Classical and quantum probabilities as truth values (Q2861735)

This paper deals with a problem how probabilities can be treated as truth values in suitable sheaf topoi. For the specific presheaf topoi used in the reformulation of quantum theory, a truth value is a lower set in the set \({\mathcal V}({\mathcal H})\) of commutative sub-algebras of the algebra \({\mathcal B}({\mathcal H})\) of all bounded operators on a Hilbert space \({\mathcal H}\).NEWLINENEWLINE A truth value is therefore a collection of classical perspectives from which a given proposition is true. In this paper, the authors only consider non-trivial commutative von Neumann sub-algebras \(V \subset {\mathcal B}({\mathcal H})\) that contain the identity operator.NEWLINENEWLINE In a series of papers, the authors and others have shown how quantum theory can be re-expressed as a type of classical physics in the topos of presheaves (= set-valued contravariant functors) on the partially ordered set of all commutative von Neumann sub-algebras. This article deals with aspects of probability as it shows up in both classical and quantum physics. In both cases the usual probabilistic description can be absorbed into the logical framework supplied by topos theory. In fact, for pure quantum states, probabilities are replaced by truth values which are given by the structure of the topos itself.NEWLINENEWLINE Peculiar features of this work consists in the following three points. (i) Their topos approach is different from any other existing schemes. As a matter of fact, there is no problem in defining logical connectives as these are given uniquely by the theory itself. (ii) In the topos approach, truth values are not only multi-valued, but are also contextual in a way that is deeply tied to the underlying quantum theory. (iii) In both classical and quantum physics, probability can be faithfully interpreted using the intuitionistic logic associated with truth values in sheaf topoi. As for other related works, see, e.g. [\textit{J. Butterfield} and \textit{C. J. Isham}, Int. J. Theor. Phys. 41, No. 4, 613--639 (2002; Zbl 1021.81002); \textit{A. Döring} and \textit{C. J. Isham}, J. Math. Phys. 49, No. 5, 053518, 29 p. (2008; Zbl 1152.81411)]. For other related monographs, see also [\textit{A. Döring} and \textit{C. J. Isham}, Lect. Notes Phys. 813, 753--937 (2011; Zbl 1253.81011); \textit{S. Mac Lane} and \textit{I. Moerdijk}, Sheaves in geometry and logic: a first introduction to topos theory. New York etc.: Springer-Verlag (1992; Zbl 0822.18001)].