{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,19]],"date-time":"2026-03-19T06:40:52Z","timestamp":1773902452622,"version":"3.50.1"},"reference-count":23,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2021,9,2]],"date-time":"2021-09-02T00:00:00Z","timestamp":1630540800000},"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 Dirac equation with chiral symmetry is derived using the irreducible representations of the Poincar\u00e9 group, the Lagrangian formalism, and a novel method of projection operators that takes as its starting point the minimal assumption of four linearly independent physical states. We thereby demonstrate the fundamental nature of this form of the Dirac equation. The resulting equation is then examined within the context of spacetime and CPT symmetries with a discussion of the implications for the general formulation of physical theories.<\/jats:p>","DOI":"10.3390\/sym13091608","type":"journal-article","created":{"date-parts":[[2021,9,6]],"date-time":"2021-09-06T23:55:22Z","timestamp":1630972522000},"page":"1608","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Chiral Dirac Equation and Its Spacetime and CPT Symmetries"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8631-2253","authenticated-orcid":false,"given":"Timothy","family":"Watson","sequence":"first","affiliation":[{"name":"Department of Physics, University of Texas at Arlington, Arlington, TX 76019, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1975-9298","authenticated-orcid":false,"given":"Zdzislaw","family":"Musielak","sequence":"additional","affiliation":[{"name":"Department of Physics, University of Texas at Arlington, Arlington, TX 76019, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,9,2]]},"reference":[{"key":"ref_1","first-page":"610","article-title":"The quantum theory of the electron","volume":"117","author":"Dirac","year":"1928","journal-title":"Proc. R. Soc. Lond."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1143\/PTP.66.303","article-title":"Generalized Dirac equation with four orthogonal families of spin 1\/2 solutions","volume":"66","author":"Sogami","year":"1981","journal-title":"Prog. Theor. Phys."},{"key":"ref_3","unstructured":"Kruglov, S.I. (2006). Chiral Bargmann-Wigner Equations for Spin-1 Massive Fields. arXiv."},{"key":"ref_4","first-page":"82","article-title":"Fermion unification model based on the intrinsic SU(8) symmetry of a generalized Dirac equation","volume":"3","author":"Marsh","year":"2015","journal-title":"Front. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1844","DOI":"10.1103\/PhysRev.182.1844","article-title":"Theory of the heavy electron","volume":"182","author":"Barut","year":"1969","journal-title":"Phys. Rev."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"310","DOI":"10.1016\/0370-2693(78)90522-1","article-title":"The mass of muon","volume":"73B","author":"Barut","year":"1978","journal-title":"Phys. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1427","DOI":"10.1007\/BF02821059","article-title":"Mixed-symmetry solutions of generalized three-particle Bargmann-Wigner equations in the strong-coupling limit","volume":"108","author":"Pfister","year":"1995","journal-title":"Nuovo Cim. A"},{"key":"ref_8","first-page":"11","article-title":"On the Hamiltonian form of generalized Dirac equation for fermions with two mass states","volume":"3","author":"Kruglov","year":"2006","journal-title":"Elect. J. Theor. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"228","DOI":"10.1016\/j.physletb.2012.10.037","article-title":"Modified Dirac equation with Lorentz invariance violoation and its solutions for particles in an external magnetic field","volume":"718","author":"Kruglov","year":"2012","journal-title":"Phys. Lett. B"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"814","DOI":"10.1007\/BF02812315","article-title":"Pseudoscalar mass and its relationship to conventional scalar mass in relativistic Dirac theory of the electron","volume":"5","author":"Leiter","year":"1972","journal-title":"Lett. Nuovo Cim."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"302","DOI":"10.1016\/j.chaos.2006.06.056","article-title":"Generalized Dirac equation and its symmetries","volume":"32","author":"Nozari","year":"2007","journal-title":"Chaos Solitons Fractals"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2050189","DOI":"10.1142\/S0217751X20501894","article-title":"Chiral symmetry in Dirac equation and its effecst on neutrino masses and dark matter","volume":"35","author":"Watson","year":"2020","journal-title":"Int. J. Mod. Phys. A"},{"key":"ref_13","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. Math."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Kim, Y.S., and Noz, M.E. (1986). Theory and Applications of the Poincar\u00e9 Group, Reidel.","DOI":"10.1007\/978-94-009-4558-6"},{"key":"ref_15","unstructured":"Ryder, L.W. (1985). Quantum Field Theory, Cambridge University Press."},{"key":"ref_16","unstructured":"Frampton, P.H. (2000). Gauge Field Theories, John Wiley & Sons, Inc."},{"key":"ref_17","unstructured":"Daughty, N.A. (1990). Lagrangian Interactions, Addison-Wesley Publ. Comp., Inc."},{"key":"ref_18","first-page":"L59","article-title":"A new conservation law constructed without using either Lagrangians or Hamiltonians","volume":"27","author":"Hojman","year":"1992","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2399","DOI":"10.1088\/0305-4470\/17\/12\/012","article-title":"Symmetries of Lagrangians and of their equations of motion","volume":"17","author":"Hojman","year":"1984","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Landau, L.D., and Lifschitz, E.M. (1969). Mechanics, Pergamon Press.","DOI":"10.1007\/978-3-322-85937-2"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1007\/BF01646436","article-title":"Group-theoretical foundations of classical mechanics: The Lagrange gauge problem","volume":"12","year":"1969","journal-title":"Comm. Math. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"126642","DOI":"10.1016\/j.physleta.2020.126642","article-title":"Gauge functions and Galilean invariance of Lagrangians","volume":"384","author":"Musielak","year":"2020","journal-title":"Phys. Lett. A"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Petitjean, M. (2019). About chirality in Minkowski spacetime. Symmetry, 11.","DOI":"10.3390\/sym11101320"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/9\/1608\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:54:37Z","timestamp":1760165677000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/9\/1608"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,9,2]]},"references-count":23,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2021,9]]}},"alternative-id":["sym13091608"],"URL":"https:\/\/doi.org\/10.3390\/sym13091608","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,9,2]]}}}