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Scale-Dependent Suppression Functions and Functional Space Geometry in Renormalization

Daniel Ketels

TL;DR

This work introduces a scale-dependent regulator $\Omega(k,\Lambda)$ to regulate high-momentum modes without a hard cutoff, embedding the regulator within a dynamic RG-like framework. It shows that the suppression induces a modified metric on functional space, $ds_\Omega^2=\int \Omega(k,\Lambda)|\delta\phi(k)|^2\,dk$, leading to increasingly negative UV Ricci curvature and suggesting spectral suppression and possible dimensional reduction at high energies. The paper develops operator- and measure-theoretic foundations, proving that associated integral operators such as $T_\Omega$ can be Hilbert-Schmidt or trace-class under reasonable decay of $\Omega$, and demonstrates spectral gaps and compact embeddings in weighted spaces. Collectively, these results provide a rigorous mathematical framework for smooth regularization techniques, controlled UV behavior of function spaces, and RG-flow of curvature in renormalization settings, with potential implications for functional-integral convergence and spectral regularization.

Abstract

We analyze the effects of a scale-dependent suppression function $Ω(k, Λ)$ on the functional space geometry in renormalization theory. By introducing a dynamical cutoff scale $Λ$, the suppression function smoothly regulates high-momentum contributions without requiring a hard cutoff. We show that $Ω(k, Λ)$ induces a modified metric on functional space, leading to a non-trivial Ricci curvature that becomes increasingly negative in the ultraviolet (UV) limit. This effect dynamically suppresses high-energy states, yielding a controlled deformation of the functional domain. Furthermore, we derive the renormalization group (RG) flow of $Ω(k, Λ)$ and demonstrate its role in controlling the curvature flow of the functional space. The suppression function leads to spectral modifications that suggest an effective dimensional reduction at high energies, a feature relevant to functional space deformations and integral convergence in renormalization theory. Our findings provide a mathematical framework for studying regularization techniques and their role in the UV behavior of function spaces.

Scale-Dependent Suppression Functions and Functional Space Geometry in Renormalization

TL;DR

This work introduces a scale-dependent regulator to regulate high-momentum modes without a hard cutoff, embedding the regulator within a dynamic RG-like framework. It shows that the suppression induces a modified metric on functional space, , leading to increasingly negative UV Ricci curvature and suggesting spectral suppression and possible dimensional reduction at high energies. The paper develops operator- and measure-theoretic foundations, proving that associated integral operators such as can be Hilbert-Schmidt or trace-class under reasonable decay of , and demonstrates spectral gaps and compact embeddings in weighted spaces. Collectively, these results provide a rigorous mathematical framework for smooth regularization techniques, controlled UV behavior of function spaces, and RG-flow of curvature in renormalization settings, with potential implications for functional-integral convergence and spectral regularization.

Abstract

We analyze the effects of a scale-dependent suppression function on the functional space geometry in renormalization theory. By introducing a dynamical cutoff scale , the suppression function smoothly regulates high-momentum contributions without requiring a hard cutoff. We show that induces a modified metric on functional space, leading to a non-trivial Ricci curvature that becomes increasingly negative in the ultraviolet (UV) limit. This effect dynamically suppresses high-energy states, yielding a controlled deformation of the functional domain. Furthermore, we derive the renormalization group (RG) flow of and demonstrate its role in controlling the curvature flow of the functional space. The suppression function leads to spectral modifications that suggest an effective dimensional reduction at high energies, a feature relevant to functional space deformations and integral convergence in renormalization theory. Our findings provide a mathematical framework for studying regularization techniques and their role in the UV behavior of function spaces.

Paper Structure

This paper contains 17 sections, 8 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Momentum-Space Integral Convergence
  3. Implications for Renormalization.
  4. Functional Integrability
  5. Scale-Dependent Suppression of $\Omega(k,\Lambda)$
  6. Metric and Induced Curvature in Functional Space
  7. Operator-Theoretic Properties
  8. Definitions and Preliminaries
  9. Integral Operators and Compactness
  10. Spectral Properties and Regularization
  11. Suppression-Induced Spectral Gaps
  12. Measure-Theoretic Interpretation
  13. Weighted Measure Spaces and Absolute Continuity
  14. Integral Convergence in Weighted Spaces
  15. Compactification of Functional Space via Weighted Metric Structure
  16. ...and 2 more sections

Key Result

Proposition 1.1

Assume there exist constants $\eta>0$ and a smooth function $\epsilon:\,\mathbb{R}_{\ge0}\to(0,1]$ such that for all $\Lambda>0$ and all $k\in\mathbb{R}^d$, Then the scale-dependent suppression function satisfies In particular, $\Omega(\cdot,\Lambda)$ remains in the interval $(0,1]$ and never becomes negative.

Theorems & Definitions (21)

  • Proposition 1.1: Positivity and Boundedness of $\Omega$
  • proof
  • Remark 1.2: Analytic Properties of $\Omega$
  • Theorem 2.1: Convergence of Suppressed Integrals
  • proof
  • Theorem 2.2: Well-Defined Functional Measure
  • proof
  • Remark 3.1
  • Definition 4.1: Hilbert-Schmidt Operator
  • Definition 4.2: Trace-Class Operator
  • ...and 11 more