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Palgrave Macmillan eBooks

Quantum Probabilities

Published Nov 7, 2014• 0 citations
Publisher

Abstract

There are well known mathematical models of quantum structures ([3]) including probability aspects of the theories. One of the successful ways for the theories can be in using MV-algebras ([2]). Recall that MV-algebras play in multi-valued logics the same role as Boolean algebras in two-valued logics. The probability theory on MV-algebras was reviewed in [8] and [9]. The aim of this contribution is in using the methods of MV-algebra probability theory in some structures dierent

DOI
10.1057/9781137447159.0011

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Let [Formula: see text] be a set of states of a physical system. The probabilities [Formula: see text] of the occurrence of an event when the system is in different states [Formula: see text] define a function from [Formula: see text] to [Formula: see text] called a numerical event or, more accurately, an [Formula: see text]-probability. Sets of [Formula: see text]-probabilities ordered by the partial order of functions give rise to so-called algebras of [Formula: see text]-probabilities, in particular, to the ones that are lattice-ordered. Among these, there are the [Formula: see text]-algebras known from probability theory and the Hilbert-space logics which are important in quantum-mechanics. Any algebra of [Formula: see text]-probabilities can serve as a quantum-logic, and it is of special interest when this logic turns out to be a Boolean algebra because then the observed physical system will be classical. Boolean algebras are in one-to-one correspondence to Boolean rings, and the question arises to find an analogue correspondence for lattice-ordered algebras of [Formula: see text]-probabilities generalizing the correspondence between Boolean algebras and Boolean rings. We answer this question by defining ring-like structures of events (RLSEs). First, the structure of RLSEs is revealed and Boolean rings among RLSEs are characterized. Then we establish how RLSEs correspond to lattice-ordered algebras of numerical events. Further, functions for associating lattice-ordered algebras of [Formula: see text]-probabilities to RLSEs are studied. It is shown that there are only two ways to assign lattice-ordered algebras of [Formula: see text]-probabilities to RLSEs if one restricts the corresponding mappings to term functions over the underlying orthomodular lattice. These term functions are the very functions by which also the Boolean algebras can be assigned to Boolean rings.

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There are well known mathematical models of quantum structures ([3]) including probability aspects of the theories. One of the successful ways for the theories can be in using MV-algebras ([2]). Recall that MV-algebras play in multi-valued logics the same role as Boolean algebras in two-valued logics. The probability theory on MV-algebras was reviewed in [8] and [9]. The aim of this contribution is in using the methods of MV-algebra probability theory in some structures dierent
Let [Formula: see text] be a set of states of a physical system. The probabilities [Formula: see text] of the occurrence of an event when the system is in different states [Formula: see text] define a function from [Formula: see text] to [Formula: see text] called a numerical event or, more accurately, an [Formula: see text]-probability. Sets of [Formula: see text]-probabilities ordered by the partial order of functions give rise to so-called algebras of [Formula: see text]-probabilities, in particular, to the ones that are lattice-ordered. Among these, there are the [Formula: see text]-algebras known from probability theory and the Hilbert-space logics which are important in quantum-mechanics. Any algebra of [Formula: see text]-probabilities can serve as a quantum-logic, and it is of special interest when this logic turns out to be a Boolean algebra because then the observed physical system will be classical. Boolean algebras are in one-to-one correspondence to Boolean rings, and the question arises to find an analogue correspondence for lattice-ordered algebras of [Formula: see text]-probabilities generalizing the correspondence between Boolean algebras and Boolean rings. We answer this question by defining ring-like structures of events (RLSEs). First, the structure of RLSEs is revealed and Boolean rings among RLSEs are characterized. Then we establish how RLSEs correspond to lattice-ordered algebras of numerical events. Further, functions for associating lattice-ordered algebras of [Formula: see text]-probabilities to RLSEs are studied. It is shown that there are only two ways to assign lattice-ordered algebras of [Formula: see text]-probabilities to RLSEs if one restricts the corresponding mappings to term functions over the underlying orthomodular lattice. These term functions are the very functions by which also the Boolean algebras can be assigned to Boolean rings.
We argue that limitations arise when using non-constructive mathematics (NCM) as a foundation for quantum mechanics, since the Boolean logical substrate of NCM fits a distributive propositional structure, but fits less naturally into the non-distributive lattice of quantum propositions. We distinguish three logical structures relevant to this discussion: Boolean algebras (classical, distributive, complemented), Heyting algebras (constructive, distributive, not necessarily complemented), and orthomodular lattices (quantum, non-distributive). The linear structure of quantum state space is compatible with NCM, but the propositional structure of quantum mechanics forms a non-distributive lattice rather than a Boolean algebra. Topos theory, one of the models of constructive mathematics, offers a way forward: the Döring–Isham framework provides an explicit bridge—the daseinization map—from quantum propositions to a topos whose internal logic is intuitionistic. Although some topoi carry Boolean logic, in general, topoi are not constrained to it. We argue that this bridged framework is more suitable for the mathematical foundation of quantum mechanics than one that retains the law of excluded middle (LEM) without such a bridge. Our discussion centers on the pivotal role of the LEM.