Dynamical Correspondence in a Generalized Quantum Theory

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• 作者：Gerd Niestegge
• 关键词：Order derivations ; Positive groups ; Operator algebras ; Lie algebras ; Foundations of quantum mechanics ; 46L70 ; 81P10
• 刊名：Foundations of Physics
• 出版年：2015
• 出版时间：May 2015
• 年：2015
• 卷：45
• 期：5
• 页码：525-534
• 全文大小：155 KB
• 参考文献：1. Alfsen, EM, Shultz, FW (1998) On orientation and dynamics in operator algebras (Part I). Commun. Math. Phys. 194: pp. 87-108 CrossRef
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  10. Niestegge, G (2001) Non-Boolean probabilities and quantum measurement. J. Phys. A 34: pp. 6031-6042 CrossRef
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  12. Niestegge, G (2012) Conditional probability, three-slit experiments, and the Jordan algebra structure of quantum mechanics. Adv. Math. Phys. 2012: pp. 156573 CrossRef
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• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Physics
  Quantum Physics
  Relativity and Cosmology
  Biophysics and Biomedical Physics
  Mechanics
  Condensed Matter
  
• 出版者：Springer Netherlands
• ISSN：1572-9516

文摘

In order to figure out why quantum physics needs the complex Hilbert space, many attempts have been made to distinguish the C*-algebras and von Neumann algebras in more general classes of abstractly defined Jordan algebras (JB- and JBW-algebras). One particularly important distinguishing property was identified by Alfsen and Shultz and is the existence of a dynamical correspondence. It reproduces the dual role of the selfadjoint operators as observables and generators of dynamical groups in quantum mechanics. In the paper, this concept is extended to another class of nonassociative algebras, arising from recent studies of the quantum logics with a conditional probability calculus and particularly of those that rule out third-order interference. The conditional probability calculus is a mathematical model of the Lüders–von Neumann quantum measurement process, and third-order interference is a property of the conditional probabilities which was discovered by Sorkin (Mod Phys Lett A 9:3119-127, 1994) and which is ruled out by quantum mechanics. It is shown then that the postulates that a dynamical correspondence exists and that the square of any algebra element is positive still characterize, in the class considered, those algebras that emerge from the selfadjoint parts of C*-algebras equipped with the Jordan product. Within this class, the two postulates thus result in ordinary quantum mechanics using the complex Hilbert space or, vice versa, a genuine generalization of quantum theory must omit at least one of them.