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It is possible to tie up closed loops of string into complicated tangles which can nevertheless be untied without cutting the string. But suppose I gave you a tangled loop of string. How could you determine if it could be untied?
Conway showed us a clever trick with groups which enabled him to determine that some knotted loops could not be untied, but there were others which were not classified in this way. Conway had generalised a polynomial invariant of knots first discovered by Alexandria many years ago. The Conway Polynomial was quite a powerful tool to distinguish some knots from others, but it could not separate all. I remember thinking at the time that this was a piece of pure maths which would never have any useful applications apart from providing a way of proving that your boat cannot slip its moorings, perhaps. Mathematicians delight in this kind of problems.
Ten years later a dramatic change had taken place. Knot theory now looked like it was going to have applications to solving quantum gravity and probably other problems in condensed matter theory. Louis Kauffman had even written a substantial book called Knots and Physics (World Scientific). Conway's Knot Polynomial had been generalised and the problem of classifying knots seemed all but solved.
To summarise, I will list just a few points of interest here:
From this point on things are going to get more technical and I am going to assume that the reader knows some maths.
By way of illustration consider the following:
When Wheeler took some of the first steps in the development of canonical gravity he used the term "superspace" to refer to the three-dimensional geometry of space which describes the states of the theory. Similarly, in the early days of string theory, they discovered that space-time symmetry must be generalised to something they also called "superspace" . Are these two types of superspace related? Surely it is not a coincidence!
But, of course, it was just a coincidence. Wheeler's superspace has nothing to do with the new superspace of superstring theories. They are very different. Likewise, most string theorists hold the opinion that there is probably no connection between the loops of the loop representation of quantum gravity and the strings of string theory. The knot which the loops make in space cannot pass through each other without changing the quantum state discontinuously. On the other hand, superstrings can pass through each other and themselves without consequence. Despite this there is a small group of people such as John Baez and Lee Smolin who have suggested that there might be a connection all the same. The strings and loops both have a common origin in gauge theories and they share some mathematics such as quantum groups in their description.
The braid group is defined in the same way but with only the former relation. Put into words, this means that the braid group describes a symmetry where it does not matter in which order you exchange things but if you exchange two things then exchange them again you do not necessarily get back to where you were before.
There is a homomorphism from the braid group onto the symmetric group generated by the second relation. This means that the braid group is also a candidate for part of the universal symmetry according to the principle of event-symmetric space-time. In that case space-time events would behave like particles with fractional statistics.
Defining creation and annihilation operators for anyons is not a simple matter. Various schemes have been proposed but none seem ideal. However, here we have the advantage that our anyons are strung together. The statistics and symmetries of anyons must be described by knot theory.
The commutation relations used to generate the closed string algebra will remind anyone who knows about knot polynomials of Skein relations. This suggests a generalisation may be possible if the string connections are replaced by knotted cords which can be tied. These could be subject to the familiar Skein relations which define the HOMFLY polynomial.
In the special case where q=1 and z=0 this relation says that string can pass through itself. This is what we have for the strings which join the fermions. The crucial question is, are there generalisations of the parton commutation relations which are consistent with the general Skein relation?
One way to do it is as follows, but does this define a consistent algebra? It is not easy to say without some interpretation of what these symbols mean. A deeper understanding could guide us towards the right solution.
Perhaps that is why Weizsäcker came up with a fundamental idea which seemed completely out of touch with what anybody else was doing at the time. He proposed a bold theory of a way that space-time and physics might be constructed from a single bit of information by repeatedly applying the process of quantisation.
A binary digit or bit can take the value zero or one. You could think of a bit as about the simplest universe possible. Any amount of information can be coded using a sufficient number of bits. The universe is quantised, so quantise the bit. Now you have the quantum of spin-1/2, the spin of an electron which can take to values, spin-up or spin-down. The spin state is a unit length vector with two complex components which rotates under the action of SU(2) matrices.
This group is also a double covering of SO(3); the group of rotations in three-dimensional space. Weizsäcker wrote the two components as ur where r = 1 or 2, so he called them urs and the theory was ur-theory, but ur- is also a prefix meaning 'original' or 'primitive' in German so there is a double meaning.
