In Search of Perfect Symmetry


The ancient Greeks ordained that the planets moved in exact circular orbits, these being the most perfectly symmetrical of the conic sections. This mathematical ideal continued until the Enlightenment and indeed even Kepler originally developed his planetary theory on the bases that the orbits of the 5 known planets, were determined by perfect spheres that fitted around the Platonic solids. [The Greeks knew there were only 5 of these regular polyhedral but it required the advent of topology to prove this]. Kepler later realised the need to relax the constraints of such symmetry and to consider the ellipse, of which the circle is just one special example. Symmetry in geometry determines the shape of crystals and snowflakes, while the symmetry of equations can determine the laws of physics. Thus progress of fundamental physics, has often involved the search for new symmetries and it seems that nature allows as many symmetries as possible, rather than being restrictive in its laws. In particular it is the breaking of these symmetries that often provide the interesting manifestations that we observe around us, but which also hides the fundamental laws of nature. Indeed one of the most obvious examples of symmetry breaking is the preponderance of matter over antimatter in the early universe, possibly due to an asymmetry in the decay of the X boson in Grand Unified Theory.
Newton observed a symmetry that allowed him to identify/unify the motion of objects falling to the ground with that of the moon orbiting the Earth, while the symmetry of Minkowski space-time is evident in Einstein's special relativity(SR), which when 'relaxed' to include non inertial frames gives us a greater insight into gravity. Indeed gravity can be regarded in terms of a local (i.e. position dependent) symmetry applied to special relativity thus making it equivalent to acceleration, (velocity and therefore the space-time frame of an observer, varies locally with its 4 dimensional 'position').This explains the equivalence of gravitational mass and inertial mass (Einstein's 'most beautiful thought'). We can therefore apply classical laws of physics to an accelerating frame of reference, if we invoke a gravitational field or more importantly, a gravitational field can be eliminated locally by applying an accelerating frame of reference and hence the correct metric (curvature) of space-time that goes with it. A gravitational field causes the global Lorentz transformations of special relativity (with the invariant interval of its pseudo-Euclidean space-time) to become locally symmetric, thereby extending special relativity to that of general relativity(GR), which is described in terms of Riemann geometry. Likewise the unified electroweak field needs to be invoked, if we are to allow a local gauge transformation to act symmetrically on a family of leptons or quarks. Symmetry therefore allows us to unify phenomena, which would otherwise appear disparate and group theory allows us to study these symmetries. However Einstein's theory of GR (which deals successfully with large gravitational forces) comes into conflict with quantum theory (which deals non classically with the microcosm), when we consider the laws of physics near the singularity of a black hole or the early universe near the time of the big bang. This need to provide a quantum theory of gravity is therefore the central problem in unifying physics and has led to an even more elaborate use of symmetry.
Klein redefined geometry as the study of configurations that can be mapped into each other by means of a group of transformations and of those properties that remain invariant when these operations are applied. [Topology therefore being one of the least restrictive of these geometries] This interpretation was particularly valuable in the early days of particle physics when Special Unitary SU(n) groups, were found to describe a pattern in the plethora of observed particles (the Eightfold Way,) which led to the discovery of quarks. The fact that these particles could be given quantum numbers (e.g. strangeness, isospin, hypercharge), which were invariant under the action of hadronic processes, led to the discovery that baryons and mesons consisted of 3 or 2 quarks respectively. Each of these quarks has their own flavour, which remains conserved (invariant) under the strong nuclear interaction (quantum chromodynamics). However, these SU(n) groups have other applications in relation to the spinor theory of fermions/bosons and also regarding the unification of the fundamental interactions. These symmetries provide examples of Lie groups that involve continuous transformations and their group space therefore has topological interest. [Historical note; The Lie algebra of SU(2) has a basis consisting of the Pauli matrices who was the first to use them, when encorporating the spin of an electron into Schrodingers formulation. Later Heisenberg proposed SU(2) as a symmetry for the strong nuclear force, in which both protons and neutrons are considered as being different states of an internal isospin. Then in 1961 Gell-Mann and Ne'eman postulated that baryons and mesons could be arranged in what they called the Eight Fold way which employed an SU(3) discrete symmetry, where 3 quarks (up, down and strange ) were assigned as the fundamental representation. the up and down quarks formed the standard SU(2) isodoublet but the third quark was necessary due to the observation that a new quantum number in addition to isospin was conserved in hadronic processes, called strangeness. This new quantum number could be explained in terms of SU(3), which is a rank 2 Lie group, and its representation is consequently labelled by the third component of isospin (I3) and also a new quantum number 'hypercharge'. Couriously it therefore seems that the familiar concept of electric charge(Q) is therefore less fundamental than the concept of I3 and hypercharge. The I3 isospin of a particle describes how it transforms under SU(2) while the hypercharge (Y) describes how it transforms under SU(1). The electric charge is then computed from these two i.e. Q = I3 +Y/2 , while the electroweak theory describes the relevant particles under an SU(2)*U(1) symmetry, which under a Yang-Mills (loca)l gauge theory, yields the familiar photon, W and Z bosons].
Maxwell's laws of electromagnetism were a paradigm in that they unified electric forces with magnetism and were the first equations to be Lorentz invariant i.e they were symmetrical under the unimodular group that defined special relativity. Quantum Theory(QT) however involved wave functions that transformed under unitary transfomation (which left probability amplitudes invariant). The problem is however, that group theory states that there is no finite dimensional unitary (U) representation of a non-compact Lie group (such as the unimodular Lorentz group of SR). The solution was to invoke a quantum field theory (viz. Dirac's equation), which can combine SR and QT by resorting to a field of spinors, which therefore has infinite degrees of freedom. Such an equation involves the concept of functionals (functions with an infinite number of variables) but allows quantum functions to remain symmetrical under the Lorentz group of transformations. [Dirac's equation has a natural interpretation as representing an assembly of fermionic particles and antiparticles, together with their spin statistics i.e. a field equation] as already mentioned, Maxwell's equations are invariant under Lorentz transformations and making all of Physics compatible with these symmetries led Einstein to formulate the Theory of Relativity. Other important symmetries of these equations are conformal and gauge invariance, which have later played important roles in our understanding of phase transitions and critical phenomena, and in the formulation of the fundamental interactions in terms of gauge theories. Yet another symmetry hidden in Maxwell's equations is that of duality in which, there is an invariance of the vacuum equations under the interchange of electric and magnetic fields. In presence of matter however, the duality symmetry is not valid and in order to keep it, magnetic sources have to be introduced. [Duality. has been described in more detail above, in relation to string theory *!*]


