In the Standard Model, the weak and electromagnetic forces are two aspects of something called the `electroweak force', which is described by the group . Curiously, it turns out that the familiar concept of `electric charge' is less fundamental than the concepts of `weak isospin' and `hypercharge'. The weak isospin of a particle describes how it transforms under , while its hypercharge describes how it transforms under . The electric charge is computed in a funny way from these two!

In the following problems, you will examine how this works.
The full symmetry group of the Standard Model is

but we will focus on the electroweak force, so we'll ignore and and only think of particles as irreps of . You can read these irreps off the elementary particle chart at the end of this handout.

We begin by introducing bases for and :

- The Lie algebra consists of traceless skew-adjoint
complex matrices, so it has a basis consisting of
the matrices
, where

- The Lie algebra consists of skew-adjoint complex matrices, which are the same as imaginary numbers. For these we choose the basis vector .

In quantum theory, observables correspond to self-adjoint
operators. Given any unitary representation of
on some Hilbert space, the above basis of
gets mapped by to skew-adjoint operators on that Hilbert space.
Dividing these by we get self-adjoint operators called
the three components of **weak isospin**: and .
In short:

where we have cancelled some factors of in a slightly underhanded manner.

Similarly, given any unitary representation of
on some Hilbert space, the above basis of
gets mapped by to a skew-adjoint operator.
Dividing this by we get a self-adjoint operator called
**hypercharge**, . In short:

**Digression on History and Terminology**

*The following stuff is not strictly necessary for doing the
homework. But if you want to understand the terms `isospin'
and `hypercharge' and that weird factor of ,
read this!*

The group first showed up in physics because it is the double
cover of the rotation group . The matrices
serve as a convenient basis for the Lie algebra because they
satisfy the commutation relations

which show that is isomorphic to with its usual cross product:

While the matrices are skew-adjoint, the matrices are self-adjoint, so they correspond to observables of any quantum system with symmetry. When is used to describe the rotation symmetries of some system, the observables corresponding to the matrices are called the components of its

Later, Heisenberg proposed as a symmetry group for the strong
nuclear force. His idea was that this group would explain a symmetry
between protons and neutrons: both these particles would really be two
states of a single particle called the **nucleon**, which would
transform under the spin- representation of .
The proton would be the spin-up state:

while the neutron would be the spin-down state:

However, the `spin' in question here has nothing to do with angular momentum -- we're just reusing that word because the same group was used to describe rotation symmetries. To avoid confusion, Heisenberg needed another name for this new sort of `spin'. Since different isotopes of an element differ in how many neutrons they have in their nucleus, he coined the term `isotopic spin'. Later this got shortened to `isospin'. So, we call the observables corresponding to in this new context the three components of a particle's

Still later, Glashow, Weinberg and Salam used as a symmetry
group for the weak nuclear force, and called the observables
corresponding to
the three components of
a particle's **weak isospin**, .
There is a close relation between weak isospin and Heisenberg's
original isospin: in particular, the isospin of a nucleon is
the sum of the weak isospins of the three quarks it is
made of. However, weak isospin is now considered to be important
for the weak rather than the strong nuclear force.

Similarly, started out being used as a symmetry group for
the electromagnetic force. For each integer
there is a unitary irrep of on
called the **charge- irrep**. This is given by

for any unit complex number and any vector . Differentiating with respect to and setting , we get an irrep of the Lie algebra on with

The operator is skew-adjoint, but we can divide it by to get a self-adjoint operator

or if we cancel some factors of in a slightly underhanded way. It's easy to see that

The observable corresponding to is called **electric charge**,
and the above equation says that any state of a particle
described by the charge- irrep of has electric
charge equal to .
Using the group this way gives a nice `explanation' of the
fact that the electric charge is quantized: the charge of any
particle is an integer times some smallest charge. However, it
doesn't say what this smallest charge actually is!

For a long time people thought that the electron had the smallest
possible charge, so they said the electron has charge 1.
Actually they said it has charge : an unfortunate
convention which we can blame on Benjamin Franklin, because he was
mixed up about which way the electricity flowed in a current. But
what do you expect from someone who *flies a kite with a key
hanging on it during a thunderstorm, to attract lightning bolts?*
Dumb! But lucky: the next two people to try that experiment were killed.

Much later, people discovered that quarks have electric charges smaller than that of the electron. Measured in units of the electron charge, quark charges are integral multiples of . Mathematically it would be nicest to redefine our units of charge so the smallest possible charge is still , but people are too conservative to do this, so now the smallest charge is taken to be .

Still later, people reused the group as a symmetry group for the electroweak force, and used the term `hypercharge' for the observable corresponding to this new . Since hypercharge is closely related to charge, physicists also measure hypercharge in integral multiples of .

Here's how we accomodate this foolish factor of .
For each number with , there is a unitary
irrep of on called the **hypercharge-
irrep**. This is given by

Differentiating with respect to as before, we get an irrep of on with

The operator is skew-adjoint, but dividing it by we get a self-adjoint operator

or for short. It's easy to see that

We call the observable corresponding to

**Back to Business**

Any particle in the Standard Model corresponds to some unitary irrep
of
. This is a unitary rep of both
and of , so we get self-adjoint operators
and on this irrep, corresponding to weak isospin
and hypercharge. The observable **electric charge** is
related to these by the mystical formula

Now let's use this to work out the electric charges of all the elementary particles!

