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Gupta-Bleuler's quantization of a parity-odd CPT-even electrodynamics of the standard model extension

{Following a successfully quantization scheme previously }{{% developed {\ in Ref. \cite{GUPTAEVEN} for a}} parity-even {gauge sector} {of the SME, we}} have established the Gupta-Bleuler {% quantization {of a} {\ parity-odd} and CPT-even }electrodynamics of {the }standard model extension (SME) {without recoursing to a small photon mass regulator}. {Keeping the photons massless,} {% {we have adopted the gauge fixing condition:}} $G(A_{μ})=(\partial _{0}+κ^{0j}\partial _{j}) (A_{0}+κ^{0k}A_{k})+\partial _{i}A^{i}$% . The{\ four} polarization vectors of the gauge field are {% exactly} determined by solving an eigenvalue problem,{\ exhibiting birefringent second order contributions in the Lorentz-violating parameters}% . They allow to express the Hamiltonian in terms of annihilation and creation operators whose positivity is guaranteed by imposing a weak Gupta-Bleuler constraint, defining the physical states. Consequently, we compute the field commutation relation which has been expressed in terms of Pauli-Jordan functions modified by Lorentz violation whose light-cone structures have allowed to analyze the microcausality issue.