Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-28T01:02:22.087Z Has data issue: false hasContentIssue false
  • English
  • Français

Transfer principle in quantum set theory

Published online by Cambridge University Press:  12 March 2014

Masanao Ozawa
Masanao Ozawa*
Affiliation:
Graduate School of Information Sciences, Tôhoku University, Aoba-ku, Sendai, 980-8579, Japan, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In 1981, Takeuti introduced quantum set theory as the quantum counterpart of Boolean valued models of set theory by constructing a model of set theory based on quantum logic represented by the lattice of closed subspaces in a Hilbert space and showed that appropriate quantum counterparts of ZFC axioms hold in the model. Here, Takeuti's formulation is extended to construct a model of set theory based on the logic represented by the lattice of projections in an arbitrary von Neumann algebra. A transfer principle is established that enables us to transfer theorems of ZFC to their quantum counterparts holding in the model. The set of real numbers in the model is shown to be in one-to-one correspondence with the set of self-adjoint operators affiliated with the von Neumann algebra generated by the logic. Despite the difficulty pointed out by Takeuti that equality axioms do not generally hold in quantum set theory, it is shown that equality axioms hold for any real numbers in the model. It is also shown that any observational proposition in quantum mechanics can be represented by a corresponding statement for real numbers in the model with the truth value consistent with the standard formulation of quantum mechanics, and that the equality relation between two real numbers in the model is equivalent with the notion of perfect correlation between corresponding observables (self-adjoint operators) in quantum mechanics. The paper is concluded with some remarks on the relevance to quantum set theory of the choice of the implication connective in quantum logic.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 2 , June 2007 , pp. 625 - 648
Copyright
Copyright © Association for Symbolic Logic 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Bell, J. L., Boolean-valued models and independence proofs in set theory, 2nd ed., Oxford University Press, Oxford, 1985.Google Scholar
[2]Birkhoff, G. and Von Neumann, J., The logic of quantum mechanics, Annals of Mathematics, vol. 37 (1936), pp. 823–845.CrossRefGoogle Scholar
[3]Bratteli, O. and Robinson, D. W., Operator algebras and quantum statistical mechanics I, Springer, New York, 1979.CrossRefGoogle Scholar
[4]Cohen, P. J., The independence of the continuum hypothesis I, Proceedings of the National Academy of Sciences of the United States of America, vol. 50 (1963), pp. 1143–1148.Google Scholar
[5]Cohen, P. J., Set theory and the continuum hypothesis, Benjamin, New York, 1966.Google Scholar
[6]Eda, K., On a Boolean power of a torsion free abelian group, Journal of Algebra, vol. 82 (1983), pp. 84–93.CrossRefGoogle Scholar
[7]Greechie, G. J. and Gudder, S. P., Is a quantum logic a logic?, Helvetica Physica Acta, vol. 41 (1971), pp. 238–240.Google Scholar
[8]Gudder, S., Joint distributions of observables, Journal of Mathematics and Mechanics (Indiana University Mathematics Journal), vol. 18 (1968), pp. 325–335.Google Scholar
[9]Hardegree, G. M., The conditional in abstract and concrete quantum logic, The logico-algebraic approach to quantum mechanics, Volume II: Contemporary consolidation (Hooker, C. A., editor), D. Reidel, Dordrecht, 1979, pp. 49–108.Google Scholar
[10]Hardegree, G. M., Material implication in orthomodular (and boolean) lattices, Notre Dame Journal of Formal Logic, vol. 22 (1981), pp. 163–182.CrossRefGoogle Scholar
[11]Jech, T., Abstract theory of abelian operator algebras: an application of forcing, Transactions of the American Mathematical Society, vol. 289 (1985), pp. 133–162.CrossRefGoogle Scholar
[12]Kalmbach, G., Orthomodular lattices, Academic, London, 1983.Google Scholar
[13]Kotas, J., An axiom system for the modular logic, Studia Logica, vol. 21 (1967), pp. 17–38.CrossRefGoogle Scholar
