Topological phenomena typically govern the behavior of delocalized waves, giving rise to robust transport in electronic, photonic, and mechanical systems. Whether similar principles can directly control the motion of a localized particle, particularly one dynamically coupled to the field that guides it, has remained largely unexplored. Here we show that topology can govern the dynamics of a self-guided particle. Using a walking droplet whose motion is coupled to a self-generated wave field, we demonstrate that structuring the wave environment enables band-gap mediated particle exclusion, edge-guided transport, and chirality-dependent orbital dynamics arising from an emergent gauge structure. Unlike conventional topological systems, where topology constrains wave propagation alone, the present system allows global geometric structure to act directly on particle trajectories. These results extend topological control from waves to particles and establish a route toward directing matter through global geometric design rather than local forcing.
Topological phenomena typically govern the behavior of delocalized waves, giving rise to robust transport in electronic, photonic, and mechanical systems. Whether similar principles can directly control the motion of a localized particle, particularly one dynamically coupled to the field that guides it, has remained largely unexplored. Here we show that topology can govern the dynamics of a self-guided particle. Using a walking droplet whose motion is coupled to a self-generated wave field, we demonstrate that structuring the wave environment enables band-gap mediated particle exclusion, edge-guided transport, and chirality-dependent orbital dynamics arising from an emergent gauge structure. Unlike conventional topological systems, where topology constrains wave propagation alone, the present system allows global geometric structure to act directly on particle trajectories. These results extend topological control from waves to particles and establish a route toward directing matter through global geometric design rather than local forcing.
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 S 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
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 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.
Within a realist extension of the de Broglie–Bohm framework, this work proposes aphysical interpretation of elementary particle spin as an emergent topological property of anunderlying guiding-wave structure.In previous works, a hypothetical physical wave component associated with photons wasintroduced as a possible experimentally accessible extension of the pilot-wave concept. Thisapproach was subsequently extended to entangled photons through the hypothesis of a com-mon non-separable guiding structure WAB existing in physical three-dimensional space.The present paper develops the internal dynamics of this guiding structure. We proposethat spin does not correspond to the mechanical rotation of a point-like particle, but ratherto a topological excitation of the physical guiding wave itself. The particle is interpreted as alocalized excitation embedded within a structured field W (r, t) possessing an internal phasegeometry.A nonlinear field model is introduced in which stable localized excitations can emerge astopological soliton-like structures. The quantization of spin is associated with a phase wind-ing number characterizing the topology of the guiding field around the particle excitation.The central result of this work is a generalized guiding equation combining the transla-tional guidance induced by the energy flow of the physical wave with an additional topologicalspin current. This equation provides a unified description linking particle motion, internalfield geometry, and spin quantization within a single deterministic framework.The proposed model remains phenomenological and aims at establishing a possible phys-ical ontology for quantum spin rather than replacing the standard mathematical predictionsof quantum mechanics.
Waveguide quantum electrodynamics (wQED) has become a central platform for studying collective light-matter interactions in low-dimensional photonic environments. While conventional wQED systems rely on uniform chirality or reciprocal emitter-waveguide coupling, we propose a structured wQED framework, where the coupling directionality of each emitter can be engineered locally to control excitation transport in an atom-nanophotonic interface. For different combinations of patterned coupling directionalities of the emitters, we identify four representative configurations that exhibit distinct dynamical behaviors: centering, wave-like, leap-frog, and dispersion excitations. Spectral analysis of the effective non-Hermitian Hamiltonian reveals that these dynamics originate from interferences among subradiant eigenmodes. Variance analysis further quantifies the spreading of excitation as functions of interatomic spacing and global chirality, showing tunable localization-delocalization transitions. Including nonguided losses, we find that the transport characteristics remain robust for realistic coupling efficiencies (beta >= 0.99). These results establish structured wQED as a practical route to manipulate excitation localization, coherence, and transport through programmable directionality patterns, paving the way for controllable subradiant transport and chiral quantum information routing.
A millimetric droplet can be induced to bounce on the surface of a fluid bath by vibrating the bath near the droplet's resonant frequency (Figure 1(a) ). [superscript 1–3] The localized field of Faraday waves excited by the bouncing droplet can cause it to propel itself laterally across the surface, moving in resonance with its guiding wave field (Figure 1(b) ). [superscript 4,5] These walking droplets, or “walkers,” generally move in a straight line at constant speed; however, they can be diverted through interaction with boundaries or external forces. This hydrodynamic system represents a macroscopic realization of the pilot-wave theory of quantum dynamics proposed by Louis de Broglie, according to which microscopic particles are propelled through a resonant interaction with a wave field generated by the particle's internal vibration. [superscript 6] Coincidentally, it exhibits many behaviors once thought to be exclusive to the microscopic quantum realm, including single-particle diffraction, [superscript 7] tunneling, [superscript 8] quantized orbits, [superscript 9] and orbital-level splitting. [superscript 10]