Dr. Christopher J. Isham
Imperial College

Project Title:
Topos Quantum Theory

Co-Investigators:
Andreas Doering, Imperial College

Summary:
In fundamental physics, we are faced with the task of reconciling and uniting two extremely successful, but conceptually very different theories: general relativity, which describes the large-scale structure of the universe in the context of a theory of space and time; and quantum theory, which describes the world of atoms and smaller.

Virtually all approaches to quantum gravity try to change or adapt general relativity; quantum theory is taken more or less for granted. Our research programme aims at a novel reformulation of quantum theory using the concept of a topos - a sophisticated mathematical structure that generalizes set theory. One important feature of topos theory is that a proposition such as "the physical quantity A has a value in a certain range" need not be simply true or false: rather, there are more possibilities that are given by the intrinsic logic that is possessed by a topos. In our programme to reformulate quantum theory in this way, we have already found some deep and beautiful structures. However, much needs to be done before we can express the whole of quantum theory in a topos language. Our long-term goal is to find a topos framework in which all theories of physics can be meaningfully discussed and which will, in particular, allow significant generalizations of quantum theory when applied to quantum cosmology and quantum gravity.



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