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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Technical Abstract
Existing results on the use of topos logic in quantum physics are extended in significant ways, culminating with a new axiomatic scheme in which to formulate physical theories. The basic mathematics is that of the topos of presheaves over a category of contexts/world views; in particular, the internal logic of this topos is used to give a distributive replacement for the normal (non-distributive) quantum logic of projectors on a Hilbert space.
A key result is that the spectral presheaf sigma can be viewed as a quantum analogue of a classical space of microstates. The Kochen-Specher theorem is equivalent to the statement that sigma has no elements but, nevertheless, this object has many sub-objects that can be interpreted as generalised propositions in the quantum theory. Furthermore, these objects satisfy a distributive logic: a new quantum analogue of the Boolean logic satisfied by subsets of a classical state space. We show that conventional quantum theory can be entirely encoded in this structure, which thereby leads to a new axiomatic framework within which to formulate mathematical theories of physics. A key intention is to use this new framework for formulating, and giving physical meaning to, theories of quantum cosmology.