{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,31]],"date-time":"2025-12-31T17:51:00Z","timestamp":1767203460185,"version":"build-2238731810"},"reference-count":31,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2019,12,31]],"date-time":"2019-12-31T00:00:00Z","timestamp":1577750400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["www.mdpi.com"],"crossmark-restriction":true},"short-container-title":["Entropy"],"abstract":"<jats:p>In 2007, and in a series of later papers, Joy Christian claimed to refute Bell\u2019s theorem, presenting an alleged local realistic model of the singlet correlations using techniques from geometric algebra (GA). Several authors published papers refuting his claims, and Christian\u2019s ideas did not gain acceptance. However, he recently succeeded in publishing yet more ambitious and complex versions of his theory in fairly mainstream journals. How could this be? The mathematics and logic of Bell\u2019s theorem is simple and transparent and has been intensely studied and debated for over 50 years. Christian claims to have a mathematical counterexample to a purely mathematical theorem. Each new version of Christian\u2019s model used new devices to circumvent Bell\u2019s theorem or depended on a new way to misunderstand Bell\u2019s work. These devices and misinterpretations are in common use by other Bell critics, so it useful to identify and name them. I hope that this paper can serve as a useful resource to those who need to evaluate new \u201cdisproofs of Bell\u2019s theorem\u201d. Christian\u2019s fundamental idea is simple and quite original: he gives a probabilistic interpretation of the fundamental GA equation     a \u00b7 b = ( a b + b a ) \/ 2    . After that, ambiguous notation and technical complexity allows sign errors to be hidden from sight, and new mathematical errors can be introduced.<\/jats:p>","DOI":"10.3390\/e22010061","type":"journal-article","created":{"date-parts":[[2019,12,31]],"date-time":"2019-12-31T07:40:55Z","timestamp":1577778055000},"page":"61","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Does Geometric Algebra Provide a Loophole to Bell\u2019s Theorem?"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5821-9986","authenticated-orcid":false,"given":"Richard David","family":"Gill","sequence":"first","affiliation":[{"name":"Mathematical Institute, Faculty of Science, Leiden University, Rapenburg 70, 2311 EZ Leiden, The Netherlands"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,12,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1103\/PhysicsPhysiqueFizika.1.195","article-title":"On the Einstein Podolsky Rosen Paradox","volume":"1","author":"Bell","year":"1964","journal-title":"Physics 1"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"777","DOI":"10.1103\/PhysRev.47.777","article-title":"Can quantum-mechanical description of physical reality be considered complete?","volume":"16","author":"Einstein","year":"1935","journal-title":"Phys. Rev."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1070","DOI":"10.1103\/PhysRev.108.1070","article-title":"Discussion of experimental proof for the paradox of Einstein, Rosen and Podolsky","volume":"108","author":"Bohm","year":"1957","journal-title":"Phys. Rev."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1103\/PhysRev.77.136","article-title":"The angular correlation of scattered annihilation radiation","volume":"77","author":"Wu","year":"1950","journal-title":"Phys. Rev."},{"key":"ref_5","unstructured":"Christian, J. (2014). Disproof of Bell\u2019s Theorem: Illuminating the Illusion of Entanglement, Brown Walker Press."},{"key":"ref_6","unstructured":"Christian, J. (2007). Disproof of Bell\u2019s theorem by Clifford algebra valued local variables. arXiv, Available online: https:\/\/arxiv.org\/abs\/quant-ph\/0703179."},{"key":"ref_7","unstructured":"Christian, J. (2011). Disproof of Bell\u2019s theorem. arXiv."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2042","DOI":"10.1007\/s10773-014-2412-2","article-title":"Macroscopic Observability of Spinorial Sign Changes under 2\u03c0 Rotations","volume":"54","author":"Christian","year":"2015","journal-title":"Int. J. Theor. Phys."},{"key":"ref_9","unstructured":"Christian, J. (2014). Local Causality in a Friedmann-Robertson-Walker Spacetime. arXiv, Available online: https:\/\/arxiv.org\/pdf\/1405.2355v1.pdf."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"180526","DOI":"10.1098\/rsos.180526","article-title":"Quantum correlations are weaved by the spinors of the Euclidean primitives","volume":"5","author":"Christian","year":"2018","journal-title":"R. Soc. Open Sci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"133388","DOI":"10.1109\/ACCESS.2019.2941275","article-title":"Bell\u2019s Theorem Versus Local Realism in a Quaternionic Model of Physical Space","volume":"2019","author":"Christian","year":"2019","journal-title":"IEEE Access"},{"key":"ref_12","unstructured":"Christian, J. (2019). Eight-dimensional Octonion-like but Associative Normed Division Algebra. arXiv, Available online: https:\/\/arxiv.org\/abs\/1908.06172."