Conservation laws are powerful – often all you need to know about a process is what’s being conserved in order to make predictions. Many who did physics in school will remember calculating the final momenta of two objects that smashed into each other using nothing but conservation of momentum.

It’s tempting to think of conservation laws as fundamental rules in physics – conservation of energy seems like something that just *is*, just like Newton’s laws or the laws of thermodynamics. But actually, if you look hard enough, you’ll find that conservation laws are a product of some much deeper facets of reality. And uncovering the hidden insights that these conservation laws betray has been indispensable to modern physics, particularly particle physics.

Conservation laws are a product of the *symmetries* of nature. So when we use conservation laws to make predictions, we’re actually using these symmetries.

So what is a symmetry? Physics is all about using mathematical models to describe physical systems. We can think of two sides of the physics coin as the physical system we’re studying and the bunch of maths that we use to describe it. We say a system has a symmetry when we can rearrange the maths describing it without changing the physics. Let’s look at an example of a symmetry, and how it’s associated with a conservation law.

**Symmetry in Space and Conservation of Momentum**

Let’s imagine two physics nerds are having a game of pool. Since they’re physicists they’re probably really shit at it. However, they know how to model the game mathematically.

The first thing they need is a language to describe *where* each of the balls are. Easy, they can use coordinates. They draw a grid onto the pool table so they can describe the position of the ball according to which box of the grid it’s on.

Fig. 1: A ball’s position being quantified with coordinates

By doing this, we’ve made the table act like a frame of reference for the ball. Where the ball lies on the table decides where it is “mathematically”, i.e. what numbers we use to describe it’s position.

One of the nerds hits the ball, and it moves along at a constant speed in a straight line (assuming no friction or the like). Such a path has *conservation of momentum*. The momentum of a thing tells you both the speed and the direction of travel of that thing. So on this path, since neither speed nor direction change, we know the momentum doesn’t change throughout time. In other words, it’s conserved.

It will follow such a straight path regardless of where it is on the table. Imagine the ball taking exactly the same path through space, but the table is in a different position.

Fig. 2: 3 cases where the ball takes the same trajectory through space, but the table (therefore the way we quantify the trajectory) is different.

In each of these three cases, the “physics” is identical (the ball is taking the same trajectory through space), but the numbers the nerds use to describe it are different, since the ball goes through different squares on the grid.

We can think of the above picture as a symmetry – the position of the table (therefore the numbers describing the ball’s path) do not change the ball’s path. Seems obvious, but it’s actually an important observation.

Now I’m going to make a claim, then I’ll justify it to you. The ball has conservation of momentum (i.e. straight path of constant speed) *because* it doesn’t care where the table is, and what numbers are used to describe it.

To justify this, let’s imagine that the table is no longer flat, but warped. It has bumps and slants all over it. Now imagine moving the table around like before, does it still not affect the ball’s motion? No – now the ball cares about where it is on the table, and therefore what numbers are used to describe it’s position. As a result of this loss of symmetry, it no longer moves in a straight line, so no longer experiences conservation of momentum.

Fig. 3: Now the table is warped, and the ball cares about where it is on the table.

So, conservation of momentum was a result of this symmetry.

Of course, we don’t need a pool table to define a coordinate system, but the pool table served as an apt analogy. The point is that in the above case momentum was conserved only when the ball didn’t care what coordinate system was being used to describe it. An object that cares about the coordinate system will behave differently in different places, causing its momentum to change as it moves through space.

Symmetry: Physics does not care how you set up your coordinate system.

Resulting Conservation Law: Total momentum does not change over time.

We don’t need to know everything about the ball’s situation to derive momentum conservation, the height or colour of the table, only the symmetry (i.e. the fact that the table is flat). But momentum conservation tells you everything about the ball’s motion. So all we needed was the symmetry to make predictions.

This may seem like a super obtuse way of thinking about conservation of momentum. I mean, we all *know* that when we hit a pool ball it’s going to move in a straight line, that’s just common sense. And *of course* the laws of physics don’t change when we change coordinate system. If we just take momentum conservation as a fundamental law then we can do all the predicting we need. But if we understand the general relationship between symmetries and conservation laws, it’ll provide an extremely useful connection for more abstract situations.

With that in mind, let’s have a look at some other examples of this kind of connection.

**Symmetries of things and Conservation of things
**

Now let’s think about how we measure *time*. When we do some experiment that depends on measuring the time of certain events, those events don’t care about the reading on the clock. We could start the clock just before the experiment, or in 1970, and we’d still be describing the same physics. All that can ever matter is the difference between two times during the experiment. This is another symmetry.

This is the symmetry that produces conservation of energy.

Symmetry: Physics does not care how you measure time.

Resulting Conservation Law: Total energy does not change over time.

Another example is conservation of electric charge. Imagine a charge going around a circuit. The particle’s motion is dependant on the *voltage* across the circuit. The concept of a volt has a symmetry that causes the conservation of charge.

That symmetry is that you can change the number of volts something has without changing the physics. The only thing about volts that affect the motion of the charge in the circuit is the difference in volts between two points in a circuit, for example the two ends of a battery. If there are 5V at one end, and 15V at the other, the resulting voltage difference across the circuit would be 10V. That difference is the only thing that affects the motion of the charges, so it would be just as valid to say the one end of the battery had 25V and the other had 35V.

Symmetry: Physics does not change upon increasing or decreasing all voltages by the same amount.

Resulting Conservation Law: Total electric charge does not change over time.

This relationship between conservation laws and symmetries has been beautifully formulated in very general terms. Emmy Noether, the most influential mathematician you’ve never heard of, is responsible for it. She produced what could be thought of as a universal law of conservation. *Noether’s theorem* states that every symmetry in nature has a corresponding conservation law, and provides a simple mathematical formula for relating them.

Conservation laws are often all you need to make predictions. The pool ball’s motion in a straight line at constant speed could be predicted using just conservation of momentum. Now we know that those conservation laws come from symmetries, so actually, often *all we need are symmetries to make predictions. *This comes in very handy when you’re trying to study something very difficult to learn anything about, say, the entire universe on its most fundamental level…

**The Universe is Made of Symmetries
**

Gradually throughout the 20th century, the powerful tool of Noether’s theorem was used to explain all of the fundamental forces of nature with a simple list of symmetries, known as *gauge symmetries*. Let’s go through the forces and see what symmetries they correspond to. Things may get a teeny bit abstract, but don’t panic.

The first force to be truly understood was *electromagnetism. *This is the combination of electricity and magnetism, both of these are manifestations of this single force. It was realized that all of the behaviours of electromagnetism was a consequence of the following symmetry. The electromagnetic field can be modelled by a swarm of particles, the famous *photons*. Photons by themselves aren’t strictly observable, so in a sense are just mathematical constructs. In contrast, the electromagnetic field that emerges from the photons is a physical thing that can be measured. Photons are the maths and the electromagnetic field is the physics.

Since it’s only the electromagnetic field that can be measured, we have a freedom to move a bunch of the photons around, or remove some, or add some in, as long as those actions don’t affect the field that they describe. This is a symmetry. All of the properties of the electromagnetic force, including how your computer and phone works, how light works, all of it, can be derived from this symmetry. This symmetry is also connected the to voltage symmetry described above, since the density of photons is intimately related to voltages.

Fig. 4: Two different configurations of photons (left), both create identical electric field (right).

It turned out that it was not just the presence of the symmetry that’s important, it’s also the structure. What do I mean by structure? Photons behave in such a way that, if you pile some large number of them onto one point in space, the overall effect is the same as if there were no photons there at all. The density of photons at some given point in space then, as opposed to being just a number, should be represented by an angle, since if you pile “360 degrees” worth of photons into one position you end up back at the start, i.e., no photons.

Fig. 5: Quantifying the density of photons with an angle.

Therefore, the act of rearranging the photons is mathematically just like a rotation. If you rotate something 360 degrees it ends up back at the start, and if you rotate the photon filed 360 degrees you also end up with the same number of photons. The symmetry is mathematically the same as a rotation.

Let’s recast that fact into a simpler language. What else is unchanged when it gets rotated? A circle. If you rotate a circle any number of degrees, it will look completely unchanged. So the density of photons has the same symmetry properties as a circle. The understanding of this structure in the symmetry was important for understanding the other forces.

This brings us to the other forces. Once the symmetry responsible for the electromagnetic force was understood, it was wondered if other forces could come from similar looking symmetries. What if we imagined a swarm of particles similar to photons, that instead of having the symmetry properties of a circle, had the symmetry properties of a different shape? If you start from the assumption that your particles have the symmetry of a *sphere *instead of a circle, you produce a model for a new, more complicated force. In fact, this model exactly reproduces the behaviour of another known force, the *weak nuclear force*! That’s the force that governs radioactive decay.

This previously mysterious and poorly understood force becomes a simple modification of electromagnetism when we understand the symmetries that cause it.

The third force on our list is the *strong nuclear force, *the force responsible for holding the nucleus of the atom together. This too can be described by a symmetry, this time it’s the symmetry of a six-dimensional sphere. Similarly to before, without this symmetry the force’s behaviour seems bizarre, arbitrary and difficult to understand. But it all makes perfect sense if you explain it with this straightforward symmetry principle.

There’s one fundamental force left, gravity. Einstein’s theory of general relativity is a beautiful mathematical marvel that elegantly explains gravity. Seriously it’s totally worth going through a physics degree to understand this thing properly, it’s the most incredible idea anyone has ever thought. Anyway. General relativity fundamentally is a product of a special type of symmetry called *diffeomorphism invariance*. This quite philosophical symmetry deserves a whole post by itself to really get down to what it means, so won’t go into it here.

**The Universal Pool Table
**

Time for a summary. Conservation laws are powerful, they’re often all you need to make predictions. Conservation laws come from symmetries, so all of the physics boils down to symmetries. Turns out the most fundemental laws that govorn our universe comes from a simple list of symmetries.

Fig. 6: Mathematical names for the list of symmetries that govern the universe. U(1) is the symmetry of a circle and creates electromagnetism, SU(2) causes the weak force, SU(3) causes the strong force, and Diff(M) causes gravity.

But what is the *thing* that has these symmetries? Knowing these symmetries is like knowing that the pool table is flat, but what about all the other things? What is the height, colour, etc of this proverbial pool table that objects in the universe roll along? Sure, I was talking about photons before, but we don’t know anything about the photons besides their symmetry – where they came from or what they want. There are deep mysterious truths about the universe that we don’t need to know about to make predictions. The fact that we don’t need to know is entangled with another fun idea from theoretical physics, *renormalization.* I wrote an explanation of this, and you can do a read of it here.

Consider yourself learned.

]]>It’s a fair question. Society collectively pays for us to build big colliders and devote trillions of hours of computing time to finding the next fundamental particle. Particle physics is one of a number of fields one could call *blue skies research*, research that has no obvious useful application. Blue skies research mostly relies on funding from taxpayers, so in general people deserve an answer to “what is the point of all this?” The following is my attempt at an answer.

Firstly, I’ll just state that there is no argument about whether research *in general* is worthwhile. Your phone, computer, clothes, medicines, car, are all products of the collective elbow grease of the millions of scientists that innovated before you. If you want something more quantitative, most studies estimate investment in research by governments to return 20-30%, twice the average return on the equivalent investment is stocks [1].

Secondly, in arguing for blue skies research I’m not going to appeal to all this “noble quest to find the true nature of reality” bullshit. It’s nice that people see fundamental science that way, and it’s certainly an inspiring way to frame things, but it doesn’t count as a real argument for investment in such research. What is the value of finding the true nature of reality? It’s impossible to decide. Things like technological progress and the improvement of people’s lives however, is something one can realistically estimate a value for.

Also, to be clear, I’m not trying to say that blue skies is the *most* important type of research, rather that it’s one of many necessary parts of the science ecosystem. Since the share of funding for blue skies research is currently falling in general [2,3], blue skies fields are those most in need of advocacy.

I’ll go into three broad arguments below; the necessity for basic knowledge in innovation, our inability to predict the directions of investigation that turn out to be useful, and what makes blue skies research more important now than it has ever been.

**1: Innovation depends on Blue Skies Research**

The science of engineering is in a sense the backbone of civilization. And engineering is underpinned by the work of Isaac Newton. Newton’s laws, the laws that govern how object behave when acted on by forces, is an extremely general way of modelling any physical system. Newton’s work is quintessential blue skies; driven only by his desire to understand nature in a deeper way. His work did not result from efforts to build a bridge, yet most bridges build in the recent past incorporated Newton’s work [4].

The understanding of chemistry has been used on some level to produce everything we eat, wear and use. Similar to engineering, the world would look very different without it. Much of modern chemistry is deeply dependant on quantum mechanics, the product of an array of discoveries and formulations driven by curiosity.

If you’d like a more direct and concrete application of quantum mechanics, I give you computers. The fundamental building block of a computer is a transistor, something that would not have existed if it weren’t for our understanding of quantum mechanics.

I could go on indefinitely. The general point is that all applied research relies on the more fundamental work that preceded it. Pick up any piece of technology, medicine or whatever, and investigate all the knowledge that was necessary for its creation, and you will discover a manifold of insights born of curiosity.

A common analogy is that science is like a forest, and each tree is a specific discipline [5]. The base of the tree is the underpinnings of that field, and usually a result of curiosity. For example, Newton’s laws could live at the base of the engineering tree, or the theory of evolution at base of the biology tree. The branches represent more applied areas, and the leaves represent the benefit of the application, the real-world benefit to people’s lives. It’s the most important part, but could not exist without the branches or the base.

Some work has been done to quantify the impact of fundamental research on applied sciences. Let’s take biomedical science as an example field. One study [6] focussed on a number of medical breakthroughs from 1945-1975, and identified an “essential body of knowledge” that if absent, these breakthroughs could not have taken place. This body of knowledge was around 62% comprised of fundamental, curiosity-driven research. Another study [7], using different methods, produced a lower estimate of 21%. While one can quarrel about which to believe, both imply that some amount of blue skies research was essential to the breakthroughs in question.

Less investigation of this type has been done in other sciences like physics or engineering. But if you need some hard evidence for the fact that engineering relies on fundamental research, open an engineering textbook.

Science has followed this pattern of applications following fundamental developments throughout history, and there’s no reason to expect this to change in the future. The blue skies scientists of today are planting new trees that will grow leaves in the future. Fundamental research is a long-term investment, it will not produce immediate applications, but its value will be realized at some point down the line.

You may protest however that this only shows that *some *fundamental research is important, while much of it may end up being a waste of time. The next section addresses this worry.

**2: The Scattergun**

Imagine you lived in the year 1860, and you are tasked with designing a new type of candle. The candle must be brighter, more efficient and longer lasting. You start playing around with different mixtures of wax and different candle shapes. All the obvious stuff. Perhaps you do end up with an optimized candle somehow. But, a decade or so later, the light bulb will burst onto the scene, presenting a far better solution to your problem. That light bulb did not come from a rival trying to work on the same problem as you, it came due to far more general studies of the phenomenon of electricity.

Electricity was not invented to make better candles. Nuclear physics was not created to generate power. The fact is,it’s often impossible to predict where the best solution to a problem will come from. There will always be a huge forest of different disciplines, besides the most obvious, that may hold the solution to your problem amongst its leaves.

In the last section I referenced a paper that estimated 62% of essential prior work needed for medical discoveries that were curiosity-driven [8]. This study also estimated that 41% of those papers were not even clinically orientated research.

One way of looking at this is the fact that all of the disciplines are connected to some extent. The effect of a breakthrough in one field can shoot out to a number of other parts of science in a difficult to predict way. The quantum revolution in physics produced new innovations and discoveries in chemistry, engineering, computer science, and recently even biology [9].

Scientists in general have recently been becoming more interested in these connections, with more and more inter-disciplinary work being done involving a number of fields [10].

Areas that we think of as deeply abstract often turn out to have connections to applied science. What do set theory, number theory, graph theory, combinatorics, and abstract algebra all have in common? They are all indispensable to computer scientists [11]. No matter how removed from the real world some PhD project may seem, it likely either already has connections to more applied fields, or connections will emerge in the future.

In order to solve the world’s current problems, it’s not enough to just focus directly on those problems. We need to explore every avenue, every nook and cranny of the manifold of systems to study and ideas to explore. We need a scattergun of science. This has always been true to an extent, but is becoming even truer today, and will become truer in the future. This brings us to the final section.

