- The orthogonal groups SO(N)
- The unitary groups SU(N)
- The symplectic groups Sp(N)
- Exceptional Groups G2 F4 E6 E7 E8
*Asymptotic Freedom and the Emergence of QCD*, David Gross*Kaluza-Klein Theory in Perspective*, Michael Duff*The Life and Times of Emmy Noether*, Nina Byers*Exact Electromagnetic Duality*, David Olive*The Status of Supersymmetry*, Jonathon Bagger*The Unreasonable Effectiveness of Quantum Field Theory*, R. Jackiw

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Part 5 of 9

We don't have to examine nature very closely to see its
beauty. A bird, a forest or a galaxy has a form of beauty
which is typical of complex organised systems. A
snowflake has another element to its beauty which is
also very common in nature but which is often only
evident on close inspection. We call it *symmetry*.

The snowflake begins its life as a minute hexagonal crystal forming in a cloud. During its passage from there to the ground, it experiences a sequence of changes in temperature and humidity which cause it to grow at varying rates. Its history is recorded in the variations of thickness in its six petals as it grows. This process ensures that each petal is virtually identical and accounts for the snowflakes symmetry.

When a snowflake is rotated through an angle of 60 degrees
about its centre, it returns to a position where it looks
the same as before. It is said to be *invariant*
under such a transformation and it is invariance which
characterises symmetry. The shape of the snowflake is also
invariant if it is rotated through 120 degrees. It is
invariant again if it is turned over. By combining rotations and
turning over it is possible to find 12 different
transformations (including the identity transformation which
does nothing). We say that the *order* of the
snowflakes symmetry is 12.

Consider now the symmetry of a regular tetrahedron. That is a solid shape in the form of a pyramid with a triangular base for which all four faces are equilateral triangles. The shape of a regular tetrahedron is invariant when it is rotated 120 degrees about an axis passing through a vertex. It is also invariant when rotated 180 degrees about an axis passing through the midpoints of opposite edges. If you make a tetrahedron and experiment with it you will find that it has a symmetry of order 12. But the symmetry of the tetrahedron is not quite the same as that of a snowflake because the snowflake has a transformation which must be repeated six times to restore it to its original position and the tetrahedron does not.

Mathematicians have provided precise definitions of
what I meant by *not quite the same*. The
invariance transformations of any shape form an
algebraic structure called a *group* when you
consider composition of transformations as
multiplication. Two groups are isomorphic if there
is a one-to-one mapping between them which respects
the multiplication. Groups can be considered to be
a mathematical abstraction of symmetry and mathematicians
have spent a great deal of effort in classifying them
but with only partial success. One spectacular achievement
is the complete classification of finite simple groups
which culminated in the discovery of the monster group which has
808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000
elements. It is a subject of great beauty in itself.

There are also infinite order symmetries described by infinite groups. The simplest example is the group of rotations in a plane which describes the symmetry of a circle. Mathematicians have also succeeded in classifying an important class of infinite dimensional groups known as semi-simple Lie groups.

Symmetry is important in physics because there are all kinds of transformations which leave the laws of physics invariant. For example, we know that the laws of physics are the same everywhere. I.e. we can detect no difference in the results of any self contained experiment which depends on where we do it. Another way to say the same thing is that the laws of physics are invariant under a translation transformation. The infinite dimensional group of translation transformations is a symmetry of the laws of physics.

The next important example is rotation symmetry. The
laws of physics are invariant under rotations in space
about any axis through some origin. An important
difference between the
translation symmetry and the rotation symmetry is that
the former is *abelian* while the latter is
*non-abelian*. An Abelian group is one in
which the order of multiplication does not matter,
they *commute*. This is true of translations
but is not true of rotations about different axis.

If the laws of physics are invariant under both
rotations and translations then they must also be
invariant under any combination of a rotation and
a translation. In this way we can always combine
any two symmetries to form a larger one. The smaller
symmetries are contained within the larger one. Note
that the symmetry of a snowflake is already contained
within rotation symmetry. Mathematicians say that the
invariance group of the snowflake is a *subgroup*
of the rotation group.

Symmetry in physics is not always evident at first sight. When we are comfortably seated on the ground we notice a distinct difference between up and down, and between the horizontal and the vertical. If we describe the motion of falling objects in terms of physical laws which have the concept of vertical and horizontal built in then we do not find the full rotational symmetry in those laws. Many ancient philosophers thought that the Earth marked a special place at the centre of the universe. In such a case we could not say that the laws of physics were invariant under translations.

