A Toy Model of Renormalization and Reformulation

TL;DR

This paper explores a simplified mechanical model to understand renormalization, revealing how correcting fundamental constants restores physical accuracy and exposes deeper physics of coupled constituents, despite initial modeling errors.

Contribution

It demonstrates the application of renormalization in a toy mechanical model, highlighting how to correct for modeling errors and uncover deeper physical insights.

Findings

Renormalization restores correct physical properties.

Incorrect self-interaction models can be corrected through renormalization.

Exact equations can sometimes accidentally match correct physics.

Abstract

I consider a specially designed simple mechanical problem where "particle acceleration" due to an external force creates sound waves. Theoretical description of this phenomenon should provide the total energy conservation. To introduce "small radiative losses" into the phenomenological mechanical equation, I advance first an "interaction Lagrangian" similar to that of the Classical Electrodynamics (kind of a self-action ansatz). New, "better-coupled" mechanical and wave equations manifest unexpectedly wrong dynamics due to changes of their coefficients (masses); thus this ansatz fails. I show how we make a mathematical error with advancing a self-interaction Lagrangian. I show, however, that renormalization of the fundamental constants in the wrong equations works - the original inertial properties of solutions are restored. The exactly renormalized equations contain only physical fundamental constants, describe well the experimental data, and reveal a deeper physics - that of permanently coupled constituents. The perturbation theory is then just a routine calculation giving small and physical corrections. I demonstrate that renormalization is just illegitimately discarding harmful corrections compensating this error, that the exactly renormalized equations may sometimes accidentally coincide with the correct equations (a fluke), and that the right theoretical formulation of permanently coupled constituents can be fulfilled directly, if realized.