Convolutions and Gaussians in Renormalization
Raymond Puzio, Sam McCrosson
TL;DR
This work reinterprets renormalization through an algebraic lens that foregrounds Gaussian convolution and the oscillator group $\mathsf{Osc}(V)$. By constructing $\mathsf{Osc}(V)$ from $\mathrm{GL}(V)$ and additive structures, the authors define a convolution-based action on $C^{\infty}(V)$ and show that renormalization can be realized as a one-parameter submonoid of $\mathrm{GL}(V)\ltimes \mathrm{Sym}_2^+(V)$ acting on the space of interactions. Gaussian distributions emerge as fixed points under this action, and their closure under convolution is established via $\mathcal{N}(\mathbf{A})*\mathcal{N}(\mathbf{B})=\mathcal{N}(\mathbf{A}+\mathbf{B})$, providing a rigorous backbone for the renormalization steps (block formation, coarse graining, and rescaling) in a finite-dimensional setting. The framework connects the familiar ideas of Wilsonian renormalization with concrete algebraic structures, setting the stage for extending to infinite dimensions and relating to Polchinski-type flow equations in future work.
Abstract
The Kadanoff-Wilson-Fisher approach to renormalization is based upon studying the renormalization transform, which may be described as an action of the monoid $\mathbb{R}^{\times}_{\geq 1}$ on a suitable space of interactions. It is typically computed by manipulating the path integral or the perturbation series. Here we will present an alternative algebraic description of the renormalization transform. We treat the space of interactions as a semigroup under convolution and act on it with a Lie group associated with the quantum harmonic oscillator.