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Interactions and Effective Forces Between Topological Defects in Emergent Schrödinger Dynamics

Jakob Viñas Solé

Published Jan 24, 2026• 0 citations
Publisher

Abstract

Title:Interactions and Effective Forces Between Topological Defects in Emergent Schrödinger Dynamics Publication type:Preprint / Working Paper Description: This paper investigates the emergence of particle interactions in a classical wave system through topological defects. Building on previous work establishing an emergent Schrödinger equation in classical elastic media, we show how multiple particle-like excitations—identified as topological phase singularities—interact via phase-mediated mechanisms without introducing fundamental force laws. Key findings: Defect interactions arise automatically from the shared global wavefield, with no explicit interaction terms in the underlying wave equation. The interaction is long-ranged and geometry-driven, with like topological charges repelling and opposite charges attracting in a Coulomb-like manner. Inertia emerges from the distributed kinetic energy of the wavefield, yielding an effective mass that depends on the defect structure and system geometry. The dynamics exhibits deterministic scattering and bound-state behavior governed entirely by wave-mediated topology. Despite structural similarities to quantum mechanics, the framework remains fully classical and deterministic, with no superposition of trajectories, entanglement, or quantum statistics. Theoretical framework: The analysis is formulated within classical continuum mechanics, where an effective Schrödinger equation governs the envelope dynamics of elastic waves. Particle-like excitations correspond to phase singularities whose motion is constrained by a guidance equation. Multi-defect configurations are encoded within a single global wavefield whose phase topology captures all interactions. Significance: This work demonstrates how concepts typically regarded as fundamental—particles, mass, forces, and interactions—can emerge from continuous classical fields through topological constraints. The model provides a concrete and physically transparent example of emergence in physics, with implications for understanding: the relationship between wave and particle descriptions, geometric origins of effective forces, classical analogs of quantum-like behavior, and the role of topology in dynamical systems. Mathematical approach: The paper employs tools from: topological field theory (phase singularities, winding numbers), differential geometry (phase gradient and velocity fields), classical mechanics (effective mass, scattering dynamics), and continuum mechanics (elastic wave equations). Relation to existing work: This study completes a self-consistent framework connecting wave dynamics, particle motion, and interactions within classical physics. It relates to vortex dynamics in fluids, pilot-wave hydrodynamics, and emergent phenomena in condensed matter systems, while maintaining clear conceptual and physical distinctions from quantum mechanics.

