Quantum Field Theories with Symmetries in the Wilsonian Exact Renormalization Group

TL;DR

This thesis develops a framework for implementing symmetries within the Wilsonian Exact Renormalization Group, enabling analysis of gauge, chiral, and supersymmetric theories while preserving their fundamental symmetries.

Contribution

It introduces a method to incorporate and analyze symmetries in the ERG, including the derivation of Slavnov-Taylor identities and handling of anomalies and supersymmetry.

Findings

Derived Slavnov-Taylor identities at any momentum scale.

Showed how to obtain chiral anomalies within ERG.

Extended ERG to supersymmetric gauge theories with preserved supersymmetry.

Abstract

The purpose of the present thesis is the implementation of symmetries in the Wilsonian Exact Renormalization Group (ERG) approach. After recalling how the ERG can be introduced in a general theory (i.e. containing both bosons and fermions, scalars and vectors) and having applied it to the massless scalar theory as an example of how the method works, we discuss the formulation of the Quantum Action Principle (QAP) in the ERG and show that the Slavnov-Taylor identities can be directly derived for the cutoff effective action at any momentum scale. Firstly the QAP is exploited to analyse the breaking of dilatation invariance occurring in the scalar theory in this approach. Then we address SU(N) Yang-Mills theory and extensively treat the key issue of the boundary conditions of the flow equation which, in this case, have also to ensure restoration of symmetry for the physical theory. In case of a chiral gauge theory, we show how the chiral anomaly can be obtained in the ERG. Finally, we extend the ERG formulation to supersymmetric (gauge) theories. It is emphasized regularization is implemented in such a way that supersymmetry is preserved.