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This is the fourth in a series of papers on developing a formulation ofquantum mechanics in non-inertial reference frames. This formulation isgrounded in a class of unitary cocycle representations of what we have calledthe Galilean line group, the generalization of the Galilei group to includetransformations amongst non-inertial reference frames. These representationsshow that in quantum mechanics, just as the case in classical mechanics, thetransformations to accelerating reference frames give rise to fictitiousforces. In previous work, we have shown that there exist representations of theGalilean line group that uphold the non-relativistic equivalence principle aswell as representations that violate the equivalence principle. In theseprevious studies, the focus was on linear accelerations. In this paper, weundertake an extension of the formulation to include rotational accelarations.We show that the incorporation of rotational accelerations requires a class of\emph{loop prolongations} of the Galilean line group and their unitary cocyclerepresentations. We recover the centrifugal and Coriolis force effects fromthese loop representations. Loops are more general than groups in that theirmultiplication law need not be associative. Hence, our broad theoretical claimis that a Galilean quantum theory that holds in arbitrary non-inertialreference frames requires going beyond groups and group representations, thewell-stablished framework for implementing symmetry transformations in quantummechanics.

Quantum Mechanics in non-inertial reference frames: time-dependent rotations and loop prolongations