If you read the thing, you may recall that I justified renormalization with the argument that physics at different scales mostly don’t effect each other. Gallileo’s pendulum wasn’t effected by quantum mechanics or the gravitational pull of Jupiter.

There is an outstanding problem in particle physics at the moment that, if not resolved, may send that whole philosophy down the toilet. The problem has been around for a while, but it has got a lot worse in the last two or three years, sending particle physics into a bit of a crisis.

I speak of the *hierarchy problem*. Buckle your seatbelts and all that.

**We Need to Talk About Mass**

The hierarchy problem has its origin in interpreting the mass of the recently discovered *Higgs boson*. To get down to what the problem is about, we have to first think about mass more generally.

If you only know one equation from physics, it’s probably *E=mc ^{2}*. This says that energy and mass are basically the same thing, just in different forms. An object of mass

The total energy of an object is the energy contained in its mass plus the energy associated with its motion, i.e., its *kinetic energy*. When the object is at rest it has no kinetic energy, so all of its energy can be associated with its mass. Flipping this argument on its head, you can say that *the energy E inside an object at rest tells you its mass m, via m=E/c ^{2}.*

This may seem like an obvious and redundant thing to say, but consider the following example. A proton, one of the constituents of an atomic nucleus, is not simply a single particle but can be thought of as three smaller particles (called *quarks*) bound together. The quarks are in general wobbling around, moving in relation to each other, so they contain some kinetic energy. Quantum field theory tells us that the quarks interact by emitting and absorbing other particles called *gluons*, which are very similar to photons. Gluons can spontaneously create new quarks, then destroy them again an instant later. The motion of all these extra particles contribute to the overall energy of the proton.

Since the mass is given by the total energy it contains when it’s at rest, it includes all of this extra energy due to the motion and interactions. As a consequence, the mass of the proton is larger than just the sum of the quark masses. In fact, the quark masses only account for around 1% of the total proton mass!

Fig. 1: The inner workings of a proton. The mass of the proton is given by all of the energy enclosed by the dotted line (divided by *c ^{2}*)

A similar effect occurs for individual particles. Namely, working out the mass of the Higgs boson requires an analogous consideration.

The Higgs can both emit and absorb many different types of particle, including quarks, electrons, you name it. It could emit a quark, which exists for a tiny period of time, then absorb it again before it gets the chance to go anywhere. The result is that the Higgs is covered with a cloud of extra particles popping in and out of existence. The mass and motion of these particles all contribute to the overall energy of the Higgs, therefore enhancing its mass.

Fig. 2: The Higgs, dressed with emissions. The effective Higgs mass is given by all the energy enclosed in the dotted line (divided by *c ^{2}*)

Similar things occur for other particles, like electrons, but not to the extent that it happens to the Higgs. To get into the reasons for this difference requires some deep discussions about *symmetries* in particle physics, a subject I should really do a post about at some point. But I won’t go into it here.

From this point of view the Higgs really has two masses, the *apparent *mass *m* which is measured in an experiment, and the *bare *mass *m _{0}*, the mass of the Higgs if it wasn’t coated in emissions.

*m _{0}* is the more fundamental of the two, a parameter of the underlying theory. However, only

*m = m _{0} + E/c^{2}*

But how do we work out *E*? We can make an approximation according to the following argument.

Just like in the Feynman diagrams in the previous article, the cloud of particles surrounding the Higgs can have any momentum, so the energy gets contributions from emissions with all possible momenta. But recall that, in order to make sure probabilities can’t become infinite, we need to restrict particles from having momentum above Λ. This corresponds to ignoring scales below 1/Λ. So we only need to consider emissions having momentum up to Λ. Most of the bonus energy in this case comes from the most energetic possible particles, the ones with momentum Λ. Assuming this to be large, we can say that most of their energy is kinetic, and can ignore the energy due to their masses. The kinetic energy of the most energetic allowed particles then are roughly Λ, leading to an overall bonus energy for the Higgs to be round about Λ. So we end up with

*m ≈ m _{0} + *Λ

This equation is where the hierarchy problem comes from.

**One in a Hundred Million Billion Trillion Trillion
**

Imagine the scene. We’ve measured the mass of the Higgs *m*, to be the famous number 125GeV (GeV is just a unit of mass particle physicists use). Looking at the above equation, you can see that if we decide to set Λ at some value, we then have to tune the value of *m _{0}* in order to produce the observed 125GeV for

What are the possibilities for choosing Λ? Λ is meant to be chosen to cut out effects at scales where we don’t know what’s going on, so we can choose Λ such that 1/Λ is anything down to scales where “new physics” appears.

What if there was no new physics at all, our current model is valid at all scales? Then we could take 1/Λ to be the theoretically smallest possible length – the Planck length *L*_{P}. In this case, we have that Λ = 1/*L*_{P}*, *leading to a new equation:

*m ≈ m _{0} + *1

*L*_{P} is a very very small number, the smallest possible length. As a result, this new bonus mass 1*/ L*

For this theory to be consistent with the observed Higgs mass *m*,* m _{0}_{ }*needs to be a number which when added to this huge number 1

Imagine you changed the 33rd decimal place of *m*_{0}, in other words, the number was shifted up by a hundred-million-billion-trillion-trillionth of its size. The value of *m* would go from being 125GeV to double that size, a huge change. If you increased *m*_{0} at just the 3rd decimal point, *m* would still become a million billion trillion trillion times bigger. And so on. This is referred to as the *fine-tuning* of *m*_{0}.

Fig. 3: Above equation visualized. The towers* *Λ*/ c^{2}* and

The universe would be radically different if that value of *m*_{0} was changed even a tiny bit. The Higgs particle is what gives mass to all the other particles, and the mass of all the other particles is decided in part by the Higgs mass. If *m* was billions of times larger, all the other particles would become billions of times heavier also. We certainly couldn’t have stars, planets and all that, the universe would be too busy collapsing in on itself. It seems like, to generate a universe remotely like the one we live in, nature needs to decide on a parameter *m*_{0}, highly tuned to 33 decimal places.

