  We canonically quantize a spin-less non-relativistic point particle in a
rigidly rotating cylinder symmetric reference system. The resulting quantum
mechanics is investigated in the case of both a two-dimensional cylindrical
shell and in the case of the full rotating cylinder; in both cases energy
eigenstates exist which exhibit rotation induced negative energies. Based on a
reanalysis of the classical Sagnac effect a novel way to compute the quantum
Sagnac phase shift is deduced. It is pointed out that certain states on both
the cylindrical shell and in the bulk give rise to a novel anomalous quantum
Sagnac effect.
