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

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.

Non-local property of single-photon states: Illustration with spontaneously emitted photon from a Hydrogen atom

“…The ability to produce pulse-shaped single-photons [1] has brought back the importance of their description in position space [2,3]. Our recent works show that instead of defining photons through the decomposition of the field into plane waves, one can write the theory with bosonic creation-annihilation operators directly on pulses of arbitrary shape…”

“…where 𝜓 ⃗ is a classical complex representation of the pulse in position space [2,3]. The dynamics associated to this single-photon pulse is then given directly by 𝐵 ̂𝜓 ⃗⃗⃗ (𝑡) † |∅ >…”