3.1 The SU(5) GUT

The grand unified theory appeared in a 1974 paper by Howard Georgi and Sheldon Glashow [12]. It was the first grand unified theory, and is still considered the prototypical example. As such, there are many accounts of it in the physics literature. The textbooks by Ross [31] and Mohapatra [21] both devote an entire chapter to the theory, and a lucid summary can be found in a review article by Witten [39], which also discusses the supersymmetric generalization of this theory.

In this section, we will limit our attention to the nonsupersymmetric version of theory, which is how it was originally proposed. Unfortunately, this theory has since been ruled out by experiment; it predicts that protons will decay faster than the current lower bound on proton lifetime [26]. Nevertheless, because of its prototypical status and intrinsic interest, we simply must talk about the theory.

The core idea behind the grand unified theory is that because the Standard Model representation is 32-dimensional, each particle or antiparticle in the first generation of fermions can be named by a 5-bit code. Roughly speaking, these bits are the answers to five yes-or-no questions:

- Is the particle isospin up?
- Is it isospin down?
- Is it red?
- Is it green?
- Is it blue?

We can flesh out this scheme by demanding that the operation of taking antiparticles correspond to switching 0's for 1's in the code. So the code for the antiparticle of , the `antired right-handed antidown antiquark', is . This is cute: it means that being antidown is the same as being up, while being antired is the same as being both green and blue.

Furthermore, in this scheme all antileptons are `black' (the particles with no color, ending in 000), while leptons are `white' (the particles with every color, ending in 111). Quarks have exactly one color, and antiquarks have exactly two.

We are slowly working our way to the theory. Next let us bring Hilbert spaces into the game. We can take the basic properties of being up, down, red, green or blue, and treat them as basis vectors for . Let us call these vectors . The exterior algebra has a basis given by wedge products of these 5 vectors. This exterior algebra is 32-dimensional, and it has a basis labelled by 5-bit strings. For example, the bit string corresponds to the basis vector , while the bit string corresponds to .

Next we bring in representation theory. The group has an obvious representation on . And since the operation of taking exterior algebras is functorial, this group also has a representation on . In the grand unified theory, this is the representation we use to describe a single generation of fermions and their antiparticles.

Just by our wording, though, we are picking out a splitting of into : the isospin and color parts, respectively. Choosing such a splitting of picks out a subgroup of , the set of all group elements that preserve this splitting. This subgroup consists of block diagonal matrices with a block and a block, both unitary, such that the determinant of the whole matrix is 1. Let us denote this subgroup as .

Now for the miracle: the subgroup is isomorphic to the Standard Model gauge group (at least modulo a finite subgroup). And, when we restrict the representation of on to , we get the Standard Model representation!

There are two great things about this. The first is that it gives a concise and mathematically elegant description of the Standard Model representation. The second is that the seemingly ad hoc hypercharges in the Standard Model must be exactly what they are for this description to work. So, physicists say the theory explains the fractional charges of quarks: the fact that quark charges come in units the size of electron charge pops right out of this theory.

With this foretaste of the fruits the theory will bear, let
us get to work and sow the seeds. Our work will have two
parts. First we need to check that

where is some finite normal subgroup that acts trivially on . Then we need to check that indeed

as representations of .

First, the group isomorphism. Naively, one might seek to build
the theory by including
as a subgroup of .
Can this be done?
Clearly, we can include
as block diagonal
matrices in :

but this is not enough, because also has that pesky factor of , related to the hypercharge. How can we fit that in?

The first clue is that elements of must commute with the
elements of
. But the only elements of
that commute with everybody in the
subgroup are
diagonal, since they must separately commute with
and
, and the only elements doing so are
diagonal. Moreover, they must be scalars on each block. So, they have
to look like this:

where stands for the identity matrix times the complex number , and similarly for in the block. For the above matrix to lie in , it must have determinant 1, so . This condition cuts the group of such matrices from down to . In fact, all such matrices are of the form

where runs over .

So if we throw in elements of this form, do we get
? More precisely, does this map:

give an isomorphism between and ? It is clearly a homomorphism. It clearly maps into into the subgroup . And it is easy to check that it maps onto this subgroup. But is it one-to-one?

The answer is no: the map has a kernel,
.
The kernel is the set of all elements of the form

and this is , because scalar matrices and live in and , respectively, if and only if is a sixth root of unity. So, all we get is

This sets up a nerve-racking test that the theory must pass for it to have any chance of success. After all, not all representations of factor through , but all those coming from representations of must do so. A representation of will factor through only if the subgroup acts trivially.