**3. Challenges in the 21st Century
**

There are a lot of pressing and complicated problems on humanity’s horizon, and we really really need to get our arses into gear with solving them. None of the major carbon-emitting countries are on-target for fulfilling the Paris climate agreement [12,13,14,15]. There is no plan for mitigating for the coming rise of superbacteria or understanding of how automation will change society. Pandemics, dying bees, dwindling resources, asteroids, the list goes on and it’s scary as shit.

The problems of today are more complicated and challenging than the problems of the past [16]. Science suffers from diminishing returns; if any of these problems were easily and directly soluble, they would already have been solved. The problems of today are so complex that it’s more likely that connections to other fields of science, and strong contribution from fundamental research, will be necessary.

Take the problem of fusion power for example. If we can sustain a fusion reaction that can be exploited for a net energy gain, this would solve a lot of issues. There are indeed a bunch of technical engineering challenges there, but there are also some very fundamental problems, namely in plasma physics, that stand between us and fusion power [17]. It may also require new innovations from totally unexpected corners of science.

Another new facet of today’s looming problems in comparison to the past is their long-term nature. Problems like climate change are anything but immediate, the changes will happen over decades, and solutions will take decades to implement. Solutions will require the long-term investment that is fundamental research, research that may not have immediate uses but will improve our options 20 or 30 years down the line.

The climate is a very difficult problem for society to fully appreciate due to the insidious and abstract way it is creeping into our lives. It’s just as difficult to appreciate the importance of the subtle ways various scientific advancements will contribute to an overarching long-term effort against climate change, via the trickling of knowledge from fundamental to applied areas. The story is to some extent the same for all the other problems I have mentioned.

We must fire the scattergun of science everywhere in order to hit the solutions. We will need innovations from all areas, including blue skies research, to overcome the unimaginably complex web of challenges our world faces.

** **

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Example time. Many moons ago, Descartes showed that every problem in geometry can be reformulated as an equivalent problem in algebra. A geometry problem could be solved using algebra, or vice versa. The proof of a theorem in geometry may seem very complicated and arbitrary, but when expressed in terms of algebra, the proof becomes obvious and natural. The problem of *“can angles be trisected using a ruler or compass?**“* was solved using a straightforward proof in the language of algebra.

This problem was known for over a thousand years before it was solved. Throughout most of this time, algebra had not yet been developed – the natural language for the proof was not available. This is probably why it took so long for a proof to be found. It wasn’t *impossible* for a proof to be found using geometry, but in such a context, the proof is so far from obvious that it was never thought of.

There are countless more examples. In this post, I’ll show you a fun example of such a connection that is used a lot in particle physics. The connection is that *quantum mechanics* has the same underlying structure as *statistical mechanics**, *a completely different field of physics used for completely different things. Using this connection, the physics of *quantum fields, *the stuff that makes up everything in the universe, can be drastically reformulated to look like something completely different. In doing this, we can uncover some natural explanations to previously unnatural seeming phenomena.

**Quantum & Statistical Mechanics**

First of all, what is quantum mechanics? Quantum mechanics is the study of stuff that’s really small, like, way smaller than a safety pin or a grape.

At a bigger scale, like that of an Andy Murray, everything is in principle predictable. If the Andy Murray hits a tennis ball with his tennis racket, a physicist could predict the trajectory that ball would follow, using *classical* laws (classical just means *not* quantum). The trajectory of the ball is called its *classical path. *If the Andy Murray and his balls were shrunk to way smaller than a grape against his will, then it would be a different story. At tiny scales, there is no longer certainty of which trajectory the ball will take. There are other possible paths the ball can take. The new paths tend to be close to the classical path. These new paths are called *quantum fluctuations*. I go into a bit more detail about this here.

Fig. 1, Top: A conventionally sized Andy Murray hitting a normal tennis ball. The ball will follow a unique, predictable path. Bottom: A quantum-sized Andy Murray hitting a quantum tennis ball, now the ball may take a number of paths.

This is all we need to know about quantum mechanics. So what about statistical mechanics? Wikipedia tells me that “statistical mechanics is a branch of theoretical physics that uses probability theory to study the average behaviour of a mechanical system whose exact state is uncertain.”

Statistical mechanics is usually applied to large messy systems that are so complicated that it’s impractical for us to predict exactly how it will behave. Things like a car or jet engine, power generation processes, air conditioning, the understanding of all of these are underpinned by statistical mechanics. A nice easy example of a system studied in statistical mechanics is the air in a room. Any room contains trillions of air molecules, so we could never work out each of their trajectories individually. But we can apply statistical mechanics to understand averaged out quantities that describe the air like temperature, pressure etc.

In statistical mechanics, we accept a degree of uncertainty, since it’s too difficult to work out the system exactly. This isn’t strictly the same as quantum mechanical uncertainty, we make a practical choice to not retain all information rather than an intrinsic property of nature.

This uncertainty manifests itself as us not being certain which *state *the gas of molecules is in. In other words, we couldn’t predict with certainty something like the energy contained in the air. There is, however, a special case where that uncertainty disappears. To explain this, consider the following thought experiment.

If Andy Murray was in a room, and the temperature suddenly shot down to absolute zero, Andy Murray would die. Also, the air would become solid. The air is in it’s *lowest energy state*. At absolute zero, there is no longer uncertainty about which state the air is in, since there’s no energy around to help it turn into any state with higher energy.

Heat the room up a bit, however, and some of the air could potentially start to melt. A little bit of uncertainty has been injected. Andy Murray is still dead though. Most of the molecules are probably still in their lowest energy state, but the little bit of temperature means some may fluctuate away from that. These fluctuations away from the lowest energy state are called *thermal fluctuations*.

Fig. 2, Left: Air molecules frozen into a crystalline structure. At absolute zero, the molecules are fixed in place, one knows for certain the position of each molecule. Right: The air after cranking the temperature up a little. Now since some could be moving around, there is no longer certainty of where each molecule is.

**Quantum Mechanics = Statistical Mechanics**

You’ve probably already noticed some resemblances here. Both quantum and statistical mechanics describe fluctuations and uncertainty, so maybe they’re the same thing deep down? Problem is: while statistical mechanics is concerned with states of a system, quantum mechanics is concerned with how a system evolves over time.

To turn quantum mechanics into statistical mechanics, we must perform something called a *Wick rotation. *A Wick rotation is when you reach into all the equations of your theory and replace the time variable *t* with a new direction in space. This turns a universe with 3 directions in space and one in time, into a totally static world with 4 directions in space. (For those familiar with complex numbers, all it takes is multiplying *t* by *i*).

This has the effect of replacing a *path* through time that a system can take, with a *state*. The many paths that the tennis ball can take in quantum mechanics are replaced with a bunch of allowed states for our 4-dimensional system. The ball’s classical path turns into the lowest energy state of the 4D system, and quantum fluctuations around that path become thermal fluctuations due to the temperature.

Quantum mechanics = statistical mechanics. Pretty dope.

Now, what would happen if we apply this connection to the universe as a whole? If you do so, you’ll discover that the universe can be described by a huge 4-dimensional magnet of infinite extent in each direction.

**A Universe and a Magnet **

We can describe everything that makes up a universe using *fields*. A field exists at every point in space, at each point it has a certain strength, and you can assign a number quantifying that strength at each point. For example, the strength of a magnetic field at position *x* tells you how much a magnet placed in position *x* would feel the magnetic force. All other forces can also be described by fields. And the stuff that makes up matter, like electrons and all that, they too can be described by a field. This is because, loosely speaking, the strength of, say, the *electron field* tells us the probability of there being an electron at that point.

The state of the entire universe and all of its contents can be fully described using a bunch of numbers at each point in space, one number per field. I go into a bit more detail about this stuff here.

These are *quantum fields* by the way. They obey the laws of quantum mechanics, so they do quantum fluctuations and all that.

Now let’s get onto a magnet. Much of the important behaviour of a magnet (by which I mean a permanent magnet, the type that sticks things to your fridge) can be captured in a simplified model called the *Ising model. *The Ising model is a classic example from statistical mechanics, in a nutshell it describes the overall behaviour of magnets at different temperatures. The model can be pictured as a grid of points, at each point we specify a variable that we’ll call a *spin*. A spin can be either “up” or “down”. The spin represents the direction of the magnetic field being created by an atom at that point, we assume the strength of the field is the same for each atom, but the field can either be pointing up or down.

Fig. 3: The Ising Model. Each arrow (*spin*) represents the direction of a magnetic field created by an atom at that point.

