Some Remarks on the Gupta–Bleuler and Dirac–Fock Quantization Schemes

The classical Gupta–Bleuler quantization scheme in quantum electromagnetic field theory, based on the so-called “weakened” Lorenz subsidiary condition and indefinite metric on the reduced Fock space, is revisited within the context of the canonical Dirac–Fock quantization approach. The latter, as applied to the Fermi–Fock–Podolsky Lagrangian, allows for both satisfaction of the strong Lorenz subsidiary condition on the Fock space and its averaged invariance in time to be shown, as well as the positive energy density definiteness of quantum photon states. The time non-invariance of the averaged "weakened" Lorenz subsidiary condition within the Gupta and Bleuler approach is also demonstrated, resulting in energy density positive non-definiteness, which is later replaced by the so-called BRST cohomology condition, imposed on a suitable construction of the Fock space. We also sketch the corresponding variational approach of N.N. Bogolubov (Sr.) to constructing a deformed strong Lorenz-type operator condition, both within the perturbative Scwinger’s S-matrix approach and the Fock multi-temporal paradigm. A short supplement is devoted to analysis of the classical Lorenz-type subsidiary condition and the related temporal invariance of the electric potential energy, providing a derivation of the complete classical system of Maxwell’s equations.