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International Workshop on Hadron Physics

Relativistic Landau levels via Feynman-Gell-Mann formulation

V. B. Mendrot, Antonio S. de Castro

Published Aug 1, 2022• 0 citations
Publisher

Abstract

Relativistic energy levels for a spin-1/2 fermion in a stationary and homogeneous magnetic field, namely the Landau levels, are obtained via the Feynman-Gell-Mann formulation of the Dirac equation using the axially symmetric gauge, such that the radial functions obey a similar equation to the one of a singular harmonic oscillator in the nonrelativistic theory.

DOI
10.22323/1.408.0073

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Relativistic energy levels for a spin-1/2 fermion in a stationary and homogeneous magnetic field, namely the Landau levels, are obtained via the Feynman-Gell-Mann formulation of the Dirac equation using the axially symmetric gauge, such that the radial functions obey a similar equation to the one of a singular harmonic oscillator in the nonrelativistic theory.
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We show that a linear superposition of mesoscopic states in relativistic Landau levels can be built when an external magnetic field couples to a relativistic spin 1/2 charged particle. Under suitable initial conditions, the associated Dirac equation produces unitarily superpositions of coherent states involving the particle orbital quanta in a well-defined mesoscopic regime. We demonstrate that these mesoscopic superpositions have a purely relativistic origin and disappear in the nonrelativistic limit.
In this work we obtain the exact solution for relativistic Landau problem plus oscillator potential in a complex symmetric gauge field in a non-commutative complex space, using the algebraic techniques of creation and annihilation operators. It is shown that the relativistic Landau problem in a complex symmetric gauge field in a non-commutative complex space is similar behavior to the Pauli equation in the presence of a symmetric gauge field in a commutative ordinary space. We derive the exact non-commutative Landau and oscillator energy levels, while the non-relativistic limit of the energy spectrum is obtained. We show that the energy is not degenerate and is splitted into two levels, as in the Zeeman effect.
Abstract The (2+1)-dimensional Dirac equation for an electron in a constant magnetic field is considered. Exact solutions of this equation along with the corresponding energy spectrum are derived in a concise form. Among these solutions, we noticed that the lowest Landau level wavefunction corresponds to an anti-electron (an electron with a negative energy). The latter appears because special relativity allows solutions with negative energies. This means that a relativistic electron (matter) in a uniform magnetic field, at the lowest Landau level, is mutated to an anti-electron (anti-matter). This is a good finding for those that are interested in anti-matter applications.