With regards to such spinor space, this is not generally a faithful representation of the rotational group and leads to the result that 2 complete rotations are needed to make the spinor function return to its original state. However, although such first rank spinors (e.g. electrons) are a non-faithful representation of the rotation group (as can be shown from studying unimodular group space), higher rank spinors (e.g. photons) can produce faithful representations. Pauli and Schwinger showed using the CPT theorem (the invariance of physical laws under reversal of charge, parity and time), that half integer spinors have operators which are anti-symmetric i.e. fermions, while integer spinors operators are bosonic. There is a great deal of group theory proving all this and it can be shown that an odd number of spinors multiplied together gives a non- faithful (fermionic) representation of the rotation group, while an even number gives a faithful representation (bosonic). [This result can be achieved by analysing higher rank spinors using what's known as the Clebsch-Gorden series].
In the context of unification, SU(N) groups have been applied to Noether's theorem which relates conservation laws to the symmetries of the Lagrangian action in higher dynamics. By employing local gauge** symmetries, Yang-Mills theory has allowed the unification of Electromagnetism with the Weak nuclear force (this has SU(2)*U(1) symmetry) and by 'attaching' SU(3) symmetry we can also obtain a description for the Strong nuclear force -- this is known as the standard model. In order for the Lagrangian to be symmetrical under such a local group transformation, we have to add another field term, which are none other than the vector bosons that mediate the interactions between the fermions described by the Lagrangian. These are the gauge fields, a term which historically refers to the invariance of Maxwell's equations under a re-gauging of the electromagnetic field. Mills and Yang showed that this field arises naturally if one requires invariance under a localised U(1) symmetry.

Other interactions that result from such symmetries are likewise referred to as gauge fields, even though general relativity is not actually a gauge invariant field theory (also as explained below, gravity has so far defied all attempts of such a quantum description). This is such an important point that it is worth restating again in more specific detail. Consider the Dirac field which is a relativistic description of an assembly of electrons/positrons. The Dirac field posses a property (common to all wavefunctions) called phase, which can be considered as an internal symmetry that can be represented as an arrow pointing in a specific direction. We can pick out a direction for the arrow in an arbitary way and regard this as the zero point, (as we would when estimating the potential energy of a book placed on the table, since we are not concerned whether the table is in the basement or on the roof). Next, we apply a global transformation on the phase of the Dirac field (which rotates the arrow equally throughout space or time). Now since the physics is unaffected by such a gauge transformation, (i.e. Nature is invariant under such a symmetry group transformation acting upon its gauge field), then Noether's theorem leads us to the fact that there must be a conservation of electric charge! What is more important however, is that if we now restrict this transformation to be local, then we need to invoke a gauge field in order to maintain the invariance of the Lagrangian, (which after all determiner the physical laws). Hence the physics is unaltered, providing we compensate for the rate of change of phase with position or time, by introducing a gauge field. For the Dark (quantum) field, the gauge field is that of the photon.

Conservation laws for energy, momentum and angular momentum are of course valid in classical (Galena and relativistic) and in quantum mechanics, however Dark was able to demonstrate right from the beginning the relation that exists between energy momentum and angular momentum on the one hand and the time and space translation and rotation operators on the other. The reason for this early transparency is that in QT, the canonical transformations are represented by unitary transformations of the vector space of the physical states. In classical mechanics instead, phase space does not have this simple structure and therefore the connection between conserved quantities and the differential operators connected to the group of canonical transformations of phase space become clear much latter

Thus, these local gauge (internal) symmetries, acting upon the Lagrangian action of a multiple particle state, implies the conservation of quantum properties but require the introduction of these so-called gauge fields, in order to ensure this invariance of the Lagrangian under such (local) transformations. The interactions can therefore be viewed on a quantum level as gauge (Yang-Mills) fields, that must be added in order to ensure that the Lagrangian of (unified) particle states are symmetric under localised 'internal' transformation. [The quanta of these gauge fields, being the photon, 8 gluons and W, Z particles of the electromagnetic, strong and weak interactions respectively.] These symmetries are mathematically described by specific groups of transformations. Indeed as Wigner showed, quanta are actually defined by their irreducible representation of the Poincare group. Specifically it is the subgroup (called little group) which leave the relativistic momenta coordinates invariant, that determines the effect of an arbitrary Lorentz transformation and for all timelike momentum this is just the rotational group!! In other words mass and spin are the two properties which characterize systems invariant under the Poincare group and that spin also corresponds to a rotation group symmetry SU(2) but only if M^2>0 i.e. momenta are timelike. In the case of M^2=0, spin is no longer described by SU(2) and this is why the polarization states of a massless particle with spin J are Jz = +- J only, for example physical photons do not exist in a Jz = 0 state, whereas massive spin 1 particles do. [We have therefore seen that SU(2) symmetry, which can be used to describe the rotation of 2 objects into each other, has been employed for a number of different purposes. Firstly it is relevant to the description of spin 1/2 particles (i.e. spinor mathematics) and secondly it was used for isospin, an internal symmetry which put protons and neutrons on an equal footing in the early (failed) attempts to describe the strong nuclear force. This discrete form of Special Unitary groups, namely SU(3), was however fruitful in predicting the quark structure of Hadrons-- the so-called eight fold way. Thirdly it has been used in weak isospin (which has nothing to do with spin or isospin), that has been successfully employed in gauge unification of the electoweak theory. Also note that while fermion particles are (irreducible) fundamental representation of an (internal symmetry) gauge group, bosons are described in terms of the adjoint representation of the (internal symmetry) gauge group i.e. SU(2) acts on the Lie algebra su(2) by conjugation]


