Introduction
Usually the electric and the magnetic fields are seen as two separate things that are somehow connected with each other. In his original papers Maxwell saw the electromagnetic field as a field of quaternions. That’s a very interesting perspective. I know quaternions form my game development background, and they are used to describe rotations. A quaternion is the combination of a scalar and a vector, it could be described as the vector given the axis for the rotation and scalar the amount of rotation around that axis. “Rotate this much around this axis!”. The four-potential description of the EM field is very similar, using a scalar for the electric potential and a vector for the magnetic field. So, the electric field is like pure energy while the vector field of the magnetic field is structure! The magnetic field is what directs the energy flow, what creates shape and curls and form.
I think that this interconnectness can create the non-linear effects required to create stable wave packets and wave configurations.
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Historical Context
- Maxwell originally formulated electromagnetism using quaternions (1865)
- Heaviside and Gibbs reformulated using vector calculus (what we teach today)
- Some claim information was “lost” in translation—this is debated
Quaternion Structure
A quaternion has four components: q = w + xi + yj + zk
- Scalar part: w
- Vector part: (x, y, z) with independent magnitude
- i,j,k are imaginary, i² = j² = k² = -1
In Maxwell’s quaternion formulation:
- Electromagnetic potential as single quaternion: Q = φ + A
- φ (scalar potential) → relates to electric field E
- A (vector potential) → relates to magnetic field B via curl (rotation)
The electric field has “potential-like” character (intensity, energy landscape), while the magnetic field has “rotational” character (shape, curl). The quaternion structure naturally unifies these as scalar + vector parts of one object.
In quaternion multiplication, the cross product emerges naturally. Curl isn’t a bolted-on operation – it’s intrinsic to how quaternions combine. B = ∇ × A literally extracts the rotational structure of the vector potential.
The Aharonov-Bohm Effect
The potentials (φ, A) are more fundamental than the fields (E, B).
- Electrons traveling around a solenoid show interference shifts
- Outside the solenoid: B = 0, but A ≠ 0
- The electrons “see” the potential, not just the field
- This is usually explained using quantum mechanics.
- But it can also be explained by treating the four-potential as physically real, not just mathematical convenience.
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I tried to improve my understanding of the EM field, i.e. in regard to its four potential representation and I stumbled upon Hamiltonian mechanics, in which the electric E component can be seen as the momentum conjugate, and the magnetic B component as potential energy. This is a very interesting concept. Especially seeing E as momentum and B (or A) as potential energy is slightly surprising, because in the four potential, the electric potential is scalar, and A is a vector. But that’s the combined four-potential, not just another notation for E and B (the magnetic field). The mathematical connection between the four potential and E and B shows that E is a time derivative, and B is the spatial derivative. That’s why E relates to changes over time (momentum) and A about spatial correlations (potential).
So you could try to image the EM field as an harmonic oscillator, but that image is also a bit misleading because in an EM wave the energy is not really alternating like a pendulum between momentum and potential, it’s equally in both, oscillating in their common intensity, and the waveform is a spatial result of the ongoing interaction, pushing the energy forward.
And surprisingly, this push can be totally unidirectional. My view of the EM field that saw it as a kind of lattice of springs, giving tension, is not quite right, at least not in the lateral direction. A collimated beam of light can perfectly remain consistent, there are some complications at its rim, but there is no need for it to disperse. Beside the unavoidable diffraction at borders and slits, such a beam can remain a parallel wavefront, it can perfectly travel long distances, even from stars to us, without losing energy or coherence. There is no tension in the field that would tear it apart. And a transverse wave (linear polarized light) doesn’t even need any rotation, it can just travel in a straight line, no helix, no corkscrew.
The energy stress tensor although shows that there is a kind of tension in the field that has real applications. In optical tweezers e.g. it can be used to manipulate tiny (macroscopic) particles with laser light.
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Speculation: Field Quantization as emergent Property
With this background, the open core question I’m pondering is whether the Planck constant h is truly fundamental or emergent and whether it’s defined by the properties of the electron and interactions between the EM field and electrons, or vice versa. Is the electron and its specific energy content emergent from h being a fundamental property of the field?
