Many colleagues have their own web pages, but I have been very slow in organising mine.
Looking at the list of my publications, you will see that a good proportion of them are joint, and that I have enjoyed working with many collaborators. Simply writing down a list of publications seemed rather cold and uninformative, so I began to write a few jottings about the background to the origins of the ideas behind the work, and how the publications came about. Although I have tried to be brief, the story has turned out to be quite long, and a few anecdotes have crept in. Writing it has unearthed many memories, but these go back over 50 years. I am sure that others will have different recollections of the same times, but I hope that my memory has not led me too far astray.
Two things that I already knew have been brought home to me by writing these notes: first, how large a role chance plays in all of our lives; and second, how fortunate I have been both in my colleagues and in my marriage and family life, the two entangled threads which have formed the main pattern of my adult life.
Research student in Cambridge (1948-51)
Research Fellow at Glasgow (1951-54)
Lecturer/Senior Lecturer at Cardiff (1954-60/62)
Harwell and the Rutherford Laboratory
Trinity College, Dublin (1963-65)
Texas A&M, Los Alamos, La Jolla and Stanford (1965-66)
Professor of Applied Mathematics, University of Kent (1965-1994)
Professor Emeritus of Applied Mathematics, University of Kent (1994-present)
In the autumn of 1948, after taking Parts 2 and 3 of the Mathematical Tripos at Cambridge, I began research into Quantum Field Theory of Elementary Particles under the supervision of Dr Nick Kemmer. The students in the group included Paul Matthews, Richard Eden, Gordon Moorhouse, Behram Kursunoglu, Angas Hurst, and from 1949, Sam Edwards and Abdus Salam. The senior academic in the group was Professor Paul Dirac, and we heard his first talk on his 1948 paper on monopoles. Several times each year, we had a seminar from one or more of Fred Hoyle, Hermann Bondi and Tom Gold; this was the age when they were hotly debating the Steady State model, and there were interesting occasions when all three were at the blackboard, arguing.
About three weeks into my first term, Nick Kemmer handed me preprints of the first two papers by Freeman Dyson on the 'new quantum field theory', saying that they 'seemed quite interesting'. The work of Feynman, Schwinger and Tomonaga did not reach Cambridge for several months, so Dyson's papers were the first information we had about this revolution. I was given the task of explaining the first paper in a seminar 3 weeks later, and the second paper in the following week. Naturally, I did not understand the papers when I reported on them. However, this was the beginning of years of work by the group in understanding and developing the new theory.
My first research in Cambridge extended the concept of Feynman graphs to a large thin shell in a space with signature (1,4), with periodic boundary conditions, and I calculated the generalisation of the Kaluza-Klein formula in this shell. In the limit of thinness, this reduced to the standard formula. In a seminar I gave on this result in autumn 1949, my new supervisor, Jim Hamilton, heavily criticised me on minor technical grounds. As a result of this discouragement, this original work was never published, and it was nearly 30 years before I returned to higher-dimensional theories.
In 1950, I began a calculation of the mesonic equivalent of Karplus and Kroll's fourth order calculation of the electron's magnetic moment. I was able to systematise an operator factorisation that they had introduced, and by discovering the 'symmetric integration' formula, I showed how the energy-momentum integration could be performed automatically for any Feynman graph. A brief account of this work was published as a letter (ref 1): on the advice of Jim Hamilton, the full account (ref 2) was buried in the Cambridge Philosophical Society proceedings. The symmetric integration method was re-discovered by Nakanishi several years later, but he kindly gave me priority in his book. The method was also one of the approaches to the analysis of Feynman matrix elements in the book by Eden, Landshoff, Olive and Polkinghorne. This work formed the basis of my doctoral thesis Calculation of S-Matrix elements and Magnetic Moments (Cambridge University, 1952); in this thesis, I also derived the first set of algorithms for the reduction of scalar products of Dirac Gamma-matrices. These two algorithms were discovered independently by Caianello and Fubini. I published them later (ref 10), when I had completed the full set of algorithms.
In 1951, I went to Glasgow as a Nuffield Research Fellow in Natural Philosophy, working under Professor John Gunn. My very lively and inventive colleague Dr Bruno Touschek suggested that virtual mesons might be the explanation of spin-orbit coupling in nuclei, and wanted to use my knowledge of Feynman's methods. We carried out a second-order calculation (ref 3) which showed that pseudoscalar mesons gave an effect of the right magnitude but wrong sign. Later, my student Ernest Laing showed that, by including self- energy terms, a more complex calculation gave the correct order of magnitude and sign for the spin-orbit coupling.
