{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,25]],"date-time":"2026-10-25T02:39:41Z","timestamp":1792895981810,"version":"4.5.0"},"reference-count":45,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2020,12,24]],"date-time":"2020-12-24T00:00:00Z","timestamp":1608768000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004663","name":"Ministry of Science and Technology, Taiwan","doi-asserted-by":"publisher","award":["109-2112-M-008-016"],"award-info":[{"award-number":["109-2112-M-008-016"]}],"id":[{"id":"10.13039\/501100004663","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004663","name":"Ministry of Science and Technology, Taiwan","doi-asserted-by":"publisher","award":["107-2119-M-008-011"],"award-info":[{"award-number":["107-2119-M-008-011"]}],"id":[{"id":"10.13039\/501100004663","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We present the formulation of a version of Lorentz covariant quantum mechanics based on a group theoretical construction from a Heisenberg\u2013Weyl symmetry with position and momentum operators transforming as Minkowski four-vectors. The basic representation is identified as a coherent state representation, essentially an irreducible component of the regular representation, with the matching representation of an extension of the group C*-algebra giving the algebra of observables. The key feature is that it is not unitary but pseudo-unitary, exactly in the same sense as the Minkowski spacetime representation. The language of pseudo-Hermitian quantum mechanics is adopted for a clear illustration of the aspect, with a metric operator obtained as really the manifestation of the Minkowski metric on the space of the state vectors. Explicit wavefunction description is given without any restriction of the variable domains, yet with a finite integral inner product. The associated covariant harmonic oscillator Fock state basis has all the standard properties in exact analog to those of a harmonic oscillator with Euclidean position and momentum operators. Galilean limit and the classical limit are retrieved rigorously through appropriate symmetry contractions of the algebra and its representation, including the dynamics described through the symmetry of the phase space.<\/jats:p>","DOI":"10.3390\/sym13010022","type":"journal-article","created":{"date-parts":[[2020,12,24]],"date-time":"2020-12-24T22:56:45Z","timestamp":1608850605000},"page":"22","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Group Theoretical Approach to Pseudo-Hermitian Quantum Mechanics with Lorentz Covariance and c \u2192 \u221e Limit"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6245-867X","authenticated-orcid":false,"given":"Suzana","family":"Bedi\u0107","sequence":"first","affiliation":[{"name":"ICRANet, P.le della Repubblica 10, 65100 Pescara, Italy"},{"name":"Physics Department, ICRA and University of Rome \u201cSapienza\u201d, P.le A. Moro 5, 00185 Rome, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3832-648X","authenticated-orcid":false,"given":"Otto C. W.","family":"Kong","sequence":"additional","affiliation":[{"name":"Department of Physics, National Central University, Chung-Li 32054, Taiwan"},{"name":"Center for High Energy and High Field Physics, National Central University, Chung-Li 32054, Taiwan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hock King","family":"Ting","sequence":"additional","affiliation":[{"name":"Department of Physics, National Central University, Chung-Li 32054, Taiwan"},{"name":"Center for High Energy and High Field Physics, National Central University, Chung-Li 32054, Taiwan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,12,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"553","DOI":"10.4236\/jhepgc.2019.53031","article-title":"Observables and Dynamics Quantum to Classical from a Relativity Symmetry and Noncommutative-Geometric Perspective","volume":"5","author":"Chew","year":"2019","journal-title":"J. High Energy Phys. Gravit. Cosmol."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"De Azc\u00e1rraga, J.A., and Izquierdo, J.M. (1995). Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics, Cambridge University Press.","DOI":"10.1017\/CBO9780511599897"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"703","DOI":"10.1070\/PU1977v020n09ABEH005459","article-title":"Generalized Coherent States and Some of their Applications","volume":"20","author":"Perelomov","year":"1977","journal-title":"Sov. Phys. Usp."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"867","DOI":"10.1103\/RevModPhys.62.867","article-title":"Coherent States: Theory and Some Applications","volume":"62","author":"Zhang","year":"1990","journal-title":"Rev. Mod. