The photon

The photon has a wave nature, which is why we can refract and diffract light. But what sort of a wave nature? When you try to find a picture, a lot of illustrations depict the photon as some kind of wave train. Even Feynman diagrams do this.

Image by bitwise, see Wikipedia commons

The photon is shown as a squiggly line, sometimes with an arrowhead, something like this: ⇝. That suggests you could split a photon lengthwise and end up with two photons, each with the same wavelength as the original, each with half the energy. That can’t be right. The photon energy E = hc/λ depends on Planck’s constant h and the wavelength lambda λ. Wavelength is inversely related to frequency f via the speed of light c, which is distance over time. Hence we can also write the photon energy as E = hf. But there is nothing in this expression to denote the number of waves in the train. And when you chop a photon in half with a beam splitter to convert it into two photons, each has twice the wavelength as the original. The photon is one of those things where when you chop it in half it’s twice as big. So it isn’t a wave train.

The photon is not a wave packet

Other illustrations depict the photon as a wave packet. You can find articles suggesting that Einstein talked about wave packets in his 1905 photoelectric paper. However he didn’t actually use the phrase, he talked about the light quantum instead. He said light quanta move without dividing and are absorbed or generated only as a whole, and that the simplest picture is one where the light quantum gives its entire energy to a single electron. That fits with the wave-packet idea. But Einstein also said “it must not be excluded that electrons accept the energy of light quanta only partially”. That’s as per Compton scattering, where some of the photon energy is transferred to an electron and the photon wavelength increases. That doesn’t fit with the wave-packet idea. And as far as we know the photon has a single wavelength, not multiple wavelengths. There is no actual evidence that a photon is some infinite set of component sinusoidal waves. On the contrary, the evidence says the photon is a single wave or pulse as per How Long Is a Photon? by Igor Drozdov and Alfons Stahlhofen dating from 2008. There are no observations of any oscillations inside a photon. There is no evidence that a photon is a wave-train like their figure1. Or a lemon-like wave-packet of waves of different amplitudes like figure 2. The evidence says the photon is more like their figure3:

Images from  How Long Is a Photon? by Igor Drozdov and Alfons Stahlhofen

That’s not to say waves don’t come in trains. We know they do. We’ve seen what happens in an earthquake. But I think it’s better to say a train of light waves is a succession of photons, not a single photon. So I think the photon must be some kind of a singleton soliton light wave. So far so good.

The photon is not an electric wave and a magnetic wave

So, a photon is a singleton soliton light wave. Light waves are electromagnetic waves. Not electric waves and magnetic waves, electromagnetic waves. James Clerk Maxwell unified electricity and magnetism a hundred and fifty years ago. The electromagnetic field is dual entity, wherein electric and magnetic fields are “better thought of as two parts of a greater whole”. See section 11.10 of John Jackson’s Classical Electrodynamics where he says “one should properly speak of the electromagnetic field Fµv rather than E or B separately”. It’s similar for electromagnetic waves. See the Wikipedia electromagnetic radiation article and note this: “the curl operator on one side of these equations results in first-order spatial derivatives of the wave solution, while the time-derivative on the other side of the equations, which gives the other field, is first order in time”. The orthogonal sinusoidal electric and magnetic waves in the depictions are somewhat misleading. The electric wave is the spatial derivative of the electromagnetic wave, whilst the magnetic wave is the time derivative. For an analogy, imagine you’re in a canoe at sea. Imagine something like an oceanic swell wave or tsunami comes at you. Let’s say it’s a 10m high sinusoidal hump of water without a trough. As the wave approaches, your canoe tilts upward.

The canoe analogy, E= tilt, B=rate of change of tilt

The degree of tilt denotes E, whilst the rate of change of tilt denotes B. When you’re momentarily at the top of the wave, your canoe is horizontal and has momentarily stopped tilting, so E and B are zero. Then as you go down the other side, the situation is reversed. The important point to note is that there’s only one wave there. Like Oleg Jefimenko said: “neither Maxwell’s equations nor their solutions indicate an existence of causal links between electric and magnetic fields. Therefore, we must conclude that an electromagnetic field is a dual entity always having an electric and a magnetic component simultaneously created by their common sources: time-variable electric charges and currents”. That’s why E and B are always in phase. Because the current in this canoe analogy is water surging up then down. It displaces you in an upward direction with a tilt and a rotation that increases then decreases. Then it displaces you back down again. Because of this It’s an alternating current rather than a direct current like a river. Note that electrical impedance is resistance to alternating current, and that there’s such a thing as wave impedance and vacuum impedance. Also note that it’s a myth that an E wave generates a B wave which generates an E wave and so on. The people who say this tend to be unaware of electromagnetic unification, and tend to say that this is why light doesn’t need a medium to travel in. It’s an incorrect assertion. We have Faraday’s law, usually written as ∇ × E = − ∂B/∂t, not because changing one field creates the other, or because one circulates round the other. The equals sign is an “is”. The curl of E is the time rate of change of B. Because they’re two aspects of the same thing, the electromagnetic wave.

