Gauge Theory and the Circular Economy

Can some ideas from particle physics help us think about the circular economy?  How does the very tiny affect the very big?

Twenty years ago I studied theoretical physics and I developed a mental picture of the underlying gauge theory that bizarrely is very similar to the mental picture I have developed recently when I think about the circular economy.

The concept that links the two is the idea of a “phase”.

Sub-atomic particles have a special property called a phase which is defined at each point in spacetime. I’ll explain that in a moment.

In the circular economy every substance on Earth has a cycle – like the water cycle.  There is a carbon cycle. And a nitrogen cycle.  Every atom of each element is always at a certain point in its cycle. And we could call that point its “phase”.

You could represent how far through a recycling loop an atom is with a one handed clock. When the hand is at 9 o’clock water is in the ocean. At 12 o’clock it’s evaporated and in the air as clouds. At 3 o’clock its raining, at 6 o’clock it’s in the rivers and at 9 o’clock it’s back in the ocean.

This works because the water molecule stays as a water molecule the whole way through the cycle. It’s just its relationship with other water molecules and its location that is described by the phase.

So you couldn’t have, say, a “ketchup cycle” because the ketchup doesn’t stay as ketchup the whole time. In fact the objects that stay as they are more or less the whole time are the elements (ignoring radioactive decay), such as carbon and nitrogen etc.

So we can think about each of the 118 elements as having a cycle and each atom of each element as having a phase angle associated with it describing where it is in that cycle. So you could imagine every atom has a little clock associated with it telling you where in its cycle it is.  That’s a lot of little clocks spread out all around the Earth.

So far so good. What about the gauge theory bit?


As the centuries have rolled on by physicists have realised that to accurately describe all the properties of a subatomic particle, like an electron, we need to use a mathematical object to describe the particle that is called a “complex field”.  And the way these fields behave is described by “field theory”, and there’s one type of field theory called a gauge theory.

In gauge theory the complex field describing each particle (one field for every particle) extends throughout the entire universe.  At each point in spacetime we can assign a complex number to the field, which is just like a normal number except that it has two components, one real and one imaginary. The real part is just a normal number between minus infinity and plus infinity.  And the imaginary part extends from negative infinity to positive infinity just like real numbers but it’s just a different kind of number!

We usually plot the imaginary part on a vertical axis and the real part on a horizontal axis so you can draw cute little pictures of complex numbers on what is known as an Argand diagram.

The simplest way of thinking about a complex number is to think of it as a one handed clock (surprise!).  As the hand goes around the amount of real and imaginary parts change. So at 12 o’clock it’s all imaginary. At 3 o’clock it’s all real. At 6 o’clock it’s all imaginary again (but negative) and at 9 o’clock it’s all real and negative. Then it’s back to 12 o’clock…

The size of the hand tells you how big the complex number is. The angle of the hand tells you its “phase” and it is this special property called phase that you need to correctly describe the quantum aspects of particles, such as interference patterns.


So, in particle physics we can imagine a little one handed clock at each point in spacetime. Which is a bit like the picture of a little clock for each atom spread over the Earth that we talked about before, except here we’re talking about lots of little clocks for just one particle, and the field spreads across the whole universe!

Now we have this basic mental picture we can begin to describe how the particle behaves in terms of the huge number of clocks that represent it.

All the clocks for a given particle rotate at the same rate and the rate of rotation tells you something about how the particle is moving through time – i.e. how much energy it has.   You could also imagine the length of the hand as telling you the probability of the particle being at that point. So in most places the length of the hand is very tiny – vanishingly close to zero length. But in other places the length of the hand is big! That is probably where the particle is!  And you can play that game lots more describing properties like momentum and position. The information stored in the field tells you everything you can know about the particle, and to extract information from the field you have to perform “operations” on the field. E.g. to actually find the probability of where the particle is you have to multiply the field by its complex conjugate, so the imaginary parts cancel out and you get just a real number. But that’s a technicality.

What’s really, really cool though is to think about how the angles of the clock hand are related at different points in the field.  You could imagine that they started all pointing straight up! and rotating in synchronicity.  Or you could imagine that they were all pointing in different directions at any time, but always going round at the same rate…

Here’s a thought:  maybe some clocks are going faster than others? wierd. That would mean the particle has more energy in some places than it does at others?  That would be like time dilation…? Maybe that’s what gravity is! When lots of particles get together they speed up their clocks…

You can also think about how the hands grow and shrink in time. Remember the length of the hand is the probability of where to find the particle physically.  So as well as picturing all the clock hands rotating. You can imagine then growing and shrinking in length too.  So you get waves!  A bit like a mexican wave at a stadium. People effectively get taller when they stand up and shorter when they sit down again. The clock hands at each point in the field grow and shrink as the particle passes that region of space.

