Skip to main content
SpringerLink
Log in

Topos theory and consistent histories: The internal logic of the set of all consistent sets

  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript

Abstract

A major problem in the consistent-histories approach to quantum theory is contending with the potentially large number of consistent sets of history propositions. One possibility is to find a scheme in which a unique set is selected in some way. However, in this paper the alternative approach is considered in which all consistent sets are kept, leading to a type of ‘many-world-views’ picture of the quantum theory. It is shown that a natural way of handling this situation is to employ the theory of varying sets (presheafs) on the spaceB of all nontrivial Boolean subalgebras of the orthoalgebraUP of history propositions. This approach automatically includes the feature whereby probabilistic predictions are meaningful only in the context of a consistent set of history propositions. More strikingly, it leads to a picture in which the ‘truth values’ or ‘semantic values’ of such contextual predictions are not just two-valued (i.e., true and false) but instead lie in a larger logical algebra—a Heyting algebra—whose structure is determined by the spaceB of Boolean subalgebras ofUP. This topos-theoretic structure thereby gives a coherent mathematical framework in which to understand the internal logic of the many-world-views picture that arises naturally in the approach to quantum theory based on the ideas of consistent histories.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Bell, J. L. (1988).Toposes and Local Set Theories: An Introduction, Clarendon Press, Oxford.

    Google Scholar 

  • Butterfield, J. (1995). Words, minds and quantun,Proceedings of the Aristotelian Society,69, 113–158.

    Google Scholar 

  • Crane, L. (1995). Clocks and categories: Is quantum gravity algebraic?Journal of Mathematical Physics,36, 6180–6193.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • Dowker, H. F., and Kent, A. (1995). Properties of consistent histories.Physical Review Letters,75, 3038–3041.

    Article  ADS  Google Scholar 

  • Dowker, H. F., and Kent, A. (1996). On the consistent histories approach to quantum mechanics,Journal of Statistical Physics,82, 1575–1646.

    Article  MATH  MathSciNet  Google Scholar 

  • Dummett, M. (1959). Truth,Proceedings of the Aristotelian Society,59, 141–162.

    Google Scholar 

  • Foulis, D. J., Greechie, R. J., and Rüttimann, G. T. (1992). Filters and supports in orthoalgebras,International Journal of Theoretical Physics,31, 789–807.

    Article  MATH  MathSciNet  Google Scholar 

  • Gell-Mann, M., and Hartle, J. (1990a). Alternative decohering histories in quantum mechanics, inProceedings of the 25th International Conference on High Energy Physics, Singapore, August, 1990, K. K. Phua and Y. Yamaguchi, eds.), World Scientific, Singapore.

    Google Scholar 

  • Gell-Mann, M., and Hartle, J. (1990b). Quantum mechanics in the light of quantum cosmology, inComplexity, Entropy and the Physics of Information, W. Zurek, ed., Addison-Wesley, Reading, Massachusetts.

    Google Scholar 

  • Goldblatt, R. (1984).Topoi: The Categorial Analysis of Logic, North-Holland, Amsterdam.

    Google Scholar 

  • Griffiths, R. B. (1984) Consistent histories and the interpretation of quantum mechanics,Journal of Statistical Physics,36, 219–272.

    Article  MATH  MathSciNet  Google Scholar 

  • Griffiths, R. B. (1993).Foundations of Physics,23, 1601.

    Article  MathSciNet  Google Scholar 

  • Griffiths, R. B. (1996). Consistent histories and quantum reasoning [quant-ph/9606004].

  • Halliwell, J. (1995). A review of the decoherent histories approach to quantum mechanics, inFundamental Problems in Quantum Theory, D. M. Greenberger and A. Zeilinger, eds., New York Academy of Sciences, New York.

    Google Scholar 

  • Hardegree, G. M., and Frazer, P.J. (1982). Charting the labyrinth of quantum logics: A progress report, inCurrent Issues in Quantum Logic, E. G. Beltrametti and B. V. van Frassen, eds., Plenum Press, New York.

    Google Scholar 

  • Hartle, J. (1991).The quantum mechanics of cosmology, inQuantum Cosmology and Baby Universes, S. Coleman, J. Hartle, T. Piran, and S. Weinberg, eds. World Scientific, Singapore.

