What question did this study set out to answer?

The central aim is to investigate how the electron can be modeled as a finite-energy configuration of a nonlinear field.

March 18, 2026Open Access

Physical Scaling and Phenomenology of a Finite-Energy Electron Core

Key Points

The central aim is to investigate how the electron can be modeled as a finite-energy configuration of a nonlinear field.
Analyzed stationary localized solution of nonlinear field equations
Mapped dimensionless configuration to physical units using a scaling parameter κ
Derived estimates for core size and deformation modes
Applied phenomenological constraints from experimental data
Identified five collective modes organized into dipolar and quadrupolar components
Defined the scaling parameter κ, with κ ≳ 10⁷ for effective point-like behavior
Provided a coherent interpretation supporting localized electron core model
Established consistency with experimental observations of the electron's nature

Abstract

This work presents the third stage of the finite-energy electron core research program, which investigates the possibility that the electron may be described as a localized finite-energy configuration of a nonlinear field. In earlier stages of this project, a stationary localized solution of the underlying nonlinear field equations was constructed numerically, and the spectral structure of the Hessian operator around this configuration was analyzed. That analysis revealed a compact deformation sector consisting of five collective modes with an approximate geometric organization into three dipolar and two quadrupolar components. The present study focuses on the physical scaling and phenomenological interpretation of these numerical results. The dimensionless stationary configuration and its deformation spectrum are mapped to physical units through a single scaling parameter κ, which determines the characteristic length and energy scales of the localized electron core. Using the dimensionless observables extracted from the stationary configuration and the deformation sector, the work derives estimates for the physical core size, the spatial extent of the deformation modes, and the associated deformation energy scale. Phenomenological constraints from experimental bounds on the electron size, precision measurements of the electron magnetic moment, and the absence of observed excited electron states lead to the condition κ ≳ 10⁷. Within this regime, the electron remains effectively point-like at currently accessible experimental scales while the model admits a localized finite-energy core endowed with a compact set of collective deformation modes at much smaller length scales. The results provide a coherent phenomenological interpretation of the finite-energy electron core framework and illustrate how particle-like behavior may arise from localized solutions of nonlinear field equations while remaining consistent with the experimentally observed point-like nature of the electron.

Physical Scaling and Phenomenology of a Finite-Energy Electron Core

Key Points

Abstract

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