Just as bits can be combined to make volumes of information, urs can be combined by tensor products to define higher-dimensional state spaces. It is also possible to quantise a second time, each ur of the quantum bit is replaced with a creation and annihilation operator, just as when a harmonic oscillator is quantised. This defines a more structured object which includes the symmetries of space-time. Just as quantisation of a field generates a multi-particle theory, the urs can be quantised again. This third quantisation generates a primitive form of field theory. Perhaps further quantisation can produce more of the structures of physics but the work remains incomplete.
But spin-networks turned out to be significant for four-dimensional quantum gravity too. Using the canonical quantisation methods which had led to the loop representation, relativists discovered that spin-networks should define a base of states for quantum gravity. If only they could discover the correct dynamics the breakthrough would be complete. There has already been much progress towards a four-dimensional theory of spin foams.
An interesting aspect to this story which makes it relevant here, is a remarkable parallel between the spin-network program started by Penrose and the ur-theory of Weizsäcker. Both are based in properties of SU(2) spinors. In ur-theory these spinors are regarded as the first quantisation of a bit, and are then quantised twice more. Spin networks are also derived by quantising SU(2) twice, but in rather different ways. SU(2) is first quantised to give the quantum group SUq(2), an algebraic deformation of the original group which was discovered in the 1980s. Then in 1992 Boulatov showed how you could define a quantisation of functions on quantum groups which formed spin networks. This achieved the same end as Weizsäcker but in a mathematically more powerful form.
What all this suggests is that multiple quantisation is of some fundamental importance to physics. It had been known since nearly the beginning of quantum theory that second quantisation was the way to construct quantum field theory, but this has always been regarded as a quirk rather than a fundamental feature. The first quantisation is often seen as a mistake of little significance. Some physicists even want to get rid of the term second quantisation because they dislike that interpretation so much. It is possible that they will turn out to be utterly wrong and Weizsäcker's multiple quantisation will be seen as a great insight many years ahead of its time when he first wrote about it in 1955.
The operators act on a state wavefunction 
which evolves according to a general form of the Schrödinger equation.
If Planck's constant h were zero this would merely mean that all operators commute like real numbers, which is what happens in classical mechanics. Quantum mechanics is said to be a deformation because it reduces to classical mechanics as a special case.
It is rather curious that this process of quantisation exists. We now think of classical mechanics as just an approximation to the real quantum mechanics. The fact that it is possible to derive the quantum mechanics from the classical approximation through a process of quantisation is just a handy trick of nature to which we should attach no great significance, or should we?
The fact that we have to do a second quantisation to get field theory is also just a curiosity, after all, it only works exactly for a simple non-relativistic system of non-interacting electrons. In the real world the Schrödinger equation must be modified to make it relativistic and gauged to introduce forces between the first and second quantisation. This certainly mucks up the procedure. Then again, it is very curious that things should work that way at all. Could multiple quantisation as we now understand it nevertheless be an echo of some deep feature of the final theory which just happens to become messed up as that theory is reduced to the approximation we know of it?
In modern times the term quantisation has been used to mean things other than what Dirac and Feynman meant. A symmetry from a classical matrix group like SU(N) can be quantised to give a quantum group SUq(N). Here quantisation is another type of deformation. q is a complex number parameter and in the special case where q = 1 the quantum group reduces to the classical one. This is not quite the same process as Dirac's quantisation but the analogy goes further than just borrowing the terminology. There is a real sense in which quantising a group with q=exp(ih) is very similar to quantising a system of mechanics. The suggestion is that there is some much more general algebraic process of quantisation of which both these things are a special case. We do not yet know what that general process is.
Since Dirac's first formulation, other equivalent ways to quantise a classical system were found. The most revealing of those was Feynman's path integral. Again you could in principle take any classical system with an action and quantise it using the path integral to define how the wave function evolves. Mathematicians have found ways in which quantum groups can arise through path integration too, but it is less direct.
Path integrals may give a clearer picture of what quantisation really is. Quantising a system which has different states seems to have something about all the different ways of going from A to B which are two different states of the system. In quantum mechanics these ways are the possible time evolutions of the system between the two states but it may be possible to generalise the concept further. In quantum field theory multiple particle systems are a derived consequence of quantising a classical field theory.