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Klauder, J.R. (2010). A Modern Approach to Functional Integration, Birkh\u00e4user.","DOI":"10.1007\/978-0-8176-4791-9"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Palmer, T.W. (2001). Banach Algebras and the General Theory of *-Algebras Vol II, Cambridge University Press.","DOI":"10.1017\/CBO9780511574757"},{"key":"ref_7","unstructured":"Pedersen, G.K. (1979). C*-algebras and their Automorphism Groups, Academic Press."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Dubin, D.A., Hennings, M.A., and Smith, T.B. (2000). Mathematical Aspects of Weyl Quantization and Phase, World Scientific.","DOI":"10.1142\/9789812813459"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1016\/0034-4877(84)90008-9","article-title":"Quantum Mechanics in Phase Space","volume":"19","author":"Hansen","year":"1984","journal-title":"Rep. Math. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"869","DOI":"10.1063\/1.528200","article-title":"Algebras of Distributions Suitable for Phase-space Quantum Mechanics I","volume":"29","year":"1988","journal-title":"J. Math. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Zachos, C.K., Fairlie, D.B., and Curtright, T.L. (2005). Quantum Mechanics in Phase Space: An Overview with Selected Papers, World Scientific.","DOI":"10.1142\/9789812703507"},{"key":"ref_12","unstructured":"Kong, O.C.W., and Liu, W.-Y. (2019). Noncommutative Coordinate Picture of the Quantum Phase Space. arXiv."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"103606","DOI":"10.1016\/j.rinp.2020.103636","article-title":"A Geometric Picture of Quantum Mechanics with Noncommutative Values for Observables","volume":"19","author":"Kong","year":"2020","journal-title":"Results Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"4395918","DOI":"10.1155\/2017\/4395918","article-title":"A Quantum Space Behind Simple Quantum Mechanics","volume":"2017","author":"Chew","year":"2017","journal-title":"Adv. High Energy Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1605","DOI":"10.1063\/1.1664490","article-title":"Possible Kinematics","volume":"9","author":"Bacry","year":"1968","journal-title":"J. Math. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1016\/j.physletb.2008.05.060","article-title":"A deformed relativity with the quantum \u210f","volume":"665","author":"Kong","year":"2008","journal-title":"Phys. Lett. B"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1803","DOI":"10.1007\/s10773-019-04075-x","article-title":"The First Physics Picture of Contractions from a Fundamental Quantum Relativity Symmetry Including All Known Relativity Symmetries, Classical and Quantum","volume":"58","author":"Kong","year":"2019","journal-title":"Int. J. Theor. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"764","DOI":"10.1063\/1.1704991","article-title":"Unitary Representations of the Lorentz Group on 4-Vector Manifolds","volume":"7","author":"Zmuidzinas","year":"1966","journal-title":"J. Math. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1755","DOI":"10.1103\/PhysRev.181.1755","article-title":"Position Operators and Proper Time in Relativistic Quantum Mechanics","volume":"181","author":"Johnson","year":"1969","journal-title":"Phys. Rev."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"045009","DOI":"10.1103\/PhysRevD.79.045009","article-title":"Relativistic Harmonic Oscillator Revisited","volume":"79","author":"Bars","year":"2009","journal-title":"Phys. Rev. D"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"43","DOI":"10.4236\/jhepgc.2020.61005","article-title":"The Case for a Quantum Theory on a Hilbert Space with an Inner Product of Indefinite Signature","volume":"6","author":"Kong","year":"2020","journal-title":"J. High Energy Phys. Gravit. Cosmol."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Bedi\u0107, S., and Kong, O.C.W. (2020). Analysis on Complete Set of Fock States with Explicit Wavefunctions for the Covariant Harmonic Oscillator Problem. Symmetry, 12.","DOI":"10.3390\/sym12010039"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1098\/rspa.1942.0023","article-title":"The Physical Interpretation of Quantum Mechanics","volume":"180","author":"Dirac","year":"1942","journal-title":"Proc. R. Soc. Lond. A"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1103\/RevModPhys.15.175","article-title":"On Dirac\u2019s New Method of Field Quantization","volume":"15","author":"Pauli","year":"1943","journal-title":"Rev. Mod. Phys."