The rules governing how the clock hands rotate and grow or shrink in a field are the Laws of Physics. And you can’t just have any laws. Only the laws that give the same answers as experiments.  Just because you can imagine it in this picture doesn’t mean it is real! Like that idea I had about different clocks for the same particle going at different speeds being something to do with gravity. That’s just a thought about how particles could behave.

You can choose a behaviour for the clocks yourself! and that probably describes something, but not necessarily a real particle. In fact that’s what theoreticians do; they think of sets of rules for these fields and try to find behaviours that correctly govern the zoo of all 200+ subatomic particles in one coherent theory! It’s called the standard model.

The principle that physicists use to derive the correct rules for the behaviour of the clocks that accurately describe real particles is actually very simple. We use symmetry and we use the axiom that all the laws of physics must always be the same throughout spacetime.

That’s it!

With those two principles you can say things like:  Well. If I rotate this clock hand here – if I force it manually -then all the other clock hands for that particle everywhere else must rotate to compensate for it so that when I operate on the field I still get the same answer that I get from an experiment.   We call this “local gauge invariance”.  It doesn’t matter where the clock hands are pointing initially, as long as they all change according to the same rules!

So how can I manually change the setting of a clock? Well. You need another particle. And that’s when things start to get really funky!

Now I have two fields.  And maybe one particle gives some of its energy to the other particle?  So the clocks related to one particle might slow down and the other one speed up? Interesting. Or maybe they exchange information about the size of their hands, thus modifying the propability of where they are.

You can imagine what happens when two of these “mexican waves” in probablity encounter each other. Do they bounce off each other? or just pass straight through each other like waves on the sea? Maybe they change direction?

Who knows! There’s lots of things they could do! And physicists know the right equation to describe what they are allowed to do! (The dirac and schrodinger equations tell us what rules the waves must obey).

And now the really clever bit happens:

Any changes to the first particle’s field caused by the interaction with the second particle must be compensated for in the second particle’s field and vice versa, so that the total energy doesn’t change. Or momentum. Certain physical properties are conserved in the interaction between the two particles.

So that means that we can now imagine a third field of clocks which contains the information about the changes I must add to first field and subtract from the second field when the two particles interact.  This auxiliary field is also a field of clocks! So it must be a particle too! and it is! A very special particle all by itself called a gauge boson.  This boson is what a force is.  A force is the change in clock information given from one particle to another. And it is represented by a particle!

(NB the gauge boson is a boson but it’s not the famous Higgs boson. The Higg’s boson tells you about the “clock information” that the raw vacuum imparts to particles as they move about. That’s what tells each particle how heavy it is (i.e. how much energy they need to move at a certain speed through the spacetime)).

In electromagnetism the gauge boson is a photon. Although it’s not a normal freely moving photon. We say it’s called a virtual photon.  i.e. The rules governing the exchange of clock information between electrons (and positrons) are the same as the rules obeyed by electromagnetic waves…

For the other two forces in physics – the weak and strong forces – things get even more funky still!!   Instead of having one hand on the clock, the weak force has two hands and the strong force has three hands!

So how does that work? Well they obey more complex symmetries!  The relationship between the positions of the multiple hands on the clock tells you what kind of particle the field represents. i.e. there are internal angles between the hands which are always the same for the same type of particle, but different types of particles have different internal angles.  The total length of all the hands is constrained (we say it’s normalised).

You can do some really funky stuff now, like change the angles between the hands, or change the length of one hand and shrink another. This effectively convert particles from one type to another type to make it something different.That’s what happens in particle accelerators like CERN. As long as the total energy, momentum, charge and mass are conserved you can do what you like to the hands.

Some particles feel all the forces! So you need to have three clocks to describe them! A one handed clock, a two handed clock and a three handed clock.   Einstein spent the last bit of his life trying to find a single five (or more) handed clock that was big enough to explain all three of these forces, but he failed – that’s what my final year project as an undergraduate was all about – Grand Unification!   I didn’t succeed either. I don’t think I tried any where near as hard as Einstein. But he believed he could do it. Whereas I didn’t have the confidence of someone of Einsteins standing! (plenty of arrogance though!).

Ok. So that’s enough about gauge theory.

Let’s go back to the other topic: the circular economy!

Now in my mind I have a similar mental picture. Instead of a field governing one sub-atomic particle across spacetime, we have a single field for each element across the Earth.

Imagine all the carbon atoms on Earth are spead out and each atom has a clock representing its phase in the recycling loop of the carbon atoms. There is a blanket of clocks spread out over the surface of the Earth.  This superficially  looks in my mind a bit like the field for a particle in spacetime. But the rules governing the behaviour of these atom clocks are very, very different!!

