Some Aspects of Classical Gauge Theories and First Quantization Schemes.

Electromagnetism is a U(1) gauge theory. If regions of singularity of the field are removed from Minkowski space, a homotopically non-trivial space M is obtained. The integrality of the magnetic or electric charge is shown to be equivalent to the existence of a U(1) vector bundle of fields with connection and a U(1) invariant product over M. In an SO(3) gauge theory, it is shown that a well known singular gauge transformation is not at all singular if the correct coordination of the homotopically non-trivial space is used. A method is detailed whereby the standard quantization based on the Heisenberg group may be generalized to quantizations based on other Lie groups. The case of nilpotent Lie groups is considered for generalization.