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 , shortly after his paper with Glashow on the theory. However, Georgi has said that he conceived of the theory first. See Zee , 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,
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
Standard Model rep have dimension
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
More importantly, there is increasing indirect evidence from experimental particle physics that right-handed neutrinos do exist. For details, see Pati . 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
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
We can use this method to get a rep of
that extends the rep of on this space.
In fact, quite generally
. Then, because
To see this, we use operators on
called `creation and annihilation operators'. Let
standard basis for
. Each of these gives a creation operator:
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:
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
are adjoint. That is, they satisfy
These operators satisfy the following relations:
As an algebra,
is generated by the standard basis vectors of
. Let us call the elements of
corresponding to these
. From the definition of
the Clifford algebra, is easy to check that
Now for we may define
to be the universal cover of
, with group structure making the covering map
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
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
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
Now, it is easy to guess a formula for in terms of
creation and annihilation operators. After all, the
elementary matrix satisfies
Now, the annihilation operators are a lot like derivations: they are
antiderivations. That is, if
So, can really be expressed in terms of annihilation and
creation operators as above. Checking that
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: