2.3.1 Isospin and SU(2), Redux

The tale we told of isospin in Section 2.1 only concerned the strong force, which binds nucleons together into nuclei. We learned about an approximation in which nucleons live in the fundamental rep of the isospin symmetry group , and that they interact by exchanging pions, which live in the complexified adjoint rep of this group, namely .

But this tale is mere prelude to the modern story, where weak isospin,
defined in Section 2.2.2, is the star of the show. This story
is not about the strong force, but rather the weak force. This story
parallels the old one, but it involves left-handed fermions instead of nucleons.
The left-handed fermions, with
, are paired up into
fundamental representations of , the weak isospin symmetry
group. There is one spanned by left-handed leptons:

and one spanned by each color of left-handed quarks:

The antiparticles of the left-handed fermions, the right-handed antifermions, span the dual representation .

Because these particles are paired up in the same representation,
physicists often write them as doublets:

with the particle of higher written on top. Note that we have suppressed color on the quarks. This is conventional, and is done because acts the same way on all colors.

The particles in these doublets then interact via the exchange of bosons,
which are the weak isospin analogues of the pions. Like the pions, there are
three bosons:

They span the complexified adjoint rep of , , and they act on each of the doublets like the pions act on the nucleons, via the action of on . For example,

Again, Feynman diagrams are the physicists' way of drawing intertwining operators. Since all the 's are acted on by the same , they can interact with each other via boson exchange. For example, quarks and leptons can interact via 's:

The fact that only left-handed particles are combined into doublets reflects
the fact that only they take part in weak interactions. Every right-handed
fermion, on the other hand, is trivial under . Each one spans the
trivial rep, . An example is the right-handed electron

Physicists call these particles singlets, meaning they are trivial under . This is just the representation theoretic way of saying the right-handed electron, , does not participate in weak interactions.

In summary, left-handed fermions are grouped into doublets (nontrivial representations of on ), while right-handed fermions are singlets (trivial representations on ). So, the left-handed ones interact via the exchange of bosons, while the right-handed ones do not.

The First Generation of Fermions --
Representations |
|||

Name | Symbol | rep | |

Left-handed leptons | |||

Left-handed quarks | |||

Right-handed neutrino | |||

Right-handed electron | |||

Right-handed up quark | |||

Right-handed down quark | |||

2010-01-11