I also published a paper (ref 4) with Tony de Borde, on the fundamentals of statistical mechanics. On the basis of this work, we set about writing a student textbook on Statistical Mechanics, and approached Robert Maxwell of Pergamon Press about publication. We were pleased to have our proposal accepted, and I visited Maxwell in London to discuss the contract. He was most pleasant, and spent an hour making sure that I understood every clause in the contract. Tony de Borde and I went through the contract carefully, and thought it very reasonable, so we signed. All was well until we found that, on overseas sales, we received 3.33% royalties instead of the 6.67% that we had expected - but this was perfectly legal. We had to console ourselves with the thought that the book (booklist 1) did not sell very well anyway!
Monty and I were married in the summer of 1953, and she came north to share the freezing fog of the Glasgow winter, and to experience teaching in some poor areas of Scotland. I had no guarantee of employment beyond the summer of 1954, but I was fortunate to land one of the few University appointments that came up in 1954, in a very congenial College in an attractive part of the country.
In 1954, I was appointed Lecturer in applied Mathematics at University College, Cardiff. The Head of the Mathematics department was Professor Lionel Cooper, and the senior applied mathematician was Dr Rosa Morris. There were no other particle theorists at Cardiff, but I kept in touch with the subject by attending conferences and visiting the Harwell (later, the Rutherford Laboratory) Theory Group most summers.
I continued work started in Glasgow, and showed that the field theory based on neutral pseudoscalar mesons with pseudovector interactions was equivalent, after renormalisation, to the theory with exponential interactions (ref 5). With my research student Milo Dixon, I extended these ideas to the more realistic charge-symmetric PS-PV theory, but this paper (ref 6) was a great deal more complicated. Milo Dixon also carried out a very beautiful piece of work, examining in detail how Feynman graphs in neutral PS- PV theory combined to become Feynman graphs in exponential theory. His report on this work earned him a valuable Studentship at Cambridge. However, he felt that this work was incomplete, and should be extended to symmetric theory before publication. I urged him repeatedly to complete his doctoral thesis and to publish his very significant analysis of the neutral theory, but I was not able to persuade him to do this. I have known other occasions, sadly, when a student, ambitious to complete a very difficult task, has not realised that some limited results are extremely valuable and interesting.
In a more general study of equivalent field theories (ref 8), I showed that almost local changes of variable in renormalised field theories left the physical content unchanged, provided that the free-particle Lagrangian was unchanged. This work was related to Haag's theorem, and I benefited from discussions on the Reimann-Lebesgue theorem with Lionel Cooper and Milo Dixon.
At Cardiff, I lectured on a wide variety of topics to mathematics, science and engineering students. Around 1958, Dr Roger Blin-Stoyle, acting for North-Holland Press, asked me whether I would write a textbook on mathematical methods for scientists and engineers. This seemed to be a huge task for one person. I knew Rosa Morris to be a very clear, careful and experienced expositor, and she agreed to share the task. This was acceptable to Blin-Stoyle, so we divided the writing of chapters between us. Some material was new to both of us, and we each tackled some of these areas of ignorance; my study of the basics of statistics and probability came in useful later. We continually discussed details of the work, and checked each other's material carefully. Writing and production took us several years, and the book (booklist2) was well over 700 pages long. The effort was worth it: in the late 1960's, North-Holland told us that we had broken their publication record for technical books. We even made a little money. Around 1984, we were asked whether we would update the book, but neither of us felt that we could undertake what would be a very substantial re-write, including new material on computation and numerical analysis.
One apparently small piece of work I did was to review a book (ref 9) on the foundations of statistical mechanics. In fact, the job took five months, and I learnt a great deal from the study. My thoughts turned to the definition of entropy, and I realised, as others have done, that entropy is a relative concept. I invented the parable of a 19th and a 20th century physicist each observing a simple experiment:
A box is partitioned into two equal halves, and the two halves contain two isotopes of a gas at STP, indistinguishable except that one isotope is radioactive and the other is not. The partition is raised, mixing the gases, and each physicist is asked whether the entropy has changed. The 19th century physicist detects no change in entropy; but the 20th century physicist, possessing a Geiger counter, can detect the radioactivity of one isotope, and so knows that the entropy gain is R ln2.
