  A fundamentally different approach to path integral quantum mechanics in
curved space-time is presented, as compared to the standard approaches
currently available in the literature. Within the context of scalar particle
propagation in a locally curved background, such as described by Fermi or
Riemann normal co-ordinates, this approach requires use of a constructed
operator to rotate the initial, intermediate, and final position ket vectors
onto their respective local tangent spaces, defined at each local time step
along some arbitrary classical reference worldline. Local time translation is
described using a quantum mechanical representation of Lie transport, that
while strictly non-unitary in operator form, nevertheless correctly recovers
the free-particle Lagrangian in curved space-time, along with new
contributions. This propagator yields the prediction that all probability
violating terms due to curvature contribute to a quantum violation of the weak
equivalence principle, while the remaining terms that conserve probability also
correspondingly satisfy the weak equivalence principle, at least to
leading-order in the particle's Compton wavelength. Furthermore, this
propagator possesses an overall curvature-dependent and gauge-invariant phase
factor that can be interpreted as the gravitational Aharonov-Bohm effect and
Berry's phase.
