{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,11]],"date-time":"2026-06-11T16:44:33Z","timestamp":1781196273088,"version":"3.54.1"},"reference-count":31,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2024,1,26]],"date-time":"2024-01-26T00:00:00Z","timestamp":1706227200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Research, Development and Innovation Office (NKFIH)","award":["K137852"],"award-info":[{"award-number":["K137852"]}]},{"name":"National Research, Development and Innovation Office (NKFIH)","award":["2019-2.1.7-ERA-NET-2021-00028"],"award-info":[{"award-number":["2019-2.1.7-ERA-NET-2021-00028"]}]},{"name":"National Research, Development and Innovation Office (NKFIH)","award":["TKP2021-NVA-16"],"award-info":[{"award-number":["TKP2021-NVA-16"]}]},{"name":"Ministry of Innovation and Technology and NKFIH within the Quantum Information National Laboratory of Hungary","award":["K137852"],"award-info":[{"award-number":["K137852"]}]},{"name":"Ministry of Innovation and Technology and NKFIH within the Quantum Information National Laboratory of Hungary","award":["2019-2.1.7-ERA-NET-2021-00028"],"award-info":[{"award-number":["2019-2.1.7-ERA-NET-2021-00028"]}]},{"name":"Ministry of Innovation and Technology and NKFIH within the Quantum Information National Laboratory of Hungary","award":["TKP2021-NVA-16"],"award-info":[{"award-number":["TKP2021-NVA-16"]}]},{"name":"Ministry of Innovation and Technology of Hungary from the National Research, Development and Innovation Fund","award":["K137852"],"award-info":[{"award-number":["K137852"]}]},{"name":"Ministry of Innovation and Technology of Hungary from the National Research, Development and Innovation Fund","award":["2019-2.1.7-ERA-NET-2021-00028"],"award-info":[{"award-number":["2019-2.1.7-ERA-NET-2021-00028"]}]},{"name":"Ministry of Innovation and Technology of Hungary from the National Research, Development and Innovation Fund","award":["TKP2021-NVA-16"],"award-info":[{"award-number":["TKP2021-NVA-16"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Through the discussion of three physical processes, we show that the Klein\u2013Gordon equations with a negative mass term describe special dynamics. In the case of two classical disciplines\u2014mechanics and thermodynamics\u2014the Lagrangian-based mathematical description is the same, even though the nature of the investigated processes seems completely different. The unique feature of this type of equation is that it contains wave propagation and dissipative behavior in one framework. The dissipative behavior appears through a repulsive potential. The transition between the two types of dynamics can be specified precisely, and its physical meaning is clear. The success of the two descriptions inspires extension to the case of electrodynamics. We reverse the suggestion here. We create a Klein\u2013Gordon equation with a negative mass term, but first, we modify Maxwell\u2019s equations. The repulsive interaction that appears here results in a charge spike. However, the Coulomb interaction limits this. The charge separation is also associated with the high-speed movement of the charged particle localized in a small space domain. As a result, we arrive at a picture of a fast vibrating phenomenon with an electromagnetism-related Klein\u2013Gordon equation with a negative mass term. The calculated maximal frequency value \u03c9=1.74\u00d71021 1\/s.<\/jats:p>","DOI":"10.3390\/sym16020144","type":"journal-article","created":{"date-parts":[[2024,1,26]],"date-time":"2024-01-26T04:05:46Z","timestamp":1706241946000},"page":"144","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Symmetry Breaking and Dynamic Transition in the Negative Mass Term Klein\u2013Gordon Equations"],"prefix":"10.3390","volume":"16","author":[{"given":"Ferenc","family":"M\u00e1rkus","sequence":"first","affiliation":[{"name":"Department of Physics, Budapest University of Technology and Economics, M\u0171egyetem rkp. 3, H-1111 Budapest, Hungary"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Katalin","family":"Gamb\u00e1r","sequence":"additional","affiliation":[{"name":"Department of Natural Sciences, Institute of Electrophysics, K\u00e1lm\u00e1n Kand\u00f3 Faculty of Electrical Engineering, \u00d3buda University, Tavaszmez\u0151 u. 17, H-1084 Budapest, Hungary"},{"name":"Department of Natural Sciences, National University of Public Service, Ludovika t\u00e9r 2, H-1083 Budapest, Hungary"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,1,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/0550-3213(74)90355-1","article-title":"Supergage Transformations in Four Dimensions","volume":"70","author":"Wess","year":"1974","journal-title":"Nucl. Phys. B"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0550-3213(74)90112-6","article-title":"Supergage Invariant Extension of Quantum Electrodynamics","volume":"78","author":"Wess","year":"1974","journal-title":"Nucl. Phys. B"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3316","DOI":"10.1103\/PhysRevD.32.3316","article-title":"Generalized Klein-Gordon equations in d dimensions from supersymmetry","volume":"32","author":"Bollini","year":"1985","journal-title":"Phys. Rev. D"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF02724327","article-title":"Classical Tachyons and Possible Applications","volume":"9","author":"Recami","year":"1986","journal-title":"Riv. Nuovo Cimento"},{"key":"ref_5","first-page":"345","article-title":"Simplified Deduction of the Field and the Forces of an Electron Moving in Any Given Way","volume":"7","author":"Sommerfeld","year":"1904","journal-title":"K. Akad. Wet. Amst. Proc."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"718","DOI":"10.1119\/1.1941773","article-title":"\u201cMeta\u201d Relativity","volume":"30","author":"Bilaniuk","year":"1962","journal-title":"Am. J. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1089","DOI":"10.1103\/PhysRev.159.1089","article-title":"Possibility of Faster-Than-Light Particles","volume":"159","author":"Feinberg","year":"1967","journal-title":"Phys. Rev."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1923","DOI":"10.1103\/PhysRevD.26.1923","article-title":"Tachyons and the Radiation of an Accelerated Charge","volume":"26","author":"Agudin","year":"1982","journal-title":"Phys. Rev. D"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"603","DOI":"10.1007\/BF02787229","article-title":"The Tachyon Propagator","volume":"106","author":"Barci","year":"1993","journal-title":"II Nuovo C. A"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"4169","DOI":"10.1142\/S0217751X94001692","article-title":"Higher Order Equations and Constituent Fields","volume":"23","author":"Barci","year":"1994","journal-title":"Int. J. Mod. Phys. A"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1016\/S0034-4877(08)80027-4","article-title":"A Simple Mechanical Model to Demonstrate a Dynamical Phase Transition","volume":"62","year":"2008","journal-title":"Rep. Math. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2065","DOI":"10.1007\/s10773-010-0392-4","article-title":"Wheeler Propagator of the Lorentz Invariant Thermal Energy Propagation","volume":"49","year":"2010","journal-title":"Int. J. Theor. Phys."},{"key":"ref_13","unstructured":"Ma, S.-K. (1982). Modern Theory of Critical Phenomena, Addison-Wesley."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Wei, C., and Li, A. (2019). Existence and multiplicity of solutions for Klein\u2013Gordon\u2013Maxwell systems with sign-changing potentials. Adv. Diff. Eq., 72.","DOI":"10.1186\/s13662-019-2020-9"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"175","DOI":"10.12700\/APH.17.1.2020.1.10","article-title":"A Repulsive Interaction in Classical Electrodynamics","volume":"17","author":"Rocca","year":"2020","journal-title":"Acta Polytechn. Hung."},{"key":"ref_16","unstructured":"Morse, P.M., and Feshbach, H. (1953). Methods of Theoretical Physics, McGraw-Hill."},{"key":"ref_17","unstructured":"Courant, R., and Hilbert, D. (1966). Methods of Mathematical Physics, Interscience."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"344","DOI":"10.1126\/science.1225244","article-title":"Anticipating Critical Transitions","volume":"338","author":"Scheffer","year":"2012","journal-title":"Science"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"4613","DOI":"10.1103\/PhysRevLett.84.4613","article-title":"Unusually High Thermal Conductivity of Carbon Nanotubes","volume":"84","author":"Berber","year":"2000","journal-title":"Phys. Rev. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2256","DOI":"10.1143\/JPSJ.72.2256","article-title":"Specific heat of Single-Walled Carbon Nanotubes: A Lattice Dynamics Study","volume":"72","author":"Cao","year":"2003","journal-title":"J. Phys. Soc. Jpn."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"073407","DOI":"10.1103\/PhysRevB.69.073407","article-title":"Thermal Conductivity of Zigzag Single-Walled Carbon Nanotubes: Role of the Umklapp Process","volume":"69","author":"Cao","year":"2004","journal-title":"Phys. Rev. B"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2122","DOI":"10.1016\/j.physa.2009.02.002","article-title":"Time Evolution of Thermodynamic Temperature in the Early Stage of Universe","volume":"388","year":"2009","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_23","unstructured":"Jackson, J.D. (1999). Classical Electrodynamics, John Wiley & Sons."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"2877","DOI":"10.1023\/A:1026605612226","article-title":"Wheeler Propagator","volume":"37","author":"Bollini","year":"1998","journal-title":"Int. J. Theor. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"777","DOI":"10.1023\/A:1026675811685","article-title":"Coupling of Tachyons to Electromagnetism","volume":"38","author":"Bollini","year":"1999","journal-title":"Int. J. Theor. Phys."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"553","DOI":"10.1073\/pnas.14.7.553","article-title":"An Interpretation of Dirac\u2019s Theory of the Electron","volume":"14","author":"Breit","year":"1928","journal-title":"Proc. Nat. Acad. Sci. USA"},{"key":"ref_27","first-page":"418","article-title":"\u00dcber die kr\u00e4ftefreie Bewegung in der relativistischen Quantenmechanik","volume":"24","year":"1930","journal-title":"Sitzungsber. Preuss. Akad. Wiss. Phys. Math. Kl."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1016\/0375-9601(89)90897-9","article-title":"On the Derivation of Zitterbewegung","volume":"137","author":"Roy","year":"1989","journal-title":"Phys. Lett. A"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"17007","DOI":"10.1209\/0295-5075\/89\/17007","article-title":"Study of Zitterbewegung in Graphene Bilayer with Perpendicular Magnetic Field","volume":"89","author":"Wang","year":"2010","journal-title":"Europhys. Lett."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"073011","DOI":"10.1088\/1367-2630\/15\/7\/073011","article-title":"Direct Observation of Zitterbewegung in a Bose\u2013Einstein Condensate","volume":"15","author":"LeBlanc","year":"2013","journal-title":"New J. Phys."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"126","DOI":"10.1038\/s41377-023-01162-x","article-title":"Observation of Zitterbewegung in Photonic Microcavities","volume":"12","author":"Lovett","year":"2023","journal-title":"Light Sci. Appl."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/2\/144\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T13:49:43Z","timestamp":1760104183000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/2\/144"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,1,26]]},"references-count":31,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2024,2]]}},"alternative-id":["sym16020144"],"URL":"https:\/\/doi.org\/10.3390\/sym16020144","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,1,26]]}}}