[Paper Review] A new perspective on renormalization: the scattering transformation

Michael E. Glinsky|arXiv (Cornell University)|Jun 22, 2011

Particle physics theoretical and experimental studies3 citations

TL;DR

This paper introduces a novel non-perturbative renormalization framework using the scattering transformation—originally developed in signal analysis—by reinterpreting it as a simultaneous transformation of the field and the time coordinate in quantum field theory. The method maps the original field to a function of the inverse time scale (renormalization parameter), yielding a form-invariant S-matrix and an effective action expressed as a functional of the classical field and a scale-dependent transfer matrix, enabling renormalization without ad hoc regularization and offering potential for quantum gravity and strongly coupled QCD.

ABSTRACT

The scattering transformation developed by Mallat is put into the perspective of field theory. It is shown to be a simultaneous transformation of the field and the "time" parameter explicitly used in the definition of path integrals central to the Lagrangian approach and in the definition of the "time" ordered products used in the Hamiltonian (or canonical) approach. This transformation preserves the form of the S-matrix as "time" ordered products. The transformed "time" coordinate is the inverse "time" scale. This is traditionally the UV cutoff or renormalization parameter in standard approaches to renormalization. The critical calculation will be the determination of the form of the effective action in this transformed coordinate system. This action will now be expressed as an integral of a Lagrangian density that is a function of the renormalization parameter or transformed "time". Other symmetries of the action can be explicitly built into the transformation. It will be demonstrated on a simple 1D $ϕ^4$ field theory. This non-perturbative approach has great potential in possibly being used to renormalize quantum gravity and obtaining expressions for the strongly coupled limit of QCD.

Motivation & Objective

• To provide a non-perturbative, symmetry-preserving approach to renormalization in quantum field theory.
• To resolve longstanding challenges in regularization that break symmetries or require ad hoc procedures.
• To unify the path integral and Hamiltonian approaches via a transformation that redefines time as the inverse renormalization scale.
• To enable the calculation of the S-matrix and effective action directly in terms of scale-dependent classical fields and transfer matrices.
• To extend the applicability of renormalization to strongly coupled systems like quantum gravity and QCD in the strong coupling limit.

Proposed method

• The scattering transformation is applied to map the field $ f(x) $ and the time coordinate $ x $ to a new representation $ \psi(\lambda) $, where $ \lambda $ is the inverse time scale (UV cutoff or renormalization parameter).
• The transformation simultaneously redefines the field and the time ordering, preserving the form of the S-matrix as a time-ordered product in the new coordinate system.
• The effective action is derived in the transformed coordinates, expressed as a functional of the classical field $ \varphi_0(\lambda) $ and the inverse transfer matrix $ S_2^{-1}(\lambda, \lambda') $, which encodes scale-dependent mass and dynamics.
• The method incorporates additional symmetries of the action directly into the scattering operator, ensuring they are preserved under renormalization.
• The approach is validated in a 1D $ \phi^4 $ theory, showing that the physics is encoded in $ \varphi_0(\lambda) $ and $ S_2^{-1}(\lambda, \lambda') $, with solutions corresponding to self-similar or multifractal behavior depending on the transfer matrix.
• The generating functional $ Z[J(\lambda)] $ is constructed to leading order in $ \hbar $, using the classical action and quantum fluctuation corrections as functions of the renormalization scale.

Experimental results

Research questions

• RQ1 How can the scattering transformation from signal analysis be systematically applied to quantum field theory to achieve non-perturbative renormalization?
• RQ2 Can the S-matrix be preserved under a simultaneous transformation of the field and the time coordinate, with time reinterpreted as the inverse renormalization scale?
• RQ3 What is the form of the effective action in the transformed coordinate system, and how is it related to the classical field and scale-dependent transfer matrix?
• RQ4 How can global and gauge symmetries of the original action be explicitly built into the scattering transformation?
• RQ5 Can this framework yield renormalized solutions for strongly coupled theories such as quantum gravity or QCD in the strong coupling limit?

Key findings

• The scattering transformation redefines the time coordinate as the inverse renormalization scale, eliminating UV divergences and singularity issues inherent in conventional field theory calculations.
• The S-matrix remains invariant under the transformation, preserving its physical interpretation as a scattering amplitude.
• The effective action is expressed as a functional of the classical field $ \varphi_0(\lambda) $ and the inverse transfer matrix $ S_2^{-1}(\lambda, \lambda') $, which captures the scale evolution of the system.
• For self-similar physics, the transfer matrix is constant, leading to scale-invariant behavior; for multifractal behavior, it depends on the ratio of initial to final scales.
• The method provides a compact, non-perturbative representation of the physics, with only two key functions—$ \varphi_0(\lambda) $ and $ S_2^{-1}(\lambda, \lambda') $—encoding the full dynamics to leading order in $ \hbar $.
• The framework enables extrapolation of scale behavior from partial S-matrix data and provides a natural metric for comparing textures in complex systems, as established by Mallat’s original work.