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

TL;DR

This paper explores a first-quantized approach to the free Klein-Gordon equation incorporating Lorentz violation, highlighting how such violations affect the theory's structure and the definition of key quantum operators.

Contribution

It extends the Klein-Gordon framework to include Lorentz-violating effects, analyzing their impact on the Hilbert space and position operator in a first-quantized setting.

Findings

Lorentz violation influences the inner product structure.

The position operator depends explicitly on Lorentz-violating terms.

The theory remains consistent for free particles despite Lorentz violation.

Abstract

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.