  Stochastic hybrid systems involve a coupling between a discrete Markov chain
and a continuous stochastic process. If the latter evolves deterministically
between jumps in the discrete state, then the system reduces to a piecewise
deterministic Markov process (PDMP). Well known examples include stochastic
gene expression, voltage fluctuations in neurons, and motor-driven
intracellular transport. In this paper we use coherent spin states to construct
a new path integral representation of the probability density functional for
stochastic hybrid systems, which holds outside the weak noise regime. We use
the path integral to derive a system of Langevin equations in the
semi-classical limit, which extends previous diffusion approximations based on
a quasi-steady-state reduction. We then show how in the weak noise limit the
path integral is equivalent to an alternative representation that was
previously derived using Doi-Peliti operators. The action functional of the
latter is related to a large deviation principle for stochastic hybrid systems.
