Longitudinal Electrodynamics and Scalar Waves
Whittaker's real mathematics, the Aharonov–Bohm effect, and the longitudinal fields Maxwell already permits.
This thread starts on solid ground and reaches a long way. The ground is mathematics from 1903 that is still taught. Above it sit measured effects that surprise people the first time they meet them, and above those sit proposals nobody has measured yet. This chapter climbs that stack from the bottom.
Start with the effect, not the mathematics. An electron travels where the magnetic field is exactly zero, and its interference fringes still move. Reach for the first control and watch them go.
The interferometer with the field switched off
Split an electron beam, send the two halves either side of a magnetic flux that is completely confined and completely shielded, and bring them back together. On the paths themselves the magnetic field is exactly zero — the shaded corridor below is field-free the whole way. The vector potential there is not zero, and the fringes move. Dial the enclosed flux and watch the pattern slide sideways by exactly one fringe for every flux quantum. Then try the second control. It changes the gauge — the potential itself, at every point on both paths, as the arrows show — and the fringes do not move at all. Both controls are live; only one of them changes anything a detector can see.
- Upper path
- Lower path
- Confined flux
- Vector potential on the paths
What is happening: The enclosed flux is a whole number of quanta, so the phase difference is a whole number of turns and the pattern sits exactly where it started. This is the periodicity that makes the effect a measurement rather than a drift.
- Magnetic field on the electron paths
- zero, at every setting
- Vector potential on the upper path, in this gauge
- 0.00 nWb/m
- Vector potential on the lower path, in this gauge
- 0.00 nWb/m
- Phase the gauge adds to each half
- 0.00 radians — the same for both halves, so it cancels
- Phase difference between the two halves
- 0.00 radians
- Fringe shift
- 0.00 fringe widths
- What no choice of gauge can change
- the field on the paths, the phase difference, the fringe shift
The precision rule this instrument exists to protect: the vector potential controls phase. No energy is stored in it, and none needs to be — the phase difference is set by the flux the two paths enclose between them, which is exactly the quantity you are dialling. And the folklore reading — that the effect proves the potential is physically real at a point — is exactly what the second control is here to foreclose: the honest observable is the loop integral, never the potential at a point.
Nothing happening is the result here, not a broken control. The potential is not unique: for any smooth function χ you may replace A by A + ∇χ, which changes the potential at every point and changes the magnetic field nowhere, because the curl of a gradient is zero. This instrument performs that substitution for real and then integrates the full potential, gauge term included, around the closed loop the two halves of the beam make between them. Each half does pick up a phase from the gauge term — exactly 2λ radians in this family — and both halves pick up the same one, so it cancels out of their difference. What survives is the loop integral, which is the enclosed flux, and that is the observable.
Tonomura's 1986 experiment answered the obvious objection by construction: the flux sat inside a toroidal magnet wrapped in a superconducting shield, so stray field was not merely small on the electron wave but excluded from it. The shift was still there, and it was still one fringe per flux quantum.
Scope: a drawn two-path interferometer with a perfectly confined flux, which is the geometry of the published experiment. Beam energy, path length and magnet size are not modelled, and no absolute intensity is asserted. Φ₀ = h/e ≈ 4.136e-15 Wb.
The real mathematics: Whittaker's potentials
In 1903 and 1904 the mathematician E. T. Whittaker published two short papers. He showed that any solution of the relevant field equations can be written using just two scalar potential functions, and that a scalar potential splits neatly into a pair of waves running opposite ways. This is correct, lasting mathematics: the ancestor of the Debye potentials and Hertz vectors in textbooks like Stratton's. Physicists use it every day.
Where potentials genuinely matter: Aharonov–Bohm
You often hear that potentials are more fundamental than fields. Mainstream physics agrees, in one precise and beautiful place: the Aharonov–Bohm effect, predicted in 1959 and confirmed by Tonomura's team in 1986. An electron travels where the magnetic field is exactly zero. It still picks up a measurable shift in its quantum phase, set by the potential alone.
It is measured, quantised and reproducible, and it anchors the whole vector-potential thread on this site. A potential can act on matter where no force acts.
That the effect is real is Tonomura's business; that it is inevitable is Berry's. In 1984 Michael Berry showed that a quantum system carried slowly round a closed loop in the space of its own parameters keeps a phase set by the shape of that loop alone — and then derived the Aharonov–Bohm phase as a special case of it, the charge times the enclosed flux divided by the reduced Planck constant, using only single-valued wavefunctions. That last clause is the point. The standing objection to the elementary presentation is that the wavefunction it builds is not single-valued; Berry's route never builds one, so the objection has nothing to bite on. The same paper gives the spin result this book keeps meeting — a spin carried slowly round a loop in a magnetic field keeps the solid angle the field direction swept out — and it is also where the effect's own history is set down, Ehrenberg and Siday having anticipated it in 1949 and Chambers having seen it in 1960.
