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Oxford Quantum Talks Archive

On model theory, noncommutative geometry and physics

Boris Zilber, University of Oxford
Categories, Logic and Foundations of Physics VI, March 2010, University of Oxford

Studying possible relations between a mathematical structure and its description in a formal language Model Theory developed a hierarchy of a 'logical perfection'. On the very top of this hierarchy we discovered a new class of structures called Zariski geometries. A joint theorem by Hrushovski and the speaker (1993) indicated that the general Zariski geometry looks very much like an algebraic variety over an algebraically closed field, but in general is not reducible to an algebro-geometric object. Later the present speaker established that a typical Zariski geometry can be explained in terms of a possibly noncommutative 'co-ordinate' algebra. Moreover, conversely, many quantum algebras give rise to Zariski geometries and the correspondence 'Co-ordinate algebra - Zariski geometry' for a wide class of algebras is of the same type as that between commutative affine algebras and affine varieties. General quantum Zariski geometries can be approximated (in a certain model-theoretic sense) by quantum Zariski geometries at roots of unity. The latter are of a finitary type, where Dirac calculus has a well-defined meaning. We use this to give a mathematically rigorous calculation of the Feynman propagator in a few simple cases.

[video] [streaming video]