Steve Vickers: Quantum Topos theory (for MGS)

Quantum Topos Theory

Course for the Midland Graduate School in Foundations of Computer Science, Sheffield 2010.

Despite the title "Quantum topos theory", this is not about any kind of "quantum toposes". However, there are recent ideas by which topos theory is being applied to the foundations of quantum physics. They originated with Chris Isham at Imperial College, first in collaboration with Jeremy Butterfield and subsequently with Andreas Doering. Later, similar ideas, but with some significant technical differences, were explored by a group at Nijmegen: Klaas Landsman, Bas Spitters and Chris Heunen.

Isham's essential idea starts from the fact that the classical idea of "state", determining the value of any measurement that might be made on a system, fundamentally does not work in quantum systems. His approach uses a topos whose points give partial "classical points of view" on the system. Those classical points of view are mutually inconsistent (more or less because of Heisenberg's uncertainty principle), but the topos has its own internal logic, which in a certain sense deals with them all simultaneously.

At Birmingham I now have a project to explore the extent to which a particular logical side of toposes, using the so-called "geometric logic", can be applied to this topos approach to quantum physics.

My aim in these lectures is to give a conceptual overview of how these ideas work. I shall start with some background in topology and continuity, then move on to toposes of sheaves, C*-algebras, and how they fit together in quantum theory.

An important idea will be that of "bundle", in which a space is thought of as not fixed, but varying continuously with a parameter taken from another space. What corresponds to classical state space is not a single space, but a bundle over a space of classical points of view.


  1. Topology via test maps: Topology really is structure in terms of which we can explain continuity. One way to understand it is that we choose a test space T, and then describe the topological structure on X by specifying the continuous "test functions" from X to T. A map f: X -> Y is then continuous if, by composition, it transforms continuous test functions on Y to ones on X. A key part of this is to know what operations on T are continuous, for this algebraic structure should then be lifted to the sets of test functions.
    The prime example is with T the Sierpinski space, its continuous operations being finite meets and arbitrary joins. This directly gives the usual definition and axiomatization of topological space (as collection of opens).
    Other examples of T we shall find relevant are 2 (the 2-element space with discrete topology), C (complex numbers; the reals can also be used in a similar way) and Sets (which leads to toposes).
  2. Spaces as algebras: We shall also need to apply the fundamental idea of point-free topology, that the algebra of test functions is more fundamental than the set of points. We shall see various dualities working this way.
    • T = Sierpinski, algebras = frames (locale theory), test function = open
    • T = 2, algebras = Boolean algebras (Stone's representation theorem), test function = clopen
    • T = complex numbers, algebras = C*-algebras (Gelfand-Naimark duality)
    • T = sets, algebras = Grothendieck toposes, test function = sheaf
  3. Toposes: Toposes can be understood as arising from the idea T = Sets, the "continuous operations" being the categorical ones of finite limits and arbitrary colimits. The "continuous test functions" to Sets are the sheaves, and I shall discuss how this matches the formulation of sheaf as local homeomorphism.
    Sheaves and opens give rise to the same notion of continuity, but with sheaves one can describe a vast generalization (due to Grothendieck) of topological space that includes, as prime example, the "space of sets".
    An important feature of "T = Sets" is that the algebras (the Grothendieck toposes) can also be used as generalized universes of sets and hence support an "internal" mathematics. When point-free topology is done internally, it turns out - rather remarkably - to correspond externally to bundles over the topos as base space. With suitable ("geometric") restrictions to the internal logic, the internal constructions correspond to fibrewise constructions on the bundles. A good example of such a geometric, fibrewise construction is that of valuation space, which takes any point-free space Y and gives a space V(Y) of valuations (regular measures) on it.
  4. Quantum: The case "T = C" leads to Gelfand-Naimark duality for commutative C*-algebras, each of which has a compact Hausdorff spectrum. But quantum systems rely on non-commutative C*-algebras (or, more narrowly, von Neumann algebras).
    I shall briefly discuss how they relate to the foundations of quantum theory. Classical physics presumes an underlying classical state space, a pure state determining the results of all possible measurements. Probabilistic accounts are then based on distributions (mixed states) over the classical state space. In quantum physics the Kochen-Specker theorem rules out such a classical state space. The quantum pure states, taken from a Hilbert space, are already probabilistic in nature, following the Born rule.
  5. Quantum+topos: Chris Isham at Imperial (with Butterfield and then Doering) has instigated a topos-theoretic approach using a topos whose points include commutative sub-C*-algebras of a non-commutative C*-algebra (more specifically for them: a von Neumann algebra) taken as "classical points of view" of a quantum system. This idea was also taken up, but with important technical differences, by Klaas Landsman's group at Nijmegen, involving Bas Spitters and Chris Heunen.
    In this approach there is a classical state space in the internal mathematics of the topos, but not externally - Kochen-Specker says that the different "classical points of view" cannot be consistently reconciled. However, the internal valuation space does have external points, and their probabilistic nature can be seen through the quantum pure states.
    I shall outline the Imperial and Nijmegen approaches and their differences, and also discuss possible extensions from their presheaf toposes to sheaf toposes.


Sunday, 4:30--5:30, LT6: lecture
Monday, 10:00--11:00, LT8: exercises; 4:30--5:30, LT6: lecture
Tuesday, 10:00--11:00, LT8: exercises; 4:30--5:30, LT6: lecture
Wednesday, 10:00--11:00, LT8: exercises; 4:30--5:30, LT6: lecture
Thursday, 10:00--11:00, LT8: exercises

Slides and exercise sheets

  1. Slides: Topology via test functions; Exercises
  2. Slides: Toposes; Exercises
  3. Slides: Quantum physics; Exercises
  4. Slides: Toposes in quantum foundations; Exercises


Most of my own papers are available on my web pages.

  1. Mac Lane and Moerdijk: Sheaves in Geometry and Logic. Springer-Verlag, 1992.
    A good introduction to sheaves and toposes.
  2. Vickers: Locales and Toposes as Spaces. Chapter 8 (pp. 429--496) in Handbook of Spatial Logics (ed.Aiello, Pratt-Hartmann and van Benthem), Springer, 2007.
    An introduction to toposes that stresses the point-free approach to topology using algebras.
  3. Banaschewski and Mulvey: A Globalisation of the {G}elfand Duality Theorem. Annals of Pure and Applied Logic 137 (2006), pp.62--103.
    A topos-theoretic treatment of commutative C*-algebras as algebras of test functions.
  4. Doering and Isham: What is a Thing?: Topos Theory in the Foundations of Physics. arXiv:0803.0417v1, 2008. To appear in "New Structures in Physics", ed R. Coecke.
  5. Heunen, Landsman and Spitters: A Topos for Algebraic Quantum Theory. Communications in Mathematical Physics 291 (1) (2009), pp. 63--110; doi:10.1007/s00220-009-0865-6
  6. Mermin: Quantum Computer Science. Cambridge University Press, 2007.
    Not directly relevant to the topos approach, but an excellent introduction to the theory of quantum computation.