Topos theory and consistent histories: The internal logic of the set of all consistent sets
 C. J. Isham
 … show all 1 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
A major problem in the consistenthistories 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 ‘manyworldviews’ 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 twovalued (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 topostheoretic structure thereby gives a coherent mathematical framework in which to understand the internal logic of the manyworldviews picture that arises naturally in the approach to quantum theory based on the ideas of consistent histories.
 Bell, J. L. (1988).Toposes and Local Set Theories: An Introduction, Clarendon Press, Oxford.
 Butterfield, J. (1995). Words, minds and quantun,Proceedings of the Aristotelian Society,69, 113–158.
 Crane, L. (1995). Clocks and categories: Is quantum gravity algebraic?Journal of Mathematical Physics,36, 6180–6193. CrossRef
 Dowker, H. F., and Kent, A. (1995). Properties of consistent histories.Physical Review Letters,75, 3038–3041. CrossRef
 Dowker, H. F., and Kent, A. (1996). On the consistent histories approach to quantum mechanics,Journal of Statistical Physics,82, 1575–1646. CrossRef
 Dummett, M. (1959). Truth,Proceedings of the Aristotelian Society,59, 141–162.
 Foulis, D. J., Greechie, R. J., and Rüttimann, G. T. (1992). Filters and supports in orthoalgebras,International Journal of Theoretical Physics,31, 789–807. CrossRef
 GellMann, 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.
 GellMann, M., and Hartle, J. (1990b). Quantum mechanics in the light of quantum cosmology, inComplexity, Entropy and the Physics of Information, W. Zurek, ed., AddisonWesley, Reading, Massachusetts.
 Goldblatt, R. (1984).Topoi: The Categorial Analysis of Logic, NorthHolland, Amsterdam.
 Griffiths, R. B. (1984) Consistent histories and the interpretation of quantum mechanics,Journal of Statistical Physics,36, 219–272. CrossRef
 Griffiths, R. B. (1993).Foundations of Physics,23, 1601. CrossRef
 Griffiths, R. B. (1996). Consistent histories and quantum reasoning [quantph/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.
 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.
 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.
 Hartle, J. (1995). Spacetime quantum mechanics and the quantum mechanics of spacetime, inProceedings of the 1992 Les Houches School, Gravitation and Quantization, B. Julia and J. ZinnJustin, eds., Elsevier Science, Amsterdam.
 Isham, C. J. (1994). Quantum logic and the histories approach to quantum theory.Journal of Mathematical Physics,35, 2157–2185. CrossRef
 Isham, C. J. (1995). Quantum logic and decohering histories, inTopics in Quantum Field Theory, D. H. Tchrakian, ed., World Scientific, Singapore.
 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. CrossRef
 Isham, C. J., and Linden, N. (1995). Continuous histories and the history group in generalized quantum theory.Journal of Mathemtical Physics,36, 5392–5408. CrossRef
 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. CrossRef
 Kent, A. (1996). Consistent sets contradict [grqc/9604012].
 Kripke, S. (1963). Semantical considerations on modal logic,Acta Philosophica Fennica,16, 83–94.
 Lawvere, F. W. (1975). Continuously varible sets: Algebraic geometry=geometric logic, inProceedings Logic Colloquium Bristol 1973, NorthHolland, Amsterdam.
 Loux, M. J. (1979).The Possible and the Actual, Cornell University Press, Ithaca, New York.
 MacLane, S., and Moerdijk, I. (1992).Sheaves in Geometry and Logic: A First Introduction to Topos Theory, SpringerVerlag, Berlin.
 Omnès, R. (1988a). Logical reformulation of quantum mechanics. I. Foundations,Journal of statistical Physics,53, 893–932. CrossRef
 Omnès, R. (1988b). Logical reformulation of quantum mechanics. II. Interferences and the EinsteinPodolskyRosen experiment,Journal of Statistical Physics,53, 933–955. CrossRef
 Omnès, R. (1988c). Logical reformulation of quantum mechanics. III. Classical limit and irreversibility,Journal of Statistical Physics,53, 957–975. CrossRef
 Omnès, R. (1989). Logical reformulation of quantum mechanics. IV. Projectors in semiclassical physics.Journal of Statistical Physics,57, 357–382. CrossRef
 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. CrossRef
 Omnès, R. (1992). Consistent interpretations of quantum mechanics,Reviews of Modern Physics,64, 339–382. CrossRef
 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 [grqc/9508064].
 Wittgenstein, L. (1966).Tractatus LogicoPhilosophicus, Routledge & Kegan Paul, London.
 Title
 Topos theory and consistent histories: The internal logic of the set of all consistent sets
 Journal

International Journal of Theoretical Physics
Volume 36, Issue 4 , pp 785814
 Cover Date
 19970401
 DOI
 10.1007/BF02435786
 Print ISSN
 00207748
 Online ISSN
 15729575
 Publisher
 Kluwer Academic PublishersPlenum Publishers
 Additional Links
 Topics
 Industry Sectors
 Authors

 C. J. Isham ^{(1)}
 Author Affiliations

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