Foreword

The intention of this paper is to propose an idea that uses a classical approach yet complements modern physics. It offers a model that gives a tangible form or structure to the electron, a form that is in complete agreement with physical reality. Hopefully, this may simplify and unify our understanding of nature.

The paper is aimed at the general science reader in the hope that as wide an audience as possible may consider these ideas. After all, ideas may not always change the world - but they can change our view of it.

It would not have been possible without the help and thoughts of many people, some of whom are listed in the Acknowledgements section. Two especially have shown me key concepts which I have used to hopefully build a coherent theory. They are Qui-Hong Hu in his 2005 paper "The Nature of the Electron" [1], and Claudio Rebbi who wrote the article "Solitons" for Scientific American in February 1979 [4]. Of course, the internet itself deserves much praise by providing a wealth of information. No scientist before us had such a vast library and resource.

My friends and family have also provided much needed support as well as constructive criticism. To them I am truly thankful.

If I have pondered and seen some things it's because so many brilliant people have shone their light. I merely looked.

Structural Electrodynamics - SED

The Electron                                                Rowan Kramer BSc (Physics)

Abstract

A new dynamic model of the electron is postulated based on the structure of its fields. The electron is a soliton or standing wave that does not disperse, created by the superposition of two orthogonal circular waves. To interact, two photons can spin with the speed of light simultaneously in two orthogonal axes. The first is that of the normal E and B fields of all photons. However, rather than moving in a straight line at speed c, this forward motion is translated into a circular twist or spin with the same frequency and speed. The B field initiates this turn and creates a new curved path that turns back on itself, allowing these fields to remain localised in space and to portray the mass, charge and magnetic moment of the electron. This process is supersymmetric in that it transforms bosons into fermions. Antimatter is simply twist or helicity in the opposite direction. Other fundamental properties such as half-integer spin, the two quantum spin states, the Bohr Magneton and the magnitudes of the integrands of the internal E and B fields, can be derived from this model. Lastly, a new explanation of the origin of the Fine Structure Constant is discussed.

Introduction

Beginning with Planck’s paper in 1900, there have been many amazing human endeavours to advance physics. Our ideas and technology bring new ways to perceive the world. Quantum Mechanics has given us the most accurate tools to predict experimental outcomes at the subatomic level. However, it has ignored attempts to visualize or grasp the underlying structure of this world. Perhaps we can.

Matter is energy enfolded and trapped in a soliton. A soliton is a standing wave or quantity of energy that can move from place to place but does not spread out – like a soap bubble, it has nowhere else to go [4]. With this model, the equations of energy and Maxwell alone, correctly describe the electron’s mass, charge, magnetic moment and spin. Also, of fundamental importance is the quantum that is Plank’s constant – a minimum of angular momentum.

We will only consider a free electron in isolation - not one that exists in an atom. A free electron is not influenced by external forces or events: there is no power transfer, so we can concentrate on the electron itself and its own spin motion. Orbital motion as found inside the atom, has a complex range of energy levels but that is not considered here, so the electron's interaction with light photons need not be investigated. We will find though that there is much to be revealed.

A photon, like all bosons, is field energy both spinning and moving forward at the speed of light. Being purely energy transfer with no volume, it has zero rest-mass and according to Maxwell's equations, continually self-creates its transverse E and B fields in a regular oscillation. The photon's path is always a straight line unless curved due to gravitational or external E-M fields.

In this model, an electron is field energy in a 3-dimensional cycle spinning simultaneously in two orthogonal axes, giving it a volume that is localised in space. It can be thought of as a super energetic photon consisting of two E and B fields both orthogonal to its direction of motion. When the Big Bang, with its extreme temperatures, supplied this photon with sufficient additional energy, it twisted or spun on a second axis, either in a left or right-hand helix, until it no longer moved forward in a straight line at c. Energy still flowed within the soliton at c, but having now acquired mass meant its localised speed must always be less. The left-hand helix became the electron and the right-hand the positron. They are identical other than spin direction.