DOI
10.5281/zenodo.18357258

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Title:Interactions and Effective Forces Between Topological Defects in Emergent Schrödinger Dynamics Publication type:Preprint / Working Paper Description: This paper investigates the emergence of particle interactions in a classical wave system through topological defects. Building on previous work establishing an emergent Schrödinger equation in classical elastic media, we show how multiple particle-like excitations—identified as topological phase singularities—interact via phase-mediated mechanisms without introducing fundamental force laws. Key findings: Defect interactions arise automatically from the shared global wavefield, with no explicit interaction terms in the underlying wave equation. The interaction is long-ranged and geometry-driven, with like topological charges repelling and opposite charges attracting in a Coulomb-like manner. Inertia emerges from the distributed kinetic energy of the wavefield, yielding an effective mass that depends on the defect structure and system geometry. The dynamics exhibits deterministic scattering and bound-state behavior governed entirely by wave-mediated topology. Despite structural similarities to quantum mechanics, the framework remains fully classical and deterministic, with no superposition of trajectories, entanglement, or quantum statistics. Theoretical framework: The analysis is formulated within classical continuum mechanics, where an effective Schrödinger equation governs the envelope dynamics of elastic waves. Particle-like excitations correspond to phase singularities whose motion is constrained by a guidance equation. Multi-defect configurations are encoded within a single global wavefield whose phase topology captures all interactions. Significance: This work demonstrates how concepts typically regarded as fundamental—particles, mass, forces, and interactions—can emerge from continuous classical fields through topological constraints. The model provides a concrete and physically transparent example of emergence in physics, with implications for understanding: the relationship between wave and particle descriptions, geometric origins of effective forces, classical analogs of quantum-like behavior, and the role of topology in dynamical systems. Mathematical approach: The paper employs tools from: topological field theory (phase singularities, winding numbers), differential geometry (phase gradient and velocity fields), classical mechanics (effective mass, scattering dynamics), and continuum mechanics (elastic wave equations). Relation to existing work: This study completes a self-consistent framework connecting wave dynamics, particle motion, and interactions within classical physics. It relates to vortex dynamics in fluids, pilot-wave hydrodynamics, and emergent phenomena in condensed matter systems, while maintaining clear conceptual and physical distinctions from quantum mechanics.
This paper shows that particle-like excitations can emerge naturally within classical wave physics as topological defects of an effective wavefield. Focusing on a two-dimensional elastic medium whose envelope dynamics reduces to an effective Schrödinger equation, we demonstrate that phase singularities—points where the wave amplitude vanishes—behave as robust, localized objects carrying conserved topological charge. These defects follow deterministic trajectories governed by a guidance equation formally identical to that of de Broglie–Bohm pilot-wave theory, yet derived entirely from classical continuum mechanics without invoking quantum postulates. The central result is that defect cores move with velocityv=(ℏeff/meff)∇S,\mathbf{v} = (\hbar_{\mathrm{eff}}/m_{\mathrm{eff}})\nabla S,v=(ℏeff/meff)∇S,where SSS is the phase of the emergent wavefunction. This wave-guided dynamics arises as a kinematic constraint: the defect must remain a zero of the evolving field. The paper presents an asymptotic analysis of defect structure and motion, and discusses both the parallels with quantum particle dynamics and the fundamental limitations of the analogy, including the absence of superposed trajectories, entanglement, and quantum statistics. Overall, the work provides a concrete example of how topology and dispersion in classical systems can give rise to quantum-like behavior without quantization, illustrating a physically transparent route to emergent particle dynamics. Keywords: emergent dynamics; topological defects; phase singularities; pilot-wave theory; classical analogs of quantum mechanics; Schrödinger equation; guidance equation; elastic waves
This paper examines the foundational implications of a trilogy demonstrating that particle-like behavior, effective forces, and guidance dynamics can emerge from classical wave systems through topological constraints. Particles are identified as phase singularities (topological defects), interactions arise from geometric phase distortions, and mass emerges from field reconfiguration costs. The work clarifies which quantum-like features (guidance equations, discrete charges) can arise classically from wave topology, and which genuinely quantum aspects (superposition, entanglement) require nonclassical principles. This perspective shifts emphasis from postulation to emergence, suggesting that some apparently fundamental features—particles, forces, quantization—may reflect universal patterns arising from wave dynamics and topological constraints in continuous media.
The mathematical framework of quantum mechanics is conventionally founded upon a set of abstract axioms governing a Hilbert space. This work investigates whether this formalism can be derived as the effective description of a deeper, deterministic physical reality. We propose a mechanical model wherein a fundamental particle is treated as a 1-dimensional elastic defect (a string-like oscillator) topologically anchored to a background medium. By analyzing the dynamics of this system in the linear regime, we demonstrate that the core tenets of quantum theory emerge as necessary theorems of classical wave mechanicssubject to geometric constraints. Specifically, we show that: (1) the quantization of energy levels is a direct consequence of boundary conditions and resonance; (2) the Hilbert space structure is the natural mathematical arena for describing the superposition of stable vibrational modes; and (3) the momentum and energy operators, along with the Schrödinger equation, arise uniquely from the definition of observables and the conservation of energy. This approach grounds the quantum formalism in the mechanics of a continuous medium, suggesting that quantum behavior is not a break from classical physics, but the natural manifestation of wave dynamics in a confined topology.
