Is gauge symmetry vacuous or physical ? : Lessons from the Landau problem as a solvable quantum gauge theoretical system

Abstract: The gauge symmetry is one of the most important concepts in modern physics, but there are two conflicting views on its meaning or interpretation. The standard view is that local gauge symmetry is the basis of the pursue of fundamental particles and forces in nature. Another view is that the gauge symmetry is not a symmetry of nature but just a redundancy in description. Naturally, both statements are nothing wrong, but one might feel that there is a slight conceptual conflict between the two points of view. Due to the subtlety of the subject, however, little literature exists that discusses the root of such an anxiety. In the present paper, by making full use of the analytically solvable nature of the quantum Landau problem, we argue that the familiar gauge principle plays a critical role in unraveling a subtle mismatch between the two viewpoints above. We reveal that there exist two types of quantities in gauge theories, which should clearly be discriminated. The first are quantities, which look seemingly gauge-invariant but actually not, whereas the second are genuinely gauge-invariant quantities, which correspond to direct experimental observables.