The Ising model was created to describe the phenomenon that makes magnets magnetic. What makes magnets magnetic? It’s when the magnetic field of each atom lines up, and each magnetic field adds up to one huge magnetic field – this is the field that lets the magnet stick to fridges and all that. In terms of our Ising model, the lowest energy state needs to be one where all the spins are in the same direction. Accordingly, the model is set up so that a pair of neighbouring spins will have some interaction energy if they point in opposite directions, and contain no energy if they point in the same direction. Then, the lowest energy state is that with no interaction energy – all spins are lined up.

Fig 4: Spins next to each other that point in the opposite direction will contain some interaction energy.

This is a low energy state so the spins are all aligned only at a low temperature. Imagine you heated up the magnet to some higher temperature. This would cause some thermal fluctuations around the state of lowest energy. Some neighbouring spins would become misaligned. Heat the magnet up enough, and the nice uniformity of spins all pointing in the same direction will be gone, each will point in a basically random direction. All of the small magnetic fields will cancel each other out, and the overall effect will be the magnet falling off the fridge. Indeed, if you heated your fridge magnet to around 800 degrees C, it would no longer be magnetic. This is an example of a *phase transition*, just like when water freezes into ice or boils into steam.

Fig. 5: Two phases of the Ising model – Magnetic phase on the left, non-magnetic on the right.

**A Magnet Universe**

Now let’s modify the Ising model to turn it into a universe. Let’s assign a number at each site instead of a spin. Now our model looks a little bit like a field, except the picture is somewhat pixelated. We’re only specifying the field on a lattice of discrete points rather than in a continuous space. But with a small enough spacing between sites, the picture looks almost the same as continuous space. To include more than one field in our model, we just assign more than one number at each point, each representing the strength of a field.

Now let’s add an extra dimension to the model so we’re in 4 dimensions. Via the Wick rotation, we know that 4-dimensional statistical mechanics is the same as 3-space and 1-time dimensional quantum mechanics. And since our 4D statistical model can describe a number of fields, we know that it also describes quantum fields in 3 space and 1 time dimension – just like our universe. We’ve taken a model that describes a fridge magnet and turned it into a model of the universe.

Great! But what is this useful for? There are a bunch of applications of our magnet universe model. Quite a large chunk of particle physics depends on it actually. Below is just one fun example.

**Phase Transitions of the Universe
**

Now that we can apply the tools of statistical mechanics to the universe, we can ask a question like – can the universe go through phase transitions, like water freezing into ice? Yes – it looks like the universe has a number of possible phases, and it’s gone through maybe a number of phase transitions throughout its history.

To understand the universe’s phase transitions, we can apply our intuition from the magnet universe to the most famous of fundamental fields – the Higgs field. The Higgs field is the field that decides the masses of the fundamental particles. Loosely speaking, the strength of the Higgs field at a point decides the masses of any particles that are at that point.

Our Ising model that represents the universe is currently in it’s “magnetic” phase when it comes to the Higgs field. By this I mean the Higgs field has the same strength at every point in space, just like when the magnet had all spins pointing in the same direction. This leads to particles having nice regular unvarying masses, regardless of where they are. We can always rely on an electron having the same mass, and by extension a 10kg weight will always weigh 10kg, and Andy Murray will always be the weight of one Andy Murray.

This is a low-temperature phase that the universe cooled into over time. Billions of years ago the universe was hot and dense, and accordingly it was in a different phase. This was it’s “non-magnetic” phase, the Higgs field wasn’t regular, it just fluctuated wildly from point to point. A particle’s mass would vary through time and depend on where in the universe it was. As you would expect, a different phase of the universe is totally at odds with all of our experience from the phase *we* live in, unimaginable.

This also means the universe must have gone through a phase transition, similar to water freezing. It’s expected that these phase transitions would have left clues in the form of certain patterns of radiation, or even gravitational waves. Particle physicists and astronamers are currently working together to find evidence of such phase transitions.

Everything in maths, and by extension physics, is to some extent connected. Some see the connection between quantum and statistical mechanics as a useful tool, while others see it as a clue to some deeper truths about nature. Finding and understanding these connections that bind together the field will be key to solving many of the outstanding mysteries in physics today. I reckon so at least.

]]>The wave-particle duality is often loosely stated: “an electron is both a particle and a wave”. But what does that really mean? The following is an explanation of what a particle physicist means when they refer to a subatomic particle.

**Particles**

The best way to approach this is to follow the history of the concept of a particle. The idea of a “fundamental building block” most likely arose first around the 4th or 5th century BC. The philosophy that everything is made of discrete building blocks, known as *atomism*, came in contrast to the prevailing assumption that matter is continuous. You could chop a piece of string in half, then chop those two halves to get quarters, then eighths, and you could keep chopping indefinitely until you had an infinite number of string segments. The atomists believed that at some point, you would reach some kind of dead end on this venture, there would be a smallest possible unit of string.

Fast forward 15 hundred years to the 19th century. It’s my article so I can totally just do that. The 19th century was when the debate between atomists and non-atomists was finally settled. Chemists worked out that matter must be made of indivisible units, suitably called atoms. Not so long after that, it turned out atoms weren’t indivisible at all, they were totes divisible. They’re made up of smaller particles – electrons, protons and neutrons.

At this point, people imagined the electrons, protons and neutrons as little billiard balls. The balls could attract or repulse each other via forces like the electric force.

**Waves**

The electric force between the billiard balls could be described by a *field.* A field exists at every point in space, at each point it has a certain strength. The strength of the field at point *x* tells you how much a charged particle placed at point *x* would feel the electric force. The electric field is strong close to an electric charge (call it charge A). So if you put another charge (charge B) close to it, it will feel the electric force, leading it to being attracted or repulsed.

Figure 1: The electric field around a charged particle. The field is strongest in the dark areas, and weakest in the light areas.

If you were to move charge A, the electric field would also change. But it wouldn’t be a simultaneous change everywhere. Since no information can move faster than the speed of light, it takes time for the strength of the electric field far away to ‘update’ according to the motion of the charge. If you moved the charge then quickly moved it back to where it was, the field around it would have to change, then change back in quick succession. This change would move out from the charge, like a ripple on water, at the speed of light. Congratulations, you’ve just made an electromagnetic wave.

Figure 2: Ripple in electric field due to wobble of electric charge. Try out this applet to get more of a feel.

The opposite of this can also occur, a pulse moving through the electric field can hit an electron and cause it to wobble. This is the physical underpinning of radio transmission. A charge (the transmitter) is shaken to create a wave. The wave travels along and nudges another charge (the receiver). That nudge can then be translated into a message.

In the 19th century, two things made up the universe. Billiard-ball like particles (e.g. electrons), and forces they feel, described by fields (e.g. the electric field), in which waves could propagate. They were two totally separate things, that slotted together nicely to make a satisfying view of nature. Everyone was happy with this description of the universe, so no more discoveries were required. The end.

**Waves that look like Particles**

Then in 1905, Einstein ruined it all by explaining the *photoelectric effect.* It’s usually explained in a way that distracts from the central point it uncovers, getting all tied up in discussions about electron shells and the work function of materials. The main point behind it is this: remember those ripples you can send through an electric field by shaking a charge? There is a smallest possible ripple you can make.

If you tried to shake the charge half as vigorously as you did before, you would make a ripple half the size. Then half your efforts again, and you have a ripple a quarter of the original size. You would naively expect that you can keep doing this indefinitely, making smaller and smaller ripples, right? No way Jose. Einstein found that there is a lower limit to the size of a wave you can send through the electric field. Before, ripples in a field were considered to be like a continuous substance, but now it seems that all of the various wobbles and shapes that you get in the electric field are ultimately built up of indivisible packets of wobbliness. Thus, the *photon *was born.

The experimental setup included what was essentially a device that detected the light (a.k.a. ripples in the electric field) in a clever way. It turned out that this detector was being hit, not by a continuous stream of light, but by individual units of it. One way of looking at this could be that the detector was being bombarded by an array of little billiard balls called photons that collectively created the impression of light.

**Particles that look like Waves**

Around the same time this mindfuck was happening to the physics community, a problem was also found with seeing the electron as a billiard ball. The *electron double slit experiment* showed that a beam of electrons will behave like waves as they travel, even though when they reach the detector they are measured as individual particles.

Figure 3: Left – the double slit experiment if electrons travelled like particles. Right – if electrons travelled like waves. The dark patches on the detector represent where most of the electrons are hitting.

