  This is the fourth in a series of papers on developing a formulation of
quantum mechanics in non-inertial reference frames. This formulation is
grounded in a class of unitary cocycle representations of what we have called
the Galilean line group, the generalization of the Galilei group to include
transformations amongst non-inertial reference frames. These representations
show that in quantum mechanics, just as the case in classical mechanics, the
transformations to accelerating reference frames give rise to fictitious
forces. In previous work, we have shown that there exist representations of the
Galilean line group that uphold the non-relativistic equivalence principle as
well as representations that violate the equivalence principle. In these
previous studies, the focus was on linear accelerations. In this paper, we
undertake 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 cocycle
representations. We recover the centrifugal and Coriolis force effects from
these loop representations. Loops are more general than groups in that their
multiplication law need not be associative. Hence, our broad theoretical claim
is that a Galilean quantum theory that holds in arbitrary non-inertial
reference frames requires going beyond groups and group representations, the
well-stablished framework for implementing symmetry transformations in quantum
mechanics.
