  While it is well-known that quantum mechanics can be reformulated in terms of
a path integral representation, it will be shown that such a formulation is
also possible in the case of classical mechanics. From Koopman-von Neumann
theory, which recasts classical mechanics in terms of a Hilbert space wherein
the Liouville operator acts as the generator of motion, we derive a path
integral representation of the classical propagator and suggest an efficient
numerical implementation using fast fourier transform techniques. We then
include a first quantum correction to derive a revealing expression for the
semi-classical path integral, which augments the classical picture of a single
trajectory through phase space with additional wave-like spreading.
