In the previous post in this series, I discussed the internal U(1) symmetry of the standard model of particle physics. This is the simplest symmetry in the standard model, and it gives rise to charge conservation and the existence of the electromagnetic field. But does the recognition of this symmetry tell us anything we didn't already know about the electromagnetic field and electrodynamics? No, not really. Local U(1) symmetry is a beautiful and economical way to describe the electromagnetic field, but it makes no new predictions about experimental behavior.

We'll now turn our attention to the second symmetry of the standard
model, SU(2). We will find that the combination of U(1) and SU(2),
which we write U(1) x SU(2), together with the concept of **spontaneous
symmetry breaking**, leads to a unified description of the
electromagnetic field and the weak field and their interactions with
leptons (electrons and neutrinos.) This unified description, first
published by Glashow,
Weinberg, and Salam in a series of papers in the 1960s, predicted
the existence of two new fundamental particles, the W and Z bosons,
which were experimentally observed in 1973.

We'll begin by examining a symmetry very similar to SU(2), which is
known as SO(3). SO(3) is simply the rotational symmetry of the sphere.
There are numerous representations of SO(3), but all have in common
that a particular rotation is specified by three continuous parameters.
For example, a rotation can be specified by its three Euler angles.
A more intuitive representation is a vector^{1} directed along
the axis of rotation with a length equal to the angle of rotation.

SO(3) has an important property that U(1) does not: It is not commutative. It doesn't matter in which order you apply a set of rotations in a two-dimensional plane. Rotating by 30 degrees, then 70 degrees, gives the same result as rotating by 70 degrees, then 30 degrees: 30+70 = 70+30. This is not true of arbitrary rotations in three-dimensional space. You can do an experiment right at your terminal to see that this is so.

Hold up your favorite CD case so that the front is facing you and the title is facing up. Now rotate the CD by 90 degrees around an axis pointed at your face. Then rotate by 90 degrees around an axis pointed directly to your left. You should get this result:

Now reverse the order of the rotations. Start with the CD facing you and the title up, as before. Now rotate by 90 degrees around an axis pointed directly to your left. Then rotate by 90 degrees around an axis pointed at your face. You should get this result:

The order in which the rotations are applied matters. We say that U(1) is a commutative group, while SO(3) is a noncommutative group. Sometimes you will hear these described as Abelian or non-Abelian groups, after the Norwegian mathematician Niels Henrik Abel.

In the previous post, we skimmed over a couple of points that we need to look at more closely before proceeding further.

The standard model of particle physics is a theory of *fields*.
These include gauge fields like electromagnetism, which we are already
accustomed to talking about as fields. But particles like electrons or
quarks are *also* treated as fields. A single particle
corresponds to a single excitation of the field. This treatment of
particles as fields is called **second quantization**.
(As if the first wasn't bad enough.)

In our discussion of U(1) and the electromagnetic field, we assumed
the existence of particle fields that were locally and *nontrivially*
invariant under a U(1) transformation. Our geometrical model was a
point moving in a circle in the U(1) plane. Such a point is described
by two coordinates, and these two coordinates correspond (in a slightly
nonobvious way) to a charged particle and its oppositely-charged
antiparticle. Such a field is described as a **complex scalar
field** because its simplest and most obvious representation is
as a complex
number. If U(1) is to be a local symmetry, then there must be a
gauge field with two independent polarization states, which we
recognize as the electromagnetic field. We have a total of four fields,
corresponding to the original particle and its antiparticle and the two
polarization states of the massless gauge field.

It is also possible to have a field that has only one component -- a
**real scalar field**. This field can be thought of as
sitting at the origin of the U(1) plane, where it is completely
unaffected by rotations in U(1). This field is *trivially*
invariant under U(1) and corresponds to a neutral particle, which is
its own antiparticle.

So far as U(1) is concerned, each kind of particle is represented by an independent field. Each neutral particle is represented by a real scalar field that is trivially symmetric under U(1), and each charged particle and its antiparticle is represented by a complex scalar field that is nontrivially symmetric under U(1). All are coupled to the two polarizations of the electromagnetic field.

