A Periodic Structural Model for the Electron Can Calculate its Intrinsic Properties to an Accuracy of Second or Third Order

Scientific Paper
Title	A Periodic Structural Model for the Electron Can Calculate its Intrinsic Properties to an Accuracy of Second or Third Order
Read in full	Link to paper
Author(s)	Horace R Drew
Keywords	finite models for electron and muon magnetism, helix, fine structure, spin
Published	2002
Journal	Apeiron
Volume	9
Number	4
No. of pages	42
Pages	25-66

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Abstract

In two previous papers, the electron was described in terms of a periodic structural model, namely a four-dimensional 'helix' or 'stationary wave' of spin ½ symmetry. That specific model generates most first-order properties of the electron as observed, and is stable in a Casimir sense where inward vacuum pressure balances outward inertial motion. It predicts a large electrical self-repulsion equal to 1/137 of mc² across the helical diameter 2r, or a small electrical self-repulsion equal to 1/(137 × 2π) of mc² along the curved helical path 4πr.  Here it will be shown how those two finite electrical self-repulsions, when used together, can explain the magnetic moments of an electron or muon to second or third order in powers of 1/(137 × π). The small self-repulsion of 1/(137 × 2π) represents a stable part of the electron mass, and accounts for a first-order Lamb shift in atoms. By contrast, the large self-repulsion of 1/137 contributes only temporarily to electron mass, and accounts for the probability of any electron to emit or absorb light. A periodic structural model may also explain the quantized nature of magnetism in atoms, on the hypothesis that a bound electron can only join to itself using an integral number of spin  ½ double-turns. The electron paths can then be considered as resonant, non-radiating rings whose net angular momenta explain the magnetic energies s, p, d, f of atomic fine-structure spectra.

Overview

This is the third instalment of Horace R. Drew's programme, following his 1999 Physics Essays paper "The electron as a four-dimensional helix of spin ½ symmetry" and his 2000 Apeiron paper on Thomas precession. The electron is not treated as a dimensionless point that jiggles inside a probability cloud, but as a continuous structure that rotates simultaneously in two mutually orthogonal planes of a four-dimensional space — for example the xy and zt planes — with the radii about the "major" and "minor" planes differing by a factor of two. Drew notes that his word "helix" is imperfect and that "four-dimensional stationary electromagnetic scalar wave", the language of Volodimir Simulik and Krivsky, would do as well. The essential postulate is the double rotation, which he identifies with the four-dimensional symmetry 2²; from it he claims spin ½, the non-classical g = 2, and the squaring of the wave function follow without further assumption.

The novel claim of this paper is quantitative. Where the earlier work reproduced first-order properties, Drew here attempts the numbers that are usually regarded as the crown jewels of quantum electrodynamics: the anomalous magnetic moments of the electron and muon to third and fourth order, and the Lamb shift of hydrogen. He does so without renormalisation, arguing that the point-like electron generates its infinities precisely because r = 0, and that a structure of finite radius yields finite self-energies from the outset. The fine-structure constant, added by postulate in QED, is on this view a consequence of the geometry rather than an input. The paper is thus a direct challenge to the claim that only QED can produce ten-digit agreement with experiment.

The argument

Two finite self-energies

The model's whole arithmetic rests on two self-repulsions. Across the helical diameter 2r, through space, the electron repels itself by 1/137 of mc²; along the curved helical path 4πr, through time, it repels itself by 1/(137 × 2π) of mc². Only parts of the rotating structure differing in phase by an integral 360° contribute, which is why just these two terms survive to first order. Drew assigns them different roles: the small self-repulsion through time is a stable part of the electron mass, while the large one through space is only temporary and represents the electron's probability of emitting or absorbing light.