It was the Copernican revolution that changed all that. Newton discovered a law of gravity which could at the same time account for falling objects on Earth and the motion of the planets in the Solar system. From that point on it could be seen that the laws of physics are invariant under rotations and translations. It was a profound revelation. Whenever new symmetries of physics are discovered the laws of physics become more unified. Newton's discovery meant that it was no longer necessary to have different theories about what was happening on Earth and what was happening in space.

Once the unifying power of symmetry is realised and combined with the observation that symmetry is not always recognised at first sight, the great importance of symmetry is revealed. Physicists have discovered that as well as the symmetries of space transformations, there are also more subtle internal symmetries which exist as part of the forces of nature. These symmetries are important in particle physics. In recent times it has been discovered that symmetry can be hidden through mechanisms of spontaneous symmetry breaking. Such mechanisms are thought to account for the apparent differences between the known forces of nature. This increases the hope that there are other symmetries not yet found. Ultimately we may discover the universal symmetry which combines all other symmetries of physics.

During the centuries which followed Newton's work physicists and mathematicians came to realise that there is a deep relationship between symmetry and conservation laws in physics. The law of conservation of momentum is related to translation invariance, while angular momentum is related to rotation invariance. Conservation of energy is due to the invariance of the laws of physics with time.

The relationship was finally established in a very
general mathematical form known as Noether's theorem.
Mathematicians had discovered that classical laws of
physics could be derived from a philosophically pleasing
*principle of least action*. Noether showed that
any laws of this type which have a continuous symmetry
would have a conserved quantity which could be derived
from the action principle.

Although Noether's work was based on classical Newtonian notions of physics. The principle has survived the quantum revolution of the twentieth century. In quantum mechanics we find that the relationship between symmetry and conservation is even stronger. There are even conservation principles related to discrete symmetries.

An important example of this is *parity*. Parity
is a quantum number which is related to symmetry of
the laws of physics when reflected in a mirror. Mirror
symmetry is the simplest symmetry of all since it has
order two. If the laws of physics were indistinguishable
from their mirror inverse then according to the rules
of quantum mechanics parity would be conserved. It was
quite a surprise to physicists when they discovered that
parity is not conserved in weak nuclear interactions.
Because these interactions are not significant in our
ordinary day-to day life, we do not normally notice
this asymmetry of space.

Simple laws of mechanics as well as those of gravity and electrodynamics are symmetric under mirror inversion. They are also invariant under time reversal. This is a little surprising because our everyday world does not appear to be symmetric in this way, there is a clear distinction between future and past. Time reversal is also broken by the weak interaction but not enough to account for the perceived difference. There is a combined operation of mirror inversion and time reversal and a third operation which exchanges a particle with its antiparticle image. This is known as CPT. Again the universe does not appear to realise particle-antiparticle symmetry macroscopically because there seems to be more matter than anti-matter in the universe. However, CPT is an exact symmetry of all interactions, as far as we know.

There is another symmetry which is found in ordinary mechanics. If you are travelling in a modern high speed train like the French TGV, on a long straight segment of track, it is difficult to tell that you are moving without looking out of the window. If you could play a game of billiards on the train, you would not notice any effects due to the speed of the train until it turned a corner or slowed down.

This can be accounted for in terms of an invariance of
the laws of mechanics under a *Galilean transformation*
which maps a stationary frame of reference onto one which
is moving at constant speed.

When you examine the laws of electrodynamics discovered by
Maxwell you find that they are not invariant under a
Galilean transformation. Light is an electrodynamic wave
which moves at a fixed speed *c*. Because *c*
is so fast compared with the speed of the TGV, you could
not notice this on a train. However, towards the end of
the nineteenth century, a famous experiment was performed
by Michelson and Morley. They hoped to detect changes in
the speed of light due to the changing direction of the
motion of the Earth. To everyone's surprise they could not
detect the difference.

Maxwell believed that light must propagate through some medium which he called ether. The Michelson-Morley experiment failed to detect the ether. The discrepancy was finally resolved by Einstein when he discovered special relativity. The Galilean transformation, he realised, is just an approximation to a Lorentz transformation which is a perfect symmetry of electrodynamics. The correct symmetry was there in Maxwell's equations all along but symmetry is not always easy to see. In this case the symmetry involved an unexpected mixing of space and time co-ordinates. Minkowski later explained that Einstein had unified space and time into one geometric structure which was thereafter known as space-time.

