One of the joys of studying the science of the universe is using lots of lovely equations that can tell you basically anything you want to know. How far can I kick this football? How long will my phone battery last? How many copies of Chalkdust can stop this bullet?
A slightly more fundamental question might be asking how all of these formulae came about. Perhaps even more fundamentally, why do we expect certain equations to work as they do? Most people would be satisfied giving the answer ‘they were introduced in my high school physics class, and therefore they work!’ I am not most people.
This article aims to answer the latter question by unifying maths and physics via group theory. Group theory is built on the idea of mathematical symmetries, whereas physics is the study of describing and explaining observations using equations. One might naturally ask what happens when we observe a symmetry and want to put some equations behind it.
What is a symmetry?
The first of many beautiful theorems a mathematical physicist learns in their career is Noether’s theorem: any observed symmetry in your favourite physical system has a 1:1 correspondence with a specific conservation law. From this, we can build equations that describe the motion of everything in that system: fields, particles, classical balls, you name it.
But what is a symmetry? Fear not; this is not a mathematical term we throw around lightly and try to link it to something completely unrelated—like rings or holes. A mathematical symmetry is a transformation that leaves an object completely unchanged. Take a simple square; what transformations can you perform on this square that leave it unchanged?
You could rotate the square 90° and it would look exactly the same. Rotating it another 90° leaves it visually unchanged too. And the same for two more 90° rotations. This leaves us with four rotational symmetries of the square:
But there are more! If we also consider reflections of the square, there are four ways in which this can occur. This leaves us with four reflectional symmetries:
Therefore, the collection of symmetries of the square has eight transformations: four rotational symmetries and four reflectional symmetries. Now that we have discovered all these symmetries, we can collect them into a mathematical group.
Groups in mathematics are collections of symmetries where combinations of symmetries should also exist in the group. In the case of squares, a rotation followed by a reflection is just another type of reflection. Evidently this is in the group, and so this condition is automatically satisfied. We call a group a Lie group if it behaves smoothly.
The case of squares forms the dihedral group of degree 4 (since a square has four corners), denoted $D_4$. In fact, the symmetries of any regular $n$-gon form the dihedral group of degree $n$, $D_n$, which will always have $2n$ elements. The dihedral groups are unfortunately not Lie groups—this can be seen by the fact that squares have corners and the above transformations would not be smooth. The rotational symmetries of a circle, however, do form a Lie group: it is denoted $\operatorname{U}(1)$.
Enough about shapes though. We’re here to talk about the universe. The universe exhibits symmetries of its own. In the context of physics, these symmetries leave the laws of physics unchanged. If you perform an experiment at one place at a certain time, you should hope to obtain the same observations at a different location at a different time.
Classical symmetries
We would first like to interpret a physical symmetry in an empty universe; one in which we assume no spacetime curvature or air resistance. We know nothing about the laws that describe motion in this space, but we should hope to find them using our intuitions. Suppose I throw a ball away from me while I am standing in one fixed position. It will travel in a straight line, at a constant rate, never interfering with anything and thus never stopping.
Throwing a ball from two platforms in an otherwise empty universe.
It should come as no surprise that if I move the platform I am standing on to a new position and repeat the experiment of throwing the ball away from me, the same exact motions will occur: the ball will travel in a straight line, never stopping. This tells us that the physical laws governing the movement of the ball, whatever they are, should be invariant under a spatial translation. That is, if I translate my initial position through space, the ball still has precisely the same governing motion.
Another way of viewing this is by considering each time step of the ball’s movement. A time step is a regular measurement of time; whether that be a minute, an hour, an aeon, or just one unit. When I release the ball it is in its zeroth time step. After one unit, it is in its first time step and has moved position according to its motion law. The same thing happens for the second time step. If we compare the first and second time steps, these are identical situations in which the ball has moved its position and nothing else.
But remember what we said before: the equations of motion don’t change as the ball changes position. So, at every point in time, the ball keeps its motion. This observation is what Noether identified as the conservation of momentum—any physical system with movement will have a total amount of momentum, usually labelled by a number, and this number is kept the same throughout\hfill the entire time evolution of the system. Momentum cannot be created nor destroyed and is a direct result of the spatial translation symmetry of our empty universe.
Gaining momentum on a swing.
