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.
Mathematics Subject Classification (2010). Primary 81P99; Secondary 06A11, 18B25, 46L10.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
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).
M. Caspers, C. Heunen, N.P. Landsman, B. Spitters. Intuitionistic quantum logic of an n-level system. Found. Phys. 39, 731–759 (2009).
B. Coecke, K. Martin. A partial order on classical and quantum states. In New Structures for Physics, ed. B. Coecke, Springer, Heidelberg (2011).
A. Döring. Kochen-Specker theorem for von Neumann algebras. Int. Jour. Theor. Phys. 44, 139–160 (2005).
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).
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).
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).
A. Döring. The physical interpretation of daseinisation. In Deep Beauty, ed. Hans Halvorson, 207–238, Cambridge University Press, New York (2011).
A. Döring, and C.J. Isham. A topos foundation for theories of physics: I. Formal languages for physics. J. Math. Phys 49, 053515 (2008).
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).
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).
A. Döring, and C.J. Isham. A topos foundation for theories of physics: IV. Categories of systems. J. Math. Phys 49, 053518 (2008).
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).
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).
C. Flori. A topos formulation of consistent histories. Jour. Math. Phys 51 053527 (2009).
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).
C. Heunen, N.P. Landsman, B. Spitters. A topos for algebraic quantum theory. Comm. Math. Phys. 291, 63–110 (2009).
C. Heunen, N.P. Landsman, B. Spitters. The Bohrification of operator algebras and quantum logic. Synthese, in press (2011).
C. Heunen, N.P. Landsman, B. Spitters. Bohrification. In Deep Beauty, ed. H. Halvorson, 271–313, Cambridge University Press, New York (2011).
C.J. Isham. Topos methods in the foundations of physics. In Deep Beauty, ed. Hans Halvorson, 187–205, Cambridge University Press, New York (2011).
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).
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).
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).
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).
C.J. Isham and J. Butterfield. Some possible roles for topos theory in quantum theory and quantum gravity. Found. Phys. 30, 1707–1735 (2000).
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).
K. Martin, P. Panangaden. A domain of spacetime intervals in general relativity. Commun. Math. Phys. 267, 563-586 (2006).
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).
D.S. Scott. Continuous lattices. In Toposes, Algebraic Geometry and Logic, Lecture Notes in Mathematics 274, Springer, 93–136 (1972).
D.S. Scott. Formal semantics of programming languages. In Lattice theory, data types and semantics, Englewood Cliffs, Prentice-Hall, 66–106 (1972).
S. Willard. General Topology. Addison-Wesley Series in Mathematics, Addison Wesley (1970).
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).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer Basel AG
About this chapter
Cite this chapter
Döring, A., Barbosa, R.S. (2012). Unsharp Values, Domains and Topoi. In: Finster, F., Müller, O., Nardmann, M., Tolksdorf, J., Zeidler, E. (eds) Quantum Field Theory and Gravity. Springer, Basel. https://doi.org/10.1007/978-3-0348-0043-3_5
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0043-3_5
Published:
Publisher Name: Springer, Basel
Print ISBN: 978-3-0348-0042-6
Online ISBN: 978-3-0348-0043-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)