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Journal of Mathematical Physics

Operational statistics. II. Manuals of operations and their logics

Charles H. Randall, David J. Foulis

Published Oct 1, 1973• 115 citations
Publisher

Abstract

In the first paper of this series, the notion of a manual of operations and its (generalized) sample space was introduced. For each such manual of operations, a so-called logic was constructed. In the present paper, some of the algebraic properties of such logics are identified with the corresponding operational properties of the manual.
DOI
10.1063/1.1666208

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The recent study of operational statistics (see [2; 4; 5; 10 ; 11 ; 12 ; 13 ; 14 ; 15 ; 16]) describes a generalized sample space which represents the set of all possible outcomes of a collection of coherently related operations (experiments). This approach to probability generalizes the classical notion of a sample space due to A. N. Kolmogorov [8], and it gives the concept somewhat wider applicability. For instance in [4] and [14], D. J. Foulis and C. H. Randall set out the start of a program wherein a generalized sample space (hereafter called a GSS) and its affiliated partially ordered set of generalized propositions could be a framework within which a genuinely operational interpretation of the so called “logic” of quantum mechanical systems may be found.
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A manual of operations is a model for a collection of physical experiments, or a collection of (perhaps overlapping) sample spaces. To understand this theory, it is essential to understand the relations between manuals. Hence, in this dissertation, we study the morphisms (and, more specifically, the interpretations) of manuals. This is followed by an investigation of the categorical properties of the tensor product of two manuals. We define biinterpretations and prove a universal mapping theorem. We then prove that the tensor product is associative, and define, using an inductive limit, an infinite tensor product. In the final chapter, we develop a language and theory that is broad enough to encompass much of the work which has been done under the general title of "quantum logic."
Introduction. Operational quantum logic: An overview B. Coecke, et al. I: Algebras. Algebraic aspects of orthomodular lattices G. Bruns, J. Harding. Observables in the logico-algebraic approach P. Ptak. Test spaces and orthoalgebras A. Wilce. Representations on unigroups D.J. Foulis. Free extensions of group actions, induced representations, and the foundations of physics D.J. Foulis, A. Wilce. II: Categories. Short introduction to enriched categories F. Borceux, I. Stubbe. Operational Galois adjunctions B. Coecke, D. Moore. Operational axiomatics and compound systems F. Valckenborgh. Quantales J. Paseka, J. Rosicky. III: Languages. Quantales and observational semantics P. Resende. Quantum languages S. Gudder. Indices. Concept index. Citation index.