Abstract
The notion of relative compability is introduced, according to which compatibility is construed as relative to individual quantum states. The compatibility domain of two observablesA, B is defined to be the set com(A, B) of states relative to whichA andB are compatible. Three basic categories of relative compatibility are then defined according to the character of com(A, B): absolute compatibility (ordinary compatibility), absolute incompatibility, and partial compatibility. Then com(A, B) is seen to be a subspace of Hilbert space invariant underA andB; being a subspace, it corresponds to a quantum binary observable (projection), denotedc(A, B). IfA andB are themselves binary observables, thenc(A, B) is expressible as a quantum logical compound ofA andB using the lattice operations meet, join, and orthocomplement. This suggests extending the notion of relative compatibility to the theory of orthomodular lattices, as well as to more general lattice-theoretic formulations of quantum mechanics.
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References
J. von Neumann,Mathematical Foundations of Quantum Mechanics (Princeton University Press, Princeton, 1955).
J. L. Park and H. Margenau,Int. J. Theoret. Phys. 1, 205 (1968).
V. S. Varadarajan,Comm. Pure Appl. Math. 15, 189 (1962).
J. M. Jauch,Foundations of Quantum Mechanics (Addison-Wesley, Reading, Mass., 1968).
H. Reichenbach,Philosophic Foundations of Quantum Mechanics (U. Cal. Press, Berkeley, 1946).
B. van Fraassen, inBoston Studies in the Philosophy of Science XIII, Cohen and Wartofsky, eds. (D. Reidel, Dordrecht, 1974), p. 224.
P. Halmos,Trans. Amer. Math. Soc. 144, 381 (1969).
S. P. Holland, inTrends in Lattice Theory, Abbott, ed. (van Nostrand, New York, 1970).
D. M. Topping,Lectures on von Neumann Algebras (van Nostrand, New York, 1971).
P. A. Fillmore,Proc. Amer. Math. Soc. 16, 648 (1965).
A. Lenard,J. Functional Analysis 10, 410 (1972).
G. M. Hardegree, Relative Compatibility in Orthomodular Lattices, to appear.
G. M. Hardegree, Relative Compatibility and the Lattice Approach to Quantum Mechanics, to appear.
C. R. Putnam,Commutation Properties of Hilbert Space Operators and Related Topics (Springer-Verlag, New York, 1967).
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Hardegree, G.M. Relative compatibility in conventional quantum mechanics. Found Phys 7, 495–510 (1977). https://doi.org/10.1007/BF00708865
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DOI: https://doi.org/10.1007/BF00708865