In this thesis the features that arise from the interference of simple coherent waves are described, namely the topological defects of phase known as phase vortices permeating such wavefields. Phase vortices are associated with the rotation of energy and mass and arise naturally in regions containing coherent light and/or matter. The simplest possible system exhibiting these defects is that of three linearly interfering plane waves, described using the classical theory of monochromatic complex scalar wavefields. We explore the related case of three outgoing waves produced by point sources, such as a laser-illuminated screen that has been punctured by three pinholes. We develop a description of the resulting vortex structure in the optical far-field, and relate this to the source arrangement and parameters. This allows an estimate of the number and positions of the vortices, and leads to connections to particular results from information theory and number theory. Numerical simulations of the near-field defect structure are also presented. Findings from this study are then applied to the time evolution of a nonlinear Bose-Einstein condensate of atoms, which shares the property of coherence that enables its description by a related model of interfering Gaussian wavepackets. Numerical modelling of the nonlinear system is performed for both trapped and untrapped Bose-Einstein condensates (BECs) in two and three spatial dimensions, for BEC pieces initially positioned at the three corners of an equilateral triangle. We demonstrate the production of a distorted honeycomb vortex-antivortex lattice, due to wavepacket interference. The wavepacket expansion is halted by the presence of a magnetic trap, causing the lattice to melt and providing a population of mobile vortices and antivortices. The addition of nonlinearity in this system over the linear case leads to the formation of vortex clusters, whose dynamics we investigate. A model of a Laguerre-Gauss (LG) laser beam, which carries a phase vortex along its axis, is studied following its interaction with a ground-glass plate. We present preliminary results showing that the beam acquires interference-generated vortices whose number density evolves as a function of propagation distance and beam topological charge, due to interaction with the roughened glass. Beyond the nonlinear coherent wavefields applied to the description of Bose-condensed gases, a pseudo-spinor system described as a nonlinear coupled system of two complex scalar fields is studied. Such systems admit the existence of texture defects similar to those that exist in ferromagnetic systems. We numerically model a pseudo-spinor system that we arrange to generate carefully aligned honeycomb vortex-antivortex lattices in the coupled components. We find that this corresponds to a texture defect lattice and identify its constituent texture defects as merons and a type of planar defect. Following the analogy with the single component BEC, we also study texture defects and clusters in the trapped pseudo-spinor system.
Phase Differential Theory (PDT) proposes that quantum mechanics emerges as an effective description of a deeper dynamical phase manifold. In this framework, four scalar phase fields define a locked background whose long‑wavelength envelope dynamics reproduce the Schrödinger equation. The Born rule arises as the natural conserved probability density compatible with locality and continuity of phase flow, and measurement is interpreted as phase synchronisation between subsystems rather than wavefunction collapse. This paper develops a controlled derivation of the Schrödinger equation as an envelope approximation, identifies the conserved probability current associated with phase flow, and shows that |ψ|² is the natural density selected by locality, positivity, and conservation. Measurement is modelled as a dynamical locking of relative phase between system and apparatus, yielding discrete outcomes without introducing non‑unitary dynamics at the fundamental level. All results are obtained within a low‑energy, long‑wavelength effective regime. The microscopic equations of motion for the phase manifold are intentionally deferred, as the effective envelope theory presented here is self‑contained and does not require explicit microscopic specification.
This work proposes a structural theory of wave dynamics in which quantum, statistical, and cosmological phenomena emerge from a single underlying principle: whether a wave process achieves phase closure or remains phase-open. Within a discrete conservative framework, wave evolution is shown to admit two qualitatively distinct regimes. Phase-closed processes generate stationary normal modes and localized particle-like states, whereas phase-open processes produce dispersive envelope dynamics and persistent background fluctuations. In this framework, Schrödinger-type evolution is not postulated as a fundamental law but arises universally as the effective continuum description of structurally incomplete (phase-open) wave processes. A central result of this paper is that Heisenberg-type uncertainty relations emerge as a structural consequence of phase openness rather than measurement postulates or operator non-commutativity. Persistent background fluctuations, including vacuum noise and cosmological background phenomena, are likewise interpreted as unavoidable residuals of incomplete phase closure in conservative wave systems. The theory further introduces a probabilistic structural law governing phase closure, predicting that particle-like states are rare events in a dominantly phase-open universe. This leads to falsifiable scaling relations connecting microscopic phase dynamics to macroscopic cosmological matter fractions. The framework is integrated with the JS–SH discrete geometry and the SRCD ratio coordinate, providing a unified structural interpretation of quantum mechanics, particle formation, and cosmic background structure. Unlike conventional approaches, this work does not modify quantum mechanics by additional axioms or stochastic assumptions. Instead, it derives quantum and cosmological phenomena from structural properties of discrete wave dynamics, offering a unified conceptual and mathematical foundation linking Schrödinger evolution, uncertainty, and cosmic background fluctuations.