Dirac fermions in a spinning conical Gödel-type spacetime
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2024
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| _version_ | 1866914953154265088 |
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| author | Oliveira, R. R. S. |
| author_facet | Oliveira, R. R. S. |
| contents | In this paper, we determine the relativistic and nonrelativistic energy levels for Dirac fermions in a spinning conical Gödel-type spacetime in $(2+1)$-dimensions, where we work with the curved Dirac equation in polar coordinates and we use the tetrads formalism. Solving a second-order differential equation for the two components of the Dirac spinor, we obtain a generalized Laguerre equation, and the relativistic energy levels of the fermion and antifermion, where such levels are quantized in terms of the radial and total magnetic quantum numbers $n$ and $m_j$, and explicitly depends on the spin parameter $s$ (describes the ``spin''), spinorial parameter $u$ (describes the two components of the spinor), curvature and rotation parameters $α$ and $β$ (describes the conical curvature and the angular momentum of the spinning cosmic string), and on the vorticity parameter $Ω$ (describes the Gödel-type spacetime). In particular, the quantization is a direct result of the existence of $Ω$ (i.e. $Ω$ acts as a kind of ``external field or potential''). We see that for $m_j>0$, the energy levels do not depend on $s$ and $u$; however, depend on $n$, $m_j$, $α$, and $β$. In this case, $α$ breaks the degeneracy of the energy levels and such levels can increase infinitely in the limit $\frac{4Ωβ}α\to 1$. Already for $m_j<0$, we see that the energy levels depends on $s$, $u$ and $n$; however, it no longer depends on $m_j$, $α$ and $β$. In this case, it is as if the fermion ``lives only in a flat Gödel-type spacetime''. Besides, we also study the low-energy or nonrelativistic limit of the system. In both cases (relativistic and nonrelativistic), we graphically analyze the behavior of energy levels as a function of $Ω$, $α$, and $β$ for three different values of $n$ (ground state and the first two excited states). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_11334 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dirac fermions in a spinning conical Gödel-type spacetime Oliveira, R. R. S. General Relativity and Quantum Cosmology High Energy Physics - Theory In this paper, we determine the relativistic and nonrelativistic energy levels for Dirac fermions in a spinning conical Gödel-type spacetime in $(2+1)$-dimensions, where we work with the curved Dirac equation in polar coordinates and we use the tetrads formalism. Solving a second-order differential equation for the two components of the Dirac spinor, we obtain a generalized Laguerre equation, and the relativistic energy levels of the fermion and antifermion, where such levels are quantized in terms of the radial and total magnetic quantum numbers $n$ and $m_j$, and explicitly depends on the spin parameter $s$ (describes the ``spin''), spinorial parameter $u$ (describes the two components of the spinor), curvature and rotation parameters $α$ and $β$ (describes the conical curvature and the angular momentum of the spinning cosmic string), and on the vorticity parameter $Ω$ (describes the Gödel-type spacetime). In particular, the quantization is a direct result of the existence of $Ω$ (i.e. $Ω$ acts as a kind of ``external field or potential''). We see that for $m_j>0$, the energy levels do not depend on $s$ and $u$; however, depend on $n$, $m_j$, $α$, and $β$. In this case, $α$ breaks the degeneracy of the energy levels and such levels can increase infinitely in the limit $\frac{4Ωβ}α\to 1$. Already for $m_j<0$, we see that the energy levels depends on $s$, $u$ and $n$; however, it no longer depends on $m_j$, $α$ and $β$. In this case, it is as if the fermion ``lives only in a flat Gödel-type spacetime''. Besides, we also study the low-energy or nonrelativistic limit of the system. In both cases (relativistic and nonrelativistic), we graphically analyze the behavior of energy levels as a function of $Ω$, $α$, and $β$ for three different values of $n$ (ground state and the first two excited states). |
| title | Dirac fermions in a spinning conical Gödel-type spacetime |
| topic | General Relativity and Quantum Cosmology High Energy Physics - Theory |
| url | https://arxiv.org/abs/2405.11334 |