In short: the GUT is doomed unless acts trivially on every fermion. (And antifermion, but that amounts to the same thing.) For this to be true, some nontrivial relations between hypercharge, isospin and color must hold.

For example, consider the left-handed electron

For any sixth root of unity , we need

to act trivially on this particle. Let us see how it acts. Note that:

- acts on as multiplication by ;
- acts on as multiplication by ;
- acts trivially on .

The action is indeed trivial--precisely because is a sixth root of unity.

Or, consider the right-handed quark:

How does act on this? We note:

- acts on as multiplication by ;
- acts trivially on the trivial representation ;
- acts on as multiplication by .

Again, the action is trivial.

For to work, though, has to act trivially on every fermion. There are 16 cases to check, and it is an awful lot to demand that hypercharge, the most erratic part of the Standard Model representation, satisfies 16 relations.

Or is it? In general, for a fermion with hypercharge , there are four distinct possibilities:

Hypercharge relations |
||||

Case | Representation | Relation | ||

Nontrivial , nontrivial | ||||

Nontrivial , trivial | ||||

Trivial , nontrivial | ||||

Trivial , trivial |

Hypercharge relations |
||||

Case | Representation | Relation | ||

Left-handed quark | ||||

Left-handed lepton | ||||

Right-handed quark | ||||

Right-handed lepton |

Hypercharge relations |
||

Case | Relation | |

Left-handed quark | ||

Left-handed lepton | ||

Right-handed quark | ||

Right-handed lepton |

Now it is easy to check this indeed holds for every fermion in the standard model. passes the test, not despite the bizarre pattern followed by hypercharges, but because of it!

By this analysis, we have shown that
acts trivially on the Standard Model
rep, so it is contained in the kernel of this rep. It is better than just a
containment though:
is the entire kernel. Because of this, we could say
that
is the `true' gauge group of the Standard Model. And because
we now know that

it is almost as though this kernel, lurking inside this whole time, was a cryptic hint to try the theory.

Of course, we still need to find a representation of that extends the Standard Model representation. Luckily, there is a very beautiful choice that works: the exterior algebra . Since acts on , it has a representation on . Our next goal is to check that pulling back this representation from to using , we obtain the Stadard model representation

As we do this, we will see another fruit of the theory ripen. The triviality of already imposed some structure on hypercharges, as outlined in above in Table 3. As we fit the fermions into , we will see this is no accident: the hypercharges have to be exactly what they are for the theory to work.

To get started, our strategy will be to use the fact that, being representations of compact Lie groups, both the fermions and the exterior algebra are completely reducible, so they can be written as a direct sum of irreps. We will then match up these irreps one at a time.

The fermions are already written as a direct sum of irreps, so we need
to work on
. Now, any element
acts as automorphisms
of the exterior algebra
:

where . Since we know how acts on the vectors in , and these generate , this rule is enough to tell us how acts on all of . This action respects grades in , so each exterior power in

is a subrepresentation. In fact, these are all irreducible, so this is how breaks up into irreps of . Upon restriction to , some of these summands break apart further into irreps of .

Let us see how this works, starting with the easiest cases. and are both trivial irreps of . There are two trivial irreps in the Standard Model representation, namely and its dual , where we use angle brackets to stand for the Hilbert space spanned by a vector or collection of vectors. So, we could select and , or vice versa. At this juncture, we have no reason to prefer one choice to the other.

Next let us chew on the next piece: the first exterior power,
. We have

as vector spaces, and as representations of . But what is as a representation of ? The Standard Model gauge group acts on via the map

Clearly, this action preserves the splitting into the `isospin part' and the `color part' of :

So, let us examine these two subrepresentations in turn:

- The
part transforms in the hypercharge 1 rep of :
that is, acts as multiplication by . It
transforms according to the fundamental representation of
, and the trivial representation of . This
seems to describe a left-handed lepton with hypercharge 1.
- The part transforms in the hypercharge rep of : that is, acts as multiplication by . It transforms trivially under , and according to the fundamental representation of . Table 1 shows that these are the features of a right-handed down quark.

In short, as a rep of
, we have

and we have already guessed which particles these correspond to. The first summand looks like a left-handed lepton with hypercharge 1, while the second is a right-handed quark with hypercharge .

