U(1) x SU(2) x SU(3), part 4

Part 3 can be found here.

In the previous posts of this series, I discussed the U(1) x SU(2) sector of the Standard Model, which is known as electroweak theory. This post will wrap up by discussing the SU(3) sector of the Standard Model, then

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briefly sampling some ideas that go beyond the Standard Model.

As I explained in the earlier posts, U(1) x SU(2) x SU(3) is a description of the internal symmetries of the Lagrangian density of the Standard Model. U(1) is the symmetry of a circle lying in a plane, which is unchanged by rotations in the plane. SU(2) is harder to visualize, but it is closely related to SO(3), the symmetry of a sphere in three dimensions.

Noether's Theorem tells us that every continuous symmetry of the Lagrangian density corresponds to some conservation law. We find that the original U(1) symmetry corresponds to the conservation of electric charge, and that the SU(2) symmetry introduces three more U(1) symmetries that correspond to conservation of electron, muon, and tau lepton numbers. The SU(2) symmetry itself corresponds to conservation of a quantity called "weak isospin."

The Standard Model requires that these symmetries be local symmetries; that is, the Lagrangian density should be unchanged by a local U(1) or SU(2) transformation, one that smoothly varies across spacetime. This requires that there be additional fields, called gauge fields, that transform under local U(1) or SU(2) transformations in a way that precisely cancels the transformation of the original particle fields. These gauge fields must be massless vector fields, like the electromagnetic field, and in fact we find that U(1) symmetry produces a field closely resembling the electromagnetic field.

However, observations show that the three gauge fields associated with the weak force are massive, not massless, fields. Glashow, Weinberg, and Salam explained this by showing that spontaneous breaking of the SU(2) symmetry changes the original massless U(1) gauge field and three massless SU(2) gauge fields into a single massless vector field (corresponding to electromagnetism) and three massive vector fields (corresponding to the W and Z bosons of the weak interaction.) The Glashow-Weinberg-Salam electroweak theory matches all observations to within experimental error.

All of these ideas, which I described in more detail in the earlier posts in this series, can be applied in a straightforward way to the remaining symmetry of the Standard Model, the SU(3) symmetry. However, we will find that the resulting color forces have some novel properties.

SU(2) is mathematical shorthand for "the special unitary group of rank 2." It's closely related to SO(3), the symmetry of the sphere in three dimensions. Both the elements of SU(2) and the elements of SO(3) are described by three parameters, corresponding to the three Euler angles of a three-dimensional rotation. SU(2) can also be visualized as a special subset of the possible rotations of a sphere in four dimensions, SO(4).

SU(3) is harder to visualize. One might guess that it is closely related to SO(4) in the same way that SU(2) is closely related to SO(3), but this turns out not to be the case. SU(3) is described by eight independent parameters, while SO(4) is described by six independent parameters, so that no correspondence is possible. However, SU(3) can be visualized as a special subset of SO(6), the set of possible rotations in six dimensions, in the same way that SU(2) is a special subset of SO(4). But perhaps it is best to admit that, at this point, we are beyond anything that we can visualize geometrically.

SU(3) does share an important characteristic with SU(2): It is a non-commutative group. In other words, the ordering of elements of SU(3) is significant: If A and B are elements of SU(3), then AB is not the same as BA, as would be the case if A and B were ordinary numbers. We will see shortly that this has important consequences.

When we promote SU(3) to a local symmetry of the Lagrangian density, we obtain a gauge field for each parameter describing an element of SU(3). There are eight such parameters, and therefore eight such gauge fields. At present, it is thought that there is no spontaneous breaking of the SU(3) symmetry, so these eight gauge fields remain massless. The particles corresponding to these fields are called gluons, and they are responsible for the color force that holds quarks together.

