Mechanical Analogy for the Wave-Particle: Helix on a Vortex Filament
| Scientific Paper | |
|---|---|
| Title | Mechanical Analogy for the Wave-Particle: Helix on a Vortex Filament |
| Read in full | Link to paper |
| Author(s) | Valery P Dmitriyev |
| Keywords | quantum physics, ideal fluid, line vortex, soliton. |
| Published | 2001 |
| Journal | Apeiron |
| Volume | 8 |
| Number | 2 |
| No. of pages | 31 |
| Pages | 1-31 |
Read the full paper here
Abstract
The small amplitude-to-thread ratio helical configuration of a vortex filament in the ideal fluid behaves exactly as a de Broglie wave. The complex-valued algebra of quantum mechanics finds a simple mechanical interpretation in terms of differential geometry of the space curve. The wave function takes the meaning of the velocity, with which the helix rotates about the screw axis. The helices differ in type of the screw — right or left-handed. Two kinds of the helical waves deflect in the inhomogeneous fluid vorticity field in the same way as spin particles in the Stern-Gerlach experiment. The helix represents the low curvature asymptotics of a loop-shaped soliton, the latter being governed by the nonlinear Schroedinger equation. The length of the redundant segment, needed in order to form a curvilinear configuration on the originally straight vortex filament, measures the mass of a particle. The unique size of the loop on the vortex filament can be determined by the balance between the energy of the redundant segment and the energy due to the curvature of the loop. The translational velocity of the soliton has the maximum at a value, which is inversely proportional to the length of the redundant segment. Insofar as the maximal velocity of a soliton is restricted from the above by the speed of the perturbation wave in the turbulent medium (i.e. the speed of light in vacuum), there must be a minimal redundant segment. Its length correlates with the Planck's constant. In the stochastic environs a loop-shaped soliton disintegrates into the collection of the elementary asymptotic helices. An asymptotic helix obeys the linear Schroedinger equation with no dependence on mass. The mass of the particle appears explicitly when we describe the motion of the whole ensemble of the elementary splinters.
Overview
Valery P. Dmitriyev of Lomonosov University, Moscow, published this in Apeiron in April 2001 as one instalment of a long-running project to build "a regular mechanical analogy of physical fields and particles". The framework is a substratum: a universal medium modelled mesoscopically as a turbulent ideal fluid, whose turbulent perturbations were shown by Troshkin (1990) to reproduce Maxwell's Equations exactly, and whose voids give dilatational inclusions modelling charged particles. Microscopically the substratum is a vortex sponge — an ideal fluid pierced in all directions by straight hollow vortex tubes, a construction Dmitriyev takes from E. M. Kelly's 1976 work. This paper takes a one-dimensional slice: a single vortex filament, and the small-amplitude helical waves that run along it.
The claim is that no quantum postulates are needed to get quantum behaviour out of this. The self-induction motion of a bent vortex filament — classical Kelvin/Arms hydrodynamics — turns into the linear Schrödinger equation once the two transverse displacements are written as a single complex number. The wave function is then not an abstraction but a velocity: the rate at which the helix rotates about the screw axis. The de Broglie wave, the wave packet, Spin and its two-valuedness, the appearance of Planck's constant, and the emergence of Mass all follow from properties of the filament. This is squarely in the vortex tradition of Kelvin and the nineteenth-century ether modellers, and Dmitriyev is careful to derive rather than assert: the mathematics is standard fluid dynamics, drawing on Batchelor and on Hasimoto's 1972 discovery of the soliton on a vortex filament.
The argument
The filament and its self-induction
A vortex filament is described as a space curve r(l,t) in the Frenet–Serret frame, with tangent e = ∂r/∂l, principal normal n, curvature κ and torsion τ. Motion without stretching obeys the Arms equation u = ∂r/∂t = νκe × n, i.e. ∂r/∂t = ν(∂r/∂l) × (∂2r/∂l2), where ν is the coefficient of local self-induction. A bent filament drifts sideways, perpendicular to both its own direction and its curvature vector.
Schrödinger's equation as filament geometry
For small transverse disturbances of a filament along the x axis, the x component of the cross product vanishes identically and the equation reduces to a two-component transverse system. Writing φ = y(x,t) + iz(x,t) turns it into
- i∂φ/∂t = ν ∂2φ/∂x2
— the free linear Schrödinger equation, with ν in the role usually taken by ħ/2m. Dmitriyev stresses that the imaginary unit here has a concrete geometric job: multiplication by i is the 90° counterclockwise rotation about the x axis that carries the principal normal into the self-induction velocity. The complex algebra of Quantum mechanics is thus the plane vector mechanics of the curve.
