Article
A Representation of Quantum Measurement in OrderUnit Spaces
Foundations of Physics (Impact Factor: 1.03). 01/2010; 38(9). DOI: 10.1007/s107010089236y
Source: arXiv
ABSTRACT
A certain generalization of the mathematical formalism of quantum mechanics beyond operator algebras is considered. The approach is based on the concept of conditional probability and the interpretation of the Lueders  von Neumann quantum measurement as a probability conditionalization rule. A major result shows that the operator algebras must be replaced by orderunit spaces with some specific properties in the generalized approach, and it is analyzed under which conditions these orderunit spaces become Jordan algebras. An application of this result provides a characterization of the projection lattices in operator algebras. Comment: 12 pages, the original publication is available at htt://www.springerlink.com
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 "A quantum logic with a sufficiently rich state space as postulated by (UC1) can be embedded in the unit interval of an orderunit space. In the present section, it will be shown that the existence and the uniqueness of the conditional probabilities postulated by (UC2) give rise to some important additional structure on this orderunit space, which was originally presented in [14]. A partially ordered real vector space A is an orderunit space if A contains an Archimedean order unit 1 [1], [4], [10]. "
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ABSTRACT: Most quantum logics do not allow for a reasonable calculus of conditional probability. However, those ones which do so provide a very general and rich mathematical structure, including classical probabilities, quantum mechanics as well as Jordan algebras. This structure exhibits some similarities with Alfsen and Shultz's noncommutative spectral theory, but these two mathematical approaches are not identical. Barnum, Emerson and Ududec adapted the concept of higherorder interference, introduced by Sorkin in 1994, into a general probabilistic framework. Their adaption is used here to reveal a close link between the existence of the Jordan product and the nonexistence of interference of third or higher order in those quantum logics which entail a reasonable calculus of conditional probability. The complete characterization of the Jordan algebraic structure requires the following three further postulates: a HahnJordan decomposition property for the states, a polynomial functional calculus for the observables, and the positivity of the square of an observable. While classical probabilities are characterized by the absence of any kind of interference, the absence of interference of third (and higher) order thus characterizes a probability calculus which comes close to quantum mechanics, but still includes the exceptional Jordan algebras. 
Article: Sorkin's thirdorder interference term in quantum logics with unique conditional probabilities
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ABSTRACT: Considering not only the wellknown twoslit experiment, but also experiments with three slits, Sorkin introduced the thirdorder interference term I3 and discovered that the absence of thirdorder interference (I3=0) is typical of quantum mechanics where only secondorder interference occurs. In the present paper, the interference term I3 is ported to the quantum logics with unique conditional probabilities. In this framework, the identity I3=0 does not hold in general and its consequences are analysed. A first result reveals a close link between this identity and the existence of a product in the orderunit space generated by the quantum logic. In the general case, this product is neither commutative nor associative. By a second result, the orderunit space becomes a Jordan algebra, if each element behaves like one would expect from an observable (i.e., its square is positive and there is a polynomial functional calculus). Almost all such Jordan algebras can be represented as operator algebras on a Hilbert space, and a reconstruction of quantum mechanics up to this point is thus achieved from the absence of thirdorder interference and a few other principles. Besides the identity I3=0, two further interesting properties of quantum mechanics distinguishing it from more general theories are studied. These are a novel bound for quantum interference and a symmetry condition for the conditional probabilities.  [Show abstract] [Hide abstract]
ABSTRACT: In the quantum mechanical Hilbert space formalism, the probabilistic interpretation is a later adhoc addon, more or less enforced by the experimental evidence, but not motivated by the mathematical model itself. A model involving a clear probabilistic interpretation from the very beginning is provided by the quantum logics with unique conditional probabilities. It includes the projection lattices in von Neumann algebras and here probability conditionalization becomes identical with the state transition of the Lueders  von Neumann measurement process. This motivates the definition of a hierarchy of five compatibility and comeasurability levels in the abstract setting of the quantum logics with unique conditional probabilities. Their meanings are: the absence of quantum interference or influence, the existence of a joint distribution, simultaneous measurability, and the independence of the final state after two successive measurements from the sequential order of these two measurements. A further level means that two elements of the quantum logic (events) belong to the same Boolean subalgebra. In the general case, the five compatibility and comeasurability levels appear to differ, but they all coincide in the common Hilbert space formalism of quantum mechanics, in von Neumann algebras, and in some other cases. Comment: 12 pages