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As we ascend a mountain the scenery changes. We may pass from grassy pastures to harsher slopes, through alpine forest, up rocky cliffs till beyond the snow line we find the summit. As we climb the mountain of scientific truth our experience is similar. What will remain of our familiar surroundings when we reach the top, if indeed there is a top. When we passed from the land of classical certainty to the indeterminism of quantum mechanics Einstein said it was like the ground had been pulled out from under us leaving nothing to stand on. He was left behind as others climbed on. As we rise higher space-time is fading from our grasp and we have even less to hold on to.
A philosopher would tell you that the only thing which remains at the top is the realm of our perceptions. According to the storyteller's paradigm the universe is no more than the sum total of all possible experiences which can be perceived. This is realised in the multiverse of quantum mechanics described by Feynman's path integral. Thus some remnant of quantum mechanics should be valid on at least the final slopes. All else must emerge further down the levels of thought. Indirectly we apprehend events and the relations between them. According to a dictionary an event is anything which happens, but to a physicist an event is also a point of space-time; a place and a moment where something could happen.
Events are also what the physicist sees in his experiments when particles come together and interact. Particle physics, both theoretical and experimental is the pursuit of the most basic events and the rules which join them. Space-time is made of events but events are more fundamental than their when and where. Space-time forms out of the relationships between events.
In 1925 Alfred North Whitehead, philosopher of science, asked us to regard events as primordial. Space-time is constructed by us from the prehension of events. A physics based on events is sometimes called Whiteheadian but the origins of such philosophy can be traced back through the monadology of Leibniz to the atomistic doctrine of space and time in the Kalám of tenth century Baghdad, and perhaps beyond to the ancient Greeks.
With heavy irony John Archibald Wheeler described a universe constructed out of events as "a bucket of dust". He sought a pregeometry for space-time but felt that starting from the set of events is premature. A deeper guiding principle must be found.
I imagined what might happen if the fixed linkage structure of the lattice was discarded. It could be made dynamic allowing any site to link to any other nearby site at random. Why not even allowing linkage to any site no matter how far away? For maximum simplicity each site should have no preferences for which other sites it likes to link to. When doing lattice gauge theory calculations, the path integral of quantum mechanics becomes a sum over different configurations of the field variables weighted by a factor related to the action. Dynamic links changing at random fit into the sum quite naturally. It now includes a sum over all the ways of linking up the lattice sites as well as a sum over the values of the field variables. You can even look for interesting physics in models where there are no field variables, just random links between events.
This paints a rather strange image of the universe. Events and links between events would be fundamental objects but there would be no built in structure to space-time, no continuity, no dimension. The dynamics would be determined by the form chosen for the action as a function of the way the events were linked up. It might take into account the number of links meeting at each event, the number of triangles which form and other similar quantities which depend on the network of connections. For the right choice of action, lattices with a four-dimensional structure might be favoured and the structure of space-time could be determined dynamically. In some appropriate limit a continuum might emerge. If it could be done it would show how the laws of physics, including the nature of space-time, could be derived from much simpler equations than those normally used to specify them.
Such speculations are often naive and unlikely to work out right, which is why Wheeler likened such models to a bucket of dust. Nevertheless you have to try these things out because if you do not make a few mistakes you never learn anything. The attractive thing about the idea for me was that you could simulate such systems on a computer and watch what happened. The results I got were not overly encouraging. There is no simple and natural way to specify the dynamics of the lattice so that it tends to form structures like space-time, unless you build in some preference for which sites want to join up. To go further it would be necessary to think more carefully about how space-time is expected to behave.
Topology change is found to be an important feature of superstring theory, so again string theorists seem to be on the right track. When they try to understand together the concepts of topology change and universal symmetry they come up against a strange enigma known as Witten's Puzzle after the much cited string theorist, Ed Witten, who first described it.
The difficulty is that both diffeomorphism invariance and internal gauge symmetry are strictly dependent on the topology of the space. Different topologies lead to non-equivalent symmetries. The diffeomorphism group of smooth mappings on a sphere is not isomorphic to the diffeomorphism group on a torus. The same applies to internal gauge groups. If topology change is permitted then it follows that the universal symmetry must, in some fashion, contain the symmetry structures for all allowable topologies at the same time. Witten admitted he could think of no reasonable solution to this problem.
An old maxim of theoretical physics says that once you have ruled out reasonable solutions you must resort to unreasonable ones. As it happens there is one unreasonable but simple solution to Witten's puzzle. It can already be identified as a property of the universal lattice where any event has no preference for which other events it connects to. This implies a simple permutation symmetry on events.
