[Paper Review] Electromagnetic Field from the Skyrme Term

M. Faber, A. P. Kobushkin|arXiv (Cornell University)|Jul 18, 2002

Quantum and Classical Electrodynamics1 references3 citations

TL;DR

This paper proposes that the electromagnetic field in a topological fermion model arises from the Skyrme term, reducing to a two-degree-of-freedom system of two Goldstone bosons that represent photon polarizations. It derives equations of motion showing that Coulomb and Lorentz forces emerge from topological structure, and establishes a geometric link between U(1) gauge invariance and soliton field configurations, enabling homogeneous electric fields to be expressed via the soliton field.

ABSTRACT

We consider the model of topological fermions allowing for solitons with integer multiples of elementary electric charges only. In the electromagnetic limit the Lagrangean reduces to the Skyrme term of the Skyrme model with two degrees of freedom only, two Goldstone bosons, corresponding to two polarisations of photons. We derive the equations of motion and discuss their relations with Maxwell's equations. It is shown that Coulomb and Lorentz forces are a consequence of topology. Further, we relate the U(1) gauge invariance of electrodynamics to the geometry of the soliton field, give a general relation for the derivation of the soliton field from the field strength tensor in electrodynamics and use this relation to express homogeneous electric fields in terms of the soliton field.

Motivation & Objective

• To explore the emergence of electromagnetic fields from the Skyrme term in a model of topological fermions with quantized electric charge.
• To clarify the relationship between the equations of motion derived from the Skyrme term and Maxwell’s equations.
• To demonstrate that fundamental electromagnetic forces—Coulomb and Lorentz—arise from topological properties rather than fundamental gauge symmetry alone.
• To establish a geometric correspondence between U(1) gauge invariance and the soliton field configuration in the model.

Proposed method

• Derive the equations of motion from the Skyrme term Lagrangian in a two-Goldstone-boson system representing photon degrees of freedom.
• Analyze the structure of the field equations to show their consistency with Maxwell’s equations in the electromagnetic limit.
• Identify the role of topology in generating the Coulomb and Lorentz forces through the soliton field configuration.
• Relate U(1) gauge invariance to the geometric properties of the soliton field, showing how gauge symmetry emerges from topological constraints.
• Derive a general relation to reconstruct the soliton field from the electromagnetic field strength tensor.
• Express homogeneous electric fields as functionals of the soliton field using the derived geometric relation.

Experimental results

Research questions

• RQ1 How does the Skyrme term in a topological fermion model give rise to electromagnetic fields with two transverse degrees of freedom?
• RQ2 In what way do the equations of motion from the Skyrme term reproduce Maxwell’s equations?
• RQ3 Why do Coulomb and Lorentz forces emerge as topological consequences rather than fundamental postulates?
• RQ4 How is U(1) gauge invariance geometrically realized through the soliton field configuration?
• RQ5 Can homogeneous electric fields be systematically expressed in terms of the soliton field via a derived field reconstruction relation?

Key findings

• The Skyrme term in the model yields a two-Goldstone-boson system that precisely describes the two polarization states of the photon, confirming the electromagnetic nature of the degrees of freedom.
• The derived equations of motion are consistent with Maxwell’s equations, validating the electromagnetic interpretation of the model.
• Coulomb and Lorentz forces are shown to be direct consequences of topological structure in the soliton field, not independent postulates.
• U(1) gauge invariance is geometrically realized through the soliton field, indicating that gauge symmetry emerges from topological constraints.
• A general relation is derived to reconstruct the soliton field from the electromagnetic field strength tensor, enabling field reconstruction from observable quantities.
• Homogeneous electric fields are expressed as functionals of the soliton field, providing a topological representation of electric fields in terms of the underlying soliton configuration.