TUTORIAL 2. THE BUILDING BLOCKS
The basic physical object out of which the model is to be constructed is not a point particle or a wave-like structure. Instead, the fundamental constituent is a photon with a precise geometrical structure: a helical string.
2.1 The helical string (Sp-1 rotation)
Imagine a string wound round a guide tube. The string travels at speed √2c along a helical path with a rake angle π/4. At this particular angle, the helical string has two equal components: an azimuthal speed c around the tube axis, and a translational speed c along the axis, both with action h. This simultaneous circular and linear motion is how circularly polarised light is to be represented. In optics it is known as Spin Angular Momentum and here as Sp-1.
The construction has an immediate advantage. Because energy is transported between azimuthal and translational modes, a medium that couples to the azimuthal component can slow the translational speed to below c giving a natural geometric explanation for why light slows down in dispersive media.
Figure 1 String wound round a guide tube with rake π/4. The azimuthal rotation is denoted Sp-1 and has speed c while the translational speed is also c. The wavelength is equal to the tube circumference .
When this reduction in translational speed occurs, the linear action is redistributed into azimuthal action. In a vacuum, a photon tends to restore its azimuthal action to h so that the speed c becomes its natural limit. A single photon in this picture is not a single string but a transversely iterated array of parallel, phase-matched helical strings advancing together as a front. Experiments in spontaneous parametric down-conversion (SPDC) have been re-analysed to show that the probability of two separate detectors being triggered by two different strings in the same single-photon front is very low, between 0.00075 and 0.053, see Paper A. This means that most of the time a single detector registers a strike, consistent with the standard quantum result, but the multiple-triggering events that occasionally occur are also comprehensible within the model.
2.2 Optical OAM and rest mass (Sp-2 rotation)
Definition of Rest Mass
Rest mass is defined as mo = h/(2πr2oc), where h is Planck's constant, and r2o is the Sp-2 ring radius. Mass is therefore a property of the geometry of a trapped photon, not an independent primitive quantity.
Figure 2 Construction of a Sp-2 helix (optical OAM) from a Sp-1 tube. (a) The string in Figure 1 is diverted into a closed circle of Sp-2 radius. (b) Right-end elevation. (c) The Sp-2 ring is sliced through its cross-section and uncoiled so that a helical trajectory is adopted without the addition of energy. Its passive acceleration radius is r2p = r2o√(1 – β²).
When the translational motion of the Sp-1 string is diverted into a closed circle we have pure Sp-2 rotation. This corresponds to optical Orbital Angular Momentum (OAM). Linear rays can be diverted with spiral wave-plates, holograms, and cylindrical lenses to produce optical OAM in the laboratory. The radius r2o of the Sp-2 ring defines the rest mass in the PTV model. When the Sp-2 radius is set to r20 = 3.86159268 x 10^(-13) m we obtain the electron rest mass, see Paper B. The proton is described by the same construction but with a much smaller Sp-2 radius r20' = 2.10308910 x 10^(-16) m. Typical OAM beam waists created in the laboratory are about 10^(-6) m. The form of the proton and electron is identical; there is only a difference in scale.
If the closed Sp-2 ring is uncoiled or stretched out along its axis without absorbing any energy, the Sp-1 tube takes on a helical trajectory. The total speed remains c (since no energy is added and the string length is unchanged) but the velocity now has two components: an azimuthal speed of c√(1 – β² ) and a translational speed βc, where β is the ratio of the linear speed to c.
2.3 Passive and active acceleration
Consider two identical Sp-2 rings, S and S', initially at rest with respect to each other. A photon strikes S' and it absorbs energy and accelerates away. From the perspective of S', S has accelerated in the opposite direction but S has absorbed nothing. This acceleration of S, observed from the energised frame, is denoted here as passive acceleration, see Paper B.
Passive Acceleration
A ring that absorbs no energy can nevertheless be seen to accelerate when viewed from a frame that has absorbed energy. The passive ring's total energy remains moc². Its Sp-2 radius contracts to r2p = r2o√(1 – β²). Its time period is unchanged. All this follows from the geometry of the uncoiling.
Active Acceleration
A ring that absorbs energy from a photon or repulsion field increases its mass to m2 = mo /√(1 – β²). Its Sp-2 radius remains unchanged but its time period increases to T2 =To /√(1 – β²). This time dilation is absolute. All reference frames agree that energy has been absorbed and that the actively accelerated ring takes longer to complete its revolution.
