Abstract
Starting from an abstract setting for the Lüders-von Neumann quantum measurement process and its interpretation as a probability conditionalization rule in a non-Boolean event structure, the author derived a certain generalization of operator algebras in a preceding paper. This is an order-unit space with some specific properties. It becomes a Jordan operator algebra under a certain set of additional conditions, but does not own a multiplication operation in the most general case. A major objective of the present paper is the search for such examples of the structure mentioned above that do not stem from Jordan operator algebras; first natural candidates are matrix algebras over the octonions and other nonassociative rings. Therefore, the case when a nonassociative commutative multiplication exists is studied without assuming that it satisfies the Jordan condition. The characteristics of the resulting algebra are analyzed. This includes the uniqueness of the spectral resolution as well as a criterion for its existence, subalgebras that are Jordan algebras, associative subalgebras, and more different levels of compatibility than occurring in standard quantum mechanics. However, the paper cannot provide the desired example, but contribute to the search by the identification of some typical differences between the potential examples and the Jordan operator algebras and by negative results concerning some first natural candidates. The possibility that no such example exists cannot be ruled out. However, this would result in an unexpected new characterization of Jordan operator algebras, which would have a significant impact on quantum axiomatics since some customary axioms (e.g., power-associativity or the sum postulate for observables) might turn out to be redundant then.
Similar content being viewed by others
References
Baez, J.C.: The octonions. Bull. (New Ser.) Am. Math. Soc. 39, 145–205 (2001)
Hanche-Olsen, H., Størmer, E.: Jordan Operator Algebras. Pitmann, Boston (1984)
Iochum, B., Loupias, G.: Banach-power-associative algebras and J-B algebras. Ann. Inst. H. Poincaré 43, 211–225 (1985)
Jordan, P., von Neumann, J., Wigner, E.: On an algebraic generalization of the quantum mechanical formalism. Ann. Math. 35, 29–64 (1934)
Niestegge, G.: An approach to quantum mechanics via conditional probabilities. Found. Phys. 38, 241–256 (2008)
Niestegge, G.: A representation of quantum measurement in order-unit spaces. Found. Phys. 38, 783–795 (2008)
Schafer, R.: An introduction to nonassociative algebras. Academic Press, New York (1966)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Niestegge, G. A Representation of Quantum Measurement in Nonassociative Algebras. Found Phys 39, 120–136 (2009). https://doi.org/10.1007/s10701-008-9264-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10701-008-9264-7