We have studied three different grand unified theories: the , and theories. The and theories were based on different visions about how to extend the Standard Model. However, we saw that both of these theories can be extended to the theory, which therefore unites these visions.
The theory is all about treating isospin and color on an equal footing: it combines the two isospins of with the three colors of , and posits an symmetry acting on the resulting . The particles and antiparticles in a single generation of fermions are described by vectors in . So, we can describe each of these particles and antiparticles by a binary code indicating the presence or absence of up, down, red, green and blue.
In doing so, the theory introduces unexpected relationships
between matter and antimatter. The irreducible representations of
But the Standard Model has another
does respect. This is the distinction between left- and
right-handedness. Remember, the left-handed particles and
antiparticles live in the even grades:
This characteristic of the theory lives on in its extension to . There, the distinction between left and right is the only distinction among particles and antiparticles that knows about, because and are irreducible. This says the theory unifies all left-handed particles and antiparticles, and all right-handed particles and antiparticles.
In contrast, the theory was all about adding a fourth `color', , to represent leptons, and restoring a kind of symmetry between left and right by introducing a right-handed that treats right-handed particles like the left-handed treats left-handed particles.
Unlike the theory, the
-gradings in the Standard Model: the matter-antimatter
grading, and the right-left grading.
The reason is that
, and we have:
When we extend
. Then the
using addition in
This sounds rather technical, but it is as simple as ``even + odd = odd'':
Furthermore, the two routes to the
theory that we have described,
one going through and the other through
are compatible. In other words, this cube commutes:
So, all four theories fit together in an elegant algebraic pattern. What this means for physics--if anything--remains unknown. Yet we cannot resist feeling that it means something, and we cannot resist venturing a guess: the Standard Model is exactly the theory that reconciles the visions built into the and theories.
What this might mean is not yet precise, but since all these
theories involve symmetries and representations, the `reconciliation' must
take place at both those levels--and we can see this
in a precise way.
First, at the level of symmetries, our Lie groups are related by the
commutative square of homomorphisms:
To see this, first recall that the image of a group under
a homomorphism is just the quotient group formed by
modding out the kernel of that homomorphism. If we do this for each of our
homomorphisms above, we get a commutative square of inclusions:
As a step towards showing this, first consider what happens when we pass from
the spin groups to the rotation groups. We can accomplish this by modding out
by an additional
above. We get another commutative square of
Now, let us show:
Proof. We can prove this in the same manner that we showed, in
Section 3.1, that
To begin with, the group is the group of orientation-preserving symmetries of the 10-dimensional real inner product space . But is suspiciously like , a 5-dimensional complex inner product space. Indeed, if we forget the complex structure on , we get an isomorphism , a real inner product space with symmetries . We can consider the subgroup of that preserves the original complex structure. This is . If we further pick a volume form on , i.e. a nonzero element of , and look at the symmetries fixing that volume form, we get a copy of .
Then we can pick a splitting on
subgroup of that also preserves this is
We can also reverse the order of these processes. Imposing a splitting on yields a splitting on the underlying real vector space, . The subgroup of that preserves this splitting is : the block diagonal matrices with and orthogonal blocks and overall determinant 1. The connected component of this subgroup is .
The direct summands in
came from forgetting the complex
. The subgroup of
preserving the original complex structure is
and the subgroup of this that also fixes a volume
This group is connected, so it must lie entirely
in the connected component of the identity, and we get the inclusions:
It follows that
is precisely the subgroup of
that preserves a complex structure on
, a chosen volume form on the
resulting complex vector space, and a splitting on this space. But this
splitting is the same as a compatible splitting of
, one in which each summand is a complex vector subspace as well as a
real subspace. This means that
From this, a little diagram chase proves our earlier claim:
Proof. By now we have built the following commutative diagram:
The kernel of consists of two elements, which we will
simply call . Since
, we know
In short, the Standard Model has precisely the symmetries shared by both the theory and the theory. Now let us see what this means for the Standard Model representation.
We can `break the symmetry' of the theory in two different ways. In the first way, we start by picking the subgroup of that preserves the -grading and volume form in . This is . Then we pick the subgroup of that respects the splitting of into . This subgroup is the Standard Model gauge group, modulo a discrete subgroup, and its representation on is the Standard Model representation.
We can draw this symmetry breaking process in the following diagram:
We can also break the symmetry of in a way that uses the theory as a halfway house. We do essentially the same two steps as before, but in the reverse order! This time we start by picking the subgroup of that respects the splitting of as . This subgroup is modulo a discrete subgroup. The two factors in this subgroup act separately on the factors of . Then we pick the subgroup of that respects the -grading and volume form on . This subgroup is the Standard Model gauge group, modulo a discrete subgroup, and its representation on is the Standard Model representation.
We can draw this alternate symmetry breaking process in the following