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Aidan Randle-Conde | Université Libre de Bruxelles | Belgium

View Blog | Read Bio

The (seemingly) fractal nature of matter

Matter and energy have a very curious property. They interact with each other in predictable ways and the more energy an object has, the smaller length scales it can interact with. This leads to some very interesting and beautiful results, which are best illustrated with some simple quantum electrodynamics (QED).

QED is the framework for describing the interactions of charged leptons with photons, and for now let’s limit things to electrons, positrons and photons. An electron is a negatively charged fundamental particle, and a positron is the same particle, but with a positive charge. A photon is a neutral fundamental particle of light and it interacts with anything that has a charge.

That means that we can draw a diagram of an interaction like the one below:

An electron radiating a photon

An electron radiating a photon

In this diagram, time flows from left to right, and the paths of the particles in space are represented in the up-down direction (and two additional directions if you have a good enough imagination to think in four dimensions!) The straight line with the arrow to the right is an electron, and the wavy line is a photon. In this diagram an electron emits a photon, which is a very simple process.

Let’s make something more complicated:

An electron and positron exchange a photon

An electron and positron make friends by exchanging a photon

In this diagram the line with the arrow to the left is a positron, and the electron and positron exchange a photon.

Things become more interesting when we join up the electron and positron lines like this:

An electron and positron annihilate

An electron and positron get a little too close and annihilate

Here an electron and positron annihilate to form a photon.

Now it turns out in quantum mechanics that we can’t just consider a single process, we have to consider all possible processes and sum up their contributions. So far only the second diagram we’ve considered actually reflects a real process, because the other two violate conservation of energy. So let’s look at electron-positron scattering. We have an electron and a positron in the initial state (the left hand side of the diagram) and in the final state (the right hand side of the diagram):

What happens in the middle?  According to quantum mechanics, everything possible!

What happens in the middle? According to quantum mechanics, everything possible!

There are two easy ways to join up the lines in this diagram to get the following contributions:

Two possible diagrams for electron-positron scattering

Two possible diagrams for electron-positron scattering

There’s a multiplicative weight (on the order of a percent) associated with each photon interaction, so we can count up the photons and determine the contribution each process has. In this case, there are two photon interactions in each diagram, so each one contributes roughly equally. (You may ask why we bother calculating the contributions for a given pair of initial and final states. In fact what we find interesting is the ratio of contributions for two different pairs of initial and final states so that we can make predictions about rates of interactions.)

Let’s add a photon to the diagram, just for fun. We can connect any two parts of electron and positron lines to create a photon, like so:

Taking up the complexity a notch, by adding a photon

Taking up the complexity a notch, by adding a photon

A fun game to play in you’re bored in a lecture is to see how many unique ways you can add a photon to a diagram.

So how do we turn this into a fractal? Well we start off with an electron moving through space (now omitting the particle labels for a cleaner diagram):

A lonely electron :(

A lonely electron 🙁

Then we add a photon or two to the diagram:

An electron with a photon

An electron with a photon

An electron hanging out with two photons

An electron hanging out with two photons

An electron going on an adventure with two photons

An electron going on an adventure with two photons

Similarly let’s start with a photon:

A boring photon being boring

A boring photon being boring

And add an electron-positron pair:

Ah, that's a bit more interesting

Ah, that’s a bit more interesting

This is all we need to get started. Every time we see an electron or positron line, we can replace it with a line that emits and absorbs a photon. Every time we see a photon we can add an electron-positron pair. We can keep repeating this process as much as we like until we end up with arbitrarily complex diagrams, each new step adding more refinement to the overall contributions:

A very busy electron

A very busy electron

At each step the distance we consider is smaller than the one before it, and the energy needed to probe this distance is larger. When we talk about an electron we usually think of a simple line, but real electrons are actually made of a mess of virtual particles that swarm around the central electron. The more energy we put into probing the electron’s structure (or lack of structure) the more particles we liberate in the process. There are many diagrams we can draw and we can’t pick out a single one of these diagrams as the “real” electron, as they all contribute. We have to take everything to get a real feel of what something as simple as an electron is.

As usual, things are even more complicated in reality than this simple picture. To get a complete understanding we should add the other particles to the diagrams. After all, that’s how we can get a Higgs boson out of proton- in some sense the Higgs boson was “already there” inside the proton and we just liberated it by adding a huge amount of energy. If things are tricky for the electron, they are even more complicated for the proton. Hadrons are bound states of quarks and gluons, and while we can see an individual electron, it’s impossible to see an individual quark. Quarks are always found in groups, so have the take the huge fractal into account when we look inside a proton and try to simulate what happens. This is an intractable problem, so need a lot of help from the experimental data to get it right, such as the dedicated deep inelastic scattering experiments at the DESY laboratory.

The view inside a proton might look a little like this (where the arrows represent quarks):

The crazy inner life of the proton

The crazy inner life of the proton

Except those extra bits would go on forever to the left and right, as indicated by the dotted lines, and instead of happening in one spatial dimension it happens in three. To make matters worse, the valence quarks are not just straight lines as I’ve drawn them here, they meander to and fro, changing their characteristic properties as they exchange other particles with each other.

Each time we reach a new energy range in our experiments, we get to prober deeper into this fractal structure of matter, and as we go to higher energies we also liberate higher mass particles. The fractals for quarks interact strongly, so they are dense and have high discovery potential. The fractals for neutrinos are very sparse and their interactions can spread over huge distances. Since all particles can interact with each other directly or through intermediaries, all these fractals interact with each other too. Each proton inside your body contains three valence quarks, surrounded by a fractal mess of quarks and gluons, exactly the same as those in the protons that fly around the LHC. All we’ve done at the LHC is probe further into those fractals to look for something new. At the same time, since the protons are indistinguishable they are very weakly connected to each other via quantum mechanics. In effect the fractals that surround every valence particle join up to make one cosmological fractal, and the valence particles just excitations of that fractal that managed to break free from their (anti-)matter counterparts.

The astute reader will remember that the title of the post was the seemingly fractal nature of matter. Everything that has been described so far fulfils the requirements of any fractal- self similarity, increased complexity with depth and so on. What it is that makes matter unlike a fractal? We don’t exactly know the answer to that question, but we do know that eventually the levels of complexity have to stop. We can’t keep splitting space up into smaller and smaller chunks and finding more and more complex arrangements of the same particles over and over again. This is because eventually we would reach the Planck scale, which is where the quantum effects of gravity become important and it becomes very difficult to keep track of spatial distances.

Meanwhile, deep inside an electron, something weird happens at the Planck scale

Meanwhile, deep inside an electron’s fractal, causality breaks down and something weird happens at the Planck scale

Nobody knows what lies at the Planck scale, although there are several interesting hypotheses. Perhaps the world is made of superstrings, and the particles we see are merely excitations of those strings. Some models propose a unification of all known forces into a single force. We know that the Planck scale is about fifteen orders of magnitude higher in energy than the LHC, so we’ll never reach the energy and length scales needed to answer these questions completely. However we’ve scratched the surface with the formulation of the Standard Model, and so far it’s been a frustratingly good model to work with. The interactions we know of are simple, elegant, and very subtle. The most precise tests of the Standard Model come from adding up just a handful of these fractal-like diagrams (at the cost of a huge amount of labour, calculations and experimental time.)

I find it mind boggling how such simple ideas can result in so much beauty, and yet it’s still somehow flawed. Whatever the reality is, it must be even more beautiful than what I described here, and we’ll probably never know its true nature.

(As a footnote, to please the pedants: To get a positron from an electron you also need to invert the coordinate axes to flip the spin. There are three distinct diagrams that contribute to the electron positron scattering, but the crossed diagram is a small detail might confuse someone new to these ideas.)


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  • Amir Livne Bar-on

    Do we have an indication that QFT as we know it actually extends up to the Planck scale, perhaps with new particles and forces added along the way? (I’m a physics layman)

  • Kudzu

    Given the wave nature of matter, how accurate is this viewpoint? What I see here is a bunch of point masses all moving about, vanishing and reappearing randomly. As if I could take a snapshot and add up the particles at a particular length\energy scale.

    But waves can interfere with each other and there’s the exclusion principle and suchlike. For any given real particle I can break its waveform down into a series of waves with perfectly defined wavelengths\energies which when added produce that particle. Can’t I thus do the same with any virtual particle? And can I not take two virtual particles that are in (within uncertainty) the same volume of space and combine their waveforms into a single one?

    In other words, how grainy is the virtual landscape?

  • FartingFoulFarts this noise:-

    Ha! I get it.
    You’ve got to think of the Universe as a computer with
    one planck time being one Universal clock cycle. In one planck time
    frame, the Universe is in a fixed state. As the next planck time frame
    occurs, the physical state of the Universe changes. If you then think of
    spacetime as an invisible, enigmatic, mysteriously energetic
    “substance” that spins or vibrates when initialised, then it’s a bit
    easier to get. When tiny pieces of space spin, it emanates in-formation
    about it’s current formation which we record and call a particle. If we
    record spae vibrating, we call it electro-magnetic energy, and so on and
    so forth.

    It seems to me, that different pieces of space
    suddenly starts spinning at each planck time iteration, whereas others
    continue to spin. A similar model is a charge in a computers memory
    units. At each computer clock cycle, the memory changes it’s physical
    state. This could be like particles switching on from hitherto
    unmeasured “dark” energy (matter).

    Particles have a life span,
    like everything else. They have been, are be-ing, and could be. At each
    planck time frame, different ones switch on, others go off (disappear)
    and a relatively few others stay on.

    A bit like a row of lights
    one planck apart (for example’s sake). As the lights consecutively swich
    on or off, it looks like the light is moving across the row.

    it is with matter, moving through space. As these tiny particles blink
    in and out of 4D spacetime existence, they are behaving like everything
    else that has existed, is existing and could exist. Including life.

    spins and forms what we perceive as matter. Matter forms chemical
    behaviours founded on and are the surface of subatomic effects. Chemisty
    forms biochemical behaviour, which itself forms, organis-ms, which
    fractally self replicate into organised multicellular organisms, from
    which sprouted super-organised beings like people.