This pursuit of unification has highlighted 2 important considerations. One of these is the necessity to incorporate a spontaneous symmetry breaking mechanism (the Higgs field), which bestows mass upon the mediating W and Z vector bosons and hence explains the short range of the weak nuclear force [The first ever act of symmetry breaking was therefore the separating of gravity, perhaps by the mechanism of a super Higgs field 'immediately' after the big bang, when many of the universes dimensions became compactified. This was soon followed by a brief period of rapid inflation in which the universe expanded exponentially and the strong force became distinct from the electroweak.] Spontaneous symmetry breaking means, that although the symmetry of the Hamiltonian and commutator relations of Quantum Theory are preserved, the actual physical states (especially the vacuum state) no longer posses these symmetries. The other considerations, is the incompleteness of the endeavour, since to obtain full unification, these groups should not be merely 'attached' to each other but be subsumed in a larger group. This has led to the bold attempt to jump straight to a 'Theory of Everything', which will also include gravity, a force that has so far refused to conform with QT, since the important technique known as renormalisation, fails to remove infinities in the field of gravitons (one of the problems is that gravitons themselves 'gravitate'). Such an equation will be symmetrical under a master group, which will contain U(1) SU(2) and SU(3) as its sub groups, while its spectrum will contain all known particles and no doubt, other more exotic quantum states. It is hoped that such a complete unification will cause the embarrassing infinities of the gravitational field, to be cancelled by those of the other fields, thereby obviating the need for the 'trick' of renormalisation. [In supersymmetry mentioned below, it is hoped that that the infinities in the zero point energies of the particles will be cancelled by those of their superpartners, with any small excess resulting in an acceptable value for the cosmological constant i.e a small negative value for the energy of the vacuum. The observed 'flatness' of the universe can be accounted for, if the cosmological constant*** contributes approximately 70% to Einstein's field equation, compared to the ~ 5% observable matter, together with ~25% dark matter that we know exists from the study of the dynamics of galaxies and galaxy clusters. This cosmological constant would be in agreement with the recently observed acceleration in the expansion of the universe, however other notions such as the introduction of a 'quintessence' field have been considered, wich allows more flexibility in explaining this 5th force of repulsion.]
We therefore need to find a master symmetry group, which contains the smaller symmetry gauge groups that are associated with the fundamental interactions. However we have the same problem as before, in that such a quantised application of Noether's theorem must transform under a (master) unitary group, which cannot contain the non-compact unimodular group that describes GR. In other words we cannot find a gauge group that can combine the particle spectrum with quantum gravity, since a group that nontrivially combines both the Lorentz group and a compact Lie group cannot have finite dimensional unitary representations. Fortunately, there is hiatus to this theorem, in that it is only valid for real numbers and if we resort to supernumbers we can avoid this no-go theorem. [Just as Dirac had discovered a new CPT symmetry in his relativistic quantum field theory, which required a doubling of the number of particles (so as to include anti-particles of same mass) so too, the new super symmetry also requires a doubling of the number in order to accomodate heavier super particles] In other words, it is not possible to satisfactorily incorporate gravity (which is governed by the non-compact Poincare group) in what is known as a unitary representation (that dictates the other 3 quantum interactions), unless one resorts to supernumbers, which combines both fermions and bosons via supersymmetry. [Supersymmetry may also solves what is known as the hierarchy problem and not only overcomes the problem of renoralization of a gravitational field but in the case of a localised supersymmetry actualy invokes gravity! The hierarchy problem results from the fact that there is a huge difference, producing the so called 'Desert', between the electroweak scale and the Planck scale (~1Tev compared to ~10^16 Tev), when quantum effects should in fact drag the electroweak scale up to the Plank scale. It arises in GUT's without gravity since the weak Higg's field that is responsible for providing the mass (and short range) of the W and Z particles, is very different from strong Higg's field that provides the enormously larger mass that is needed to explain the observed stability of the proton. Yet these 2 Higg's field have to be united at high energies of the GUT's. Mass renormalization involves interactions between all particle species in the theory and it is difficult to see why the masses of similar species should not turn out to have comparable values. Indeed the large mass differences can be achieved in GUT's but only by (exceedingly) fine tuning various adjustable parameter and this, it in turn has to be altered at each energy scale.]

Supersymmetry, was originally invoked in the early study of string theory but although this has received a recent surge in popularity, supersymmetry itself does not require a string formulation. [Such a Lagrangian, is that associated with the surface being swept out by a string, rather than that of a point like particle which, as it moves through time sweeps out a curve.] Strings employ an additional symmetry -- a Weyl transformation -- in which the metric that defines the surface of the string is allowed to undergo a localised change of scale( i.e. a conformal symmetry). Localized Supersymmetry is called supergravity as it requires invoking a gravitational field since a repeated application of its generators causes a spacetime translation. Quantized (closed) string theory on the other hand requires a massless spin 2 boson (the graviton) even without supersymmetry being applied. However even the combined power of superstring theory cannot as yet explain why the cosmological constant (vacuum energy) is so small.
Hence modern unification deals with supermanifolds, which have anticommuting/fermionic coordinates (Grassmann variables), as well as the more usual bosonic/commuting numbers. Supersymmetry hopes to put the quanta of both 'particle' and 'field' (i.e. fermions and bosons), on an equal footing, while the latest development -- string theory, hopes to explain what symmetries are allowed, which in turn determine the conservation laws of physics. [However when spontaneous symmetry breaking is applied, we cannot put gauge fields and matter fields in the same multiplet since they transform differently under the isospin group; that is the fermions belong to the fundamental representation, while the gauge (Yang-Mills) fields belong to the adjoint representation. Therefore we must introduce superpartners for both the gauge and matter fields e.g. bosonic squarks and sleptons transforming under the fundamental representation and fermionic gauginos transforming under the adjoint representation as well as the Higgsinos and Goldstinos.] As an extra bonus, putting fermions and bosons in the same super multiparticle state, necessitates the introduction of a gravitational field when the supersymmetry is localised [The supersymmetric transformations invoke a field that produces the localised space-time translations that is indicative of gravity]. Incidentally another indication of the problem with gravity is due to a theorem by Cartan which, from topological considerations, shows that spinor equations cannot be incorporated into Riemannian techniques. Hence Dirac's equation cannot be extended from flat space-time of SR to that of the curved space-time of GR that is associated with gravitational fields. [Also a quantum theory of gravity implies quantised space-time!)
The group space of spinors is actually simpler than that of Euclidean space (the former is simply connected while the latter is doubly connected) it just that our minds find it easier to think in terms of space and time rather than the complex numbers of the covering manifolds of spinors (or twisters). In order to analyze which groups may be of use to particle theory, physicists often resort to irreducible representation of these groups and utilize such tools as the Clebsch-Gordan series, Adjoin representations, Young's tableaux and Dynkin diagrams etc. [In particular the Cartan group E6 is studied in some Grand Unified Theories, while E8 plays a prominent role in string theory.] In studying the dimensions and topology of the supermanifolds themselves, a useful tool is cohomology theory, where the Betti number in either combinatorial or de Rham theory allows us to calculate the Euler characteristic of the space (which is determined by the number of holes or genus of the space). This is particularly important in string theory where half the Euler characteristic is equal to the number of generations of quarks that are permitted! In twister theory, which is written in the language of 2-Spinors, sheaf cohomology is indispensable in analysing the properties of any equations written in this complex number manifold. Hence, we need to study the symmetry properties of the manifolds in which the equations exist (which is affected by the number of dimensions and how they are compactified), since this determines the allowed groups of transformations, the permitted form of the Lagrangian action and hence the spectra of quanta. These gauge transformations are topological and therefore they have a group space whose properties are important when deciding which are relevant to the real world of particles. [So far none of the predicted superparteners e.g. sleptons, photinos, have been detected].

