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arXiv · 1706.05740

On the renormalization group of quantum field theory (after R. Bocherds)

G. A. Palau
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Abstract

The aim of this thesis is to analyze part of the results presented by R.E. Borcherds in his article on renormalization and quantum field theories. To facilitate this, the first chapter presents the basic facts on category theory, sheaf theory and the theory of distributions on a manifold. In the particular case of distributions on manifolds we differentiate between the H-distribution (introduced by H\"ormander) and the D-distributions (described in the book of Dieudonn{\'e}) and extend the product of distributions defined by H\"ormander only for H-distibution, to an action from H-distributions over D-distributions. Each one of these facts will be used along the thesis in order to exhibit more detailed descriptions and explanations for the results. In Chapter 2 the concepts of spacetime, fields, Lagrangians, propagators and Feynman measure are defined. The notion of Feynman measure is introduced with a more detailed explanation of the Gaussian condition and its relation with the cut propagator. In the last chapter we define the renormalization group and immediately characterize it using the results from Chapter 1. The algebraic structure of this group is described in detail. We present a descending filtration of the renormalization group by normal subgroups, compute the corresponding quotients and commutators, and show that any element of the renormalization group can be written as an infinite product of elements each one living in one of the aforementioned subgroups. Lastly we show that an action from the renormalization group over the set of Feynman measures is well defined and prove that this action is transitive over the Feynman measures associated with a given cut local propagator.

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G. A. Palau. 2017-06-18. On the renormalization group of quantum field theory (after R. Bocherds). https://arxiv.org/abs/1706.05740

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