In a similar fashion, this empty universe admits a rotational symmetry, which means the laws of physics are invariant when the entire system is rotated about an arbitrary angle. Imagine now you are spinning the ball on the end of a string; almost as if it is orbiting your head, for example. No matter which orientation or which direction you decide to initially set the ball spinning, you observe that it carries on spinning for all eternity. Remember, there is no gravity or air resistance. The key point is that the equations of motion governing the ball’s motion are the same at each point in its orbit.
To see this, we can use the same timestepping argument. After spinning the ball, it enters its first time step and rotates according to its motion law. After the second time step, it obeys the same law and continues orbiting. As mentioned above, the system admits a rotational symmetry, so having the ball in the second time step means it keeps its motion law from the first. At\linebreak every point in time, the ball continues spinning.
Another conservation law has appeared—the conservation of angular momentum. This is also represented by some number that is unchanged throughout the time evolution of the system. Angular momentum cannot be created nor destroyed and is a direct result of the rotation symmetry of our empty universe.
The main point here is that the translation symmetry allows us to perform experiments in different locations and obtain the same results, and an angular momentum symmetry lets us do these experiments on a rotating object (like the Earth) and obtain the same results.
Spinning basketballs at different angles.
Group actions
It turns out that you’re a genius; knowing nothing about this universe, you have seemingly contrived some beautiful conservation laws that will allow you to harness the true power of physics. But alas, nobody will accept your claim without a mathematical formulation. So we’re going to do exactly that.
As with the dihedral groups laying out the symmetries of polygons, there is a group-based way to represent the symmetries of what we should call space. An easy example is via the two-dimensional Cartesian plane, which will act as our sandbox for this universe. If we have a figure on it, whether that be a graph or tear marks, we would like to preserve all of its angles and distances after rotating it. This is analogous to taking our ball and spinning it around without deforming it.
Polar coordinates
Any point $(x,y)$ on the plane can be represented in polar coordinates $(r,\theta)$ measuring its absolute distance from the origin and how far anticlockwise it is rotated from the $x$-axis. This makes further rotation rather easy; to rotate by an arbitrary angle $\alpha$, we transform the coordinates by \[(r,\theta)\mapsto(r,\theta+\alpha).\] This can be noted by drawing a triangle and doing some quick trigonometry to find \[(x,y)=(r\cos\theta,r\sin\theta).\] Then, by moving by some angle $\alpha$, we arrive at the point \[(x’,y’)=(r\cos(\theta+\alpha),r\sin(\theta+\alpha)).\]
Linear finite groups in mathematics are often described using matrices, which is helpful in our case since that’s what we’re working with. One can use some trigonometric identities to show that rotation in Cartesian space is nothing but matrix multiplication: \[\begin{pmatrix} r\cos(\theta+\alpha)\\r\sin(\theta+\alpha) \end{pmatrix}=\begin{pmatrix} \cos\alpha&-\sin\alpha\\\sin\alpha&\cos\alpha \end{pmatrix}\begin{pmatrix} r\cos\theta\\r\sin\theta \end{pmatrix}.\] If the laws of physics are to remain invariant after any rotation on this plane, then all possible rotations ($\alpha$ going from 0$^{\circ}$ to 360$^{\circ}$) are admitted. By putting all of these together we have uncovered the special orthogonal group $\operatorname{SO}(2)$, denoting the group of rotations on a 2D system. This group behaves just as nicely as the group of circular symmetries (for a somewhat obvious reason!), and we thus crown it a Lie group.
This also generalises to higher dimensions. We have rotation groups in higher dimensions denoted the ‘special orthogonal’ groups of degree $n$, $\operatorname{SO}(n)$. These are Lie groups that consist of rotations around the $n$ axes and all of their combinations. In a physical sense, any of these rotations can be undone via a rotation in the opposite direction. This kind of backwards rotation is an inverse.
Local transformations
Everything I’ve talked about so far is good for mathematical theory. Secretly we were describing global transformations: ones that affect every point in spacetime in the same way. But we’re here to describe interactions within the universe. For that, we need to understand local transformations that depend on where you are in spacetime.
Why is that? The way we described conserved quantities earlier assumed a really boring universe. If you want to put one of these quantities to work by making it interact with other things in nature, while still keeping it conserved, local transformations are the way to do this.
To paint an intuitive picture, imagine that we are in our flat Cartesian land again and at each coordinate we place a clock, or dial, tuned to some time that depends on that specific coordinate. A global transformation could, for example, turn the dials of each clock exactly 90$^{\circ}$—or turn time by three hours. A local transformation would instead turn one clock by three hours, another by five, and perhaps even one back by four.