We have therefore seen that the same symmetry groups have been applied in different ways. For example SU(2) has been used to describe the Strong 'internal' Isospin by Heisenberg (now replaced by the SU(3) of QCD), Weak isospin (in the electroweek gauge theory of Salam Glashow and Weinberg) and the 'intrinsic' spinor theory of fermions. Also note that in regards to the Eight Fold Way and Isospin, the symmetries are global and also are not exact (since the quark masses are all different and the 8 parameter of the SU(3) group are truly constants only when we neglect the other interactions). Indeed even the symmetries of parity and strangeness are not preserved in the weak interaction, however the SU(3) symmetry of the strong interaction is quite accurate due to the smallness of the quark masses (even though the strange mass is about 20 times that of the down quark). Symmetry principles first made their appearance with Einstein's identification of the invariance group of space and time. Later, Heisenberg enshrined isospin symmetry as one of the fundamental principles underlying his unified field theory of elementary particles, but it is now seen instead as an accidental consequence of QCD and the smallness of the up and down quark masses. A major step forward was therefore the realisation that these unitary symmetries could be productively applied to non-Abelian gauge theories, the prototype of which was proposed by Yang and Mills (1954). In particular a version based on the local gauge group SU(2)*SU(1) gives a unified divergence free description of the weak and electromagnetic interactions. In such theories, the basic structure is no longer the 4 dimensional spac-time but a fibre bundle, whose basis is the space-time manifold and whose fibre is the SU(2)*SU(1) group acting on the isospin and the hypercharge variables. The gauge group that acts on this enlarged space is (like the group of GR and the Abelian group of the electromagnetic interaction), an infinite-dimensional group whose elements are the local transformations of both space-time and internal variables. Invariance under the non-Abelian gauge group expresses the arbitrariness of the choice of space-time as well as of the internal variables and represents the non-Abelian generalization of the arbitrariness in the choice of the phase of the charged particle fields this mathematical scheme therefore generalizes the one introduced by Einstein to describe gravity and allows us to connect the fields associated to the electromagetic and weak interaction, to the metric properties of the fiber bundle. The equations of motion are therefore in many respects similar to those of GR (they are intrinsically non linear and thus give rise to stringent relations between the field equations and the equations of motion of their sources.) Invariance under these invariance groups (i.e. non-Abelian gauge groups), allows us to deduce not only the form of the interaction (in this case the weak and electromagnetic) but also the local conservation laws (whose integral form must correspond to the conservation of isospin and hypercharge. It also needs to be noted that the topological properties of the space of these theories gives rise to an entirely new type of conservation laws, which have become significant in recent theories (although these types of invariants might still be inclusive in Klein's system of geometrical classification).

It has even been suggested by some, that symmetry is not a fundamental property of the laws of physics but that it appears only at an infra-red level i.e. low energy. If we consider the most general renormalisable Poincare action for a spinless field then by dimensional counting, renormalisibility requires that there is no coefficient in the Lagrangian having dimension of mass to a negative power. [This requirement prevents the occurrence of terms with gradients other than the usual kinetic energy term]. It then follows that the action must be invariant under the parity operation. The existence of a renormalisable field theory for the spinless fields therefore plays a crucial role in the above derivation of parity symmetry. If however the spinless particles were not really fundamerntal but only bound states, then our Lagrangian would become an effective Lagrangian not constrained by renormalisibility. The allowed existence of certain effective terms, would mean that the parity operation was only an approximate symmetry at low energies but not one which is intrinsic to the theory or observable at high eneries! In other words more complicated Lagrangian terms involving bound states, may allow more flexibility than the renormalisibility constraint but the symmetries are not obeyed except at low energy.

Such an argument based on renormalisibility might also be applied to the more realistic Yang-Mills theories, providing an explanation of the many of the symmetries of the standard SU(3)*SU(2)*U(1) model of particle physics The study of Renormalisation groups have shown that theories that are asymmetric at high energies appear to be symmetric at lower energies, in an analogous way to which a slab of glass appears symmetric on a macroscopic scale whereas in actual fact it is very asymmetric and amorphous on a microscopic scale. [Sugar and its mirror image have identical elastic properties despite the parity non invariance of its molecules.] Specifically a theory of non covariant QED, can be shown to exhibit Lorentz invariance at low energies (only), as a result of the properties of the beta function of the renomatisation group and this has also been demonstrated for Yang Mills gauge theories. Indeed some success has even been achieved in showing that gauge symmetry has a chance of arising spontaneously even if nature is not gauge invariant at the fundamental level. The upshot of this approach, is that we may not need to insist on genera/Lorentz covariance when searching for a gauge quantum theory of gravity, since this symmetry may instead be just 'low' energy phenomena, that is not intrinsic to a 'unified theory' or quantum gravity. [The renormalisation group in not actually a group but it has served as a good constraint for what an acceptable gauge theory should look like. For example the only Lorentz invariant and gauge invariant renormalizable Lagrangian for photons and electrons is the Dirac Lagrangian of QED].

