Consequences of the topology of the configuration space of a Hamiltonian dynamical system are considered for a coherent system of trajectories. It is shown that when the space is multiply-connected and therefore the action integral is multivalued, the allowed states of motion (labeled by the initial data) are constrained to a discrete set by the requirement that the action be single-valued. One thus obtains a quantum-like discretization of allowed states of motion even in classical mechanics. Such discrete “allowed” and “forbidden” states have indeed been observed in the classical mechanical system of charged particles in a magnetic field. The relationship of this formalism with a Schrödinger-like formalism for the latter problem given earlier is discussed.
Consequences of the topology of the configuration space of a Hamiltonian dynamical system are considered for a coherent system of trajectories. It is shown that when the space is multiply-connected and therefore the action integral is multivalued, the allowed states of motion (labeled by the initial data) are constrained to a discrete set by the requirement that the action be single-valued. One thus obtains a quantum-like discretization of allowed states of motion even in classical mechanics. Such discrete “allowed” and “forbidden” states have indeed been observed in the classical mechanical system of charged particles in a magnetic field. The relationship of this formalism with a Schrödinger-like formalism for the latter problem given earlier is discussed.
In section 6.5 we looked at the problem of evaluating quantum mechanical path integrals in curvilinear co-ordinates. For the model considered there (a point particle in two dimensions) the choice of curvilinear co-ordinates was capricious, but the use of curvilinear co-ordinates is natural in describing constrained systems. For example, suppose the particle in two dimensions was forced to move on a ring of radius R . The problem of path roughness would then go away, since the scalar curvature would be constant along the path. However, we acquire a new problem in that the configuration space for the particle (the circle) is no longer simply connected – a notion that we shall define later. This, in turn, can induce a new term in the action. We stress that this term is in no way a replacement for that due to roughness. For one thing, it is of order ħ , rather than of order ħ 2 . Most importantly it is not unique , leading to an ambiguity in quantisation. This ambiguity is a general property of quantisation on non-simply-connected manifolds. In this chapter we shall sketch how such O( ħ ) terms (topological ‘charges’) arise naturally in the path integral formalism which, being an ‘integral’ over paths in the configuration space of the system, is ideally suited for handling non-trivial configuration spaces.
This paper discusses an application to the study of dynamics of the typical overcomplete, non-independent sets of unit vectors that characterize continuous-representation theory. It is shown in particular that the conventional, classical Hamiltonian dynamical formalism arises from an analysis of quantum dynamics restricted to an overcomplete, nonindependent set of vectors which lie in one-to-one correspondence with, and are labeled by, points in phase space. A generalized ``classical'' mechanics is then defined by the extremal of the quantum-mechanical action functional with respect to a restricted set of unit vectors whose c-number labels become the dynamical variables. This kind of ``classical'' formalism is discussed in some generality, and is applied not only to simple single-particle problems, but also to finite-spin degrees of freedom and to fermion field oscillators. These latter cases are examples of an important class of problems called exact, for which a study of the classical dynamics alone is sufficient to infer the correct quantum dynamics.
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.
Almost everything that happens in classical mechanics also shows up in quantum mechanics when we know where to look for it. A phenomenon in classical mechanics involves topological changes in action-angle loops as a result of passage around a "monodromy circuit." This phenomenon is known by the short name "Hamiltonian monodromy" (or, more ponderously, "nontrivial monodromy of action and angle variables in integrable Hamiltonian systems"). In this paper, we show a corresponding change in quantum wave functions: These wave functions change their topological structure in the same way that the corresponding classical action-angle loops change.
We report here experimental results which point to a rather unexpected, astonishing behavior of charged particle motion in a magnetic field in the presence of a retarding electric potential. An electron beam, when energy scanned with respect to its ‘parallel’ energy using a retarding potential, the response of the transmitted current as a function of the retarding potential is found to be oscillatory as against the monotonic response, expected according to the standard initial value paradigm of classical mechanics. The unexpected ‘peaks’ and ‘dips’ in the transmitted current plots are interpreted as “allowed” and “forbidden” energy states of the system. The peaks (or dips) are found to fit very well with a theoretical relationship which motivated the search for such a behavior; also “quantum numbers” for the peaks have been identified. A very astonishing aspect of the relationship is the dependence of the energies of the allowed states on the length of the box.
Different classical limits of quantum mechanics are discussed in the setting of a simple example, the two-dimensional motion of a spinless electron in a uniform magnetic field. Both the time-independent energy eigenstates and the time-dependent states developing from initial localized states are exactly solvable. In the first example, an ℏ→0 limit with fixed energy leads to a simple ensemble of classical orbits. In the second, ℏ→0 with fixed initial mean position and momentum leads to a unique classical orbit.