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 free spinless particles. Moreover, usual may be extended to describe Lorentz-violating behavior, same types that exist can in second-quantized scalar field theories. However, because construction requires restriction positive-energy modes, Hilbert space inner product and position operator depend explicitly on form Lorentz violation.