On the Nonrelativistic Quantum-Mechanical Hamiltonian with $1/c^2$ terms. Transverse current-current interaction

TL;DR

This paper extends the solid-state quantum Hamiltonian by including $1/c^2$ transverse current-current interactions, clarifying its relation to non-relativistic QED and highlighting potential implications for magnetic field theories like the Meissner effect.

Contribution

It introduces a Hamiltonian with $1/c^2$ terms in Coulomb gauge, differing from the Darwin Hamiltonian, and clarifies its equivalence to non-relativistic QED without photons.

Findings

Hamiltonian includes transverse current-current interactions.

Shows equivalence to non-relativistic QED without photons.

Highlights potential relevance for magnetic field theories.

Abstract

We extend the standard solid-state quantum mechanical Hamiltonian containing only Coulomb interactions between the charged particles by inclusion of $1/c^2$ terms representing (transverse) current-current interaction. For its derivation we use the classical formulation of Landau-Lifshitz, however consequently in the Coulomb gauge. Our Hamiltonian does not coincide with the Darwin Hamiltonian and we emphasize the mathematical inconsistency in its derivation. We show, that the quantized version of our Hamiltonian is equivalent to the non-relativistic QED considering only states without photons and retaining only terms of order $1/c^2$. The importance of this extended Hamiltonian lies in the possibility to distinguish external from internal magnetic fields. This aspect may be relevant for theories of the Meissner effect.