April 13, 1998

This Week's Finds in Mathematical Physics (Week 119)

John Baez

I've been slacking off on This Week's Finds lately because I was busy getting stuff done at Riverside so that I could visit the Center for Gravitational Physics and Geometry here at Penn State with a fairly clean slate. Indeed, sometimes my whole life seems like an endless series of distractions designed to prevent me from writing This Week's Finds. However, now I'm here and ready to have some fun....

Recently I've been trying to learn about grand unified theories, or "GUTs". These were popular in the late 1970s and early 1980s, when the Standard Model of particle interactions had fully come into its own and people were looking around for a better theory that would unify all the forces and particles present in that model - in short, everything except gravity.

The Standard Model works well but it's fairly baroque, so it's natural to hope for some more elegant theory underlying it. Remember how it goes:

                            GAUGE BOSONS
         photon                     W+                  8 gluons 


        LEPTONS                                     QUARKS

electron        electron neutrino         down quark        up quark
muon            muon neutrino             strange quark     charm quark
tauon           tauon neutrino            bottom quark      top quark

                        HIGGS BOSON (not yet seen)

The strong, electromagnetic and weak forces are all described by Yang-Mills fields, with the gauge group SU(3) x SU(2) x U(1). In what follows I'll assume you know the rudiments of gauge theory, or at least that you can fake it.

SU(3) is the gauge group of the strong force, and its 8 generators correspond to the gluons. SU(2) x U(1) is the gauge group of the electroweak force, which unifies electromagnetism and the weak force. It's not true that the generators of SU(2) corresponds to the W+, W- and Z while the generator of U(1) corresponds to the photon. Instead, the photon corresponds to the generator of a sneakier U(1) subgroup sitting slantwise inside SU(2) x U(1); the basic formula to remember here is:

Q = I3 + Y/2

where Q is ordinary electric charge, I3 is the 3rd component of "weak isospin", i.e. the generator of SU(2) corresponding to the matrix

(1/2   0)
(0  -1/2)
and Y, "hypercharge", is the generator of the U(1) factor. The role of the Higgs particle is to spontaneously break the SU(2) x U(1) symmetry, and also to give all the massive particles their mass. However, I don't want to talk about that here; I want to focus on the fermions and how they form representations of the gauge group SU(3) x SU(2) x U(1), because I want to talk about how grand unified theories attempt to simplify this picture - at the expense of postulating more Higgs bosons.

The fermions come in 3 generations, as indicated in the chart above. I want to explain how the fermions in a given generation are grouped into irreducible representations of SU(3) x SU(2) x U(1). All the generations work the same way, so I'll just talk about the first generation. Also, every fermion has a corresponding antiparticle, but this just transforms according to the dual representation, so I will ignore the antiparticles here.

Before I tell you how it works, I should remind you that all the fermions are, in addition to being representations of SU(3) x SU(2) x U(1), also spin-1/2 particles. The massive fermions - the quarks and the electron, muon and tauon - are Dirac spinors, meaning that they can spin either way along any axis. The massless fermions - the neutrinos - are Weyl spinors, meaning that they always spin counterclockwise along their axis of motion. This makes sense because, being massless, they move at the speed of light, so everyone can agree on their axis of motion! So the massive fermions have two helicity states, which we'll refer to as "left-handed" and "right-handed", while the neutrinos only come in a "left-handed" form.

(Here I am discussing the Standard Model in its classic form. I'm ignoring any modifications needed to deal with a possible nonzero neutrino mass. For more on Standard Model, neutrino mass and different kinds of spinors, see "week93".)

Okay. The Standard Model lumps the left-handed neutrino and the left-handed electron into a single irreducible representation of SU(3) x SU(2) x U(1):

L, eL)                                 (1,2,-1)
This 2-dimensional representation is called (1,2,-1), meaning that it's the tensor product of the 1-dimensional trivial rep of SU(3), the 2-dimensional fundamental rep of SU(2), and the 1-dimensional rep of U(1) with hypercharge -1.

