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

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, ${\mathrm{\partial }}_{\mu }{A}^{\mu }=0$, 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.