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Elementary Particles
John C. Baez

April 29, 2003

Also available in PDF and Postscript formats.

To specify a quantum field theory on 3+1-dimensional Minkowski spacetime, we first need to list the elementary particles. These are irreducible representations (or irreps) of the group

ISpin(3,1) × G

where ISpin(3,1) is the universal cover of the identity component of the Poincaré group, while G is some compact Lie group called the internal symmetry group. Then we need to list the interactions, which are intertwining operators between tensor products of these irreps.

The Standard Model is the currently best accepted theory of elementary particles and their interactions, taking quantum theory and special relativity into account, but not general relativity -- i.e., not gravity. There is a huge amount of experimental evidence for this theory, but its mathematical structure remains complicated and mysterious. Nobody really knows `why' nature likes to work this way! However, the Standard Model displays many tantalizing patterns, which are probably important clues. In the following exercises you will ponder the elementary particles of the Standard Model. We will not discuss the interactions yet.

In the Standard Model, the internal symmetry group is

G = SU(3) × SU(2) × U(1).

An irrep of a product of a bunch of groups is the same as a tensor product of irreps of these groups. Thus, to specify an elementary particle in the Standard Model, we just need to specify irreps of ISpin(3,1), SU(3), SU(2) and U(1).

The groups SU(3), SU(2) and U(1) roughly correspond to the three forces other than gravity: strong, weak, and electromagnetic. The strong force holds quarks together in particles like protons and neutrons. The weak force lets one sort of quark or lepton turn into another, which is the process responsible for many radioactive decays in nuclei, and which also occurs in nuclear fusion. The electromagnetic force is the most familiar of the lot.

However, there's a sneaky twist. Specifying an irrep of SU(3) says how an elementary particle interacts with the strong force. Specifying an irrep of U(1) says how it interacts with the electromagnetic force. In particular, the unitary irreps of U(1) are labelled by an integer called electric charge: for each integer Q there is an irrep of U(1) on in which any element α of U(1) acts on a vector x in to give αQx, and we say this irrep describes an elementary particle of charge Q. But, the U(1) group describing electromagnetism is not the obvious U(1) subgroup of SU(2) × U(1)! It's actually some other U(1) subgroup of SU(2) × U(1), which we will describe later. What this means is that the weak and electromagnetic forces are described using SU(2) × U(1) in a tangled-up, "unified" way. For this reason, physicists say that specifying an irrep of SU(2) × U(1) says how an elementary particle interacts with the "electroweak force".

It would take quite a while to explain this better, and it's impossible without describing the interactions. For now, let's just see which irreps of

ISpin(3,1) × SU(3) × SU(2) × U(1)

correspond to elementary particles in the Standard Model.

Again, to get an irrep of this group, we just need to take irreps of ISpin(3,1), SU(3), SU(2), and U(1) and tensor them all together. The following chart lists what these irreps are for every particle in the Standard Model:

 type of particle ISpin(3,1) irrep SU(3) irrep SU(2) irrep U(1) irrep GAUGE BOSONS gluons (SU(3) force carriers): massless spin-1 SU(2) force carriers: massless spin-1 U(1) force carrier: massless spin-1 HIGGS BOSON Higgs: massless spin-0 and its antiparticle! FIRST GENERATION FERMIONS Leptons: left-handed electron neutrino and electron: left-handed massless spin-1/2 right-handed electron neutrino: right-handed massless spin-1/2 right-handed electron: right-handed massless spin-1/2 and their antiparticles! Quarks: left-handed up and down quarks: left-handed massless spin-1/2 right-handed up quark: right-handed massless spin-1/2 right-handed down quark right-handed massless spin-1/2 and their antiparticles! SECOND GENERATION FERMIONS Leptons: left-handed mu neutrino and muon: left-handed massless spin-1/2 right-handed mu neutrino: right-handed massless spin-1/2 right-handed muon: right-handed massless spin-1/2 and their antiparticles! Quarks: left-handed charm and strange quarks: left-handed massless spin-1/2 right-handed charm quark: right-handed massless spin-1/2 right-handed strange quark right-handed massless spin-1/2 and their antiparticles! THIRD GENERATION FERMIONS Leptons: left-handed tau neutrino and tau: left-handed massless spin-1/2 right-handed tau neutrino: right-handed massless spin-1/2 right-handed tau: right-handed massless spin-1/2 and their antiparticles! Quarks: left-handed top and bottom quarks: left-handed massless spin-1/2 right-handed top quark: right-handed massless spin-1/2 right-handed bottom quark right-handed massless spin-1/2 and their antiparticles!

Here are some conventions used in the above chart:

Now for some problems:

2. Ignoring ISpin(3,1) for a moment, all the fermions of the first generation correspond to complex irreps of

G = SU(3) × SU(2) × U(1).

Let's call the direct sum of all these irreps the first-generation fermion representation of G, or for short.

What is the dimension of ? If you know about Clifford algebras and spinors, what does this instantly make you wonder? If you were trying to describe using the normed division algebras and , what might you try?

3. contains a one-dimensional trivial representation of G as a subrepresentation, so we have

where stands for this trivial representation.

What particle corresponds to this trivial representation? This particle don't interact at all with the strong, weak or electromagnetic forces, so it's very hard to see. Physicists have only become convinced of its existence in the last couple of years. Before this, they used a different version of the Standard Model, where instead of they thought the first-generation fermion representation was . What is the dimension of ? How would this make you feel if you really liked Clifford algebras and/or normed division algebras?

4. If we define second-generation and third-generation fermion representations as we did for the first generation, what are these representations like? The direct sum of all three of these representations could be called the fermion representation of G. What is the dimension of this representation?

5. Again ignoring ISpin (3,1), all the gauge bosons correspond to real irreps of G. Let's call the direct sum of all these irreps the gauge boson representation of G, or G for short. What would mathematicians call this representation? What is the dimension of this representation?

6. Again ignoring ISpin(3,1), the Higgs boson corresponds to a complex irrep of G = SU(3) × SU(2) × U(1). Let's call this irrep the Higgs representation of G, or H for short. What is the dimension of this representation?

7. Can you think of any interesting patterns involving , G and ? It would be especially cool if they suggested a way to combine F G and H into some mathematically interesting structure.

Hints: G is a real vector space, while F and H are complex vector spaces. Thinking about the dimensions of F, G and H may be helpful, especially if you're into numerology. You can think of them all as real vector spaces, but then the dimensions of F and H double. You might want to keep the ISpin(3,1) irreps in mind when trying to combine F, G and H into something interesting. You may or may not want to bring all three generations into the picture. Finally, don't forget the fact that F and H come along with their conjugate reps and , corresponding to antiparticles.

There are some fairly interesting solutions to Problem 7, but I'll be satisfied with any honest attempt. A really good solution could earn you a Nobel prize!

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