The soliton, trapping this energy in the form of mass occupying space, can persist for billions of years. Doing so creates a single standing wave of field energy flow. A tiny volume with a certain sized curved path that turns back on itself - a special curve consisting of two loops, each one half of the full E-M cycle. It is called a Hubius Helix or 2-turn helix. Appropriately, it has the shape of the lemniscate or infinity symbol, ∞, draped over a hemisphere whose radius is about 137 times that which was classically attributed to the electron, re, the thickness of this path The wavelength of this soliton is equal to ƛe (lambda bar-e), the reduced Compton wavelength of the electron. The ratio of this classical electron radius, re to ƛe is exactly α (alpha) - the enigmatic Fine Structure Constant.

Figure 1 - Dimensions for the electron shown on a flat Hubius Helix (left) and the bent Hubius Helix from the front (right)

The Hubius Helix can be drawn flat to show the relative sizes of the overall field path and each loop (figure 1, left). Its equation, in polar form, is then: r2 = ƛe2Cos2θ. Before it twisted or curled, the photon was a flat circle with a radius of one wavelength, ƛe. In the actual soliton, each loop is folded down into the page at 90° to each other, creating a volume with the crossover at the top as in figure 1, right.

The path is draped over a sphere of diameter ƛe√2. The electron is a soliton of field energy embedded on this sphere, like a figure-8 lying on a northern hemisphere with its crossover on the north pole. It slices through the sphere with two curved planes at 90°, each touching the equator and intersecting at the pole, forming the two loops. The soliton captures energy and portrays it as mass by its rotational inertia. The path is made with two equal circular frequencies, creating a 3-D fundamental standing wave as shown in Fig 2 below. Note that the electron has a size through its wavelength but no hard surface or edge. It appears to be a point, taking up no spatial extent, precisely because it is not a solid particle with mass. It is trapped field energy that possesses both wave and particle properties. The changing field that is the electron fills all surrounding space-time and that is how it travels all possible paths (theoretically). Through all possible slits (maybe).

Figure 2 - A 3D view of the Hubius Helix (Thanks to Qiu-Hong Hu) [1]

This dynamic structure gives rise to quantum behaviour and finally settles the wave-particle dilemma. The electron is a localised pattern of field energy - only energy flowing on an endless tiny path cut into the surface of a hemisphere. As discussed later, this energy can appear concentrated in the soliton's core in a region of space equal to re. However, because of its momentum, the uncertainty principle forbids giving sense to any size less than ƛe/2, half the diameter of each loop which we will write as r0

Like the Higgs field, our soliton has its minimum energy at a constant non-zero field strength. This minimum is known as the vacuum state and surrounds the soliton, trapping it inside. Later it will be shown that this vacuum state corresponds to a minimum of electrical potential energy, and according to Coulombs law, lies on the sphere with a radius r0√2. Inside this radius we find field energy rapidly rises above this minimum. Theory also shows that for a soliton to exist we need its field magnitude to be zero at some point, but that the energy carried in this field must have a definite non-zero value at that time. This occurs twice in our model at each crossover when both fields are momentarily zero. Now, because the intrinsic energy of a free electron always remains constant at mc2 = hf, we have both of the necessary conditions to allow a soliton to form [4].

So, our soliton is an energy quantum forming an object that does not decay. Many other examples exist where the wave system sustains itself. Since the energy in this soliton is fixed (through E = ћω), then its mass, wavelength and intrinsic properties are also fixed in order to maintain its rotational speed at c, and keep angular momentum constant at ћ.

A Model of the electron - the Hubius Helix

A model is a simplified (usually physical) explanation of some complex phenomena. All useful scientific models need to meet at least two criteria:

1.       They must be self-consistent

2.       For every input given to the model the output must be real

The model proposed here is both a shape and a process, consisting of a field of electromagnetic energy, flowing at speed c on a path that is infinite yet enfolds a tiny space - the electron.

The Hubius or 2-turn helix is the path that forms the structure of the electron. It can be created in a number of ways as in the examples described below, these three being demonstration models. Familiarity with this shape and how it arises is vital to understanding the model and this theory.

1.       A thin strip of semi rigid material e.g. a woven plastic strap about 1 cm x 60 cm is twisted one full rotation or 360°, then the ends are joined at the crossover, creating a curved infinity symbol as in Figure 3. This can be marked-up to show field flow direction and distinguish inside/outside surfaces. Note: This is not a Möbius strip which has only a 180° twist and forms one continuous surface.