Now let us turn our attention back to the SU(2) symmetry. SU(2)
closely related to, but not identical with, the SO(3) symmetry
describing rotations of the sphere. However, SU(2) is versatile,
operating on either a single complex number, a pair of complex numbers,
or a set of three real numbers encoded in a particular way.^{2}
A complex number is trivially invariant under an SU(2) transformation,
which leaves the number unchanged. We will see an example of this in
the standard model a little later.

When SU(2) operates on three encoded real numbers, the effect is exactly the same as rotation in three-dimensional space of a vector whose components are equal to the three encoded real numbers. In fact, each element of SO(3) corresponds to a pair of elements of SU(2), and the combination of two elements of SU(2) corresponds to a combination of the corresponding elements of SO(3). It is the existence of two distinct elements of SU(2) for every element of SO(3) that distinguishes SU(2) from SO(3) mathematically.

The most interesting case, for purposes of the present discussion, is when SU(2) operates on a pair of complex numbers. Each complex number has two components, so there is a total of four components per particle field in the corresponding particle model. SU(2) can be thought of as performing rotations in four-dimensional space, where these four components form a four-dimensional vector. However, an arbitrary rotation in four-dimensional space, SO(4), is described by six Euler angles, whereas SU(2) is described by just three parameters. So only a subset of the possible rotations is permitted.

To describe this subset, we must first understand how a rotation is defined in higher spatial dimensions. A given rotation is defined by two distinct directions in the plane of rotation. In 2-D, of course, there are only two directions to choose from, and only the amount of rotation is in question. In 3-D, one can rotate in the x-y plane, in the y-z plane, or in the z-x plane, or combinations of these. A rotation in the x-y plane leaves the z direction alone, and so on.

In four dimensions, *two *directions are left alone by a
given rotation. Let's call the four dimensions a, b, c, and d. Then a
rotation in the a-b plane leaves both the c and d directions alone.
This means that rotations in the a-b and c-d planes can occur
independently. In particular, the two rotations commute; it makes no
difference in what order you perform them.

SU(2) is built out of three combinations of rotations in 4-dimensional space. The first is a rotation in the a-b plane combined with an equal rotation in the c-d plane. The second is a rotation in the a-c plane combined with an equal rotation in the b-d plane. And the third is a rotation in the a-d plane combined with an equal rotation in the b-c plane. The pattern should be clear. Its significance is not, at least to me. I'll let you know if I ever figure it out.

Suppose we have a particle field consisting of two complex numbers, which can be operated on by both U(1) and SU(2). If we require the Lagrangian density to be unchanged by local transformations belonging to both U(1) and SU(2), we find that the four components of the particle field correspond to two particles and their antiparticles. We also must have a massless gauge field with two polarizations, like the electromagnetic field, for each parameter describing the symmetry. U(1) contributes one such gauge field; SU(2) contributes three more, for a total of eight gauge field components. The noncommutative nature of SU(2) requires that the three gauge fields interact with the particle with equal strength; U(1) can couple with a different strength. Our theory predicts the existence of a total of 8 gauge field components, plus two particles and their antiparticles per primordial field.

If a particle field consists of a single complex number, it is
trivially symmetric under SU(2). It contributes a single charged
particle and its antiparticle, but is unaffected by the SU(2) gauge
fields. If a particle field consists of two real numbers, it is
trivially symmetric under U(1). It contributes two neutral particles
that interact through the SU(2) gauge fields. And if a particle field
consists of a single real number, it is a single neutral particle that
is also unaffected by the SU(2) gauge fields. We do not necessarily
expect to find examples of all these kinds of particles in nature, but
if SU(2) is an internal symmetry of nature, then we should see *some*
of them.