    Google Scholar 

  • Hartle, J. (1995). Spacetime quantum mechanics and the quantum mechanics of space-time, inProceedings of the 1992 Les Houches School, Gravitation and Quantization, B. Julia and J. Zinn-Justin, eds., Elsevier Science, Amsterdam.

    Google Scholar 

  • Isham, C. J. (1994). Quantum logic and the histories approach to quantum theory.Journal of Mathematical Physics,35, 2157–2185.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • Isham, C. J. (1995). Quantum logic and decohering histories, inTopics in Quantum Field Theory, D. H. Tchrakian, ed., World Scientific, Singapore.

    Google Scholar 

  • Isham, C. J., and Linden, N. (1994). Quantum temporal logic and decoherence functionals in the histories approach to generalized quantum theory.Journal of Mathematical Physics,35, 5452–5476.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • Isham, C. J., and Linden, N. (1995). Continuous histories and the history group in generalized quantum theory.Journal of Mathemtical Physics,36, 5392–5408.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • Isham, C. J., Linden, N., and Schreckenberg, S. (1994). The classification of decoherence functionals: An analogue of Gleason's theorem,Journal of Mathematical Physics,35, 6360–6370.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • Kent, A. (1996). Consistent sets contradict [gr-qc/9604012].

  • Kripke, S. (1963). Semantical considerations on modal logic,Acta Philosophica Fennica,16, 83–94.

    MATH  MathSciNet  Google Scholar 

  • Lawvere, F. W. (1975). Continuously varible sets: Algebraic geometry=geometric logic, inProceedings Logic Colloquium Bristol 1973, North-Holland, Amsterdam.

    Google Scholar 

  • Loux, M. J. (1979).The Possible and the Actual, Cornell University Press, Ithaca, New York.

    Google Scholar 

  • MacLane, S., and Moerdijk, I. (1992).Sheaves in Geometry and Logic: A First Introduction to Topos Theory, Springer-Verlag, Berlin.

    Google Scholar 

  • Omnès, R. (1988a). Logical reformulation of quantum mechanics. I. Foundations,Journal of statistical Physics,53, 893–932.

    Article  MATH  MathSciNet  Google Scholar 

  • Omnès, R. (1988b). Logical reformulation of quantum mechanics. II. Interferences and the Einstein-Podolsky-Rosen experiment,Journal of Statistical Physics,53, 933–955.

    Article  MathSciNet  Google Scholar 

  • Omnès, R. (1988c). Logical reformulation of quantum mechanics. III. Classical limit and irreversibility,Journal of Statistical Physics,53, 957–975.

    Article  MathSciNet  Google Scholar 

  • Omnès, R. (1989). Logical reformulation of quantum mechanics. IV. Projectors in semiclassical physics.Journal of Statistical Physics,57, 357–382.

    Article  MathSciNet  Google Scholar 

  • Omnès, R. (1990). From Hilbert spaceto common sense: A synthesis of recent progress in the interpretation of quantum mechanics,Annals of Physics,201, 354–447.

    Article  MathSciNet  ADS  Google Scholar 

  • Omnès, R. (1992). Consistent interpretations of quantum mechanics,Reviews of Modern Physics,64, 339–382.

    Article  MathSciNet  ADS  Google Scholar 

  • Rovelli, C. (1996). Relational quantum theory,International Journal of Theoretical Physics, in press.

  • Smolin, L. (1995). The Bekenstein bound, topological quantum field theory, and pluralistic quantum cosmology [gr-qc/9508064].

  • Wittgenstein, L. (1966).Tractatus Logico-Philosophicus, Routledge & Kegan Paul, London.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Blackett Laboratory, Imperial College of Science, Technology and Medicine, SW7 2BZ, London, UK

    C. J. Isham

Authors
  1. C. J. Isham
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Isham, C.J. Topos theory and consistent histories: The internal logic of the set of all consistent sets. Int J Theor Phys 36, 785–814 (1997). https://doi.org/10.1007/BF02435786

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02435786

Keywords

Search

Navigation