Similarly, the left-handed up and down quarks fit together as:

(uL, uL, uL, dL, dL, dL)              (3,2,1/3)
Here I'm writing both quarks 3 times since they also come in 3 color states. In other words, this 6-dimensional representation is the tensor product of the 3-dimensional fundamental rep of SU(3), the 2-dimensional fundamental rep of SU(2), and the 1-dimensional rep of U(1) with hypercharge 1/3. That's why we call this rep (3,2,1/3).

(If you are familiar with the irreducible representations of U(1) you will know that they are usually parametrized by integers. Here we are using integers divided by 3. The reason is that people defined the charge of the electron to be -1 before quarks were discovered, at which point it turned out that the smallest unit of charge was 1/3 as big as had been previously believed.)

The right-handed electron stands alone in a 1-dimensional rep, since there is no right-handed neutrino:

eR                                         (1,1,-2)
Similarly, the right-handed up quark stands alone in a 3-dimensional rep, as does the right-handed down quark:
(uR, uR, uR)                             (3,1,4/3) 

(dR, dR, dR)                             (3,1,-2/3)
That's it. If you want to study this stuff, try using the formula

Q = I3 + Y/2

to figure out the charges of all these particles. For example, since the right-handed electron transforms in the trivial rep of SU(2), it has I3 = 0, and if you look up there you'll see that it has Y = -2. This means that its electric charge is Q = -1, as we already knew.

Anyway, we obviously have a bit of a mess on our hands! The Standard Model is full of tantalizing patterns, but annoyingly complicated. The idea of grand unified theories is to find a pattern lurking in all this data by fitting the group SU(3) x SU(2) x U(1) into a larger group. The smallest-dimensional "simple" Lie group that works is SU(5). Here "simple" is a technical term that eliminates, for example, groups that are products of other groups - these aren't very "unified". Georgi and Glashow came up with their "minimal" SU(5) grand unified theory in 1975. The idea is to stick SU(3) x SU(2) into SU(5) in the obvious diagonal way, leaving just enough room to cram in the U(1) if you are clever.

Now if you add up the dimensions of all the representations above you get 2 + 6 + 1 + 3 + 3 = 15. This means we need to find a 15-dimensional representation of SU(5) to fit all these particles. There are various choices, but only one that really works when you take all the physics into account. For a nice simple account of the detective work needed to figure this out, see:

1) Edward Witten, Grand unification with and without supersymmetry, Introduction to supersymmetry in particle and nuclear physics, edited by O. Castanos, A. Frank, L. Urrutia, Plenum Press, 1984.

I'll just give the answer. First we take the 5-dimensional fundamental representation of SU(5) and pack fermions in as follows:

(dR, dR, dR, e+R, nubarR)             5 = (3,1,-2/3) + (1,2,-1)

Here e+R is the right-handed positron and nubarR is the right-handed antineutrino - curiously, we need to pack some antiparticles in with particles to get things to work out right. Note that the first 3 particles in the above list, the 3 states of the right-handed down quark, transform according to the fundamental rep of SU(3) and the trivial rep of SU(2), while the remaining two transform according to the trivial rep of SU(3) and the fundamental rep of SU(2). That's how it has to be, given how we stuffed SU(3) x SU(2) into SU(5).

Note also that the charges of the 5 particles on this list add up to zero. That's also how it has to be, since the generators of SU(5) are traceless. Note that the down quark must have charge -1/3 for this to work! In a sense, the SU(5) model says that quarks must have charges in units of 1/3, because they come in 3 different colors! This is pretty cool.

Then we take the 10-dimensional representation of SU(5) given by the 2nd exterior power of the fundamental representation - i.e., antisymmetric 5x5 matrices - and pack the rest of the fermions in like this:

 (     0      ubarL  ubarL    uL    dL   )       10 = (3,2,1/3) + 
 (  -ubarL     0     ubarL    uL    dL   )            (1,1,2)   +
 (  -ubarL  -ubarL    0       uL    dL   )            (3,1,-4/3)
 (   -uL     -uL    -uL       0     e+L  )
 (   -dL     -uL    -dL     -e+L    0    )
Here the u-bar is the antiparticle of the up quark - again we've needed to use some antiparticles. However, you can easily check that these two representations of SU(5) together with their duals account for all the fermions and their antiparticles.

