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A few animals and many flowers have more than bilateral symmetry. A daisy or a starfish has radial symmetry from its centre. Crystals also form symmetrical shapes such as octahedra and cubes. A snowflake is a crystal of ice with 6-fold radial symmetry and it is particularly elegant. How does it acquire its shape?
The snowflake begins its life as a minute hexagonal crystal forming in a cloud. The origins of this structure lie in a lattice arrangement of the water molecules which form the ice. During its passage from the clouds 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 almost identical to any other and accounts for the snowflake's 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. Its shape is said to be invariant (meaning unchanged) under such a transformation. It is invariance which characterises symmetry in mathematics. 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 which leave its shape invariant (including the identity transformation which does nothing). We say that the order of the snowflake's 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 and the centre of the opposite face. 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 also has a symmetry of order 12. But the symmetry of the tetrahedron is not quite the same as that of a snowflake. 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 or isometries of any shape form an algebraic structure called a group. You can consider composition of transformations as a kind of multiplication. For example, two isometries of the snowflake are a rotation of 60 degrees clockwise (call it a) and a reflection about the vertical axis (call it b). The transformations are composed by doing one and then the other, a followed by b. The result is a reflection about a different axis set at 30 degrees to vertical which is also an isometry (call it c). This composition is expressed algebraically as ab = c, as if it were a multiplication. The algebraic structure defined by these elements of symmetry is the group. The order of the symmetry is the number of elements in the group. Two groups are isomorphic if there is a one-to-one mapping between them which respects the multiplication. Two groups which are isomorphic are often regarded as essentially the same thing. The symmetry group of a snowflake is not isomorphic to that of a tetrahedron but it is isomorphic to that of a hexagon. Groups can be considered to be a mathematical abstraction of symmetry. Many of them have symbolic names. The symmetry group of the snowflake and hexagon is called D6 while that of the tetrahedron is called A4.
The historical origins of group theory can be traced back to tragic events of May 30th 1832. That morning a young Frenchman named Évariste Galois died in a dual. At 21 years old his life was already a tale of rejection and failure as a mathematician, yet the night before he met his death he wrote a letter which brought about a revolution in abstract thought. Galois developed a theory about which polynomial equations could be solved exactly using simple arithmetic operations such as addition, multiplication and square roots. Polynomials up to degree four could be solved in this way but quintic equations had been proven insoluble by the Norwegian mathematician Niels Abel in 1823. Galois found that the answer lay in the group of permutations of the solutions of the equations. A permutation is a way of rearranging or shuffling an ordered set of objects. Suppose, for example, that there are six numbered objects in numerical order 1, 2, 3, 4, 5, 6. A possible permutation would be 3, 4, 1, 6, 5, 2. It can be shown as a diagram,
It is not really the numbers which are important. It is the arrows which permute them. There are 720, (6! = 1 x 2 x 3 x 4 x 5 x 6) different possible permutations of six objects.
A rotation of a snowflake can be regarded as a permutation of its arms. Number them clockwise and look at the 60 degree rotation.
This is a permutation of the arms
Likewise a reflection about the vertical axis is another permutation
Any of the twelve transformations which leave the shape of the snowflake invariant can be shown as a permutation. To appreciate the algebraic structure of the group formed by the transformations we need to see how they can be combined. This is how it works for the rotation followed by the reflection
So combining permutations by joining the arrows is equivalent to performing one isometry followed by another. This is the same as multiplication in the group of isometries.
Like ordinary multiplication of numbers this kind of multiplication is associative, i.e. a(bc) = (ab)c for any three transformations a, b and c, but unlike ordinary multiplication it is not always commutative (ab) != (ba). There is always an identity which has the property, a1 = 1a = a. Each element has an inverse, aa-1 = a-1a = 1. These algebraic rules are taken as the definition of a group.
Permutations, symmetry and groups all go together. A permutation is just a one-to-one mapping from some set to itself. A symmetry is a subset of permutations which leaves something (like shape) invariant. The algebraic structure of symmetries and the ways they combine is a group. To complete the triangle any group can also be seen as a collection of permutations of its own elements because multiplication by any element of the group is a one-to-one mapping onto itself.