When I related this parable to Professor Rudolph Peierls in 1964, he immediately added, 'Yes, and he can get work out of it'.
On one of my summer visits to Harwell, Dr Tony Skyrme encouraged me to work with Dr Euan Squires on the Brueckner t-matrix, about which I knew nothing. Over a 2-week period, I asked Euan about 50 questions, most of them trivial. He was able to answer 48 of these, but the other two led us to extend the theory (ref 7): this involved isolating a Delta-function singularity in certain singular linear integral equations, and showing how this singularity showed up in numerical computations. On this visit, I also got to know John Bell; our friendship continued until his death.
In 1961, Dr Walter Marshall, then Head of the Theoretical Group at the Rutherford laboratory, invited me to visit. Dr John Gammel, visiting from Los Alamos, wanted to use my Feynman graph computational methods to evaluate fourth-order matrix elements, and to sum the mesonic strong-coupling perturbation series using Pade approximants. When we met, John Gammel launched into a 7-minute introduction to Pade approximants (of which I knew nothing), finishing, in his terse Texan way, with 'Well, there it is'. In 7 minutes, he had convinced me of the potential of the Pade method, and it was the basis of much of my work for the next sixteen years. What is surprising is that John and I have never collaborated on a paper.
At coffee one morning at the Rutherford, I was struck by a revelation. In quick succession, two smartly dressed physicists of around my age arrived, each with a small suitcase. They were warmly greeted by the company, and it turned out that each had been visiting Universities and Institutes in far-flung places such as Bombay, Rome, New York and California. I asked myself why on earth I had not travelled around, and immediately collected from the secretariat a form of application to visit CERN in Geneva for a year. My application was successful, and Cardiff UC gave me leave of absence for the academic year 1962-63.
In the summer of 1962, I was offered a chair of Natural Philosophy (meaning applied maths/theoretical physics) at Trinity College, Dublin. Nobel Prizewinner Professor Cecil Powell of Bristol gave me very strong advice to put CERN before Trinity College, saying 'Going to CERN will change your life, and that of your family', a prophetic utterence. However, TCD offered to let me start with a year on leave of absence. I was very dubious about moving to Dublin, but after weeks of thought, my wife Monty and I decided that I should leave Cardiff, and go to TCD in 1963. It was a finely-balanced decision, and if we had known all the facts, I would probably not have accepted the Chair. But we cannot say that the decision was wrong, because we can never know what would have happened in the 'parallel world' in which I declined the offer.
We left Cardiff in the summer of 1962 with children aged 3 (Carol) and 1 (David), and set off for a 5-week holiday on the Mediterranean, where we camped for the first time. Monty and Carol were ill when we arrived in Geneva, but we were fortunate in having arranged with Roger Phillips (of the Rutherford Laboratory) to take over his apartment in Annemasse for a few weeks. By asking around, I was able to rent a small chalet at Mies belonging to the remarkable Family Steffen. We stayed there happily for the rest of the year, and were most kindly and interestingly treated by the Steffens.
My early impression of CERN was that everyone was very busy at his or her own task, and it was slightly overwhelming to suddenly be at a world centre for particle physics. I finished off a piece of work started at Cardiff, showing that the (n,n) Pade approximant, formed from the power series solution of a Fredholm integral equation with kernel of rank n, was the exact solution of the equation (ref 10); it followed that the n-infinity limit gives the solution of a Fredholm equation with compact kernel.
My interest in summing divergent power series led me to discover something about 'band- wagons'. The latest theoretical fashion early in 1963 was 'peratisation', Pais's method of summing divergent series by including only the dominant term of each order. A young French theorist had been asked to give two seminars on the topic, and after the first I realised that the method was nonsensical. At the end of his second lecture I gave this 2- minute counter-example:
The nth term of the series to be summed is the sum of the nth powers of X and Y, where X is almost unity, and Y is large. The peratisation rule eliminates the powers of X, but the dominant term in the sum is in fact 1/(1-X), the sum of these powers.
I wrote the series on the board. Not only was there no response from the audience, but many of them had their eyes half-closed and glazed over, as though they did not really want to see what I had written. Work continued on the subject for several months after these seminars.