Consider diffeomorphisms to begin with. A diffeomorphism is a suitably smooth one to one mapping of a space onto itself. The set of all such mappings form a group under composition which is the diffeomorphism group of the space. A group is an algebraic realisation of symmetry. One group which contains all possible diffeomorphism groups as a subgroup is the group of all one-to-one mappings irrespective of how smooth or continuous they are. This group is the symmetric group on the manifold. Unlike the diffeomorphism groups, the symmetric groups on two topologically different space-times are algebraically identical. A solution of Witten's puzzle would therefore be for the universal group to contain the symmetric group acting on space-time events.
This is called The Principle of Event Symmetry which states that: The universal symmetry of the laws of physics includes the symmetric group acting on space-time events.
The principle of event symmetry is realised by the universal lattice, but it is more general. The universal lattice is a naive model of space-time whereas event symmetry is a deep principle which solves the puzzle of combining symmetry and topology change. There are also philosophical reasons for holding to the principle of event symmetry. According to the storyteller's paradigm, the multiverse describes all ways of putting together events. The events are taken from a heap within which they are not ordered. If something is not ordered then it does not matter how its contents are mixed up. They can be permuted without consequence. The symmetric group is a symmetry of the heap.
In its simplest form, event symmetry is realised in a heap of discrete events. The universal lattice is a good example. But the symmetric group can be a subgroup of a larger group allowing the individuality of events to be blurred. There are other ways of including event symmetry within larger symmetries. You can have a mapping from a larger symmetry onto a smaller one which preserves its structure. This is called a homomorphism. You can also deform symmetries by introducing a more general symmetry structure with a deformation parameter which reduces to something containing the symmetric group for one special case of that parameter. I will describe examples of all of these. The beauty of event symmetry is revealed in the ways it can become part of the full universal symmetry.
A more direct reason to doubt the principle would be that there is no visible or experimental evidence of such a symmetry. The principle suggests that the world should look the same after permutations of space-time events. It should even be possible to swap events from the past with those of the future without consequence. This does not seem to accord with experience. Event symmetry cannot be a principle of nature unless it is well hidden. Since the symmetric group acting on space-time can be regarded as a discrete extension of the diffeomorphism group in general relativity, it is worth noting that the diffeomorphism invariance is not all that evident either. If it were then we would expect to be able to distort space-time in ways reminiscent of the most bizarre hall of mirrors without consequence. Everything around us would behave like it is made of liquid rubber. Instead we find that only a small part of the symmetry which includes rigid translations and rotations is directly observed on human scales. The rubbery nature of space-time is more noticeable on cosmological scales where space-time can be distorted in quite counterintuitive ways.
If space-time is event-symmetric then we must account for space-time topology as it is observed. Topology is becoming more and more important in fundamental physics. Theories of magnetic monopoles, for example, are heavily dependent on the topological structure of space-time. To solve this problem is the greatest challenge for the event-symmetric theory.
To get a more intuitive idea of how the event symmetry of space-time can be hidden we use an analogy. Anyone who has read popular articles on the Big Bang and the expanding universe will be familiar with the analogy in which space-time is compared to the surface of an expanding balloon. The analogy is not perfect since it suggests that curved space-time is embedded in some higher-dimensional flat space, when in fact, the mathematical formulation of curvature avoids the need for such a thing. Nevertheless, the analogy is useful so long as you are aware of its limitations.
We can extend the balloon analogy by imagining that space-time events are like a discrete set of particles populating some higher-dimensional space. The particles might float around like a gas of molecules interacting through some kind of forces. In any gas model with just one type of molecule the forces between any two molecules will take the same form dependent on the distance between them and their relative orientations. Such a system is therefore invariant under permutations of molecules. In other words, it has the same symmetric group invariance as that postulated in the principle of event-symmetric space-time, except that it applies to molecules rather than events.
Given this analogy we can use what we know about the behaviour of gases and liquids to gain a heuristic understanding of event-symmetric space-time. For one thing we know that gases can condense into liquids and liquids can freeze into solids. Once frozen, the molecules stay fixed relative to their neighbours and form rigid objects. In a solid the symmetry among the forces still exists but because the molecules are held within a small place the symmetry is hidden.
Another less common form of matter gives an even better picture. If the forces between molecules are just right then a liquid can form thin films or bubbles. This is familiar to us whenever we see soap suds. A soap film takes a form very similar to the balloon which served as our analogy of space-time for the expanding universe. The permutation symmetry of the molecular forces is hidden and all that remains is a surface. The same idea works in higher dimensions so it is possible that four-dimensional space-time may condense out of something like a gas of events, just like the formation of a soap bubble. Curvature of space-time is similar to the curvature of the surface of the soap film.
