Yes, there’s a hole in the heart of quantum electrodynamics because it describes the interaction between light and matter, but not the interaction between light and light. That’s the interaction that creates matter in gamma-gamma pair production. QED misses the crucial point that waves interact. Even though we’ve all seen waves interact, down on the beach. Imagine a big wave is coming towards you. You make a little wave with your hand and send it scooting towards the big wave:
The little wave rides up and over the big wave, and carries on going. If you didn’t see the little wave ride up and over the big wave, you might conclude that waves don’t interact. You might conclude that water waves pass through each other without any interaction. You might also conclude that electromagnetic waves pass through each other without any interaction. But look again. We aren’t just talking about superposition here. The little wave rode up and over the big wave. As it did so, it changed direction. The little wave was originally moving horizontally, then it changed direction gradually so it was going up at an angle, then down at an angle, and so on. Eventually it was moving horizontally again. If you hadn’t watched, you might have thought there was no interaction. There was no lasting interaction, but there was an interaction. And the essence of that interaction is this: when a wave interacts with a wave, it changes direction.
Tying light in knots
There’s plenty of physicists who know that light interacts with light. That’s why there’s plenty of arXiv papers with photon-photon interaction in the title. There’s also people like Kirk McDonald of SLAC, see the 1998 PhysicsToday article gamma rays create matter just by plowing into laser light. It’s about gamma-gamma pair production, and it’s good stuff:
Image by Gil Eisner from the November 1997 issue of Photonics Spectra above a brief note about the SLAC experiment
But it’s from 20 years ago, and since then not much has happened. It’s almost as if it’s been “studiously ignored” by particle physicists. It’s as if the physicists who are interested are working in optics or photonics or electromagnetism. They’re the people who feature in media articles about light interacting with light. For example see the 2018 Imperial article experiments underway to turn light into matter. It’s about the Breit-Wheeler process which dates from 1932. Also see the 2013 physicsworld article physicists tie light in knots. Alternatively see the 2010 Bristol article tying light in knots. That’s where Miles Padgett of the Glasgow optics group said this: “The sophisticated hologram design required for the experimental demonstration of the knotted light shows advanced optical control, which undoubtedly can be used in future laser devices”. That’s fair enough, but look closer and you’ll perhaps pick up on the rather cryptic comment by Mark Dennis of Bristol: “the study of knotted vortices was initiated by Lord Kelvin back in 1867 in his quest for an explanation of atoms. This work opens a new chapter in that history”. He was referring to the vortex theory of atoms. That’s where Lord Kelvin said this: “after noticing Helmholtz’s admirable discovery of the law of vortex motion in a perfect liquid – that is, in a fluid perfectly destitute of viscosity (or fluid friction) – the author said that this discovery inevitably suggests the idea that Helmholtz’s rings are the only true atoms”. Lord Kelvin was also William Thomson of Thomson and Tait fame. Peter Tait was the physicist who got the Edinburgh chair in competition with Maxwell, and the man who drew up the knot zoo. Thomson and Tait wrote a treatise on natural philosophy. They also coined the phrase spherical harmonics.
That’s what Mark Dennis was referring to, but the new chapter isn’t about atoms. See David St John’s essay on vortex particles dating from 2011. The vortex theory of atoms was way ahead of its time, because subatomic particles hadn’t been discovered in 1867. It was JJ Thomson (no relation) who discovered the electron in 1897. It was JJ’s son George Thomson who discovered the electron’s wave properties with Andrew Reid via electron diffraction in 1927, at the same time as Davisson and Germer. And it was Mark Dennis who co-authored the 2016 Nature Communications paper vortex knots in tangled quantum eigenfunctions. The abstract says “the results suggest that knotted vortex structures are generic in complex three-dimensional wave systems”. In the PhysOrg reportage you can read that all matter has a wave nature. Dennis was also one of the organisers of the 2011 ABB50/25 conference in Bristol. That’s where Qiu-Hong Hu had a poster entitled the electron, twisted photon, and knotted light. Qiu-Hong told me he talked to Sir Michael Atiyah about that, and more. Atiyah was of course involved in the development of topological quantum field theory: “although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things, knot theory”. Ed Witten was also involved in TQFT, see his knots and quantum theory. I am reminded of Buffalo Bill Menasco’s circular history of knot theory. What goes round comes round.
