Abstract: The Bargmann-Wigner program for grounding quantum mechanics in the irreducible representations of space-time symmetry groups is extended to accelerating quantum systems.  For non-relativistic quantum theory this results in the notion of synthetic magnetic fields for rotationally accelerated systems and violations of the quantum equivalence  principle for linearly accelerated systems.  For relativistically accelerated systems the main consequence is a formulation for coupling quantum systems to an external gravitational field.