Enlargement of Symmetry Groups in Physics: A Practitioner’s Guide
Wigner’s classification has led to the insight that projective unitary representations play a prominent role in quantum mechanics. The physics literature often states that the theory of projective unitary representations can be reduced to the theory of ordinary unitary representations by enlarging t...
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2024-12-01
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| author | Lehel Csillag Julio Marny Hoff da Silva Tudor Pătuleanu |
| author_facet | Lehel Csillag Julio Marny Hoff da Silva Tudor Pătuleanu |
| author_sort | Lehel Csillag |
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| description | Wigner’s classification has led to the insight that projective unitary representations play a prominent role in quantum mechanics. The physics literature often states that the theory of projective unitary representations can be reduced to the theory of ordinary unitary representations by enlarging the group of physical symmetries. Nevertheless, the enlargement process is not always described explicitly: it is unclear in which cases the enlargement has to be conducted on the universal cover, a central extension, or a central extension of the universal cover. On the other hand, in the mathematical literature, projective unitary representations have been extensively studied, and famous theorems such as the theorems of Bargmann and Cassinelli have been achieved. The present article bridges the two: we provide a precise, step-by-step guide on describing projective unitary representations as unitary representations of the enlarged group. Particular focus is paid to the difference between algebraic and topological obstructions. To build the bridge mentioned above, we present a detailed review of the difference between group cohomology and Lie group cohomology. This culminates in classifying Lie group central extensions by smooth cocycles around the identity. Finally, the take-away message is a hands-on algorithm that takes the symmetry group of a given quantum theory as input and provides the enlarged group as output. This algorithm is applied to several cases of physical interest. We also briefly outline a generalization of Bargmann’s theory to time-dependent phases using Hilbert bundles. |
| format | Article |
| id | doaj-art-b17f179f1bad4ebeb642b5d13f645464 |
| institution | Kabale University |
| issn | 2218-1997 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Universe |
| spelling | doaj-art-b17f179f1bad4ebeb642b5d13f6454642024-12-27T14:57:16ZengMDPI AGUniverse2218-19972024-12-01101244810.3390/universe10120448Enlargement of Symmetry Groups in Physics: A Practitioner’s GuideLehel Csillag0Julio Marny Hoff da Silva1Tudor Pătuleanu2Faculty of Mathematics and Computer Science, Transilvania University, Iuliu Maniu Street 50, 500091 Brașov, RomâniaDepartamento de Física, Universidade Estadual Paulista, UNESP, Av. Dr. Ariberto Pereira da Cunha, 333, Guaratinguetá 12516-410, SP, BrazilDepartment of Physics, West University of Timișoara, Bd. Vasile Pârvan 4, 300223 Timișoara, RomâniaWigner’s classification has led to the insight that projective unitary representations play a prominent role in quantum mechanics. The physics literature often states that the theory of projective unitary representations can be reduced to the theory of ordinary unitary representations by enlarging the group of physical symmetries. Nevertheless, the enlargement process is not always described explicitly: it is unclear in which cases the enlargement has to be conducted on the universal cover, a central extension, or a central extension of the universal cover. On the other hand, in the mathematical literature, projective unitary representations have been extensively studied, and famous theorems such as the theorems of Bargmann and Cassinelli have been achieved. The present article bridges the two: we provide a precise, step-by-step guide on describing projective unitary representations as unitary representations of the enlarged group. Particular focus is paid to the difference between algebraic and topological obstructions. To build the bridge mentioned above, we present a detailed review of the difference between group cohomology and Lie group cohomology. This culminates in classifying Lie group central extensions by smooth cocycles around the identity. Finally, the take-away message is a hands-on algorithm that takes the symmetry group of a given quantum theory as input and provides the enlarged group as output. This algorithm is applied to several cases of physical interest. We also briefly outline a generalization of Bargmann’s theory to time-dependent phases using Hilbert bundles.https://www.mdpi.com/2218-1997/10/12/448projective representationcentral extensionuniversal coverlifting problem |
| spellingShingle | Lehel Csillag Julio Marny Hoff da Silva Tudor Pătuleanu Enlargement of Symmetry Groups in Physics: A Practitioner’s Guide Universe projective representation central extension universal cover lifting problem |
| title | Enlargement of Symmetry Groups in Physics: A Practitioner’s Guide |
| title_full | Enlargement of Symmetry Groups in Physics: A Practitioner’s Guide |
| title_fullStr | Enlargement of Symmetry Groups in Physics: A Practitioner’s Guide |
| title_full_unstemmed | Enlargement of Symmetry Groups in Physics: A Practitioner’s Guide |
| title_short | Enlargement of Symmetry Groups in Physics: A Practitioner’s Guide |
| title_sort | enlargement of symmetry groups in physics a practitioner s guide |
| topic | projective representation central extension universal cover lifting problem |
| url | https://www.mdpi.com/2218-1997/10/12/448 |
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