Two years later the world went through a new revolution in information technology: the internet. Its impact on science rivals the introduction of the printing press into Europe in the fifteenth century. The internet had already existed for some time. I had used it myself as a research student in 1984 when I used to control computers in Germany from my base in the University of Glasgow. But in 1993 the internet came out of academic institutes into the wider world, where I was then working as a programmer in France. I gained access to usenet and the world wide web and I regained access to what was happening in physics. I could download the latest papers in physics which appeared as electronic pre-prints each day. I could search databases of papers compiled over the previous twenty years. Best of all, I could write my own papers and circulate them on the internet. In April 1994 my first tentative paper about event-symmetric space-time emerged and drew no response.
I decided that it would be prudent to find out who else had done similar work in the past. Using on-line databases I searched the literature for papers with titles that had anything to do with discrete space-time and then followed their hyperlinked references and citations to find other relevant papers. I discovered the work on Wheeler, Finkelstein and others which I had not heard of before. There were, in fact, just a few examples of such work which dared to speculate about the small scale structure of space-time with models not unlike my universal lattice. Some of what I found was more mathematically sophisticated, yet not one example expressing the principle of event symmetry came to light. I continued my work. A couple of years later a contact on the internet told me about a book which discussed ideas similar to mine. It was not a physics book. It was 'Permutation City', a science fiction novel by Greg Egan, but it was a science fiction novel with more interesting things to say about the philosophy of physics than many physicists or philosophers.
In 2045 the protagonist, Paul Durham, programs a simulation of himself into a computer. Applying the strong AI hypothesis, the story line continues from the point of view of the copy. It is another invocation of the storyteller's paradigm. A computer simulation can be regarded as a sophisticated way of recounting a story. As the storyteller told us, there is no need to distinguish between the story and reality. Durham performs some experiments with his copy, now referred to as Paul, in the simulation. He divides the program up and changes the order in which states are computed. The events of Paul's simulated life are permuted but he does not experience anything different from normal.
Paul tries to understand what is happening to him in terms of the theory of general relativity. Relativity declares that points of view of different observers are equally valid, but only observers whose reference frames can be related by continuous co-ordinate transformations. The mapping between the events of Paul's existence and the events of space-time outside the computer were discontinuous. In relativity influences have to be localised travelling from point to point at a finite velocity. Paul thought that if you chop up space-time and rearrange it, then causal structure would fall apart.
Finally Paul appreciates the principle of event symmetry, or as Egan calls it; the dust theory. It would be a new principle of equivalence, a new symmetry between observers. Relativity threw out absolute space and time but it did not go far enough. Absolute cause and effect must go too.
Permutation City was first published in 1994 and parts were adapted from a story called 'Dust' which was first published in Isaac Asimov's Science Fiction Magazine, July 1992.
The principle of event symmetry is stronger, in a sense, than diffeomorphism symmetry because it is larger, but it also allows for more general models of space-time as discrete sets. Einstein was able to assume that space-time was a continuous manifold with one temporal and three spatial dimensions. We no longer have such a restriction and consequently there are too many possible ways to devise event-symmetric theories. Event symmetry on its own is not very powerful. To go further the symmetry must be extended.
So far we have seen how the principle of event-symmetric space-time allows us to retain space-time symmetry in the face of topology change. Beyond that we would like to find a way to incorporate internal gauge symmetry into the picture too. It turns out that there is an easy way to embed the symmetric group into matrix groups. This is interesting because, as it happens, matrix models are already studied as simple models of string theory. String theorists do not normally interpret them as models on event-symmetric space-time but it would be reasonable to do so in the light of what has been said here.
To see how event-symmetry leads naturally to matrices consider how the universal random lattice may be represented. Each event could be labelled with an index i. For each pair of events (i, j) there may or may not be a link joining them in the lattice. This could be represented by a matrix of variables aij each of which is zero or one. One indicates that events i and j are linked, and zero indicates that they are not linked.
To put a model of a gauge theory on this lattice, field variables 
i can be associated with each event and gauge variables Uij with each link. The field variables form a column vector 
and the gauge variables can again be collected together in a matrix A. If it is a Z2 gauge theory, the elements of the matrix are now always zero or plus or minus one. The matrix A can be symmetric but it may be more convenient to make it antisymmetric since the diagonal elements are then necessarily zero without imposing an extra condition. Gauge invariant quantities which could be used in an action for this model can be expressed in matrix notation e.g.
A gauge transformation can be effected as a similarity transformation on the matrix and vector. That is,
For the Z2 gauge transformation T is a diagonal matrix with 1 and -1 down the diagonal. For example,
All of this generalises easily to other gauge groups. For an SO(N) gauge transformation T is a block diagonal matrix with blocks of N by N orthogonal matrices down the diagonal.