The Planck constant shows up in many relations, again and again. It defines the energy content of a photon as E = h*f. But it also defines the Compton wavelength, the De Broglie wavelength and is part of the connection between the elementary charge e (the electron charge) and the finestructure constant ⍺.
⍺ = e² / (4 * PI * ε₀ * c * ) (with = h / 2 )
or transformed for h:
h = e² / ( 2 * ε₀ * c * ⍺)
or transformed for e:
e = √(2 * ε₀ * h * c * ⍺)
ε₀ and μ₀ are the fundamental properties for the EM field, and c is derived as c = 1 / √(ε₀ * μ₀).
So, h, e and ⍺ are somehow connected. But which one “came first”, which one is the defining property and which one is derived?
I assume the fine structure constant alpha to have a topological source, based on the structure of the electron, which is probably also self-confined wave configuration of the EM field (see: EM – The Toroidal Electron). I see the fact that this is a dimensionless constant (unlike ε₀, μ₀ and h) as a strong confirmation for this idea.
Regarding e and h … is the electron charge derived by h as also being a fundamental property of the EM field? Or is h derived from the properties of the electron?
The interaction between light and matter do always involve the electron, and maybe the quantization of the interactions is only caused by the quantization of the stable states that the electron can have, such as certain orbitals.
But then, when the electron is just a special EM wave configuration, why does the electron have exactly the energy content it has? Or, put differently, why is it apparently not possible to put any other amount of energy in the shape of an electron?
If h was fundamental, electron charge would be defined through the other fundamental properties (and alpha).
Q = φ + A
The Aharonov-Bohm effect shows that the four potential (φ + A) is more fundamental than the E and B vector representation.
I noticed that φ, the scalar value of the electric field is coupled to the magnetic field by ε₀, and the magnetic field is coupled to the electrical field by μ₀. But the magnetic vector field A has another internal degree of freedom, vector magnitude and direction. So, maybe there is another coupling between the magnitude and the direction of the magnetic field, within the A field itself. That would create a kind of constraint of a working field configuration and might give rise to quantization of the field itself. Maybe this is the third fundamental property of the EM field explaining h.
E and B can be derived from the four potential, but this is not a lossless conversion. E and B are just pseudo vector fields that can be derived from underlying quaternion, and usually the Lorenz gauge is used for this conversion. (An addition to the Maxwell equations by Ludvig Lorenz, not to be confused with Hendrik Lorentz. Even more confusing: The Lorenz gauge makes the formulation of electromagnetism Lorentz invariant.)
Yang-Mills proposed a different “gauge theory” that involves a factor “g” defining a quaternionic self-coupling. [https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_theory]
This internal coupling of the A field could be seen as a form of “directional stiffness”.
The Yang-Mills theory, a quantum field theory that aims to unify the weak force (causing nuclear decay) with the electromagnetic force and that is seen as a basis for understanding the Standard Model of particle physics, contains the following field intensity tensor:
F = dA + gA∧A
That A∧A term is exactly the self-coupling of the direction of A with itself. The coupling constant g controls how strongly the direction of A at one point influences nearby points. This is literally “directional stiffness.”
In this view:
- ε₀ governs the response of the field to scalar (electric) sources
- μ₀ governs the response to vector (magnetic/current) sources
- The quaternionic self-coupling (the A∧A term) governs the internal directional coherence of the field
The third coupling would set the scale at which the field’s own directional stiffness forces configurations into discrete topological classes. And the quantization scale – h – would emerge from this.
It describes how strongly a twist in the field direction at one point pulls on neighboring points. In a sense, g is just a property of the medium, like ε₀ and μ₀ are.
And here’s something interesting: ε₀, μ₀, and c are not independent. c = 1/√(ε₀μ₀). If g is the third fundamental field property, then you’d have three parameters (ε₀, μ₀, g) describing the EM field, from which both c and h emerge:
- c emerges from ε₀ and μ₀ (propagation speed)
- h emerges from g combined with ε₀ and μ₀ (quantization scale)
So, the core question is whether this can help to explain photon confinement. Can it help to explain why photons seem not to be able to have arbitrarily weak intensities? Why is there this direct relationship between wavelength and energy?
