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

TL;DR

This paper rigorously constructs a physical Hilbert space for the Dirac-Maxwell model in the Lorenz gauge, eliminating unphysical photon components and ensuring a positive semi-definite metric.

Contribution

It provides a mathematically rigorous method to define a physical subspace with positive metric in the Lorenz gauge, ensuring the Hamiltonian is self-adjoint on this space.

Findings

Established a positive semi-definite physical subspace.

Proved the induced Hamiltonian is essentially self-adjoint.

Ensured Lorentz covariance in the quantization process.

Abstract

We study the Dirac-Maxwell model quantized in the Lorenz gauge. In this gauge, the space of quantum mechanical state vectors inevitably be an indefinite metric vector space 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-definit metric, and naturally induces a physical Hilbert space with a positive definite metric. The original Dirac-Maxwell Hamiltonian naturally defines an induced Hamiltonian on the physical Hilbert space which is essentially self-adjoint.