That experiment is the instrument at the top of this chapter, and it has two controls. The shaded corridor is field-free: the magnetic field on the electron paths is zero at every setting. The first control changes the flux the two paths enclose between them — dial it, and the fringes move, one whole fringe for every flux quantum, and back to where they started at the next one. The second changes the gauge, which is the freedom in the potential itself: the arrows on the paths swing and stretch, because the potential really has changed at every point, and the fringes do not move at all. Two controls, and only one of them changes anything a detector can see. The observable is the loop integral — the flux the two paths enclose — and never the potential at a point.
A loop integral is easiest to build as a toroid, and that is why the vector potential keeps arriving in this book wound on a ring. Here is one from the outside.
- Geometry computed from
- An exact circular core wound poloidally with one continuous wire, one turn per equal step of arc length on a parallel-transported frame. The confined flux lines are copies of the core curve offset inside the tube, because the field in a toroidal solenoid runs along the core and closes on itself. The curves outside are exact as topology: closed loops in the plane normal to the core, every one of them linking the ring, which are the integral curves of the exterior potential and the paths whose loop integral is the observable
- Numbers from
- Aharonov and Bohm's 1959 prediction and Tonomura and colleagues' 1986 measurement, in this site's library, where a toroidal magnet sealed inside a superconductor and a copper shield still shifted the electron's phase; the single-winding toroid as a field-propulsion geometry is Froning's, read on air from his 1997 conference presentation, with the published Froning and Barrett record of 1998 held in the library as a summary
- Schematic parts
- The scale is a normalisation, not a measurement: the sources describe the shape in words and give no dimension for it, so the major radius is exactly one metre by construction and every other length is a ratio to it. The turn count, the core tube radius, and the number and radii of the exterior loops are chosen so that the geometry reads. How the potential falls away with distance is not drawn, because a curve carries a direction and a linkage but no magnitude.
What Maxwell already permits
Longitudinal fields — fields pointing along the direction of travel — are not exotic. Maxwell's equations give you them in three familiar places.
- The near field. Within about a wavelength of an antenna, the field has a component along the direction of travel. Every antenna engineer works with it. It stays bound to the source instead of radiating away.
- Plasmas. A plasma carries Langmuir waves. The electrons slosh back and forth along the direction of travel, the way sound moves through air. These are everyday, measured longitudinal waves.
- Guided surfaces. A Zenneck surface wave runs along the boundary between air and a lossy conductor such as ground or seawater. It carries a field component along its direction of travel and clings to the surface instead of spreading into space.
That is generous ground. Longitudinal electric fields exist, they carry energy where they are bound or guided, and engineers build with them already.
Free space is the harder case. In quantum electrodynamics the photon's longitudinal and timelike polarisations are bookkeeping modes, and the Gupta–Bleuler formalism cancels them out of free radiation. So a free-space longitudinal wave carrying energy would be new physics, not a corollary of Whittaker. The guided cases are the obvious place to hunt for something engineerable.
The proposals: scalar interferometry and the MEG
The modern thread runs through Tom Bearden and is repeated in independent-researcher streams. He argued that two beams can be arranged so their interference draws energy from the vacuum, and that the motionless electromagnetic generator (MEG, US6362718B1) already does it. The patent is public and the claims are there to read.
The test is clean and cheap. Poynting's theorem is the energy ledger of any electromagnetic device. Power out equals power in, minus what the fields store, minus what the device dissipates. Meter every input line on a bench and the ledger either closes or it does not. No independent laboratory has published that measurement yet.
Open question: does any scalar device close its energy ledger? What to watch next: a metered bench test of the MEG, published with the raw traces.
Free-space longitudinal energy transport, and Bearden-style vacuum tapping, are well-posed claims with named experiments and no measurement yet.
“Whittaker showed that scalar potentials underlie the fields, and the Aharonov–Bohm effect shows potentials are real where fields vanish. So longitudinal scalar energy waves should be real too.”
The mathematical decomposition and the measured phase effect answer different questions. The step that still needs work is the one from a bounded, measured phase effect to energy carried across empty space. In quantum electrodynamics the longitudinal and timelike photon modes are bookkeeping, and Gupta–Bleuler cancels them out of free radiation. Where longitudinal fields do carry energy is the near field, the plasma and the guided surface wave — all three engineerable today, and the more promising place to push. Watch for a controlled detection of a longitudinal wave carrying energy across a vacuum gap, or a metered bench test with the ledger closed.
What the field added — July to September 2026
The vector potential had its season. Charles Chase, late of Lockheed's Skunk Works, described the A potential as more fundamental than the fields. It can change the phase of matter waves with no energy exchange. That is the Aharonov–Bohm effect this chapter already teaches. Chase described a coherent matter-wave beam built on that principle. Froning and Barrett's 1998 paper on "conditioned" fields was read on screen. A beam's symmetry is raised from ordinary U(1) to SU(2), and the field gains structure that ordinary radiation lacks. It is now in the sources as the framework for engineering such fields. The aether returned in two forms. One was the historical case, told through Ørsted, Faraday, Maxwell and Hertz. The other was a citizen-science vertical interferometer, worth repeating on a controlled bench. Dirac's 1951 "Is there an Æther?" gave the question its most distinguished voice. Research log. For the full story of the vector potential — from a flow map anyone can picture to the matter-wave beam — take the vector-potential drill-down course.