Potential is more fundamental than field

Or they’re two aspects of an electromagnetic field-variation if you prefer. I prefer the former myself, because I think the photon electromagnetic wave is more fundamental than the electron electromagnetic field. But perhaps neither is ideal, because potential is said to be more fundamental than field. There’s an unattributed remark in the Wikipedia Aharonov-Bohm article: it says Feynman wished he’d been taught electromagnetism from the perspective of electromagnetic potential rather than electromagnetic fields. Yes, there’s an unfortunate ambiguity in that the use of the word fields as opposed to field suggests we’re talking about the electric field and the magnetic field, not the electromagnetic field. And things can get confusing in that electromagnetic four-potential is also described as the gauge field of quantum electrodynamics. But remember that as per the canoe analogy, the orthogonal sinusoidal electric and magnetic waves are the spatial and time derivatives of the real wave. The real wave is the integral of either sine wave. Now think of that hump of water, and the potential is the height of it. It’s there because the water is there. The electromagnetic wave is the exterior derivative of potential, the shape of the hump. The electric field is the slope of the hump at some point, and the magnetic field is the time-rate-of-change of slope. The important point is that without ten metres of extra water underneath you, there would be no hump, no slope, and no change of slope either. That’s why potential is more fundamental than field. However it can be difficult to detect. If you were canoeing on Lake Superior, you might not realise you’re 600ft above sea level. It tends to be a local change in potential that you can readily detect. Like at Niagara Falls.

The photon takes many paths

Mind you, it might not be quite as local as you might think. When you look at the sea, you see waves that are perhaps a metre high. It’s tempting to think that’s the size of them, but it isn’t. If you take a look at wind waves on Wikipedia, you can see what lies beneath. The wave isn’t something that’s a metre high. It extends deep into the ocean:

GNUFDL image by Kraaiennest, see Wikipedia Commons

If you could pick up the whole ocean and place it upside down on top of another ocean, you would appreciate that a wave running through it isn’t just a metre high. The displacement might be a metre in extent at its maximum, but the wave itself might be much more extensive. It’s similar with a shear wave in a solid. Think of a seismic S-wave travelling West to East from A to B across a plain. It isn’t just the houses sitting on top of the AB line that shake. Houses five miles north and south of the AB line shake, albeit less. Houses ten miles north or south shake too, albeit even less. People can still feel that earthquake a hundred miles away from the AB line. Seismometers can still detect it a thousand miles away. The point to note is that a seismic wave doesn’t just take the direct AB path like it’s some point-particle. Even if it goes straight as a die from A to B it effectively takes many paths. It’s similar for a photon, because it has a wave nature, and that’s what waves are like:

However electromagnetic waves aren’t exactly like water waves on the ocean. Water waves are also known as surface gravity waves, and they’re trochoidal rather than sinusoidal. In addition the speed of an ocean wave depends on wavelength, and the speed of an electromagnetic wave does not. An electromagnetic wave is more like the seismic S-wave in that the speed depends on the medium through which it moves, not on wavelength. But it isn’t exactly like the S-wave. As per Richard Beth’s 1936 paper on the mechanical detection and measurement of the angular momentum of light, the photon’s angular momentum is either ħ or -ħ depending on whether it has a left or right circular polarization. It’s orthogonal to the angular momentum of the trochoidal wave.

The quantum nature of light

See Leonard Susskind talking about Planck’s constant in the YouTube video demystifying the Higgs boson. At 2 minutes 50 seconds he rolls his whiteboard marker round saying angular momentum is quantized. Think like this: ”roll your marker round fast or slow, but roll it round the same circumference, because Planck’s constant of action h is common to all photons regardless of wavelength”. The quantum nature of light isn’t just some slope on a photoelectric graph. It’s something real. As for what, the dimensionality of action can be expressed as energy multiplied by time, or momentum multiplied by distance. As for what distance, take a look at some pictures of the electromagnetic spectrum. Note how the wave height is always the same regardless of wavelength:

Electromagnetic spectrum image thanks to NASA

Yes, it’s only a picture, but I think it’s important, because I think the quantum nature of light is hiding in plain sight. It isn’t anything to do with energy being conveyed in lumps. A photon can have any frequency you like, and therefore any E = hf energy you like. You can vary the energy smoothly, so it isn’t anything to do with quantum lumps. It’s to do with this: the amplitude of all light waves is the same regardless of frequency. What amplitude? The answer to that depends on another question, which is this: what waves? To answer that, we need to ask another question: do you know of any waves where something doesn’t wave? Because I don’t.

