Fundamental dynamical equations for spinor wave functions. I. Levy-Leblond and Schrodinger equations

TL;DR

This paper investigates fundamental Galilean invariant equations for spinor wave functions, showing no such equations exist for two-component spinors and deriving the unique four-component Levy-Leblond equation, relating it to the Schrödinger and Pauli equations.

Contribution

It proves the non-existence of Galilean invariant two-component spinor equations and derives the Levy-Leblond equation for four-component spinors, establishing its uniqueness.

Findings

No Galilean invariant equations for two-component spinors.

Derivation of the Levy-Leblond equation for four-component spinors.

Relation of these equations to Schrödinger and Pauli equations.

Abstract

A search for fundamental (Galilean invariant) dynamical equations for two and four-component spinor wave functions is conducted in Galilean space-time. A dynamical equation is considered as fundamental if it is invariant under the symmetry operators of the group of the Galilei metric and if its state functions transform like the irreducible representations of the group of the metric. It is shown that there are no Galilean invariant equations for two-component spinor wave functions. A method to derive the L\'evy-Leblond equation for a four-component spinor wave function is presented. It is formally proved that the L\'evy-Leblond and Schr\"odinger equations are the only Galilean invariant four-component spinor equations that can be obtained with the Schr\"odinger phase factor. Physical implications of the obtained results and their relationships to the Pauli-Schr\"odinger equation are discussed.