However it must be said that today, symmetry is by far the most important guiding principle in fundamental physics, as has been stressed by this article. The interaction amongst elementary particles is well described at low energies by a gauge theory based on the group U(1)*SU(2)*SU(3) and the prevailing philosophy is to assume that this is a remnant of a larger, group (SU(5), SO(10) or E6 etc.), which is spontaneously broken at very high energies. In other words, we assume that the apparent symmetry will increase at higher energies, where we will achieve Grand Unification. There is however an unorthodox approach which postulates that that at 'low' energies one see the maximum symmetry, which gradually disappears into a chaotic theory at high energies. More explicitly, attempts have been made to prove that the symmetric theory is an infra-red attractive fixed point to a large class of theories, which do not posses this symmetry and that this property will be true only for the small unitary groups that we observe.

General Relativity also exhibits a diffeomorphic symmetry (reparameterisation invariance), which does not exist in Special Relativity. A natural generalization of the work of Kaluza and Klein, shows that diffeomorphism symmetry of a higher than 4 dimensional space-time manifold, can give rise to the gauge symmetry of a Yang-Mills theory. [This effect can also be obtained from special diffeomorphic symmetries associated with the 2 dimensional string world sheet. Also there are some special cases of diffeomorphic symmetry, which are transformed into Poincare symmetries.] So by using the Kaluza-Klein idea we may 'derive' gauge symmetries from the geometrical symmetries of a higher dimensional space-time. Part of the diffeomorphism symmetry of gravity in higher dimensions becomes a gauge symmetry in four dimensions. In some sense, this just means that one assumed symmetry is interpreted as transformed into another one. Five-dimensional anti-de Sitter space (AdS), has a boundary which is four-dimensional, and in a certain limit looks like flat space-time with one time and three space directions. The AdS/CFT correspondence ( known as the Maldacina conjecture), states that the physics of gravity in this five dimensional anti-de Sitter space, is equivalent to a certain supersymmetric Yang-Mills theory (a Conformal Field Theory without gravity), which is defined on the boundary of AdS. The AdS/CFT correspondence is a type of duality, which states that two apparently distinct physical theories are actually equivalent. On one side of this duality is the physics of gravity in a spacetime known as anti-de Sitter space (AdS). while on the other is a supersymmetric Yang-Mills gauge theory that lives on the boundary of the AdS.

On another front, some theoreticians are working on hypercomplex numbers, of which the 8-Dimensional OCTONIONS seem to be the most favoured. Hypercomplex numbers are extensions of the more familiar complex numbers and include 4-D Quarterions which were invented(discovered) by Hamilton. These must have dimensions that are a power of 2 (e.g Sedions, 8-D and Voudons, 256-D hyper numbers whose name relates to the 256 binary entities at the bases of the traditional Voodoo pantheon). Octonions are of paticular interest since the symmetry illustrated by their Dynkin diagrams, are that of Cartan's Exceptional group G2. This group explains the compactification that occurs in the Calabi-Yau manifold that is a strong candidate in the study of string theory, since it nicely curls up the 10 dimensions into the 4 dimensions of space-time that we directly observe -- the other 6 dimensions have to be compacted on a Ricci flat manifold. [The main Lie groups are the Rotational groups SO(n) and SO(2n+1) the special Unitary groupSU(n) and the Symplectic group SP(2n), but there are also 5 exceptional groups identified by Cartan, viz. G2, E6,E7, E8 and F4.] However, string theory lacks a general principle which could help point us in the right direction, whereas in SR we had the principle of covariance, in GR the Principle of Equivalence, and in the 'standard model' we had gauge symmetry to guide us. Incidentally the number of unbroken supersymmetries surviving compactification, in a Kaluza-Klein theory, depends on the holonomy group of the extra dimensions and in order to obtain the observed chirality in particle physics we nedd to ensure that N=1

M theory is a recent revolution in our understanding of sting theory*! and has demonstrated a symmetry/duality between the 5 competing formulations, namely Type I, IIa, IIb, heterotic-O and heterotic-E as well as supergravity. This is achieved by increasing the number of dimensions to 11 and which also allows for the existence of higher dimensional 'branes' as well as the more usual strings, which represent our universe. Under this scheme it is found that the competing string theories can paired off under what is known as T-duality. For example when the extra dimension is curled up into a circle, M-theory yields the type IIA string, which when observed at low energy, can behave like type IIB theory under high energy. Similarly, if the extra dimension shrinks to a line segment, M-theory reduces to a heteroic-E string, which is conected to the heterotic-O string by dualities! [T- duality can itself also be shown to be a duality of S-duality which relates a gauge charge e, such as that of an electron with its topological soliton counterpart with charge1/e --- the magnetic monopole!]. Here again it becomes important to study the symmetries (and hence topologies) of the types of p-dimensional branes that can exist in such an exotic manifold. Some of the most recent discoveries include the fact that the weakness of gravity can be shown to be a result of it leaking from a parallel membrane into our own (it is consequently diluted by the extra dimensions that we do not directly observe). Also it suggests that the big bang can be formulated in terms of an initial violent collision between 2 such membranes. This has arison partly from the study of an important property of 10-dimensional string theory, that of boundary conditions on open strings. Unlike closed strings, open strings have endpoints, and this means that in defining the configuration space of the string, we have to specify the boundary conditions on these endpoints. The most natural choice would be to allow these endpoints to be located anywhere in space. Indeed, this is the only choice compatible with translation invariance and Lorentz invariance in 10 dimensions. If we "anchored" the endpoints in any way, it would distinguish some of the 9 spatial dimensions from the others. Mathematically, the boundary conditions that allow the endpoint of the string to be located anywhere in space, and hence satisfy Lorentz invariance, are called "Neumann" boundary conditions, familiar from the study of differential equations on spaces with boundaries. The alternative "Dirichlet" boundary conditions, which constrain the string endpoints to lie on definite surfaces in space, are clearly incompatible with translation and Lorentz invariance, and may preserve at most a part of these
invariances.

However, Neumann boundary conditions (which for a long time were considered to be the only reasonable ones for open strings) turn out to be too restrictive and miss some very profound dynamical phenomena in string theory. The reason is as follows. Consider an ordinary quantum field theory of pointlike particles. Although the underlying theory has translation invariance and Lorentz invariance, the individual states of the theory do not. For example, while the vacuum state of such a theory is translation invariant, the theory also has one-particle and multi-particle states in its spectrum. These states involve particles located at fixed positions (for example, imagine the state of a single particle at rest at a definite point). It is no surprise that this state breaks translation invariance.