This disturbs a lot of people because it is very *unnatural*. It seems like an incredible coincidence that *m*_{0} ended up with the exact value it did, the exact value needed for a universe where stars could burn, planets could form and life could frolic. It’s a bit like saying someone dropped a pencil and it landed on its point and stayed there, perfectly balanced. Except in this case to get the same degree of coincidence, the pencil would have to be as long as the solar system and have a millimeter wide tip [source].

This is concerning by itself, but its consequences go further. It represents a breakdown of our assumption of physics at different scales being mostly independent. *m*_{0} is a parameter of the theory which describes physics down to the scale of *L*_{P}, so includes whatever physics is happening at the Planck length. In that case, *m*_{0} in a sense is decided by whatever is happening at the Planck length. Physics at large scales seem to be incredibly strongly dependent on *m*_{0} which comes from the Planck scale.

Before, we thought that physics at very small scales shouldn’t strongly effect physics at larger scales, but this changes all that. Is renormalization valid if this is the case?

**Supersymmetry to the Rescue
**

In constructing the hierarchy problem above, we made an assumption that our current theory of particle physics is valid all the way down to the Planck length. This may be true, but it may not be. There may be new unknown laws of physics that appear as you go down to smaller scales, before you get anywhere near the Planck length.

If we assume some new physics appears at a new length scale we’ll call *L*_{N}, then our current theory is only valid at scales larger than this, and can only contain particles of momentum smaller than 1/*L*_{N}. This changes the bonus Higgs mass, changing the above equation to:

*m ≈ m _{0} + *1

If the scale *L*_{N} is much bigger than the Planck length, *L*_{P}, then 1*/L*_{N}*c ^{2}* is much smaller, and

Still, if 1*/L*_{N}*c ^{2}* is only a million times the size of

It is for this reason that a popular candidate theory of smaller scales, *supersymmetry*, is hoped to become apparent at length scales not much smaller than what we’ve already tested. This would solve our problem, as 1*/L*_{N}*c ^{2}* would end up being roughly the same size as

Since the LHC at CERN started bashing together protons at higher momenta then ever before, we’ve been keeping an eye out for signs of supersymmetry. We’ve now searched for signs at lengths scales quite a lot smaller than where we discovered the Higgs. Unfortunately, no evidence of supersymmetry’s existence has appeared. With every year of experiments that pass, and supersymmetry isn’t found, the possible scale where supersymmetry appears gets pushed to smaller and smaller, making *L*_{N} smaller and smaller. The smaller *L*_{N} gets, the more fine-tuned *m*_{0 }needs to be.

People are starting to worry. Even if supersymmetry is found tomorrow, it looks like it’ll only become important at scales where 1*/L*_{N}*c ^{2}* is a hundred times the size of the Higgs mass. So a tuning of one part in a hundred… Is that already too much of a coincidence? The further up the energy scale we have to go to find supersymmetry, the less power it has to resolve the issue.

The Hierarchy problem is one of the biggest driving forces in particle physics research today, giving hints that there is more physics to be found at scales close to us. If supersymmetry is not found at the LHC, we’re going to have to do a proper re-think about our philosophy of renormalization. Could there be something wrong with our understanding of scales? And could the stars, planets and life really exist on merit of a massive coincidence?

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What if I was to tell you that us humans have been creating universes on computers, taking into account the most fundamental of physics, detailed to some of the smallest length scales that we understand? They’re not quite the size of our universe, or even something smaller like a planet, current computers would struggle somewhat. They’re only about 10 femtometers across, smaller than an atom. But it’s a start!

They’re called Lattice simulations, and belong to a subgenre of particle physics called *Lattice Gauge Theory.*

To illustrate what this is and the drive behind it, let’s consider a very simple and general problem in physics. Working out the trajectory of, say, an electron. Deducing a trajectory is to be able to say where the electron is at any given point in time.

In classical mechanics (a.k.a how the world looked before quantum mechanics became a thing), given all the forces acting on the electron, along with the initial conditions (i.e. its position and velocity) there exists one unique trajectory the particle can take. One can plug the initial conditions into an equation of motion (like Newton’s 2nd law, *F=ma*) and solve it to deduce with certainty the position of that particle after some arbitrary period of time.

Taking quantum mechanics into account the water becomes muddied. The electron is no longer bound to follow the unique trajectory, but can take other trajectories which disobey its classical equation of motion. Before, the probability of the electron following the classical path was 1, and following any other path was 0. Now, each path has some non-binary probability between 0 and 1.

Not even the most versed physicist can predict with certainty where the particle will be after a period of time. As a consolation prize, it is possible to deduce the probability of the particle arriving at a certain point in space at a given time.

To do this, a physicist would basically sum up the probabilities of each of the many trajectories that result in the electron arriving at your chosen location at your chosen time. This is called a *path integral, *the sum of all probabilities of a particle taking each possible path between two points. In general there is an infinite number of possible paths. The classical path is always the most likely, paths that are close to the classical path have a smaller probability but still contribute, and completely deviant paths that go to jupiter and back are incredibly unlikely and basically don’t contribute.

One of the reasons quantum mechanics is ‘hidden’ at sizes bigger than an atom is that the perturbed paths become so unlikely that the classical path is basically the only path the particle can take.

Fig.1: Particle travelling from A to B. Rightmost- a particle on the quantum scale, e.g. an electron. Leftmost- a particle on the classical scale, e.g. a baseball. Solid lines are “very likely paths”, and dotted are less likely.

Now let’s complicate the picture further by moving from quantum mechanics to *quantum field theory*. This takes into account the possibility of the electron emitting and absorbing particles, or decaying into different particles only to reappear somewhere down the line before reaching its destination. Things become more complicated, but the principle of the path integral still holds, with the new feature that the bunch of paths we need to add up now include all combinations of emissions and decays. I’ll refer to a ‘path’ as a trajectory + any specific interaction including other particles.

