Energy can never be created or destroyed. Heard that one before, aye? This is an example of a conservation law, the statement that some quantity will not change over time. In this case, that quantity is the total energy in the universe. Physics is full of conservation laws, conservation of energy, conservation of momentum, of angular momentum, of charge, the list goes on.

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.

more on Noether’s theorem



the mathematical details of gauge symmetry

details of diffeomorphism invariance: the cause of gravity