Categories, Logic and the Foundations of Physics 1

Imperial College London, Wednesday January 9, 2008

The full schedule and announcement is given here.

12.00-13.00 Chris Isham, Imperial College London
Topos theory in the formulation of theories of physics
Video, Slides
(no abstract provided)

13.00-13.30 Chris Heunen, University of Nijmegen
A topos for algebraic quantum theory
Video, Slides
Motivated by Bohr's idea that the empirical content of quantum physics is accessible only through classical physics, we show how a C*-algebra A induces a topos in which the amalgamation of all its commutative subalgebras comprises a single commutative C*-algebra. According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum X in the topos, which plays the role of a quantum phase space of the system. States on A become probability integrals on X, and self-adjoint elements of A define functions from X to the pertinent internal real numbers (the interval domain), allowing for a state-proposition pairing. Thus the quantum theory defined by A is turned into a classical theory by restriction to its associated topos.

13.30-14.15 Lunch

14.15-15.15 Samson Abramsky, University of Oxford
Categorical quantum mechanics: The "monoidal" approach
Video, Slides
(no abstract provided)

15.15-15.45 Ross Duncan, University of Oxford
Classical structures, MUBs, and pretty pictures
Video, Slides
(no abstract provided)

15.45-16.30 General discussion session

16.30-17.00 Coffee break

17.00-17.30 Jamie Vicary, Imperial College London
A categorical framework for the quantum harmonic oscillator
Video, Slides
I will describe a categorical approach to the construction of symmetric Fock space, the state space of the quantum harmonic oscillator. Many of the conventional mathematical tools used to study this system — such as raising and lowering operators, and coherent states — emerge naturally from the category theory, and satisfy the usual equations. However, the formalism is more general than the conventional approach, and I will describe how to construct an infinite variety of 'exotic' Fock spaces. I will finish with the question: "Where has the 'quantumness' come from?"

17.30-18.00 Louis Crane, Kansas State University
Relational topology and quantum gravity
We explore arguments for replacing the absolute point set by a sheaf over the site of observation as a foundation for quantum gravity. Time permitting, we consider apparent geometry as a formulation for relational geometry.

18.15-19.00 Pub session

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