3.2 The Spin(10) Theory

We now turn our attention to another grand unified theory. Physicists call it the ` theory', but we shall call it the theory, because the Lie group involved is really , the double cover of . This theory appeared in a 1974 paper by Georgi [10], shortly after his paper with Glashow on the theory. However, Georgi has said that he conceived of the theory first. See Zee [40], Chapter VII.7, for a concise and readable account.

The GUT has helped us explain the pattern of hypercharges in the
Standard Model, and thanks to the use of the exterior algebra,
, we
can interpret it in terms of a binary code. This binary code explains another
curious fact about the Standard Model. Specifically, why is the number of
fermions a power of 2? There are 16 fermions, and 16 antifermions, which
makes the
Standard Model rep have dimension

With the binary code interpretation, it could not be any other way.

In actuality, however, the existence of a right-handed neutrino (or its antiparticle, the left-handed antineutrino) has been controversial. Because it transforms trivially in the Standard Model, it does not interact with anything except perhaps the Higgs.

The right-handed neutrino certainly improves the aesthetics of the
theory. When we include this particle (and its antiparticle),
we obtain the rep

which is all of , whereas without this particle we would just have

which is much less appealing--it wants to be , but it comes up short.

More importantly, there is increasing indirect evidence from experimental particle physics that right-handed neutrinos do exist. For details, see Pati [27]. If this is true, the number of fermions really could be 16, and we have a ready-made explanation for that number in the binary code.

However, this creates a new mystery. The
works nicely with the representation
, but does
not require this. It works just fine with the smaller rep

It would be nicer to have a theory that required us to use all of . Better yet, if our new GUT were an extension of , the beautiful explanation of hypercharges would live on in our new theory. With luck, we might even get away with using the same underlying vector space, . Could it be that the GUT is only the beginning of the story? Could unification go on, with a grand unified theory that extends just as extended the Standard Model?

Let us look for a group that extends and has an irrep whose dimension is some power of 2. The dimension is a big clue. What representations have dimensions that are powers of 2? Spinors.

What are spinors? They are certain representations of ,
the double cover of the rotation group in dimensions, which do not factor
through the quotient . Their dimensions are always a power of two.
We build them by exhibiting as a subgroup of a
Clifford algebra. Recall that the **Clifford algebra** is
the associative algebra freely generated by
with relations

If we take products of pairs of unit vectors in inside this algebra, these generate the group : multiplication in this group coincides with multiplication in the Clifford algebra. Using this fact, we can get representations of from modules of .

We can use this method to get a rep of
on
that extends the rep of on this space.
In fact, quite generally
acts on
. Then, because

becomes a representation of , called the Dirac spinor representation.

To see this, we use operators on
called `creation and annihilation operators'. Let
be the
standard basis for
. Each of these gives a **creation operator**:

We use the notation because is a Hilbert space, so is the adjoint of some other operator

which is called an

In physics, we can think of the basis vectors as particles.
For example, in the binary code approach to the
theory we imagine five particles from which the observed
particles in the Standard Model are composed: *up, down, red, green*
and *blue*. Taking the wedge product with `creates a particle'
of type , while the adjoint `annihilates a particle' of type .

It may seem odd that creation is the adjoint of annihilation, rather than its inverse. One reason for this is that the creation operator, , has no inverse. In some sense, its adjoint is the best substitute.

This adjoint does do what want, which is to delete any particle of type
. Explicitly, it deletes the `first' occurence of from any basis
element, bringing out any minus signs we need to make this respect the
antisymmetry of the wedge product:

And if no particle of type appears, we get zero.

Now, whenever we have an inner product space like , we get an inner product on . The fastest, if not most elegant, route to this inner product is to remember that, given an orthonormal basis for , the induced basis, consisting of elements of the form , should be orthonormal in . But choosing an orthonormal basis defines an inner product, and in this case it defines an inner product on the whole exterior algebra, one that reduces to the usual one for the grade one elements, .

It is with respect to this inner product on
that and
are adjoint. That is, they satisfy

for any elements . Showing this from the definitions we have given is a straightforward calculation, which we leave to the reader.

These operators satisfy the following relations:

where curly brackets denote the anticommutator of two linear operators, namely .