My interest in Feynman graph computations was revived by meeting Dr Mike Levine, who was an expert programmer. In the summer of 1963, we started to calculate the sixth-order contribution to the magnetic moment of the electron. Sadly, we both left CERN after a few months. I went to a very difficult and onerous job in Dublin; Mike went on a 3-year Fellowship at La Jolla. With very little help from me, he struggled on for a further year with the calculation, but gave up on the advice of Professor Norman Kroll, who told him to do something simpler in order to get some publications - 'publish or perish' is not new. However, my interest was revived in algorithms for scalar products of Dirac gamma-algebra, and I completed the set of three basic algorithms (ref 11); although the first two were independently discovered by Caianello and Fubini, my name has been attached to the full set. I also began working out similar formulae for scalar products of Pauli matrices, but these were not completed for a few years (ref 12). Mike Levine eventually re-started the sixth-order moment calculation with Jon Wright, and they successfully completed it about eight years later; Kinoshita carried out the same calculation independently.
The seminars and discussions at CERN were very helpful to me, and, later, I shall tell a strange story involving a seminar given by Cabibbo. I enjoyed many hours in what John Bell called discussions, but really, I was just listening to John arguing things out for himself: this was the time when he was working out what was wrong with von Neumann's logic of measurement, and his arguments - often, it seems, starting with 'consider the harmonic oscillator' - were very educational for me. Around Easter 1963, John put his head round my door and asked 'do you want to stay another year?' I felt that it would be unfair to Trinity College to ask for a second year of leave of absence, which might well have been refused, so I declined the invitation with great reluctance. Another parallel universe!
Our two young children were involved in an incident over a CERN preprint. As a preliminary, I must explain that, before we set off for Switzerland, my son had injudiciously put his finger through the wire netting of a rabbit's cage in a pet shop, resulting in a lot of blood and screams. At CERN, old preprints were available as scrap paper, and my daughter Carol, just 4 years old, became fascinated by Feynman diagrams; I explained that they represented the tracks of fundamental particles, and she used to work hard copying them. My former supervisor Jim Hamilton visited soon after we settled in Mies. As we talked, he happened to look over Carol's shoulder, finding her hard at work on a paper by Amati, Fubini and Stanghellini on the Multiperipheral Model, which contained many elaborate Feynman graphs. He was so surprised that he took his pipe out of his mouth; he asked Carol what she was doing. 'I'm drawing Feynman diagrams', she replied enthusiastically. 'Do you like Feynman diagrams?', asked Jim. 'Yes, I like them very much.' My son David, not quite two years old, was standing nearby, so Jim asked him very kindly, 'And do you like Feynman diagrams, David?'. David, clearly aggrieved, replied firmly,'No. When I was a little boy, I was bitten by a Feynman diagram.'
Our two years in Ireland were difficult. Alison was born just before Christmas, and was ill for a long time. Unusually, we had several other quite serious illnesses and injuries, and my father died in 1964. We had not realised that the ancient religious apartheid still dominated Irish life, and we were excluded from Catholic squash clubs and butchers! Our worst crime was to send our two young children to the best school in south Dublin - attached to a progressive Dominican convent: we and our children were regarded by some as traitors to the Protestant Ascendency. Also, I felt that I was in a continual fight with TCD authorities for resources which I had insisted on at my interview, and for reasonable conditions for running the School of Mathematics; this feeling was shared by my successive professorial colleagues Heini Halberstam and Gabor Dirac.
Despite these difficulties and a very high teaching load, I managed to do some research in Dublin. My main result was an analytic theorem on convergence of sequences of Pade approximants (ref 13); I learnt that the main problem was to pin down the location of singularities of the approximants. I also completed the paper on Pauli spin matrices (ref 12), and kept up a technical correspondance with Mike Levine at La Jolla.
Early in 1965, I decided that I would leave TCD, and took up an invitation from John Gammel, now Professor of Theoretical Physics at Texas A&M University, to visit there for a year. I also applied for posts elsewhere. I was very pleased to be offered a full Professorship in Mathematics at Toronto, and, ten days before we left for Texas, the Chair of Applied Mathematics at the new University of Kent at Canterbury. It would have been a great honour to be a colleague of Coxeter and other eminent mathematicians in Toronto, but we decided that we would like to bring our family up in England. We have never regretted our choice to move to Canterbury.
