Farewell to Reality (26 page)

We can think about this another way. Suppose we discover mathematical relationships that describe the energy of a physical system and we find that these relationships do not change with time. Then we will also find that the energy so described is a quantity that is conserved — it is a quantity that can be converted from one form to another, but it cannot be created or destroyed.

A snooker player lines up a shot. He judges the speed of the cue ball and the angle of impact required to send the black into the corner pocket. He strikes the cue ball and pots the black (to ripples of applause).
His judgement and his technical skills as a snooker player are based on the principle of the conservation of linear momentum.

The momentum of an object is simply its mass multiplied by its speed in a given direction. When the cue ball strikes the black, its momentum is distributed between the cue ball and the black ball. The cue ball recoils, its mass unchanged but now with a different speed and direction. As a result of the impact, the black ball is propelled in the direction of the pocket at a certain speed. The total momentum of the cue ball and the black ball after the impact is the same as the momentum of the cue ball and the black ball prior to impact. Total momentum is conserved.

Noether found the equations describing changes in momentum to be invariant to continuous
translations in space.
The laws do not depend on location. They are the same here, there and everywhere.

An Olympic figure skater concludes her medal—winning performance. She enters a spin with arms and leg outstretched. As she draws her arms and leg back towards her centre of mass, she reduces her size as a spinning object as measured from her centre of rotation. Her linear momentum increases in compensation, and she spins faster. This is the conservation of angular momentum in action. For angular momentum, defined as motion in a circle at a constant speed, the equations are invariant to rotational symmetry transformations. They are the same irrespective of the
angle of direction
measured from the centre of the rotation.

Once the connection had been established, the logic of Noether's theorem could be turned on its head. Suppose there is a physical quantity which appears to be conserved but for which the laws governing its behaviour have yet to be worked out. If the physical quantity is indeed conserved, then the laws — whatever they are — must be invariant to a specific continuous symmetry transformation. If we can discover what this symmetry is, then we are well on the way to figuring out the mathematical form of the laws themselves.

Physicists found that they could use Noether's theorem to help them find a short cut to a theory, thus avoiding a lot of hit-and-miss speculation.

So, I can now reveal, the standard model of particle physics is built out of quantum field theories that are invariant to different continuous symmetry transformations. Consequently they respect the conservation
of certain physical properties. For example, QED is invariant to symmetry transformations within something called the U(1) symmetry group. It really doesn't matter too much what this means, but we can think of the U(1) symmetry group as describing transformations synonymous with continuous rotations in a circle.
²

Another way of representing U(1) is in terms of continuous transformations of the
phase angle
of a sine wave. Different phase angles correspond to different amplitudes of the wave in its peak—trough cycle. In the case of the electron, symmetry is preserved if changes in the phase of its wavefunction are matched by changes in its accompanying electromagnetic field. This ties the electron and the electromagnetic field together in an intimate embrace. The upshot is that, as a direct result of this phase symmetry, electric charge is conserved.

Although it is often difficult to follow the rather esoteric arguments from symmetry, make no mistake: this is powerful stuff. Electro-weak theory was constructed from an SU(2) quantum field theory ‘multiplied' by the U(1) quantum field theory of electromagnetism, written SU(2) × U(1). Quantum chromodynamics is an SU(3) quantum field theory.
³
The standard model, which we discussed previously as the product QCD × QFD × QED, is more often written in physics books and journal articles in terms of the symmetry groups to which these theories refer: as SU(3) × SU(2) × U(1).

The numbers in brackets refer to the number of variables, dimensions or ‘degrees of freedom' associated with the symmetry group. These are not the familiar dimensions of spacetime. They are abstract mathematical dimensions associated with the properties of the different symmetry groups.
^*

But although they are abstract, the dimensions have important consequences which are reflected in the properties and behaviours of particles and forces in our physical world. At this stage it's useful just to note that a U(1) quantum field theory
describes a force carried by a single force particle, the photon. An SU(2) quantum field theory describes a force carried by three force particles, the ‘heavy photons' W
⁺
, W
^-
and Z
⁰
. Finally, an SU(3) quantum field theory describes a force carried by eight different force particles, the coloured gluons. The higher the symmetry, the greater the number of particles that tend to be involved.

Symmetry and the search for grand unification

The standard model is a triumph. But don't be misled. It is not a unified theory of the fundamental atomic and subatomic forces. We should interpret the terminology SU(3) × SU(2) × U(1) literally. This rather clumsy notation tells us that the standard model is actually a collection — a product — of a set of distinct quantum field theories which describe the different forces. It doesn't explain where these forces come from or why they have the strengths that they have. And it doesn't explain why the elementary particles that are acted on by and carry these forces have the masses that they possess.

The attempts that were made in the mid—1970s to construct a grand unified theory, or GUT, were based on the search for an appropriate symmetry group. The SU(3) × SU(2) × U(1) structure of the standard model would spring naturally from the
breaking
of this higher symmetry, requiring more Higgs—like fields and Higgs particles. In the context of the standard model of big bang cosmology, this is thought to have happened towards the end of the grand unified epoch, about a trillionth of a trillionth of a trillionth of a second (10
^-36
seconds) after the big bang.

In 1974, Sheldon Glashow and Howard Georgi thought they had found the symmetry group on which a grand unified theory could be constructed. It was the symmetry group SU(5).

It seems perfectly sensible to look for unification at higher and higher symmetries, and Glashow and Georgi's symmetry group SU(5) appeared to fit the bill. It has five ‘dimensions'.
^*
Three are needed to accommodate the SU(3) field theory of QCD and two are needed for the SU(2) × U(1) theory of QFD × QED. SU(5) is therefore the minimum symmetry required to fit everything in.