Casimir stability

Why should the charge follow a closed periodic path at all? Drew invokes Casimir's 1953 model. A closed structure supports fewer modes of electrical self-energy than the surrounding zero-point vacuum, so an inward vacuum pressure folds the electron back on itself — an analogy is drawn with van der Waals forces folding polypeptides into coils. Treating the four-dimensional electron projected into space as a shell of radius r, the inward pressure from a zero-point wave of frequency f = c/2πr gives E(in) = hf/2 = hc/4πr, while outward inertial motion gives E(out) = 137 × (e²/2r). Using hc/2π = 137 × e² the two are equal. Casimir's original shell was unstable because its self-repulsion was a full mc²; reducing it to 1/137 of mc² makes the balance work. Drew adds that 2r = 3.8 × 10⁻¹³ m is the smallest distance over which the screened charge 1/137 still applies, the bare charge below it being 1/129.

Anomalous magnetic moments

Magnetic moments are expressed, as in QED, as 1 + C₁/(137 × π) + C₂/(137 × π)² + C₃/(137 × π)³ + … Drew's replacement for the QED Feynman-diagram sum is a series of harmonics along the helical path: self-repulsions of 1/(137 × 2π), 1/(137 × 4π), 1/(137 × 8π) and so on, each with half the frequency of its predecessor, "just as for the same note played at successively lower octaves". Odd multiples such as 6π or 10π are excluded because their predecessors carry an odd phase of n × 180°. Summing ½ + ¼ + ⅛ + … and converting total energy e²/r to net energy e²/2r gives C₁ = +0.50000, identical to QED.

Second order comes from two processes: a second self-repulsion cancelling the first 180° out of phase (calculated from the sum of squares, doubled because the second photon may sit one turn ahead or behind, giving −0.334108 after a small correction), and a first-order photon dissociating temporarily into an electron–positron pair (+0.005745). The total C₂ = −0.328363 against QED's −0.32848, agreeing to within 0.03 %. Third order counts eight positions by a formula C_n−1 × (n − 1) × 2, plus half-frequency cross-terms giving a factor of twelve, plus photon–photon scattering; the resulting C₃ = +1.12 against QED's +1.18, and the running total 1.0011596522 matches the measured last three digits (522) exactly. C₄ = −0.8 matches QED.

For the muon the extra ingredient is that internal light may convert to electron–positron pairs whose magnetism is 206.77 times stronger than that of muon–antimuon pairs. The raw estimate overshoots, and Drew corrects it by the increased mass of a bound pair, 2m′ = 2.1534 rather than 2.0000, giving C₂ = +0.773 (QED: +0.766) and, after third- and fourth-order terms and contributions from pion and kaon pairs, a muon moment of 1.00116590 against a measured 1.00116592.

The Lamb shift

If 1/(137 × 2π) really is the stable, electrically derived fraction of the electron's mass, that fraction should be lost when the electron overlaps the nucleus and its self-repulsion through time is destroyed. Drew computes the probability of an electron lying within its own radius r_e = 1.92 × 10⁻¹³ m of a point nucleus as P(1s) = 1/(6 × 137³), P(2s) = 1/(48 × 137³) and P(2p) = 1/(1280 × 137⁵). The last is essentially zero, matching the observation that the Lamb shift affects s but not p orbitals. The raw prediction for hydrogen 2s is 1157 × 10⁶ against 1085 × 10⁶ measured — too large by 10 % at Z = 1 and 30 % at Z = 3. Introducing a vacuum-polarisation screening factor 1/(1 + 2/5π) = 0.887 brings the predictions to 1026, 910 and 807 against data of 1046–1085, 905 and 802.

