Conserved Current Densities, Localization Probabilities, and a New Global Gauge Symmetry of Klein-Gordon Fields

Abstract

For free Klein-Gordon fields, we construct a one-parameter family of conserved current densities $J_a^μ$, with $a\in(-1,1)$, and use the latter to yield a manifestly covariant expression for the most general positive-definite and Lorentz-invariant inner product on the space of solutions of the Klein-Gordon equation. Employing a recently developed method of constructing the Hilbert space and observables for Klein-Gordon fields, we then obtain the probability current density ${\cal J}_a^μ$ for the localization of a Klein-Gordon field in space. We show that in the nonrelativistic limit both $J_a^μ$ and ${\cal J}_a^μ$ tend to the probability current density for the localization of a nonrelativistic free particle in space, but that unlike $J_a^μ$ the current density ${\cal J}_a^μ$ is neither covariant nor conserved. Because the total probability may be obtained by integrating either of these two current densities over the whole space, the conservation of the total probability may be viewed as a consequence of the local conservation of $J_a^μ$. The latter is a manifestation of a previously unnoticed global gauge symmetry of the Klein-Gordon fields. The corresponding gauge group is U(1) if the parameter $a$ is rational. It is the multiplicative group of positive real numbers if $a$ is irrational. We also discuss an extension of our results to Klein-Gordon fields minimally coupled to an electromagnetic field.