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