All of the particles in our universe are quantum-oriented. In quantum mechanics, objects like these particles are described by a quantity called its wavefunction $\psi(x,y)$ that assigns a complex number to every point $(x,y)$ in two-dimensional space. There are many different representations of complex numbers, but for quantum mechanics it is often useful to choose polar form where we interpret the radius as the wavefunction’s modulus, $\vert\psi\vert$, and $\theta$ measures the phase of rotation. If we relate this to clocks on a grid, one could imagine that the placement of the clock describes the value of $\vert\psi\vert$ and the ‘time’ describes the phase.
It might then be sensible to suggest that after a local transformation, where we changed the phase rotation differently at each point, we would change the physics we are trying to describe. But this is not true! This is only a mathematical representation of quantum physics and therefore is not actually physical. The wavefunction itself is unobservable and (most—we’re looking at you, Aharonov–Bohm effect) physical predictions only depend on $\vert \psi\vert$, which is left unchanged by a local transformation.
A wavefunction changing its phase.
This change-of-phase invariance is an intrinsic symmetry of the system. Mathematically, this system is locally $\operatorname{U}(1)$-invariant because the action of an element inside $\operatorname{U}(1)$ (complex circular rotation) does not change the laws of physics.\[\operatorname{U}(1)=\{\mathrm{e}^{\mathrm{i}\alpha}:\alpha\in[0^{\circ},360^{\circ}]\}.\] How might we describe this phenomenon mathematically? All this time we have discussed the motion of objects in flatspace, and motion is governed by derivatives. For observable physics we have to make sure that the derivatives themselves obey a local $\operatorname{U}(1)$ transformation so that we can include them in our equations. One of these transformations on $\psi$ looks like $\mathrm{e}^{\mathrm{i}\alpha(\boldsymbol{x})}\psi$, where one should note that $\alpha(\boldsymbol{x})$ is parametrised because it is a local transformation. When we differentiate,\[\partial_{\mu}\left(\mathrm{e}^{\mathrm{i}\alpha(\boldsymbol{x})}\psi\right) = \mathrm{e}^{\mathrm{i}\alpha(\boldsymbol{x})}\left(\partial_{\mu}+\mathrm{i}\partial_{\mu}\alpha(\boldsymbol{x})\right)\psi,\] which tells us that differentiation does not obey local $\operatorname{U}(1)$ transformations. This is an issue because one would really like to describe the motion of particles, and this requires derivatives. But we cannot do this with local transformations alone. A change in physics must be made.
Contracted differentiation
The term $\boldsymbol{x}$ or $x^{\mu}$ represents the 4-dimensional vector $(x^0,x^1,x^2,x^3)$, in which the numbers are used as coordinate indexing as opposed to powers. These values represent a point in space and time and are sometimes written as $(ct,x,y,z)$, where $c$ is the speed of light. The shorthand $\partial_\mu$ is used for $\partial/\partial x^{\mu}$, the partial derivative with respect to each coordinate.
Charge conservation
We’ve seen the general rule of thumb for how local $\operatorname{U}(1)$ transformations give structure to certain mathematical objects: wavefunctions behave nicely, but their dynamics do not. We should probably also come up with a name for this theory that isn’t ‘locally $\operatorname{U}(1)$-symmetric mathematical physics’. To fix this, we need to dive into the existing theory of electromagnetism.
In early days of scientific experimentation, society thought that the theories of electricity and magnetism were completely separate. An astounding result was formulated in the late 19th century, when Maxwell and notable others proved that these were two aspects of the same formulae. How do we understand this? As with previous reasoning, we can package it in a matrix. In 1908, Minkowski introduced the field strength tensor $F_{\mu\nu}$ as \[ F_{\mu\nu}:=\begin{pmatrix} 0&-E_1&-E_2&-E_3\\E_1&0&B_3&-B_2\\E_2&-B_3&0&B_1\\E_3&B_2&-B_1&0 \end{pmatrix},\] that encodes information about electric and magnetic quantities $\boldsymbol{E}$ and $\boldsymbol{B}$. For those interested, the metric signature here is $(-,+,+,+)$ and the speed of light has been normalised. Maxwell’s equations of electromagnetism can be formed by taking derivatives of various components of this tensor in a neat way.