The geometric symmetries of the snowflake and tiger are just one type of symmetry which leaves the shape of an object invariant. The permutations on a set of n objects also form a group which is called the symmetric group of the set or Sn for short. All these things are very important in physics but the theory of groups and symmetries also has its own intrinsic power and beauty which makes it interesting to mathematicians.
Permutations are not only applied to finite sets. There are also infinite order symmetries described by infinite groups and permutations of infinite numbers of objects. The simplest example is the group of rotations in a plane about the origin. It describes part of the symmetry of a circle and is known as U(1).
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 (ab = ba). This is true of translations but it 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. They are both subgroups of the full group of permutations of points of space which leave the distance between any two points invariant.
Such symmetry is important because we can use it to test new theories of physics. Once we have accepted that certain symmetries are exactly observed in nature we can check that any set of equations looks the same after applying the transformations under which physics is supposed to be invariant. If they are not then they cannot form any part of the laws of physics. Mathematicians often go much further than this and work out all possible forms the laws of physics might take to respect the symmetry. Given translational and rotational symmetry we know that the equations can be expressed using scalars, vectors, tensors and spinors; quantities which can be combined in certain ways such as using vector and scalar multiplications. Nature has been kind to physicists. With these rules they waste much less time dreaming up useless theories of physics than they would if there was no symmetry. The more symmetry they know about, the better physicists can do. This is one of the secret of their success.
It was the Copernican revolution that changed all that. Nicolaus Copernicus described a cosmology in which the Earth had no special place and initiated a new freedom of thought taken up by Galileo. Newton, in response to Galileo, discovered his 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. If the moon was subject to Earthly forces why did it not fall down like objects do on Earth.
Newton's answer was that the moon does fall, but it moves horizontally fast enough to keep it from coming down. 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 the heavens.
Once the unifying power of symmetry is realised and combined with the observation that symmetry is hidden and not always recognised at first sight, the unique importance of symmetry is clear. 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 such as spontaneous symmetry breaking. Such mechanisms are thought to account for the apparent differences between the known forces of nature. This increases the hope that other symmetries remain to be found.
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 the philosophically pleasing principle of least action. In 1918 Emmy Noether showed that any laws of this type which have a continuous symmetry, like translations and rotations, 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. This is the case for electromagnetism, gravity and the strong nuclear force. It was quite a surprise to physicists when they discovered that parity is not conserved in the rare 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 involving the forces of gravity and electricity are invariant under time reversal as well as mirror reflection. If you could freeze every particle in the universe and then send them on their way with exactly reversed velocity, they would retrace their history in reverse. 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. In the primary laws of physics time reversal is also only broken by the weak interaction but not enough to account for the perceived difference. There is an important combined operation of mirror inversion, 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.
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. Galileo used this symmetry to explain how the Earth could be moving without us noticing it but he used a ship at sea rather than a train to demonstrate the principle.
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 the 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 and Poincaré when they independently discovered special relativity in 1905. The Galilean transformation, they 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 relativity had unified space and time into one geometric structure which was thereafter known as space-time. Symmetry was again a unifying principle.
It seems that Einstein was more strongly influenced by symmetry 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. Symmetry is an important part of this method of attack. Of course these principles are still based on observations and empiricism serves as a check on the correctness of the theory afterwards, yet by using symmetry it is possible to leap ahead of where you would get to using just simple induction.
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 might just as easily have been corrected by some small modification to Newtonian gravity or even by some more mundane effect due to the shape of the sun. Einstein knew that Newton's description of gravity was inconsistent with special relativity. 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 with weights dropped from the leaning tower of Pisa, it was known that inertial mass is equal to gravitational mass. Otherwise objects of different mass would fall at different rates even in the absence of air resistance. 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 knew 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. The form of the equations which express the laws of physics must be the same when transformed from one space-time co-ordinate system to another no matter how curvilinear the transformation equations are.
The symmetry of general relativity is a much larger one than any which had been observed in physics before Einstein. We can combine rotation invariance, translation invariance and Lorentz invariance to form the complete symmetry group of special relativity which is known as the Poincaré group. The Poincaré 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. Already we have passed from finite order symmetries like that of the snowflake, to symmetries which are of infinite order but finite dimensional like translation symmetry. Now we have moved on to infinite-dimensional symmetries and we still have a long way to go.