I'll do an example: consider the left-handed
electron neutrino . As indicated in
the chart at the end of this handout,
this is the first member of the standard basis of the irrep
of
:

Note that

Since the eigenvalue is , a physicist reading this equation will say `the left-handed electron neutrino has '.

In the chart at the end of this handout, the hypercharge-
irrep of is denoted .
As explained in the Digression, the hypercharge operator acts
as multiplication by the number on any vector in this representation.
Since the left-handed electron lives in the
hypercharge- rep, it follows that

Now that we know and for the left-handed electron
neutrino, we can use the magic formula to work out its electric charge:

Since the eigenvalue is 0, the left-handed electron neutrino has electric charge 0. And indeed, this particle is neutral!

1. Use this idea to fill out as much of the following chart as you can. If you know enough representation theory you can do it all! It may help to reread the list of conventions in the previous homework on elementary particles.

type of particle eigenvalue of: |
I_{3} |
Y |
Q |

GAUGE BOSONS |
|||

HIGGS BOSON |
|||

FIRST GENERATION
FERMIONS |
|||

Leptons: |
|||

Quarks: |
|||

SECOND GENERATION FERMIONS |
|||

Leptons: |
|||

Quarks: |
|||

THIRD GENERATION FERMIONS |
|||

Leptons: |
|||

Quarks: |
|||

2. In Problem 1 of the previous homework you may have noticed that for leptons and quarks, the average of the hypercharge of the right-handed ones is equal to the hypercharge of the left-handed one. Use your new-found knowledge to say more about the significance of this fact.

3. What is the sum of the hypercharges of all the fermions in a given generation? To do this right you have to sum over all 16 basis vectors of the fermion rep, e.g. , , , , ,, ,,, ,,, ,,.

4. What is the sum of the eigenvalues of over all the fermions in a given generation?

5. What is the sum of the electric charges of all the fermions in a given generation?

The answers to questions 3-5 are very important in **grand unified
theories**. These are theories where
is embedded as a Lie subalgebra of some **simple** Lie algebra like
or : i.e., a Lie algebra with no nontrivial ideals.
The fermion rep can only extend to a rep of a simple Lie algebra
if the answers to questions 3-5 take a certain special form!

type of particle | irrep | irrep | irrep | irrep |

GAUGE BOSONS |
||||

gluons ( force carriers): | ||||

massless spin-1 | ||||

force carriers: | ||||

massless spin-1 | ||||

force carrier: | ||||

massless spin-1 | ||||

HIGGS BOSON |
||||

Higgs: | ||||

massless spin-0 | ||||

and its antiparticle! | ||||

FIRST GENERATION
FERMIONS |
||||

Leptons: |
||||

left-handed electron neutrino and electron: | ||||

left-handed massless spin-1/2 | ||||

right-handed electron neutrino: | ||||

right-handed massless spin-1/2 | ||||

right-handed electron: | ||||

right-handed massless spin-1/2 | ||||

and their antiparticles! |
||||

Quarks: |
||||

left-handed up and down quarks: | ||||

left-handed massless spin-1/2 | ||||

right-handed up quark: | ||||

right-handed massless spin-1/2 | ||||

right-handed down quark | ||||

right-handed massless spin-1/2 | ||||

and their antiparticles! |
||||

SECOND GENERATION
FERMIONS |
||||

Leptons: |
||||

left-handed mu neutrino and muon: | ||||

left-handed massless spin-1/2 | ||||

right-handed mu neutrino: | ||||

right-handed massless spin-1/2 | ||||

right-handed muon: | ||||

right-handed massless spin-1/2 | ||||

and their antiparticles! |
||||

Quarks: |
||||

left-handed charm and strange quarks: | ||||

left-handed massless spin-1/2 | ||||

right-handed charm quark: | ||||

right-handed massless spin-1/2 | ||||

right-handed strange quark | ||||

right-handed massless spin-1/2 | ||||

and their antiparticles! |
||||

THIRD GENERATION
FERMIONS |
||||

Leptons: |
||||

left-handed tau neutrino and tau: | ||||

left-handed massless spin-1/2 | ||||

right-handed tau neutrino: | ||||

right-handed massless spin-1/2 | ||||

right-handed tau: | ||||

right-handed massless spin-1/2 | ||||

and their antiparticles! |
||||

Quarks: |
||||

left-handed top and bottom quarks: | ||||

left-handed massless spin-1/2 | ||||

right-handed top quark: | ||||

right-handed massless spin-1/2 | ||||

right-handed bottom quark | ||||

right-handed massless spin-1/2 | ||||

and their antiparticles! |

** Next:** The
True Internal Symmetry Group of the Standard Model
** Previous:** Elementary Particles

© 2003 John Baez - all rights reserved.

baez@math.removethis.ucr.andthis.edu