Electromagnetic waves travel through space

Electromagnetic waves travel through space, which Einstein said was a something rather than a nothing. He is said to have dispensed with the luminiferous aether, but in his 1920 Leyden Address he referred to space as the ether of general relativity. See the Robert Laughlin quote in the Wikipedia aether theories article: “It is ironic that Einstein’s most creative work, the general theory of relativity, should boil down to conceptualizing space as a medium when his original premise [in special relativity] was that no such medium existed”. That might comes as a surprise, but remember what Cao and Schweber said about the plenum assumption of the bare vacuum – the vacuum isn’t some state of nothingness, it’s a polarizable medium. Hence Schwinger’s 1949 paper quantum electrodynamics II : vacuum polarization and self-energy. The speed of a shear wave is given as c = √(G/ρ), and the speed of a light wave is given as c = 1/√(ε0μ0). I do not believe this similarity is mere coincidence. Particularly since the Maxwell stress tensor is a part of the electromagnetic stress–energy tensor. Particularly since in his 1929 history of field theory Einstein described a field as a state of space. That might sound archaic, but see what Steven Weinberg said on page 25 of Dreams of a Final Theory: “a field like an electric or magnetic field is a sort of stress in space”. Also note that the stress-energy-momentum tensor “describes the density and flux of energy and momentum in spacetime, and it includes a shear stress term. Shear stress is something that only a solid can sustain. Gases and liquids cannot, which is why there are no transverse waves in the air or the sea. So in a way, the stress-energy-momentum tensor treats space like some kind of gin-clear ghostly elastic solid. It’s not totally unlike the continuum-mechanics Cauchy stress tensor.

Space waves

So, when we ask what waves, I think the answer is clear. After all, when an ocean wave moves through the sea, the sea waves. When a seismic wave moves through the ground, the ground waves. So, what waves when a light wave moves through space? There can only be one answer, and that answer must be space. Space waves. Some might dispute that, and say the electromagnetic field waves instead. But if you’ve ever read Einstein trying to unify the electromagnetic and gravitational field I think you would take a different view. Especially if you’ve also read LIGO articles which say “gravitational waves are ‘ripples’ in the fabric of space-time”. Then when you know that gravity is “not the curvature of space, but of spacetime”, you know that a gravitational field is not a place where space is curved. Instead it’s a place where space is inhomogeneous, in a non-linear way. It’s like a spatial-energy density-variation, and it must be the same for a gravitational wave but in a dynamical fashion. A gravitational wave is said to be a quadrupole wave, with alternate transverse and longitudinal compression. To keep it simple I’ll show only the latter:

It’s rather like a sound wave. The grid lines aren’t curved. Space isn’t curved where a gravitational field is, or where a gravitational wave is. So where is it curved? Come with me to a cliff by the sea.

Where space is curved

Imagine you’re standing on a headland overlooking a flat calm sea near an estuary. The water is saltier on the left than on the right. You see a single ocean wave, and notice that its path curves left a little because of the salinity gradient. The sea is an analogy for space. The salinity gradient is an analogy for a gravitational field. The ocean wave is an analogy for a photon. Now look at the surface of the sea where the wave is. It’s curved. It’s curved in a far more dramatic fashion than the curved path of the wave. This  observation might sound radical, but see what Percy Hammond said in the 1999 Compumag: “We conclude that the field describes the curvature that characterizes the electromagnetic interaction”. See what Schrödinger said on page 18 of his 1926 paper quantization as a problem of proper values, part II: “classical mechanics fails for very small dimensions of the path and for very great curvature”. Also see what Maxwell said when he was talking about displacement current in 1861: “light consists of transverse undulations in the same medium that is the cause of electric and magnetic phenomena”. Where is space curved? Where the photon is. Because space waves.

Photon spin

Hence I would say a photon is a region of curved space. But I’d also say it’s like when you play classical guitar: you work the fret with your left hand changing the wavelength, but the amplitude of your pluck is always the same. When you do pluck, you make a wave, and the string is then curved. Try plucking a washing line whilst sighting your eye along it to watch the wave race away. Then try plucking in orthogonal directions with two different hands. I say this because we have linearly-polarized light, but we have circular-polarized light too, and the latter is thought to be more fundamental. I don’t know why. It’s as if in space there’s nothing to brace against. As if guitar strings are like infinite washing lines, and you have to pluck your guitar string with two plectrums at once, one vertical, one horizontal, with a linear separation between them. Then there’s a rotation of sorts, we have experimental proof of the spin of the photon. But the photon itself isn’t actually spinning. It has no magnetic dipole moment. The circular-polarized light wave is depicted as two orthogonal 90° out-of-phase electromagnetic waves propagating linearly through space with a combined left-or-right handedness, and so a spin of ħ or ‑ħ. The net electric vector is helical, but the circular-polarized photon isn’t going round and round:

