Event-Symmetric Space-Time
Quantum Gravity
The scientific revolution of the twentieth century has constructed two major pillars; General Relativity and Quantum Theory, on which all known physics is built. Through general relativity we have come to understand the force of gravity, while through quantum field theory we understand the other three known forces in particle physics. Physicists are now trying to complete the final step, to put together these two theories into one theory of quantum gravity.
To do so is proving to be a difficult task because General Relativity and Quantum Theory seem to be utterly incompatible if we try to put them together in the most obvious ways. It is now generally believed that a full theory of quantum gravity will not be possible without a radical revision in the way we think of space-time.
Symmetry in String Theories
In 1981 a new theory of quantum gravity known as superstring theory was born. According to this idea, all fundamental particles are different resonance modes of tiny loops of string. Many theorists believe that superstring theory is going to turn out to be a Theory Of Everything, by which they mean a complete unified theory of the fundamental laws of physics to which all other physical laws can be reduced. This hope is rather frustrated by the fact that they don't have a completely self consistent and elegant set of equations to define it. All they do have are some perturbative calculation schemes and classical limits.
One of the discoveries which have enabled physicists to get so far towards a complete understanding, is the importance of the role played by symmetry in physics. In general relativity it is a form of symmetry known as diffeomorphism invariance which is important, while in particle physics it is internal gauge symmetry.
If symmetry is so important in both general relativity and quantum theory then we should surely expect it to be of at least as much importance in a combined theory of quantum gravity. In superstring theory we do indeed find all kinds of symmetries. There are clues in the mathematics of string theory which seem to suggest that at a very high temperature, known as the Hagedorn Temperature, a huge order symmetry manifests itself. This could be the universal symmetry which includes all known symmetries unified together. The problem is that we don't know what that symmetry is. If we did then we might understand superstring theory much better.
Topology Change
Back in 1958 John Wheeler suggested that when general relativity and quantum theory were put together there would be astonishing things going on at the very small length scale known as the Planck length (about 10-35 metres). If we could look down to such distances we would see the topology of space-time changing through quantum fluctuations. The structure of this space-time foam has been a mysterious area of research ever since.
Witten's Puzzle
Topology change is found to be an important part of superstring theory, so again string theorists seem to be on the right track. But, 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 ran across it.
The difficulty is that both diffeomorphism invariance and internal gauge symmetry are strictly dependent on the topology of the space-time on which they act. Different topologies lead to non-equivalent symmetries. 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.
Consider diffeomorphisms to begin with. A diffeomorphism is a suitably smooth one to one mapping of a space-time onto itself. The set of all such mappings form a group under composition which is the diffeomorphism group of space-time. 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 known as 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 Symmetric Space-Time
which states that
The universal symmetry of the laws of physics includes a mapping onto the symmetric group acting on space-time events.
Richard Feynman
Hidden Symmetry
There are a number of reasons why this principle may seem unreasonable. For one thing it suggests that we must treat space-time as a discrete set of events. In fact there are plenty of reasons to believe in discrete space-time. Theorists working on quantum gravity in various forms agree that the Planck scale defines a minimum length beyond which the Heisenberg uncertainty principle makes measurement impossible. In addition, arguments based on black hole thermodynamics suggest that there must be a finite number of physical degrees of freedom in a region of space.
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.
To counter this it is only necessary to point out that symmetries in physics can be spontaneously broken. The most famous example of this is the Higgs Mechanism which is believed to be responsible for breaking the symmetry between the electromagnetic and weak interactions. Could space-time symmetry be hidden through an analogous mechanism of spontaneous symmetry breaking?
There are, in fact, other ways in which symmetries can be hidden without being broken. 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.
The Soap Film Analogy
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.
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.
More Symmetry
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.
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 corresponds 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.
Identical Particles
Theorists often talk about unifying the gauge symmetries which are important to our understanding of the four natural forces. There are, however, other symmetries in nature which are rarely mentioned in the context of unification. These symmetries take the form of an invariance under exchange of identical particles. For example, every electron in the universe is the same, they all have the same charge, mass etc. If we swap one electron in the universe with another the universe will carry on as before.
