QM IV — QM IV: Interaction Dynamics of Defect States

Reader summary

Particles in PDT are not point-like. They are stable knots in the phase manifold. So where do forces come from? QM IV answers that. When one defect moves or wobbles, it disturbs the phase fields around it. Another defect feels that disturbance. Working through the maths, the answer takes a form physicists already know: a source, a propagator, another source. Massive disturbances give short-range Yukawa forces. Massless disturbances give long-range Coulomb-like forces. The familiar structure of relativistic field theory comes out of the soliton picture, without putting it in by hand.

Abstract

Phase Differential Theory (PDT) identifies particle states with topologically stabilised defects of a dynamical phase manifold. This paper develops the interaction dynamics of multiple defect states within a controlled effective-field-theory regime. Using a collective-coordinate formulation, we construct the residual source currents generated by time-dependent defect configurations and determine the response of the phase manifold through the linearised fluctuation operator. Integrating over mediator modes yields an explicit effective interaction energy between defects. The resulting interaction energy has a universal bilinear form in the induced sources. Massive mediator modes generate Yukawa-type forces, while massless modes produce long-range gauge-like interactions. The structure matches the interaction form obtained in relativistic field theory, providing an effective soliton-based interaction framework consistent with known interaction structures. Coefficient-level mapping and nonlinear corrections are deferred to subsequent work.

Falsifiable predictions

01
Inter-defect interactions take the universal bilinear form V_int = (1/2) ∫ J G J. Any leading-order interaction that cannot be cast in this source-propagator-source form falsifies the framework.
02
Massive mediator modes give Yukawa potentials V(r) = Q₁Q₂ e^(−mr)/(4πr); massless modes give Coulomb potentials V(r) = Q₁Q₂/(4πr). No long-range non-Coulombic forces are predicted at leading order.
03
Three effective interaction channels arise: scalar (Yukawa), vector (gauge-like), tensor (gravitational). No fourth channel of comparable strength is allowed at leading EFT order.

Full paper

1. Introduction

In conventional quantum field theory, interactions between particles arise through the exchange of mediator fields. PDT offers a different starting point: particle states correspond to extended, topologically stabilised defects embedded in a dynamical phase manifold. Interactions must therefore emerge from the response of the phase manifold to defect motion and internal excitation.

This paper derives the interaction structure explicitly within an EFT framework. Starting from the microscopic phase-field action, we construct the multi-defect configuration, identify the residual source generated by defect motion, and solve the linearised fluctuation operator to obtain the mediator propagator. Substituting the resulting field perturbation back into the action yields a universal quadratic interaction energy.

Assumptions

- **A1.** Stable, finite-energy topological defects. - **A2.** Defect motion slow compared to internal relaxation. - **A3.** Linearised response around static defect background. - **A4.** Large separation compared to defect size. - **A5.** Leading-order EFT truncation.

2–3. Multi-defect ansatz and collective-coordinate Lagrangian

For N well-separated defects Φ(x,t) ≈ Σ Φ_i(x − X_i(t), Ω_i(t)). The collective Lagrangian is L = Σ [(1/2) M_i Ẋ_i² + (1/2) I_i Ω_i²] − V_int.

4–5. Residual source and linearised operator

Static defects make the action stationary. Time-dependent collective coordinates generate a residual source J(x,t). To leading order J ≈ Σ [−M_i Ẍ_i · ∇Φ_i + internal]. Expanding the action to quadratic order gives a fluctuation operator L = −∂_t² + ∇² − V'(Φ_static).

6–7. Propagator and effective interaction

The Green function satisfies L G(x,x') = δ⁴(x−x'), with momentum-space form G(k) = 1/(k² − m² + iε). Substituting back gives S_eff = S_static + (1/2) ∫ J G J. The interaction potential is the spatial bilinear V_int = (1/2) ∫ J G J.

8–10. Mediator modes, Yukawa and gauge-like

The operator decomposes into scalar, vector, and tensor mediator modes. For a massive mediator, G(r) = exp(−mr)/(4πr) gives V(r) = Q₁Q₂ exp(−mr)/(4πr) with effective charges Q_i = ∫ J_i ψ_α. For a massless mediator, V(r) = Q₁Q₂/(4πr).

11–12. Correspondence and channels

Integrating out a mediator field in standard field theory gives an identical bilinear form S_int = (1/2) J G J. Three effective channels arise: scalar (Yukawa), vector (gauge-like), tensor (gravitational).

13–15. Validity, limitations, conclusion

The derivation assumes large separation, nonrelativistic motion, linearised fluctuations, and leading-order EFT. Relativistic corrections, nonlinear multi-defect interactions and SM coefficient mapping are deferred. Interactions between defects reproduce Yukawa and Coulomb forms and match relativistic field theory.

BibTeX

@article{fincham_hilton_qm_iv,
  author  = {Fincham, Graham and Hilton, Daniel},
  title   = {Phase Differential Theory and the Interaction Dynamics of Defect States: Mediator Modes, Collective Coordinates, and Emergent Forces},
  journal = {Phase Differential Theory Series, QM IV},
  year    = {2025}
}