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Hier finden Sie wissenschaftliche Publikationen aus den FraunhoferInstituten. Dynamical correspondence in a generalized quantum theory
 Foundations of physics 45 (2015), No.5, pp.525534 ISSN: 00159018 (Print) ISSN: 15729516 (Online) 

 English 
 Journal Article 
 Fraunhofer ESK () 
Abstract
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 JBWalgebras). 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 thirdorder interference. The conditional probability calculus is a mathematical model of the Lüdersvon Neumann quantum measurement process, and thirdorder interference is a property of the conditional probabilities which was discovered by Sorkin (Mod Ph ys Lett A 9:31193127, 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.