Connecting in the Dirac Equation the Clifford Algebra of Lorentz Invariance with the Lie Algebra of <i>SU</i>(<i>N</i>) Gauge Symmetry
In this paper, we study possible mathematical connections of the Clifford algebra with the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mi>u</mi><mo>(</mo>...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-03-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/13/3/475 |
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| author | Eckart Marsch Yasuhito Narita |
| author_facet | Eckart Marsch Yasuhito Narita |
| author_sort | Eckart Marsch |
| collection | DOAJ |
| description | In this paper, we study possible mathematical connections of the Clifford algebra with the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mi>u</mi><mo>(</mo><mi>N</mi><mo>)</mo></mrow></semantics></math></inline-formula>-Lie algebra, or in more physical terms the links between space-time symmetry (Lorentz invariance) and internal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>U</mi><mo>(</mo><mi>N</mi><mo>)</mo></mrow></semantics></math></inline-formula> gauge-symmetry for a massive spin one-half fermion described by the Dirac equation. The related matrix algebra is worked out in particular for the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>U</mi><mo>(</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula> symmetry and outlined as well for the color gauge group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>U</mi><mo>(</mo><mn>3</mn><mo>)</mo></mrow></semantics></math></inline-formula>. Possible perspectives of this approach to unification of symmetries are briefly discussed. The calculations make extensive use of tensor multiplication of the matrices involved, whereby our focus is on revisiting the Coleman–Mandula theorem. This permits us to construct unified symmetries between Lorentz invariance and gauge symmetry in a direct product sense. |
| first_indexed | 2024-03-10T13:15:07Z |
| format | Article |
| id | doaj.art-0c1e567d96b248a184899f33229b1c1b |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T13:15:07Z |
| publishDate | 2021-03-01 |
| publisher | MDPI AG |
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| series | Symmetry |
| spelling | doaj.art-0c1e567d96b248a184899f33229b1c1b2023-11-21T10:27:36ZengMDPI AGSymmetry2073-89942021-03-0113347510.3390/sym13030475Connecting in the Dirac Equation the Clifford Algebra of Lorentz Invariance with the Lie Algebra of <i>SU</i>(<i>N</i>) Gauge SymmetryEckart Marsch0Yasuhito Narita1Institute for Experimental and Applied Physics, Christian Albrechts University at Kiel, Leibnizstraße 11, 24118 Kiel, GermanySpace Research Institute, Austrian Academy of Sciences, Schmiedlstraße 6, A-8042 Graz, AustriaIn this paper, we study possible mathematical connections of the Clifford algebra with the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mi>u</mi><mo>(</mo><mi>N</mi><mo>)</mo></mrow></semantics></math></inline-formula>-Lie algebra, or in more physical terms the links between space-time symmetry (Lorentz invariance) and internal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>U</mi><mo>(</mo><mi>N</mi><mo>)</mo></mrow></semantics></math></inline-formula> gauge-symmetry for a massive spin one-half fermion described by the Dirac equation. The related matrix algebra is worked out in particular for the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>U</mi><mo>(</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula> symmetry and outlined as well for the color gauge group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>U</mi><mo>(</mo><mn>3</mn><mo>)</mo></mrow></semantics></math></inline-formula>. Possible perspectives of this approach to unification of symmetries are briefly discussed. The calculations make extensive use of tensor multiplication of the matrices involved, whereby our focus is on revisiting the Coleman–Mandula theorem. This permits us to construct unified symmetries between Lorentz invariance and gauge symmetry in a direct product sense.https://www.mdpi.com/2073-8994/13/3/475extended Dirac equationisospinClifford algebra<i>SU</i>(<i>N</i>) symmetry |
| spellingShingle | Eckart Marsch Yasuhito Narita Connecting in the Dirac Equation the Clifford Algebra of Lorentz Invariance with the Lie Algebra of <i>SU</i>(<i>N</i>) Gauge Symmetry Symmetry extended Dirac equation isospin Clifford algebra <i>SU</i>(<i>N</i>) symmetry |
| title | Connecting in the Dirac Equation the Clifford Algebra of Lorentz Invariance with the Lie Algebra of <i>SU</i>(<i>N</i>) Gauge Symmetry |
| title_full | Connecting in the Dirac Equation the Clifford Algebra of Lorentz Invariance with the Lie Algebra of <i>SU</i>(<i>N</i>) Gauge Symmetry |
| title_fullStr | Connecting in the Dirac Equation the Clifford Algebra of Lorentz Invariance with the Lie Algebra of <i>SU</i>(<i>N</i>) Gauge Symmetry |
| title_full_unstemmed | Connecting in the Dirac Equation the Clifford Algebra of Lorentz Invariance with the Lie Algebra of <i>SU</i>(<i>N</i>) Gauge Symmetry |
| title_short | Connecting in the Dirac Equation the Clifford Algebra of Lorentz Invariance with the Lie Algebra of <i>SU</i>(<i>N</i>) Gauge Symmetry |
| title_sort | connecting in the dirac equation the clifford algebra of lorentz invariance with the lie algebra of i su i i n i gauge symmetry |
| topic | extended Dirac equation isospin Clifford algebra <i>SU</i>(<i>N</i>) symmetry |
| url | https://www.mdpi.com/2073-8994/13/3/475 |
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