Single-Particle Quantum Mechanics of the Free Klein-Gordon Equation with Lorentz Violation

In spite of its problems with interactions, the first-quantized Klein-Gordon equation is a satisfactory theory of free spinless particles. Moreover, the usual theory may be extended to describe Lorentz-violating behavior, of the same types that exist can in second-quantized scalar field theories. However, because the construction of the theory requires a restriction to positive-energy modes, the Hilbert space inner product and the position operator depend explicitly on the form of the Lorentz violation.