In ordinary Maxwell theory, any field configuration is allowed. You can have a wave of any amplitude, any wavelength, any shape. You can superpose them freely.
There’s nothing preventing an arbitrarily weak, arbitrarily spread-out disturbance. This is the linear field and this is why confinement is so hard to explain.
If A couples to itself (the A∧A term), then the field equation is no longer linear.
If “g” is real, and the A field has some self-influence that is not covered by the Maxwell equations (or to be more precise, the Maxwell-Heaviside equations for a 3D vector field), this could introduce a non-linearity in the EM field, explaining both the emergence of “h” as a fixed (preferred) relationship between energy and frequency, and the confinement of particle-like wave configurations.
When Planck defined his “Hilfsgröße” (that’s what the “h” stands for) he was thinking about the harmonic oscillators in the black body walls [https://en.wikipedia.org/wiki/Planck%27s_law#Trying_to_find_a_physical_explanation_of_the_law].
Instead of corpuscular oscillators, we have a vacuum EM field. Does it make sense to connect this “vector field stiffness” to “harmonic oscillators”?
This is exactly what g provides.
The self-coupling term in Yang-Mills theory effectively gives the A field a directional stiffness. When the field direction varies across space, the self-coupling term contributes an energy that depends on how much the direction deviates from its neighbors, not just how the field magnitude varies. The A∧A term in the field strength tensor couples the field to itself at the same point, meaning that spatial variation in field direction encounters a restoring force from the field’s own structure.
So:
- ε₀, μ₀ → spatial stiffness (spring constant) → determines propagation speed c
- g → directional stiffness→ determines the energy cost of spatial variation in field direction
If the EM field vacuum is an infinite collection of oscillators, each with its dynamics determined by ε₀ and μ₀ (which provide the complementary electric and magnetic responses that drive oscillation), then g adds a directional stiffness that constrains which oscillation modes can propagate. Only modes where the energy-to-curvature ratio matches the natural action density set by all three parameters together can resonate with the field. That natural action density is h.
This reframes h very concretely:
h is not a mysterious quantum of action imposed from outside. It’s the product of the field’s spatial stiffness and its directional stiffness, in the same way that the characteristic action of a mechanical oscillator is determined by its inertia and spring constant.
In a classical, mechanical harmonic oscillator (a weight on a spring), the combination of stiffness (spring constant) and inertia of the weight gives the harmonic oscillator its natural frequency. This is a constraint, the amplitude can vary, but the frequency stays (more or less) constant. The EM field already has its two oscillation ingredients – ε₀ and μ₀ provide the complementary electric and magnetic responses that drive the E↔B exchange. A pure EM field built from these alone could carry any combination of frequency and amplitude. But the directional stiffness (g) of the A field adds a third ingredient as the missing constraint. Instead of fixing a natural frequency, it fixes a natural action density. So for a given amount of energy to travel in a self-confined package, only the frequency (wavelength) remains as a degree of freedom.
This is a natural, “preferred” relationship between energy density and the “action”, the curvature (curvature density) of the field. When the energy density (field intensity) and the curvature match this natural density, a resonance is found and the field resistance (impedance) drops. Any combination of energy and wavelength is “pulled” towards this specific relationship. And when they match, “h” emerges.
- Mechanical oscillator: frequency fixed, amplitude free → energy ∝ amplitude²
- Self-coupled EM field: amplitude fixed (by g), frequency free → energy ∝ frequency
E = hf is the expression of this constraint. The proportionality constant h encodes the stiffness g that locks amplitude to frequency.
What are the dimensions of g?