Fig.2: Similar to figure 1, now with the possibility of other particles being created and destroyed.

Once we’re in quantum field theory we are getting into some real fundamental shit. The standard model of particle physics, containing the recently discovered Higgs boson, is expressed in the language of quantum field theory.

In practice it’s not possible to work out probabilities for an infinite number of paths. Happily, as I discussed, there are a small number of *dominant* paths which account for the majority of the probability, the classical path and small perturbations of it. In particle physics, the way we usually work out the probability of some process is to consider only these dominant paths, and we get to a result which is pretty close to the ‘true’ answer. It can be done with just a pen, paper and the knowhow. The method is referred to as *perturbation theory.*

This doesn’t work for everything. Namely, if we were trying to compute the path integral for a quark rather than an electron. The electron interacts mostly with the electromagnetic force (electricity+magnetism). Quarks feel not only the electromagnetic force, but also the *strong nuclear force, *it’s horrendously more complicated cousin. The strong force is ‘purely quantum’ in the sense that there isn’t really a dominant path and subdominant perturbations of the path, there are many different dominant paths and there is no good way to order them in terms of probability.

It’s possible to use perturbation theory on quarks, but since it’s difficult to find all the dominant paths, uncertainties usually lie at around 10% (i.e., the true answer could be the answer we worked out give or take 10% of that answer). Compare this with what one can achieve with the electron, with uncertainties dipping well below 1%.

**The solution: lattice simulations!**

Simulate a small period of time playing out on a small patch of space on a powerful computer, give the patch a little prod so it has enough energy for a quark to appear, and let your tiny universe play out all the possible paths, decays, interactions, whatever.

Inside the patch it’s necessary to approximate space and time as a *lattice* of discrete points. Each point has some numbers attached to it, signifying the probability of a quark being at that point, the probability of the presence of other quarks, and the strength of the strong force. With this little universe, we can forget about which paths are dominant and which aren’t since all paths occur automatically in the simulation.

Like perturbation theory, it is also an approximation. Spacetime is not discrete (as far as we know), and not inside a finite patch. People often will perform the simulation at many different *‘lattice spacings’ *(the distance between each discrete point), and look at the trend of these numbers to extrapolate the answer to a zero lattice spacing, representing a continuous space. Similarly in the real world there are no ‘walls’ like there are on the edges of the patch. So folk will use a range of sizes of patch, and extrapolate results to an infinite size where there are no walls.

Uncertainties in lattice simulations are in many cases a lot smaller than perturbation theory, at about the 1% level. The method has proven shockingly effective in understanding how quarks bind together to form *mesons. *A meson is like the little brother of the proton and neutron, while these contain 3 bound quarks, a meson contains 2. Lattice people have their sights on making a simulation big enough that a whole proton can fit inside its walls, but we’re not quite there yet.

I think lattice gauge theory is still only in its ‘calibration phase’. The motivation of a lot of the work lattice people do is to show it works, by matching its predictions to experiments. As computers become faster, our methods become more efficient, and our understanding of the physics improves, the lattice could end up being the tool which uncovers the next big discovery in particle physics. Watch this space.

*Path integrals in quantum mechanics*

*What if spacetime really is discrete?*

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I always found the popular science description of entropy as ‘disorder’ as a bit unsatisfying.

It has a level of subjectivity that the other physical quantities don’t. Temperature, for example, is easy- we all experience low and high temperatures, so can readily accept that there’s a number which quantifies it. It’s a similar story for things like pressure and energy. But no one ever said ‘ooh this coffee tastes very disordered.’

Yet entropy is in a way one of the most important concepts in physics. Among other things, because it’s attached to the famous *second law of thermodynamics, *with significance* *towering over the other laws of thermodynamics (which are, in relation, boring as shit). It states that the entropy of a closed system can only increase over time.

But what does that mean?! What is entropy really? If you dig deep enough, it has an intuitive definition. I’ll start with the most general definition of entropy. Then, applying it to some every day situations, we can build up an idea of what physicists mean when they say ‘entropy’.

**Missing Information**

Fundamentally, entropy is not so much the property of a physical system, but a property of our *description* of that system. It quantifies the difference between the amount of information stored in our description, and the total quantity of information in the system, i.e. the maximum information that could in principle be extracted via an experiment.

Usually in physics, it’s too difficult to model things mathematically without approximations. If you make approximations in your model, or description, you can no longer make exact predictions of how the system will behave. You can instead work out the probabilities of various outcomes.

Consider the general setup of an experiment with *n* possible outcomes, which one can label outcome 1, outcome 2*, *… outcome *n.* Each outcome has a probability *p*_{1},*p*_{2}, *p*_{3} …*p*_{n} assigned to it. Each *p* has a value between 0 (definitely won’t happen) and 1 (definitely will happen). The *Gibbs entropy* *S* one assigns to the description of a system is a function of these *p*‘s (see mathsy section at the end for the explicit definition).

Imagine we thought up a perfect theory to describe the system under study using no approximations, so we could use this theory to predict with certainty that outcome 1 would occur. Then *p*_{1}* = *1 and all other *p*‘s would be zero. One can plug these probabilities into the Gibbs entropy, and find that in this case, *S = *0. There is no missing information. In contrast, if we had no information at all, then all probabilities will have the same value. How could we say one outcome is more or less likely than any other? In that case, *S* ends up with its maximum possible value.

There’s a classic example that’s always used to describe the Gibbs entropy – tossing a coin. There are two possible outcomes- heads or tails. Usually we consider the outcome to be pretty much random, so we say that they’re equally likely: *p(heads) =* 1/2,* p(tails) = *1/2. This description contains no information, a prediction using these probabilities is no better than a random guess. What if we discovered that one of the sides of the coin was weighted? Then one outcome would be more likely than the other, we can make a more educated prediction of the outcome. The entropy of the description has been reduced.

