  Weak coherent states share many properties of the usual coherent states, but
do not admit a resolution of unity expressed in terms of a local integral. They
arise e.g. in the case that a group acts on an inadmissible fiducial vector.
Motivated by the recent Affine Quantum Gravity Program, the present article
studies the path integral representation of the affine weak coherent state
matrix elements of the unitary time-evolution operator. Since weak coherent
states do not admit a resolution of unity, it is clear that the standard way of
constructing a path integral, by time slicing, is predestined to fail. Instead
a well-defined path integral with Wiener measure, based on a continuous-time
regularization, is used to approach this problem. The dynamics is rigorously
established for linear Hamiltonians, and the difficulties presented by more
general Hamiltonians are addressed.
