Poincaré Symmetry from Heisenberg’s Uncertainty Relations
It is noted that the single-variable Heisenberg commutation relation contains the symmetry of the <inline-formula> <math display="inline"> <semantics> <mrow> <mi>S</mi> <mi>p</mi> <mo>(</mo> <mn>2</mn> <mo>)</...
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MDPI AG
2019-03-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/11/3/409 |
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| author | Sibel Başkal Young S. Kim Marilyn E. Noz |
| author_facet | Sibel Başkal Young S. Kim Marilyn E. Noz |
| author_sort | Sibel Başkal |
| collection | DOAJ |
| description | It is noted that the single-variable Heisenberg commutation relation contains the symmetry of the <inline-formula> <math display="inline"> <semantics> <mrow> <mi>S</mi> <mi>p</mi> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> group which is isomorphic to the Lorentz group applicable to one time-like dimension and two space-like dimensions, known as the <inline-formula> <math display="inline"> <semantics> <mrow> <mi>S</mi> <mi>O</mi> <mo>(</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> group. According to Paul A. M. Dirac, from the uncertainty commutation relations for two variables, it possible to construct the de Sitter group <inline-formula> <math display="inline"> <semantics> <mrow> <mi>S</mi> <mi>O</mi> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, namely the Lorentz group applicable to three space-like variables and two time-like variables. By contracting one of the time-like variables in <inline-formula> <math display="inline"> <semantics> <mrow> <mi>S</mi> <mi>O</mi> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, it is possible to construct the inhomogeneous Lorentz group <inline-formula> <math display="inline"> <semantics> <mrow> <mi>I</mi> <mi>S</mi> <mi>O</mi> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> which serves as the fundamental symmetry group for quantum mechanics and quantum field theory in the Lorentz-covariant world. This <inline-formula> <math display="inline"> <semantics> <mrow> <mi>I</mi> <mi>S</mi> <mi>O</mi> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> group is commonly known as the Poincaré group. |
| first_indexed | 2024-04-14T02:29:35Z |
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| language | English |
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| spelling | doaj.art-1e61eef6750a4e1db45cc2238739001c2022-12-22T02:17:44ZengMDPI AGSymmetry2073-89942019-03-0111340910.3390/sym11030409sym11030409Poincaré Symmetry from Heisenberg’s Uncertainty RelationsSibel Başkal0Young S. Kim1Marilyn E. Noz2Department of Physics, Middle East Technical University, 06800 Ankara, TurkeyCenter for Fundamental Physics, University of Maryland, College Park, MD 20742, USADepartment of Radiology, New York University, New York, NY 10016, USAIt is noted that the single-variable Heisenberg commutation relation contains the symmetry of the <inline-formula> <math display="inline"> <semantics> <mrow> <mi>S</mi> <mi>p</mi> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> group which is isomorphic to the Lorentz group applicable to one time-like dimension and two space-like dimensions, known as the <inline-formula> <math display="inline"> <semantics> <mrow> <mi>S</mi> <mi>O</mi> <mo>(</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> group. According to Paul A. M. Dirac, from the uncertainty commutation relations for two variables, it possible to construct the de Sitter group <inline-formula> <math display="inline"> <semantics> <mrow> <mi>S</mi> <mi>O</mi> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, namely the Lorentz group applicable to three space-like variables and two time-like variables. By contracting one of the time-like variables in <inline-formula> <math display="inline"> <semantics> <mrow> <mi>S</mi> <mi>O</mi> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, it is possible to construct the inhomogeneous Lorentz group <inline-formula> <math display="inline"> <semantics> <mrow> <mi>I</mi> <mi>S</mi> <mi>O</mi> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> which serves as the fundamental symmetry group for quantum mechanics and quantum field theory in the Lorentz-covariant world. This <inline-formula> <math display="inline"> <semantics> <mrow> <mi>I</mi> <mi>S</mi> <mi>O</mi> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> group is commonly known as the Poincaré group.https://www.mdpi.com/2073-8994/11/3/409Poincaré symmetry from uncertainty relationsone symmetry for quantum mechanicsspecial relativity |
| spellingShingle | Sibel Başkal Young S. Kim Marilyn E. Noz Poincaré Symmetry from Heisenberg’s Uncertainty Relations Symmetry Poincaré symmetry from uncertainty relations one symmetry for quantum mechanics special relativity |
| title | Poincaré Symmetry from Heisenberg’s Uncertainty Relations |
| title_full | Poincaré Symmetry from Heisenberg’s Uncertainty Relations |
| title_fullStr | Poincaré Symmetry from Heisenberg’s Uncertainty Relations |
| title_full_unstemmed | Poincaré Symmetry from Heisenberg’s Uncertainty Relations |
| title_short | Poincaré Symmetry from Heisenberg’s Uncertainty Relations |
| title_sort | poincare symmetry from heisenberg s uncertainty relations |
| topic | Poincaré symmetry from uncertainty relations one symmetry for quantum mechanics special relativity |
| url | https://www.mdpi.com/2073-8994/11/3/409 |
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