It seems that Einstein was more strongly influenced by symmetry principles than he was by the Michelson-Morley experiment. According to the scientific principle as spelt out by Francis Bacon, theoretical physicists should spend their time fitting mathematical equations to empirical data. Then the results can be extrapolated to regions not yet tested by experiment in order to make predictions. In reality physicists have had more success constructing theories from principles of mathematical beauty and consistency alone. Symmetry is an important part of this method of attack.

Einstein demonstrated the power of symmetry again with his dramatic discovery of general relativity. This time there was no experimental result which could help him. Actually there was an observed discrepancy in the orbit of Mercury but this could just as easily have been corrected by some small modification to Newtonian gravity. Einstein knew that Newton's description of gravity was inconsistent with special relativity, and even if there were no observation which showed it up, there had to be a more complete theory of gravity which complied with the principle of relativity.

Since Galileo's experiments on the leaning tower of Pisa, it was known that inertial mass is equal to gravitational mass. Einstein realised that this would imply that an experiment performed in an accelerating frame of reference could not separate the apparent forces due to acceleration from those due to gravity. This suggested to him that a larger symmetry which included acceleration might be present in the laws of physics.

It took several years and many thought experiments before Einstein completed the work. He realised that the equivalence principle implied that space-time must be curved, and the force of gravity is a direct consequence of this curvature. In modern terms the symmetry he discovered is known as diffeomorphism invariance. It means that the laws of physics take the same form when written in any 4d co-ordinate system on space-time.

I would like to stress that the symmetry of general relativity
is a much larger symmetry than any which had been observed in
physics before. We can combine rotation invariance, translation
invariance and Lorentz invariance to form the complete symmetry
group of special relativity which is known as the Poincare group.
The Poincare group can be parameterised by ten real numbers. We
say it has *dimension* 10. Diffeomorphism invariance, on
the other hand, cannot be parameterised by a finite number of
parameters. It is an infinite dimensional symmetry.

Diffeomorphism invariance is a hidden symmetry. If the laws of physics were invariant under any change of co-ordinates in a way which could be clearly observed, then we would expect the world around us to behave as if everything could be deformed like rubber. The symmetry is hidden by the local form of gravity just as the constant vertical gravity seems to hide rotational symmetry in the laws of physics. On cosmological scales the laws of physics do have a more versatile form allowing space-time to deform, but on smaller scales only the Poincare invariance is readily observed.

Einstein's field equations of general relativity which describe the evolution of gravitational fields, can be derived from a principle of least action. It follows from Noether's theorems that there are conservation laws which correspond to energy, momentum and angular momentum but it is not possible to distinguish between them. A special property of conservation equations derived from the field equations is that the total value of a conserved quantity integrated over the volume of the whole universe is zero, provided the universe is closed. This fact is useful when sceptics ask you where all the energy in the universe came from if there was nothing before the big bang! However, the universe might not be finite.

A final remark about relativity is that the big bang breaks diffeomorphism invariance in quite a dramatic way. It singles out one moment of the universe as different from all the others. It is even possible to define absolute time as the proper time of the longest curve stretching back to the big bang. This fact does not destroy relativity provided the big bang can be regarded as part of the solution rather than being built into the laws of physics. In fact we cannot be sure that the big bang is a unique event in our universe. Although the entire observable universe seems to have emerged from this event it is likely that the universe is much larger than what is observable. In that case we can say little about its structure on bigger scales.

What about electric charge? It is a conserved quantity so is there a symmetry which corresponds to charge according to Noether's theorem? The answer comes from a simple observation about electric voltage. It is possible to define an electrostatic potential at any point in space. The voltage of a battery is the difference in this potential between its terminals. In fact there is no way to measure the absolute value of the electrostatic potential. It is only possible to measure its difference between two different points. In the language of symmetry we would say that the laws of electrostatics are invariant under the addition of a value to the potential which is the same everywhere. This describes a symmetry which through Noether's theorem can be related to conservation of electric charge.

In fact the electric potential is just one component of the electromagnetic vector potential which can be taken as the dynamical variables of Maxwell's theory allowing it to be derived from an action principle. In this form the symmetry is much larger than the simple one parameter invariance I just described. It corresponds to a change in a scalar field of values defined throughout space-time. Like the diffeomorphism invariance of general relativity this symmetry is infinite dimensional. Symmetries of this type are known as gauge symmetries.

