2.3.2 Hypercharge and U(1)

In Section 2.2.2, we saw how to extend the notion of isospin to weak isospin, which proved to be more fundamental, since we saw in Section 2.3.1 how this gives rise to interactions among left-handed fermions mediated via bosons.

We grouped all the fermions into representations. When we did this in
Section 2.1, we saw that the representations of
particles were labeled by a quantity, the hypercharge , which relates the
isospin to the charge via the Gell-Mann-Nishijima formula

We can use this formula to extend the notion of hypercharge to weak
hypercharge, a quantity which labels the weak isospin representations. For
left-handed quarks, this notion, like weak isospin, coincides with the old
isospin and hypercharge. We have weak hypercharge
for these
particles:

But just as weak isospin extended isospin to leptons, weak hypercharge extends hypercharge to leptons. For left-handed leptons the Gell-Mann-Nishijima formula holds if we set :

Note that the weak hypercharge of quarks comes in units one-third the size of the weak hypercharge for leptons, a reflection of the fact that quark charges come in units one-third the size of lepton charges. Indeed, thanks to the Gell-Mann-Nishijima formula, these facts are equivalent.

For right-handed fermions, weak hypercharge is even simpler. Since
for these particles, the Gell-Mann-Nishijima formula reduces to

So, the hypercharge of a right-handed fermion is twice its charge. In summary, the fermions have these hypercharges:

The First Generation of Fermions -- Hypercharge | ||

Name | Symbol | |

Left-handed leptons | ||

Left-handed quarks | ||

Right-handed neutrino | ||

Right-handed electron | ||

Right-handed up quark | ||

Right-handed down quark | ||

But what is the meaning of hypercharge? We can start by reviewing our answer for the quantity . This quantity, as we have seen, is related to how particles interact via bosons, because particles with span the fundamental representation of , while the bosons span the complexified adjoint representation, which acts on any other representation. Yet there is a deeper connection.

In quantum mechanics, observables like correspond to self-adjoint
operators. We will denote the operator corresponding to an observable with a
caret, for example is the operator corresponding to . A state
of specific , like which has
, is an eigenvector,

with an eigenvalue that is the of the state. This makes it easy to write as a matrix when we let it act on the with basis and , or any other doublet. We get

Note that this is an element of divided by . So, it lies in , the complexified adjoint representation of . In fact it equals , one of the gauge bosons. So, up to a constant of proportionality, the observable is one of the gauge bosons!

Similarly, corresponding to hypercharge is an observable . This is also, up to proportionality, a gauge boson, though this gauge boson lives in the complexified adjoint rep of .

Here are the details. Particles with hypercharge span irreps
of
. Since
is abelian, all of its irreps are one-dimensional. By
we denote
the one-dimensional vector space with action of given by

The factor of takes care of the fact that might not be an integer, but is only guaranteed to be an integral multiple of . For example, the left-handed leptons and both have hypercharge , so each one spans a copy of :

or, more compactly,

where is trivial under .

In summary, the fermions we have met thus far lie in these representations:

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 | |||

Now, the adjoint representation of is just the tangent space to
the unit circle in at 1. It is thus parallel to the imaginary axis, and
can be identified as . Is is generated by . also generates the
complexification,
, though this also has other
convenient generators, like 1. Given a particle
of
hypercharge , we can differentiate the action of on

and set to find out how acts:

Dividing by we obtain

In other words, we have

as an element of the complexified adjoint rep of .

Particles with hypercharge interact by exchange of a boson, called the boson, which spans the complexified adjoint rep of . Of course, since is one-dimensional, any nonzero element spans it. Up to a constant of proportionality, the boson is just , and we might as well take it to be equal to , but calling it is standard in physics.

The boson is a lot like another, more familiar gauge boson--the photon! The hypercharge force which the boson mediates is a lot like electromagnetism, which is mediated by photons, but its strength is proportional to hypercharge rather than charge.

2010-01-11