{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,27]],"date-time":"2025-10-27T16:01:19Z","timestamp":1761580879641,"version":"build-2065373602"},"reference-count":33,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2011,4,27]],"date-time":"2011-04-27T00:00:00Z","timestamp":1303862400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Quantum theory is a probabilistic calculus that enables the calculation of the probabilities of the possible outcomes of a measurement performed on a physical system. But what is the relationship between this probabilistic calculus and probability theory itself? Is quantum theory compatible with probability theory? If so, does it extend or generalize probability theory? In this paper, we answer these questions, and precisely determine the relationship between quantum theory and probability theory, by explicitly deriving both theories from first principles. In both cases, the derivation depends upon identifying and harnessing the appropriate symmetries that are operative in each domain. We prove, for example, that quantum theory is compatible with probability theory by explicitly deriving quantum theory on the assumption that probability theory is generally valid.<\/jats:p>","DOI":"10.3390\/sym3020171","type":"journal-article","created":{"date-parts":[[2011,4,27]],"date-time":"2011-04-27T11:24:04Z","timestamp":1303903444000},"page":"171-206","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["Quantum Theory and Probability Theory: Their Relationship and Origin in Symmetry"],"prefix":"10.3390","volume":"3","author":[{"given":"Philip","family":"Goyal","sequence":"first","affiliation":[{"name":"Department of Physics, University at Albany (SUNY), 1400 Washington Avenue, Albany, NY 12222, USA"}]},{"given":"Kevin H.","family":"Knuth","sequence":"additional","affiliation":[{"name":"Departments of Physics and Informatics, University at Albany (SUNY), 1400 Washington Avenue, Albany, NY 12222, USA"}]}],"member":"1968","published-online":{"date-parts":[[2011,4,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1103\/RevModPhys.20.367","article-title":"Space-time approach to non-relativistic quantum mechanics","volume":"20","author":"Feynman","year":"1948","journal-title":"Rev. Mod. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1119\/1.1990764","article-title":"Probability, frequency, and reasonable expectation","volume":"14","author":"Cox","year":"1946","journal-title":"Am. J. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Cox, R.T. (1961). The Algebra of Probable Inference, The Johns Hopkins Press.","DOI":"10.56021\/9780801869822"},{"key":"ref_4","unstructured":"Kolmogorov, A.N. (1933). Foundations of Probability Theory, Julius Springer."},{"key":"ref_5","unstructured":"Boole, G. (1854). An Investigation of the Laws of Thought, Macmillan."},{"key":"ref_6","unstructured":"Erickson, G.J., and Zhai, Y. (2004). Bayesian Inference and Maximum Entropy Methods in Science and Engineering, Proceedings of 23rd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, American Institute of Physics."},{"key":"ref_7","first-page":"132","article-title":"Measuring on lattices","volume":"Volume 707","author":"Goggans","year":"2004","journal-title":"Bayesian Inference and Maximum Entropy Methods in Science and Engineering, Proceedings of 23rd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Knuth, K.H. (, January July). Valuations on lattices and their application to information theory. Proceedings of the 2006 IEEE World Congress on Computational Intelligence, Vancouver, Canada.","DOI":"10.1109\/FUZZY.2006.1681717"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1016\/j.neucom.2004.11.039","article-title":"Lattice duality: The origin of probability and entropy","volume":"67C","author":"Knuth","year":"2005","journal-title":"Neurocomputing"},{"key":"ref_10","unstructured":"Knuth, K.H., and Skilling, J. (2011, April 27). The foundations of inference. Available online: http:\/\/arxiv.org\/abs\/1008.4831."},{"key":"ref_11","first-page":"022109:01","article-title":"Origin of complex quantum amplitudes and feynman\u2019s rules","volume":"A81","author":"Goyal","year":"2010","journal-title":"Phys. Rev. A"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1086\/286465","article-title":"Causality and complementarity","volume":"4","author":"Bohr","year":"1937","journal-title":"Philos. Sci."},{"key":"ref_13","unstructured":"Acz\u00e9l, J. (1966). Lectures on Functional Equations and Their Applications, Academic Press."},{"key":"ref_14","unstructured":"Readers familiar with lattice (order) theory, will observe that bivaluations represent a generalization of the zeta function [31,32], which is an indicator function for the Boolean lattice where\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t           \t\t  \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\u03b6\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tX\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\u00a0\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\twhen\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tX\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\u2192\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\u03b6\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tX\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\u00a0\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\totherwise\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t      \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  noting that the arguments of the zeta function are swapped with respect to the definition of a bivaluation. Furthermore, the sum rule is to be identified with the inclusion-exclusion relation."