Gupta--Bleuler's quantization of the anisotropic parity-even and CPT-even electrodynamics of standard model extension

TL;DR

This paper develops a Gupta-Bleuler quantization method for a specific Lorentz-violating photon sector in the standard model extension, addressing gauge condition issues and confirming microcausality preservation.

Contribution

It introduces a modified Lorentz gauge condition enabling successful Gupta-Bleuler quantization of anisotropic CPT-even Lorentz-violating electrodynamics.

Findings

Positive-energy Hamiltonian derived with modified gauge condition

Microcausality preserved for small Lorentz-violating parameters

Quantization method applicable to anisotropic Lorentz-violating photon sector

Abstract

We have established the Gupta-Bleuler quantization of the photon belonging to the anisotropic parity-even sector of the CPT-even and Lorentz-violating nonbirefringent electrodynamics of the standard model extension. We first present a rule for the Maxwell electrodynamics to be successfully quantized via Gupta-Bleuler technique in the Lorentz gauge. Recognizing the failure of the Gupta-Bleuler method in the Lorentz gauge, $\partial _{\mu }A^{\mu }=0$, for this massless LV theory, we argue that Gupta-Bleuler can be satisfactorily implemented by choosing a modified Lorentz condition, $\partial _{\mu }A^{\mu }+\kappa ^{\mu \nu }\partial _{\mu }A_{\nu }=0$, where $\kappa ^{\mu \nu }$ represents the Lorentz-violation in photon sector. By using a plane-wave expansion for the gauge field, whose polarization vectors are determined by solving an eigenvalue problem, and a weak Gupta-Bleuler condition, we obtain a positive-energy Hamiltonian in terms of annihilation and creation operators. The field commutation relation is written in terms of modified Pauli-Jordan functions, revealing the preservation of microcausality for sufficiently small LV parameters.