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.
Mach's Space and Time
At this point it is worth taking a little philosophical diversion. Philosophers and scientists have often discussed what each believe to be the true nature of space and time. It seems that the way we perceive space and time is rather indirect. You can mark notches which measure distance on a ruler but you can't mark notches on space itself. Likewise, time is measured by the ticking of clocks which are assumed to be related to some passage of absolute time of the universe. Our minds have evolved a process of placing objects into a model of space and time in our heads but to what extent is this model an accurate model of the real world?
In the 17th century Leibnitz was of the opinion that space-time was some sort of illusion. Newton took a more pragmatic approach and decided to take it for granted that space and time exist as part of the arena within which the laws of physics play out their roles. This dichotomy between the preferred philosophy and the progress of science has existed ever since with philosophers such as Berkeley, Kant and Mach insisting that space and time should not be absolute despite the success of physical theories based on the contrary assumption. Ultimately, they argued, space and time would be understood in terms of relationships among physical objects which we observe more directly. Some philosophers go further and claim that physical objects themselves are illusions which can only be understood as artefacts of our own perceptions.
Most ironic of all was the part played by Einstein in the development of this debate. He was much impressed by the work of Mach and hoped that Mach's principle would follow from his theory of general relativity. Minkowski had already shown that Einstein's theory of special relativity could be interpreted as a unification of space and time into a single geometry of space-time. Space and time were not absolute if treated independently. It turned out that the correct formulation of Einstein's ideas resulting from his equivalence principle needed the mathematics of curved space-time as developed by Gauss, Riemann and others in the previous century. The result was that space-time seemed to take on a more absolute form than before. It was no longer a static background, but could itself move dynamically and have a complicated existence even in the absence of any matter at all.
The mathematical elegance of general relativity, as much as its empirical success, has led to its strong influence over the way physicists have formed their theories. They seem to be more and more contrary to the philosophical view. These days we are used to trying to find ways to describe matter as some kind of manifestation of geometry such as in Kaluza-Klein theories where matter is seen as a result of higher dimensions of space-time.
It could simply be that the philosophers are wrong. Many physicists of the mid-twentieth century, including Dirac, Feynman and Weinberg, have been openly disdainful of the ways of philosophers. More recently a slight change in the wind might be felt. As theoretical physicists themselves have moved further away from the influence of experimental data, they have themselves become a lot more philosophical. If the absoluteness of space-time is to breakdown it will probably be at the scales of quantum gravity that it is found to happen. Already we see the influence of non-commutative geometry in which space-time takes a secondary role to the matter fields defined on it as functions. Many physicists now express a wish to find a more algebraic theory for quantum gravity. In former years Einstein had been alone in expressing such wishes.
Clifford's Legacy
On its own, the principle of event symmetric space-time is not very fruitful. What is needed is a mathematical model which incorporates the principle and which gives body to some of the speculative ideas outlined above.
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.
Back to Superstrings
Since superstring theory was an important part of the motivation for proposing the principle of event-symmetric space-time in the first place. String theorists seem to believe that the subject they are studying is already the correct theory of physics, but they are probably missing the key to understanding its most natural formulation.
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 relation have a clear resemblance to the relations which define the discrete event-symmetric string algebras. This suggest that there is a generalisation of those algebras which represents strings of anyonic partons, that is to say, particles with fractional statistics.
Event-Symmetric Physics
What can this theory tell us about the universe? Since it is incomplete it is limited. The one place where a theory of quantum gravity would have most significance would be at the big bang. In the first jiffy of existence the temperature was so high that the structure of space-time would have been disrupted. It is known that in string theory there is a high temperature phase transition in which the full symmetry is realised. If the principle of event-symmetric space-time is correct then that is a much larger symmetry than people have previously imagined. At such high temperature space-time would cease to exist in the form we would know it, and only a gas of interacting strings would be left. A reasonable interpretation of this state of affairs would be to say that space-time has evaporated. The universe started from such a state, then space-time condensed and the rest is history.