Advertisement

Unsharp Values, Domains and Topoi

  • Andreas DöringEmail author
  • Rui Soares Barbosa
Chapter

Abstract

The so-called topos approach provides a radical reformulation of quantum theory. Structurally, quantum theory in the topos formulation is very similar to classical physics. There is a state object \(\underline\sum\), analogous to the state space of a classical system, and a quantity-value object \(\underline{\mathbb{R}^{\leftrightarrow}}\), generalising the real numbers. Physical quantities are maps from the state object to the quantity-value object – hence the ‘values’ of physical quantities are not just real numbers in this formalism. Rather, they are families of real intervals, interpreted as ‘unsharp values’. We will motivate and explain these aspects of the topos approach and show that the structure of the quantity-value object \(\underline{\mathbb{R}^{\leftrightarrow}}\) can be analysed using tools from domain theory, a branch of order theory that originated in theoretical computer science. Moreover, the base category of the topos associated with a quantum system turns out to be a domain if the underlying von Neumann algebra is a matrix algebra. For general algebras, the base category still is a highly structured poset. This gives a connection between the topos approach, noncommutative operator algebras and domain theory. In an outlook, we present some early ideas on how domains may become useful in the search for new models of (quantum) space and space-time.

Keywords

Topos approach domain theory intervals unsharp values von Neumann algebras. 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    S. Abramsky, A. Jung. Domain Theory. In Handbook of Logic in Computer Science, eds. S. Abramsky, D.M. Gabbay, T.S.E. Maibaum, Clarendon Press, 1–168 (1994).Google Scholar
  2. [2]
    M. Caspers, C. Heunen, N.P. Landsman, B. Spitters. Intuitionistic quantum logic of an n-level system. Found. Phys. 39, 731–759 (2009).MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    B. Coecke, K. Martin. A partial order on classical and quantum states. In New Structures for Physics, ed. B. Coecke, Springer, Heidelberg (2011).Google Scholar
  4. [4]
    A. Döring. Kochen-Specker theorem for von Neumann algebras. Int. Jour. Theor. Phys. 44, 139–160 (2005).Google Scholar
  5. [5]
    A. Döring. Topos theory and ‘neo-realist’ quantum theory. In Quantum Field Theory, Competitive Models, eds. B. Fauser, J. Tolksdorf, E. Zeidler, Birkh¨auser (2009).Google Scholar
  6. [6]
    A. Döring. Quantum states and measures on the spectral presheaf. Adv. Sci. Lett. 2, special issue on “Quantum Gravity, Cosmology and Black Holes”, ed. M. Bojowald, 291–301 (2009).Google Scholar
  7. [7]
    A. Döring. Topos quantum logic and mixed states. In Proceedings of the 6th International Workshop on Quantum Physics and Logic (QPL 2009), Oxford, eds. B. Coecke, P. Panangaden, and P. Selinger. Electronic Notes in Theoretical Computer Science 270(2) (2011).Google Scholar
  8. [8]
    A. Döring. The physical interpretation of daseinisation. In Deep Beauty, ed. Hans Halvorson, 207–238, Cambridge University Press, New York (2011).Google Scholar
  9. [9]
    A. Döring, and C.J. Isham. A topos foundation for theories of physics: I. Formal languages for physics. J. Math. Phys 49, 053515 (2008).Google Scholar
  10. [10]
    A. Döring, and C.J. Isham. A topos foundation for theories of physics: II. Daseinisation and the liberation of quantum theory. J. Math. Phys 49, 053516 (2008).Google Scholar
  11. [11]
    A. Döring, and C.J. Isham. A topos foundation for theories of physics: III. Quantum theory and the representation of physical quantities with arrows \(\breve{A}:\underline{\Sigma}\rightarrow\textit{P}\mathbb{R}\). J. Math. Phys 49, 053517 (2008).Google Scholar
  12. [12]