The simplest knot
However we’re not talking about the Gordian knot here. We’re talking about the simplest knot, the trivial knot. It’s also known as the unknot, so you might think it’s not a knot. But note that ”many useful practical knots are actually the unknot”, and that there is such a thing as the stuck unknot. Also note that as I was saying in a previous article, the ocean wave is an analogy for a photon, a singleton soliton electromagnetic wave. And that the ocean wave is a place where the surface of the sea is curved, whilst the photon is a place where space is curved. I also referred to 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”. So, what’s going to happen when an electrmagnetic wave moves through curved space? Its path is going to curve. And then what’s going to happen if you curve its path so much, that the wave starts moving through itself? If the wavelength was just right for the curvature of the wave, the wave could end up moving entirely through itself, like an electromagnetic Ouroborus. You could end up with a wave in a closed path. Not a closed string. A closed path. The simplest such path is a double loop. The trivial knot is usually depicted as a single loop, but it’s sometimes depicted as a double loop too:
GNUFDL trivial knot image by Makotoy, see Wikipedia
This double loop is reminiscent of a figure-of-eight that’s been folded over. It’s also reminiscent of a clove hitch with no free ends. And because it’s a double loop, it features a 720° rotation. It goes around and around. That rings a bell. Especially if you’ve read Is the electron a photon with toroidal topology? by John Williamson and Martin van der Mark. They talk about a circular polarized photon formed into a double loop:
Images by Martin van der Mark, from Is the electron a photon with toroidal topology? by John Williamson and Martin van der Mark
They wrote this paper in 1991, and it took them six years to get it published in a low-impact journal. Called Annales de la Fondation Louis de Broglie. Oh the irony, particularly since it’s been studiously ignored ever since. They say the electron is in essence a 511keV photon trapped in a spin ½ double loop, and I think they’re right. The dark line in the image on the right above traces out a single geodesic to highlight the double loop, in a torus that looks like a single loop. There’s a similar double loop in Qiu-Hong Hu’s 2005 paper the nature of the electron:
Image from the nature of the electron by Qiu-Hong Hu
Once you’ve seen papers like these, the scales fall from your eyes. Well I think they do, because I know that when you make a torus fatter and fatter, it tends towards a spherical symmetry. I also think it’s easy to play detective when someone has told you whodunnit. I think that once you’ve seen papers like these, there’s just no going back to point particles. Particularly since there’s also mysteries and insights of Dirac theory plus other papers by David Hestenes. And the history of the toroidal ring model, which doesn’t start with Allen or Parson or Kelvin, but with Ampère. He proposed “tiny magnetic loops of charge” in 1823.
The Dirac String Trick
The Dirac string trick also features a 720° rotation. There’s a nice animated explanation of that on Robert Gray’s web site. He says this: “It has been suggested that this motion may have something to do with the spin of a particle (like the electron or proton). See for example the paper Geometric Model for Fundamental Particles by Batty-Pratt and Racey“. In 1980 Batty-Pratt and Racey talked about spherical rotations, and said “our geometrical model demonstrates the difference not only between spin-up and spin-down states, but also between the particle and the antiparticle”.
Fair use excerpt from Geometric Model for Fundamental Particles by Batty-Pratt and Racey
Note that the wave nature of matter and electron spin means we’re talking about waves going around and around in a medium rather than a central lump of the medium spinning round bodily. And that Batty-Pratt and Racey said they’d “made the presumptive leap of assuming photons to be undulations of the space-time structure”. They also talked about a “theory of the continuum that purports to describe matter as a distortion of space in the manner first suggested by W K Clifford”. That’s William Kingdon Clifford, the Clifford algebra Clifford. You can see his 1870 space theory of matter on Wikipedia. Clifford said “I hold in fact:
(1) That small portions of space are in fact of a nature analogous to little hills on a surface which is on the average flat; namely, that the ordinary laws of geometry are not valid in them.
(2) That this property of being curved or distorted is continually being passed on from one portion of space to another after the manner of a wave.
(3) That this variation of the curvature of space is what really happens in that phenomenon which we call the motion of matter, whether ponderable or etherial.
(4) That in the physical world nothing else takes place but this variation, subject (possibly) to the law of continuity”.
That nothing else takes place strikes a chord. Clifford died of tuberculosis at the age of 33. If he had not, I suspect the world would be a different place today. Charles Misner and John Wheeler referred to Clifford in their 1957 paper Classical Physics as Geometry. Sadly they missed the trick. If they had not, I suspect the world would be a different place today.
The Dirac string trick is also known as the plate trick or the belt trick, which relates to Dirac’s belt. Kevin Brown has a mathspages article on Dirac’s belt. He refers to the Möbius strip: ”in this sense a Mobius strip is reminiscent of spin ½ particles in quantum mechanics, since such particles must be rotated through two complete rotations in order to be restored to their original state”. There’s a picture of a Möbius strip on the Wikipedia spinor article. This might remind you of something Dirac said: “thus the change in phase round a closed curve must vanish”. How do you achieve that? How do you start with a wave and make the change in phase vanish? Simple. Get a sheet of paper and a pair of scissors, and cut a sinusoidal strip of paper to emulate a wave, like this:
Let’s say the strip is 1cm wide at its widest point, and that the paper is paler on the other side. Now give it a longitudinal twist to emulate the circular-polarized photon. Then turn the ends towards one another and slide them along to make a twisting turning Möbius strip. Like this:
Is it one loop or two? It’s both. A photon is not a flat strip of course, just as a seismic wave is not a flat strip. But I think it’s a good place to start. Your Möbius strip is the same 1cm width all round. There is no width variation as you go round the strip, because at all locations round the loop there are two widths that add up to the same total width. After pair production there’s no field variation any more, because at all locations around the wave path there’s a superposition of two field variations that add to the same field value. When you wrap a sinusoidal electromagnetic field variation into a twisted double loop, the minimum and maximum field variation combine, along with all points in between, to leave you with an all-round standing field. See the Wikipedia atomic orbitals article: “electrons do not orbit the nucleus in the sense of a planet orbiting the sun, but instead exist as standing waves”. This is so true. Something else that’s true is that electrons exist as standing waves when they’re not in an orbital. That’s why we can diffract electrons. The moral of the tale is this: standing wave, standing field.