Figure 3 - Making a Hubius Helix from a strip (Based on drawing by Williamson & Van Der Mark) [2]

2.       Figure 4 below illustrates a rotating vertical disc (e.g. a CD), in its plane while simultaneously twisting it sideways in a circular motion at the same rate. The locus of a small circle (~ 1/137th of the disc & shown in pink) drawn on the disc's edge is the Hubius Helix. Significantly, rotation direction (i.e. clockwise or anticlockwise when viewed from front) reverses twice each 3600 to maintain angular momentum. There are also two, 360o turns per 720o cycle, as is a requirement of Lie groups, and consistent with the spin of fundamental particles. However, this model does not need any abstract space. This twist motion prevents the fields changing their sense as will be explored in the section on charge and electrons.

Figure 4 - Making a Hubius Helix from rotating discs

3.       Figure 5 shows how the same 2-turn helix, but in this case in a different orientation, can be created by tracing the path in space a small area on a polar great circle sweeps out, as it rotates around a horizontally revolving sphere. A bit like a plane flying a polar great circle every 24 hours. The Hubius Helix lies on the surface of the sphere in one vertical hemisphere. This has the same shape as model 2 above but may be easier to visualise.

Figure 5 - Making a Hubius Helix from a revolving sphere and a great circle

In the last two models we have used a small circle rather than a point, to form the Hubius Helix path because this helps envisage that the twist of the small circle, as it moves around its path, is essential to the creation of the soliton. With the strip model, this twist is more obvious and more easily shows how the model’s topology allows the standing E-M fields to produce an electron.

Principles of Structural Electrodynamics - SED

1.       The laws of physics are invariant regardless of the size of the system under consideration. This applies even at the scale of sub-atomic or quantum dimensions.

2.       At this subatomic level, all inherent or intrinsic motion is circular, and the minimum quantum of angular momentum is ћ/2. Linear momentum is merely the case when the radius is large.

By adhering to these two ideas and applying the laws of classical physics to our model, we will show that all the properties of the electron, together with their correct values, may be derived from first principles. The magnitudes of the EM-field within the electron are calculated and then used to determine the size of the elementary charge, thus providing self-consistency of the theory. Even the elusive fine structure constant will be shown to have a new physical basis when considered from this perspective.

The mass of the electron - me

Firstly, we need to consider:  The Origin of Mass

To do this, we need Planck and Einstein's equations:        E = hf     and        E = mc2

Combining them gives:    or    . . .   ①

Secondly, from circular motion and Planck's quantum of action we have: mcƛ = ћ

where ƛ is the wavelength or equivalently the radius of the motion, giving:  . . .   ②

An equivalent equation is Eƛ = ћc.

Next, we will do a thought experiment: Imagine a lighthouse with a rotating light sweeping out a beam once per second. How far do we have to go from this lighthouse until the sideways or tangential speed of the beam is c? The answer is about to the moon or 300,000 km (temporally ignoring the factor of 2π) because light takes just over a second to travel this distance. Now, imagine a kind of a stationary photon with this wavelength and frequency, remembering that all photons have angular momentum of ћ. Then if we reduce its size or radius to 1m, while keeping ћ and c (its circumferential speed) constant, what is its new frequency? Like an ice skater, its spin or angular frequency will increase, so if ƛ = 1 by using ωƛ = c, we get ω = 3 x 108 radians/sec. From ① above, its mass has become 3.52010787 x 10-44 kg. Finally, if we reduce this radius to ƛe (the reduced Compton wavelength of the electron), its angular frequency will increase further to ωe = 7.76344070 x 1020 radians/sec, giving it a mass of 9.10938291 x 10-31 kg - the mass of the electron me.

Now this was a photon like our soliton that remained fixed in space with zero rest-mass when infinitely large. We have kept both angular momentum constant at ћ and its rotational speed constant at c. By reducing its wavelength or radius, we must increase its frequency in direct proportion. We do this by giving the photon energy that it stores and manifests as an increase in mass (rotational inertia). So, we find that mass is created where space is compacted. In the case of the electron, this mass is a fixed constant formed inside the curl of the photon’s path from energy supplied by The Big Bang and held in a soliton.

This reasoning shows that we don't need a separate entity called mass - only energy in the form of frequency confined in a soliton. The universe becomes simpler. The soliton localises this energy and because mƛ = ћ/c = constant, increasing the mass of a photon or soliton means reducing its size and increasing its curvature. Mass (or energy) and wavelength are inversely proportional. This is why an electron is much smaller than a light photon. Its wavelength is about one millionth the size, so it is twisted into a very tight curl. It is also why a proton is smaller than an electron. For a given mass, its wavelength is fixed and vice versa. The entire electromagnetic spectrum bears this out.

Like a gyroscope, it is rotational inertia of the E-M field that we perceive as mass, - the inertia of the electric and magnetic fields constantly recreating themselves from confined energy, held inside the localised and compacted volume of space that is the soliton. The intense curvature of the path creates mass as Einstein predicted.

Charge and the electron

Empirically, we find charge, e, is always negative for the electron and positive for the positron. This model predicts that an electron is created from left-hand spin and a positron from right-hand. How does this come about?  (Note: A demonstration model such as the strip can help to explain this).

The Origin of Charge

A photon has rapidly rotating E and B fields creating each other as it moves in a straight line through space at speed c. Being self-generating, these fields need no medium: only their initial energy ћω. With the correct additional energy, however, this photon can also rotate on a second orthogonal axis becoming the soliton or standing wave of this model. There is a very interesting property associated with this motion, due to the fact that these two frequencies are the same.

It turns out that each loop is created after a sideways twist of 1800, (i.e. phase change of π), and the two E and B fields turn back on themselves, never reversing direction or sense. Overall, each 360o loop forms half of one cycle, of the full 720o E-M oscillation. Each loop is also a spinor that inverts the phase of the fields after each rotation around a loop. Therefore, their fields signs (i.e. +ve or -ve) don't change like those of the straight-line photon. They still generate each other and behave according to Maxwell's equations (see below) yet remain confined in space with a permanent sense, either positive or negative. This continues unabated for the life of the soliton.

By considering only one cycle of an E-M wave and the influence of sideways twist on the direction or sense of its fields, we can understand how charge is generated and maintained in our model and depicted in Figure 6 below. It shows that the spinning photon’s E-field creates charge and not vice versa.

Figure 6 - A photon moving in the Hubius Helix path creates charge. (Modified from Williamson and Van Der Mark) [2]

The top drawing here in Figure 6 shows the single cycle of an untwisted E-M wave, with the effect of a 3600 twist drawn below it. Due to this twist, our model shows that outside the electron, the E-field is always pointing inward at 900 to the path's motion (opposite or out for the positron), thus giving it what appears to be a negative charge, here shown in the lowest drawing as the up and down E-field arrows, both pointing inward in the hubius helix. This is divergence and is the source of negative charge in the electron. For the positron the field lines point in the opposite direction due to its opposite twist or helicity.

It means that what we call charge and magnetic moment are actually constructs of the photon’s standing E and B-fields, always oriented either in or though the soliton.

The electric field defines the force per unit charge experienced by other charged objects in space. Normally, this can vary depending on position and the inverse square law, but because of the uncertainty principle, inside the electron where electric force, Fe = Eee, both Fe and Ee portray a constant total magnitude over each half-cycle. Later we will show why this is so and then calculate these values, thereby determining the fixed value of elementary charge, e.

As stated before, the electron has left-handed twist or helicity and the positron right-handed. This comes about because opposite twist or curl cause the sense of the field to be opposite, and once created in the soliton they will not change – charge is conserved. The chance of forming an electron or positron was almost the same but somehow electrons became dominant after the early universe.

Magnetism and the electron

This model explains magnetism in the electron both qualitatively through its structure and quantitatively through its calculation of the Bohr Magneton or electron’s magnetic moment. Magnetic poles, like charge, are a consequence of the soliton’s dynamic field structure and not a cause.

The Origin of Magnetism

All magnets are dipoles and, by definition, a magnetic dipole must have opposite poles separated by a certain distance.

If we change the orientation shown in Figure 6, so that the crossover is now on top and the two loops are horizontal, like the second drawing in Figure 1, then the path of this model can be viewed both from above and end-on. Significantly, field energy flow is in opposite directions viewed from above but in the same direction when viewed end-on. See next figures.