Now this is problematic, because another glance at Table 1
reveals that there is no left-handed lepton with
hypercharge 1. The only particles with hypercharge 1 are the right-handed
antileptons, which span the representation

But wait! is unique among the 's in that its fundamental rep is self-dual:

This saves the day. As a rep of , becomes

so it describes the right-handed antileptons with hypercharge 1 and the right-handed quarks with hypercharge . In other words:

where we have omitted the color label on to save space. Take heed of this: is short for the vector space , and it is three-dimensional.

Now we can use our knowledge of the first exterior power to
compute the second exterior power, by applying the formula

So, let us calculate! As reps of
we have

Consider the first summand,
. As a rep of this space is just
,
which is the one-dimensional trivial rep, .
As a rep of it is also trivial. But as a rep of
, it is nontrivial. Inside it we are juxtaposing
two particles with hypercharge 1. Hypercharges add, just like charges,
so the composite particle, which consists of one particle and
the other, has hypercharge 2. So, as a representation of the Standard
Model gauge group we have

Glancing at Table 1 we see this matches the left-handed positron, . Note that the hypercharges are becoming useful now, since they uniquely identify all the fermion and antifermion representations, except for neutrinos.

Next consider the second summand:

Again, we can add hypercharges, so this representation of is isomorphic to

This is the space for left-handed quarks of hypercharge , which from Table 1 is:

where once again we have suppressed the label for colors.

Finally, the third summand in
is

This has isospin , so by Table 1 it had better correspond to the left-handed antiup antiquark, which lives in the representation

Let us check. The rep is trivial under . As a rep of it is the same as . But because preserves the volume form on , taking Hodge duals gives an isomorphism

so we have

which is just what we need to show

In summary, the following pieces of the Standard Model rep
sit inside
:

We are almost done. Because preserves the canonical volume form on
, taking Hodge duals gives an isomorphism between

as representations of . Thus given our results so far:

we automatically get the antiparticles of these upon taking Hodge duals,

So , as desired.

How does all this look in terms of the promised binary code? Remember, a 5-bit code is short for a wedge product of basis vectors . For example, 01101 corresponds to . And now that we have found an isomorphism , each of these wedge products corresponds to a fermion or antifermion. How does this correspondence go, exactly?

First consider the grade-one part
.
This has basis vectors called and . We have seen that
the subspace
, spanned by and , corresponds to

The top particle here has isospin up, while the bottom one has isospin down, so we must have and . Likewise, the subspace spanned by and corresponds to

Thus we must have , where runs over the colors .

Next consider the grade-two part:

Here lives in the one-dimensional rep of , which is spanned by the vector . Thus, . The left-handed quarks live in the rep of , which is spanned by vectors that consist of one isospin and one color. We must have and , where again runs over all the colors . And now for the tricky part: the quarks live in the rep of , but this is isomorphic to the fundamental representation of on , which is spanned by antired, antired and antiblue:

These vectors form the basis of that is dual to , , and under Hodge duality in . So we must have

where can be any anticolor. Take heed of the fact that is grade 2, even though it may look like grade 1.

To work out the other grades, note that Hodge duality corresponds to switching 0's and 1's in our binary code. For instance, the dual of 01101 is 10010: or written in terms of basis vectors, the dual of is . Thus given the binary codes for the first few exterior powers:

Now we can see a good, though not decisive, reason to choose . With this choice, and not the other, we get left-handed particles in the even grades, and right-handed particles in the odd grades. We choose to have this pattern now, but later on we need it.

Table 4 defines a linear isomorphism in terms of the basis vectors, so the equations in this table are a bit of a exaggeration. When we write say, , we really mean . This map is an isomorphism between representations of . It tells us how these representations are the `same'.

More precisely, we mean these representations are the same when we
identify
with
using the isomorphism
induced by . In general,
we can think of a unitary representation as a Lie group homomorphism

where is a finite-dimensional Hilbert space and is the Lie group of unitary operators on . In this section we have been comparing two unitary representations: an ugly, complicated representation of :

and a nice, beautiful representation of :

We built a homomorphism

so it is natural to wonder if is there a fourth homomorphism

such that this square:

commutes.

Indeed, we just showed this! We have seen there exists a
unitary operator from the Standard Model rep to
, say

such that the induced isomorphism of the unitary groups,

makes the above square commute. So, let us summarize this result as a theorem:

2010-01-11