Observation about the Classical Electromagnetic Gauge Transformation and Its Quantum Correspondence

Abstract

Using the Landau and symmetric gauges for the vector potential of a constant magnetic field and the quantum problem of a charged particle moving on a flat surface, we show the classical electromagnetic gauge transformation does not correspond to a one-dimensional unitary group transformation U(1) of the wave function for the quantum case. In addition, with the re-examination of the relation between the magnetic field B and its vector potential A , we found that, in order to have a consistent formulation of the dynamics of the charged particle with both expressions, we must have that B = ∇ × A if and only if B ≠ 0 .