{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:13:12Z","timestamp":1760242392313,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2017,6,23]],"date-time":"2017-06-23T00:00:00Z","timestamp":1498176000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Wigner\u2019s little groups are the subgroups of the Lorentz group whose transformations leave the momentum of a given particle invariant. They thus define the internal space-time symmetries of relativistic particles. These symmetries take different mathematical forms for massive and for massless particles. However, it is shown possible to construct one unified representation using a graphical description. This graphical approach allows us to describe vividly parity, time reversal, and charge conjugation of the internal symmetry groups. As for the language of group theory, the two-by-two representation is used throughout the paper. While this two-by-two representation is for spin-1\/2 particles, it is shown possible to construct the representations for spin-0 particles, spin-1 particles, as well as for higher-spin particles, for both massive and massless cases. It is shown also that the four-by-four Dirac matrices constitute a two-by-two representation of Wigner\u2019s little group.<\/jats:p>","DOI":"10.3390\/sym9070097","type":"journal-article","created":{"date-parts":[[2017,6,23]],"date-time":"2017-06-23T08:16:18Z","timestamp":1498205778000},"page":"97","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Loop Representation of Wigner\u2019s Little Groups"],"prefix":"10.3390","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3795-7565","authenticated-orcid":false,"given":"Sibel","family":"Ba\u015fkal","sequence":"first","affiliation":[{"name":"Department of Physics, Middle East Technical University, 06800 Ankara, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Young","family":"Kim","sequence":"additional","affiliation":[{"name":"Center for Fundamental Physics, University of Maryland College Park, Maryland, MD 20742, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Marilyn","family":"Noz","sequence":"additional","affiliation":[{"name":"Department of Radiology, New York University, New York, NY 10016, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,6,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"149","DOI":"10.2307\/1968551","article-title":"On unitary representations of the inhomogeneous Lorentz group","volume":"40","author":"Wigner","year":"1939","journal-title":"Ann. 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