In the field intensity tensor F = dA + gA∧A, the dimensions have to be consistent. In SI:
- A (four-potential) has dimensions of [V·s/m] = [kg·m/(A·s²)]
- F = dA has dimensions of [A]/[m] = [kg/(A·s²)] (which is Tesla, correct for B)
- The A∧A term has dimensions of [A]² = [kg·m/(A·s²)]²
For gA∧A to have the same dimensions as dA:
[g] = [dA] / [A²] = [A]⁻¹·[m]⁻¹ = A·s² / (kg·m²)
Connecting g to , ε₀ and μ₀
Can we derive a direct connection between g, h, ε₀ and μ₀? In the following exponent table, each row represents a physical constant expressed as a product of base SI units (kg, m, s, A) raised to powers. The numbers are those powers.
| Constant | Units | kg | m | s | A |
| ε₀ | A²s⁴/(kg·m³) | -1 | -3 | 4 | 2 |
| μ₀ | kg·m/(A²·s²) | 1 | 1 | -2 | -2 |
| g | A·s²/(kg·m²) | -1 | -2 | 2 | 1 |
| ħ | kg·m²/s | 1 | 2 | -1 | 0 |
(We’re looking for ħ, the reduced Planck constant which is just h / 2, so they have the same dimensions, but is considered to be more fundamental than h.)
We want to find exponents a, b, c such that:
ħ = ε₀ᵃ · μ₀ᵇ · gᶜ
Since the dimensions must match on both sides, the exponents of each base unit must match. This gives four equations (one per base unit), where we just add up the weighted exponents:
For kg: a·(-1) + b·(1) + c·(-1) = 1
For m: a·(-3) + b·(1) + c·(-2) = 2
For s: a·(4) + b·(-2) + c·(2) = -1
For A: a·(2) + b·(-2) + c·(1) = 0
Four equations, three unknowns. If a unique solution exists, it means there’s exactly one way to combine these constants to get the dimensions of h. If no solution exists, ħ can’t be built from them. If it’s underdetermined, there are multiple ways.
Solving (I’ll spare the substitution chain):
a = ½, b = -½, c = -2
So:
= ε₀^(1/2) · μ₀^(-1/2) · g^(-2)
Which simplifies to:
ħ = ε₀/μ₀ / g² = 1 / (g² Z₀)
since Z₀ = √(μ₀/ε₀) is the vacuum impedance (~377 Ω).
This turns a physics question (“can these constants produce h?”) into a simple linear algebra problem. And the fact that we have four equations (four base units) but only three unknowns (a, b, c) means the system is overdetermined – it could easily have no solution. The fact that it does have a unique, clean solution (½, -½, -2) is nontrivial. It means g with exactly those dimensions fits perfectly into the structure alongside ε₀ and μ₀.
If we had picked random dimensions for g, this system would almost certainly be inconsistent. So the fact that the dimensions dictated by the Yang-Mills A∧A term produce a valid solution is a meaningful consistency check for this framework.
This gives us:
- Fundamental field properties: ε₀, μ₀, g (three dimensionful constants)
- Fundamental topological constant: α (dimensionless, from electron structure)
- Derived: c = 1/√(ε₀μ₀), ħ = √(ε₀/μ₀)/g², e = √(2α)/(g·Z₀)
This is a clean and consistent structure. Three field properties plus one topological number generate all the key constants of electrodynamics and quantum mechanics.
The open challenge remains explaining α itself – which could come from the topology of the electron as a self-confined field configuration.
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Speculation Summary
The EM field has three fundamental properties: permittivity (ε₀), permeability (μ₀), and directional stiffness (g, the quaternionic self-coupling). The first two give propagation at speed c. The third creates a natural action density — a preferred relationship between field intensity and curvature.
Field excitations below this natural density cannot resonate with the field and are either suppressed, reflected or dispersed.
Field excitations at or near the natural density self-confine into topological structures, such as the “helical photon” or soliton waves. These are photons. They propagate as localized energy bundles, interact with electrons, and carry energy E = hf, where h emerges from g combined with ε₀ and μ₀.