The experiment consisted of firing a bunch of electrons at a big detector, capable of recording exactly where an electron has hit it. Between the source and the detector was a wall with two little holes in it. If the electrons were particles, the detector would be hit with electrons around two regions just across from the holes. If you carry out this experiment however, you’ll see something different. The distribution of electrons will cover the whole detector, and build up an *interference** pattern,* exactly what you’d expect from two sources of ripples interfering with each other. Throw two stones into a pond at the same time, and look at what happens at the edges of the pond when the two sets of ripples meet, this is an interference pattern.

So both the electron and the photon actually have basically the same properties. They both travel around like waves, but appear as individual blobs when they are detected. This lead to the reformulation of our description of the universe: electrons, along with all other types of particle, are just wobbles moving through various fields. There is a field not just for every force, but also every species of particle.

A ‘particle detector’ in some experiment is really only reacting to the changes in a field. When a particle seems to have been detected, it is due to the smallest possible wobble in the field hitting the detector.

We can describe nature consistently by moving the emphasis away from both the particle and the wave, and just refer to fields themselves. The *standard model of particle physics,* which is thought to be our best formulation of physics at subatomic scales to date, is basically just a list of different fields and how they interact with each other. Fields are the more fundamental way of describing nature, and one can consider a particle/wave as a phenomenon that emerges from the behaviour of the underlying field.

By changing the language we use about the constituents of nature, we get away from the paradox of particle vs wave. That’s not to say that describing the electron as a particle or a wave is useless, in many situations it’s the most efficient way of formulating a problem. But, if you want to get down to what the universe really is deep down, you’ve got to talk about the fields.

**Particle or wave, it doesn’t exist anyway
**

I’m not saying the field is more fundamental than ‘particle’ or ‘wave’ just because it’s more elegant. There is a problem deep at the heart of the concept of indivisible blobs of anything.

This problem can be explained by appealing to general relativity, another one of Einstein’s achievements. A bit of a change of gear, but you’ll be fine. One could write a gazillion blog posts explaining what general relativity is, but for our purposes all you need to hear is this: two observers can disagree on distances and shapes.

Imagine two astronauts, Alice and Bob. Say Alice was in her spaceship, and she turned on the boosters in order to leave a vapor trail. She leaves a trail that, to her, seems perfectly straight. In the mean time, Bob is watching her from some different point in space, and since his *frame of reference* is different to hers, geometry works differently for him. To Bob, the vapor trail isn’t straight, it’s all curvy. Which one of them is right? Both of them.

Figure 4: The world according to Bob (left) and the world according to Alice (right). Bob has drawn three (straight according to him) axes so he can quantify positions in space. Alice has done the same. These are their *frames of reference.*

In general relativity, frames of reference can vary such that the shapes become subjective. This leads to a straight line in one reference frame being a curvy line in the other. The example I used here is a bit extreme, the effects of general relativity are usually a lot more subtle, but the picture is nevertheless possible according to our current laws of physics.

Ok general relativity lesson over. Now say Alice detected some particles, maybe in a cosmic ray or something like that. Say she detects an electron. Each of them are manifestations of ripples in some underlying field. But Bob detects nothing. His view of geometry is different to Alice’s, so what goes on in the underlying field is different. Alice seeing that ripple was dependent on the geometry she perceives. The geometry Bob perceives is such that it cancels out the excitation, see the figure below.

Figure 5: Top – the electron field according to Alice, there is a wave going through the field so she sees an electron. Bottom – the electron field according to Bob, the field is constant so he sees no electron.

In general, the way we define what a wave/particle is is inseparably related to one’s frame of reference. Two observers can disagree about whether a particle is present or not. One says that they have differing *particle concepts*, dictated by the geometry that they are experiencing.

It’s also possible, if someone’s reference frame was to change over time, that one single observer can change between different particle concepts. It can seem that particles are bursting into existence or disappearing, for no reason. This is the source of the famous phenomenon of *Hawking radiation,* a gas of particles that appear to radiate from black holes. These particles aren’t strictly being created by the black hole. Rather the natural way to count particles is different before and after the black hole forms, since one’s frame of reference would be warped by the black hole’s presence.

So as you can see, it isn’t really enough to explain physics with particles or waves at the end of the day. You can’t describe the state of the universe by listing all the particles/waves that it contains. If you only consider the field, you can have an objective view of the universe.

**Outlook**

That brings us up to what we think the universe is made out of today. Manifestations of wobbles in underlying fields, that don’t really exist if you think about it too much. There is every chance that, in the future, the field picture also turns out to be not the whole story. Maybe fields are a manifestation of something more fundamental, like strings, foam, discrete space-time points, or little rubber ducks.

*more stuff to read:*

*The Electron Two-Slit Experiment*

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Some of the particles on this list are shy, they very rarely interact with the others. An example is the neutrino. Roughly a hundred trillion neutrinos emitted by the sun pass through your body every second. You don’t notice it because it only extremely rarely interacts with what you’re made of, i.e., the electrons, protons and neutrons that make up the atoms and molecules that make up you. It may very occasionally bump into an electron in one of your atoms, making it do a little wobble, but this happens so rarely that it has an imperceptible overall effect.

What else could be going through you right now? Maybe there are other things we haven’t discovered yet, because they’re so difficult to detect. The following is my attempt at an answer.

**Outnumbered, five to one**

Neutrinos are notoriously difficult to detect for the same reason that we don’t notice them as they surge through us. They extremely rarely have any effect on atoms, or electric fields, magnetic fields, or anything like that. They were only just directly detected in 1956 when a sensitive enough experiment was conducted. It’s not a huge leap of imagination to expect that there exists particles that have an even weaker interaction with us, that to date has never influenced any experiment.

Actually we already have indirect evidence that such a thing exists. We know that the universe is in fact dominated by a mystery fog that interacts extremely weakly with us, commonly referred to as *dark matter*.

The first clues of its existence came from the unexpected motion of spiral galaxies, which could only be explained by a huge amount of invisible matter imposing a gravitational pull on them. It’s estimated that there is roughly five times as much dark matter as there is normal matter in our universe.

What is this mystery soup is made of? If we could see it with our eyes, what would it look like? Is it just a boring gas of particles flying around in space, or could it be something more? Could it clump together, and form structures like stars or planets?

In order for something like that to happen, dark matter would have to be complicated. It would probably have to be made up of not just one, but a number of different types of particle. While each of these particle types interact very weakly with us (electrons and all that) they must interact strongly with each other.

Think back to our list of particles and the strength of interactions between them. In this hypothetical scenario, our list could be arranged into two sectors, one containing the particles we are made of (a “standard sector”), and one containing the dark matter particles (a “hidden sector”). A particle will interact strongly with other types of particle in its sector, but vanishingly weakly with particles in other sectors.

Fig. 1: A visual representation of the list of particles in the universe and their interactions with each other. Each box represents a particle species. Each line represents interactions between species.

Let’s make explicit why we would need a complicated hidden sector to form dark clumps. The reason is that it takes more than gravity to clump things together. If you start off with a gas, the gravitational pull it has on itself will cause particles to shoot towards one another. But if they don’t interact with each other in any way besides gravity, they’re just going to shoot straight through each other. There needs to be another force that sticks them together.

The prevailing view is that dark matter does *not *interact with itself, so could not form clumps. From what we can tell from how it effects galaxies and all that, it seems to be made of very spread out clouds. If dark matter was capable of sticking to itself, surely over time it would have all collapsed into something more dense?

Noone knows for sure if this is the case, since it’s difficult to accurately observe the density of dark matter at any one place. There are also people that believe the opposite, that dark matter can and does clump together.

So that would be pretty dope if dark matter could form structures. Your imagination could run wild with what wonders there are in the hidden sector. Stars? Planets? Civilizations? And they could all be arbitrarily close to us, and we’d be none-the-wiser. Forget the boring neutrinos flying through the earth, there could be a planet going through us. There could be a dark Pret A Manger zooming through you right now.

**Outnumbered, infinity to one**

Now that I’m wildly speculating, and any real particle physicists reading this have stopped in disgust, I might as well continue. First of all, who says dark matter is made of only one sector? There’s 5 times as much mass in dark matter than our sector, so there’s room for a number of disjoint sectors, connected only via their interaction with gravity.

