In the last section, we learned how Heisenberg unified the proton and neutron into the nucleon, and that Yukawa proposed nucleons interact by exchanging pions. This viewpoint turned out to be at least approximately true, but it was based on the idea that the proton, neutron and pions were all fundamental particles without internal structure, which was not ultimately supported by the evidence.
Protons and neutrons are not fundamental. They are made of particles called
quarks. There are a number of different types of quarks,
called flavors. However, it takes only two
flavors to make protons and neutrons: the up quark, ,
and the down quark, . The proton consists of two up
quarks and one down:
|Fundamental Fermions (second try)|
Yet quarks, fundamental as they are, are never seen in isolation. They are always bunched up into particles like the proton and neutron. This phenomenon is called confinement. It makes the long, convoluted history of how we came to understand quarks, despite the fact that they are never seen, all the more fascinating. Unfortunately, we do not have space for this history here, but it can be found in the books by Crease and Mann , Segrè , and Pais .
It is especially impressive how physicists were able to discover that each flavor of quark comes in three different states, called colors: red , green , and blue . These `colors' have nothing to do with actual colors; they are just cute names--though as we shall see, the names are quite well chosen. Mathematically, all that matters is that the Hilbert space for a single quark is ; we call the standard basis vectors and . The color symmetry group acts on this Hilbert space in the obvious way, via its fundamental representation.
Since both up and down quarks come in three color states, there are
really six kinds of quarks in the matter we see around us.
Three up quarks, spanning a copy of
How could physicists discover the concept of color, given that quarks are confined? In fact confinement was the key to this discovery! Confinement amounts to the following decree: all observed states must be white, i.e., invariant under the action of . It turns out that this has many consequences.
For starters, this decree implies that we cannot see an individual
quark, because they all transform nontrivially under .
Nor do we ever see a particle built from two quarks, since
no unit vectors in
are fixed by . But
we do see particles made of three quarks: namely, nucleons!
This is because there are unit vectors in
So: color is deeply related to confinement. Flavor, on the other hand, is deeply related to isospin. Indeed, the flavor is suspiciously like the isospin of the nucleon. We even call the quark flavors `up' and `down'. This is no accident. The proton and neutron, which are the two isospin states of the nucleon, differ only by their flavors, and only the flavor of one quark at that. If one could interchange and , one could interchange protons and neutrons.
Indeed, we can use quarks to explain the isospin symmetry of Section 2.1. Protons and neutrons are so similar, with nearly the same mass and strong interactions, because and quarks are so similar, with nearly the same mass and truly identical colors.
So as in Section 2.1, let act on the flavor states . By analogy with that section, we call this the isospin symmetries of the quark model. Unlike the color symmetries , these symmetries are not exact, because and quarks have different mass and charge. Nevertheless, they are useful.
The isospin of the proton and neutron then arises from the isospin of
its quarks. Define
and the isospin up and down states at which their names hint. To
find the of a composite, like a proton or neutron, add the
for its constituents. This gives the proton and neutron the right
do not span a copy of the fundamental rep of inside
. So, as with color, the equations
In physics, the linear combination required to make and work
also involves the `spin' of the quarks, which lies outside of our
scope. We will content ourselves with showing that it can be
done. That is, we will show that
really does contain a copy of the fundamental rep
. To do this, we use the fact that any rank 2 tensor can be
decomposed into symmetric and antisymmetric parts; for example,
As a representation of , we thus have
As in the last section, there is no reason to have the full of isospin states for nucleons unless there is a way to change protons into neutrons. There, we discussed how the pions provide this mechanism. The pions live in , the complexification of the adjoint representation of , and this acts on :
Pions also fit into this model, but they require more explanation, because they are made of quarks and `antiquarks'. To every kind of particle, there is a corresponding antiparticle, which is just like the original particle but with opposite charge and isospin. The antiparticle of a quark is called an antiquark.
In terms of group representations, passing from a particle to its
antiparticle corresponds to taking the dual representation.
Since the quarks live in
, a representation
, the antiquarks live in the dual
has basis vectors called up and down:
All pions are made from one quark and one
antiquark. The flavor state of the pions must therefore live in
In writing these pions as quarks and antiquarks, we have once again neglected
to write the color, because this works the same way for all pions.
As far as color goes, pions live in
Finally, the Gell-Mann-Nishijima formula also still works for quarks,
provided we define the hypercharge for both quarks to be