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Doran, C., and Lasenby, A. (2003). Geometric Algebra for Physicists, Cambridge University Press.","DOI":"10.1017\/CBO9780511807497"},{"key":"ref_14","first-page":"1","article-title":"A unified language for mathematics and physics","volume":"Volume 183","author":"Hestenes","year":"1986","journal-title":"Clifford Algebras and their Applications in Mathematical Physics"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1007\/978-94-009-4728-3_27","article-title":"Clifford algebra and the interpretation of quantum mechanics","volume":"Volume 183","author":"Hestenes","year":"1986","journal-title":"Clifford Algebras and their Applications in Mathematical Physics"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Dorst, L., Fontijne, D., and Mann, S. (2009). Geometric Algebra for Computer Science, Elsevier.","DOI":"10.1016\/B978-0-12-374942-0.00006-3"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1997","DOI":"10.1109\/ACCESS.2016.2538262","article-title":"The Vector Algebra War: A Historical Perspective","volume":"4","author":"Chappell","year":"2016","journal-title":"IEEE Access"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"880","DOI":"10.1103\/PhysRevLett.23.880","article-title":"Proposed experiment to test local hidden-variable theories","volume":"23","author":"Clauser","year":"1969","journal-title":"Phys. Rev. Lett."},{"key":"ref_19","unstructured":"Boole, G. (1854). An Investigation of the Laws of Thought, Walton and Maberly."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1007\/BF00417500","article-title":"Quantum generalizations of Bell\u2019s inequality","volume":"4","year":"1980","journal-title":"Lett. Math. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1103\/PhysRevLett.48.291","article-title":"Hidden Variables, Joint Probability, and the Bell Inequalities","volume":"48","author":"Fine","year":"1982","journal-title":"Phys. Rev. Lett."},{"key":"ref_22","unstructured":"Gill, R.D. (2013). The triangle wave versus the cosine (how to optimally approximate EPR-B Correlations by classical systems). arXiv, Available online: https:\/\/arxiv.org\/abs\/1312.6403."},{"key":"ref_23","unstructured":"Pitowsky, I. (1989). Quantum Probability\u2013Quantum Logic, Springer. Available online: https:\/\/www.springer.com\/gp\/book\/9783662137352."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"512","DOI":"10.1214\/14-STS490","article-title":"Statistics, Causality and Bell\u2019s Theorem","volume":"29","author":"Gill","year":"2014","journal-title":"Stat. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1214\/lnms\/1215091935","article-title":"Accardi contra Bell (cum mundi): The Impossible Coupling","volume":"Volume 42","author":"Gill","year":"2003","journal-title":"Lecture Notes\u2013Monograph Series"},{"key":"ref_26","unstructured":"Gill, R.D. (2003). Time, Finite Statistics, and Bell\u2019s Fifth Position. arXiv, Available online: https:\/\/arxiv.org\/abs\/quant-ph\/0301059."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Lakatos, I. (1976). Proofs and Refutations: The Logic of Mathematical Discovery, Cambridge University Press.","DOI":"10.1017\/CBO9781139171472"},{"key":"ref_28","unstructured":"Gill, R.D. (2012). Simple refutation of Joy Christian\u2019s simple refutation of Bell\u2019s simple theorem. arXiv, Available online: https:\/\/arxiv.org\/abs\/1203.1504."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1007\/s10773-015-2657-4","article-title":"Macroscopic Unobservability of Spinorial Sign Changes","volume":"55","author":"Gill","year":"2016","journal-title":"Int. J. Theor. Phys."},{"key":"ref_30","first-page":"1418","article-title":"Hidden-variable example based upon data rejection","volume":"2","author":"Pearle","year":"1970","journal-title":"Phys. Rev."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Gill, R.D. (2019). Pearle\u2019s Hidden-Variable Model Revisited. Entropy, 22.","DOI":"10.3390\/e22010001"}],"updated-by":[{"DOI":"10.3390\/e23050631","type":"correction","label":"Correction","source":"publisher","updated":{"date-parts":[[2019,12,31]],"date-time":"2019-12-31T00:00:00Z","timestamp":1577750400000}}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/1\/61\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,8,3]],"date-time":"2025-08-03T22:11:37Z","timestamp":1754259097000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/1\/61"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,12,31]]},"references-count":31,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,1]]}},"alternative-id":["e22010061"],"URL":"https:\/\/doi.org\/10.3390\/e22010061","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,12,31]]}}}