Now what about if there are sectors that don’t interact with gravity, or interact imperceptibly weakly? A sector that doesn’t even contribute to the weird behaviour of galaxies. There’s no bound on the number of sectors like that that could exist. There could be in principle an infinite number of overlapping worlds. Even if dark matter turns out to be a boring old gas, there’s nothing stopping one of the other completely invisible sectors forming dense structures.

Follow this line of thinking indefinitely, and you end up with the question – are there sectors that are *completely *separated from us, that have zero interaction with us, not even indirectly? This isn’t a scientific question, since no experiment, even in principle, could answer it. I have gone too far and should stop.

Let’s ask a more useful question – how can we ever hope to study these sectors with an experiment? One way could be via a theoretical property of the Higgs boson.

**The Higgs Portal**

Evidence for the existence of separate sectors is not just empirical. You can also apply well grounded theoretical reasoning for why other sectors can exist and why the familiar particles cannot interact strongly with those other sectors. How this reasoning works is a whole other story for another time.

But there is one known particle that is different, that in principle could interact strongly with other sectors. There’s no theoretical constraint stopping it from such interactions. This is the Higgs boson. In 2012 the Higgs was discovered, and there was much rejoicing. The Higgs is part of a mechanism that gives mass to all the particles in our sector (except the particles in our sector that we know not to have mass).

The Higgs could be strongly interacting with other sectors. It is possible that, not only does the Higgs give our own sector mass, but it gives other sectors mass also. Alternatively, there could be a family of Higgs like particles, one belonging to each sector, but interact strongly with each other.

Fig. 2: A hypothetical list of particles, including a Higgs that can interact with many sectors. Each box represents a particle species. Each line represents interactions between species.

In this case, the Higgs could be a stepping stone towards accessing the hidden sectors. It would be like a portal to other worlds. It is possible that we could study the properties of a hidden sector via the Higgs in one of our current experiments, like at the large hadron collider at CERN. Physicists are currently working on a better understanding of the Higgs, and with that better understanding may come hints of hidden sectors.

If there were other civilizations living in different sectors, could we communicate with them via the Higgs portal? They could be very close to us, so the finite speed of light wouldn’t be a problem like it is for aliens in our sector. Ok I’ve gone too far again. Time to stop.

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The LHC is arguably the best tool we have currently for discovering new particles. According to the old chestnut of *E=mc ^{2}*, energy can be converted into mass, and vice versa. When protons collide, the energy in the collision can be converted into new particles. The more energy you throw into the collision, the heavier the particle you can make. It’s likely that many of the particles that we haven’t discovered yet are heavier than the ones we know already. That’s why the LHC needed to be mighty big and mighty powerful, so the collisions could create new particles never before seen.

The new particles will only last a fraction of a second before decaying into more familiar ones. But by studying the aftermath of the proton collisions, we can find footprints of these undiscovered particles, therefore peeling back more layers of reality and peering in.

Figure 1: a cross-section of the ATLAS detector at the LHC. Think of the beam of protons to be coming out of the page. All the words will be explained below.

But what’s actually going on when those protons collide? And, more importantly, how can we use said aftermath to learn that juicy new physics? Before jumping into these two questions, we need a little primer about hadrons.

**Tell me what a Hadron is**

Yes I’ll tell you what a Hadron is. New heavy particles will often decay into hadrons, then it’s the hadrons that we detect. To explain hadrons I must first explain the *colour force.*

The colour force is the force responsible for holding together the constituents of the proton: *quarks* and *gluons*. Quarks are the meat of the proton, and gluons glue the quarks together. The colour force is more often called the *strong nuclear force*, but this is a bit of a confusing historical hangover.

There are two things I’d like you to know about the colour force:

*Thing 1*

The strength of the colour force varies depending on the energy of the particles that feel it.

Just after the big bang, the universe was extremely hot, and all the particles contained huge amounts of energy. The colour force was weak at this time. As a result the universe was made of a soup of freely moving quarks and gluons. Over time as the universe cooled, and particles started to mellow out, the colour force became strong and bounded quarks and gluons into tightly bound clumps. We call these clumps *hadrons*. Protons and neutrons are examples of hadrons, but there are many more kinds.

The behavior of quarks and gluons is well understood, but only at high energies. The theory describing their interactions at high energies is called *quantum chromodynamics *or *QCD*. It’s an extremely simple and elegant theory, explaining a veritable smörgåsbord of phenomena with just a tiny number of parameters and a single equation.

At low energies however, it becomes difficult to explain everything in terms of quarks and gluons, we don’t have a good understanding of how they behave in bound states. We can however forget about the quarks and gluons and just treat hadrons as the fundamental particles. The theory that goes with this is, in comparison to QCD, quite messy and unappealing. But hadrons are where the real physics is, since we can only ever do experiments on hadrons. If you want to do an experiment directly on quarks and gluons, the detector you design better be mighty small (smaller than a proton) or be able to withstand mighty high temperatures (like, literally a bajillion degrees).

There is a grey area between high energies and low energies, when it is neither right to explain things in terms of quarks and gluons, or in terms of hadrons. How exactly did the individual quarks and gluons turn into hadrons at the beginning of the universe? That is very poorly understood.

Some particles are completely immune to the colour force, so the above discussion does not apply to them. Particles like the humble electron exist as a concept no matter what energy it has. We refer to these particle that don’t feel the colour force as *leptons*.

*Thing 2 (a consequence of thing 1)*

At low energies, it is impossible to see a quark or a gluon by itself.

Say you went back to the big bang, harvested a single quark in a jar, and brought it back to the present day. As it cooled down, it would emit a whole bunch of gluons and other quarks, resulting in not just a single quark but a soup of quarks and gluons. Then, as the soup cools down, and the colour force becomes strong, all the inhabitants of the soup will bind together into hadrons. If you tried to pull one of the hadrons apart into its individual quarks, those individual quarks would just immediately emit more soup and form into hadrons again.

Figure 2: What would happen as a single quark cooled down after being transported to the current day.

You now know all the things about the colour force. At high energies, nature is described by individual quarks and gluons interacting. The colour force is weak so no binding together, no hadrons. At low energies, everything is bounded into hadrons, everything can be described by hadrons alone.

Now we can start to talk about what happens when protons collide at the LHC.

**The Proton Collision
**

There are millions of billions of protons passing each other every second when the LHC is turned on. Most of the time they just shoot past each other, but occasionally (on about every trillionth pass), the protons will collide.

The protons are given lots of energy when they are accelerated around the ring, so when they collide, we need to be thinking about quarks and gluons rather than hadrons. Quarks and gluons from one proton will interact with quarks and gluons from the other, which produces a bunch of other particles. If you’re lucky, one of the new particles will be one that has never been seen before.

Figure 3: A proton collision involving a short-lived Higgs boson.

A new mysterious particle won’t last very long, it’ll be around for a tiny fraction of a second before it decays into something else. This is true almost by definition, a particle we haven’t seen before can’t be something that hangs around after it is produced – otherwise there would be loads of them just lying around and getting in the way.

The new particle will inevitably decay into particles we already know. These familiar particles will shoot off away from the event, and smash into one of the detectors (which we’ll get onto later). In order to work out what happened at the proton collision, we need to be able to work out what particles emerged, and their initial trajectories. More on this later.

So what will come out of the collision? Sometimes it will create leptons, the particles resistant to the colour force. These will likely travel undisturbed until they reach the detector. Since the reaction is bursting with energy, it can also create individual quarks and gluons. In this case, it can’t be as simple as a quark shooting off and hitting a detector. An individual quark could never reach the detector, as by that time it would have cooled down to low energies, and at low energies quarks are no longer a thing. Somewhere along the way, that lonely little quark must somehow become part of a hadron.

**Jets
**

We have a single energetic quark, and before it gets anywhere near the detectors, it’s going to turn into a bunch of hadrons. As I said, the grey area between quarks/gluons and hadrons is not very well understood. We want to be able to deduce the presence of that quark from the hadrons that hit the detectors, but the presence of this grey area may give you the impression that it’s an impossible task.

Luckily, there is a something about quarks and gluons that will make this problem easy. Easyish. Consider the very first thing the quark emits: it’ll probably be a gluon. The theory of QCD can tell us the probability of the emission:

*P _{emission} ≃ 1 / E*θ

where *E* is the energy inside the gluon, and θ is the angle between the two particle’s trajectories. Look at this equation a little bit and you’ll see that the most likely angle θ of an emitted gluon is very small. In other words, the gluon most of the time ends up traveling in basically the same direction as the quark. It’s also apparent that it’s most likely for the gluon to have a small energy. The importance of this I’ll get onto in a minute.