Going further, if we modelled the whole thing properly with Newton’s laws, and knew exactly how strongly it was flipped, its initial position, etc, we could make a precise prediction of the outcome and *S* would shoot towards zero.

Fig.1: *S* for different predictions of the outcome of a coin flip.

Working with this definition, the second law of thermodynamics comes pretty naturally. Imagine we were studying the physics of a cup of coffee. If we had perfect information, and knew the exact positions and velocities of all the particles in the coffee, and exactly how they will evolve in time, then *S=*0, and would stay at 0. We always know exactly where all the particles are at all times. However, what if there was a rogue particle we didn’t have information about, then *S* is small but non-zero*. *As that particle (possibly) collides with other particles around it, we become less sure what the position and velocity of those neighbours could be. The neighbours may collide with further particles, so we don’t know their velocities either. The uncertainty would spread like a virus, and *S* can only increase. It can never go the other way.

I said before that this *S* is about a description, rather than a physical quantity. But entropy is usually considered to be a property of the stuff we’re studying. What’s going on there? This brings us to…

**Microstates and Macrostates**

In physics, we can separate models into two broad groups. The first, with “perfect” information, is aiming to produce exact predictions. This is the realm containing, for example, Newtons laws. The specification of a “state” in one of these models, contains all possible information about what its trying to describe, and is called a *microstate*.

The second group of models are those with “imperfect information”, containing only some of the story. Included in the second set is* *thermodynamics. Thermodynamics seeks not to describe the positions and velocities of every particle in the coffee, but more coarse quantities like the temperature and total energy, which only give an overall impression of the system. A thermodynamic description is missing any microscopic information about particles and forces between them, so is called a *macrostate.
*

A microstate specifies the position and velocity of all the atoms in the coffee.

A macrostate specifies temperature, pressure on the side of the cup, total energy, volume, and stuff like that.

In general, one macrostate corresponds to many microstates. There are many different ways you could rearrange the atoms in the coffee, and it would still have the same temperature. Each of those configurations of atoms corresponds to a microstate, but they all represent a single macrostate.

Some macrostates are “bigger” than others, containing lots of microstates, and some contain little. We can loosely refer to the number of ways you could rearrange the atoms while remaining in a macrostate as its *size*.

What does all this have to do with entropy? If I were to tell you that your coffee is in a certain macrostate, this gives you information. It narrows down the set of possible microstates the coffee could be in. But you still don’t know for sure exactly what’s going on in the coffee, so there is missing information, and a non-zero entropy. But if the coffee was in a smaller macrostate, our thermodynamic description would give *more* information, since we’ve narrowed down further the number of microstates the coffee could be in. Then our description contains more information, so this is a lower entropy macrostate.

Hence, the entropy of a macrostate (called the *Bolzmann entropy*) is defined to be proportional to its size. For messy thermodynamic systems like the coffee, entropy is a measure of how many different ways you can rearrange its constituents without changing its macroscopic behaviour. The Bolzmann entropy can be derived from the Gibbs entropy. It is not a different definition, but a special case of Gibbs, the case where we’re interested only in macroscopic physics.

Working with this definition, the second law of thermodynamics comes reasonably naturally. Over time, a hot and messy system like a cup of coffee will explore through microstates randomly, the molecules will move around producing different configurations. Without any knowledge of what’s going on with the individual atoms, we can only assume that each microstate is equally likely. What macrostate is the system most likely to end up in? The one containing the most microstates. Which is also the one of highest entropy.

Consider the milk in your coffee. Soon after adding the milk, it ended up evenly spread out through the coffee, since in the macrostate of ‘evenly spread out milk’ is the biggest, so has the highest entropy. There are many different ways the molecules in the milk could arrange themselves, while conspiring to present a macroscopic air of spreadoutedness.

You don’t expect all the milk to suddenly pool up into one side of your cup, since this would be a state of low entropy. There are few ways the milk molecules could configure themselves while making sure they all stayed in that side. The second law predicts that you will basically never see your coffee naturally partition like this.

Fig.2: Cups of coffee in different microstates. The little blobs represent molecules of milk.

**The Mystery Function of Thermodynamics**

When one talks about the Boltzmann entropy, naturally there is a transition between considering entropy a property of the *description* to a property of *physics*. Different states in thermodynamics can be assigned different entropies depending on how many microstates it represents.

Once we stop thinking at all about what is going on with individual atoms, we are left with a somewhat mysterious quantity, *S*.

The “original” entropy, now known as the *thermodynamic entropy,* is a property of a system related to its temperature and energy. This was defined by Clausius in 1854, before the nature of the atoms at the macroscopic level were even understood. Back then, not everyone had been convinced that atoms were even a thing.

Thermodynamic entropy is what people most commonly mean when they refer to entropy, but, since it is defined without any consideration of the microscopic world, its true meaning is obscured. I hope it’s slightly less obscured for you now.

*Why entropy is at the heart of information theory*

*All of thermodynamics can be derived from entropy*

*The arrow of time: why the second law causes a paradox*

**The Equations – for those who like maths**

A system can be in *n* possible states. The probability of it being in state *i* is a number between 0 and 1, called *pi*. The **Gibbs entropy **is defined by:

*log* is the natural logarithm, and all you need to know about it is that *log(p*i*)** =* 0* *when *p*i=1. If the system is definitely in state 1, *p*_{1}*=*1 and the rest are 0. Then the first term disappears since *log(p*1*)* becomes zero, and the rest of the terms also disappear, not because of the *log* but because of the factor of *pi *at the front is zero. We end up with *S = *0, corresponding to the fact that there is no missing information- we can perfectly predict the behaviour of the system.

In the more realistic situation, the *p’*s are a bunch of non-zero numbers, representing non-perfect information and leading to a non-zero *S*. Applying this equation to the coin flipping scenario, you end up with Fig. 1.

