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

L. Bányai|arXiv (Cornell University)|May 7, 2019

Quantum and electron transport phenomena4 citations

TL;DR

This paper derives a nonrelativistic quantum mechanical Hamiltonian including $1/c^2$-order transverse current-current interactions by extending the standard Coulomb-only Hamiltonian using the Landau-Lifshitz formalism in the Coulomb gauge. It demonstrates that the quantized version of this Hamiltonian is equivalent to the $1/c^2$ approximation of non-relativistic QED in the photon-free subspace, and crucially enables a physical distinction between external and internal magnetic fields—key for understanding the Meissner effect in superconductors.

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.

Motivation & Objective

• To derive a consistent $1/c^2$-corrected Hamiltonian for nonrelativistic quantum systems that includes transverse current-current interactions, avoiding the mathematical inconsistencies of the Darwin Hamiltonian.
• To resolve the ambiguity in gauge choice by using the Coulomb gauge, which preserves only the two physical transverse degrees of freedom of the electromagnetic field.
• To establish a direct equivalence between the derived classical and quantized Hamiltonians and the $1/c^2$ approximation of non-relativistic QED in the subspace without photons.
• To enable a physical distinction between external and internal magnetic fields, which is essential for a microscopic understanding of the Meissner effect in superconductors.

Proposed method

• Adopt the Landau-Lifshitz classical derivation of the $1/c^2$ Hamiltonian but apply it in the Coulomb gauge instead of the gauge used by Landau-Lifshitz, ensuring only physical degrees of freedom are retained.
• Use the Coulomb gauge condition $\nabla \vec{A}^{\text{ext}} = 0$ to express the external vector potential in terms of transverse current densities, avoiding longitudinal components.
• Derive the classical Hamiltonian up to $1/c^2$ order, resulting in a transverse current-current interaction term analogous to the Coulomb density-density interaction.
• Quantize the derived classical Hamiltonian and show that its second-order S-matrix elements match those of non-relativistic QED at $1/c^2$ order, with photon propagators truncated to $1/q^2$ form.
• Use Feynman diagram analysis to confirm that the current-current interaction arises from second-order electron-electron scattering via virtual photons, with the $-1/2$ factor in the Hamiltonian ensuring proper normalization.
• Demonstrate that the resulting Hamiltonian correctly describes the magnetic field response in systems where external and internal fields must be distinguished, such as in superconductors.

Experimental results

Research questions

• RQ1 Why is the Darwin Hamiltonian, commonly used for $1/c^2$ corrections, physically inconsistent in its derivation, and how can a more rigorous alternative be constructed?
• RQ2 How can the $1/c^2$ electromagnetic interaction between charged particles be consistently derived in the Coulomb gauge, preserving only the two transverse degrees of freedom of the electromagnetic field?
• RQ3 To what extent does the quantized version of the derived Hamiltonian reproduce the $1/c^2$ limit of non-relativistic QED in the absence of photons?
• RQ4 Can the extended Hamiltonian distinguish between external and internal magnetic fields, and why is this distinction critical for theories of the Meissner effect?
• RQ5 What is the role of the transverse current-current interaction in enabling a microscopic description of ideal diamagnetism in superconductors?

Key findings

• The derived classical Hamiltonian includes a transverse current-current interaction term of order $1/c^2$, which is mathematically consistent and gauge-invariant in the Coulomb gauge.
• The quantized version of the Hamiltonian is equivalent to the $1/c^2$ approximation of non-relativistic QED when restricted to the subspace of states without photons.
• The current-current interaction arises from second-order electron-electron scattering via virtual photons, with the photon propagator approximated as $1/q^2$ in momentum space.
• The $-1/2$ factor in front of the current-current interaction term ensures consistency with the S-matrix of QED, correcting for overcounting in the perturbative expansion.
• The Hamiltonian allows a clear physical separation between external and internal magnetic fields, a feature absent in the standard Coulomb-only model.
• This extended framework provides a more fundamental basis for understanding the Meissner effect, as it enables the internal magnetic field to dynamically cancel the external field without relying on self-consistent field approximations.