Nonrelativistic anti-Snyder model and some applications. (English) Zbl 1361.81048
Summary: In this paper, we examine the \((2+1)\)-dimensional Dirac equation in a homogeneous magnetic field under the nonrelativistic anti-Snyder model which is relevant to doubly/deformed special relativity (DSR) since it exhibits an intrinsic upper bound of the momentum of free particles. After setting up the formalism, exact eigensolutions 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 nmax due to the orthogonality of the polynomials and the maximum energy is truncated at nmax. 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 \(\beta\to 0\). By taking \(m\to 0\), we explore the motion of effective massless charged fermions in graphene-like material and obtained a maximum bound of deformed parameter \(\beta_{\text{max}}\). Furthermore, we consider the modified energy dispersion relations and its application in describing the behavior of neutrinos oscillation under modified commutation relations.
MSC:
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 81R20 | Covariant wave equations in quantum theory, relativistic quantum mechanics |
| 81V15 | Weak interaction in quantum theory |
| 81U30 | Dispersion theory, dispersion relations arising in quantum theory |
| 83C45 | Quantization of the gravitational field |
| 82D80 | Statistical mechanics of nanostructures and nanoparticles |
Keywords:
modified commutation relations; Snyder and anti-Snyder algebra; Dirac-Landau problem; modified energy dispersion relations; modified neutrinos oscillation; quantum gravity phenomenologyReferences:
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