Classical Topology and Quantum Phases: Quantum Mechanics

A. P. Balachandran

Michael Baer, Gert D. Billing

In this paper are discussed the quantum mechanical connection and its classical counterpart, as well as the quantum and classical topological angles. The two were calculated for a model system to see whether they show any resemblance. The most interesting result we found is that the quantum and the classical topological phases are each formed due to a single discontinuous jump that occurs at some time during the external cycle. However, whereas the quantum mechanical topological phase is formed in the usual adiabatic limit, we had to establish a new parameter which, by allowing it to approach its limiting value, forms the classical topological phase.

Jakob Viñas Solé

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.

A. P. Balachandran, Giuseppe Marmo, Bo-Sture Skagerstam +1 more

Souhaib zayed Awad alkarim alnur صهيب زائد عوض الكريم النور

This work presents a rigorous mathematical framework for phase dynamics in configuration space. All physical dynamics are derived from a smooth phase field Φ without assuming forces, particles, or fundamental time. The framework introduces the Euler–Phase equation, derives the unique quantum potential, and recovers classical mechanics in the limit κ → 0. It includes explicit mappings to Hamilton–Jacobi and Madelung formulations, gauge symmetry analysis, circulation quantization, and a comparison of Phase Dynamics with classical and quantum mechanics. This unified approach generalizes both CM and QM, allowing topologically nontrivial and multi-valued phase structures.

Giuseppe Morandi