    A. Döring, and C.J. Isham. A topos foundation for theories of physics: IV. Categories of systems. J. Math. Phys 49, 053518 (2008).Google Scholar
  13. [13]
    A. Döring, and C.J. Isham. ‘What is a thing?’: topos theory in the foundations of physics. In New Structures for Physics, ed. B. Coecke, Springer (2011).Google Scholar
  14. [14]
    T. Erker, M. H. Escard´o and K. Keimel. The way-below relation of function spaces over semantic domains. Topology and its Applications 89, 61–74 (1998).Google Scholar
  15. [15]
    C. Flori. A topos formulation of consistent histories. Jour. Math. Phys 51 053527 (2009).MathSciNetCrossRefGoogle Scholar
  16. [16]
    G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove, D.S. Scott. Continuous Lattices and Domains. Encyclopedia of Mathematics and its Applications 93, Cambridge University Press (2003).Google Scholar
  17. [17]
    C. Heunen, N.P. Landsman, B. Spitters. A topos for algebraic quantum theory. Comm. Math. Phys. 291, 63–110 (2009).MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    C. Heunen, N.P. Landsman, B. Spitters. The Bohrification of operator algebras and quantum logic. Synthese, in press (2011).Google Scholar
  19. [19]
    C. Heunen, N.P. Landsman, B. Spitters. Bohrification. In Deep Beauty, ed. H. Halvorson, 271–313, Cambridge University Press, New York (2011).Google Scholar
  20. [20]
    C.J. Isham. Topos methods in the foundations of physics. In Deep Beauty, ed. Hans Halvorson, 187–205, Cambridge University Press, New York (2011).Google Scholar
  21. [21]
    C.J. Isham. Topos theory and consistent histories: The internal logic of the set of all consistent sets. Int. J. Theor. Phys., 36, 785–814 (1997).MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    C.J. Isham and J. Butterfield. A topos perspective on the Kochen-Specker theorem: I. Quantum states as generalised valuations. Int. J. Theor. Phys. 37, 2669–2733 (1998).MathSciNetzbMATHGoogle Scholar
  23. [23]
    C.J. Isham and J. Butterfield. A topos perspective on the Kochen-Specker theorem: II. Conceptual aspects, and classical analogues. Int. J. Theor. Phys. 38, 827–859 (1999).Google Scholar
  24. [24]
    C.J. Isham, J. Hamilton and J. Butterfield. A topos perspective on the Kochen-Specker theorem: III. Von Neumann algebras as the base category. Int. J. Theor. Phys. 39, 1413-1436 (2000).MathSciNetzbMATHGoogle Scholar
  25. [25]
    C.J. Isham and J. Butterfield. Some possible roles for topos theory in quantum theory and quantum gravity. Found. Phys. 30, 1707–1735 (2000).MathSciNetCrossRefGoogle Scholar
  26. [26]
    C.J. Isham and J. Butterfield. A topos perspective on the Kochen-Specker theorem: IV. Interval valuations. Int. J. Theor. Phys 41, 613–639 (2002).MathSciNetzbMATHCrossRefGoogle Scholar
  27. [27]
    K. Martin, P. Panangaden. A domain of spacetime intervals in general relativity. Commun. Math. Phys. 267, 563-586 (2006).MathSciNetzbMATHCrossRefGoogle Scholar
  28. [28]
    D.S. Scott. Outline of a mathematical theory of computation. In Proceedings of 4th Annual Princeton Conference on Information Sciences and Systems, 169–176 (1970).Google Scholar
  29. [29]
    D.S. Scott. Continuous lattices. In Toposes, Algebraic Geometry and Logic, Lecture Notes in Mathematics 274, Springer, 93–136 (1972).Google Scholar
  30. [30]
    D.S. Scott. Formal semantics of programming languages. In Lattice theory, data types and semantics, Englewood Cliffs, Prentice-Hall, 66–106 (1972).Google Scholar
  31. [31]
    S. Willard. General Topology. Addison-Wesley Series in Mathematics, Addison Wesley (1970).Google Scholar
  32. [32]
    L. Ying-Ming, L. Ji-Hua. Solutions to two problems of J.D. Lawson and M. Mislove. Topology and its Applications 69, 153–164 (1996).Google Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Quantum Group Department of Computer ScienceUniversity of OxfordOxfordUK

Personalised recommendations