Eliminating unphysical photon components from Dirac–Maxwell Hamiltonian quantized in the Lorenz gauge

We study the Dirac–Maxwell model quantized in the Lorenz gauge. In this gauge, the space of quantum mechanical state vectors inevitably adopt an indefinite metric so that the canonical commutation relation (CCR) is realized in a Lorentz covariant manner. In order to obtain a physical subspace, in which no negative norm state exists, the method first proposed by Gupta and Bleuler is applied with mathematical rigor. It is proved that a suitably defined physical subspace has a positive semi-definite metric, and naturally induces a physical Hilbert space with a positive definite metric. Then, the original Dirac–Maxwell Hamiltonian defines an induced Hamiltonian on the physical Hilbert space which is essentially self-adjoint.