Diffeomorphism invariance is another 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. Diffeomorphisms leave the physics invariant under any amount of stretching and bending of space-time. 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 show a more versatile form allowing space-time to deform, but on smaller scales only the Poincaré 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. According to relativity there should be no absolute standard of time but we can define cosmological time since 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 than those which are observable.
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 at each event throughout space-time. Like the diffeomorphism invariance of general relativity this symmetry is infinite-dimensional. Symmetries of this type are known as gauge symmetries. The principles of gauge theories were first recognised by Herman Weyl in 1918. He hoped that the similarities between the gravitational and electromagnetic forces would herald a unification of the two. It was many years before the full power of his ideas was appreciated.
There is an analogy of gauge symmetry in the world of finance. Consider the money which circulates in an economy. If one day the government wants to announce a currency devaluation, it has to be implemented in such a way that nobody loses out. Every price can be adjusted to be one tenth of its previous value, but everybody's wage must be changed in the same way, as must their savings. If done correctly the effect would be cosmetic. The economy is invariant under a global change in the scale of currency. It is a symmetry of the system.
What about the combined system of economies of the different countries of the world? Any one currency can revalue its currency but to avoid any economic effect the exchange rates with other currencies must also reflect the change. In this larger system there is a degree of symmetry for each currency of the world.
This is analogous to a local gauge symmetry which allows a gauge transformation to take place independently anywhere in space. Prices and wages are analogous to the wave functions of matter. Exchange rates are like the gauge fields of gravity and electromagnetism. The purpose of these fields which propagate the forces of nature is to allow the gauge symmetry to change locally, just as varying exchange rates allow economies to adjust and interact. In both cases the variables change dynamically, evolving in response to market forces in the case of economy and evolving in response to natural forces in the case of physics.
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. In 1927 Fritz London noted that to implement the gauge transformation the phase of the wave function of matter fields is multiplied by a phase factor, which is a complex number of modulus one. Such factors have no physical effects since only the modulus of the wave function is observable. Through this action the transformation is related to the group of complex numbers of modulus one which is isomorphic to the rotation symmetry group of the circle, 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 force. They realised that the U(1) gauge symmetry could be generalised to gauge symmetries based on other continuous groups. As I have already said, an important class of such symmetries has been classified by mathematicians. In the 1920s Elie Cartan proved that a subclass known as semi-simple Lie groups can be described as matrix groups which fall into three families parameterised by an integer N and five other 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 developed by Chen Ning Yang and Robert Mills in 1954. Maxwell's equations for electromagnetism are a special case of Yang-Mills theory corresponding to the gauge group U(1) but there is a generalisation for any other gauge group. From tables of particles, physicists were able to conjecture that the strong nuclear interactions used the gauge group SU(3) which is metaphorically referred to as colour. This symmetry is hidden by the mechanism of confinement which prevents quarks escaping from the proton and neutron to reveal the colour charge. For the weak interaction it turned out that the symmetry was SU(2) X U(1) but that it was broken by a Higgs mechanism. There is a Higgs boson whose vacuum state breaks the symmetry at low energies. By these uses of symmetry theoretical physicists were able to construct the complete standard model of particle physics by 1972.
The rapid acceptance of gauge theories at that time was due to the discovery by 't Hooft and Veltman that Yang-Mills theories are renormalisable, even when the symmetry is broken. Other theories of the nuclear interactions were plagued with divergences when calculations were attempted. The infinite answers rendered the theory useless. These divergences are also present in Yang-Mills theory but a process of renormalisation can be used to cancel out the infinities leaving sensible consistent results. In the years that followed this discovery, experiments at the world's great particle accelerator laboratories have rigidly confirmed the correctness of the standard model. Of the four forces only gravity remains in a form which stubbornly refuses to be renormalised.
A possible catch to this hope is that fermions and bosons cannot be related by the action of a classical symmetry based on a group. One way out of this problem would be if all bosons were revealed to be bound states of fermions so that at some fundamental level only elementary fermions would be necessary. This is an unlikely solution because gauge bosons such as photons 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 new type of symmetry which cannot be described by a classical group. It is defined as a different but related algebraic structure which still has all the essential properties which make symmetry work.