Both diffeomorphism invariance and the electromagnetic symmetry are local gauge symmetries because they correspond to transformation which can be parameterised as fields throughout space-time. In fact there are marked similarities between the forms of the equations which describe gravity and those which describe electrodynamics, but there is an essential difference too. Diffeomorphism invariance describes a symmetry of space-time while the symmetry of electromagnetism acts on some abstract internal space of the components of the field.

The gauge transformation of electrodynamics acts on the matter fields of charged particles as well as on the electromagnetic fields. The phase of the matter fields is multiplied by a phase factor. Through this action the transformation is related to the symmetry group of the circle which is known as U(1).

In the 1960s physicists were looking for quantum field
theories which could explain the weak and strong nuclear
interactions as they had already done for the
electromagnetic. They realised that the U(1) gauge
symmetry could be generalised to gauge symmetries based
on other Lie groups. As I have already said, an
important class of such theories has been classified by
mathematicians. They can be described as matrix groups
which fall into three families parameterised by an
integer *N* and five exceptional groups:

The best thing about gauge symmetry is that once you have selected the right group the possible forms for the action of the field theory are extremely limited. Einstein found that for general relativity there is an almost unique most simple form with a curvature term and an optional cosmological term. For internal gauge symmetries the corresponding result is Yang-Mills field theory. From tables of particles physicists were able to conjecture that the strong nuclear interactions used the gauge group SU(3). The weak interaction was a little more difficult. It turned out that the symmetry was SU(2)xU(1) but that it was broken by a Higgs mechanism. By this use of symmetry theoretical physicists were able to construct the complete standard model of particle physics which has kept the experimentalists busy for 30 years.

Symmetry is proving to be a powerful unifying tool in particle physics because through symmetry and symmetry breaking, particles which appear to be different in mass, charge etc. can be understood as different states of a single unified field theory. Ideally we would like to have a completely unified theory in which all particles and forces of nature are related through a broken symmetry.

A possible catch is that fermions and bosons cannot be related by the action of a classical group based symmetry. One way out of this problem would be if all bosons were revealed to be bound states of fermions. but the gauge bosons appear to be fundamental.

A more favourable possibility is that fermions and bosons are related by supersymmetry. Supersymmetry is an algebraic construction which is a generalisation of the Lie-group symmetries already observed in particle physics. It is a type of symmetry which can not be described by a classical group, but it has most of the essential algebraic properties.

If supersymmetry existed in nature we would expect to find that fermions and bosons came in pairs of equal mass. In other words there would be bosonic squarks and selectrons with the same masses as the quarks and electrons, as well as fermionic photinos and higgsinos with the same masses as photons and Higgs. The fact that no such partners have been observed implies that supersymmetry must be broken if it exists.

It is probably worth adding that there may be other ways in which supersymmetry is hidden. For example, If quarks are composite then the quark constituents could be supersymmetric partners of gauge particles. Also the creation and annihilation operators of fermions define a supersymmetry. Finally, Witten has found a mechanism which allows particles to have different masses even though they are supersymmetric partners and the symmetry is not broken.

There is now some indirect experimental evidence in favour of supersymmetry, but the main reasons for believing in its existence are purely theoretical. During the 1970s it was discovered that a form of space-time supersymmetry applied to general relativity provides a perturbative quantum field theory for gravity which has better renormalisation behaviour. This was one of the first breakthroughs of quantum gravity.

The big catch with supergravity theories is that they work best in ten or eleven dimensional space-time. To explain this discrepancy with nature theorists revived an old idea called Kaluza-Klein theory which was originally proposed as a way to interpret internal gauge theories geometrically. According to this idea space-time has more dimensions than are apparent. All but four of them are compacted into a ball so small that we do not notice it. Particles are then supposed to be modes of vibration in the geometry of these extra dimensions. If we believe in supergravity then even fermions fall into this scheme.

At one point supergravity looked very promising as a theory which might unify all physics. At the time I was a student at Cambridge University where Stephen Hawking was taking up his position as new Lucasian professor. There was great anticipation of his inaugural lecture and even though I made a point of turning up early I found only standing room in the auditorium. It was an exciting talk at which Hawking made some of his most famous comments. He confidently predicted that the end of physics was in sight. But early hopes faded as the perturbative calculations in supergravity became difficult and it seemed less likely that it defined a renormalisable field theory. Hawking maintains his controversial claim.