One may therefore imagine that if we assign boundary conditions to open strings that violate translation and Lorentz invariance, we get a definite state of the theory, different from the vacuum state. This turns out to be true, and has a variety of deep and beautiful consequences. Suppose as a first example, we assign Dirichlet boundary conditions to the endpoints of an open string, along all the 9 spatial coordinates (x1,x2,...,x9). This constrains the endpoint of the string to lie at single position in space, while the rest of the string is free to fluctuate. For example, we can constrain the end point to lie at the origin in some coordinate system. Then this point becomes "special", and indeed behaves like a pointlike particle. For example, it breaks translation and Lorentz invariances in exactly the same way that a point particle state in a 10-dimensional field theory would. But this state is not one of the point particle-like excitations of the closed string that we have discussed! Quite the opposite -- it is an object defined by the property that open strings terminate on it. It turns out that one can assign a definite mass and charge to such an object, and that it behaves just like an elementary particle. It is commonly termed a "D-particle", where the "D" is a reminder that it arises by assigning Dirichlet boundary conditions to the endpoints of open strings.

So far we discussed two possible boundary conditions: Neumann in all 9 directions, or Dirichlet in all 9 dimensions. We can easily postulate a hybrid of the two, say Neumann in 2 directions and Dirichlet in the other 7. This corresponds to an open string that is "nailed" onto a two-dimensional spatial surface. Then, just as for the D-particle, we will be forced to interpret this entire surface as a dynamical object that extends in 2 spatial directions, commonly known as a "membrane". Therefore, just by assigning such boundary conditions, we have produced a quantum state of string theory that is extended in space like a membrane. It is a different issue whether this membrane is stable. It will be so if it is charged under some generalised gauge field, just as the string was stable because of its charge under the second-rank tensor field. Indeed, it is straightforward to see that amembrane can be charged under a third-rank tensor field, just because the trajectory of a membrane is a three-dimensional surface in spacetime. This happens in type IIA string theory, where indeed the spectrum contains a third-rank tensor field. Hence type IIA string theory is not just a theory of particles and strings, but also membranes. This is easily generalised. If we assign p Neumann and 9-p Dirichlet boundary conditions, the correspond state is extended in p spatial directions and is called a "Dirichlet p-brane", or Dp-brane for short. Thus we see that string theories are not just theories of strings. They contain, in a very natural way, extended objects called branes in their spectrum. Note that among other things,
D-branes provide an explanation of the role of the exotic tensor gauge fields in type II string theories: they endow
the branes with stability.

In the study of string theory, branes turn out to be just as important as strings -- in fact, one may say that fundamental strings are only a special type of p-brane with p=1. One should be careful, though, to realise that the fundamental string is not a Dirichlet brane as we have defined it. It is postulated from the beginning, and not defined in terms of something else ending on it. The Dirichlet branes are special precisely because they are defined through fundamental strings that end on them. This enables us to study them using familiar techniques of string perturbation theory. The search continues for the perfect symmetry that is obscured by the richness of the observed universe.

* * * * * * * * * * * * * * * * *

** The gauge principle which was originally applied by Yang and Mills, can be considered as follows. If a proposed Lagrangian of some matter field £ is invariant under a global (but not local) gauge transformations, alter the Lagrangian by replacing the partial derivatives by covariant derivatives (introducing a new gauge field, a connection $ or a potential A, whose transformation rule is compatible with the gauge transformation); the Lagrangian then has local gauge invariance. Then add to the resulting Lagrangian a new term proportional to the square of the "length" of the curvature d$ +1/2[$,$] of the gauge field, so that the gauge invariance is not destroyed. Variations with respect to £ yield the field equations for £ and variations with respect to $ yield the field equations for the gauge field. Then when we quantize the gauge field, the quanta of this field are identified as particles and the forces between the particles of the original matter field £ is explained by the exchange of these gauge particles

*!Aside
Briefly, a closed string has 2 sets of operators which do not interact, that is as the string propagates, it has right and left moving oscillator modes. The left moving modes are purely bosonic and exist in a 26 dimensional space, which has been compactified to 10 dimensions. The right moving modes only live in this 10 dimensional space and contain the supersymmetric part of the theory, while the compactified 16 dimensional string lives in the root lattice space of an E8*E8 isospin symmetry, which is more than large enough to contain the required spectum of particles. When the left moving half and the right moving half are put together they produce the heterotic string (meaning "hybrid vigour"). Compactification of the extra six dimensions on a Ricci flat manifold, then reduces the 10 dimensional superstring into our familiar 4 dimensional space-time.

A more formal treatment of supersymmetry can be found in the following paper

*** Einstein's theory states that nothing can exceed the speed of light, however a more recent idea known as DSR (Double Special Relativity) suggests that the speed of light itself varies according to the energy scale being considered and hence the wavelength ('colour')of light. This theory has been invoked in order to overcome certain problems in quantum gravity, ( viz. the Planck cut-off energy for photons is, due to Doppler shift, dependent on the directional and speed of the observer and would not therefore be invariant,) as well as explaining the observed preponderance of extremely high energy cosmic rays. Such ideas would also fit in with theories involving a variable cosmological constant (Lambda) that are used to explain the observed acceleration in the expansion of the universe, as well as solve the horizon, flatness and missing mass problems and these theories do not rely upon inflation occuring at the early stage of the universe. Einstein's equation shows that if the vacuum has a positive energy density (which equals lambda multiplied by the forth power of the speed of light), it will posses strange properties, such as a negative pressure (which will dominate), causing a repullsive force between points in space! In one such formulation involving a variable speed of light (c) at the birth of the universe, "c" decreases sharply and lambda is converted into matter and a Big Bang occurs. As soon as lambda becomes nondominent, the speed of light stabalizes and the universe expands as usual. However a small residual lambda remains in the background, and eventually resurfaces as an accelerated expansion (like inflation) as has recently been claimed to be observed using the results from studying supernova in distant galaxy clusters. But as lambda proceeds to dominate the universe in which matter has pushed away to produce mainly empty space, it again creates conditions for another sharp decrease in the speed of light "c"and a new Big Bang and a fractal universe is envisaged!

