Why Do the Quantum Observables Form a Jordan Operator Algebra?
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The Jordan algebra structure of the bounded real quantum observables was recognized already in the early days of quantum mechanics. While there are plausible reasons for most parts of this structure, the existence of the distributive nonassociative multiplication operation is hard to justify from a physical or statistical point of view. Considering the non-Boolean extension of classical probabilities, presented in a recent paper, it is shown in this paper that such a multiplication operation can be derived from certain properties of the conditional probabilities and the observables, i.e., from postulates with a clear statistical interpretation. The well-known close relation between Jordan operator algebras and C*-algebras then provides the connection to the quantum-mechanical Hilbert space formalism, thus resulting in a novel axiomatic approach to general quantum mechanics that includes the types II and III von Neumann algebras.
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- Alfsen, E. M., Shultz, F. W., and Størmer, E. (1978). A Gelfand–Neumark theorem for Jordan algebras. Advances in Mathematics 28, 11-56.Google Scholar
- Alfsen, E. M. and Shultz, F. W. (1979). On non-commutative spectral theory and Jordan algebras. Proceedings of the London Mathematical Society 38, 497-516.Google Scholar
- Birkhoff, G. and von Neumann, J. (1936). The logic of quantum mechanics. Annals of Mathematics 37, 823-843.Google Scholar
- Gudder, S. P. (1979). Stochastic Methods in Quantum Mechanics, North-Holland, Amsterdam.Google Scholar
- Gunson, J. (1967). On the algebraic structure of quantum mechanics. Communications in Mathematical Physics 6, 262-285.Google Scholar
- Guz, W. (1981). Conditional probability and the axiomatic structure of quantum mechanics. Fortschritte der Physik 29, 345-379.Google Scholar
- Hanche-Olsen, H. and Størmer, E. (1984). Jordan Operator Algebras, Pitmann, Boston.Google Scholar
- Jordan, P., von Neumann, J., and Wigner, E., (1934). On an algebraic generalization of the quantum mechanical formalism. Annals of Mathematics 35, 29-64.Google Scholar
- Keller, H. (1980). Ein nicht-klassischer Hilbertscher Raum. Mathematische Zeitschrift 172, 41-49.Google Scholar
- Niestegge, G. (2001). Non-Boolean probabilities and quantum measurement. Journal of Physics A: Mathematical and General 34, 6031-6042.Google Scholar
- Piron, C. (1964). Axiomatique quantique. Helvetica Physica Acta 37, 439-468.Google Scholar
- Pták, P. and Pulmannová, S. (1991). Orthomodular Structures as Quantum Logics, Kluwer, Dordrecht.Google Scholar
- Pulmannová, S. (1998). Quantum logics and convex spaces. International Journal of Theoretical Physics 37, 2303-2332.Google Scholar
- Segal, I. E. (1947). Postulates for general quantum mechanics. Annals of Mathematics 48, 930-938.Google Scholar
- Sherman, S. (1956). On Segal's postulates for general quantum mechanics. Annals of Mathematics 64, 593-601.Google Scholar
- Varadarajan, V. S. (1968/70). Geometry of Quantum Theory I and II, Van Nostrand, New York.Google Scholar