Quasi-local probability averaging in the context of cutoff regularization

Abstract: In this paper, we study the properties of averaged fundamental solutions of a special type for Laplace operators in the Euclidean space of an arbitrary dimension. We consider a class of kernels suitable for probabilistic averaging, and propose new representations for the deformed fundamental solutions and their values at zero. In addition, we give examples related to specific quantum field models in the context of studying renormalization properties.

• The paper provides explicit analytic forms for a quasi-local averaging operator, regularizing Green's functions in QFT to address divergences in the Laplace operator.
• It demonstrates the smoothness, monotonicity, and symmetric properties of the double-averaged kernel across various dimensions, ensuring well-behaved regularization.
• The study applies the method in scalar, sextic, and sigma models, offering parameter-dependent numerical bounds that guide precise renormalization techniques in quantum field theory.

Summary of "Quasi-local probability averaging in the context of cutoff regularization" (2603.28235)

Fundamental Solution Averaging and Motivation

The paper conducts a rigorous investigation of quasi-local probabilistic averaging methods as a regularization mechanism for Laplace operator fundamental solutions in $\mathbb{R}^n$, addressing applications prominent in perturbative quantum field theory (QFT). The central mathematical object is the Laplace operator $A_n(x)=-\sum_{i=1}^{n}\partial_{x_i}^2$ and its fundamental solution $G_n(|x|)$, which serves as a local approximation to QFT Green's functions, and whose regularization is often required to address divergences in perturbative expansions. The introduced quasi-local averaging operator, parameterized by kernel $\omega$ and regularization radius $\Lambda$, smears fields in a small neighborhood, with double averaging reflecting Gaussian pairings in Wick's theorem and producing regularized Green's functions $G_{n,\omega}^{\Lambda}(r)$.

Mathematical Development: Explicit Representations

The paper presents two main theorems. Theorem 1 provides explicit analytic forms for the double-averaged fundamental solution kernel, $k_n(r,s,t)$, including closed-form representations for its smoothness, monotonicity, symmetry, and behavior under the Laplace operator. Specifically:

• $k_n(r,s,t)$ transitions between $G_n(\max(t,s))$ for $r<|t-s|$, a sum $A_n(x)=-\sum_{i=1}^{n}\partial_{x_i}^2$0 in $A_n(x)=-\sum_{i=1}^{n}\partial_{x_i}^2$1, and $A_n(x)=-\sum_{i=1}^{n}\partial_{x_i}^2$2 for $A_n(x)=-\sum_{i=1}^{n}\partial_{x_i}^2$3.
• The function is continuous, symmetric, decreasing, strictly positive for $A_n(x)=-\sum_{i=1}^{n}\partial_{x_i}^2$4; its maximum occurs at $A_n(x)=-\sum_{i=1}^{n}\partial_{x_i}^2$5.
• The Laplacian $A_n(x)=-\sum_{i=1}^{n}\partial_{x_i}^2$6 is symmetric and continuous for $A_n(x)=-\sum_{i=1}^{n}\partial_{x_i}^2$7, with specified discontinuities for $A_n(x)=-\sum_{i=1}^{n}\partial_{x_i}^2$8 at $A_n(x)=-\sum_{i=1}^{n}\partial_{x_i}^2$9.

Theorem 2 gives a general class of averaging kernels $G_n(|x|)$0 admissible for probabilistic averaging, providing representations for the regularized Green's function and its value at zero. Several structural properties are established:

• For $G_n(|x|)$1, $G_n(|x|)$2 is continuous, decreasing, and strictly positive, attaining maximum at $G_n(|x|)$3.
• The value at the origin admits two equivalent integral representations, with explicit dependence on the averaging kernel.
• Under mild regularity conditions $G_n(|x|)$4, the first derivative and Laplacian of $G_n(|x|)$5 are continuous, with exact limiting behaviors for $G_n(|x|)$6 and $G_n(|x|)$7.

A corollary shows that kernels with compact support near the upper boundary yield bounds for the regularized Green's function, related to minimal mass in renormalized QFT.

Applications in Quantum Field Theory Models

The paper systematically analyzes three new examples relevant for QFT:

Sphere Averaging (Scalar Models)

Averaging over spheres of equal radius, realized via a $G_n(|x|)$8-concentrated kernel, minimizes $G_n(|x|)$9, yielding the lowest value for the regularized function and hence for the renormalized mass. The approach highlights that the kernel class for averaging extends beyond $\omega$0, with limiting procedures producing sharp cutoff kernels. Explicit finite representations in low dimensions (e.g., $\omega$1) are cited.

Three-dimensional Sextic Model

Regularization in three dimensions is detailed via an exponential family of kernels, $\omega$2, parameterized by $\omega$3. Explicit formulas are provided for $\omega$4 and its Laplacian. Key numerical results include:

• $\omega$5 is strictly decreasing in $\omega$6, interpolating between divergence (removal of regularization) and the sharp cutoff minimum.
• $\omega$7 shows non-monotonic behavior, with a unique extremum, diverging as $\omega$8.
• The results align with regularization schemes in the sextic model, demonstrating additional freedom in fixing renormalization coefficients.

Two-dimensional Sigma Model with Mixed Cutoff

The two-dimensional case explores mixed regularization: kernels are constructed by truncating the Fourier transform (momentum cutoff) in the coordinate representation, yielding a Bessel-function kernel. The analysis provides sharp estimates for functionals $\omega$9 appearing in the sigma model's renormalization:

• As $\Lambda$0, these functionals vanish, allowing exact cancellation of special-type functionals.
• Technical lemmas yield rates of convergence and bounds, employing oscillatory integral estimates and asymptotics of Fresnel and Bessel functions.

Implications and Future Directions

The theoretical analysis gives a unified framework for quasi-local probabilistic averaging as a regularization tool, supporting practical schemes in QFT computations, including higher-loop renormalization. The formal properties—smoothness, monotonicity, extremal values, and kernel flexibility—offer clear criteria for selecting regularization in scalar and sigma models. The established framework provides more degrees of freedom in parameterizing renormalization constants, crucial for precision calculations and exploring nontrivial model behaviors.

Mathematical implications extend to the theory of positive definite functions and convolution roots, suggesting future directions in inverse kernel reconstruction and non-probabilistic averaging cases. Tasks such as characterizing negative-valued kernels, reconstructing averaging kernels from regularized Green's functions, and generalizing to non-Euclidean manifolds remain compelling.

Conclusion

The paper offers a comprehensive analytic investigation into the structure of regularized fundamental solutions under quasi-local probability averaging, supplying explicit formulas, sharp numerical bounds, and practical examples in quantum field theory. The approach not only clarifies the mathematical underpinnings of cutoff regularization but also enhances the toolkit for renormalization methods in scalar and sigma models. Further exploration of inverse problems and kernel generalization is warranted for broader theoretical and computational advancement.