Standing wave, standing field
If you have a standing-wave photon in an optical cavity, it might look like there’s no motion. But when you kick away one of the mirrors, that photon is off like a shot, instantly moving at c from a standing start. But it wasn’t really a standing start. The standing wave might look static, but it was always moving at c, back and forth, back and forth. It was dynamical, but it looked static because the wavelength matched the cavity. To imagine this without a cavity, think of a seismic wave. When a seismic wave travels a linear path, your refrigerator is dynamically displaced 1m to the left and back again. But what would happen if that seismic wave went around and around in a tight Möbius path? Your refrigerator doesn’t move back and forth any more. It’s displaced by 1m, and that’s how it stays. The displacement is now static, because the seismic wave is now a standing wave, even though it’s still dynamical. As a result the wave of displacement is now wrapped up into an all-round spin ½ displacement field. Standing wave, standing field.
Twist and turn
You can of course make a second Möbius strip with a chirality that’s the opposite of the first. It twists and turns the other way. You can do something similar with light. In fact, you must, because in gamma-gamma pair production, light is the only tool in the box. You twist light one way using light. Conservation of angular momentum means the light you twisted with, twists and turns the other way. But remember that light is a wave in space, and space is not a paper strip. We are dealing with a 3-dimensional bulk. We can twist and turn it all the way to make one stable 511keV particle, and we can twist and turn it the other way to make another, with the opposite chirality and charge. But we can only do this for a given wavelength. We can only create a stable optical vortex if there’s a relationship between the wavelength and amplitude. Only then do we achieve a stable harmonic resonance. Only then do we end up with a dynamical wave wrapped up as an all-round static field.
The wavelength of a 511keV photon is 2.426 x 10-12 m. That’s the electron Compton wavelength. In the double-loop trivial-knot configuration you divide by 2π for the diameter, which is 3.861 x 10-13 m. That suggests the amplitude of all photons is 3.861 x 10-13 m. Perhaps this is what Hermann Weyl’s gauge theory was really about. The photon itself is like a change in the scale of railroad tracks, only they’re more like corkscrew rollercoaster tracks. The change creates a curvature where the wave in space is, and when that wave moves through curved space, its path curves. Get it right, and it curves around and around, all the way round, twice, like the Möbius strip. There is no width variation as you go around the strip, and there is no apparent electron phase. But there again, there is no strip either, because the electron is a wave in a 3D bulk. It isn’t 3.861 x 10-13 m high, just as an earthquake isn’t one metre wide. It isn’t a hedgehog either, but what does a hedgehog do when threatened? It curls up into a ball. What does a 511keV photon do when you put it through gamma-gamma pair production? It curls up into a ball. Because waves interact with waves, so much so that a wave can end up moving through itself, forever interacting with itself. Knit one purl one. Annihilation is the opposite process. Cast off. There are no magic creation and annihilation operators. It’s just waves in space changing direction, that’s all. You don’t twist your Feynman diagrams, you twist your photons instead, and knot them like knitting to make the matter from which we’re made. Nothing else takes place. It’s all just Einstein’s pure marble and electromagnetic geometry, and we were so close.
The geon was proposed by John Wheeler in 1955. That’s 61 years ago. He was the Wheeler in the Breit-Wheeler process, and Feynman’s supervisor. Like Feynman he associated curved space with the gravitational field, not the electromagnetic field. He was the man who said matter tells space how to curve. Hence the geon is “an electromagnetic or gravitational wave which is held together in a confined region by the gravitational attraction of its own field energy”. What a shame Wheeler didn’t think in terms of an electromagnetic wave only. Confined by an electromagnetic interaction circa 1039 times stronger than gravity. Then if he’d worked out that it had to be configured as a 720° double loop so it was moving through itself, he might have thought in terms of a folded figure 8. Then he would have been on the real eightfold way. Then he wouldn’t have called it a geon. He would have called it something else. After all, when we use pair production to confine a 511keV photon into a standing-wave spin ½ spinor path, we don’t call it a photon any more. We call it an electron.