Figure 4: a quark emitting a gluon.

This won’t just apply to the first emission. Either the quark or gluon could go on to have another emission, of a new quark or a new gluon. In that case, our equation above can be used again. This will lead you to the conclusion that the vast majority of new particles will travel in the same direction as its mother, and will have a low energy.

Figure 5: A jet.

This results in some broad statements that can be made about the end products of our original quark. To start off with, all the quark/gluon soup resulting from the quark will be concentrated in a narrow beam. Since new particles in general have smaller energies, the soup quickly approaches the low energy regime where it will clump up into hadrons, and these resulting hadrons will also be moving in that one specific direction. The result is what is called a *jet*, a narrow beam of hadrons. The detector will eventually be hit with a bunch of hadrons all clustered in a small area.

**Detectors
**

So what about these detectors then. There are a number of locations around the LHC where detectors are placed. Each is designed slightly differently and tuned to spot different things. The largest and most famous of them is called the ATLAS experiment, so we’ll focus on that as an example and well you know, fuck the rest of them.

ATLAS is a veritable onion of detectors. It’s basically a cylinder that encloses the beam of protons with a number of layers of detectors. Each layer is a different kind of detector, specialized to detecting certain types of particle.

The only things that survive long enough to get to the detectors are hadrons and leptons. The first layer samples the energy of leptons as they interact with the electrically charged particles in the detector. Successive detectors measure the energy of hadrons as they interact with the nuclei of atoms in the detector.

It’s not just the energy of the particles we can measure, we can also work out exactly where the particle hit the detector. ATLAS can be thought of as a rather expensive 100 megapixel camera.

There are some types of particle, for example the *neutrino*, that are well sneaky and will shoot straight through all the detectors unnoticed. The presence of these particles can however be deduced, using a little thing called conservation of energy. We know how much energy is in the initial protons (since we gave them that energy), and we can measure the energy in all of the gunk that hits our detectors. If a neutrino escapes detection, then there will be energy missing between the initial protons and the energy measured by the detectors, from that we can deduce the presence and energy of the neutrino.

**From Hadrons in the detector to Quarks at the collision**

So, imagine we have measured the energy and location on the detector of a bunch of hadrons and leptons that came from the proton collision. We need to deduce exactly what happened just as the protons collided, to see if anything new and exciting happened; the production of a shiny new particle perhaps.

For leptons, it’s pretty easy to trace things back. An electron doesn’t tend to do much as it flies through space, so if we know where it hit the detector and how fast it was moving, we can just extrapolate backwards to work out how it emerged from the collision.

Hadrons are, as you now know, a different story. The detectors can receive hundreds of hadrons from a single jet. We use things called *jet finding algorithms*, these are an attempt to deduce what quarks flew out the collision given the hadrons hitting the detector. It is a highly not-so-easy problem, since we don’t really understand that grey area between quarks/gluons and hadrons. The original attempts amounted to just adding up all the energy picked up from some region of the detector. The most popular algorithms of recent days involve attempting to trace back the process step by step, emission by emission.

These more recent algorithms are designed according to the equation we had for the probability of emission, *P _{emission} ≃ 1 / E*θ. Hence, the algorithm decides that two particles came from the same mother particle if they are traveling in a similar direction, and at least one of them has a low energy.

Now that we’re talking about the practicalities of applying this equation, there’s something about it I didn’t bring up before that we now have to pay attention to. You may feel slightly unnerved by the possibility of a gluon coming off a quark that has zero energy (*E=*0). Or a tad unhinged by a gluon moving in exactly the same direction as the quark (θ*=*0). In either of these cases, *P _{emission} *is infinite. It doesn’t make any sense for a probability to be infinite, probabilities must be between 0 (definitely won’t happen) and 1 (definitely will happen).

If we don’t pay attention to these infinities, they will mess up our jet finding algorithms. We need to understand what is going on. There must be some deep physics reason this is happening, right?

Let’s deal with the θ*=*0 problem first. The situation it describes is a quark emitting a gluon moving in exactly the same direction. Due to conservation of energy, the energy contained in the final quark/gluon pair is the same as the energy that was in the mother quark. Think about this in terms of where the energy is: a little point of energy (the quark) changes into a point containing the same energy (the quark and gluon). As far as we’re concerned, this is indistinguishable from the outcome of the quark emitting nothing, since the outcome will be the appearance of a hadron in exactly the same place containing exactly the same energy. So actually, this kind of event in a sense “doesn’t exist”, we don’t need to include the possibility of this happening in our jet algorithm.

It’s a similar story for the *E*=0 problem. If a gluon with no energy is emitted in the woods and no one is around to hear it, does it make a sound? In this case no, that gluon can never be seen by the detectors, and it won’t ever contribute to the creation of any hadron. So that event is just the same as no emission at all.

The algorithm must be designed so that it does not take these θ*=*0 and *E*=0 possibilities into account. It’s safe for an algorithm to completely ignore these possibilities, since they don’t exist.

The deep physics shit going on here is this – some things about quarks and gluons are unknowable. As a result, thinking too hard about the quarks and gluons inside the jets themselves leads you to nonsense. Asking something like “how many gluons are inside the jet?” is a nonsense question, there exists no answer. It’s a question we can never answer with any experiment (since we can never measure a gluon by itself), and it can never be predicted theoretically. Quarks and gluons are very much mathematical concepts, while it’s only really the hadrons that have a solid physical interpretation.

**How to Find the Higgs Boson**

Plugging the energy and location of detected hadrons into jet algorithms, we can deduce how many quarks initially emerged from the collision, their direction of travel, and their energy. By tracing back the trajectory of the leptons that hit the detector, we can also deduce the number of leptons produced by the collision, direction of travel and energy. Our task now is to translate this knowledge into information about the collision itself.

To do this we need to ask: what possible events at the proton collision could have produced these outgoing quarks and leptons? I’ll refer to a specific combination of quarks and leptons, with some given directions and energies, as the collision’s *final state*. Usually there are not one, but a number of possibilities that could have resulted in our deduced final state. Some of those possible events will only include boring old familiar particles. These possibilities are referred to as *background.* A possibility that includes the production of an undiscovered particle is called the *signal*.

We can predict how often a background event will lead to our given final state, since we already understand how all the particles in a background event work. Therefore, if our final state occurs more often than we expect, this is evidence of a new particle – the signal is shining through. The final state is appearing more often than expected because there is a “new unexpected way” that the colliding protons can produce that given final state.

Figure 6: Bump representing a new particle. “GeV” is just a measure of energy that particle physicists use.

The probability of a given final state varies with the total energy of the final state, i.e., the energy of each outgoing particle added up. This can be seen from fig. 6. We can predict the frequency of our background events at each energy, defining the dotted curve on the plot. Most of the time the observed curve from the LHC agrees with the background curve. But if there’s a bump that cannot be explained by the background, this would be evidence of a new particle.

Moreover, the energy at which the bump appears is important. To see this, consider that the probability of the final state occuring will be roughly proportional to the number of different ways the final state can be created. The creation and destruction of a new particle represents a new way that the final state could be created, so, when the right amount of energy is involved to make the new particle, the probability of the final state increases.

From fig. 6, it looks like there’s an “extra way” to create a final state of energy 125GeV, this extra way is a new particle being created and destroyed. Via *E=mc ^{2}, *we can work out what mass the new particle would need to have to create a final state of that energy. By dividing 125GeV by

Figure 6 is essentially a cartoonist impression of one of the plots used to discover the Higgs. You can have a look at the real plot in the original discovery paper, on page 10, figure 4.

The discovery of the Higgs was a huge success, but the search for new particles at the LHC is far from over. Physicists are still squinting away at plots like the above, hoping to find the next bump.

He could also use his physics knowledge to turn the clocks back, and deduce, given the state of the universe at some time, the state it had at some earlier time. If you wanted to destroy a document containing information you’d rather no one ever find out, and, say, burned it, you still wouldn’t be safe. The demon could look at the smoke coming off the flames, and use it to deduce what was on the page.

Laplace told this story in order to convey the idea that

“We may regard the present state of the universe as the effect of its past and the cause of its future.”

