Operator algebras associated with the Klein-Gordon position representation in relativistic quantum mechanics

Abstract

We initiate a mathematically rigorous study of Klein-Gordon position operators in single-particle relativistic quantum mechanics. Although not self-adjoint, these operators have real spectrum and enjoy a limited form of spectral decomposition. The associated C*-algebras are identifiable as crossed products. We also introduce a variety of non self-adjoint operator algebras associated with the Klein-Gordon position representation; these algebras are commutative and continuously (but not homeomorphically) embeddable in corresponding function algebras. Several open problems are indicated.