Applications of geometric logic to topos approaches to quantum theory
This EPSRC-funded project starts in September 2009 with Dr Bertfried Fauser as Research Fellow and Guillaume Raynaud on a PhD studentship.p>
Contents
- Summary of programme
- Background: quantum physics
- Background: the topos approach
- Using geometric logic
- Research Group
- References
A deep mystery of quantum physics is its inherent non-determinism. The outcome of a measurement on a quantum system has a randomness that cannot be explained away as representing just our uncertain knowledge of what precise state the system is in. Technically (the Kochen-Specker Theorem), there is mathematically no possibility in most quantum systems of describing "classical" states that consistently, and unequivocally, say what value every possible measurement would give.
One approach to understanding this is the "neo-realism" of Isham at Imperial College, recently with Doering, and taken up also by Landsman, Spitters and Heunen at Nijmegen. There are ways of seeing the system classically (with classical states), but none describes all possible measurements and they cannot be fitted together coherently. Isham's insight is that the resulting logic of systems, which varies according to which classical viewpoint is adopted, can be described overall as a non-standard "internal" logic arising out of a mathematical structure known as a topos - comprising the sheaves over a base space whose points in these quantum applications include those classical viewpoints. Now logic asks not "whether" something is true, but "where" - from which points of view. In the non-standard logic, the quantum system appears classical and has classical states. Withdrawing to standard logic, however, the classical states cannot consistently be retained - although their probabilistic distributions can and these are what we see in quantum physics.
The internal logic - and corresponding mathematics - of toposes can be difficult to work with. Some standard principles don't work. Also, the usual "point-set" idea of topological space (a set of points together with some subsets specified as "open") must be replaced by a "point-free" approach that describes the opens independently of points. The points are constructed subsequently, although there may be too few of them for the opens to be uniquely distinguished by their points. It was developed in pure mathematics, has been found to give excellent results with a range of non-standard logics, and has also been applied in computer science, with the opens related to theories of observations on computer programs.
Working with the point-free topologies directly in the internal logic is technical and difficult. However (Joyal/Tierney), they can equivalently be viewed as point-free "bundles" over the base space - that is to say, maps from another space to the base. In referring to a map as a bundle, one is thinking of it as a variable space - for each point of the base, we have a fibre over it, the inverse image of that point under the map, and as the point varies so too does its fibre.
Ideally, our internal reasoning about internal point-free spaces should also apply to the fibres, but this true only for a certain "geometric" fragment of the internal logic. Technically, the geometric constructions on the bundles are those that are preserved by bundle pullback, and this covers the fibres. By careful interpretation of logic, geometric reasoning also can work validly through the points of the point-free spaces, despite the possible shortage of them. Techniques of geometric reasoning have been developed by the proposer, with particular exploitation of "powerlocales" (point-free hyperspaces, or spaces of spaces).
The project aims to exploit those geometricity techniques in the topos approach to quantum physics, reexpressing it in terms of more familiar topological concepts - points, bundles, fibres - instead of internal point-free spaces. The goal is to make the topos approach more accessible to physicists and help clarify its relationship with other physics formalisms. It is also an excellent case study for testing out the general mathematical scope of geometricity.
Background: quantum physics
It was discovered in the 20th century that microscopic physical systems have an inevitably probabilistic aspect. Under the preceeding "classical" understanding of physics, any physical quantity is determined completely by a well-defined underlying state of the system. In quantum systems this is no longer so. The underlying state deternines only the range of possible values for a quantity, and the probabilities of obtaining them as the result of a measurement.
In itself, the probabilistic aspect itself is not so new. Even classical physics is used to getting probabilities arising from uncertainty and lack of knowledge. But it assumes that fully known states are possible, at least in principle, and that the probabilities represent statistical distributions over those.
In quantum physics both theory and experiment indicate that this cannot be. Threre are no deterministic states underlying the probabilistic ones. The problem comes to a head with quantities that are in some sense incompatible with each other, such as position and momentum: Heisenberg's uncertainty principle says that the more certainty there is for one, the less certainty there can be for the other. Technically, quantities are compatible if they "commute" with each other.
Background: the topos approach
Starting around 1998, physicist Prof. Chris Isham (at Imperial College) and philosopher Jeremy Butterfield investigated applying topos theory to describing quantum systems. Their idea can be summarized roughly as follows.
A quantum system is described by the measurements that can be made on it. Some of those (like the complementary quantities of position and momentum) may be non-commuting, so there is no underlying state that completely describes the values of all those measurable quantities. However, if one restricts to a commuting subset of the quantities then there are such states.
Isham and Butterfield think of such a commuting subset of quantities as a "classical point of view". Viewing the system only through those quantities, it is still possible to think classically, and get a Gelfand-Naimark spectrum of states for those quantities. But it is impossible to reconcile those different points of view to get states that cover all the quantities at the same time. A famous theoretical result - the Kochen-Specker Theorem - asserts this.
The topos trick is to take a space C whose points are those classical points of view, and then to use the topos of sheaves over C. The stalks at a given point are then what one sees from that point of view. In the internal mathematics of sheaves, the physics can be understood classically. The "neo-realist" strategy of Isham and Butterfield is to work classically in that internal mathematics (this is classical physics, not classical logic - the internal logic of a topos is normally non-classical), and then extract it to give external results.
They showed that in this way of thinking, the Kochen-Specker Theorem says that the internal spectra have no cross-sections: it is impossible to consistently choose a spectrum point for every classical point of view. However, there are cross-sections for statistical distributions of spectrum points, and these cross-sections come out as the non-deterministic quantum states already known about.
To summarize: the non-deterministic quantum states are statistical distributions of deterministic states, but only if you work internally in a topos. It is hoped that this internal working, though logically non-classical, can be made physically classical.
Since the original Isham-Butterfield work, the idea has been developed by two groups: by Isham himself, with Andreas Döring; and a group at Nijmegen led by Klaas Landsman.
Using geometric logic
My project at Birmingham will investigate the applications of geometric logic to this topos approach.
Geometric logic can be applied to the internal mathematics of toposes, and its big advantage is that it can be interpreted fibrewise (or stalkwise). What is done geometrically is comfortably close to an idea of "classical physics, parametrized by the classical point of view".
Its disadvantage is that it is weaker than strictly necessary for topos-inernal reasoning. However, there is mounting evidence that it is strong enough for large amounts of practical mathematics. The aim of the project is to demonstrate this for the mathematics relevant to the quantum applications (in particular, the theory of Gelfand-Naimark spectra) and to use its expositional advantages to give a more conceptual account that does not rely on heavy calculations of sheaves.
Two side aims are use the insights gained to further our understanding of
- spectra for non-commutating systems of quantities, and
- the general applicability of the geometric reasoning principles in settings for non-classical mathematics other than topos theory. A number of these are studied in computer science.
Research Group
The EPSRC project is running for 3 years, starting September 2009. It includes funding for Dr Bertfried Fauser as Research Fellow, and Guillaume Raynaud as PhD student.
References
Döring and Isham, 'What is a thing?': Topos theory in the foundations of physics (2008), arXiv:0803.0417v1; to appear in "New Structures in Physics".
Heunen, Landsman and Spitters (2008), A Topos for Algebraic Quantum Theory, arXiv:0709.4364v2
"Aspects of Topology", "Locales via Bundles" and "The Topos Approach in the Qubit Case" - slides from three talks of mine that refer to the physics issues.