If we’re only interested in the macroscopic nature of a system, we would model it to be in a macrostate, which contains Ω microstates. The **Boltzmann entropy** is defined by:

If we’re in a macrostate with Ω=1, there is a single microstate it can be in so we know everything about the system. Ω=1 causes the *log *, and therefore *S*, to become 0. All pretty consistent. For Ω larger than 1, the *log* increases as Ω gets bigger. This leads to *S* increasing as the number possible states increases, i.e., we become less sure about which state the system is in.

The **thermodynamic entropy** is defined by

In words, it says that when ΔQ of heat energy is added to a system, the change in entropy ΔS, will be equal to ΔQ divided by temperature T. It’s actually the same number as the Boltzmann entropy, just written in terms of purely thermodynamic quantities.

This isn’t a very illuminating equation in my opinion. The best I can offer to help is the following:* *imagine again pouring a drop of milk into your coffee. If, by some twist of fate, the cup was instantly cooled until both the milk and coffee froze, the milk would be frozen into the pretty pattern it made when it hit the coffee. This is quite a special, low entropy state.

You’re annoyed by the sudden freezing of your tea so you shove it in the microwave to add ΔQ worth of heat. As it melts, the milk is allowed to mix more and more with the tea, heading towards states of higher mixedness, ΔQ leads to ΔS. The division by T? As it approaches a state one would happily drink (T getting larger), adding more ΔQ leads to less of an increase in S. The milk is close to being fully mixed in, heating it more has less of an effect on S.

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But when it was first being established in the first half of the 20th century, it came across an apparently fatal flaw. It was plagued with infinities. And infinities don’t belong in physics. Following the rules of quantum field theory, you could end up predicting an electron having an infinite electric charge. Gasp. Its resolution lead to a revolutionised way of thinking that now underpins all of particle physics.

I may go into a little bit of maths, but don’t worry its all easy. Promise.

**Infinitely Probable Events**

Science is about making predictions, given initial conditions.

If our system is in state A at time 1, what is the probability of it being in state B at time 2?

In particle physics, we read “system” to mean the universe at its most bare-bones fundamental level. The question becomes the following:

At time 1, there exists a given set of particles, each with a particular momentum. What is the probability of a new set of particles, each with a new particular momentum, at time 2?

Quantum field theory is intended to be the machinery one can use to answer such a question. A nice simple challenge we can give it is this: given two electrons, hurtling towards each other at momenta *p*_{1} and *p*_{2}, what is the likelihood of them ricocheting off each other and coming out of the collision with the new momenta *q*_{1}* *and *q*_{2}?

Fig 1: Feynman Diagram of two electrons exchanging a photon

The easiest (most likely) way for this to happen is shown in fig. 1, this thing is called a *Feynman diagram*. Electron 1 emits a photon (the particle responsible for the electromagnetic force). This flies over to electron 2 with momentum *k*, and gets absorbed. We can use the principle of *conservation of momentum* to uniquely determine *k*. The principle states that total momentum must be the same at the beginning and end of all events. Applying this to electron 1 emitting the photon, *initial momentum = final momentum* implies *p*_{1}* =* *q*_{1}* + k. *Then, rearranging gets us to *k** = **p*_{1} – *q*_{1}. Since we’re given *p*_{1} and *q*_{1}, we can use this equation to work out exactly what *k *will be.

Quantum field theory can be used to work out the probability of each individual part of the Feynman diagram. Electron 1 emitting the photon, the photon travelling from electron 1 to 2 with momentum *k, *and electron 2 absorbing it. This produces the so-called *Feynman rules*, a translation between parts of the diagram and probabilities of each part taking place. The probability of the entire event can be found by just multiplying probabilities of each component event. The probability of the photon emission, multiplied by the probability of its travel to electron 2, multiplied by the probability of its absorption, gets you the overall probability. Nobel prizes all ’round.

But wait. This is not the only way you can put in two electrons of momenta *p*_{1} and *p*_{2} and get two electrons out with momenta *q*_{1} and *q*_{2}. There are a number of different ways the two electrons could interact, in order to produce the same outcome. For example, this:

Fig.2: Feynman Diagram of two electrons exchanging a photon which splits into an electron positron pair on the way.

The photon splits into two new particles, which recombine to return the photon. Similarly to before we know exactly what the photon momentum *k* is, using *k = p*_{1} – *q*_{1}, and the values for *p*_{1} and *q*_{1 } which we are given in the problem. But now, there is no guiding principle to decide what the momenta of the electron and the positron in the middle will have. We know that *k*_{1}* +* *k*_{2} *=* *k *from conservation of momentum, but this is one equation containing two unknowns. Compare it to how we worked out *k* in the first diagram, in which case there was only one unknown, so we could use all the other known values (*p*_{1} and *q*_{1}) to get the unknown one (*k*). If we fix *k*_{2} by saying *k*_{2} = *k* – *k*_{1}, we have one unfixed degree of freedom left, *k*_{1}, which could take on any value. *k*_{1} could even have negative values, these represent the electron moving in the opposite direction to all the other particles.

*k*1 is not uniquely determined by the given initial and final momenta of the electrons. This becomes significant when working out the overall probability of fig.2 occurring.

To work out the overall probability, one needs to use the Feynman rules to translate each part of the diagram into a probability, then combine them. The probability of electron 1 emitting photon, multiplied by the probability of photon moving to where it splits up, multiplied by the probability of photon splitting into the electron & positron, etc. But this time, since the middle electron could have any momentum, one needs to add up the probability of that part for all values of *k*_{1}. There is an infinite spectrum of possible *k*_{1} values so there are an infinite number of ways fig.2 could occur.

Let’s step back for a moment. In general, if there are lots of different events (call them *E*_{1}*, E*_{2}, ….) that could cause the same overall outcome *O* to occur, then the probability of *O*, *prob(O)*, is

*prob(O) = prob(E*_{1}*) + prob(E*_{2}*) + …*

If there are an infinite number of ways *O* could occur, then it becomes an infinite sum of probabilities, and as long as each of the probabilities are not zero, or tend towards zero, then *prob(O) *becomes infinite.

