{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:50:27Z","timestamp":1760237427520,"version":"build-2065373602"},"reference-count":7,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2020,5,15]],"date-time":"2020-05-15T00:00:00Z","timestamp":1589500800000},"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>The Lie algebra of the Lorentz group O(3,3) admits two types of SU(2) \u00d7 SU(2) subalgebras: a standard form based on spatial rotation generators and a second form based on temporal rotation generators. The units of measurement for the conserved quantity due to invariance under temporal rotations are investigated and found to be the same units of measure as the Planck constant. The breaking of time reversal symmetry is considered and found to affect the chiral properties of a temporal SU(2) \u00d7 SU(2) algebra. Finally, the symmetry between algebras is explored and pairs of algebras are found to be related by SU(2) \u00d7 U(1) symmetry, while a group of three algebras are related by SO(4) symmetry.<\/jats:p>","DOI":"10.3390\/sym12050817","type":"journal-article","created":{"date-parts":[[2020,5,18]],"date-time":"2020-05-18T11:34:14Z","timestamp":1589801654000},"page":"817","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["SU(2) \u00d7 SU(2) Algebras and the Lorentz Group O(3,3)"],"prefix":"10.3390","volume":"12","author":[{"given":"Martin","family":"Walker","sequence":"first","affiliation":[{"name":"Independent Researcher, 3958 Grandis Place, Victoria, BC V8N 4H6, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,5,15]]},"reference":[{"key":"ref_1","unstructured":"Cartan, \u00c9. (1966). The Theory of Spinors, Hermann."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Tung, W. (1985). Group Theory in Physics, World Scientific Publishing.","DOI":"10.1142\/0097"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"626","DOI":"10.3390\/sym4040626","article-title":"Dirac Matrices and Feynman\u2019s Rest of the Universe","volume":"4","author":"Kim","year":"2012","journal-title":"Symmetry"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Hall, B.C. (2015). Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer. [2nd ed.].","DOI":"10.1007\/978-3-319-13467-3"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Schwichtenberg, J. (2017). Physics from Symmetry, Springer. [2nd ed.].","DOI":"10.1007\/978-3-319-66631-0"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"901","DOI":"10.1063\/1.1704016","article-title":"A Remarkable Representation of the 3 + 2 de Sitter Group","volume":"4","author":"Dirac","year":"1963","journal-title":"J. Math. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"524","DOI":"10.1063\/1.531320","article-title":"The Dirac gamma matrices as \u201crelics\u201d of a hidden symmetry?: As fundamental representation of the algebra Sp(4,r)","volume":"36","author":"Lee","year":"1995","journal-title":"J. Math. Phys."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/5\/817\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:29:06Z","timestamp":1760174946000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/5\/817"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,5,15]]},"references-count":7,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2020,5]]}},"alternative-id":["sym12050817"],"URL":"https:\/\/doi.org\/10.3390\/sym12050817","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,5,15]]}}}