This seems like a pretty sensible way to view nature to most physicists. The universe is in principle predictable. If it wasn’t the case, what’s the point in physics?

Fig 1: Given everything that’s happening at the present, one can in principle predict the future or deduce the past.

I’m going to tell you about a recent(ish) strange discovery that causes problems with this way of thinking. It concerns the bat-shit behaviour of black holes, and is referred to as *the black hole information paradox*.

**Entanglement**

We first have to understand a wee bit of quantum mechanics. The main thing about quantum mechanics is that physical things can exist in *superposition*. This is when the system exists as a mixture of different states that we usually would consider to be mutually exclusive, i.e., it only makes sense if it’s in one or the other.

For example, consider a single particle flying along through space. It can exist as a mixture of, say, an electron and a positron (the positively charged version of the negatively charged electron). It could be just as much positron as electron, or mostly electron and only a little bit positron, or the other way around. The *quantum state* of the particle can be encapsulated in one number ψ, telling you where it lies on the spectrum between electron and positron. ψ = 0 means it’s an electron, ψ = 1 means it’s a positron, ψ = 1/2 means it’s half and half.

What do I mean when I say it’s a mixture of electron and positron? Imagine the particle hits a detector, which can be used to deduce its charge. When it hits the detector, and the reading comes up on a screen, it needs to make up its mind. The chances of the detector registering a positron is ψ, and the chances of it registering an electron is 1-ψ.

Now let’s complicate the picture a little. Let’s say there are two such particles, call them A and B, which both sprang from the breaking up of some original particle. The original particle had zero electric charge, so the charge of A and B need to add up to zero. Both are in a superposition, both a mix of electron and positron. But, the requirement that their charges add up to zero limits the quantum states they are allowed to have. If particle A is an electron (negative charge), then B must be a positron (positive charge), and vice versa. They can’t both be electron or both be positron, as that would mean the overall charge not adding up to zero.

Both particles have a number specifying their quantum state; ψ_{A} and ψ_{B}. But this time, due to the requirement of overall zero charge, ψ_{A} depends on ψ_{B} , and vice versa. You need to know what ψ_{B} is to know what ψ_{A} is. A and B are said to be *entangled. *

If you left particle B out of the picture, then the quantum state of A is not well defined. It would seem like there is information missing from its quantum state, that information is being held hostage by particle B.

Let me elaborate on this a little to show what I mean by missing information. If we told Laplace’s demon the quantum states of A and B (i.e. the values ψ_{A} and ψ_{B}), he could use the laws of quantum mechanics to predict exactly what their quantum states would be at some later time. However, what if we were only interested in particle A? What if we wanted to only tell the demon the quantum state of particle A, and ask him to deduce its quantum state at some later time? This couldn’t be done since particle A has information missing from its quantum state, so he couldn’t work out what would happen to particle A in the future. If the demon can’t see particle B, then his powers of perfect prediction are lost.

Fig 2: If you only know about particle A at time 1, this isn’t enough to predict its state at time 2. Only if you know the state of both particles at time 1 will you be able to predict either’s state at time 2.

This is kind of weird, but it doesn’t get in the way of Laplace’s belief. As long as the demon is given all the information available in the universe at a given time (which includes the states of both particles A and B) he can make perfect predictions of the future and deductions of the past. However, what if there was a way to, not just ignore the information in particle B, but physically destroy it?

**Evaporating Black Holes**

Ok, black hole 101. When a star dies, it collapses under its own gravity into a very dense and compact object. Some of the more massive ones will collapse into something that’s almost infinitely small and dense. Such a thing is called a *singularity*. Its gravitational pull will be so strong, it will prevent even light escaping from it. Get too close to it, and it becomes physically impossible to escape. You can imagine a sphere around the compact object signifying the point of no return, this is called the *event horizon*.

The weird nature of strong gravitational fields can make particles seem to be created out of nowhere. At the event horizon, particles appear in pairs. One flies outward, away from the black hole, and the other falls inwards toward the singularity. These pairs are entangled in a similar way that particles A and B were entangled. The quantum states of the particles radiating out of the hole are dependent on the state of those falling into the hole, who end up hiding behind the event horizon.

The black hole is always radiating these entangled particles, an effect referred to as *Hawking radiation*. If something is constantly throwing out energy, it will eventually run out of energy, and disappear completely. The black hole will evaporate leaving only the outgoing radiation as evidence of its existence. Information about the radiation’s quantum state, that was being held inside the black hole, has now been obliterated. Could it have somehow escaped before the black hole disappeared? No, it’s impossible for anything to cross the event horizon from the inside to the outside.

We are left with only a cloud of radiation that has a poorly defined quantum state. In fact, it is extremely poorly defined. Since it was so strongly entangled with the interior of the black hole, it contains almost no information. Compare this radiation to the radiation coming from a star (light, radio waves etc). If Laplace’s demon could collect up all the radiation from a star, it could deduce exactly the nature of all the reactions going on inside the star that led to the emission of the radiation. This is because, while the radiation seems random and messy, there is in fact subtle features hiding in it, delicate interactions between the constituent particles that can be used to deduce the nature of their origins. In this sense, the light from a star contains information.

Hawking radiation is not like this. It contains virtually no information, it doesn’t just look messy and disorganized, it is *intrinsically* messy and disorganized. The demon could collect up all the radiation left from the black hole, but he couldn’t deduce anything about the black hole from it.

**The Information Paradox**

Remember that document you really wanted destroyed, so no one could ever see, or even deduce, the information on it? Throwing it into a black hole seems a sure-fire way of doing that. Any information that falls into a black hole is permanently erased, since the end-state of a black hole is Hawking radiation containing no information.

The current laws of physics, or even any conceivable law of physics we could come up with in the future, are powerless to deduce what was going on before the black hole formation, even given the exact state of everything after the black hole evaporates.

Fig. 3: If you know everything at time 2, this will not be enough to deduce the information on the incriminating document at time 1, since all you have is Hawking radiation carrying insufficient information.

This also causes problems in the opposite direction in time. It seems likely at the moment that the fundamental laws of physics are symmetric in time, i.e., behave in the same way going both forward and backward in time. A video of the moon orbiting Earth would look just as sensible if played in reverse, since the equations governing gravity look the same if time is reversed.

If this is the case, then the laws of physics must allow the reverse of black hole evaporation to take place, i.e. fig.3 but flipped upside-down. Such a thing may never have happened in the history of the universe, and may never happen in the future, but the point is that such an event is allowed to happen in nature. This event would consist of radiation clumping together to produce a reverse-black-hole, and totally unpredictable things falling out of it. Our knowledge of the universe before the creation of the reverse-black-hole would not be sufficient to predict what would fall out of it. It could be anything, a sperm whale or a bowl of petunias for all we know, and no law of physics could ever tell us why they appeared.

Again, this type of thing may never happen, but the fact that our current laws of physics seem to allow this type of thing is deeply troubling to physicists. If information can be destroyed in a process like the above, who’s to say there isn’t a plethora of other possible processes in which information is destroyed?

Is it really true that the universe is fundamentally unpredictable? The debate has been ongoing since this problem was first uncovered in the 70s. A number of solutions to this problem have been proposed, for example, modifying the physical laws to let the information in the black hole somehow escape. So far none of the solutions have been conclusively shown to work, so the debate continues.

Some of the most notable attempts at a solution include: *black hole complementarity,* the existence of *firewalls* at the event horizon,* *an appeal to the principle of *holography *from string theory, and most recently, the theory of *supertranslations.*

We may be a long way from solving this problem, but I suspect when it is finally solved, it will come with some dramatic overturning of some of the most deep-rooted ideas in physics today.

Oh yea one more thing, this whole discussion can be formulated in terms of *entropy*. I wrote an article about entropy and its connection to information, which will appear on your screen if you click here.

An entangled particle is not just missing information from its description, it is intrinsically missing information, which is being held by its entangled partner. The particle has an associated entropy due to this missing information, called *entanglement entropy*.* *

Hawking radiation is intrinsically missing information, in fact it is missing any microscopic information, so it is a “physical macrostate*“*. It’s not just a gas that you can give a thermodynamic description of, it is literally *only* thermodynamic! Hence, all you can deduce about the black hole from its radiation is its temperature, total energy, stuff like that, nothing else. Universe be crazy.

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. Galileo’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 is 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 millimetre 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?

]]>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.

]]>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.
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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 *Boltzmann 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 Boltzmann 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 on 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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