This is what happens with our particles. Since there is an infinite number of momentum values the middle electron could have, there is an infinite number of probabilities that must be added up to get the probability of fig.2 occurring, so the probability of fig.2 is infinite.

What could that even mean? A probability should be a number between 0 (definitely won’t happen) and 1 (definitely will happen). Such predictions of infinite probabilities renders a theory useless, quantum field theory is doomed. The Higgs boson is a conspiracy invented by the Chinese.

**Renormalization or How to ignore all your problems**

This wasn’t the end of quantum field theory- since there is a way of resolving this problem. Kind of. The solution, or rather the family of solutions, are referred to as *renormalization*. It comes in many different manifestations, but it all boils down to something along the lines of the following. We pretend that *k*1, our unconstrained electron momentum, can only have a value below some maximum allowed size we’ll call Λ. Then, we don’t need to add up probabilities from situations where *k*1 goes arbitrarily high. We’re left with a finite number of possibilites, therefore a finite probability for the whole event. More generally, we can solve all problems like this by making Λ a universal maximum momentum for all particles involved in an interaction. Λ is called a *momentum cutoff.*

This solves the issue, we end up with sensible predictions for all processes. And as long as we make Λ suitably larger than the momentum of the initial and final electrons, the answer matches results of experiments to high precision. But I’ll understand if you feel a little unsatisfied by this. How come we can just ignore the possibility of electrons having momentum higher than Λ? To win you over, I’ll tell you a bit about what Λ physically means.

In quantum mechanics, an electron is both a particle and a wave. One of the first realisations in quantum mechanics was that the wavelength of an electron wave is inversely proportional to its momentum; *wavelength = 1/momentum*. A high momentum corresponds to a small wavelength, and vice versa. Ignoring particles with momentum higher than Λ, is the same as ignoring waves with wavelength smaller than 1/Λ. Since all particles can also be seen as waves, the universe is made completely out of waves. If you ignore all waves of wavelength smaller than 1/Λ, you’re effectively ignoring “all physics” at lengths smaller than 1/Λ.

Renormalization is a “coarse graining” or “pixelation” of our description of space, the calculation has swept details smaller than 1/Λ under the rug.

Making exceptions like this have in fact been a feature of all models of nature throughout history. When you’re in physics class doing experiments with pendulums, you know that the gravitational pull of Jupiter isn’t going to effect the outcome of your experiment, so broadly speaking, long-range interactions aren’t relevant. You also know that the exact nature of the bonds between atoms in the weight of your pendulum isn’t worth thinking about, so short-range interactions also aren’t relevant. The swing of the pendulum can be modelled accurately by considering only physics at the same *scale *as it, stuff happening on the much larger and much smaller scale can be ignored. In essence you are also using renormalization.

Renormalization is just a mathematically explicit formulation of this principle.

**The Gradual Probing of Scales**

Renormalization teaches us how to think about the discovery of new laws of physics.

The fact that experiments on the pendulum aren’t effected by small scales means we cannot use the pendulum to test small scale theories like quantum mechanics. In order to find out what’s happening at small scales, you need to study small things.

Since particles became a thing, physicists have been building more and more powerful *particle accelerators*, which accelerate particles to high momenta and watch them interact. As momenta increase, the wavelength of the particles get smaller, and the results of the experiments are probing smaller and smaller length scales. Each time a bigger accelerator is required in order to accelerate particles to higher speeds, and each jump is a huge engineering challenge. This race to the small scales has culminated in the gargantuan 27km ring buried under Geneva called the* Large Hadron Collider (LHC)*. This has achieved particle momenta high enough to probe distances of around 10 zeptometers (0.000000000000000000001 meters), the current world record.

Galileo didn’t know anything about quantum mechanics when he did his pioneering pendulum experiments. But it didn’t stop him from understanding those pendulums damn well. In the present day, we still don’t know how physics works at distances under 10 zeptometers, but we can still make calculations about electrons interacting.

From this point of view, it seems like we absolutely should impose a maximum momentum/minimum distance when working out the probabilities of Feynman diagrams. We don’t know what’s going on at distances smaller than 1/Λ. We need to remain humble and always have in mind that any theory of nature we build is only right within its regime of validity. If we didn’t involve this momentum cutoff, we would be claiming that our theory still works at those smaller scales, which we don’t know to be true. Making such a mistake causes infinite probabilities, which suggests that there is indeed something lurking in those small scales that is beyond what we know now…

**The road to the Planck scale**

There are currently a bunch of theories about what is going on at the small untested length scales. We can make educated guesses about what scales these prospective new features of nature should become detectable at.

Fig. 3: Length scales

There has been a fashionable theory floating around for a while called * supersymmetry, *which says, broadly, that matter and the forces between bits of matter are in a sense interchangeable. Its some well sick theoretical physics that I won’t go into here. The effects of this theory is believed to become visible at scales only slightly smaller than the ones we’ve already tested. It may even be detected at the LHC!

There’s a family of theories pertaining to even smaller sizes, called *grand unified theories**.* These claim that if we can see processes at some way smaller scale, many of the fundemental forces will be revealed to be manifestations of a single unified force. The expected scale where this happens is about a billion billion times smaller than what we’ve currently tested, so will take a billion billion times more energy to probe. Don’t hold your breath on that one.

Finally, there’s reason to believe that there exists a smallest possible scale. This is known as the *Planck length*. If any object is smaller than the planck length, it would collapse into a quantum black hole, then immediately evaporate, removing any evidence of its existence. This is the scale where the quantum nature of gravity becomes important, and if you want to test that, you’ll need a particle collider 100 billion billion times more powerful than the LHC.

If we want to learn about these mysterious smaller scales, we’re going to need some mighty big colliders. Perhaps impossibly big. Maybe we need some new innovation which makes the probing of scales easier. Maybe the challenge for the next generation of particle physicists will be a rethink of how we test particle physics all together.