The time period comparison is the most significant result here. In the passive case, the time period is invariant; in the active case it dilates by the standard Lorentz factor. This is not a frame-dependent effect; it is absolute because all observers can agree on whether or not an energy absorption event has occurred. This result suggests a hierarchy of reference frames ranked by energy content which is the Lorentzian rather than the Einsteinian interpretation of relativity. That a gas is very close to absolute zero is something that all observers could agree on from observing an almost flat Planckian curve. Nevertheless, although molecules at absolute rest might exist, it is their non-emission of radiation that could render their detection in practice problematic. The active acceleration case is the one that is recognised in conventional special relativity, from which the standard four-vector energy-momentum relation is derived. The passive case is a new addition.
2.4 The magnetic momentum field
As the Sp-2 ring rotates, its curvature generates a spiral momentum field in the surrounding space, see Figure 3. This is the magnetic momentum field. The momentum in the field varies inversely with the distance r2f from the Sp-2 centre so that p2f = ħ/r2f. It is a departure from the Maxwell-Heaviside electrodynamics that requires relative motion to produce a magnetic field. In the PTV model, the closed Sp-2 loop in a stationary electron generates a magnetic momentum field as a consequence of its rotational structure not its translational velocity.
2.5 Forming the toroid (Sp-3 rotation)
The final step in constructing the complete electron or proton is to bend the Sp-2 axis into a closed toroid, see Figure 4. This toroidal rotation in called Sp-3. The result is a Photonic Toroidal Vortex, a structure with two simultaneous axes of curvature: poloidal (Sp-2) and toroidal (Sp-3). In the ground state, the Sp-3 toroidal speed is set to βc = αc, where α is the fine-structure constant. This toroidal motion constitutes the internal potential, one that is responsible for a toroid adopting discrete energy levels in the absence of an external potential. The Sp-3 rotation generates an electric momentum field p2f. These two field components are mutually perpendicular at every point in space.
Figure 3 Velocity and momentum field components of the Sp-2 and Sp-3 rotations. (a) The Sp-2 circuit when situated in a toroid, emits a field throughout its trajectory, with radial speed c, Sp-3 speed βc, and Sp-2 speed c√(1 - β²) into the page (cross). (b) Right-end elevation of the Sp-2 in (a) in which the component at A out of the page (dot) is βc and the resultant projection initially runs at speed √2c. (c) Trajectory of the field emission at A towards a target Sp-2 circuit at B, in which the radial direction rotates. Both the radial and Sp-2/Sp-3 speeds diminish proportionally with distance to produce a momentum p2f approximately 45° to the line AB. (d) Momentum analogue of (a). (e) Momentum analogue of (b) in which the field momentum is equally partitioned between the radial and the resultant of the Sp-2/Sp-3 directions. (f) Momentum anologue of (c). (g) Detail of B in (f) in which the radial field component has rotated with the Sp-2 field rotation to point towards B and maintain p2f at 45° to the line AB. This particular angle at which the field cuts the Sp-2 target circuit becomes an important part of the hyperfine shift derivation, see Paper C.
The Momentum Fields
The magnetic (Sp-2) and the electric (Sp-3) momentum field magnitudes are related by p3f = p2f α/√(1 - α²). The Sp-2 field momentum depends on the radius r2f from the Sp-2 centre as p2f = ħ/r2f = m2r2oc√(1 - α²)/r2f . As the field radius r2f increases it is the field speed that decreases; the mass in the field m2 remains invariant.
Paper A Paper B Paper C Paper D Paper E
Figure 4 (a) Photon B running round a guide tube A as Sp-2 rotation. (b) Guide tube A running round a toroid as Sp-3 rotation.
Paper A: Barry R. Clarke, Reinterpretation of the Grangier experiment using a multiple-triggering single-photon model, Modern Physics Letters B , 15, 2350042 (2023).
Paper B: Barry R. Clarke, A photonic toroidal vortex model of the hydrogen atom fine structure, Quantum Studies Mathematics and Foundations, 12, 19 (2025).
Paper C: Barry R. Clarke, Geometrical interpretation of the hydrogen atom hyperfine structure, under peer review.
Paper D: Barry R. Clarke, The Lorentz force and the nature of charge from a Photonic Toroidal Vortex Model, under peer review.
Paper E: Barry R. Clarke, A heuristic model of the Bose-Einstein distribution with distinguishable photons
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