Maths, and heavily mathsy subjects like theoretical physics, are full of seemingly unrelated ideas that are in fact deeply connected. As a consequence, there are many statements in these fields that can be expressed in radically different ways. Often throughout history, reformulating a problem into a completely new context via connections like these has been instrumental in the problem being solved.

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.

more on quantum fluctuations

more on quantum fields



more on wick rotation

cosmic phase transitions