The SU(5) theory has lots of nice features. As I already noted, it explains why the up and down quarks have charges 2/3 and -1/3, respectively. It also gives a pretty good prediction of something called the Weinberg angle, which is related to the ratio of the masses of the W and Z bosons. It also makes testable new predictions! Most notably, since it allows quarks to turn into leptons, it predicts that protons can decay - with a halflife of somewhere around 1029 or 1030 years. So people set off to look for proton decay....

However, even when the SU(5) model was first proposed, it was regarded as slightly inelegant, because it didn't unify all the fermions of a given generation in a single irreducible representation (together with its dual, for antiparticles). This is one reason why people began exploring still larger gauge groups. In 1975 Georgi, and independently Fritzsch and Minkowski, proposed a model with gauge group SO(10). You can stuff SU(5) into SO(10) as a subgroup in such a way that the 5- and 10-dimensional representations of SU(5) listed above both fit into a single 16-dimensional rep of SO(10), namely the chiral spinor rep. Yes, 16, not 15 - that wasn't a typo! The SO(10) theory predicts that in addition to the 15 states listed above there is a 16th, corresponding to a right-handed neutrino! I'm not sure yet how the recent experiments indicating a nonzero neutrino mass fit into this business, but it's interesting.

Somewhere around this time, people noticed something interesting about these groups we've been playing with. They all fit into the "E series"!

I don't have the energy to explain Dynkin diagrams and the ABCDEFG classification of simple Lie groups here, but luckily I've already done that, so you can just look at "week62" - "week65" to learn about that. The point is, there is an infinite series of simple Lie groups associated to rotations in real vector spaces - the SO(n) groups, also called the B and D series. There is an infinite series of them associated to rotations in complex vector spaces - the SU(n) groups, also called the A series. And there is infintie series of them associated to rotations in quaternionic vector spaces - the Sp(n) groups, also called the C series. And there is a ragged band of 5 exceptions which are related to the octonions, called G2, F4, E6, E7, and E8. I'm sort of fascinated by these - see "week90", "week91", and "week106" for more - so I was extremely delighted to find that the E series plays a special role in grand unified theories.

Now, people usually only talk about E6, E7, and E8, but one can work backwards using Dynkin diagrams to define E5, E4, E3, E2, and E1. Let's do it! Thanks go to Allan Adler and Robin Chapman for helping me understand how this works....

E8 is a big fat Lie group whose Dynkin diagram looks like this:


If we remove the rightmost root, we obtain the Dynkin diagram of a subgroup called E7:

If we again remove the rightmost root, we obtain the Dynkin diagram of a subgroup of E7, namely E6:

This was popular as a gauge group for grand unified models, and the reason why becomes clear if we again remove the rightmost root, obtaining the Dynkin diagram of a subgroup we could call E5:

But this is really just good old SO(10), which we were just discussing! And if we yet again remove the rightmost root, we get the Dynkin diagram of a subgroup we could call E4:

This is just SU(5)! Let's again remove the rightmost root, obtaining the Dynkin diagram for E3. Well, it may not be clear what counts as the rightmost root, but here's what I want to get when I remove it:

This is just SU(3) x SU(2), sitting inside SU(5) in the way we just discussed! So for some mysterious reason, the Standard Model and grand unified theories seem to be related to the E series!

We could march on and define E2:


which is just SU(2) x SU(2), and E1:

which is just SU(2)... but I'm not sure what's so great about these groups.

By the way, you might wonder what's the real reason for removing the roots in the order I did - apart from getting the answers I wanted to get - and the answer is, I don't really know! If anyone knows, please tell me. This could be an important clue.

Now, this stuff about grand unified theories and the E series is one of the reasons why people like string theory, because heterotic string theory is closely related to E8 (see "week95"). However, I must now tell you the bad news about grand unified theories. And it is very bad.

The bad news is that those people who went off to detect proton decay never found it! It became clear in the mid-1980s that the proton lifetime was at least 1032 years or so, much larger than what the SU(5) theory most naturally predicts. Of course, if one is desperate to save a beautiful theory from an ugly fact, one can resort to desperate measures. For example, one can get the SU(5) model to predict very slow proton decay by making the grand unification mass scale large. Unfortunately, then the coupling constants of the strong and electroweak forces don't match at the grand unification mass scale. This became painfully clear as better measurements of the strong coupling constant came in.

