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Effective actions, cutoff regularization, quasi-locality, and gluing of partition functions

A. V. Ivanov

TL;DR

The paper develops a universal quasi-local regularization for the quantum action of a scalar field on a compact Riemannian manifold by introducing a deforming averaging operator that generalizes cutoff regularization to curved spaces. This deformation, applied at the level of the classical action, yields a quasi-local regularization of the effective action whose perturbative expansion remains finite and well-defined, with regularized Green's functions constructed via the operator H_ω^Λ. A central result proves that this regularization is compatible with the gluing of manifolds and partition functions: gluing submanifold partition functions along a common boundary reproduces the partition function on the assembled manifold, both in the regulated and renormalized settings. The framework extends to other models and supports multiplicative renormalization, offering a robust, geometrically natural approach to locality, regularization, and the coherence of composite quantum systems on manifolds.

Abstract

The paper studies a regularization of the quantum (effective) action for a scalar field theory in a general position on a compact smooth Riemannian manifold. As the main method, we propose the use of a special averaging operator, which leads to a quasi-locality and is a natural generalization of a cutoff regularization in the coordinate representation in the case of a curved metric. It is proved that the regularization method is consistent with a process of gluing of manifolds and partition functions, that is, with the transition from submanifolds to the main manifold using an additional functional integration. It is shown that the method extends to other models, and is also consistent with the process of multiplicative renormalization. Additionally, we discuss issues related to the correct introduction of regularization and the locality.

Effective actions, cutoff regularization, quasi-locality, and gluing of partition functions

TL;DR

The paper develops a universal quasi-local regularization for the quantum action of a scalar field on a compact Riemannian manifold by introducing a deforming averaging operator that generalizes cutoff regularization to curved spaces. This deformation, applied at the level of the classical action, yields a quasi-local regularization of the effective action whose perturbative expansion remains finite and well-defined, with regularized Green's functions constructed via the operator H_ω^Λ. A central result proves that this regularization is compatible with the gluing of manifolds and partition functions: gluing submanifold partition functions along a common boundary reproduces the partition function on the assembled manifold, both in the regulated and renormalized settings. The framework extends to other models and supports multiplicative renormalization, offering a robust, geometrically natural approach to locality, regularization, and the coherence of composite quantum systems on manifolds.

Abstract

The paper studies a regularization of the quantum (effective) action for a scalar field theory in a general position on a compact smooth Riemannian manifold. As the main method, we propose the use of a special averaging operator, which leads to a quasi-locality and is a natural generalization of a cutoff regularization in the coordinate representation in the case of a curved metric. It is proved that the regularization method is consistent with a process of gluing of manifolds and partition functions, that is, with the transition from submanifolds to the main manifold using an additional functional integration. It is shown that the method extends to other models, and is also consistent with the process of multiplicative renormalization. Additionally, we discuss issues related to the correct introduction of regularization and the locality.

Paper Structure

This paper contains 17 sections, 23 theorems, 102 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The main definitions
  3. Classical action
  4. Submanifolds
  5. Operators and their properties
  6. Effective action
  7. Cutoff regularization
  8. Deformation in the Euclidean case
  9. Generalization of the deformation
  10. Regularization of functionals
  11. Application to the gluing process
  12. Definitions and relations
  13. The main statement
  14. Corollaries
  15. Conclusion
  16. ...and 2 more sections

Key Result

Corollary 1

For $\Lambda\to+\infty$ the submanifold $\mathcal{M}_\Lambda$ tends to $\mathcal{M}$.

Figures (1)

  • Figure 1: The upper part we show the manifold $\mathcal{M}$ with the boundaries $Y_l$ on the left and $Y_r$ on the right, as well as the truncated (deformed) manifold $\mathcal{M}_\Lambda$, which differs from the original by the cut-off neighborhoods of the boundary with a thickness of $1/\Lambda$. Below we have two submanifolds $\mathcal{M}_{l}$ and $\mathcal{M}_{r}$, obtained by cutting the manifold $\mathcal{M}$ along $\Sigma$. Similarly, we obtain the deformed submanifolds $\mathcal{M}_{l,\Lambda}$ and $\mathcal{M}_{r,\Lambda}$, as well as the corresponding vectors of external normals.

Theorems & Definitions (73)

  • Remark 1
  • Definition 1
  • Definition 2
  • Remark 2
  • Definition 3
  • Corollary 1
  • Definition 4
  • Corollary 2
  • Definition 5
  • Definition 6
  • ...and 63 more