},{"key":"ref_25","unstructured":"Nagy, K.L. (1966). State Vector Spaces with Indefinite Metric in Quantum Field Theory, P. Noordhoff Ltd."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"681","DOI":"10.1088\/0370-1298\/63\/7\/301","article-title":"Theory of Longitudinal Photons in Quantum Electrodynamics","volume":"63","author":"Gupta","year":"1950","journal-title":"Proc. Phys. Soc. A"},{"key":"ref_27","first-page":"567","article-title":"Eine neue Methode zur Behandlung der longitudinalen und skalaren Photonen","volume":"23","author":"Bleuler","year":"1950","journal-title":"Helv. Phys. Acta"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Bogn\u00e1r, J. (1974). Indefinite Inner Product Spaces, Springer.","DOI":"10.1007\/978-3-642-65567-8"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1080\/00107500072632","article-title":"Introduction to PT-Symmetric Quantum Theory","volume":"46","author":"Bender","year":"2005","journal-title":"Contemp. Phys."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"012002","DOI":"10.1088\/1742-6596\/287\/1\/012002","article-title":"Pseudo-Hermitian Quantum Mechanics","volume":"287","author":"Das","year":"2011","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1191","DOI":"10.1142\/S0219887810004816","article-title":"Pseudo-Hermitian Representation of Quantum Mechanics","volume":"7","author":"Mostafazadeh","year":"2010","journal-title":"Int. J. Geometr. Methods Modern Phys."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Taylor, M.E. (1986). Noncommutative Harmonic Analysis, American Mathematical Society.","DOI":"10.1090\/surv\/022"},{"key":"ref_33","unstructured":"Barut, A.O., and Raczka, R. (1980). Theory of Group Representations and Applications, Polish Scientific Publishers. 2nd rev. ed."},{"key":"ref_34","unstructured":"Gilmore, R. (2005). Lie Groups, Lie Algebras, and Some of Their Applications, Dover Publications, Inc."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"510","DOI":"10.1073\/pnas.39.6.510","article-title":"On the Contraction of Groups and their Representations","volume":"39","author":"Wigner","year":"1953","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1016\/j.aop.2014.09.005","article-title":"Relativity Symmetries and Lie Algebra Contractions","volume":"351","author":"Cho","year":"2014","journal-title":"Ann. Phys."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1016\/j.cjph.2020.11.008","article-title":"Noncommutative Values of Quantum Observables","volume":"69","author":"Kong","year":"2020","journal-title":"Chin. J. Phys."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Steinacker, H. (2016). Emergent gravity on covariant quantum spaces in the IKKT model. J. High Energy Phys., 156.","DOI":"10.1007\/JHEP12(2016)156"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"075201","DOI":"10.1088\/1751-8121\/aaa2ab","article-title":"Higher spin gauge theory on fuzzy SN4","volume":"51","author":"Sperling","year":"2018","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"026031","DOI":"10.1103\/PhysRevD.98.026031","article-title":"Extending general covariance: Moyal-type noncommutative manifolds","volume":"98","author":"Bojowald","year":"2018","journal-title":"Phys. Rev. D"},{"key":"ref_41","unstructured":"Bjorken, J.D., and Drell, S.D. (1964). Relativistic Quantum Mechanics, McGraw-Hill Inc."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1169","DOI":"10.1119\/1.1976597","article-title":"The Reduction of the Dirac Equation to a Nonrelativistic Form","volume":"39","author":"Mosses","year":"1971","journal-title":"Am. J. Phys."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"4293","DOI":"10.1103\/PhysRevA.45.4293","article-title":"Classical eigenspinors and the Dirac equation","volume":"45","author":"Baylis","year":"1995","journal-title":"Phys. Rev. A"},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Horwitz, L.P. (2015). Relativistic Quantum Mechanics, Springer Science+Business Media.","DOI":"10.1007\/978-94-017-7261-7"},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Fanchi, J.R. (1993). Parametrized Relativistic Quantum Theory, Kluwer Academic Publishers.","DOI":"10.1007\/978-94-011-1944-3"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/1\/22\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:45:49Z","timestamp":1760179549000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/1\/22"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,12,24]]},"references-count":45,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,1]]}},"alternative-id":["sym13010022"],"URL":"https:\/\/doi.org\/10.3390\/sym13010022","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,12,24]]}}}