Topos-Theoretic Approaches to Quantum Theory

Starting from a naive investigation into the nature of experiments on a physical system one can argue that states of the system should pair nondegenerately with physical observables. This duality is closely related to that between space and quantity, or, geometry and algebra. In particular, it is grounded in the mathematical framework of both classical and quantum mechanics in the form of a pair of duality theorems by Gelfand and Naimark. In particular, they allow us to construct a classical phase space, a compact Hausdorff space, for an algebra of classical observables. In the case of quantum mechanics, this construction breaks down due to non-commutative nature of the algebra of quantum observables. However, we can construct a Hilbert space as the geometry underlying quantum mechanics. Although this Hilbert space approach to quantum mechanics has proven to be very effective, it does have its drawbacks. In particular, these arise when one associates propositions to observables and investigates what kind of logical structure they form. One way of doing this is by realising the propositions as certain subsets of the phase space. In classical mechanics this procedure indeed gives one the structure one would expect: a Boolean algebra. However, although the case of quantum mechanics yields a nice mathematical structure, an orthocomplemented lattice, the physical interpretation of this logic is rather subtle, due to its crude notion of truth. Recent work by Isham, Butterfield, Doering, Landsman, Spitters, Heunen et al., attempting to address these problems, has led to an alternative method for dealing with non-commutative algebras of observables and with that an alternative framework for quantum kinematics. Moreover, it stays much closer to our intuition from classical physics, in some sense, the motto being: Quantum kinematics is exactly like classical kinematics, that is, not in Set, but internal to some other topos! This review paper gives a first introduction to the subject. It attempts to provide a stepping stone towards more serious papers.