What has Nature actually come up with? There should be a massless
field with two polarizations, and every kind of particle should either
be unaffected by this field or be paired with a particle of opposite
charge. This is an accurate description of the electromagnetic field
and of antiparticles. We might also have individual particles (called
isospin singlets) that do not interact with the SU(2) gauge fields, as
well as pairs of identical particles (called isospin doublets) that
interact with the SU(2) gauge fields in opposite ways. This is *not*
evident in nature.

We have one state of the electron (the left-handed state^{3})
that seems to be connected in some way to the electron neutrino (which
exists *only* in the left-handed state), and both seem to
interact with the weak force in the same way. We have a second state of
the electron (the right-handed state) that does not interact through
the weak force. The nature of the weak force itself was poorly
understood until Glashow, Weinberg, and Salam published the work
mentioned earlier, but it was known that the weak force had a very
short range, meaning that its gauge field particles had to be very
massive - not massless as predicted for the SU(2) gauge fields.

If the right-handed electron is an isospin singlet, and the weak
force corresponds to the SU(2) gauge fields, this would explain why the
right-handed electron does not interact through the weak force. Could
the left-handed electron and the electron neutrino be components of an
isospin doublet? This would explain the similarity of their weak
interactions. But then why are the particles otherwise so unalike? And
why aren't the SU(2) gauge fields massless? The answer given by
Weinberg *et al*. was based on a phenomenon called **spontaneous
symmetry breaking.**

Spontaneous symmetry breaking is neither a new concept nor
particularly difficult to understand. It occurs whenever there is more
than one low-energy state of a system, and the system is cooled to
where it must fall into one state or another. For example, a drop of
molten iron in zero gravity is rotationally symmetric. The spins of the
electrons in the drop are aligned at random, so that the drop as a
whole has no magnetic field. But as the drop slowly cools, the iron
atoms align themselves into a crystal lattice with a particular
orientation in space. The symmetry of the original liquid lattice is
spontaneously broken as the spins of the electrons align themselves
into a low-energy state. it doesn't matter in which direction the spins
are aligned -- that much of the original rotational symmetry remains --
but the spins have to align themselves in *some* direction to
reach their lowest-energy state. The drop crystallizes and acquires a
magnetic field.

Now consider a region of space that is low in energy. Any particles
that are present represent oscillations of the particle fields around
the **vacuum state,** which is the state of lowest
energy. We are accustomed to assuming that the vacuum state is the
state where the field strengths are all zero. This is certainly true of
the electromagnetic field, which is in its state of lowest energy when
no field is measurable. Add a little energy to the vacuum, and an
oscillating electric field appears.

But there is one field, which we will call the primordial field, for
which this is not true. The primordial field is an isospin doublet,
with four components. SU(2) symmetry requires that the energy of any
isospin doublet field depends only on the sum of the squares of the
components, which is a measure of the total field strength. However, it
does not put any constraints on how the energy of the field varies with
the total field strength. Such constraints are imposed by the
requirement of renormalizability^{4}. The most general
permissible dependence of energy on field strength consists of a peak
at zero field strength, a decrease in energy as the field strength
increases, then an increase again in energy as the field strength
continues to grow. Note the remarkable thing abou this: Zero total
field strength is *not* the minimum energy state, which we call
the vacuum. In fact, there is an infinite number of vacuum states. It's
easier to visualize this two components at a time, as in the following
diagram:

*Image hosting by flickr*

This diagram shows a three-dimensional surface formed by rotating
the energy curve around the central axis. This rotational symmetry is
that part of the overall SU(2) symmetry that applies to two components
of the primordial field. Note that the energy when the field components
are zero (on the axis of the surface) is not a minimum; the minimum
energy occurs on a circle some distance from the axis of rotation. Also
note that the "hump" at zero field strength is not *required*;
it is simply permitted. In the standard model, only the primordial
field has this feature.

Now suppose we break the symmetry by picking a point on the circle of minimum energy to be the vacuum state. If you were a little bug sitting at this point, the SU(2) symmetry of the energy surface might not be obvious at all. You would find yourself at the bottom of a gently curving valley with steep slopes to either side. Oscillations of the field in the radial direction -- the direction of the steeply sloping sides -- correspond to a massive particle, while oscillations in the azimuthal direction -- the direction along the floor of the valley -- correspond to a massless particle. The two particles arise from two symmetric components of the primordial field, but after the symmetry is broken, the particles are not at all alike.

