## Abstract

Quantum logic originated with the Birkhoff-von Neumann paper of 1936.^{2} The authors argued that the transition from classical to quantum mechanics involves the transition from a propositional calculus forming a Boolean algebra to a propositional calculus which constitutes “a sort of projective geometry”.* For von Neumann, the significance of this thesis for the foundational problems of the theory lay in the possibility of regarding the projective structure as the logic of a quantum set theory -- a non-Boolean theory of sets for properties of quantum mechanical systems in which dimensionality plays the role of cardinality and transition probabilities between events are induced by automorphisms of the logic.** Thus, probabilities are logical: they are not defined, as in the classical case, by measures over events represented as sets of maximal truth value assignments, parametrized by the points of a phase space. Indeed, the quantum sets have no “points” in the general case; there are no “hidden variables” underlying the statistics associated with a quantum logic or quantum set theory. This is a radically different account of the “irreducibility” of the quantum statistics from that provided by the Copenhagen interpretation.

## Preview

Unable to display preview. Download preview PDF.

### References

- 1.G. Birkhoff, “Lattice Theory”, ( Third Edition ), American Mathematical Society, Providence (1967).MATHGoogle Scholar
- 2.G. Birkhoff and J. von Neumann, The logic of quantum mathematics, Ann. Math. 37 (1936), pp. 823–843.CrossRefGoogle Scholar
- 3.J. Bub, The measurement problem of quantum mechanids, “Problems in the Foundations of Physics”, G. Toraldo di Franca ed. North Holland, Amsterdam (1979). ( Proceedings of the International School of Physics “Enrico Fermi”, Course LXXII. )Google Scholar
- 4.I. Halperin, Introduction to von Neumann algebras and continuous geometries, Canadian Math. Bull. 3 (1960), pp. 273–288.MathSciNetMATHCrossRefGoogle Scholar
- 5.N.M. Hugenholtz, On the factor type of equilibrium states in quantum statistical mechanics, Commun. Math. Phys. 6 (1967), pp. 189–193.MathSciNetADSMATHCrossRefGoogle Scholar
- 6.J.M. Jauch and C. Piron, Helv. Phys. Acta 42 (1969), p. 842.CrossRefGoogle Scholar
- 7.I. Kaplansky, Any orthocomplemented complete modular lattice is a continuous geometry, Ann. Math. 61 (1955), pp. 524–541.Google Scholar
- 8.G. Luders, Ann, der Physik 8 (1951), p. 322.MathSciNetGoogle Scholar
- 9.E.J. Murray and J. von Neumann, On rings of operators, Ann. Math. 37 (1936), pp. 116–229; Am. Math. Soc. Trans. 41 (1937), pp. 208–248; Ann. Math. 41 (1940), pp. 148–161; Ann. Math. 44 (1943), pp. 716–808.Google Scholar
- 10.E. Schrodinger, Naturwiss. 23 (1935), pp. 807–812, 824–825, 844–849.Google Scholar
- 11.H. Umegaki, Tohoku Math. J. 8 (1956), p. 86.MathSciNetMATHCrossRefGoogle Scholar
- 12.J. von Neumann, “Unsolved Problems in Mathematics”. Address to International Mathematical Congress, Amsterdam, Sept. 2, 1954. Typescript, von Neumann Archives, Library of Congress, Washington, D.C.Google Scholar
- 13.J. von Neumann, “Continuous Geometry with a Transition Probability”. Typescript, von Neumann Archives, Library of Congress, Washington, D.C.Google Scholar
- 14.J. von Neumann, “Continuous Geometry”. American Mathematical Society Colloquium Lectures, deliverd at Pennsylvania State College, September, 1937. Typescript, von Neumann Archives, Library of Congress, Washington, D.C.Google Scholar
- 15.J. von Neumann, “Quantum Logics (Strict- and Probability-Logics)”. Summarized in J. von Neumann, “Collected Works, Vol. IV”, Macmillan, New York (1962). Typescript in von Neumann Archives, Library of Congress, Washington, D.C.Google Scholar
- 16.J. von Neumann, “Quantum Mechanics of Infinite Systems”. Manuscript and typescript (different) in von Neumann Archives, Library of Congress, Washington, D.C.Google Scholar