The Tensor Product of Operational Logics

Lock, Robin H.

Canadian journal of mathematics, Tome 38 (1986) no. 5, pp. 1065-1080

Résumé

The concept of an operational logic has been developed by Randall and Foulis ([l]-[4], [10], [11]) as a part of a larger effort to obtain a formalism suitable for expressing, comparing, and evaluating various approaches to empirical science, statistics, and in particular, quantum mechanics. The structure of these logics is similar to that of an orthomodular partially ordered set which is often used as a model for quantum logics. However, the operational logic is a more general structure which, among other features, allows for the creation of a tensor product of logics to represent the coupling of physical systems. Randall and Foulis have shown that, given certain reasonable physical constraints, such a product is not possible within the category of orthomodular posets [12].

Détail

DOI : 10.4153/CJM-1986-052-3

Bibliographie

[1] 1. Foulis, D. J. and Randall, C. H., The empirical logic approach to the physical sciences, in Foundations of quantum mechanics and ordered linear spaces, Lecture Notes in Physics 29 (Springer-Verlag, New York, 1974). Google Scholar

[2] 2. Foulis, D. J. and Randall, C. H., Manuals, morphisms, and quantum mechanics, in Mathematical foundations of quantum theory (Academic Press, New York, 1978). Google Scholar

[3] 3. Foulis, D. J. and Randall, C. H., Empirical logic and tensor products, Proceedings of the Colloquium on the Interpretations and Foundations of Quantum Theories, Fachbereich Physik der Philipps Universitat, Marburg, Germany (1979). Google Scholar

[4] 4. Foulis, D. J. and Randall, C. H., What are quantum logics and what ought they to he?, Proceedings of the Workshop on Quantum Logic, Ettore Majorana Centre for Scientific Culture, Erice, Sicily (1979). Google Scholar

[5] 5. Halmos, P. R., Lectures on Boolean algebras, 25 (Van Nostrand 1958). Google Scholar

[6] 6. Hardegree, G. and Frazer, P. J., Charting the Labyrinth of quantum logics; a progress report, Proceedings of the Workshop on Quantum Logic, Ettore Majorana Centre for Scientific Culture, Erice, Sicily (1979). Google Scholar

[7] 7. Lock, P. F., Categories of manuals, Ph.D. dissertation, University of Massachusetts (1981). Google Scholar

[8] 8. Lock, P. F. and Lock, R. H., Tensor products of generalized sample spaces, International Journal of Theoretical Physics 23 (1984). Google Scholar

[9] 9. Lock, R. H., Constructing the tensor product of generalized sample spaces, Ph.D. dissertation, University of Massachusetts (1981). Google Scholar

[10] 10. Randall, C. H. and Foulis, D. J., The operational approach to quantum mechanics, in Physical theory as logico-operational structure (D. Reidel Publishing Co., Dordrecht, Holland, 1978). Google Scholar | DOI

[11] 11. Randall, C. H. and Foulis, D. J., Operational statistics and tensor products, Proceedings of the Colloquium on the Interpretations and Foundations of Quantum Theories, Fachbereich Physik der Philipps Universitat, Marburg, Germany (1979). Google Scholar

[12] 12. Randall, C. H. and Foulis, D. J., Tensor products of quantum logics do not exist, Notices of the Amer. Math. Society 26 (1979). Google Scholar