Non-Relativistic Anti-Snyder Model and Some Applications

TL;DR

This paper explores the (2+1)-dimensional Dirac equation within a non-relativistic anti-Snyder model, revealing finite bound states, modified degeneracies, and applications to graphene and neutrino oscillations.

Contribution

It derives exact eigen solutions in momentum space using Romanovski polynomials and analyzes the implications of the anti-Snyder model on quantum systems and particle physics.

Findings

Finite maximum number of bound states due to polynomial orthogonality

Modified degeneracy of Landau levels in anti-Snyder model

Maximum bound of deformed parameter in graphene-like systems

Abstract

We examine the (2+1)-dimensional Dirac equation in a homogeneous magnetic field under the non-relativistic anti-Snyder model which is relevant to deformed special relativity (DSR) since it exhibits an intrinsic upper bound of the momentum of free particles. After setting up the formalism, exact eigen solutions are derived in momentum space representation and they are expressed in terms of finite orthogonal Romanovski polynomials. There is a finite maximum number of allowable bound states due to the orthogonality of the polynomials and the maximum energy is truncated at the maximum n. Similar to the minimal length case, the degeneracy of the Dirac-Landau levels in anti- Snyder model are modified and there are states that do not exist in the ordinary quantum mechanics limit. By taking zero mass limit, we explore the motion of effective zero mass charged Fermions in Graphene like material and obtained a maximum bound of deformed parameter. Furthermore, we consider the modified energy dispersion relations and its application in describing the behavior of neutrinos oscillation under modified commutation relations.