Froning's own book added a geometry worth remembering. Before his group conditioned any beam, they asked a prior question: could the local density of the zero-point field be changed at all? Froning reports that it could. The hardware was a wound toroid deliberately built out of round — "instead of making a perfect donut, they made it like a triangle on one end, a teardrop shape." Only after that did they polarise and twist the beam into what the programme called a gravity beam. The order matters. The asymmetry came first, and it was a shape you could machine. What to watch: one of the more buildable things in this chapter.
- Geometry computed from
- The classical teardrop curve at the exponent that reproduces both described features and nothing else: a round bulb at one end, and straight limbs meeting in a triangular vertex of about 64 degrees at the other. The curve is resampled to uniform arc length and wound with one continuous poloidal wire, one turn per equal step of arc length, which is how a wire is actually laid on a former and what keeps the turns evenly spaced around the point. The core tube merges into itself at the vertex because the two limbs converge there, so the nose is solid. The exterior curves are built exactly as in the round toroid, so the linkage is identical while the pattern is not
- Numbers from
- Froning's book Faster Than Light, read on air by Douglas Miller with Ashton Forbes on 8 September 2026, describing a wound toroid made triangular at one end and teardrop overall, built to change the local density of the zero-point field before any beam was conditioned; the published lineage is the Froning and Barrett record of 1998, held in this site's library as a summary whose abstract carries no geometry
- Schematic parts
- The shape comes from a description in words and no equation. The named curve, its exponent and its height ratio — and therefore the exact angle of the vertex — are drawing choices, chosen because they reproduce the two features described and nothing further. The scale is a normalisation, one metre major radius by construction, because no source gives either toroid a dimension; the turn count, the core tube radius and the number and radii of the exterior loops are chosen so the geometry reads. The result attributed to the shape is not modelled, not drawn and not asserted here.
Measurements to make next
The proposed test is a controlled detection of a longitudinal wave carrying energy across a vacuum gap; and, for the devices, a metered bench test with the Poynting ledger closed. Both are bench-scale.
Sources
Mathematical decompositions, a measured quantum phase effect, guided longitudinal waves and proposed tests.
Primary - the mathematics
- E. T. Whittaker (1903), "On the partial differential equations of mathematical physics," Math. Ann. 57, 333. DOI 10.1007/BF01444290.
- E. T. Whittaker (1904), "On an expression of the electromagnetic field due to electrons by means of two scalar potential functions," Proc. London Math. Soc. s2-1, 367 - the ancestor of the Debye potentials.
- J. Stratton, Electromagnetic Theory (1941) - the Debye and Hertz-vector descendants; standard working practice.
The measured anchor (beyond the source corpus)
- Y. Aharonov & D. Bohm (1959), "Significance of electromagnetic potentials in the quantum theory," Phys. Rev. 115, 485 - potentials act where fields vanish; confirmed by Tonomura et al. (1986).
- M. V. Berry (1984), "Quantal phase factors accompanying adiabatic changes," Proc. R. Soc. Lond. A 392, 45. DOI 10.1098/rspa.1984.0023 - the Aharonov-Bohm phase derived as a special case of a geometrical phase, using only single-valued wavefunctions, and the paper that records Ehrenberg and Siday's 1949 anticipation and Chambers' 1960 observation.
- QED (Gupta-Bleuler): the photon's longitudinal and timelike polarizations are bookkeeping modes that cancel from free radiation. Real longitudinal modes live in the near field, in plasmas (Langmuir waves) and in guided surface waves.
Where the physics is worked out
- Bearden-style "scalar electromagnetics" (e.g. the MEG, US6362718B1) - a public patent with a specific claim. Poynting's theorem decides it: a metered bench test with every input line accounted for.
- H. D. Froning & T. W. Barrett (1998), "Space coupling by specially conditioned electromagnetic fields," AIP Conf. Proc. 420, 1449 (STAIF-98). DOI 10.1063/1.54772 - the framework for conditioning a field's form rather than its strength. (Closed access; read from the authors' abstract.)
- H. D. Froning, Faster Than Light - the wound toroid built deliberately out of round (a teardrop rather than a perfect donut) used to change the local zero-point density before the beam was conditioned; read on air in the D. Miller ZPE All Stars interview, Hard Truths Podcast (8 September 2026; YouTube
5MIGis0S8Fc, @1:26:42-@1:27:36).
Where the replication stands: much of this thread is self-published, and the bench measurement above has not been made by an independent group. Whittaker provides a mathematical decomposition; longitudinal energy transport in near fields, plasmas and guided waves supplies the measured comparison.