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# Quantum Theory and Probability Theory: Their Relationship and Origin in Symmetry

^{1}Department of Physics, University at Albany (SUNY), 1400 Washington Avenue, Albany, NY 12222, USA

^{2}Departments of Physics and Informatics, University at Albany (SUNY), 1400 Washington Avenue, Albany, NY 12222, USA

* Author to whom correspondence should be addressed.

Received: 9 March 2011 / Revised: 6 April 2011 / Accepted: 12 April 2011 / Published: 27 April 2011

(This article belongs to the Special Issue Quantum Symmetry)

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# Abstract

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.*Keywords:*quantum theory; probability theory; foundations of quantum theory; foundations of probability theory

*This is an open access article distributed under the Creative Commons Attribution License (CC BY) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.*

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**MDPI and ACS Style**

Goyal, P.; Knuth, K.H. Quantum Theory and Probability Theory: Their Relationship and Origin in Symmetry. *Symmetry* **2011**, *3*, 171-206.

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