  We derive an exact quantum propagator for nonadiabatic dynamics in
multi-state systems using the mapping variable representation, where
classical-like Cartesian variables are used to represent both continuous
nuclear degrees of freedom and discrete electronic states. The resulting
expression is a Moyal series that, when suitably approximated, can allow for
the use of classical dynamics to efficiently model large systems. We
demonstrate that different truncations of the exact propagator lead to existing
approximate semiclassical and mixed quantum-classical methods and we derive an
associated error term for each method. Furthermore, by combining the
imaginary-time path-integral representation of the Boltzmann operator with the
exact propagator, we obtain an analytic expression for thermal quantum
real-time correlation functions. These results provide a rigorous theoretical
foundation for the development of accurate and efficient classical-like
dynamics to compute observables such as electron transfer reaction rates in
complex quantized systems.