Supergravity was quickly superseded by superstring theory. String theories had earlier been considered as a model for strong nuclear forces but, with the addition of supersymmetry it became possible to consider them as a unified theory including gravity. In fact, supergravity is present in superstring theories.

Enthusiasm for superstring theories became widespread after Schwartz and Green discovered that a particular form of string theory was not only renormalisable, it was even finite to all orders in perturbation theory. That event started many research projects which are a story for another article. All I will say now is that string theory is believed to have much more symmetry than is understood, but its nature and full form is still a mystery.

String theory has also excited some mathematicians. They
have found that in certain background space-times it has
a symmetry described by the * fake monster super Lie
algebra * which is related to the finite monster group.

We have seen how symmetry in nature has helped physicists uncover the laws of physics. Symmetry is a unifying concept. It has helped combine the forces of nature as well as joining space and time. There are other symmetries in nature which I have not yet mentioned. These include the symmetry between identical particles and the symmetry between electric and magnetic fields in Maxwell's equations of electrodynamics. Symmetry is often broken or hidden so it is quite possible that there is more.

There is plenty of good reason to think that there is more. Both general relativity and quantum mechanics are full of symmetry so it would be natural to imagine that a unified theory of quantum gravity would combine those symmetries into a larger one. String theory certainly seems to have many forms of symmetry: conformal symmetries, W-symmetries dualities etc. There is evidence within string theory that it contains a huge symmetry which has not yet been revealed.

The information paradox in the thermodynamics of black holes might be solved by a hologram mechanism. This would require that the number of effective degrees of freedom in quantum gravity can be reduced by a large gauge transformation; more evidence for a peculiar hidden symmetry in quantum gravity.

It seems that there is some universal symmetry in nature that has yet to be found. It will be a symmetry which includes the gauge symmetries and perhaps also the symmetry of identical particles. The existence of this symmetry is a big clue to the nature of the laws of physics and may provide the best hope of discovering them with little further empirical data.

The importance of the symmetry in a system of identical particles is often overlooked. The symmetry group is the permutation group acting to exchange particles of the same type. The reason why this symmetry is not considered to be as important as gauge symmetry lies in the relationship between classical and quantum physics. There is an automatic scheme which allows a classical system of field equations derived from a principle of least action to be quantised. This can be done either through Dirac's canonical quantisation or Feynman's path integral. The two are formally equivalent. In modern quantum field theory a classical field is quantised and particles arise as a result. Gauge symmetry is a symmetry of the classical field which is preserved in the process of quantisation. The symmetry between identical particles, however, does not exist in the classical theory. It appears along with the particles during the process of quantisation. Hence it is a different sort of symmetry.

But the matter can not simply be left there. In a non-relativistic approximation of atomic physics it is possible to understand the quantum mechanics of atoms by treating them first of all as a system of classical particles. The system is quantised in the usual way and the result is the Schroedinger wave equation for the atom. This time we have gone from a classical particle picture to a field theory and the symmetry between particles existed as a classical symmetry.

This observation suggests that the relationship between classical and quantum systems is not so clear as it is often portrayed and that the permutation group could also be a part of the same universal symmetry as gauge invariance. This claim is now supported by string theory which appears to have a mysterious duality between classical and quantum formulations. A further clue may be that the algebra of fermionic creation and annihilation operators generate a supersymmetry which includes the permutation of identical particles.

What will the universal symmetry look like? The mathematical classification of groups is incomplete. Finite simple groups have been classified and so have semi-simple Lie groups. But infinite dimensional groups appear in string theory and these are so far beyond classification. Further more, there are new types of symmetry such as supersymmetry and quantum groups which are generalisations of classical symmetries. These symmetries are algebraic constructions which preserve an abstract form of invariance. They turn up in several different approaches to quantum gravity including string theory so they are undoubtedly important. This may be because of their importance in understanding topology. At the moment we don't even know what should be regarded as the most general definition of symmetry let alone having a classification scheme.

There seems little doubt that there is much to be learnt in both mathematics and physics from the hunt for better symmetry. The intriguing idea is that there is some special algebraic structure which will unify a whole host of subjects through symmetry, as well as being at the root of the fundamental laws of physics.

This page was last updated 3 Jun 1996 by
Phil Gibbs.

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