The useful thing here is the imaginative leap required to picture such a scene.  The leap is to think about of all the carbon atoms on the Earth as being part of a single thing. The field of carbon. There is one such field for all the elements across Earth. This helps us to think beyond individual companies, mining this patch of metal here, or this animal eating that carbon there. We can begin to think of the Earth and all the things in it as a collection of 118 fields of elements, and certain sets of operations that can be done to these elements to advance them through their phases.

Unlike the particles in gauge theory, whose clock phases advance automatically because the particle have energy associated with them, we have to apply energy to the element fields from an external source to advance them through their phases.

In fact you could think of molecules as the “exchange bosons” between the fields for the elements. Each chemical reaction in which an atom becomes involved or freed from a molecule advances its phase.

In fact the idea of the cycle of the elements only really has a meaning when you think about the Earth as a whole and realise that it closely approximates a closed system from the point of view of an atom. In fact if we’re careful, we notice that there are actually several cycles at this length scale.  Much like having different hands on the clock for the particles that feel the more complicated strong and weak forces.

There is a geological cycle for each element. A biological cycle and an economic cycle.  So each atom actually has three hands on it! And the rules governing the behaviour of these hands are much more complicated than they are for particle physics!  (So you see that physics is relatively easy, by which I mean it’s rules are well defined, which is why we have made lots of progress in it).   But that said,. the rules governing how the elements combine – chemistry and thermodynamics – are very precise and well known too. It’s the rules at the longer length scales where these elemental cycles exist that become complicated and that is what humans are struggling with now!

That’s why I wrote this article to explore this idea of a phase angle in a circular economy to see if it was useful in helping people think about Earth as a single system. A sort of thought experiment.

The three cycles are interlinked and material moves between them, and we can think about the hands of the clock on each atom as it moves through the cycle.  In this case the length of the hands doesn’t mean anything really. It’s just the angle that matters. As various exchanges occur between the element fields, the hand of the clock move around.

Lets think about some examples:

We can think of water moving through the geological cycle, but also it moves through the biological cycle and the economic cycle too. Humans take advantage of the geological cycle, which is fairly rapid and so it usefully recycles water for us on a human timescale. Most geological cycles take a lot longer so we can’t exploit them.

When atoms comes together in a biological cell, information is encoded in the form of DNA which tells the different fields of elements how to combine.  That’s what an enzyme is. It’s a machine for connecting the fields of different elements – for doing chemistry! And a cell is a little bag of instructions for coupling all the elemental fields together at each point on Earth. Each organism is thus like a node in a network of interconnecting element fields and each organism propagates the clocks of the atoms, and moving them through the biological phases.  The material goes round and round and round, from one organism to the next driven by the organisms.

So you can’t just have any organisms! And you make organisms go extinct at your peril! To have a functioning top level cycle you can only have organisms that connect the cycles of the elements and cause the element fields to have closed cycles.  Just like in physics there are principles that govern the relationships between all the phases of the particles (like symmetry) so there are principles governing the relationships of the phases of all the cycles of all the elements.  If any of the elements get stuck at some point in their cycle then they will accumulate in a form that is not useful i.e. there is no organism to propagate that element in the biological cycle. Since the circularity is broken then by definition the atom will not be returned to the beginning of the cycle, and the organism causing the blockage will die from starvation! Of course, the atoms in a biological block will revert to the geological cycle and that may take aeons to reprocess the atom.

The circular economy is the process of creating an entirely separate human cycle that sits along side the biological and geological cycles. We can think of products and manufacturing much like the bags of cells – as nodes in a network that connect the element fields together.

Here’s the rub! and this is why gluttons and free market economics probably don’t like the circular economy.  Just like biology can’t have any old organisms – biology thrives when it has compatible networks of organisms in an ecology – so economics can’t just produce any old products it feels like!  We have to have rules governing how we make our products that allow all the atoms to go back to the beginning of their cycles. We may physically be able to make a product, but if it inhibits the cycling of the atoms it should be banned.  That is the “vector bosons” that connect the fields of elements – the organisms in nature and the products in economics – must obey careful rules if we are to avoid short lived or temporary situations.

Just as I said you could create your own rules for a particle, but that probably doesn’t lead to anything meaningful that can exist, so it is for economics. We have to develop systems that allow the fields of the elements to have complete cycles or the process that produces the incomplete cycle will die out, eventually. Perhaps all the possible gauge theories do exist and all the rules for all the particles fields do exist. And do occur. But it’s only the rules that lead to the emergence of a coherent time and space that survive long enough for us to notice them!




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