},{"key":"ref_15","first-page":"361","article-title":"Quantisation as an eigenvalue problem","volume":"79","author":"Schroedinger","year":"1926","journal-title":"Ann. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"879","DOI":"10.1007\/BF01328377","article-title":"Quantum-theoretical re-interpretation of kinematic and mechanical relations","volume":"33","author":"Heisenberg","year":"1925","journal-title":"Z. Phys."},{"key":"ref_17","unstructured":"Dirac, P. (1999). Principles of Quantum Mechanics, Oxford Science Publications. [4th ed.]."},{"key":"ref_18","unstructured":"von Neumann, J. (1955). Mathematical Foundations of Quantum Mechanics, Princeton University Press."},{"key":"ref_19","unstructured":"In practice, one would probably use a single Geiger counter to detect the electrons falling on a small patch of the screen, and then move the detector over the screen to build up the intensity pattern over the screen. However, in this thought-experiment, we help ourselves to more sophisticated equipment."},{"key":"ref_20","unstructured":"The resolution time of a detector is the smallest interval of time between two incident electrons for which two distinct output pulses will be obtained from the detector."},{"key":"ref_21","unstructured":"Alternatively, we can replace the single wire-loop detector at B with two finer-grained detectors, one placed in front of each of the slits, which are capable of indicating passage through one slit or the other."},{"key":"ref_22","unstructured":"We remark that, in the deBroglie\u2013Bohm interpretation of quantum theory, the model of an electron has two distinct components: (i) a discrete entity (an \u201cindicator particle\u201d), and (ii) a delocalized wave, which determines the motion of the discrete entity. Since the \u201celectron\u201d does not consist solely of a highly localized object in such a hybrid model, one would not infer that proposition B implies B1 \u2228 B2, and one would therefore not infer Equation (52). If, instead, one were to redefine the Bi to refer to the indicator particle alone, one could infer (52), but one would not be able to infer that Pr(C, B1|A) is unaffected by the closure of slit B2 (as the closure of B2 would be expected to affect the wave component), and hence one could not subject Equation (52) to the experimental test given above. Thus, in either interpretation of the Bi, the de Broglie\u2013Bohm model of an electron would not be ruled out by the experimental test mentioned above."},{"key":"ref_23","unstructured":"In this model, the electron waves are taken to move at speed, v, at which speed particle-like electrons would be expected to move, with the wavelength of the waves set equal to the de Broglie wavelength of the electrons which, for v \u226a c, is \u03bb = h\/mv, where h is Planck\u2019s constant, and m is the mass of the electron."},{"key":"ref_24","unstructured":"As mentioned in the Introduction, our choice of representation is inspired by Bohr\u2019s principle of complementarity. We are investigating other ways of understanding the origin of the pair representation."},{"key":"ref_25","unstructured":"This condition is inspired by another condition suggested by J. Skilling in discussion with one of us."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"823","DOI":"10.2307\/1968621","article-title":"The logic of quantum mechanics","volume":"37","author":"Birkhoff","year":"1936","journal-title":"Ann. Math."},{"key":"ref_27","unstructured":"Gudder, S. (1988). Quantum Probability, Academic Press."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1016\/j.jmp.2006.01.003","article-title":"Quantum dynamics of human decision making","volume":"50","author":"Busemeyer","year":"2006","journal-title":"J. Math. Psychol."},{"key":"ref_29","first-page":"2171","article-title":"A quantum probability model explanation for violations of \u201crational\u201d decision theory","volume":"276","author":"Pothos","year":"2009","journal-title":"Proc. R. Soc. Lond. B"},{"key":"ref_30","unstructured":"Knuth, K.H., and Bahreyni, N. (2011, April 27). A derivation of special relativity from causal sets. Available online: http:\/\/arxiv.org\/abs\/1005.4172."},{"key":"ref_31","first-page":"340","article-title":"On the foundations of combinatorial theory I. Theory of M\u00f6bius functions","volume":"2","author":"Rota","year":"1964","journal-title":"Prob. Theor. and Related Fields"},{"key":"ref_32","first-page":"204","article-title":"Deriving Laws from Ordering Relations","volume":"Volume 707","author":"Erickson","year":"2003","journal-title":"Bayesian Inference and Maximum Entropy Methods in Science and Engineering in Science and Engineering, Proceedings of 23rd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering"},{"key":"ref_33","unstructured":"van der Waerden, B.L. (1967). Sources of Quantum Mechanics, Dover Publications."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/3\/2\/171\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:55:55Z","timestamp":1760219755000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/3\/2\/171"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,4,27]]},"references-count":33,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2011,6]]}},"alternative-id":["sym3020171"],"URL":"https:\/\/doi.org\/10.3390\/sym3020171","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2011,4,27]]}}}