A Rigorous Derivation of the Functional Renormalisation Group Equation

TL;DR

This paper provides a rigorous mathematical derivation of the functional renormalisation group equation within a framework compatible with quantum field theory, addressing regularisation, reflection positivity, and convergence issues.

Contribution

It introduces a general regularisation scheme that preserves key properties and analyzes the classical limit and convergence in the renormalisation process.

Findings

Regularisation scheme retains reflection positivity and smoothness.

Classical limit is modified by regularisation, breaking translation invariance.

Conditions for convergence of theories after removing regularisation.

Abstract

The functional renormalisation group equation is derived in a mathematically rigorous fashion in a framework suitable for the Osterwalder-Schrader formulation of quantum field theory. To this end, we devise a very general regularisation scheme which retains reflection positivity as well as the infinite degrees of freedom including smoothness. Furthermore, it is shown how the classical limit is altered by the regularisation process leading to an inevitable breaking of translation invariance. We also give precise conditions for the convergence of the obtained theories upon removal of the regularisation.