Isomorphism between the Bialynicki-Birula and the Landau-Peierls Fock space quantization of the electromagnetic field in position representation

TL;DR

This paper demonstrates the complete equivalence between two different position-space quantization methods of the electromagnetic field, establishing a unitary isomorphism between their Fock spaces and confirming identical physical predictions.

Contribution

It proves the isomorphism between the Landau-Peierls and Bialynicki-Birula quantizations, clarifying their equivalence and extending the Bialynicki-Birula approach to avoid negative Hamiltonian eigenvalues.

Findings

The two quantization approaches are unitarily equivalent.

Physically measurable quantities are identical in both frameworks.

The Bialynicki-Birula method can be reformulated to exclude negative eigenvalues.

Abstract

We first present a summary of the quantization of the electromagnetic field in position space representation, using two main approaches: the Landau-Peierls approach in the Coulomb gauge and the Bialynicki-Birula approach, based on the Riemann-Silberstein vector. We describe both in a framework that starts with a classical Hamiltonian structure and builds the quantum model in a bosonic Fock space by a precisely defined principle of correspondence. We show that the two approches are completly equivalent. This is formulated by showing that there is a unitary map between the Fock spaces that makes them isomorphic. Since all the physically measurable quantities can be expressed in terms of scalar products, this implies that the two quantizations lead to exactly the same physical properties. We show furthemore that the isomorphism is preserved in the time evolutions. To show the equivalence, we use the concepts of helicity and frequency operators. The combination of these two operators provides a formulation that allows one to make the link between these two methods of quantization in a precise way. We also show that the construction in the Bialynicki-Birula quantization that avoids the presence of negative eigenvalues in the Hamiltonian, in analogy with the one for the Dirac equation for electrons and positrons, can be performed through an alternative choice of the canonical variables for Maxwell's equations.