Closed paths and fine structure

The final sections reinterpret the Sommerfeld–Dirac energy formula. Its third term carries the factor n/(1 + c), and Drew reads c as the number of times a closed electron path "crosses over itself" to relieve the torsional stress of end-to-end joining: c = 0 for an open circle (s paths), c = 1 for a figure-eight (2p(3/2)), c = 2 for a doubly crossed figure-eight (3d(5/2)). Each crossover halves or thirds the net angular momentum because momentum vectors cancel except in the end loops. He then maps this onto the topological identity Wr = Lk − Tw for a closed ribbon, with c = Wr, and speculates that entanglement is a surviving topological linkage, Lk = 0 for a photon pair that separates into −1 and +1 on measurement. Such a ring, being in resonant exchange with the proton at energies 1/137n, does not radiate. Drew closes by calling the work "a naïve and somewhat controversial starting point".

Assessment

What is genuinely attractive here is the economy of the starting point and the seriousness of the follow-through. Very few structural-electron papers attempt ten-digit comparisons with measurement at all; most stop at g = 2 and declare victory. Drew instead takes on the anomalous moments and the Lamb shift, publishes a table putting his constants beside the QED values, and states plainly where his agreement is poorer (C₃ for the muon, +19.4 against +24.1). The physical picture behind the harmonic series — self-repulsions at 2π, 4π, 8π as descending octaves along a closed path — is at least a concrete alternative to summing diagrams, and the Casimir-balance argument for why a finite structure should be stable is a real answer to an old objection. The reading of the Sommerfeld–Dirac factor n/(1 + c) as a crossover number is the paper's most original idea, and it makes a check available in principle: it predicts which orbitals may share a magnetic energy and which may not.

The difficulties are equally clear, and most of them concern the status of the derivations. The combinatorial factors are the weak point. The eight third-order positions come from a formula C_n−1 × (n − 1) × 2 that is asserted rather than derived from the geometry; the factor twelve then arrives by adding "8 + (8 × ½)" for half-frequency cross-terms, and Drew's own word for multiplying by it is "plausibly". The mass corrections in the muon calculation — 2m′ = 2.1534, then m′ = 2.92 at third order — are introduced only after the uncorrected numbers overshoot, and their magnitude is fixed by a magnetism-weighted sum whose weights are the same series being tested. The Lamb-shift screening factor 0.887 is likewise introduced after a systematic 10–30 % excess is noticed, and it is borrowed from the vacuum polarisation of the very theory the model is meant to replace. Where a fit is adjusted by a factor chosen after the discrepancy is known, agreement to three digits carries much less weight than agreement predicted in advance.

There are also internal tensions in the paper's own numbers. The final muon figure depends on pion and kaon contributions estimated at +2.3 × 10⁻⁸ and +0.8 × 10⁻⁸ against QED's +7 × 10⁻⁸ for all such pairs, so the closing match to 1.00116590 rests on a term the model reproduces only to a factor of two. Drew concedes that C₃ and C₄ "may be only approximate". More fundamentally, the calculation is not independent of QED: electron–positron pair creation, photon–photon scattering and vacuum polarisation are all imported wholesale, with only the weighting rules replaced. The paper is better described as a re-derivation of the QED coefficients from a geometric counting scheme than as a self-contained finite theory.

The framing question — Drew's claim that Lorentz-covariant special relativity is "a theory of perception rather than a theory of dynamics" and must not be applied to particle dynamics — is argued by reference to his earlier work rather than defended here, and it does real work in the paper, since it licenses treating the two rotation planes as physically rather than merely formally distinct. Readers who do not accept that the measurability of Thomas precession is genuinely in doubt will find the foundation less secure than the arithmetic built on it. Finally, the topological account of entanglement in section 11 is offered as a suggestion and is not developed to the point where it would give different predictions from standard quantum mechanics for any Bell-type measurement; as stated, it is a picture rather than a test. Within its own terms the magnetic-moment calculation is carefully done and honestly reported, and the closed-path reading of fine structure deserves the further study Drew asks for.

See also

• Horace R Drew
• Quantum Theory
• Quantum Mechanics
• Atomic Structure
• Zero Point Energy
• Special Relativity
• Louis de Broglie
• David Bohm
• Georg Galeczki
• David L Bergman
• Volodimir Simulik
• Apeiron