Group-Theoretical Classification of Orientable Objects and Particle Phenomenology
The quantum description of relativistic orientable objects by a scalar field on the Poincaré group is considered. The position of the relativistic orientable object in Minkowski space is completely determined by the position of a body-fixed reference frame with respect to the position of the space-f...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-04-01
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| Series: | Universe |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2218-1997/11/5/136 |
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| Summary: | The quantum description of relativistic orientable objects by a scalar field on the Poincaré group is considered. The position of the relativistic orientable object in Minkowski space is completely determined by the position of a body-fixed reference frame with respect to the position of the space-fixed reference frame, so that all the positions can be specified by elements <i>q</i> of the Poincaré group. Relativistic orientable objects are described by scalar wave functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>q</mi><mo>)</mo></mrow></semantics></math></inline-formula>, where the arguments <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>=</mo><mo>(</mo><mi>x</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></semantics></math></inline-formula> consist of space–time points <i>x</i> and of orientation variables <i>z</i> from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>L</mi><mo>(</mo><mn>2</mn><mo>,</mo><mi>C</mi><mo>)</mo></mrow></semantics></math></inline-formula> matrices. We introduce and study the double-sided representation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold-italic">T</mi><mrow><mo>(</mo><mi mathvariant="bold-italic">g</mi><mo>)</mo></mrow><mi>f</mi><mrow><mo>(</mo><mi>q</mi><mo>)</mo></mrow><mo>=</mo><mi>f</mi><mrow><mo>(</mo><msubsup><mi>g</mi><mrow><mi>l</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mi>q</mi><msub><mi>g</mi><mi>r</mi></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold-italic">g</mi><mo>=</mo><mo>(</mo><msub><mi>g</mi><mi>l</mi></msub><mo>,</mo><msub><mi>g</mi><mi>r</mi></msub><mo>)</mo><mo>∈</mo><mi mathvariant="bold-italic">M</mi><mo>,</mo></mrow></semantics></math></inline-formula> of the group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold-italic">M</mi></semantics></math></inline-formula>. Here, the left multiplication by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>g</mi><mrow><mi>l</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msubsup></semantics></math></inline-formula> corresponds to a change in a space-fixed reference frame, whereas the right multiplication by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>g</mi><mi>r</mi></msub></semantics></math></inline-formula> corresponds to a change in a body-fixed reference frame. On this basis, we develop a classification of orientable objects and draw attention to the possibility of connecting these results with particle phenomenology. In particular, we demonstrate how one may identify fields described by polynomials in <i>z</i> with known elementary particles of spins 0, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></semantics></math></inline-formula>, and 1. The developed classification does not contradict the phenomenology of elementary particles and, in some cases, even provides a group-theoretic explanation for it. |
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| ISSN: | 2218-1997 |