A topologist's dream

How Dirac's trick explains nature, matter, quantum theory and general relativity

The above animation by ronwnor is an example and a slight extension of Dirac’s trick. Dirac used the trick in his lectures, but never published anything about it. Imagine the strands to have Planck radius. Such strands thus are just thin enough to be unobservable in practice but thick enough to still be physical.

To be precise, Dirac’s trick shows a ħ/2 particle (or spin 1/2 particle for short), where ħ is Planck’s quantum of action. The quantum of action ħ is due to a crossing switch of strands, as this the only difference between the unrotated structure and one rotated by one turn. (Such structures are called tangles.) This was deduced by Kauffman in 1987.

The animation uses Dirac’s trick to realize and visualise spin 1/2. The rotating structure is a spinning fermion. Why? First, the structure comes back to the original state every two rotations. This is the definition of spin 1/2. Then, if you take two such structures and exchange their positions, the whole sets comes back to the original state only after two such exchanges (try it at home with two paper stripes!). This is the definition of a fermion.

This video was made by Antonio Martos. It is also found at at motionmountain.net/videos.html#fermion. Instead of a strip, you can substitute each edge by a strand, so that each strip is changed into two strands.

The animations are also interesting for a second reason: this is the only way to visualize a spin 1/2 particle. There is no other way - across the scientific literature. There are various attempts, but none succeeds in visualizing locality, spin 1/2, fermion behaviour, and particle countability. Of course, must always recall that the tethers are invisible, and that the fluctuating central structure is observable as a cloud, i.e., as a wave function.

In other words, spin 1/2 implies that particles are tethered - even though we do not see such tethers in everyday life. Spin is rotation, particles are tangles.

But the wonders go on. In Dirac’s trick, the little central triangle is chiral. So the structure represents a chiral fermion. Now imagine that the spinning particle is surrounded by a heap of untangled strands. (They represent the vacuum.) The chiral core will act like a screw or a maple seed, and advance. This visualizes a chiral fermion moving through vacuum.

If we imagine a little arrow attached to the central triangle, while advancing, the arrow will trace out a helix. The wavelength of that helix is the wavelength of a quantum particle. Dirac’s trick thus implies the wave-particle duality.

A particle with high momentum has many rotations per length. Every rotation of the arrow corresponds to a Planck quantum of action ħ. Dirac’s trick thus yields de Broglie’s wavelength! The energy E of such a particle then is the angular frequency 𝜔 times the Planck quantum of action ħ. This is Planck’s relation!

Now let us take Dirac’s trick and not forget the random shape fluctuations. We then only get and see a fuzzy cloud. The oriented crossing density of that cloud is what is usually called the wave function. If instead, we imagine the strands to be infinitely thin and the chiral rotating region to be as small as a point, recalling that the strands are unobservable, we get Feynman’s path integral description: we get an advancing point with a rotating arrow.

In both cases, combining the wave properties with the relations for energy and momentum E=p²/2m yields Schrödinger’s equation. But even more follows from Dirac’s trick.

The relation for energy and momentum allow defining an effective inertial mass value of the chiral fermion. Mass describes how frequently and how rapidly Dirac’s trick takes place when a particle moves. When the strands fluctuate, the motion of the animation at the top is in fact quite unlikely. The rotations is thus slow. Therefore, the tangle structure has an effective mass that is much smaller than the Planck mass. Dirac’s trick thus solves the mass hierarchy problem, the question why elementary particles have such tiny mass values. But there is more.

The curved regions in Dirac’s trick are, as a deeper investigation shows, the virtual gravitons surrounding the central mass. Dirac’s trick thus describes both inertial mass and gravitational mass. Therefore, the two are equal. This is the equivalence principle that Galileo and Einstein made popular. It forms the basis of general relativity.

And yes, the fermion can also advance by rotating through curved space. And yes, it follows geodesics. (Empty space is itself made of strands as well, but in contrast to particles, the strands are untangled.)

All this follows from Dirac’s trick. In simple words, Dirac’s trick implies quantum theory and general relativity. For strands of Planck radius, the specific structure in the animation above, with the chiral central triangle, is well known. It is usually called an electron. And yes, every electron in our body is connected to the border of space with unobservable strands. This is the strand tangle model - the tiny theory of physics.

In simple words, if you take seriously the ideas of Dirac and Kauffman, Dirac’s trick contains the main wonders of nature and of physics: both quantum physics and general relativity follow - from topology. Nature is a topologist’s dream.

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A popular introduction to the mentioned dream, the tiny theory of physics, is available here.
A short technical preprint is researchgate.net/publication/397264142.
A long technical preprint is researchgate.net/publication/361866270.
The preprints show that Dirac’s trick also implies the Dirac equation, the electric charge of the electron, and black hole rotation. Mathematically speaking, Dirac’s trick visualises the double cover of SO(3) by SU(2) and the behaviour of the unit quaternions.