Hostname: page-component-745bb68f8f-mzp66 Total loading time: 0 Render date: 2025-02-11T17:25:33.526Z Has data issue: false hasContentIssue false
  • English
  • Français

On the lattice structure of quantum logic

Published online by Cambridge University Press:  17 April 2009

P.D. Finch
P.D. Finch
Affiliation:
Monash University, Clayton, Victoria.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A weak logical structure is defined as a set of boolean propositional logics in which one can define common operations of negation and implication. The set union of the boolean components of a weak logical structure is a logic of propositions which is an orthocomplemented poset, where orthocomplementation is interpreted as negation and the partial order as implication. It is shown that if one can define on this logic an operation of logical conjunction which has certain plausible properties, then the logic has the structure of an orthomodular lattice. Conversely, if the logic is an orthomodular lattice then the conjunction operation may be defined on it.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 1 , Issue 3 , December 1969 , pp. 333 - 340
Copyright
Copyright © Australian Mathematical Society 1969

References

[1]Birkhoff, Garrett, Lattice theory, (Amer. Math, Soc. Colloquium Publications, Vol. 25 3rd edition, 1967).Google Scholar
[2]Finch, P.D., ‘On the structure of quantum logic’, J. Symbolic Logic 34 (1969), 275–282.CrossRefGoogle Scholar
[3]Finch, P.D., ‘Sasaki projections on orthocomplemented posets’, Bull. Austral. Math. Soc. 1 (1969), 319–324.CrossRefGoogle Scholar
[4]Foulis, David J., ‘Baer *–semigroups’, Proc. Amer. Math. Soc. 11 (1960), 648–654.Google Scholar
[5]Foulis, David J., ‘A note on orthomodular lattices’, Portugal. Math. 21 (1962), 65–72.Google Scholar
[6]Gericke, Helmuth, Lattice theory (Harrap, London, 1966).Google Scholar
[7]Gunson, J., ‘On the algebraic structure of quantum mechanics’, Comm. Math. Phys. 6 (1967), 262–285.CrossRefGoogle Scholar
[8]Jauch, Joseph M., Foundations of quantum mechanics (Addison-Wesley, Reading, Massachusetts, 1968).Google Scholar
[9]Jauch, J.M. and Piron, C., ‘Can hidden variables be excluded in quantum mechanics?’, Helv. Phys. Acta. 36 (1963), 827–837.Google Scholar
[10]Kunsemüller, H., ‘Zur logischen Deutung der Quantenmechanik’, (Diss. Phil. Univ. Hamburg, 1959).Google Scholar
[11]Mackey, George W., Mathematical foundations of quantum mechanics (Benjamin, New York, 1963).Google Scholar
[12]Piron, C., ‘Axiomatique quantique’, Helv. Phys. Acta. 37 (1964), 439–468.Google Scholar
[13]Varadarajan, V.S., Geometry of quantum theory (Van Nostrand, Princeton, New Jersey, 1968).Google Scholar
[14]Zierler, Neal, ‘Axioms for non-relativistic quantum mechanics’, Pacific J. Math. 2 (1961), 1151–1169.CrossRefGoogle Scholar