For a right-hand helix y = acos(x/b), z = asin(x/b) with amplitude a much less than pitch b, the curvature is κ = a/b2 and the torsion τ = 1/b, so a ≪ b is the same as κ ≪ τ. Substituting into the Arms equation, the helix rotates rigidly about the x axis with angular velocity ω = ντ2, and the solution is exactly aexp[i(τx − ντ2t)]. Superposing torsions over a narrow band gives a wave packet whose hump translates at u = 2ντ0. Dmitriyev underlines the mechanism: the packet advances not because anything moves longitudinally, but because a screw rotating in a nut screws itself forward. Since ω ~ 1/b2 and the advance per turn is b, the translation speed goes as 1/b — the de Broglie relation, obtained from a bolt.
The loop soliton and the redundant segment
Hasimoto's exact solution of the full nonlinear filament equation is a loop-shaped soliton with x = l − atanh η, y + iz = a sech η exp(iθ), whose curvature is the bell-shaped 2κ̂ sech η and which translates at u = 2ντ. Differentiating the motion law and substituting gives the nonlinear Schrödinger equation i∂Φ/∂t − ν∂2Φ/∂l2 = ½|Φ|2Φ, which linearises to the free equation exactly when κ ≪ τ. The asymptotic helix is therefore the low-curvature limit of a soliton.
The central physical notion is the redundant segment. To bend a straight filament into a curve one needs extra filament length; integrating gives that excess as exactly 2a. Dmitriyev then computes two energies: the kinetic energy of the fluid due to distortion, ε = ½ρν2∫κ2dl = 4ρν2κ̂, which is conserved by the continuity equation and is identified as the disturbance's self-energy and hence its mass mε; and, in the asymptotic limit where κ̂ = τ2a/2, the identity ε = 2ρν2τ2a which coincides with the soliton's kinetic energy Et = 2mεν2τ2 provided
- mε = ρa
That is, the mass of the particle is the mass of fluid in the redundant segment. The energy integral of motion is thereby identified with real fluid energy and the soliton's mass with real fluid mass.
Why the particle has a unique size
The redundant segment brings background energy 2aξ (with ξ the energy per unit length), while the loop's distortion energy is ρν2/8a. The sum 2aξ + ρν2/8a has a minimum at a = (ρν2/2ξ)1/2, fixing a singular loop size. The same calculation for a vortex ring of radius R (obtained from the loop by reconnection) gives 2πξR + πρν2/2R, showing that a has the meaning of the loop's diameter. Since the filament is a hollow tube, a curvilinear configuration is the inclusion of a redundant void in the vortex sponge — consistent with the author's earlier mesoscopic model of a particle. He flags an unresolved factor of two: the mass 2aρ of the redundant segment is twice the mass ρa computed for the asymptotic helix.
A plane loop cannot split into smaller plane loops without energy input, since 1/α + 1/(1−α) > 1 — a mechanical account of stability. It can split into non-planar solitons, i.e. waves, which by momentum conservation move in opposite directions; and in the asymptotic limit the distortion energy becomes additive, so the helix "can be split as a classical mass body".
Planck's constant from a speed limit
The soliton's translational velocity is bounded, u ≤ 2ν/a, and that bound cannot exceed the speed c of perturbation waves in the turbulent medium — the speed of light in this model. Hence a ≥ 2ν/c, giving a minimal redundant segment a0 = 2ν/c.
Under stochastic agitation ("thermalization") a soliton splits into m identical elementary helices of size a0, each obeying the linear Schrödinger equation with no mass in it. Composing the many-body equation and passing to the centre-of-mass coordinate x̄ = (1/m)Σxn gives i∂ψ/∂t = (ν/m)∂2ψ/∂x̄2. The phase sums to κx̄ − νκ2t/m with κ = mτ. Then m measures the mass (mε = ρa = ρma0), κ measures the momentum (p = 2ρν2a0κ), the frequency term is the kinetic energy — and the constant playing the role of ħ is identified as
- 2ρνa0 = ħ
Mass, in other words, is absent from the microscopic equation and appears only when the ensemble of splinters is described collectively.
Collapse and spin
Adding a pairwise δ-function attraction between splinters, with strength set so each fragment carries 1/m of the original self-energy, and solving in the Hartree approximation, Dmitriyev recovers the nonlinear Schrödinger equation for the recollected soliton — a mechanical picture of wave-function collapse, triggered by a fluctuation of fluid pressure. He notes that hydrodynamically a pressure decrement equals an increase in energy density, so the probability of collapse at a place should go as |ψ|2, and explicitly leaves the choice of collapse site to "a more general model of the measurement".
