________________________________________

The concept of symmetry has played an important role both in the arts and in the sciences over many centuries. The formulation of the laws of Modern Physics is strongly guided by symmetry considerations. The famous physicist Richard Feynman described symmetry as follows : if something is done to a system and it is impossible to tell the difference between the system before and after the change then a symmetry is present. For example, if a perfect cube lying on a table is rotated by ninety degrees about a vertical axis it would not be possible to tell if anything has changed by looking at it before and after the rotation. This is anexample of a geometrical symmetry. The concept of geometrical symmetry maybe extended further to embrace a wider set of possibilities. If a mathematicalequation is unaltered by a change, (i.e) left invariant by a transformation, then this is also a symmetry. In the early years of the 20th century it was proved by Emmy Noether that corresponding to every symmetry operation there is an associated conservation law and vice versa. For example, the conservation of linear momentum is associated with the homogeneity of space which in turn is related to the invariance of the laws of Physics under a translation of the coordinate system.

Symmetries and conservation laws also manifest themselves in the spectral characteristics of the physical system. This link between invariance under a transformation and conservation laws has proved very useful in unravelling the spectra of atoms, molecules and elementary particles. Some physical systems such as a simple harmonic oscillator (a mass connected to an idealised spring) and an isolated proton interacting with an isolated electron by the Coulomb force law (an idealised hydrogen atom) exhibit extra symmetries in addition to the geometrical symmetries. These additional symmetries are related to the nature of the particular force and are referred to as dynamical symmetries.

Until the late 1960’s it was widely believed that all the geometrical and dynamical symmetries had been classified and that the list of symmetries was complete. There was even a theorem claiming to prove this. However, there was a subtle error in the proof of the theorem. The proof had tacitly assumed that the physical system was to be described by a set of commuting numbers, the ordinary numbers we have all been using for millennia. Another possibility had been explored by the mathematicians.

The properties of a set of anticommuting objects, subject to an algebra, had been studied earlier in the 20th century. With ordinary numbers the orders of mathematical operations does not matter - for example 3 times 4 is the  same as 4 times 3. For the anticommuting Grassmann numbers the order of the mathematical operations is important. In the early 1970’s it was shown that if a physical system were to be described by ordinary numbers and also Grassmann numbers then it was possible to have one further symmetry, over and above the geometrical and dynamical symmetries - a Supersymmetry. This symmetry opened the possibility of studying fermions (spin half objects such as all the elementary particles: electrons, quarks etc.) and bosons (integral spin systems such as photons, gluons etc. which mediate in carrying the forces between elementary particles) together by embedding them in a larger group and using a supersymmetric transformation to provide a link between the two sets. This was an exciting possibility because it had hitherto been believed that fermions are fermions, bosons are bosons and never the twain shall meet! Over the last 30 years various field theories incorporating supersymmetry have emerged, (superstring theory, the most well known among them), which have attempted to provide a unified theory of elementary particles. The initial excitement is somewhat tempered now because the beautiful unification comes at a great price: the supersymmetric theories predict a whole host of new particles none of which have been experimentally observed until now.

The optimists hope that the new high energy colliders coming into operation at CERN in Geneva in the next few years will provide some clues. All is not lost! From a Supersymmetric Quantum Field Theory of interacting elementary particles it is possible to extract an effective single particle Supersymmetric Quantum Mechanics (SUSYQM) and this approach has already proved very useful in explaining how apparently different forces in atomic and nuclear physics can lead to the same boundstate properties such as spectra and scattering properties such as phase shifts. A large part of my research has been concerned with working out the consequences of SUSYQM. Recently I have found a link between these ideas and a corner of Mathematics which has been studied for three centuties.

The power series expansion of functions such as exponentials, sines and  cosines are familiar to most students of Mathematics and Physics. However, the power series expansions of tangent, secant and cosecant functions are not so well known. The coefficients which appear in the series expansion of tangent and secant numbers had been studied by Bernoulli and Euler nearly three centuries ago and are named after these famous mathematicians. The Bernoulli numbers are also related to the Riemann Zeta function which plays a fundamental role in number theory. The Euler and Bernoulli numbers are related to some integers. Various algorithms to generate these numbers exist. Together with the mathematician Andrew Hodges at Wadham College I have been studying the tangent and secant numbers which are related to the Bernoulli and Euler numbers. The tangent and secant numbers are related to a certain classification of the permutation of numbers which we call the zigzag classification. For example, for n = 4, of the 24 permutations of (1234) just 5 of them have the zigzag property , (i.e), rise and fall, namely: (3412), (2314), (2413), (1324), (1423) , and this in turn is related to the Euler number E4. Counting the zigzag for n = 4 is easy. The problem is to find an algorithm for counting the zigzag for arbitrary n. Amazingly the problem of counting the zigzag may be mapped on to a problem involving a quantum algebra with an underlying supersymmetry, a picture in which the permutation classifications corresponding to the Euler and Bernoulli numbers correspond to the ’bosonic’ and ’fermionic’ sectors of a ’supersymmetric’ system. This is an unexpected connection which has led to a number of interesting results.

The physicists use the word ’parity’ to describe ’mirror’ symmetry. In the firsthalf of the twentieth century it was believed that all the forces of nature exhibited mirror symmetry, (i.e) the outcome of an actual experiment viewed in a mirror should have a definite correspondence to the outcome viewed directly. In the 1950’s experiments studying the nuclear deacy of an isotope of Cobalt clearly showed that the weak nuclear force did not exhibit parity symmetry, vindicating a suggestion by the Chinese physicists Lee and Yang. The parity violating effects are now considered to be linked to the form of the unified electro-weak force. I have been studying how the quantum algebra used to ’unify’ the Euler and Bernoulli numbers can be modified to understand the occurrence of parity violating effects. In thisscheme the fundamental commutation relation between operators used to construct the quantum algebra permits the inclusion of a parameter which in turn leads to spectral features with a clear parity dependence. The idea is quite general and should apply to nuclei and molecules which have an underlying octupole symmetry in the charge distribution. Already a nucleus, an isotope of the rare gas Radon, which exhibits such a symmetry has been identified. There are grounds for hoping that organic molecules exhibiting the same symmetry may be identified. Despite the lack of experimental evidence to suggest that a Supersymmetric Quantum Field Theory can successfully describe the physical world, there is clear evidence that Supersymmetric Quantum Mechanics is a very useful tool. SUSYQM has already been used to explain many aspects of inverse scattering theory, (i.e), the inference of an underlying force from the observation of the spectra and the measurement of phaseshifts. Quantum algebras with an explicit anticommuting character enable the building of bridges between areas of mathematics such as permutation groups, number theory and graph thoery, and areas of Theoretical Physics.


____________________________________________________________
 

I was educated in India and USA. After several post-doctoral positions at Manchester, Oxford and Berlin, I finally settled on a tenured position at Oxford nearly 20 years ago.I am a Fellow of Wadham College, University of Oxford. Wadham College was founded in the year 1610, before Newton was born. My research interests are in the general area of Theoretical Physics and I have published research papers on problems in Atomic Physics, Nuclear Physics, Supersymmetric Quantum Mechanics, Mathematical Physics and Pure Mathematics.