Abstract
If one supposes a quantum logicL to be a σ-orthocomplete, orthomodular partially ordered set admitting a set of σ-orthoadditive functions (called states) fromL to the unit intervals [0, 1] such that these states distinguish the ordering and orthocomplement onL, then the observables onL are identified withL-valued measures defined on the Borel subsets of the real line. In this structure (and without the aid of Hilbert space formalism) the author shows that (1) the spectrum of an observable can be completely characterised by studying the observable (A−λ)−1, and (2) corresponding to every observableA there is a spectral resolution uniquely determined byA and uniquely determiningA.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bennett, M. K. (1968). A Finite Orthomodular Lattice which does not admit a Full Set of States. To be published.
Foulis, D. J. (1962). A Note on Orthomodular Lattices.Portugaliae mathematica,21, Fasc. 1, 65–72.
Gudder, S. P. (1966). Uniqueness and Existence Properties of Bounded Observables.Pacific Journal of Mathematics, Vol. 19, No. 1, 81–93.
Halmos, P. R. (1950).Measure Theory. D. van Nostrand Co., New York.
Mackey, G. W. (1963).The Mathematical Foundations of Quantum Mechanics. W. A. Benjamin, Inc., New York.
Ramsay, Arlan (1966). A Theorem on Two Commuting Observables.Journal of Mathematics and Mechanics, Vol. 15, No. 2, 227–234.
Varadarajan, V. S. (1962). Probability in Physics and a Theorem on Simultaneous Observability.Communication on Pure and Applied Mathematics,15, 189–217.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Catlin, D.E. Spectral theory in quantum logics. Int J Theor Phys 1, 285–297 (1968). https://doi.org/10.1007/BF00668669
Issue Date:
DOI: https://doi.org/10.1007/BF00668669