External gauge field coupled quantum dynamics: gauge choices, Heisenberg algebra representations and gauge invariance in general, and the Landau problem in particular

Even though its classical equations of motion are then left invariant, when an action is redefined by an additive total derivative or divergence term (in time, in the case of a mechanical system) such a transformation induces nontrivial consequences for the system's canonical phase space formulation. This is even more true and then in more subtle ways for the canonically quantised dynamics, with in particular an induced transformation in the unitary configuration space representation of the Heisenberg algebra being used for the quantum system. When coupled to a background gauge field, such considerations become crucial for a proper understanding of the consequences for the system's quantum dynamics of gauge transformations of that classical external background gauge field, while under such transformations the system's degrees of freedom, abstract quantum states and quantum dynamics are certainly strictly invariant. After a detailed analysis of these different points in a general context, these are then illustrated specifically in the case of the quantum Landau problem with its classical external background magnetic vector potential for which the most general possible parametrised gauge choice is implemented herein. The latter discussion aims as well to clarify some perplexing statements in the literature regarding the status of gauge choices to be made for the magnetic vector potential for that quantum system. The role of the global space-time symmetries of the Landau problem and their gauge invariant Noether charges is then also emphasized.