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Matrix Integrals: Calculating Matrix Integrals Using Feynman Diagrams
Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Theoretical Physics.
2014 (English)Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
Abstract [en]

In this project, we examine how integration over matrices is performed. We investigate and develop a method for calculating matrix integrals over the set of real square matrices. Matrix integrals are used for calculations in several different areas of physics and mathematics; for example quantum field theory, string theory, quantum chromodynamics, and random matrix theory. Our method consists of ways to apply perturbative Taylor expansions to the matrix integrals, reducing each term of the resulting Taylor series to a combinatorial problem using Wick's theorem, and representing the terms of the Wick sum graphically with the help of Feynman diagrams and fat graphs. We use the method in a few examples that aim to clearly demonstrate how to calculate the matrix integrals.

Abstract [sv]

I detta projekt undersöker vi hur integration över matriser genomförs. Vi undersöker och utvecklar en metod för beräkning av matrisintegraler över mängden av alla reell-värda kvadratiska matriser. Matrisintegraler används för beräkningar i ett flertal olika områden inom fysik och matematik, till exempel kvantfältteori, strängteori, kvantkromodynamik och slumpmatristeori. Vår metod består av sätt att applicera perturbativa Taylorutvecklingar på matrisintegralerna, reducera varje term i den resulterande Taylorserien till ett kombinatoriellt problem med hjälp av Wicks sats, och att representera termerna i Wicksumman grafiskt med hjälp av Feynmandiagram. Vi använder metoden i några exempel som syftar till att klart demonstrera hur beräkningen av matrisintegraler går till.

Place, publisher, year, edition, pages
2014. , 39 p.
Keyword [en]
fat graph, feynman diagram, matrix integral, perturbative expansion, wick's theorem
National Category
Physical Sciences
URN: urn:nbn:se:uu:diva-227928OAI: oai:DiVA.org:uu-227928DiVA: diva2:731610
Educational program
Bachelor Programme in Physics
Available from: 2014-07-02 Created: 2014-07-02 Last updated: 2014-07-02Bibliographically approved

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