Abstract
The Kochen–Specker theorem has been discussed intensely ever since its original proof in 1967. It is one of the central no-go theorems of quantum theory, showing the non-existence of a certain kind of hidden states models. In this paper, we first offer a new, non-combinatorial proof for quantum systems with a type I n factor as algebra of observables, including I∞. Afterwards, we give a proof of the Kochen–Specker theorem for an arbitrary von Neumann algebra \({\cal R}\) without summands of types I1 and I2, using a known result on two-valued measures on the projection lattice \({\cal{P(R)}}\). Some connections with presheaf formulations as proposed by Isham and Butterfield are made.
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References
Aarnes, J. F. (1969). Physical states on a C∗-algebra. Acta Mathematica 122, 161–172.
Aarnes, J. F. (1970). Quasi-states on C∗-algebras. Transactions of American Mathematical Society 149, 601–625.
Bell, J. S. (1966). On the Problem of Hidden Variables in Quantum Mechanics. Reviews of Modern Physics 38, 447–452.
Christensen, E. (1982). Measures on projections and physical states. Communications in Mathematical Physics 86, 529–538.
Christensen, E. (1985). Comments to my paper: Measures on projections and physical states, (unpublished).
de Groote, H. F. (2001). Quantum Sheaves, preprint, http://de.arxiv.org/abs/math-ph/0110035.
Fine, A. (1974). On the Completeness of Quantum Theory. Synthese 29, 257–289.
Gleason, A. M. (1957). Measures on the closed subspaces of a Hilbert space. Journal of Mathematics and Mechanics 6, 885–893.
Gunson, J. (1972). Physical states on quantum logics. I. Annales de l Institut Henri Poincaré, Section B: Calcul des Probabilities es Statistique A 17, 295–311
Hamhalter, J. (1993). Pure Jauch-Piron states on von Neumann algebras, Annales de l Institut Henri Poincaré, Section B: Calcul des Probabilities es Statistique 58(2), 173–187.
Hamhalter, J. (2003). Quantum Measure Theory, Kluwer Acedemic Publishers, Dordrecht, Boston, London.
Hamhalter, J. (2004). Traces, dispersions of states and hidden variables. Foundations of Physics Letters 17(6), 581–597.
Hamilton, J., Isham, C. J. and Butterfield, J. (2000). A topos perspective on the Kochen–Specker theorem: III. Von Neumann algebras as the base category. International Journal of Theoretical Physics 39, 1413–1436.
Isham, C. J. and Butterfield, J. (1998). A topos perspective on the Kochen–Specker theorem: I. Quantum states as generalised valuations. International Journal of Theoretical Physics 37, 2669–2733.
Isham, C. J. and Butterfield, J. (1999). A topos perspective on the Kochen–Specker theorem: II. Conceptual aspects, and classical analogues. International Journal of Theoretical Physics 38, 827–859.
Isham, C. J. and Butterfield, J. (2002). A topos perspective on the Kochen–Specker theorem: IV. Interval valuations. International Journal of Theoretical Physics 41(4), 613–639.
Kadison, R. V. and Ringrose, J. R. (1997). Fundamentals of the Theory of Operator Algebras I, AMS, Providence, Rhode Island. Kadison, R. V., and Ringrose, J. R. (1997). Fundamentals of the Theory of Operator Algebras II, AMS, Providence, Rhode Island.
Kochen, S. and Specker, E. P. (1967). The problem of hidden variables in quantum mechanics. Journal of Mathematics and Mechanics 17 (1967), 59–87.
Maeda, S. (1990). Probability measures on projections in von Neumann algebras. Review of Mathematical Physics 1(2), 235–290.
Malley, J. D. (2004). All quantum observables in a hidden-variable model must commute simultaneously. Physics Review A 69, 022118.
Peres, A. (1993). Quantum Theory: Concepts and Methods Kluwer, Dordrecht, Boston, London.
Redhead, M. (1987). Incompleteness, Nonlocality and Realism, Clarendon, Oxford.
Saito, K. (1985). Diagonalizing normal operators in properly infinite von Neumann algebras and the continuity of probability measures (unpublished).
Takesaki, M. (1979). Theory of Operator Algebras I, Springer, Berlin, Heidelberg, New York.
von Neumann, J. (1932). Mathematische Grundlagen der Quantenmechanik, Springer, Berlin, Heidelberg, New York.
Yeadon, F. J. (1983). Measures on projections in W∗-algebras of type II1. Bulletin of London Mathematical Society 15, 139–145.
Yeadon, F. J. (1984). Finitely additive measures on projections in finite W∗-algebras. Bulletin of London Mathematical Society 16, 145–150.
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Döring, A. Kochen–Specker Theorem for von Neumann Algebras. Int J Theor Phys 44, 139–160 (2005). https://doi.org/10.1007/s10773-005-1490-6
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DOI: https://doi.org/10.1007/s10773-005-1490-6