{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,24]],"date-time":"2026-09-24T22:02:34Z","timestamp":1790287354288,"version":"4.1.0"},"reference-count":20,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,1,14]],"date-time":"2021-01-14T00:00:00Z","timestamp":1610582400000},"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 Stern\u2013Gerlach experiment is notoriously counter-intuitive. The official theory is that the spin of a fermion remains always aligned with the magnetic field. Its directions are thus quantized: It can only be spin-up or spin-down. However, that theory is based on mathematical errors in the way it (mis)treats spinors and group theory. We present here a mathematically rigorous theory for a fermion in a magnetic field, which is no longer counter-intuitive. It is based on an understanding of spinors in SU(2) which is only Euclidean geometry. Contrary to what Pauli has been reading into the Stern\u2013Gerlach experiment, the spin directions are not quantized. The new corrected paradigm, which solves all conceptual problems, is that the fermions precess around the magnetic-field just as Einstein and Ehrenfest had conjectured. Surprisingly, this leads to only two energy states, which should be qualified as precession-up and precession-down rather than spin-up and spin-down. Indeed, despite the presence of the many different possible angles \u03b8 between the spin axis s and the magnetic field B, the fermions can only have two possible energies m0c2\u00b1\u03bcB. The values \u00b1\u03bcB thus do not correspond to the continuum of values \u2212\u03bc\u00b7B Einstein and Ehrenfest had conjectured. The energy term V=\u2212\u03bc\u00b7B is a macroscopic quantity. It is a statistical average over a large ensemble of fermions distributed over the two microscopic states with energies \u00b1\u03bcB, and as such not valid for individual fermions. The two fermion states with energy \u00b1\u03bcB are not potential-energy states. We also explain the mathematically rigorous meaning of the up and down spinors. They represent left-handed and right-handed reference frames, such that now everything is intuitively clear and understandable in simple geometrical terms. The paradigm shift does not affect the Pauli principle.<\/jats:p>","DOI":"10.3390\/sym13010134","type":"journal-article","created":{"date-parts":[[2021,1,15]],"date-time":"2021-01-15T01:33:29Z","timestamp":1610674409000},"page":"134","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["The Exact Theory of the Stern\u2013Gerlach Experiment and Why it Does Not Imply that a Fermion Can Only Have Its Spin Up or Down"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8283-2756","authenticated-orcid":false,"given":"Gerrit","family":"Coddens","sequence":"first","affiliation":[{"name":"Laboratoire des Solides Irradi\u00e9s, Institut Polytechnique de Paris, UMR 7642, CNRS-CEA- Ecole Polytechnique, Route de Saclay, F-91128 Palaiseau CEDEX, France"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Coddens, G. (2015). From Spinors to Quantum Mechanics, Imperial College Press.","DOI":"10.1142\/p990"},{"key":"ref_2","unstructured":"Coddens, G. (2020, August 29). Spinors for Everyone. Available online: https:\/\/hal.archives-ouvertes.fr\/cea-01572342v1."},{"key":"ref_3","unstructured":"Cartan, E. (1981). The Theory of Spinors, Dover."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Ballentine, L.E. (1998). Quantum Mechanics, A Modern Development, World Scientific. [2nd ed.].","DOI":"10.1142\/3142"},{"key":"ref_5","unstructured":"Coddens, G. (2020, August 29). A Linearly Polarized Electromagnetic Wave as a Swarm of Photons Half of Which Have Spin \u22121 and Half of Which Have Spin +1. Available online: https:\/\/hal.archives-ouvertes.fr\/hal-02636464v3."},{"key":"ref_6","unstructured":"Coddens, G. (2020, June 02). A Proposal to Get Some Common-Sense Intuition for the Paradox of the Double-Slit Experiment. Available online: https:\/\/hal.archives-ouvertes.fr\/cea-01383609v5."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"454","DOI":"10.1103\/RevModPhys.29.454","article-title":"\u201cRelative State\u201d Formulation of Quantum Mechanics","volume":"29","author":"Everett","year":"1957","journal-title":"Rev. Mod. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"166","DOI":"10.1103\/PhysRev.85.166","article-title":"A Suggested Interpretation of Quantum Theory in Terms of \u201cHidden\u201d Variables I","volume":"85","author":"Bohm","year":"1952","journal-title":"Phys. Rev."},{"key":"ref_9","first-page":"795","article-title":"The transactional interpretation of quantum mechanics","volume":"58","author":"Cramer","year":"2009","journal-title":"Rev. Mod. Phys."},{"key":"ref_10","unstructured":"Coddens, G. (2020, July 09). A Solution of the Paradox of the Double-Slit Experiment. Available online: https:\/\/hal.archives-ouvertes.fr\/cea-01459890v3."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1036","DOI":"10.1088\/0031-8949\/23\/6\/002","article-title":"Klein\u2019s Paradox and Its Resolution","volume":"23","author":"Hansen","year":"1981","journal-title":"Phys. Scripta"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1007\/BF01329580","article-title":"Der experimentelle Nachweis des magnetischen Moments des Silberatoms","volume":"8","author":"Gerlach","year":"1921","journal-title":"Z. Physik"},{"key":"ref_13","unstructured":"Villani, C. (2020, August 19). La th\u00e9orie Synth\u00e9tique de la Courbure de Ricci, 1\/7. Available online: https:\/\/www.youtube.com\/watch?v=xzVk56EKBUI."},{"key":"ref_14","unstructured":"Coddens, G. (2020, August 29). On Magnetic Monopoles, the Anomalous G-factor of the Electron and the Spin-Orbit Coupling in the Dirac Theory. Available online: https:\/\/hal.archives-ouvertes.fr\/cea-01269569v2."},{"key":"ref_15","unstructured":"Torre, C.G. (2008). Quantum Mechanics, Utah State University. Available online: http:\/\/www.physics.usu.edu\/torre\/QuantumMechanics\/6210_Spring_2008\/."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1007\/BF01328398","article-title":"Quantentheoretische Bemerkungen zum Experiment von Stern und Gerlach","volume":"11","author":"Einstein","year":"1922","journal-title":"Z. Physik"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Mezei, F. (1980). Neutron Spin Echo. Proceedings of a Laue-Langevin Institut Workshop Grenoble, Grenoble, France, 15\u201316 October 1979, Springer. Springer Lecture Notes in Physics 128.","DOI":"10.1007\/3-540-10004-0"},{"key":"ref_18","unstructured":"Purcell, E.M. (2020, August 09). The Thomas Precession. Available online: https:\/\/aapt.scitation.org\/doi\/10.1119\/1.1987061."},{"key":"ref_19","first-page":"263","article-title":"Zur Quanten-mechanik des Magnetischen Elektrons","volume":"37","author":"Pauli","year":"1926","journal-title":"Z. Physik"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Feynman, R.P., and Weinberg, S. (1987). Elementary Particles and the Laws of Physics, Cambridge University Press.","DOI":"10.1017\/CBO9781107590076"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/1\/134\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:11:18Z","timestamp":1760159478000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/1\/134"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,14]]},"references-count":20,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,1]]}},"alternative-id":["sym13010134"],"URL":"https:\/\/doi.org\/10.3390\/sym13010134","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1,14]]}}}