International Journal of Theoretical Physics

, Volume 43, Issue 1, pp 35–46 | Cite as

Why Do the Quantum Observables Form a Jordan Operator Algebra?

  • Gerd Niestegge
Article

Abstract

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.

foundations of quantum mechanics quantum probability quantum logic Jordan operator algebras 

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Copyright information

© Plenum Publishing Corporation 2004

Authors and Affiliations

  • Gerd Niestegge
    • 1
  1. 1.MuenchenGermany

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