Categories, Logic and the Foundations of Physics 2

Imperial College London, Wednesday May 14, 2008

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10.05-10.45 Fay Dowker, Imperial College London
Dynamical logic
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Despite the high regard in which physicists hold General Relativity, the spacetime nature of reality has not yet fully been taken to heart in addressing the question of the interpretation of quantum mechanics. Partial progress was made by Dirac and Feynman by casting the dynamical content of quantum theory in terms of a Sum Over (spacetime) Histories (SOH). Recently it has been suggested that this SOH is part of an interpretive framework in which the rules of inference that are used to reason about physical reality are themselves subject to dynamical law. Just as General Relativity showed that geometry is not fixed and absolute, so Quantum Mechanics may be telling us that “logic” is not a fixed background but part of physics.

10.55-12:35 Basil Hiley, Birkbeck College London
Towards a quantum geometry: groupoids, Clifford algebras and shadow manifolds
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I will present some ideas, which although not yet formulated in terms of categories, uses the spirit of category theory especially as articulated by Lawvere. Starting from first principles, I follow Kauffman in making distinctions between two aspects of each indivisible sub-process and then order these processes to form a groupoid, which is then generalised into what I call an algebra of process. The resulting algebra is shown to be isomorphic to a hierarchy of orthogonal Clifford algebras, which, of course, include the Pauli and Dirac algebras. I then exploit the minimal left and right ideals to map from the algebra of process to a vector space inducing a light cone structure thus inverting the usual approach since here the vector space inherits its structure from the fundamental processes. Although I am working with the algebra of process, all of this is still within the conceptual framework used by Clifford himself, namely, classical physics. I then present an argument to generalise this structure to include symplectic Clifford algebras enabling us to introduce the Heisenberg group. Using the techniques applied to orthogonal CAs, I am able to show that the idempotents of this algebra map onto the points of an underlying manifold. Since the symplectic group acts in (x, p) phase space algebra, quantum mechanics demands that this structure is non-commutative and it is this feature that produces shadow manifolds. I will discuss the significance of these results.

12.45-13.35 Lunch break

13.40-14.40 Peter Johnstone, University of Cambridge
Topos-theoretic models of the continuum
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We survey a number of different approaches to modelling the continuum in a topos, with particular reference to such questions as whether the real line should be viewed as a space or a locale, the ring-theoretic properties of the continuum, and whether it is sensible to require that all real functions should be continuous or even smooth.

14.50-15.30 Bruce Bartlett, University of Sheffield
Aspects of duality in 2-categories
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The notion of ‘duality’ plays an important role in quantum algebra and topological quantum field theory, as has been particularly emphasized by Baez and Dolan. One aspect of this is the idea of duals for morphisms in a 2-category, which is a generalization of the idea of rigidity in monoidal categories. I will introduce the notion of an ‘even-handed structure’ on a 2-category as a coherent means of turning right adjoints into left adjoints, and explain how this works in various examples such as fusion categories, braided monoidal categories, 2-Hilbert spaces and derived categories having a ‘trivial Serre functor’.

15.40-16.30 Discussion session
16.30-17.00 Coffee break

17.00-17.40 Paul-André Melliès, Université Paris Diderot
Quantum groupoids and logical dualities
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17.50-18.30 Paolo Bertozzini, Thammasat University
Categories of spectral geometries
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In A. Connes' non-commutative geometry, "spaces" are described "dually" as spectral triples. We provide an overview of some of the notions that we deem necessary for the development of a categorical framework in the context of spectral geometry, namely: (a) several notions of morphism of spectral geometries, (b) a spectral theory for commutative full C*-categories, (c) a tentative definition of strict-n-C*-categories, (d) spectral geometries over C*-categories. If time will allow, we will speculate on possible applications to foundational issues in quantum physics: categorical covariance, spectral quantum space-time and modular quantum gravity.

19.00 Discussion in the pub

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