Quantum logics with lattice state spaces
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- by Jiří Binder and Mirko Navara PDF
- Proc. Amer. Math. Soc. 100 (1987), 688-693 Request permission
Abstract:
Let $L$ be a quantum logic and let $S(L)$ denote the set of all states on $L$. (By a state we mean a nonnegative bounded $\sigma$-additive measure, not necessarily normalized.) We ask whether every logic whose state space is a lattice has to be Boolean. We prove that this is so for finite logics and "projection logics." On the other hand, we show that there exist even concrete non-Boolean logics with a lattice state space (in fact, we prove that every countable concrete logic can be enlarged to a logic with a lattice state space). In the appendix we shortly consider the lattice properties of the set of observables and correct the paper [10].References
- Erik Christensen, Measures on projections and physical states, Comm. Math. Phys. 86 (1982), no. 4, 529–538. MR 679201
- R. J. Greechie, Orthomodular lattices admitting no states, J. Combinatorial Theory Ser. A 10 (1971), 119–132. MR 274355, DOI 10.1016/0097-3165(71)90015-x
- Stanley P. Gudder, Stochastic methods in quantum mechanics, North-Holland Series in Probability and Applied Mathematics, North-Holland, New York-Oxford, 1979. MR 543489
- Richard V. Kadison, Order properties of bounded self-adjoint operators, Proc. Amer. Math. Soc. 2 (1951), 505–510. MR 42064, DOI 10.1090/S0002-9939-1951-0042064-2
- Pavel Pták, Exotic logics, Colloq. Math. 54 (1987), no. 1, 1–7. MR 928651, DOI 10.4064/cm-54-1-1-7
- Pavel Pták and John D. Maitland Wright, On the concreteness of quantum logics, Apl. Mat. 30 (1985), no. 4, 274–285 (English, with Russian and Czech summaries). MR 795987
- Robert M. Solovay, Real-valued measurable cardinals, Axiomatic set theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) Amer. Math. Soc., Providence, R.I., 1971, pp. 397–428. MR 0290961
- V. S. Varadarajan, Geometry of quantum theory. Vol. I, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1968. MR 0471674
- F. J. Yeadon, Finitely additive measures on projections in finite $W^{\ast }$-algebras, Bull. London Math. Soc. 16 (1984), no. 2, 145–150. MR 737242, DOI 10.1112/blms/16.2.145
- Neal Zierler, Order properties of bounded observables, Proc. Amer. Math. Soc. 14 (1963), 346–351. MR 145863, DOI 10.1090/S0002-9939-1963-0145863-X
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 100 (1987), 688-693
- MSC: Primary 81B10; Secondary 03G12, 06C15, 46L60
- DOI: https://doi.org/10.1090/S0002-9939-1987-0894439-1
- MathSciNet review: 894439