The Landau Problem and non-Classicality

TL;DR

This paper develops a phase space representation of quantum field theories using an extended Galilei group, deriving equations for spin-zero and spin-1/2 particles, and investigates the non-classicality of the Landau problem.

Contribution

It introduces a novel phase space framework for field theories based on the extended Galilei group, including covariant equations and analysis of non-classicality.

Findings

Recovered Landau Levels for an electron in an external field

Constructed covariant Pauli-Schr"odinger and Klein-Gordon-like equations in phase space

Analyzed the negativity parameter as a measure of non-classicality

Abstract

Exploring the concept of the extended Galilei group G. Representations for field theories in a symplectic manifold have been derived in association with the method of the Wigner function. The representation is written in the light-cone of a de Sitter space-time in five dimensions. A Hilbert space is constructed, endowed with a symplectic structure, which is used as a representation space for the Lie algebra of G. This representation gives rise to the spin-zero Schr\"odinger (Klein-Gordon-like) equation for the wave functions in phase space, such that the dependent variables have the content of position and linear momentum. This is a particular example of a conformal theory, such that the wave functions are associated with the Wigner function through the Moyal product. We construct the Pauli-Schr\"odinger (Dirac-like) equation in phase space in its explicitly covariant form. In addition, we analyze the gauge symmetry for spin 1/2 particles in phase space and show how to implement the minimal coupling in this case. We applied to the problem of an electron in an external field, and we recovered the non-relativistic Landau Levels. Finally, we study the parameter of negativity associated with the non-classicality of the system.