Addressing Infinities in the L\'evy-Leblond Hamiltonian

TL;DR

This paper investigates the non-relativistic limit of the Dirac equation, focusing on the Levy-Leblond equation, and proposes methods to handle infinities associated with negative energy states, linking it to Lorentz-violating terms.

Contribution

It introduces a novel approach to isolate infinite energy states in the Levy-Leblond Hamiltonian and connects it to Lorentz-violating modifications of the Dirac equation.

Findings

Negative energy states are buried under infinities in the Levy-Leblond Hamiltonian.

A method to isolate and interpret infinite energy states is proposed.

The Levy-Leblond equation is linked to Lorentz-violating terms in the Dirac equation.

Abstract

We attempt to shed light on the following question: What happens to the negative energy states when we take the non-relativistic limit of the Dirac equation? The Levy-Leblond equation is the non-relativistic limit of the Dirac equation and describes fermions in the non-relativistic limit. The Levy-Leblond equation includes singular matrices and an attempt to write the Hamiltonian appears to show that the negative energy states are "buried" under an infinity. We attempt to isolate the infinite energy states and also present an equivalent way of viewing the Schrodinger dispersion relation. We propose that the Levy-Leblond equation can also be seen as resulting from the contribution of enhanced Lorentz violating terms to the Dirac equation.