Strangely, there are other types of particle which appear as solutions of some classical systems. They are called solitary waves or just solitons. A special kind of soliton was discovered to be a solution of classical non-abelian gauge theories and they are interpreted as magnetic monopoles. What makes these especially strange is that they exist in the classical system and yet there may be a duality between monopoles and the electrically charged particles which only appear in the quantum field theory. The duality mixes up classical and quantum. There could be no clearer signal that the role of quantisation in physics is more special than it has often been given credit for.
The fermionic operators which are strung together in the discrete string model form a Heisenberg Lie superalgebra when the strings are removed. The universal enveloping algebra of this is then a Clifford algebra. I would like to repeat the string construction starting from a general Lie superalgebra. To keep things simple I will begin with just an ordinary Lie algebra A.
As before, the elements of the Lie-algebra can be strung together on strings but this time the commutation relations will look like this,
The commutation relations can be shown to be consistent with the Jacobi relations provided the functors satisfy the following associativity relationship,
and also the similar coassociativity relationship upside down. In this way we can take out Lie algebra A and generate a new Lie algebra L(A). The process can be generalised to a Lie superalgebra. In the case where A is a Heisenberg superalgebra there is a homomorphism from L(A) onto the discrete string algebra which I defined previously. So this process can be regarded as a generalisation.
The interesting thing to do now is look at what happens if we apply the L ladder operator to the string algebra. This can be visualised by circling the discrete strings around the network so that they are replaced with tubes. The interpretation is that we generate a supersymmetry algebra as string world sheets. The ladder operator can be applied as many times as desired to generate higher-dimensional symmetry algebras. Furthermore. There is always a homomorphism from L(A) back onto A. This makes it impossible to apply the ladder operator an infinite number of times to generate a single algebra which contains all the previous ones.
This last observation raises some interesting mathematical puzzles. The algebra formed by applying the ladder operator an infinite number of times will have the property that it is isomorphic to the algebra formed by applying the ladder operator to itself. It is certainly of interest to ask whether this situation actually arises after just a finite number of steps of the ladder. Would it be too daring to conjecture that the algebra becomes complete after only 26 steps in the ordinary Lie algebra case and 10 steps in the supersymmetric case?
To progress further it will be necessary to study more general categories like those defined by Skein relations. Mathematical physicists such as Louis Crane have looked at ways to construct n-categories by stepping up a ladder of dimensions. The symmetries I have described here could be a related to such structures. The hope is that a full theory of quantum gravity and string theory can be constructed algebraically in such a fashion.
It is possible to go down the scale of dimensions by compactifying space-times. From M-theory in 11 dimensions or F-theory in 12 dimensions it is possible to construct the important critical string theories in 10 dimensions. The strings themselves arise by winding membranes round the compactified dimensions so embedded objects can also be reduced in dimension. To construct such theories from first principles it may be necessary to go the other way and open up hidden dimensions but what is the process which performs this operation?
The suggestion of this chapter is that it is quantisation which allows us to go back up the dimensional ladder. This is supported in string theory by the observation that second quantised string theory in 10 dimensions is first quantised M-theory in 11 dimensions. In general we should expect a k-times quantised D-dimensional theory to correspond to a (k-1)-times quantised theory in (D+1) dimensions.
The ultimate theory may have the property that it is equivalent to itself under quantisation. In other words, quantisation acts as a symmetry on the theory. This is consistent with the observation of classical/quantum dualities in compactified string theories. Invariance under quantisation may be a fundamental principle which explains p-brane democracy.
Quantisation raises the dimensions of objects as well. Quantisation of a p-brane generates a (p+1)-brane. Everything is ultimately built out of instantons and the process of composition is multiple quantisation, but instantons too can be regarded as higher-dimensional objects which have been compactified so the process has no bottom as well as no top.
This dream of a structured theory of p-branes invariant under quantisation will only be realised if a suitable definition of quantisation can be found. It must be an algebraic definition which can be applied recursively. The best candidate for a mathematical discipline in which such a definition may be possible is category theory and its generalisation to n-category theory. Category theory is a way to describe objects and morphisms between them. n-categories permit higher-dimensional processes which map between morphisms. It is known that n-categories are related to n-dimensional topological quantum field theories but there is still much about them which is not understood.
Mathematical physicists such as John Baez have been studying their properties which relate beautifully to quantum theory and geometry. If the process of quantisation could be defined as a constructive mapping from an n-category to an (n+1)-category the link between dimension and quantisation would be established. A complete theory may be defined as the 
-category which is equivalent to itself under quantisation.