Figure 7 - View from above

 

Figure 8 - End-on or side view

If we look at our model from above, field energy flow is clockwise in one loop and anti-clockwise in the other. From this perspective, the right-hand rule for magnetism means that their poles are opposite or Nth in one and Sth in the other. Each loop is a magnetic monopole but cannot exist independently. We always have two. Further, using geometry and our front view, the radius of a loop being r0 means that the centres of each loop are separated in space by r0√2. Thus, we have a dipole. The electron is a permanent magnet.

Viewed end-on, field energy flow is in the same direction for both loops, similar to a solenoid, but here there is no point-source of current flowing. The Uncertainty Principle forbids this. Only field energy in the form of a wave moves in this path. This structure creates the magnetism that our instruments can measure outside the electron.

Magnetic attraction from opposite poles makes the path bend at its centre with a specific angle of 90° between the loops (as in the 3D helix shown in the Introduction). This is a limit because reducing this angle further means reducing the polar attraction. If the loops were to become close parallel circles, the field energy flow in each loop has the same direction, thus making identical poles which no longer attract each other but rather repel.

We can now use our model to derive values for the properties of the electron and test its validity.

The spin of the electron: s = ћ/2

Electron spin has always been something that only quantum mechanics could account for. After all, if the quantum of angular momentum is ћ, how can anything have half this amount? Further, why do photons and bosons have a spin of ћ but fermions with their mass have a spin of ћ/2? It seemed nonsensical - and what would be spinning anyway? This model provides an answer.

The key lies in the twist or curl the photon acquired from energy supplied by the extreme temperatures after The Big Bang. As energy was added, causing its frequency to increase and wavelength or path radius to decrease, the fundamental law of conservation of angular momentum was never broken. This continued until its spin or twist and the angular frequency of its E and B fields equalled one particular value. This frequency was exactly that which kept it stable and gave the electron its mass and charge. The electron is a design that stores energy.

Like all elements of the Lie Group, this process is supersymmetric, performing two turns of 360o per cycle. It turns flat bosons into fermions with mass that become localised and occupy space with their volume. The behaviour of this model perfectly explains intrinsic spin of fundamental particles without the need for the abstract concept of spin-space, as in the Standard Model. Regular space will suffice. The electron has spin, with real physical angular momentum.

Normally, angular momentum, L, manifests through circular motion and is calculated from the product of a spinning object's mass, tangential velocity and radius. Thus L = mvr, in units of ћ. (Following convention, we will write s rather than L for the electron's spin angular momentum).

Our soliton is no different and moves in two almost circular paths at the speed of light as energy jumps from loop to loop. First one then the other, not both simultaneously.

Figure 10 - Energy flows from loop to loop.

In each loop we have: s = mecr0  = ћ/2

The twist causing this 2-turn helix to form is created by energy, but it does not change total angular momentum. Like all photons this remains constant at ћ. However, the path of each loop now has a radius half the photon's original radius or wavelength, which was ƛe. Moving sequentially from loop to loop, the photon's angular momentum at any one time is now half. Thus spin, s = mecr0  = mec ƛe/2 = ћ/2. When we measure or calculate its instantaneous spin value the answer is ћ/2. Total angular momentum is shared between two loops. This structure explains spin value and its physical reality.

Spin, measurement and quantum uncertainty

We know that total spin for elementary particles is always ћ, a tiny but absolute amount. For fermions with their mass due to compressed space, this is ћ/2 at any instant, divided between the two loops with their reduced radius. Further, angular momentum is a 2-dimensiopnal vector property and can be aligned in any direction either clockwise or anti-clockwise. A free electron has no preference. The act of measurement is an external influence and causes the electron to align or anti-align with the measuring device’s magnetic field. It forces one direction, either positive or negative, to be chosen and retained if there is no further external interaction. This is why it has spin in only one axis, either x, y or z, and knowledge of one means zero knowledge of the other two – they don’t exist. Again, spin is 2-dimensional, and according to this model its axis is real and can be visualised even if its direction and sense are unknown before measurement. There is no mystery or paradox due to quantum uncertainty of spin about more than one axis at a time.