    People are fractal images of subatomic particles as appearing in the consciousness of the human observer.

    when most people see stuff moving through space, they see the object
    translocating. Only it isn’t really. It’s like the light bulbs going on
    and off. When a plane moves across the sky, a few of the supatomic
    particles it did consist of in the last planck time frame are no longer
    part of the plane. They have been left behind. However new subatomic
    particles of the same nature spin on. Like a light moving across a row
    of lights. An OFF light bulb represents space being seemingly still and
    measured in metres. An ON bulb represents a sum of spinning space that
    is manifest matter.

    If you look at a table, it looks still – but
    it isn’t. It is screaming through space at thousands of miles per
    hour… Apparently.

    Matter and Electromagetism is like the crest
    of a wave screaming through the sea of space. The perceivable crest is
    ON, the calm sea is spacetime OFF (in a dark, unapparent state) How
    long a particle effect remains part of the passing mass effect, is it’s
    life span. Once the particle is no longer energised, it becomes empty
    space you measure as distance.

    When Teslas said, “you’ve got to think of every-thing as a wave”

    He wasn’t kidding.

  • ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


    ● Space-time is a lattice type string particle field in space.
    ● Empty space is completely empty / null / void.

    There is a big difference.

    Space-time must be made out of something.
    Space-time is NOT empty space.

    You can easily fold up, distort and curve Space-time, but you are NOT going to do anything to the empty space it resides in.

    To sum it up: What Einstein calls “Space-time” is a particle field in empty space.
    The particle field is made from individual yet connected particles completely filling space.
    The field is NOT fixed in space, it moves-along-with / is-held-in-place-by the largest mass in proximity.
    It’s something like the way gravity works (it’s actually responsible for gravity), relative strength due to size and proximity.
    It’s all made from the same particles.

    Part of the field is surrounding and moving with you.
    You are completely immersed in the Earths field.
    The Earth field moves with the Earth and is inside of the Suns field.
    The Suns field encompasses the entire solar system (plus more) and moves with the Sun.
    A Galaxy of course has a particle field and it moves with the Galaxy (as a whole and with the movement of individual stars and systems).

    There is an all encompassing string particle field (not the string theory type) in space (and everywhere).
    The field is made from individual yet connected particles and conforms to whatever shape it is surrounding. So light traveling through a curved field (like the Earth or Sun) will of course curve.

    Is gravity curving the field? No! The field itself is what creates gravity (gravity is field tension).
    Does this invalidate any of Einsteins equations? Of course not, it is just another way to look at it. Einstein has field equations and this is the field.

    The particles are connected — that creates a field. The field has tension on it so vibrations can easily travel through it on the strings .
    That’s what light is…


    There is a high tension string particle field in space (not the string theory type). Everything is connected by the particle field and it moves along with largest mass in proximity (something like what gravitational fields would be doing).
    A good 2-D model would be something like a spiders web (individual string lengths are approximately one Ångström).
    Now imagine an infinite 3-D spiders web. If a vibration was set off in it, it would travel forever and the speed the vibrations travel (through the net) is the speed of light (that’s actually what light is, a vibration traveling through a string particle field)
    The speed vibrations travel through the particle field is the speed of light “c”

    The particle field strings have a certain amount of tension, length and mass. That makes ‘c’ the speed it is. If the tension, length or mass changed so would ‘c’

    Here is a regular string tension formula…

    Tension = velocity squared x mass / Length.

    If we plug c in and rearrange we get…
    TL = mc^2

    Both sides of the equation are in joules or energy… equivalent to “E”.
    It means the Tension of the strings in space times their length is equal to their energy.

    This is why the speed of light is involved in Einsteins mass energy equivalence equation…

    E = mc^2

    …and actually why light travels at the speed of light…
    I always wondered why… now I know.
    It had to be something mechanical… tension and string lengths!

    So, you can arrive at Einsteins famous formula from completely different directions.
    You can think energy is contained in mass and released.

    E = mc^2

    Or you can think there is a particle field of strings and mass is inert, the energy is only potential… released (actually pulled) by tension on the strings.

    TL = mc^2

    They are equivalent. Which is correct? You do not know.

    Tesla was correct…
    “There is no energy in matter other than that received from the environment.” – Nikola Tesla

    Mnemonic memory device…
    E for Einstein: E = mc^2
    TL for Tesla: TL = mc^2

    NOTE: if you think MM is correct go to “THE GENIUS OF MICHELSON-MORLEY” comment and pick “A”


    If you were going to test if there is a medium for the conveyance of light, would you…

    A) Test if the Earth is rushing through the medium.
    2) Test if the Earth is NOT rushing through the medium.
    C3) Both of the above (same as: just test for medium, no constraints)

    Here is your chance to agree with those great men and pick “A”, everything you think you know is based on that.

    NOTE: The correct answer is of course “C3” but modern physics is based on Michelson-Morley experiment and they picked “A”
    Michelson-Morley picked “A” and everything you think you know is based on that. Michelson-Morley “confirmed” there is no medium with their experiment (it’s actually a pillar of modern science)
    The only problem is if “2” is happening they are completely in the dark about it.