Even without a varying speed of light, if the universe does have a non zero cosmological constant, it implies that the missing mass is not as great as it would otherwise be. The flatness of the universe depends on 3 factors; the amount of visible mass and energy, the value of lambda, and the amount of dark matter. Some of this dark matter is known to exist, from the study of galaxy rotations and galaxy cluster dynamics, however it is not known for sure as to the nature of this invisible source. Suggested candidates include WHIMPS (weakly interacting massive particles) MACHOS (massive compact objects) or a small but significant mass being associated with the abundant neutrino (this would be classified as hot dark matter, and is supported by the observation that neutrinos can change flavour). It is not however known if this dark matter is enough to actually produce a flat universe since as already stated this depends on the value of lambda (the larger the cosmological constant is, the smaller the missing mass contribution). The most favorite candidate for the missing mass is however the axion which is predicted by the QCD. Without invoking an axial field with mass, strong interactions would indicate a large violation of CP (charge - parity) which is not observed (for example the neutron electric dipole is known to be very small). [A weak violation in CP has been observed in the decay of the neutral Kaon particle and it is a necessary effect in some Grand Unified Theories, in order to explain the asymmetry of matter over anti-matter].The axion field arises because in QCD there is a term in the Lagrangian that includes a constant 'Theta' that is expected to be ~1, which in turn would lead to a large CP violation. Theta arises as a constant in a term that arises due to instantons (i.e. 3-D solitons) that allow topologically distinct vacua to tunnel into each other. [The true vacuum is thus regarded as a supposition of the various vacua, each belonging to some different homotopy class]. The instanton solves one problem that is associated with an axial U(1) symmetry, since it explains why we do not observe the associated Nambu-Goldstone bosons around the mass the pion (the instantons obviates this problem by preventing the U(1) symmetry from appearing) however its very existence indicates that T and hence CP is violated! This strong CP problem can be resolved by introducing a new Peccei-Quinn symmetry, a near symmetry that can explain how strong interactions can maintain the symmetry between particles and antiparticles, while weak interactions do not. When this symmetry is spontaneously broken, it results in the appearance of a new spin -0 particle with mass -- the axion. The P-Q symmetry forces theta ~0 at low temperatures today but in the early universe the axion field was free to role around the 'Mexican hat' potential (Higgs field). The axion field motion in the angular direction is called theta and since the (vertical) curvature of the potential is zero, the axion at the high temperature was massive. However, when the universe cooled the axion field 'tilts' due to QCD instanton effects and the axion begins to oscillate round the minimum, like a marble in the rim of a tilted Mexican hat. The minimum of the potential forces the average value of theta to zero, solving the strong CP problem and the curvature of the potential means the axion now has a mass. There is no damping mechanism for the axion oscillation, so the energy density which goes into oscillations remains until today as a coherent field condensate filling the universe. This is a zero momentum condensate and so constitutes cold dark matter. One can identify this energy density as a bunch of axion particles which can later become the dark matter halo of galaxies!

Historically, the notion of quantized energy states was first applied to electromagnetic radiation and gave rise to the photon of energy [E=hf] but although it correctly accounted for black body radiation, this was still a semi classical theory and did not embrace the probabilistic uncertainty associated with full blown QM. Bohr realized the need to apply quantum theory to matter as well as radiation and elaborating on the notion of the de Broglie wavelength, he endeavored to produce a quantized description of the atoms and explain the periodic table. The first fully quantized formulation was however due to Schrodinger and Heisenberg who independently produced their wave and matrix mechanics. The original formulation of a governing set of rules/equations therefore related to the description of a particle. From this initial quantization it became evident that, after one had set up a quantum theory for a single particle state and so introduced numbers representing a state of the particle,one can make these numbers into linear operators satisfying with their conjugate complexes the correct commutational relations and one then has the appropriate mathematical basis for dealing with an assembly of the particles (in the initial instance a set of oscillators corresponding to a field of bosons, there being a corresponding procedure for fermions). In this way we have arrived at a viable way of formulating a quantum description of a field (although it sufferers from not being relativistic) and indeed this description is more profound one applicable to fermions such as an electron and supersedes Schrodingers description for a single particle. Instead we use Dirac's relativistic equation which together with the correct commutational relations, has a natural interpretation as a field theory. Likewise, we can use the Hamiltonian description of Maxwell's field to produce a relativistic field equation. (together with its respective commutational relations). Such developments therefore make the term Second quantization redundant, since we now regard that there is only one method of quantization that can either be applied to a particle or a field (the latter being interpreted as the probability of observing a given assembly of particles). The creation and annihilation operators that are part and parcel of relativistic quantum field theory are hence suitable for describing how matter can interact with their fields.

 

 

Spinor Theory Slides

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Historical Aside upon gauge fields and vector bunles

In the 1910s the ideas of Lie and Killing were taken up by the French mathematician Élie-Joseph Cartan, who simplified their theory and rederived the classification of what came to be called the classical complex Lie algebras. The simple Lie algebras, out of which all the others in the classification are made, were all representable as algebras of matrices, and in a sense Lie algebra is the abstract setting for matrix algebra. Connected to each Lie algebra there were a small number of Lie groups, and there was a canonical simplest one to choose in each case. The groups had an even simpler geometric interpretation than the corresponding algebras, for they turned out to describe motions that leave certain properties of figures unaltered. For example, in Euclidean three-dimensional space, rotations leave unaltered the distances between points; the set of all rotations about a fixed point turns out to form a Lie group, and it is one of the Lie groups in the classification. The theory of Lie algebras and Lie groups shows that there are only a few sensible ways to measure properties of figures in a linear space and that these methods yield groups of motions leaving the figures, which are (more or less) groups of matrices, unaltered. The result is a powerful theory that could be expected to apply to a wide range of problems in geometry and physics.
The leader in the endeavours to make Cartan's theory, which was confined to Lie algebras, yield results for a corresponding class of Lie groups was the German-American mathematician Hermann Weyl. He produced a rich and satisfying theory for the pure mathematician and wrote extensively on differential geometry and group theory and its applications to physics. Weyl attempted to produce a theory that would unify gravitation and electromagnetism. His theory met with criticism from Einstein and was generally regarded as unsuccessful; it has been only in the last quarter of the 20th century that similar unified field theories have met with any acceptance. Nonetheless, Weyl's approach demonstrates how the theory of Lie groups can enter into physics in a substantial way.