For all of us, the transition from Dublin to College Station, Texas, was like time-travel over several centuries. When we asked where we could shop, we were dumbstruck by the answer 'Oh, Piggly-Wiggly is just down the road': it was the first supermarket, and the first American-style name, that we had encountered. Although the USA was seriously troubled by the Vietnam war, it was rich and powerful, and everyday life was very easy and convenient. 'Yu'all are welcome 'were the hospitable words we heard everywhere. What sometimes astonished us was that some these of same people could just as easily come up with 'Just drop an atom bomb on Hanoi'. Texas was very sure of itself, and we only heard a few underground rumblings of suppressed political and social dissent: Berkeley was a long way away.
John Gammel had arranged that I be invited as a Distinguished Visiting Professor.The American salary ensured that we were comfortably off and could travel around in our Country Sedan, which had a playroom in the back. John Gammel was building up a small group of very able staff and students; these included Dr John Nuttall, who was extremely bright. We had regular group discussions on different topics. When Professor Kenneth Watson visited for a few weeks from California, he and John Nuttall gave a series of seminars on many-body dynamics. I contributed to this by studying a paper by Omnes on the three-body problem. These seminars were written up as a book, to which I contributed a chapter (ref 15). Dr Ray Cowan, a mathematician, tried to explain Helgason's book on differential geometry to John Nuttall and me, but we failed to cope with the language and notation. Since that time, I have become more familiar with mathematicians' language; I understand it better now, but I still think that some of their differential geometric notation can be very ambiguous and misleading.
Our travels included three visits to Los Alamos, 800 miles away, in the beautiful state of New Mexico. We enjoyed staying in La Fonda in Santa Fe, and then in the famous 'schoolhouse' in Los Alamos. Dr George Baker Jr, who was John Gammel's partner in bringing the Pade method to the notice of physicists in 1961, had remained at Los Alamos. On my first short visit, I discussed with George a simple model field theory by Peres, suggesting that the Pade method might be applied effectively to it. We showed that this was true, and published a paper suggesting that more serious field theory perturbation expansions, with zero radii of convergence, might give good numerical results through Pade approximants (ref 14). This was really just expanding on John Gammel's original suggestion to me, but our 'tiny model' had considerable effect elsewhere.
Dr Daniel Bessis worked in a group at Saclay, Paris, and was collaborating with Professor Dino Pusterla of Padova on field theory computations. Our tiny model convinced Bessis and Pusterla that the Pade method might work for realistic strong coupling calculations, and over the next few years they, with colleagues Basdevant and Zinn-Justin, carried out several large calculations. Their results, like those of the Gammel group, were mixed. The essential problem was that the only singularities of Pade approximants are poles, so, when a matrix element was evaluated near to a branch cut, the approximants did not have appropriate analytic structure. In their original papers, Baker and Gammel had appreciated the need for adapting the Pade method, by matching the analytic structures of the approximants and the function approximated. Later, this led to a whole new field of research: standard approximants were operated on in various ways to define new approximants with suitable matching analytic structures. George Baker and I said that one could not hope to calculate scattering matrix elements from the perturbation series, because they had to be evaluated on a branch cut: later on, other colleagues and I showed how to do this!
In mid-May, 1966, we left College Station and spent ten days travelling west, visiting the Grand Canyon, Las Vegas (which suffered the loss of 5 dollars at our hands), Death Valley (which was hot), and La Jolla, where we were very well looked after for a week by Professor Norman Kroll and his colleagues. Through Dr Tony Hearn, who was an expert on algebraic computation, I had been invited to join the famous Stanford Theoretical Physics group led by Professor Sid Drell. Most of the time, we rented the delightful home of Bob Moulton, chief administrator of SLAC, just a few yards from 'our' private Olympic swimming pool, which was a delight for our young family. I did not produce any papers there, but I could not help working quite hard in the Drell school. For lunch, the group assembled with enormous sandwiches, and assigned members of staff and students had to report on the progress of their research during the past two weeks - this pressure was the basis of the success of the group, but it still makes me feel quite exhausted. We have happy memories of Stanford, San Francisco, Berkeley, Santa Monica and the Redwoods, from those optimistic days of the mid-sixties.
On leaving from San Francisco airport, we were able to hand over our Country Sedan to Mike and Jaff Newton, who had been in College Station with us. We flew to New York and joined up with our piles of luggage shipped from Texas, and boarded the 'New Amsterdam' for a 5-day crossing of the Atlantic. This was an interesting interlude. The children were happy to spend a lot of time in their activities room while we took part in other activities. Each afternoon, we went to ballroom dancing classes given by the European Professional Champions, and were the only takers: so we each had an excellent partner to dance with. We impressed the family by each becoming the table-tennis champion; the trick was to spin the ball so that it went to one side, just as the roll of the ship took the table in the opposite direction. The ship was stopped just off Cobh, and we had the excitement of seeing our car from Ireland loaded on board from a tender. We just managed to get everyone and all our goods into it at Southampton, and drove off to Canterbury.