Image courtesy of Rod Nave’s hyperphysics

It’s like the circular-polarized photon is an arrow with one set of flights set behind the other. But this arrow isn’t spinning like a bullet. Your washing line isn’t spinning on its long axis like some drive shaft. For a better picture imagine you could lean out of an upstairs window with a whip in your hand. Move the handle quickly in a growing circle that then diminishes to make a wave that corkscrews down the whip, something like this:

I think the photon is something like this. Like you, it’s left handed or right handed, just like a screw thread. Because of the screw nature of electromagnetism, associated with the right hand rule. We’ll come on to that later.

What a photon is

But for now a picture is emerging. A picture that’s doubtless imperfect, but better than no picture at all. A picture of a photon that is a singleton soliton electromagnetic wave. A wave like half a length-wise lemon, with a twist. A corkscrew soliton in space, with a common amplitude that underlies Planck’s constant h. As to why we have this common amplitude, I’m afraid I don’t know. Perhaps I’ll have to settle for that’s the way space is. Perhaps animations of waves in a lattice will tell us something about it. And about solitons, and how they deform the lattice and alter the motion of solitons through the lattice. But for now, I think of the quantization of electromagnetic change written in 1994 by Robert Kemp. That’s the quantization of electromagnetic change, not charge. Kemp talks about saturation and maximum upper and lower potentials, and about the photon having a maximum electromagnetic amplitude in space, such that electromagnetic saturation is the cause of Planck’s constant. He also says this is what Vernon Brown predicted in his photon theory: “Brown predicted that the electromagnetic amplitude or saturation constant would be a constant from which Planck’s constant derives”. It’s as if space has an elastic nature described by permittivity and permeability, and an elastic limit, and so acts like its own waveguide. Because space is not nothing. It’s a polarizable medium, like some kind of ghostly gin-clear elastic. When an ocean wave moves through the sea, the sea waves. When a seismic wave moves through the ground, the ground waves. When an electromagnetic wave moves through space, space waves. And because of this, space is curved wherever that wave is. Not just in one dimension, but in two, because of the circular polarization. So space is curled, like your whip. Are Feynman’s wavy little arrows really corkscrews, showing this curvature? I don’t think so. But I do think that this says something about how pair production works. Because what does a 511keV photon do in gamma-gamma pair production? Or to put it another way: what does a hedgehog do when threatened?

Hedgehog image © Warren photographic


This Post Has 4 Comments

  1. Cracking article John. I shall leave some constructive comments when I have more time. I like your insight that photons are not wave trains. You can turn that around and say that the wave trains idea arises from us observing collections of photons . The train is multiple photons end to end. Split any individual photon in a given train and a photon twice the length will leave, but also, the gap will be precisely filled by the increase to twice the length of the remaining photon. Split the photon by introducing a photon in antiphase to the original that cancels one half of the photons energy and you now have an experiment I would like to see.

    1. Thanks Andy. But I don’t think that experiment is doable, because the photon is a soliton wave of 4-potential. In the middle of it E and B are zero, but the potential is at a maximum. The energy isn’t cancelled at all, it’s just not so obvious any more. But it isn’t totally hidden. Take a whole lot of photons and add a whole lot more in antiphase, and what you’ve got is a gravitational field. Look at the lattice cells up the centre of this picture and compare them with this. PS: try rotating the lattice cells and think spin 1 and spin 2.

  2. I shall have to look up the 4-potential soliton. Not something I have studied. However, just one point regarding you pointing out that all electromagnetic waves have the same potential magnitude. i.e. all photons have the same voltage magnitude. From a lecture at Manchester on my electrical engineering course I seem to remember the voltage of an EM wave being quoted as 3^.5/c or sqrt(3)/c. I have never found a reference to that anywhere else. I just wondered if you have. It is a number I have found useful.

    1. It doesn’t ring a bell Andy. But it sounds like a golden nugget, because it chimes with the guitar analogy where the amplitude of your pluck is always the same. I’ll look into this. Many thanks. E=hf, E=Fs, and E=VQ for a reason.
      PS: I googled on “voltage of a light wave” and found The Duality of Matter and Waves by Frank Znidarsic which led me to the Z theory of everything where I can see The elastic limit of space and the quantum condition. Unfortunately he says the gravitational force establishes the quantum condition.

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