The Art of Choosing A: From a Vector Problem to Gauge Freedom

Gauge freedom is often introduced by observing that A and A + nabla psi producethe same magnetic field, but this leaves open why the potential is not unique and why one gauge choice may be preferable to another. We begin with the elementary vector equation axu = c, which separates the questions of solvability and non-uniqueness. For a uniform magnetic field we then use a linear vector potential, A(r) = M r, and show that the field fixes the antisymmetric part of M, while the symmetric part generates a gradient and hence the gauge freedom. The symmetric and Landau representatives make different symmetries useful in the corresponding quantum problem, and the same construction provides a simple illustration of residual freedom within Coulomb gauge. The common origin of these ambiguities is the nontrivial kernel of the relevant operator.