Kok, Willem Adriaan, de (2021) The relation between Feynman diagrams and Kontsevich graphs in non-commutative star products. Master's Thesis / Essay, Mathematics.
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Abstract
We discuss two approaches to go from classical Poisson mechanics to non-commutative associative geometries. In regular quantum mechanics (after Heisenberg, Dirac and many others), the associative commutative algebra of coordinate functions is replaced by the non-commutative associative algebra of linear operators that act on the Hilbert space of wave functions. In contrast, using ideas by Weyl and Wigner, Groenewold and Moyal suggested to deform the old algebra of coordinate functions on phase space, i.e. a symplectic Poisson manifold, to an associative non-commutative algebra by deforming the old product to a star product. This is known as deformation quantization. After a breakthrough result by Kontsevich (1997): all Poisson brackets can be deform-quantized with a star product - Cattaneo and Felder (2000) rediscovered a remarkable Poisson sigma model (from quantum field theory (QFT)): its action combines a given Poisson structure with Lobachevsky hyperbolic geometry. The calculation of correlation functions in that model reproduces the perturbative expansion (in hbar) of the Kontsevich star product. This calculation relates the Feynman diagram technique from QFT to the oriented Kontsevich graphs in deformation quantization. We illustrate the relation by calculating the Kontsevich star product perturbatively with Feynman’s QFT-technique up to and including order hbar^3, and thereby how selection rules reduce the big set of Feynman diagrams to the set of Kontsevich graphs.
| Item Type: | Thesis (Master's Thesis / Essay) |
|---|---|
| Supervisor name: | Kiselev, A.V. and Boer, D. |
| Degree programme: | Mathematics |
| Thesis type: | Master's Thesis / Essay |
| Language: | English |
| Date Deposited: | 31 Aug 2021 11:59 |
| Last Modified: | 31 Aug 2021 11:59 |
| URI: | https://fse.studenttheses.ub.rug.nl/id/eprint/25793 |
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