Emergence in Action — Part 4 – The Naturalistic Outlook

The Naturalistic Outlook

Emergence in Action — Part 4

From Interactions Between Classical Fields to Interactions Between Particles

In the previous parts, we saw how continuous fields can give rise to objects displaying several properties usually associated with particles: localized energy, momentum, inertia, and relativistic dynamics.

This fourth part takes the next step: what happens when the fields interact with one another?

The approach is deliberately unusual. Several of the ideas discussed here are normally introduced within quantum field theory. Here, however, we begin by exploring them using entirely classical fields, with the aim of building physical intuition before moving to the quantum framework.

The chapter follows a sequence of increasingly particle-like phenomena: field-mediated interactions, the role of mass in determining interaction range and excitation thresholds, the emergence of a conserved charge in a complex field, the distinction between matter and antimatter, soliton formation, collisions, annihilation, and pair creation.

A central part of the chapter is devoted to classical solitons: localized and relatively stable field structures that provide a much closer classical analogue of particles. Their dynamics make it possible to illustrate phenomena resembling fusion, radiation, matter–antimatter annihilation, and pair production through parametric resonance.

The final section returns to relativity. Once accelerated, the solitons display a contraction of their spatial width and a slowing of their internal dynamics close to the Lorentz predictions. The model therefore provides a concrete illustration of how relativistic behaviour can emerge at the continuum level from microscopic models that are not themselves fundamentally relativistic.

Interactive animations

The article is accompanied by several interactive simulations that allow the reader to explore the models directly:

principle of stationary action :