Theoretical particle physics never really recovered from this crushing blow. In a sense, particle physics gradually retreated from the goal of making testable predictions, drifting into the wonderland of pure mathematics... first supersymmetry, then supergravity, and then superstrings... ever more elegant theories, but never yet a verified experimental prediction. Perhaps we should be doing something different, something better? Easy to say, hard to do! If we see a superpartner at CERN, a lot of this "superthinking" will be vindicated - so I guess most particle physicists are crossing their fingers and praying for this to happen.

The following textbook on grand unified theories is very nice, especially since it begins with a review of the Standard Model:

2) Graham G. Ross, Grand Unified Theories, Benjamin-Cummings, 1984.

This one is a bit more idiosyncratic, but also good - Mohapatra is especially interested in theories where CP violation arises via spontaneous symmetry breaking:

3) Ranindra N. Mohapatra, Unification and Supersymmetry: The Frontiers of Quark-Lepton Physics, Springer-Verlag, 1992.

I also found the following articles interesting:

4) D. V. Nanopoulos, Tales of the GUT age, in Grand Unified Theories and Related Topics, proceedings of the 4th Kyoto Summer Institute, World Scientific, Singapore, 1981.

5) P. Ramond, Grand unification, in Grand Unified Theories and Related Topics, proceedings of the 4th Kyoto Summer Institute, World Scientific, Singapore, 1981.

Okay, now for some homotopy theory! I don't think I'm ever gonna get to the really cool stuff... in my attempt to explain everything systematically, I'm getting worn out doing the preliminaries. Oh well, on with it... now it's time to start talking about loop spaces! These are really important, because they tie everything together. However, it takes a while to deeply understand their importance.

O. The loop space of a topological space. Suppose we have a "pointed space" X, that is, a topological space with a distinguished point called the "basepoint". Then we can form the space LX of all "based loops" in X - loops that start and end at the basepoint.

One reason why LX is so nice is that its homotopy groups are the same as those of X, but shifted:

πi(LX) = πi+1(X)

Another reason LX is nice is that it's almost a topological group, since one can compose based loops, and every loop has an "inverse". However, one must be careful here! Unless one takes special care, composition will only be associative up to homotopy, and the "inverse" of a loop will only be the inverse up to homotopy.

Actually we can make composition strictly associative if we work with "Moore paths". A Moore path in X is a continuous map

f: [0,T] → X

where T is an arbitrary nonnegative real number. Given a Moore path f as above and another Moore path

g: [0,S] → X

which starts where f ends, we can compose them in an obvious way to get a Moore path

fg: [0,T+S] → X

Note that this operation is associative "on the nose", not just up to homotopy. If we define LX using Moore paths that start and end at the basepoint, we can easily make LX into a topological monoid - that is, a topological space with a continuous associative product and a unit element. (If you've read section L, you'll know this is just a monoid object in Top!) In particular, the unit element of LX is the path i: [0,0] → X that just sits there at the basepoint of X.

LX is not a topological group, because even Moore paths don't have strict inverses. But LX is close to being a group. We can make this fact precise in various ways, some more detailed than others. I'm pretty sure one way to say it is this: the natural map from LX to its "group completion" is a homotopy equivalence.

P. The group completion of a topological monoid. Let TopMon be the category of topological monoids and let TopGp be the category of topological groups. There is a forgetful functor

F: TopGp → TopMon

and this has a left adjoint

G: TopMon → TopGp

which takes a topological monoid and converts it into a topological group by throwing in formal inverses of all the elements and giving the resulting group a nice topology. This functor G is called "group completion" and was first discussed by Quillen (in the simplicial context, in an unpublished paper), and independently by Barratt and Priddy:

6) M. G. Barratt and S. Priddy, On the homology of non-connected monoids and their associated groups, Comm. Math. Helv. 47 (1972), 1-14.

For any topological monoid M, there is a natural map from M to F(G(M)), thanks to the miracle of adjoint functors. This is the natural map I'm talking about in the previous section!

© 1998 John Baez