[14]Kusraev, A. G. and Kutateladze, S. S., Nonstandard methods of analysis, Springer, Berlin, 1994.CrossRefGoogle Scholar
[15]Kusraev, A. G. and Kutateladze, S. S., Boolean valued analysis, Springer, Berlin, 1999.CrossRefGoogle Scholar
[16]Nishimura, H., Boolean valued Lie algebras, this Journal, vol. 56 (1991), pp. 731–741.Google Scholar
[17]Ozawa, M., Boolean valued analysis and type I AW*-algebras, Proceedings of the Japan Academy. Series A, vol. 59 (1983), pp. 368–371.Google Scholar
[18]Ozawa, M., Boolean valued interpretation of Hilbert space theory, Journal of the Mathematical Society of Japan, vol. 35 (1983), pp. 609–627.CrossRefGoogle Scholar
[19]Ozawa, M., A classification of type I AW*-algebras and Boolean valued analysis, Journal of the Mathematical Society of Japan, vol. 36 (1984), pp. 589–608.CrossRefGoogle Scholar
[20]Ozawa, M., Nonuniqueness of the cardinality attached to homogeneous AW*-algebras, Proceedings of the American Mathematical Society, vol. 93 (1985), pp. 681–684.Google Scholar
[21]Ozawa, M., A transfer principle from von Neumann algebras to AW*-algebras, The Journal of the London Mathematical Society, vol. 32 (1985), no. 2, pp. 141–148.Google Scholar
[22]Ozawa, M., Boolean valued analysis approach to the trace problem of AW*-algebras, The Journal of the London Mathematical Society, vol. 33 (1986), no. 2, pp. 347–354.Google Scholar
[23]Ozawa, M., Boolean valued interpretation of Banach space theory and module structures of von Neumann algebras, Nagoya Mathematical Journal, vol. 117 (1990), pp. 1–36.CrossRefGoogle Scholar
[24]Ozawa, M., Forcing in nonstandard analysis, Annals of Pure and Applied Logic, vol. 68 (1994), pp. 263–297.CrossRefGoogle Scholar
[25]Ozawa, M., Scott incomplete Boolean ultrapowers of the real line, this Journal, vol. 60 (1995), pp. 160–171.Google Scholar
[26]Ozawa, M., Perfect correlations between noncommuting observables, Physics Letters. A, vol. 335 (2005), pp. 11–19.CrossRefGoogle Scholar
[27]Ozawa, M., Quantum perfect correlations, Annals of Physics, vol. 321 (2006), pp. 744–769.CrossRefGoogle Scholar
[28]Scott, D., Boolean models and nonstandard analysis, Applications of model theory to algebra, analysis, and probability (Luxemburg, W. A. J., editor), Holt, Reinehart and Winston, New York, 1969, pp. 87–92.Google Scholar
[29]Scott, D. and Solovay, R., Boolean-valued models for set theory, unpublished manuscript for Proc. AMS Summer Institute on Set Theory, Los Angeles, Univ. Cal., 1967.Google Scholar
[30]Smith, K., Commutative regular rings and Boolean-valued fields, this Journal, vol. 49 (1984), pp. 281–297.Google Scholar
[31]Takesaki, M., Theory of operator algebras I, Springer, New York, 1979.CrossRefGoogle Scholar
[32]Takeuti, G., Two applications of logic to mathematics, Princeton University Press, Princeton, 1978.Google Scholar
[33]Takeuti, G., Boolean valued analysis, Applications of sheaves (Fourman, M. P., Mulvey, C. J., and Scott, D. S., editors), Lecture Notes in Mathematics, vol. 753, Springer, Berlin, 1979, pp. 714–731.CrossRefGoogle Scholar
[34]Takeuti, G., A transfer principle in harmonic analysis, this Journal, vol. 44 (1979), pp. 417–440.Google Scholar
[35]Takeuti, G., Quantum set theory, Current issues in quantum logic (Beltrametti, E. G. and van Fraassen, B. C., editors), Plenum, New York, 1981, pp. 303–322.Google Scholar
[36]Takeuti, G., C*-Algebras and Boolean valued analysis, Japanese Journal of Mathematics, New Series, vol. 9 (1983), pp. 207–245.Google Scholar
[37]Takeuti, G., Von Neumann algebras and Boolean valued analysis, Journal of the Mathematical Society of Japan, vol. 35 (1983), pp. 1–21.CrossRefGoogle Scholar
[38]Takeuti, G., Boolean simple groups and Boolean simple rings, this Journal, vol. 53 (1988), pp. 160–173.Google Scholar
[39]Titani, S. and Kozawa, H., Quantum set theory, International Journal of Theoretical Physics, vol. 42 (2003), pp. 2575–2602.CrossRefGoogle Scholar
[40]Urquhart, A., Review, this Journal, vol. 48 (1983), pp. 206–208.Google Scholar
[41]von Neumann, J., Mathematical foundations of quantum mechanics, Princeton UP, Princeton, NJ, 1955, English translation of Mathematische Grundlagen der Quantenmechanik (Springer, Berlin, 1932).Google Scholar