Wick Ordering and Kinetic Energy Renormalization for L\'evy White Noise Fields

TL;DR

This paper investigates the mathematical formulation of kinetic energy for Levy white noise fields on a torus, employing Wick ordering and renormalization techniques, and finds limitations in renormalizability for certain models.

Contribution

It introduces a generalized Wick ordering approach to define the kinetic energy as an $L^1$ function for Levy white noise fields and analyzes renormalization challenges.

Findings

Wick ordering helps define the kinetic energy as a distribution.

Higher order renormalization is needed but the model appears non-renormalizable.

The approach extends to Gamma fields but does not fully eliminate divergences.

Abstract

Let $S=\mathbb{T}^d$ be a torus and $\mu$ the probability distribution of a L\'evy white noise field $x:S\rightarrow\mathbb{R}$. Using projective limit measures we address the problem of making sense of $\mathrm{e}^{-T(x)}$, where $T(x) = \int_S \lvert\nabla x(s)\rvert^2 \mathrm{d}s$ is the kinetic energy, as a function in $L^1(\mu)$. We start by making sense of $T(x)$ itself as a sort of distribution, which is achieved by a generalization of Wick ordering. Then we specify to the case of a $\Gamma$ field, finding that Wick ordering does not eliminate all divergences. Higher order renormalization would be required, but the model seems to be non-renormalizable.