Spacetime Symmetries in Elementary Particle Physics

I would like to talk about one of the most beautiful successes of mathematics in physics, in theoretical physics or mathematical physics in some sense. Which is the reduction of the problem of the classification of elementary particles to a purely mathematical problem. Very abstract in some sense. With the use of spacetime symmetries. For any theoretical physicist, one of the things that we learn is that in some sense physics is the place where the universe expresses its beauty in mathematics terms. That’s even one way that we can define physics, in some sense. It has always been a matter of surprise and wonder and inspiration that mathematics applies so well to describing the physical universe.

I will talk mostly about the results that have been done by Eugene Wigner, a Hungarian physicist who got the Nobel prize in 1963 for his contribution to the applications of symmetry principles in both atomic physics and elementary particles. Eugene Wigner wrote a paper about the unreasonable effectiveness of mathematics in natural sciences. He even called it a miracle, a gift from the universe that we neither understand nor deserve, but that we should really enjoy and appreciate fully.

So, in practice, I would like to talk about the classification of elementary particles via spacetime symmetry. Maybe I should start by reminding you what are the symmetries of spacetime. More precisely of empty spacetime in the case where I have a single particle moving in spacetime. First of all, we have translation symmetries in spacetime which means that all points in spacetime, in empty spacetime, are equal with respect to each other. So, that’s translation symmetry in spacetime. Then there’s another important symmetry which is rotation symmetry in space, all directions in space are the same. And also very important, is that all directions in spacetime in a certain sense are equal as well. More precisely this is called the connectivity principle that we can take two observers which are in uniform motion with respect to each other, they experience, they measure, they observe the same laws of physics when they do experiments.

This is the setting, these are the spacetime symmetries, now how to apply them. First of all, we should realise that we should combine two important descriptions of modern theoretical physics, one is microscopic world. It is governed by the laws of quantum mechanics. And two important things in quantum mechanics are the following; the first important thing is wave particle duality. Particles are also waves. Of course that’s a very abstract wave. It’s not a sort of wave like the wave on a surface of water or things like this, or the acoustic wave in the air that provides a sound. It’s something more abstract but still it’s a wave. An important property of linear waves is what is called the superposition principle. If two waves meet each other they just superpose each other so it’s the superposition principle. In more technical terms, more mathematical terms, we call it ‘linearity principle’. That’s the first important thing. Then there is also another very important fact in quantum mechanics is the fact that the laws of quantum mechanics are intrinsically probabilistic. There is no determinism, we cannot predict what will happen, the best we can do is compute the probabilities of what can happen. That’s a very important thing, probabilities are the core of quantum mechanics. They will not produce the position of particles in the future, just the probability of observing particles in a given place at a given time.

These are the two important properties of quantum mechanics, now what are the important properties of the microscopic world? In that case, the ones we need are just the spacetime symmetries that I mention before. The idea of Wigner is to try to combine the principles of quantum mechanics, so the microscopic world, with the principles of relativity and the other spacetime symmetries, so the properties of the macroscopic world. Concretely with mathematics how can you combine these two properties, both the laws of quantum mechanics and the spacetime symmetries. This is a complicated story. It makes use of what is called, in mathematical jargon, ‘group theory’ and ‘presentation theory’. In practice, just to say a few words for the ones that might have already heard it once, you reduce the problem of classifying particles to the mathematical problem of classification of unitary irreducible representations. Let’s give one idea of what does that mean. The first property, that is the linearity property is that if you sum waves the sum of these two is still a good wave. This is the linearity, the linear world. You have what is called a linear representation.

Another important property is that you want to preserve probabilities when you do spacetime symmetries. So, the probability to find a particle somewhere in the universe should be conserved. It can be at a different point on the spacetime symmetry but the sum of probabilities should be preserved. And this property in mathematical terms in this context is called unitarity. So, this is why I said unitary representations. Linearity is for representation, unitarity is to conserve probability. Then I also used the word ‘irreducible’. So, irreducible is that fact that you want to consider just elementary particles. By elementary particles we mean particles which are not made of more elementary particles, more fundamental particles. We know for instance protons are not elementary, they are made from quarks. But quarks themselves supposedly are elemental, they are not made of something. Again, there is a mathematical way to phrase this in a very very precise way. This property of irreducibility. Which by the way is not exactly equivalent, it’s even more subtle than the fact of being elementary of not elementary, it’s more well defined. Let us see a bit more concretely what does it mean for elementary particles in more familiar terms. What does this classification by Wigner, both mathematics and the reduction of the problem of mathematics is due to Wigner, but what does it mean maybe more concretely?

There are two important things that may apply the spacetime symmetry. The first important way to apply spacetime symmetries are if you start with a particle here and now, by using translation symmetry you can map it to a particle, the same particle but it’s there at a different point at a different time. This means by making use of spacetime symmetries we can describe the motion of particles at later times at a different place. You can relate all these different probabilities of finding the particles at different places and times just by using spacetime symmetries. This is a radical simplification, by making use of spacetime symmetries. That was the first one. But then if you remember, a very important thing about the symmetries is not only what changes it’s also what remains invariant.

If you want to classify a particle, you should also look for what is invariant. Consider for instance a particle that goes at a speed, not the speed of light, just another speed lesser than the speed of light. We call them massive particles. Why? Because if you consider a particle moving in empty spacetime at a given speed, you can always consider the observer that goes exactly at the same speed in the same direction along with this particle. If you look at this particle, for him, the particle is at rest. Like two trains that go at the same speed they look at rest with respect to each other. This means that there is always what we call a rest frame, the lab frame sometimes called, where you see the particle at rest. In that case you can measure the energy of the particle in this frame. If you make known of the relation, well know e=mc2 that tells you energy is related to the mass then by measuring energy in this frame you can measure what is called the rest mass, the mass for the particle when the particle is at rest.

That’s one property. But it’s not over because still in this rest frame you can do rotations and still the particle is invariant, it is not changing. This means now that you should look at the different representations, again it’s something a bit more abstract, the classifications of how the rotations will be described. In that case you are led to another property of particles, much more mysterious if you are not in quantum mechanics, which is called the spin. The intrinsic angular momentum, the say that it rotates in some sense, around itself in this rest frame. This is the spin.

Now you may wonder, but there are also particles that go at the speed of light, like photons, light, the particles which light is made of. In that case there is no such rest frame since under the relativity principle, part of the relativity principle, is the speed of light is the same for everyone. No observer can see a photon at rest, even two photons along each other see each other going at the speed of light. In other words, there is no rest mass, there is no notion of rest mass for a photon particle going as the speed of light. In that case they are massless, in some sense the rest mass is zero. So, that’s one property and then since you cannot go in the rest frame the best you can do is to go along the particle motion. The only rotations you can do are along the directions, so just along the transverse plane. In that case you have to characterise, take an example, you can characterise particles like you could characterise cards by the order of the rotations symmetry of the plane of the card. We would say here that if I do a half turn the card remains invariant and for the particle it would be the same, then we would say that this particle has spin 2. Like the hypothetical constituent of gravity and gravitons are supposed to be of spin two. If you have to do a complete turn this would be a photon because it has order 1, spin 1, helicity we call it

As we can say, all particles, elementary particles are characterised by two numbers, their mass and their spin, sometimes called the helicity in this context. This is one of the remarkable successes of mathematical physics and the unreasonable effectiveness of the mathematics in physics.