*More on drawing Feynmann Diagrams*

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Ask a particle physicist this, and they may be tempted to drag you to the nearest blackboard, write down four lines of maths, and stare at you expectantly as if you’re meant to understand what on earth it means. These four lines represent the *standard model of particle physics*, which is our most up-to-date attempt to mathematically describe the fundamental constituents of matter, and the forces with which they interact. By matter, I mean what makes up everything around us- planets, stars, life, all composed of atoms, which in turn are composed of smaller particles like electrons and quarks. These are the particles that the standard model governs.

Fig.1: The Standard Model

The standard model has been highly successful at predicting the outcome of experiments, for example in the *Large Hadron Collider* (LHC) at CERN, and in fact it has withstood basically every test that has been thrown at it. It explains the nature of matter on a very fundamental level. But it’s actually pretty useless at explaining how all that matter got here in the first place.

So what do these equations have to say about how matter came into existence? I hope you’ll agree with me that electrons definitely exist. How is an electron created? The standard model says it can come into existence during for example, photon decay [ref. fig 2]. But in this process necessarily another particle, a *positron* must also appear at that same place and time. The positron is the electron’s antiparticle, meaning it has the same mass as the electron but oppositely charged. Such an event is not the only way electrons can be created, but whatever the process, the electron must always emerge with it’s oppositely charged sibling. The universe seems to run a two for one deal on fundamental particles.

Fig.2: The only way to get your hands on an electron.

It seems like if the standard model is the true description of reality, it would imply that there are just as many positrons in the universe as electrons. We know there are plenty of electrons present today; each atom has a bunch of electrons orbiting them. But positrons are a rather rare spectacle; they appear in cosmic rays but not many other places. We have arrived at a paradox.

Perhaps there is some way the positrons could have been destroyed but the electrons survived. The only way you can destroy a positron, according to the standard model, is to annihilate it with an electron, e.g. the process in fig. 2, but in reverse.

The only way to create or destroy electrons or positrons is via processes similar to the above. This is not just the case for electrons; the same applies to all particles in the standard model that make up matter.

While a proton is made up of three bounded quarks, there can exist anti-protons consisting of three antiquarks. With anti-protons, anti-neutrons and positrons, anti-atoms can form, which, just like atoms, can bind in all the ways necessary to end up with anti-stars, anti-planets, anti-life, the whole shebang. It’s all referred to as antimatter, which is a little misleading since it behaves rather a lot like matter, just with charges reversed.

If we had the same number of antiprotons as protons and antineutrons as neutrons, we would end up with the same number of anti-atoms, anti-stars and anti-planets.

Where is all of this antimatter? In 1998 an experiment was sent into space (the *Alpha Magnetic Spectrometer*, or AMS) to compare the density of helium and anti-helium in cosmic rays. It detected at least three million helium atoms per anti-helium.

Perhaps matter and antimatter have become separated somehow. Perhaps we just live in some huge region of space containing matter, and there are antimatter regions elsewhere in the universe.

Could we detect the presence of an antimatter region, say, by finding an anti-galaxy? To detect normal galaxies, we rely on nuclear fusion in stars, to emit light that can be picked up by our telescopes.

Since anti-fusion in anti-stars would emit the same frequency of radiation as normal fusion, we can’t really tell whether or not a galaxy is made of matter or antimatter. However, we know that galaxy collisions are reasonably common. The collision between galaxy and anti-galaxy would result in essentially annihilation between each particle-antiparticle pair, which come into contact. The reverse of figure 2 on a grand scale.

This would be a truly awesome event, making the black hole merger recently detected via gravitational waves look like a sneeze. The energy emitted would be E = mc^{2} where m is the combined mass of the two galaxies. In a back-of-the-envelope calculation, one would simply add the masses of the two galaxies and multiply by c^{2} to find an energy output 10^{20,000} (1 followed by 20,000 zeros) times more than your average supernova. Suffice to say we definitely would have noticed this.

Regardless of how you theorise how matter and antimatter have been distributed, the inevitable annihilations of objects and anti-objects would create a lot more radiation coming from the skies than we observe. It seems matter rules our universe.

Is it curtains for the standard model? Not quite. There have been a number of results from experiments throughout the years that may account for the matter/antimatter imbalance. The first of which was concerning a strange particle called the Kaon. Similar to a proton made of three quarks bound together, a Kaon consists of only two quarks. It’s possible for the Kaon to transform into an anti-Kaon spontaneously. It was originally thought that the probability of such a transition was the same as that of the reverse, an anti-Kaon turning into a Kaon, so overall the matter/antimatter balance would be preserved.

In 1964 it was discovered that in fact the two probabilities are different; the particles prefer to be Kaons rather than anti-Kaons. The technical expression for phenomena like these is “*CP-violation*” if you want to do some hardcore technical reading. These types of processes, combined with some extra conditions about thermodynamics in the early universe, could be enough to explain the imbalance we see.

For some time Kaons seemed to be the only particle which could behave in such a CP-violating way. However in 1983, CP-violation made a comeback when it was found that a whole new family of particles, called the *B mesons*, was capable of a plethora of such processes.

The imbalance caused by Kaons and B mesons is still nowhere near enough to explain the matter-dominated universe. The race to find more of these processes, and understand the mysterious underlying mechanisms that cause them, is still happening today. Experiments around the world like the LHCb detector at CERN are dedicated to this goal. In tandem, theoretical physicists are searching for extensions to the standard model that induce imbalance of the extent we need.

It looks like the search may have only just begun, and could lead us to some new and profound realisations about why the universe is the way it is.

*I still don’t get what this antimatter stuff is*

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Noone can truly picture how quantum particles interact via quantum forces, so we’re limited to analogies. The following is an analogy that I think exposes some of the key features. Consider the asteroid belt that lies between mars and jupiter…

**Quantum**** forces and the asteroid belt**

The only way they can interact (one rock effecting the trajectory of another) is by exchanging *momentum* via collisions*. *

Image one of these rocks breaking apart into two. Due to conservation of momentum, they move in opposite directions. Perhaps one is a lot larger than the other, so we can loosely think of the bigger one as the continuing story of the original rock (call it Rock A), and the smaller one as a bit of shed skin. The *emission* of the little rock causes the rock A to be deflected slightly in the opposite direction.