As an algebra,
is generated by the standard basis vectors of
. Let us call the elements of
corresponding to these
basis vectors
. From the definition of
the Clifford algebra, is easy to check that

In other words, the elements are anticommuting square roots of . So, we can turn into a -module by finding linear operators on that anticommute and square to . We build these from the raw material provided by and . Indeed, it is easy to see that

do the trick. Now we can map to these operators, in any order, and becomes a -module as promised.

Now for we may define
to be the universal cover of
, with group structure making the covering map

into a homomorphism. This universal cover is a double cover, because the fundamental group of is for in this range.

This construction of
is fairly abstract. Luckily, we can realize
as the multiplicative group in
generated by products
of pairs of unit vectors. This gives us the inclusion

we need to make into a representation of . From this, one can show that the Lie algebra is generated by the commutators of the . Because we know how to map each to an operator on , this gives us an explicit formula for the action of on . Each changes the parity of the grades, and their commutators do this twice, restoring grade parity. Thus, preserves the parity of the grading on , and does the same. This breaks into two subrepresentations:

where is the direct sum of the even-graded parts:

while is the sum of the odd-graded parts:

In fact, both these representations of
are irreducible, and
acts faithfully on their direct sum
. Elements of these
two irreps of
are called **left- and right-handed
Weyl spinors**, respectively, while elements of
are called
**Dirac spinors**.

All this works for any , but we are especially interested in the case . The big question is: does the Dirac spinor representation of extend the obvious representation of on ? Or, more generally, does the Dirac spinor representation of extend the representation of on ?

Remember, we can think of a unitary representation as a group homomorphism

where is the Hilbert space on which acts as unitary operators. Here we are concerned with two representations. One of them is the familiar representation of on :

which acts as the fundamental rep on and respects wedge products. The other is the representation of on the Dirac spinors, which happen to form the same vector space :

Our big question is answered affirmatively by this theorem, which can be found in a classic paper by Atiyah, Bott and Shapiro [2]:

Proof. The complex vector space has an underlying real vector space of dimension , and the real part of the usual inner product on gives an inner product on this underlying real vector space, so we have an inclusion . The connected component of the identity in is , and is connected, so this gives an inclusion and thus . Passing to Lie algebras, we obtain an inclusion . A homomorphism of Lie algebras gives a homomorphism of the corresponding simply-connected Lie groups, so this in turn gives the desired map .

Next we must check that makes the above triangle commute.
Since all the groups involved are connected, it suffices to check
that this diagram

commutes. Since the Dirac representation is defined in terms of creation and annihilation operators, we should try to express this way. To do so, we will need a good basis for . Remember,

If denotes the matrix with 1 in the th entry and 0 everywhere else, then the traceless skew-adjoint matrices have this basis:

For example, has the basis

and our basis for simply generalizes this.

Now, it is easy to guess a formula for in terms of
creation and annihilation operators. After all, the
elementary matrix satisfies

and acts the same way on . So, we certainly have

on the subspace . But do these operators agree on the rest of ? Remember, preserves wedge products:

for all . Differentiating this condition, we see that must act as derivations:

for all . Derivations of are determined by their action on . So, will be given on all by the above formulas if we can show that

are derivations.

Now, the annihilation operators are a lot like derivations: they are
antiderivations. That is, if
and
, then

However, the creation operators are nothing like derivations. They satisfy

because acts by wedging with , and moving this through introduces minus signs. Luckily, this relation combines with the previous one to make the composites into derivations for every and . We leave this for the reader to check.

So, can really be expressed in terms of annihilation and
creation operators as above. Checking that

commutes is now a straightforward but somewhat tedious job, which we leave to the dedicated reader.

This theorem had a counterpart for the GUT--namely,
Theorem 1. There we saw a homomorphism that
showed us how to extend the Standard Model group
to , and
made this square commute:

Now says how to extend further to , and makes this square commute:

We can put these squares together to get this commutative diagram:

This diagram simply says that is a GUT: it extends the Standard Model group in a way that is compatible with the Standard Model representation, . In Section 3.1, all the hard work lay in showing the representations and of were the same. Here, we do not have to do that. We just showed that extends . Since already extended , extends that, too.

2010-01-11