Quarks resemble leptons in certain respects. Both leptons and quarks are massive spin-1/2 particles that interact through gravitation and the electroweak force. However, only quarks interact with gluons. Left-handed electrons/muons/taus and electron/muon/tau neutrinos form SU(2) doublets, while right-handed electrons/muons/taus are SU(2) singlets. The families of quarks can be organized the same way. For example, left-handed up and down quarks form an SU(2) doublet while right-handed up and down quarks are each SU(2) singlets. Each family is further broken down by color; for example, there are three colors of up quark that form an SU(3) triplet. (The color property of quarks has nothing to do with the conventional concept of color; it is simply an arbitrary and somewhat whimsical name for a property of quarks that plays a similar role with the color force that electrical charge plays with the electromagnetic force.)

One might guess that the color force and the electromagnetic force would closely resemble each other. After all, both arise from massless vector gauge fields. In fact, the color force does resemble the electromagnetic force at short distances, even to the extend of obeying an inverse-square law. But the forces do not resemble each other at all at larger distances. The electromagnetic force has an infinite range, obeying the inverse-square law to arbitrarily large distances, while the color force is subject to confinement: Quarks and gluons exist only in combinations with no net color charge, and any attempt to pry these apart and produce isolated quarks and gluons fails, because the energy involved is great enough to produce new colorless combinations of quarks and gluons.

It is the fact that SU(3) is a noncommutative group that is responsible for confinement. Recall that a gauge field subject to a local SU(3) transformation yields terms that exactly cancel the terms produced by the action of this transformation on the original particle fields. One of these terms is of the form AB-BA, where A and B are any two of the gauge fields. For U(1), where there is only one gauge field, this term does not exist. But for SU(3), which has eight noncommutative fields, AB does not always equal BA and this term is nonzero. It corresponds to a direct interaction between the gauge fields: The gluons stick to each other (hence their name.) The consequence of all this stickiness is confinement.

Of course, the SU(2) symmetry is also noncommutative, but we do not see confinement with the weak force, and it is possible to produce leptons and the W and Z bosons in isolation. The difference here is that the spontaneous breaking of the SU(2) symmetry gives the W and Z bosons mass, and the absolute strength of the weak force is much less than the absolute strength of the color force. Because the W and Z bosons are massive and only slightly sticky, the weak interaction dies out at much smaller distances than those at which we would expect to see confinement.

We know that the Standard Model cannot be complete, because it does not include gravitation.

Gravitation is the gauge field corresponding to the SL(2,c) symmetry of Special Relativity. However, the quantum version of this gauge field theory is not renormalizable. This is interpreted as being a consequence of the fact that, at the shortest distance scales, spacetime becomes a kind of chaotic foam that defies analysis.

The best present candidate for a quantum theory of gravity is superstring theory. Particles are replaced by tiny vibrating strings whose oscillation modes correspond to different kinds of particles. It is hoped that some version of this theory will prove to be renormalizable, but the mathematics of superstrings are simply appalling, and progress is slow.

If we ignore gravity for the present, is it possible to simplify the Standard Model by unifying the electroweak and color forces?

Theories that seek to do so are known as Grand Unified Theories, or GUTs. There are many such theories and none is without its problems. Most predict that the U(1) x SU(2) x SU(3) structure of the Standard Model is the result of a spontaneous breaking of the original GUT symmetry very early in the history of the universe. This symmetry breaking is thought to take place at an energy of 1016 GeV, corresponding to a temperature of about 1029 degrees. Such energies and temperatures are far beyond anything we can imagine ever exploring in the laboratory. Thus, GUT theories must be tested indirectly, by looking for such things as proton decay or magnetic monopoles left over from the Big Bang. None of these things has ever been observed, leaving us pretty much in the dark.

In fact, some physicists have begun to voice a fear that our theories have raced far ahead of any observations we can ever hope to make, so that the ultimate theory of the universe will prove to be forever beyond our grasp. Other physicists seem to hope that we can reason our way to a single consistent theory, proving that all other candidates for a Theory of Everything have fatal inconsistencies that rule them out, without the need for observational tests. One wonders whether, in the end, the line between theoretical physics and theology will prove to be all that sharp.