At the time, this seemed like progress. A couple of the parameters of the standard model — those relating to the relative strengths of the subatomic forces — could now be tied to each other and so lost some of their apparent arbitrariness. But that was about as far as it went. The unified theory didn't resolve any of the fundamental problems of the standard model and had nothing to add to the story of the spontaneous breaking of the symmetry of the electro-weak force.

Opening up the number of ‘dimensions' of the symmetry group also had consequences beyond simply accommodating the known forces and their particles. One rather obvious consequence of a world which becomes more and more symmetrical at higher and higher energies is that the tremendous diversity we see in nature at ‘everyday' energies tends to disappear. The world becomes much more symmetrical. The result is really rather bland, with every particle having some kind of symmetrical relationship with every other. The now massless matter and force particles become virtually impossible to distinguish one from another, and the world is ruled by a single electro-nuclear force carried by massless bosons, called X bosons.

This means that in theories involving a higher symmetry, new symmetry relationships are established between the particles of the theory. These relationships persist when the symmetry is broken.

In the case of Glashow and Georgi's SU(5) GUT, these relationships are embodied through a series of particle ‘multiplets'. The symmetry group demands a series of fundamental particle quintuplets (groups of five) and a series of decuplets (groups often). For the first—generation quarks and leptons, one quintuplet consists of the red, green and blue down quarks, the positron and the electron anti—neutrino (five particles). The decuplet consists of red, green and blue up quarks, anti-up quarks, anti-down quarks and the electron (ten particles). Similar particle multiplets are established for the second- and third-generation quarks and leptons.

The X boson (thought to acquire a mass of the order of ten million billion — 10
¹⁶
— GeV as a consequence of breaking the SU(5) symmetry) can now mediate transitions between particles in each of these multiplets. The fact that these contain both quarks and leptons means that quarks and leptons can in principle be interconverted.

Imagine a proton consisting of red and green up quarks and a blue down quark. In the Georgi—Glashow model a green up quark in a
proton can be converted into an anti-red anti-up quark, and a blue down quark can be converted into a positron. As a consequence, a proton can now be transformed into a meson formed from a red up quark and anti-red anti-up quark (actually, this is a neutral pion) and a positron. The pion goes on to decay into two photons.

‘And then I realized that this made the proton, the basic building block of the atom, unstable,' Georgi said. ‘At that point I became very depressed and went to bed.'
⁴

Neutrons are known to be unstable due to the actions of the weak force. However, protons are observed to be rather more indomitable. Unstable neutrons in the nuclei of certain isotopes might make these isotopes radioactive, but life itself is incompatible with radioactive protons. The continued existence of life forms can be combined with the kinds of threshold levels of radioactivity that they can tolerate to deduce that the half—life of the proton must be at least a billion billion (10
¹⁸
) years. Any less than this and life as we know it would not be possible.
^*

But whilst the quark—lepton transformations made possible in the Georgi—Glashow model exposed the proton to some instability, this was hardly a harbinger of doom. The X boson is thought to possess a huge mass, and the probability of proton decay is related to the fourth power of this mass.
^**
Consequently, the model predicts a half—life for the proton of about a million trillion trillion (10
³⁰
) years, about a hundred million trillion (10
²⁰
) times the present age of the universe.

Now don't get confused. This long half—life doesn't mean that we would necessarily have to wait this long to observe a proton decaying. Radioactive decay is a spontaneous, random process thought to be driven by quantum fluctuations of the ‘vacuum', and so it can in principle happen at any time. However, the long half-life does suggest that proton decays, if they occur at all, would be very, very rare.
Nevertheless, collect 10
³⁰
or so protons together in a big tank, and there's a chance you might catch one in the act of decaying.

The prediction led to a rush to establish vast underground laboratory experiments to do just this. Such experiments involve the study of large tanks of water.
^*
The tanks are insulated from the effects of cosmic rays and other stray particles.

For example, the Super-Kamioke Nucleon Decay Experiments (Super-Kamiokande or Super-K) is a laboratory located in a mine about a kilometre beneath Mount Kamioke, near the city of Hida in Japan. The laboratory contains a stainless-steel tank holding 50,000 tonnes of ultra-pure water, surrounded by instruments tuned to detect the telltale signals of dying protons. There are similar laboratories in Kamataka, India, beneath the Alps in France and in mines in Minnesota and Ohio.

These experiments have concluded that the proton must have a half-life of the order of at least ten billion trillion trillion (10
³⁴
) years, about ten thousand times longer than the predictions of the Georgi—Glashow model.
^**

Georgi had good reasons to be depressed. This early example of a GUT yielded predictions for the stability of the proton that were not upheld by experiment. Other symmetries and algebras were tried, but no real solutions were forthcoming. An annual scientific conference on the subject of GUTs established in 1980 did not survive beyond 1989, as physicists turned their attentions elsewhere.

SUSY and the symmetry between fermions and bosons

Perhaps the problem is that SU(3) × SU(2) × U(l) is far from the full story, and we're reaching for a unified theory before we know quite what it is that we're supposed to be unifying. After all, the standard model is riddled with problems of its own. Instead of expecting these
to be solved through the development of GUTs, perhaps we should try to address these problems first and then look at how unification might then be achieved.

It was natural to continue to look to symmetry relationships for solutions to some of these problems, such as the hierarchy problem and its implications for the theoretical evaluation of the Higgs mass. Perhaps all the fine-tuning is more apparent than real. As American physicist Stephen Martin put it:

The systematic cancellation of the dangerous contributions to [the Higgs mass — in other words, the hierarchy problem] can only be brought about by the type of conspiracy that is better known to physicists as a symmetry.
⁵