Spin comes from handedness. Helices come in right- and left-hand screw versions (τ > 0 and τ < 0). Both rotate in the same sense — counter to the filament's vorticity — but they translate in opposite directions relative to it: the right-hand helix travels the way the vorticity points, the left-hand one against it. In the three-dimensional sponge the rotating soliton appears macroscopically as a centre of torsion, corresponding to a magnetic dipole μ, and fluid vorticity curl u corresponds to the magnetic field, so the interaction energy is −μ·curl u. Two helices travelling the same way therefore carry oppositely oriented μ, and an inhomogeneous vorticity field deflects them in opposite directions — a helix changing course by jumping to an adjacent filament of slightly different vorticity. These are the conditions of the Stern–Gerlach experiment, reproduced without a spin postulate.
Assessment
This is the strongest kind of mechanical-analogy paper: it derives rather than asserts, and its mathematics is not its own. The Arms/Kelvin self-induction law, the Frenet–Serret formulae, and Hasimoto's 1972 soliton are established fluid dynamics, and the reduction of the small-disturbance filament equation to the linear Schrödinger equation via φ = y + iz is a real and well-known transformation, not a numerical coincidence. What Dmitriyev adds is interpretation, and some of it is genuinely illuminating. The geometric reading of the imaginary unit — multiplication by i as the 90° rotation carrying the principal normal to the self-induction velocity — gives the complex algebra of quantum mechanics a concrete job to do, which is more than most interpretations manage. The screw explanation of packet propagation is the paper's best moment: it exhibits a system in which phase velocity and group velocity are mechanically distinct, and in which the de Broglie relation between wavelength and speed emerges from the pitch of a thread rather than being imposed. The mass identification mε = ρa is not fitted but forced by equating two independently computed energies. The stability argument 1/α + 1/(1−α) > 1 is a small, clean result. And the paper's honesty is notable: it states outright in the concluding remark that the fixed strength of the intrinsic vortex tube "is taken as a postulate", and it flags its own factor-of-two discrepancy between 2aρ and ρa.
The difficulties are nonetheless decisive if the model is read as physics rather than analogy. Most fundamentally, the Schrödinger equation obtained is the free one-dimensional equation along a single filament, with the two transverse coordinates playing the roles of the real and imaginary parts of ψ. A genuine quantum theory needs ψ over three-dimensional configuration space, and the paper's route to more than one degree of freedom — the many-body equation (11.1) composed "formally" from m one-dimensional splinter equations, then reduced by centre-of-mass separation — is presented by Dmitriyev himself as "rather a formal result". No mechanism is given by which the substratum would produce entanglement between distinct particles, and the whole thrust of the model is against it: the splinters here are pieces of one soliton on one filament. The measured violation of Bell inequalities in the Aspect and later loophole-free experiments is the specific difficulty, and it is not addressed.
The identification 2ρνa0 = ħ is a definition, not a derivation. It fixes one combination of three unknown substratum parameters (ρ, ν and the energy density ξ) but predicts nothing, because none of the three is independently determined; the paper offers no number for the elementary segment a0, no scale for ν, and no way to check the relation. Similarly, the singular loop size a = (ρν2/2ξ)1/2 "ensures the discreteness of a nonlinear configuration", but a single such size cannot account for the observed spectrum of particle masses — there is one loop size in the model and many masses in nature, and the paper does not say how the difference arises.
The Stern–Gerlach account is suggestive but not equivalent to spin. Two handednesses give a two-valued deflection, which is the right qualitative outcome, but the deflection here is a classical consequence of a dipole in a gradient, and the paper produces no quantised magnitude — no ħ/2, no g-factor, and no account of why measurement along a second, rotated axis again yields exactly two outcomes rather than a continuum. That last feature is the actual content of spin and the thing the classical picture historically fails to reproduce. The magnetic moment of the electron is known to twelve significant figures, and the model neither computes it nor indicates how it might.
The collapse mechanism is candid about its incompleteness ("the competence of a more general model of the measurement"), and the Born rule is motivated by an analogy — pressure decrement equals energy-density increase, hence probability proportional to |ψ|2 — rather than derived. Finally, the model is non-relativistic throughout: c enters only as a bound on soliton speed, there is no Lorentz covariance, and no account is offered of why a preferred substratum rest frame is undetectable, which is the standing objection to every fluid-ether model since Michelson–Morley.
Taken as what its title says — a mechanical analogy — the paper succeeds, and it is one of the more mathematically disciplined examples of the genre. Taken as a replacement for quantum mechanics it is incomplete in ways the author largely acknowledges, and the incompleteness is concentrated in exactly the places (multi-particle states, quantised spin, relativistic invariance) where classical mechanical models of the quantum have always broken down.