The University of Kent had a number of rented houses available for staff arriving at the University, and I had booked one of these. When we arrived, it was packed with all of our furniture from Ireland, but we were able to get on with the job of settling children into schools, and I could immediately begin starting a new department, arranging courses and lecturers, and meeting up with a host of new colleagues. Before long, we had arranged to have a house built for us. In the first few years, everyone in the University had to help with a vast number of jobs, sitting on innumerable appointing committees and planning committees for various buildings. I have always been interested in sports, and found myself in charge of setting up the Sport and Recreation Committee and staffing up the new sports centre. We were fortunate in recruiting an excellent Director of Physical recreation, George Popplewell, who believed, as I did, in getting as many staff and students as possible involved in physical recreation, and in building excellence from the ground up. I continued as Chairman of the S&RC for 13 years,
Alan Common, whom I had appointed as Lecturer in Dublin in 1964, joined me in Canterbury in 1966. The other new Lecturer was Peter Graves-Morris, who had just completed his PhD degree. Both were excellent appointments, and they each contributed a great deal over the years, in teaching, research and administration. When I was in Geneva in 1963, I had a friendly meeting over coffee with a Major from the US Army Air Force, which resulted in my being funded for Research Fellowships until 1971. The first Fellow in Canterbury, appointed in 1966, was Dick Hughes Jones. He transferred to a Lecturership in 1967, when John McEwan was also appointed Lecturer. The five of us remained the core of Applied Mathematics until Peter Graves-Morris took up a Chair at Bradford in the 1980s. Martin Oliver was appointed Lecturer a year or two later, and Alan Genz became a Research Fellow in 1969. In 1971, Gordon Makinson was appointed as Senior Lecturer to head a numerical analysis group, and Alan Genz supported him as Lecturer.
George Baker and John Gammel planned to edit a book on Pade Approximants, and asked me to write a chapter on Pade Approximants and Linear Integral equations, which had been applied to Potential Theory by a number of physicists (ref 16). In 1968, we had had a series of lectures on generalised functions by Werner Guttinger of Munich, supported by the USAAF. Alan Common and I contributed to the Baker-Gammel book a further paper (ref 17), in which we combined generalised functions and Pade Approximants to give meaning to infinite series of derivatives of Dirac delta-functions. We showed that certain series, as expected, represented an object with point support; rather surprisingly, less convergent series validly represented a function with non-compact support; physically this meant that a series of pole, dipole, quadrupole, and so on, at a point can represent a linearly extended charge distribution.
When I was at CERN in 1962-63, we started skiing. In the late sixties and through the seventies, we went on annual skiing holidays, and for some of them linked up with Alan and Ellen Common. On one occasion, while we were all having family supper after a day of skiing, Alan and I started talking about Pade Approximants, discussing how important and how useful they were. Our children tolerated this admirably, and Alison, aged six or seven, wanted to take an interest: 'What are you talking about?', she asked. We gave the totally inadequate reply, 'Pade Approximants'. Alison, still trying in her very practical way to understand, asked 'Have we got one at home?' How to encourage your children to take an interest in your work??
My interest in the calculation of magnetic moments led me to invent further gamma-algebra algorithms, now for reducing products of matrices in vertex parts with zero photon momentum (ref 18). I realised that, for electromagnetic field scattering, it would be useful to extend this study to general electromagnetic vertex parts. In a paper that pleased me greatly, I showed that the gamma-matrix products in all such vertex parts could be reduced by a set of algorithms involving Chebyshev polynomials whose argument was a simple function of the photon 4-momentum (ref 19). Much of this paper was worked out during visits to the Rutherford Laboratory. Also, I frequently returned to CERN for a week or two, and we became true Europeans, taking the family to St Maxime each summer. In the spring of 1969, when we had finished skiing in Cervinia, we dropped down the Italian side of the Alps to the European Physical Society meeting in Florence: this was a well-organised scientific cultural experience that blended with the great artistic culture of the City.
