Quantum mechanics of Klein–Gordon fields I: Hilbert Space, localized states, and chiral symmetry

Abstract

We derive an explicit manifestly covariant expression for the most general positive-definite and Lorentz-invariant inner product on the space of solutions of the Klein–Gordon equation. This expression involves a one-parameter family of conserved current densities J a μ , with a ∈ (−1, 1), that are analogous to the chiral current density for spin half fields. The conservation of J a μ is related to a global gauge symmetry of the Klein–Gordon fields whose gauge group is U (1) for rational a and the multiplicative group of positive real numbers for irrational a. We show that the associated gauge symmetry is responsible for the conservation of the total probability of the localization of the field in space. This provides a simple resolution of the paradoxical situation resulting from the fact that the probability current density for free scalar fields is neither covariant nor conserved. Furthermore, we discuss the implications of our approach for free real scalar fields offering a direct proof of the uniqueness of the relativistically invariant positive-definite inner product on the space of real Klein–Gordon fields. We also explore an extension of our results to scalar fields minimally coupled to an electromagnetic field.