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 should 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, superstring theorist Ed Witten has found a mechanism which allows particles to have different masses even though they are supersymmetric partners and the symmetry is not spontaneously broken.
Supersymmetry unifies more than just fermions and bosons. It also goes a long way towards unifying internal gauge symmetry with space-time gauge symmetry. If gravity is to be unified with the electromagnetic and nuclear forces there should be a larger symmetry which contains diffeomorphism invariance and internal gauge invariance. In 1967 Coleman and Mandula proved a theorem which says that any group which contained both of these must separate in to a direct product of two parts each containing one of them. In other words, they simply could not be properly unified, or at least, not with classical groups. The algebraic structure of supersymmetry is a supergroup which is a generalisation and a classical group and is not covered by the Coleman-Mandula theorem, so supersymmetry provides a way out of the problem. There are still a limited number of ways of unifying gravity with internal gauge symmetry using supersymmetry and each one gives a theory of supergravity.
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 supergravity provides a perturbative quantum field theory which has better renormalisation behaviour than gravity on its own. 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 unify electromagnetism with gravity 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. Yang-Mills theory emerges from space-time curvature in the compacted dimensions so Kaluza-Klein theory is an elegant way to unify internal gauge symmetry with the diffeomorphism invariance of general relativity. If we believe in supergravity then even fermions fall into this scheme.
Supergravity theories were popular around 1980 but it was found to be just not quite possible to have a version with the right structure to account for the particle physics we know about. The sovereign theory of supergravity lives in 11 dimensions and nearly manages to generate enough particles and forces when compactified down to 4 dimensions, but unfortunately it was not possible to get the left-right asymmetry in that way. It was also realised eventually that these field theories could not be perfectly renormalisable. 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 John Schwarz and Michael 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 chapter. 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 are still a mystery.
Let us look again at the symmetry we have seen so far. There is the SU(3)XSU(2)XU(1) internal gauge symmetry of the of the strong, weak and electromagnetic forces. Since these groups are gauged there is actually one copy of the group acting at each event of space-time so the group structure is symbolically raised to the power of the number of points in the space-time manifold 
. The symmetry of the gravitational force is the group of diffeomorphisms on the manifold which is indicated by diff(). However, the combination of the diffeomorphism group with the internal gauge groups is not a direct product because diffeomorphisms do not commute with internal gauge transformations. They combine with what is known as a semi-direct product indicated by |X. The known symmetry of the forces of nature is therefore:
There is plenty of good reason to think that this is not the full story. This group will be the residual subgroup of some larger one which is only manifest in circumstances where very high energies are involved, such as the big bang. 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 which have been explored mathematically. There is evidence within string theory that it contains a huge symmetry which has not yet been revealed. Whether or not string theory is the final answer, 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 others such as the symmetry of identical particles and electromagnetic duality. 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 if experiments are not capable of supplying much more empirical data.
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. Furthermore, 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 do not even know what should be regarded as the most general definition of symmetry let alone having a classification scheme.
But the matter cannot 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 Schrödinger wave equation for the atom. This is known as first quantisation because it was discovered before the second quantisation of the Schrödinger wave equation which became a part of quantum field theory. In the first quantisation 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 aspects. A further clue may be that the algebra of fermionic creation and annihilation operators generate a supersymmetry which includes the permutation of identical particles. This opens the door to a unification of particle permutation symmetry and gauge symmetry.
There are two possible solutions that I know of. The first is the principle of event symmetry which is the central theme of this book. It says that we must simply forget the topology of space-time at the most fundamental level and regard the space-time manifold as just a set of discrete space-time events. The diffeomorphism group of any manifold is a subgroup of the symmetric group of permutations on the set of points in the manifold. The internal gauge symmetries also fall into this pattern. This solution to the puzzle generates many new puzzles and in later chapters I will describe them and start to resolve them.
The second solution to this puzzle is to generalise symmetry using the mathematical theory of categories. A category can describe mappings between different topologies and a group is a special case of a category. If the concept of symmetry is extended further to include more general categories it should be possible to incorporate different topologies in the same categorical structure. How should we interpret these two solutions to a difficult problem when at first one solution seemed difficult to find? Is only one right, or are they both different aspects of the same thing?
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.