In any physical theory the endeavour is to make sense of observations. Different observers make different observations. If they differ in choice and direction of their coordinate axes, they give different coordinates to the same points, and so on. Yet the observers agree on certain consequences of their observations: in Newtonian physics and Euclidean geometry they agree on the distance between points. Special relativity explains how observers in a state of uniform relative motion differ about lengths and times but agree on a quantity called the interval. In each case they are able to do so because the relevant theory presents them with a group of transformations that converts one observer's measurements into another's and leaves the appropriate basic quantities invariant. What Weyl proposed was a group that would permit observers in nonuniform relative motion, and whose measurements of the same moving electron would differ, to convert their measurements and thus permit the (general) relativistic study of moving electric charges.

In the 1950s the American physicists Chen Ning Yang and Robert L. Mills gave a successful treatment of the so-called strong interaction in particle physics from the Lie group point of view. Twenty years later mathematicians took up their work, and a dramatic resurgence of interest in Weyl's theory began. These new developments, which had the incidental effect of enabling mathematicians to escape the problems in Weyl's original approach, were the outcome of lines of research that had originally been conducted with little regard for physical questions. Not for the first time, mathematics was to prove surprisingly or, as the Hungarian-born American physicist Eugene Wigner said, "unreasonably effective" in science.
Cartan had investigated how much may be accomplished in differential geometry using the idea of moving frames of reference. This work, which was partly inspired by Einstein's theory of general relativity, was also a development of the ideas of Riemannian geometry that had originally so excited Einstein. In the modern theory one imagines a space (usually a manifold) made up of overlapping coordinatized pieces. On each piece, one supposes some functions to be defined, which might in applications be the values of certain physical quantities. Rules are given for interpreting these quantities where the pieces overlap. The data are thought of as a bundle of information provided at each point. For each function defined on each patch, it is supposed that at each point a vector space is available as mathematical storage space for all its possible values. Because a vector space is attached at each point, the theory is called the theory of vector bundles. Other kinds of space may be attached, thus entering the more general theory of fibre bundles. The subtle and vital point is that it is possible to create quite different bundles which nonetheless look similar in small patches. An example of this is the cylinder and the Möbius band, which look alike in small pieces but are topologically distinct, since it is possible to give a standard sense of direction to all the lines in the cylinder but not to those in the Möbius band. Both spaces can be thought of as one-dimensional vector bundles over the circle, but they are very different. The cylinder is regarded as a "trivial" bundle, the Möbius band as a twisted one.

In the 1940s and '50s a vigorous branch of algebraic topology established the main features of the theory of bundles. Then, in the 1960s, work chiefly by Grothendieck and the English mathematician Michael Atiyah showed how the study of vector bundles on spaces could be regarded as the study of cohomology theory (called K theory). More significantly still, in the 1960s Atiyah, the American Isadore Singer, and others found ways of connecting this work to the study of a wide variety of questions involving partial differentiation, culminating in the celebrated Atiyah-Singer theorem for elliptic operators. (Elliptic is a technical term for the type of operator studied in potential theory.) This is a sophisticated theorem but can briefly be summarized as follows. In physics, the kernel of an operator is related to the number of zero modes. Now the integral of the (Gaussian) curvature of the tangent bundle of a ordinary space manifold can be related to the number of zero modes of differential operators constructed from this bundle, giving an index theorem viz. the Gauss Bonnet theorem, in which the integral of the curvature is equal to the Euler number (which equals the alternating sum of the Betti numbers of the manifold, which in turn can be derived from differential operators). Replacing this bundle by more abstract bundles such as gauge spaces, using higher dimensional curvatures and replacing the Laplacian with other elliptical operators associated with the bundle in question, generalize this to the Atiyah-Singer theorem. [For a given vector bundle, the index of differential equations formed from eliptical operators, is equal to the number of solutions minus the number of constraints and is a topological invariant.] An application of this was used to help show, that the number of generations of quarks in string theory, is equal to half the Euler number of the compactified (6D) space. [The Dirac index counts the number of zero modes which, acting on chiral spinors, is the number of generations of fermions and by using the Atiyah-Singer theorem this index can be shown to equal half the Euler characteristic of the manifold.] There are remarkable implications for the study of pure geometry, and much attention has been directed to the problem of how the theory of bundles embraces the theory of Yang and Mills, which it does precisely because there are nontrivial bundles, and to the question of how it can be made to pay off in large areas of theoretical physics. These include the theories of superspace and supergravity and the string theory of fundamental particles, which involves the theory of Riemann surfaces in novel and unexpected ways.

So in general the Atiyah-Singer theorem can be written as the integral over the product of 2 curvature forms, one for the gravitational (space-time) part and one for the gauge part. The gauge field can be regarded as a bundle, together with its 'spin connection' and the simplest topological invariant is represented by their characteristic classes (determined by the cohomology of the manifold), which are used to determine various Indexes, as implied by the Atiyah-Singer theorem.(e.g. the Ponttrjagin index, the Dirac index for a spin complex or Euler characteristic of a real space ). Examples of these characteristic classes are the Chern classes, Pontrjagin classes, Euler classes, Todd class and the Stiefel-Whitney classes. For example given a manifold M and a vector (tangent) bundle V on M, the natural gauge field is the spin connection, which will be the Christoffel coefficients and the gauge invariant field strength is the Riemann tensor. [Note that 'spin connections' are a general way of allowing description of parallel displacement in fibre bundles, such as the VIERBEIN used when using spinors etc ]. For the tangent bundle just as for any other vector bundle, the differential forms Trace R^R and more generally Trace R^2k are closed; their cohomology classes are topological invariants, independent of the choice of M. The cohomology class of TraceR^2k is, according to our general discussion, the kth Pontryagin class of M. Likewise the Euler class of the tangent bundle can be integrated over M to give the well known Euler number. [In this manner the tangent field is analysed from the point of view as being a fibre bundle --- a technique which can be productively employed to gauge fields also!]. So the the Lagrangian of the gauge field can determine the relevant characteristic class, which in tern determines which cohomology group it is linked to, while the topological nature of the base manifold M, will then determine what will be the rank of each particular cohomology group. In this way the integral of the 'curvature' of the gauge field will be equal to the number of zero modes of the relevant operators that produces this (de Rham) cohomology. This number is called the index and is therefore equated to the number of solutions to the differential equation minus the number of constraints. (this relates to the counting of the number of closed and exact differential forms and the corresponding Betti number)

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