Nekovar S, Sahlmann H (2014)
Publication Language: English
Publication Type: Thesis
Publication year: 2014
Gaussian path integrals play an important role for free quantum field theories, and for the perturbative treatment of interacting quantum field theories. These path integrals are defined via measures on linear spaces. For loop quantum gravity, a framework for path integrals over spaces of connections was developed. Some examples of what one could call Gaussian measures are known. They are interesting, among other things, because they give the connections finite quantum mechanical fluctuations.
In the present thesis, we give a general definition of a Gaussian measure over (Abelian) connections, and study their properties. In particular, we define the holonomy-flux Weyl algebra for U(1). Then we investigate the absolute continuity of Gaussian measures with respect to representations of the Weyl algebra and the unitary implementation of gauge transformations. This shows that for a large subclass, the lengthlike Gaussian measures, there are no representations in which the exponentiated fluxes are implemented unitarily. We also determine some structural properties of Gaussian measures which are invariant under Euclidean transformations of the base manifold. As a side result, we discuss how new methods from analysis can be employed to show, how results obtained with well known algebraic techniques can be reproduced and strengthened using recent results on the convergence of Fourier series.
Nekovar, S., & Sahlmann, H. (2014). Gaussian Measures and Representations of the Holonomy-Flux Algebra (Master thesis).
Nekovar, Stefan, and Hanno Sahlmann. Gaussian Measures and Representations of the Holonomy-Flux Algebra. Master thesis, 2014.