Maxwell’s equations predict that light itself is an electromagnetic wave. It propagates through all of spacetime in little packets we like to call photons. Because of this, we can describe them with wavefunctions. And we already know that a change of wavefunction phase does not affect physics: it seems as though $\operatorname{U}(1)$ transformations and electromagnetism are linked. But Maxwell’s equations involve derivatives—and derivatives, as we have seen, are not locally $\operatorname{U}(1)$ invariant: $\partial_{\mu}\psi\;\not\mapsto\;\mathrm{e}^{\mathrm{i}\alpha(\boldsymbol{x})}\partial_{\mu}\psi$. There is an extra term involving the derivative of $\alpha(\boldsymbol{x})$ that we would like to remove. What we need to do is introduce a mathematical tool that, under a local $\operatorname{U}(1)$ transformation, would transform as this extra term:
\[A_{\mu}\mapsto A_{\mu}+\partial_{\mu}\alpha(\boldsymbol{x}).\] This $A_\mu$ is a mathematical tool that mediates the electromagnetic field and allows us to describe other vector fields, like $\boldsymbol{E}$ and $\boldsymbol{B}$, using its local rotation properties. We can call it the electromagnetic potential. Some may call it the photon field, as it naturally arises when discussing photons.
Instead of using the standard derivative, we now use this difference $\partial_{\mu}-\mathrm{i} A_{\mu}$ which transforms as \[\partial_{\mu}\psi-\mathrm{i} A_{\mu}\psi\mapsto\mathrm{e}^{\mathrm{i}\alpha(\boldsymbol{x})}\left(\partial_{\mu}\psi-\mathrm{i} A_{\mu}\psi\right).\] Result! With just group transformations, complex numbers and derivatives, we have discovered something wonderful. Real-world physics gives us $\boldsymbol{E}$ and $\boldsymbol{B}$, and the maths behind it uses tools such as the electromagnetic potential.
This is where Noether’s theorem comes into play, specifically the second part that deals with local transformations. Remember that any symmetry in a physical system has a determined conservation law. By observation, a local $\operatorname{U}(1)$ symmetry requires the interaction with some potential field $A_{\mu}$, which was the field encoding electric and magnetic information. But in order to interact with the electromagnetic field, our particles must have some charge associated with them. From there, to ensure quantum fluctuations don’t spin out of control, we must have conservation of electric charge.
This immediately gives us a physical representation of what a $\operatorname{U}(1)$ transformation is: the symmetry of charge. Dirac in 1928 used this to say that for every charged particle, there must exist its antiparticle counterpart with the opposite charge. Therefore, electric charge cannot be created nor destroyed and is a direct result of the $\operatorname{U}(1)$ symmetry of our particles.
Where do we go from here?
The standard model is one of the greatest feats of mathematical physics in the 20th century. It is the best quantum description we have of the fundamental forces that govern our entire universe. The best part? It is built by imposing certain symmetries on quantum particles.
More specifically, the standard model is the unification of the Lie groups \[\operatorname{U}(1)\times \operatorname{SU}(2)\times \operatorname{SU}(3),\] with the special unitary groups of degrees 2 and 3 now included. We already saw that $\operatorname{U}(1)$ could be thought of as rotation around a unit circle but is physically charge conservation. Similarly, actions of $\operatorname{SU}(2)$ are movements along the surface of a 4D hypersphere, and actions of $\operatorname{SU}(3)$ are movements along the surface of some other hyperobject. These are physically the conservation of weak isospin and colour charge, thanks to Noether.
When combined, the standard model as a whole exhibits charge–parity–time (CPT) symmetry. The action of swapping all particles for their antiparticles, exchanging chirality, and reversing time keeps the standard model invariant—but only when done together. There also exists the concept of symmetry breaking, whereby some particular symmetries make some massless particles (like the photon) massive. They have to be fixed in a nontrivial way.
These three groups mediate the electromagnetic force and the strong and weak nuclear forces among all known particles. Because of that, the equation for their dynamics is huge: it contains 93 terms. You can buy it on a T-shirt at Cern.
The story, however, does not stop there. At the time of writing, there is a huge amount of research going into a new type of symmetry called supersymmetry, which has the potential to relate all types of quantum particles together in a supergroovy way. This symmetry will present us with the conservation of supercharge (no, I am not making that up). We haven’t yet measured it, but when and if we do, it is bound to change the course of mathematical physics forever.