So it is with the SU(2) symmetry of the standard model of particle physics. We start with four identical components of the primordial field and with two polarizations each of three identical massless SU(2) gauge fields. This is a total of ten fields. After we break the symmetry of the primordial field, all that is left of it is a single extremely massive particle called the Higgs boson. The other three components of the primordial field are absorbed into the SU(2) gauge fields, which no longer correspond to massless particles. Each now has a mass and a longitudinal polarization state that was not possible for the original massless field. Thus, the original ten fields have been transformed by symmetry breaking into ten new fields -- one Higgs boson plus three polarization states of three gauge fields.

What about the two polarizations of the U(1) gauge field? This field is not directly affected by the breaking of the SU(2) symmetry, so it remains a massless field with two polarizations. However, part of it is absorbed into one of the original SU(2) gauge fields, giving it a slightly higher mass. In place of the original three massless SU(2) gauge fields, we now have one massive electrically neutral gauge field, corresponding to the Z boson, and two slightly less massive gauge fields of opposite electrical charge corresponding to the W boson and its antiparticle.

It is important to understand that, at the time Glashow, Weinberg, and Salam first published their papers, the W and Z bosons had not been detected. It was known that the bosons responsible for the weak force were very massive, and it was believed that they were electrically charged, because all weak processes known at the time involved an exchange of electrical charge. The new theory, known as the U(1) x SU(2) electroweak theory, predicted masses for the W bosons and predicted the existence, hitherto unsuspected, of the Z boson and of weak processes that did not exchange charge -- the so-called neutral current processes. All these predictions were subsequently confirmed experimentally, and the three authors of the electroweak theory received the Nobel Prize for the success of their theoretical predictions.

According to the electroweak theory, the right-handed electron is a singlet of the SU(2) symmetry and does not interact through the weak force. The left-handed electron and electron neutrino form an SU(2) doublet and interact through the weak force in similar ways. The differences in their masses and electrical charges are consequence of the spontaneous symmetry breaking of the primordial field, with which they interact through the Higgs boson. There are similar electroweak singlets and doublets among the quarks, and the Higgs boson is responsible for the differences in their masses and charges as well.

Many questions remain. For example, there is a parameter in the theory that must be "tuned" to cause the electrical charge of the neutrino to be exactly zero. There is another parameter that must be "tuned" to cause the right-handed electron to have exactly the same charge as the left-handed electron. Tunable parameters are always disquieting. Particle physicists assume that the "tuning" will someday be explained by symmetries of Nature beyond those found in the standard model of particle physics.

Meanwhile, we still have the SU(3) symmetry of the Standard Model. This is responsible for the color force that binds quarks together. SU(3) will be the subject of the next post in this series.

^{1}A vector is a quantity with both magnitude and
direction, such as velocity. It is defined by its **components**,
equal in number to the dimension of the space in which the vector
lives. For example, a three-dimensional vector at the Earth's surface
could have a east component, a north component, and an up component.

^{2}As a rank-2 Hermitian matrix, for those of my readers
for whom this means something.

^{3}A particle whose spin is aligned in the same direction
as its velocity is called a **right-handed** particle. A
particle whose spin is aligned in the opposite direction is called a **left-handed**
particle. Unlike the "up" or "down" spin states, the handedness does
not depend on the frame of reference in which the particle is observed,
and it seems to be a more fundamental description of the spin state of
the particle.

^{4}All quantum field theories include certain kinds of
processes that appear to be unlimited in their action. However, a
properly constructed quantum field theory isolates these processes so
that they have no observable consequences. The mathematical procedure
that isolates these processes is known as **renormalization**,
and a quantum field theory for which such a mathematical procedure is
possible is called a **renormalizable theory**.