The little rock may then collide with another, call it rock B, and stick to it, be *absorbed* by it, this will cause its host’s movement to be deflected a little. The overall result is; rocks A and B have been deflected in the opposite direction to each other. If your telescope was strong enough to see rocks A and B but not the messenger that travelled between them, it would look almost as if there was some repulsive *force *pushing them apart. It’s not a mysterious invisible influence like gravity pushing them apart, it was due to an exchange of a thing that carried momentum between them. Such a thing one could call a *force carrier.*

The italicised words above are all borrowed from particle physics. The electric repulsion of two electrons is because they are exchanging force carriers in a similar way to the asteroids. In this case the force carrier the beloved *photon*, the particle responsible for the electric force. This is the way all fundamental forces work (except maybe Gravity, no one knows shit about Gravity), the exchange of force carrying particles.

Less intuitive and connected to the asteroids is the case where a force is *attractive. *Try to imagine the little messenger rock carrying “negative momentum” and when it hits rock B, it pulls it towards the direction it came from instead of away. I never said the analogy was perfect.

Fig.1: asteroids exchanging a rock, causing an effective repulsion between them. Left- close up, right- from afar.

**The strength**** of a force**

Consider again the asteroid belt. Could we work out, on average, how much of an influence rocks have on each other? In other words, what is the *strength* of the force between them? The property of the rocks one should look at is their crumbliness, the propensity for them to shed little rocks, and accept little rocks to stick to them.

In analogy, the strength of the electric force should be proportional to the probability at any point in time that an electron will emit, or absorb, a photon. Then, the total probability of two electrons “interacting” via a photon, would be proportional to the probability of photon emission from electron A, times that of the photon being absorbed by electron B.

Could we put a number on this fundamental property of the electron? There are nuances to this question, but loosely speaking, I can give you a number. It is roughly 1:137. Consider any electron in the universe, and you can rest assured there is always a 1 in 137 chance that it is emitting or absorbing a photon (this is modulo the complicated relationship between probabilities, quantum mechanical amplitudes, phase spaces, other caveat sources I haven’t thought of, and factors of π).

As a fraction, it’s expressed as α = 1/137, this is called the *fine structure constant. *It quantifies the strength of the electric force. (Actually, it dictates the strength of both the electric and magnetic forces, since they’re essentially two sides of the same coin. From now on I’ll refer to as the *electromagnetic *force).

Fig.2: Probability of an electron emitting a photon = α = 1/137

If α was different, the energy shells in atoms would go haywire, chemistry would change at its core, the frequency of light coming from stars would shift, and change α too much, electrons could escape the atom all together. The value of α has a strong bearing on the world around us.

Imagine God, or some cosmic architect, had created reality and was responsible for maintaining it, so had to sit in a control room full of levers and switches and the like. α would be the label under one of the little knobs on the control panel of the universe.

There are other little knobs required to control the other forces of nature. For example the *strong nuclear force* binds the building blocks of the nucleus (*quarks*) together. It has a strength which is proportional to the probability of a quark emitting or absorbing a *gluon*, the corresponding force carrier. This probability is called α_{s}, and is much bigger than α, with a value close to 1. Hence it’s name, it’s hella strong compared to the electromagnetic force.

**Fiddling with the knobs**

Imagine what could happen if you could reach out and turn one of these knobs, changing the fundamental parameters of the universe. The results would be something reserved for our wildest imaginations. Right?

In fact, this *has* happened in the history of the universe.

The fundamental parameters have changed on the cosmological time scale. In the past, not so long after the big bang, α was much larger than it is today, and α_{s} was much smaller. Atoms could not form, as the strong nuclear force couldn’t live up to it’s name. The universe was a different place that followed different rules. We don’t know much about this chapter of cosmic history, but we know it was hot, energetic, and chaotic, certainly with no sensible “structures” like stars, planets, or asteroid belts.

The universe has “phases” in the same way H20 does. At different temperatures, H20 exists as vapour, water or ice. Water freezing into ice is referred to as a *phase transition;* when the macroscopic nature of the substance changes.

Over time the universe has cooled from this rather unimaginable plasma into what we know today, via (probably) many phase transitions. For example, there must have been a transitional period when α_{s} became strong enough to pull previously free quarks together and bound them into nuclei.

**A Grand Unified Knob**

While I’m rolling with this analogy, there’s one last thing that is defo worth a mention. Physics today has managed to describe three (of a total four) fundamental forces of nature using this picture of force carriers and all that. Electromagnetism, the strong nuclear force, and the weak nuclear force. There are three corresponding numbers which quantify their relative strengths, α, α_{s} we’ve already mentioned, and a third that controls the weak nuclear force that I’ll call α_{w}.

All three varied throughout the age of the universe. We understand reasonably well how each varied. In the past α_{s} was weaker, and α, α_{w} were stronger. We can extrapolate the behaviour of these three numbers into the past using what we know about the forces and how they interact. Something special happens to the three numbers if we go all the way back to 0.0000000000000000000000000000000000001 seconds after the big bang.

At this time, all three numbers head towards the same single value. All the forces were the same strength. There is no a priori reason to expect this to happen, on the face of it it seems like a total accident. But physicists tend to think this is too special an outcome to be an accident. There must be something deeper afoot.

It’s led the particle theory community to ponder if all three of the forces originate from a single force, governed by what is called a *grand unified theory, *boiling down to a single number that controls it all. Perhaps the control panel is in fact a single dial that just says ‘universe’ on it. No direct evidence of such a theory has been found yet, but such evidence may be waiting just around the corner.

Fig.3: Strength of forces throughout history of the universe.

*Why do the strengths of forces change?*

*Could the strength of forces vary through space as well?*

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