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Existence theorem on the UV limit of Wilsonian RG flows of Feynman measures

Andras Laszlo, Zsigmond Tarcsay, Jobst Ziebell

TL;DR

This work establishes a rigorous UV-completion framework for Euclidean QFT via Wilsonian RG flows of Feynman measures: under mild conditions there exists a unique UV-limit measure $\mu$ on distribution-sense fields such that every UV-regularized instance $\mu_{\eta}$ is a marginal of $\mu$, i.e., $\mu_{\eta}=(C_{\eta})_{*}\mu$. It also proves that when two RG flows have a relative interaction potential at some scale, a UV-limit relative potential $V$ exists with $\mu=e^{-V}\gamma$, preserving lower bounds, and with a constant field-renormalization factor possible (dimensional transmutation). The results carry over to a range of regulator schemes (continuous, bandlimited) and are illustrated by case studies including $\varphi^{4}$-type models and bounded-density potentials, where some models emerge as nonperturbatively renormalizable while others are disfavored. Collectively, the findings provide a mathematically solid bridge between Wilsonian RG flows and their UV limits, clarify the role of relative potentials, and offer insights into which interactions can yield well-defined UV completions in continuum Euclidean settings.

Abstract

In nonperturbative formulation of Euclidean signature quantum field theory (QFT), the vacuum state is characterized by the Wilsonian renormalization group (RG) flow of Feynman measures. Such an RG flow is a family of Feynman measures on the space of ultraviolet (UV) regularized fields, linked by the Wilsonian renormalization group equation. In this paper we show that under mild conditions, a Wilsonian RG flow of Feynman measures extending to arbitrary regularization strengths has a factorization property: there exists an ultimate Feynman measure (UV limit) on the distribution sense fields, such that the regularized instances in the flow are obtained from this UV limit via taking the marginal measure against the regulator. Existence theorems on the flow and UV limit of the corresponding action functional are also discussed.

Existence theorem on the UV limit of Wilsonian RG flows of Feynman measures

TL;DR

This work establishes a rigorous UV-completion framework for Euclidean QFT via Wilsonian RG flows of Feynman measures: under mild conditions there exists a unique UV-limit measure on distribution-sense fields such that every UV-regularized instance is a marginal of , i.e., . It also proves that when two RG flows have a relative interaction potential at some scale, a UV-limit relative potential exists with , preserving lower bounds, and with a constant field-renormalization factor possible (dimensional transmutation). The results carry over to a range of regulator schemes (continuous, bandlimited) and are illustrated by case studies including -type models and bounded-density potentials, where some models emerge as nonperturbatively renormalizable while others are disfavored. Collectively, the findings provide a mathematically solid bridge between Wilsonian RG flows and their UV limits, clarify the role of relative potentials, and offer insights into which interactions can yield well-defined UV completions in continuum Euclidean settings.

Abstract

In nonperturbative formulation of Euclidean signature quantum field theory (QFT), the vacuum state is characterized by the Wilsonian renormalization group (RG) flow of Feynman measures. Such an RG flow is a family of Feynman measures on the space of ultraviolet (UV) regularized fields, linked by the Wilsonian renormalization group equation. In this paper we show that under mild conditions, a Wilsonian RG flow of Feynman measures extending to arbitrary regularization strengths has a factorization property: there exists an ultimate Feynman measure (UV limit) on the distribution sense fields, such that the regularized instances in the flow are obtained from this UV limit via taking the marginal measure against the regulator. Existence theorems on the flow and UV limit of the corresponding action functional are also discussed.

Paper Structure

This paper contains 7 sections, 16 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Wilsonian renormalization group (RG) flows
  3. Existence theorem of the UV limit measure
  4. Existence of UV limit of relative interaction potential
  5. Case studies on example models
  6. Concluding remarks
  7. On the supports of Gaussian measures

Key Result

Lemma 4

Let $X$ and $Y$ be Hausdorff locally convex topological vector spaces, and suppose that $X$ is a countable union of weakly compact sets. Then for every continuous map $u:\,X\to Y$ and weakly closed set $A\subset Y$, the set $u[X]+A$ is a Borel set in $Y$ with respect to the weak and the original top

Theorems & Definitions (50)

  • Definition 1
  • Definition 2
  • Remark 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Remark 6
  • Remark 7
  • Lemma 8
  • ...and 40 more