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Regarding the impedance drop when the natural actions density is matched, I first thought that the vacuum impedance outside the match is 377 ohms (Z₀), and that there is a small well at this match. But looking at the relationships between e, c , g and h, I think it’s vice versa! For the harmonic match (the natural action density) we get Z₀, and a much higher value outside! This is a much better explanation for what we observe. For macroscopic effects such as radio waves, we always see 377 ohms, because they cause endless small harmonic wavelets carrying the energy. And photons and electrons interact smoothly, because they are like impedance matched transducers, that’s why the exchange of energy is so lossless. It might be that it’s very hard or even impossible to create any wave that’s outside of that match, only static fields. That would explain a lot!
So, from this point of view, Z₀ = 377 Ω is not just a property of the vacuum – it’s the signature of the field’s self-coupling resonance. All propagating EM configurations exist at (or extremely near) the natural action density because the impedance for non-matched configurations is prohibitively high.
This is why quantization appears universal: not because it’s imposed axiomatically, but because the field won’t carry anything else.
This also gives a clean physical picture for why E = hf appears exact in all measurements – you literally can’t create a propagating excitation that violates it, because the field won’t support it. The “attractor” isn’t just strong, it’s essentially the only propagating solution.
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So, the field is quantized. Didn’t we already know that?
Yes and no. We observe the quantization in photons, particles and the interactions, but the Maxwell equations cannot explain this. The quantization is put on top, added as an external requirement. A wave of light is a spatially extended EM field configuration and why this field configuration should have a certain match between energy (intensity) and frequency is not clear. Classically, any combination of frequency, amplitude and spatial extension should be allowed. That’s why the particle model of light and matter has been added, but with no clear explanation how these two can coexist.
By adding this third fundamental property to the EM field itself, the quantization emerges from the field. What we observe in photons is a natural match between energy and frequency that the field enforces. This does not only state the fact that the field is quantized, it provides an explanation as to why.
More Connections,
Photon spin: ±ħ. One complete twist, one quantum of angular momentum. Directly from the helical photon model — one full rotation of the field around the propagation axis.
Orbital angular momentum: electrons in atoms have L = nħ. If the electron is a self-confined field configuration orbiting a nucleus, each orbit must accumulate an integer number of action quanta. A half-integer orbit would leave the field configuration out of phase with itself – destructive self-interference.
Spin-½ particles: electrons have spin ħ/2. This is the one that’s interesting – it requires a 4π rotation (two full twists) to return to the same state. In this framework, this would directly reflect the topology of the electron as a field configuration. A structure that needs two full rotations to close is topologically distinct from one that closes in one rotation. This connects to the toroidal electron model – a toroidal structure naturally has this double-rotation property (rotation around the ring axis vs rotation of the ring cross-section).
So, from g alone we get:
- ħ = √(ε₀/μ₀) / g² → natural action per radian
- Photon spin = ħ → one twist
- Boson angular momentum = nħ → integer twists
- Fermion angular momentum = nħ/2 → topology requiring double rotation
- Orbital quantization = nħ → self-consistency of orbiting field configurations
All of these are postulated separately in QM. In this framework, they all trace back to one field property: the rotational self-coupling g.
Caveats
Although this might be a good explanation for the quantization of the field, this is NOT an explanation for some other observed quantum effects. If photons are really indivisible quanta of energy – what about single photon self interference? How can a single photon travel multiple paths (in a double slit experiment or a Zehnder-Mach interferometer), get a phase shift, and interfere with itself?
I don’t know.
From the light-as-continuous-wave perspective, these experiments are trivial to explain. From a particle (or wavelet) perspective, they are a mystery. Somehow, the EM field is able to perfectly define the probability to detect photons, yet the energy transferred is always quantized.
Some ideas:
- Droplet walkers? (https://www.youtube.com/watch?v=WIyTZDHuarQ)
- The EM field as a “guiding field”?
- …?
In the double slit experiment, the amount of photons observed is the same, with or without interference. There is no destructive or constructive interference within a single photon, only their distribution is changed. So, if light is transferred in quanta, somehow, somewhere, they HAVE to be redirected or “guided”. How this could work is the subject of further research and speculations.