Singularities of two-point functions in Quantum Field Theory

by Michal Wrochna
Doctoral thesis
Date of Examination:2013-08-16
Date of issue:2013-08-28
Advisor:Prof. Dr. Dorothea Bahns
Referee:Prof. Dr. Dorothea Bahns
Referee:Prof. Dr. Ingo Witt
Persistent Address: http://dx.doi.org/10.53846/goediss-4017

 

 

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Abstract

English

The main topic of the present thesis is the study of singularities of two-point functions of spin-0 and spin-1/2 quantum fields, possibly set on curved spacetime or in the presence of smooth, external electromagnetic potentials. The first part reviews the results necessary for the construction of neutral and charged non-interacting quantum fields on globally hyperbolic spacetimes, and supplements the arguments needed in the case when no charge symmetry exists. In this general situation, the Hadamard condition, which refers to the singularities of the two-point functions, is discussed, and its relation to the theory of distinguished parametrices of Duistermaat and Hoermander is explained. Additionally, similarities between the spin-0 and spin-1/2 case are exhibited by considering a two-component form of the Klein-Gordon equation. It is then used in the static case to reformulate the classical dynamics as an evolution equation whose generator is self-adjoint in the sense of Krein spaces. By methods of spectral theory in Krein space, we construct Hadamard two-point functions in the spin-0 case for a class of strong electric potentials which possess no ground state. The second part is concerned with renormalisation of interacting fields in the approach of Epstein and Glaser. We focus on the problem of recovering symmetries, possibly lost in the process of extending singular distributions on R^n\{0} to R^n. In our approach, this is done by imposing that the extended distributions are in the kernel of a given set of (differential) operators. The symmetries are then recovered using a map, which in typical applications turns out to be linear. The same method is applied to derive the relation between off-shell and on-shell time-ordered products for a scalar theory on Minkowski space.
Keywords: Quantum Field Theory; mathematical physics; Hadamard states; Epstein-Glaser method; renormalisation; microlocal analysis; Krein spaces
 
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