  The original intent of the Koopman-von Neumann formalism was to put classical
and quantum mechanics on the same footing by introducing an operator formalism
into classical mechanics. Here we pursue their path the opposite way and
examine what transfer operators can say about quantum mechanical evolution. To
that end, we introduce a physically motivated scalar wavefunction formalism for
a velocity field on a 4-dimensional pseudo-Riemannian manifold, and obtain an
evolution equation for the associated wavefunction, a generator for an
associated weighted transfer operator. The generator of the scalar evolution is
of first order in space and time. The probability interpretation of the
formalism leads to recovery of the Schrodinger equation in the
non-relativistic limit. In the special relativity limit, we show that the
scalar wavefunction of Dirac spinors satisfies the new equation. A connection
with string theoretic considerations for mass is provided.
