining coherence enabling interference phenomena. double slit experiment electrons Bohm-Aharonov effect are analyzed illustrative examples. Furthermore, surprising relationship instantons over an analytically continued ("Wick rotated") complex Feynman-type integrals found. play prominent recent work towards Picard-Lefschetz theory applicable oscillatory resurgence program.