What about event symmetry? A permutation of events is also a symmetry of an action expressed in matrix notation as above. Columns and rows of the matrix and vector are permuted. This can also be effected by a similarity transformation T which is a permutation matrix. I.e. T has a single element equal to 1 in each row and column and all other elements equal to zero. For example,
Now that we have put internal gauge symmetry and event-symmetry into similar forms it is tempting to unify them. In both cases the similarity transformations are orthogonal matrices. If the elements of and A are allowed to be any real numbers the matrix action has a full symmetry of orthogonal matrix transformations which includes the gauge transformations and event permutations as special cases. The same can be done with other gauge groups using orthogonal or unitary matrix models.
In these models the total symmetry of the system is a group of rotation matrices in some high-dimensional space. The number of dimensions corresponds to the total number of space-time events in the universe, which may be infinite. Permutations of events now correspond to rotations in this space which swap over the axes.
So does this mean that the universal symmetry of physics is an infinite-dimensional orthogonal matrix? The answer is probably no since an orthogonal matrix is too simple to account for the structure of the laws of physics. For example, orthogonal groups do not include supersymmetry which is important in superstring theories. The true universal symmetry may well be some much more elaborate structure which is not yet known to mathematicians.
Before moving on it is worth taking note of how the amount of symmetry has increased in going over to matrix models. In conventional gauge theory there are a few degrees of symmetry for each event so the symmetry is of dimension N; the number of space-time events. With the matrix model there is a degree of symmetry for each independent element of the matrix so the symmetry is of dimension N2. This is just the first step towards the much larger symmetries which may be present in the universe.
The symmetry involved here is described by the symmetric groups, just like event-symmetric space-time. Obviously we should ask ourselves whether or not there is any connection between the two. Could the symmetric group acting to exchange identical particles be part of the symmetric group acting on space-time events? If it were, then that would suggest a deep relation between space-time and matter. It would take the process of unification beyond the forces of nature towards a more complete unification of matter and space-time.
As we shall see it is natural to combine the permutation symmetry of particles and event-symmetry and it will imply a unification of particle statistics and gauge symmetries which has now become apparent in superstring theories.
It turns out that such a model can be constructed using Clifford algebras. These algebras are very simple in principle but have a remarkable number of applications in theoretical physics. They first appeared to physicists in Dirac's relativistic equation of the electron. They also turn out to be a useful way to represent the algebra of fermionic annihilation and creation operators.
If we regard a Clifford algebra as an algebra which can create and annihilate fermions at space-time events then we find we have defined a system which is event-symmetric. It can be regarded as an algebraic description of a quantum gas of fermions.
This is too simple to provide a good model of space-time but there is more. Clifford algebras also turn out to be important in construction of supersymmetries and if we take advantage of this observation we might be able to find a more interesting supersymmetric model.
The definition of Clifford Algebras is very simple. It is an algebra generated by a set of elements 
such that
with k < N and can be interpreted as a field variable for a k-simplex with the k events as vertices. In comparison with the matrix model which had a field variable for each event and each pair of linked events, a model using Clifford algebras will have these plus a variable for each triplet of events, each quadraplet etc.
The situation seems to parallel Maxwell's theory of electromagnetism as it was seen at the end of the 19th century. Many physicists did not accept the validity of the theory at that time. This was largely because of the apparent need for a medium of propagation for light known as the ether, but experiment had failed to detect it. Einstein's theory of special relativity showed why the ether was not needed. He did not have to change the equations to correct the theory.
Instead he introduced a radical change in the way space and time were viewed. It is likely that the equations we have for string theory are also correct, although they are not as well formed as Maxwell's were. To complete the theory it is again necessary to revise our concept of space-time and remove some of its unnecessary structure just as Einstein removed the ether.
It would be natural to search for an event-symmetric string model. We might try to generalise the fermion model described by Clifford algebras to something which was like a gas of strings. A string could be just a sequence of space-time events connected in a loop. The most significant outcome of the event-symmetric program so far is the discovery of an algebra which does just that. It is an algebraic model which can be interpreted as an algebra of strings made of closed loops of fermionic partons.
The result is not sophisticated enough to explain all the rich mathematical structures in string theory but it may be a step towards that goal. Physicists have found that new ideas about knot theory and deformed algebras are important in string theory and also in the canonical approach to quantisation of gravity. This has inspired some physicists to seek deeper connections between them. Through a turn of fate it appears that certain knot relations have a clear resemblance to the relations which define the discrete event-symmetric string algebras. This suggests that there is a generalisation of those algebras which represents strings of anyonic partons, that is to say, particles with fractional statistics.