{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:00:53Z","timestamp":1760058053611,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,3,17]],"date-time":"2025-03-17T00:00:00Z","timestamp":1742169600000},"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>This work aims to explore the algebraic and spectral properties of a novel parametric Hermitian non-Pauli spin basis. The primary motivation for this work is to introduce an alternative to the Pauli spin basis for investigating systems where conventional quantum statistics no longer apply. This study explores the symmetries, commutation relations, Lie algebra structures, and ladder operators of the proposed spin matrices. Additionally, it provides discussions on key findings, potential applications, and concludes with some final remarks. An example for modeling the spectrum of a quantum system is provided.<\/jats:p>","DOI":"10.3390\/sym17030450","type":"journal-article","created":{"date-parts":[[2025,3,17]],"date-time":"2025-03-17T15:24:56Z","timestamp":1742225096000},"page":"450","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Algebraic and Spectral Analysis of a Novel Hermitian Spin Basis"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6485-4097","authenticated-orcid":false,"given":"Timothy","family":"Ganesan","sequence":"first","affiliation":[{"name":"Department of Physics & Astronomy, University of Calgary, Calgary, AB T2N 1N4, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8227-2621","authenticated-orcid":false,"given":"Zeeshan","family":"Yousaf","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of the Punjab, Lahore 54590, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6286-1135","authenticated-orcid":false,"given":"M. Z.","family":"Bhatti","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of the Punjab, Lahore 54590, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,3,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Riazat, M., Azizi, A., Naderi Soorki, M., and Koochakzadeh, A. (2023). Robust consensus in a class of fractional-order multi-agent systems with interval uncertainties using the existence condition of Hermitian matrices. Axioms, 12.","DOI":"10.3390\/axioms12010065"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Xu, Y., Zhu, Y., and Song, Z. (2024). A new extended target detection method based on the maximum eigenvalue of the Hermitian matrix. Remote Sens., 16.","DOI":"10.3390\/rs16091488"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1007\/s42484-021-00048-8","article-title":"Compiling basic linear algebra subroutines for quantum computers","volume":"3","author":"Zhao","year":"2021","journal-title":"Quantum Mach. Intell."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"108530","DOI":"10.1016\/j.sigpro.2022.108530","article-title":"Distributed algorithms to determine eigenvectors of matrices on spatially distributed networks","volume":"196","author":"Emirov","year":"2022","journal-title":"Signal Process."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1007\/s11785-023-01440-x","article-title":"Some new characterizations of a Hermitian matrix and their applications","volume":"18","author":"Tian","year":"2024","journal-title":"Complex Anal. Oper. Theory"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1002\/jgt.22057","article-title":"Hermitian adjacency matrix of digraphs and mixed graphs","volume":"85","author":"Guo","year":"2017","journal-title":"J. Graph Theory"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1016\/j.laa.2019.09.024","article-title":"A new kind of Hermitian matrices for digraphs","volume":"584","author":"Mohar","year":"2020","journal-title":"Linear Algebra Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"112798","DOI":"10.1016\/j.disc.2022.112798","article-title":"Hermitian adjacency matrix of the second kind for mixed graphs","volume":"345","author":"Li","year":"2022","journal-title":"Discrete Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"113254","DOI":"10.1016\/j.disc.2022.113254","article-title":"The k-generalized Hermitian adjacency matrices for mixed graphs","volume":"346","author":"Yu","year":"2023","journal-title":"Discrete Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2135","DOI":"10.1137\/24M1652234","article-title":"Eigenvalues of dual Hermitian matrices with application in formation control","volume":"45","author":"Qi","year":"2024","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1007\/s40324-020-00229-8","article-title":"Numerical methods for accurate computation of the eigenvalues of Hermitian matrices and the singular values of general matrices","volume":"78","year":"2021","journal-title":"SeMA J."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"e2286","DOI":"10.1002\/nla.2286","article-title":"Non-Hermitian perturbations of Hermitian matrix sequences and applications to the spectral analysis of the numerical approximation of partial differential equations","volume":"27","author":"Barbarino","year":"2020","journal-title":"Numer. Linear Algebra Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/j.laa.2020.12.030","article-title":"The spectral spread of Hermitian matrices","volume":"616","author":"Massey","year":"2021","journal-title":"Linear Algebra Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"238","DOI":"10.1016\/j.laa.2023.11.009","article-title":"Hermitians in matrix algebras with operator norm and associated Lie algebras","volume":"682","author":"Duncan","year":"2024","journal-title":"Linear Algebra Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"376","DOI":"10.1017\/S0017089521000173","article-title":"Hermitians in matrix algebras with operator norm\u2014II","volume":"64","author":"Duncan","year":"2022","journal-title":"Glasg. Math. J."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"280","DOI":"10.1017\/S0017089520000178","article-title":"Hermitians in matrix algebras with operator norm","volume":"63","author":"Crabb","year":"2021","journal-title":"Glasg. Math. J."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Ganesan, T. (2023). Exotic particle dynamics using novel Hermitian spin matrices. Axioms, 12.","DOI":"10.20944\/preprints202310.0076.v1"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1007\/JHEP03(2022)002","article-title":"Quantum mechanism of extremely high energy processes at neutron star collapse and of quasar luminosity","volume":"2022","author":"Jacak","year":"2022","journal-title":"J. High Energy Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"092","DOI":"10.1088\/1475-7516\/2022\/10\/092","article-title":"Quantum contribution to luminosity of quasars","volume":"2022","author":"Jacak","year":"2022","journal-title":"J. Cosmol. Astropart. Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"035006","DOI":"10.1088\/1361-6382\/ad1a53","article-title":"Topology-induced quantum transition in multiparticle systems in the vicinity of a black hole","volume":"41","author":"Jacak","year":"2024","journal-title":"Class. Quantum Gravity"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1103\/RevModPhys.23.21","article-title":"The factorization method","volume":"23","author":"Infeld","year":"1951","journal-title":"Rev. Mod. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1272","DOI":"10.1063\/1.467819","article-title":"Exact solutions of regular approximate relativistic wave equations for hydrogen-like atoms","volume":"101","author":"Baerends","year":"1994","journal-title":"J. Chem. Phys."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"632","DOI":"10.1016\/0009-2614(95)01156-0","article-title":"The ZORA formalism applied to the Dirac-Fock equation","volume":"246","author":"Faas","year":"1995","journal-title":"Chem. Phys. Lett."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"9634","DOI":"10.1039\/D1CP05268C","article-title":"A theoretical investigation of the hydrolysis of uranium hexafluoride: The initiation mechanism and vibrational spectroscopy","volume":"24","author":"Lutz","year":"2022","journal-title":"Phys. Chem. Chem. Phys."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/3\/450\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:55:13Z","timestamp":1760028913000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/3\/450"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,3,17]]},"references-count":24,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2025,3]]}},"alternative-id":["sym17030450"],"URL":"https:\/\/doi.org\/10.3390\/sym17030450","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2025,3,17]]}}}