Generalized gauge transformations in phase space picture of quantum mechanics: Kirkwood representation

Here, the ordinary gauge transformations in configuration space are generalized to the phase space. The corresponding gauge functions are separable in terms of those of the configuration and the momentum spaces. Furthermore, we consider the gauge transformations in phase space for which their gauge functions are not, in general, separable. It is shown that, the assumption of the general global gauge symmetry in phase space immediately yields the quantization of the electric charge. We obtain this result without assuming the concept of the magnetic monopole that Dirac did in his approach to the problem of electric charge quantization and which is not detected yet.