Late in 1969, I was surprised to be asked to be a plenary speaker at a Colloquium on Computational Physics to be held at CNRS, Marseille. Professor Tony Visconti had decided to get together those who were involved in performing higher order Feynman graph calculations. I qualified because of my PhD work and subsequent contributions to algorithmic methods. The first meeting was held in 1970, followed by a second in 1971. At that time. I was working with Alan Genz and an MSc student Glenys Rowlands, using the Pade method to accelerate sequences of numerical approximations to single integrals. I reported on this work both at the 1970 and the 1971 Colloquia. The reports were published in the two Proceedings (refs 20, 21), and we also published the results in the Journal of Computational Physics (ref 22). This work showed that the Pade method, used judiciously, often did accelerate convergence; but the technique we used was very simple, and more sophisticated methods, generalising Pade approximants, were already being worked out elsewhere.
In 1970 and 1971, I undertook lecture and research tours of a variety of establishments in the USA. During the second visit, I spent a week at Salt Lake City, where Tony Hearn was now a Professor. We discussed the beautiful work of Kahane, who had used my earlier scalar product algorithms to give a purely topological rule for reducing a string of gamma- matrices. We set about extending Kahane's rule to the most general form of gamma- structure, and were delighted to virtually complete the work in the week. We were tidying up our results when I was asked to give a paper at the First European Conference on Computational Physics in Geneva in April. 1972. A sad misunderstanding over attribution arose because a letter of mine to Tony was wrongly sent by sea-mail. The paper I presented made clear that the work was joint, but Tony declined to put his name on the published paper (ref 23) This was a great pity, since the collaboration was such a pleasure and so fruitful. It turned out that I was unable to attend the Geneva conference, and Professor Tini Veltman kindly presented the paper.
Around this time, a meeting on Analysis was held at UKC. Several of our friends were attending, and we had a small party at our house. Paul Erdos came, and it was the only time that we met this delightful, brilliant and enigmatic man. He was very charming to everyone, but his real interest was in our three children; when he discovered that they had a 'shed' at the end of the garden, he disappeared off to it with them. The question naturally arose, what was my Erdos Number? Through a George Baker and John Gammel chain, I could claim that it was 4.
In 1971-2, our department was working towards a double-headed Science Research Council Symposium, consisting of a Summer School and then a Colloquium on Pade Approximants. These events were also supported by NATO and the Institute of Physics. The backing of Professor Leslie Fox was very important in obtaining funding. I was aware that the subject was being actively pursued in many different institutions, and felt that it was a good time to bring everyone together, and Alan Common and I directed both events. With John McEwan and our three wives Ellen, Pauline and Monty, we arranged a social programme for the participants; this was boosted by a lively bar billiards competition in Keynes College in the evenings. Peter Graves-Morris edited the proceedings of both events, which were published by Academic Press and the Institute of Physics. When we had fixed the dates for the two events, I heard from Professor Bill Jones of Boulder, Colorado, that he and Wolf Thron had planned to hold a conference on the closely related subject, Continued Fractions, in precisely the same week as our symposium. Very kindly, they offered to postpone their meeting until the following week. As many participants, including me, wished to be at both meetings, we had a very busy and fruitful time at the events in Canterbury and Boulder. Continued Fractions are a very old concept, but these two meetings, bringing together mathematicians and physicists with different slants on the subject, and emphasising the importance of the Pade approach to the subject, initiated a new international phase for the subject. My own contributions to the Summer School and to the Symposium were a 5- lecture review of the mathematical properties of Pade Approximants (ref 24), and a review of their convergence properties (ref 25). I also reported on our numerical integration work at the Boulder conference (ref 26)
Many of those working in the Pade field (sorry about that!) were thinking about possible generalisations of the approximants. For about nine years, off and on, I had thought about defining approximants from series in two or more variables. After the dust of the 1972 conferences had settled, I sat down one Monday and listed a set of desirable properties of approximants defined from double power series: Gammel-Baker homographic invariance, reduction to Pade approximants when one variable is zero, reciprocal invariance, and so on. With clear principles established, it took me only a few hours to define unambiguous diagonal approximants (equal powers in numerator and denominator). I wrote a paper and had it typed and submitted for publication on the Tuesday morning. I realised that my approximants were capable of wide generalisation, and that a whole range of convergence and other theorems might be established. So I put a copy of my paper on the desks of all of my colleagues, and invited them to contribute to the research. Also, I applied for a Research Grant to support a 3-year research fellowship for this work, and Dr David Roberts took up this post in 1974.
Shakespeare's 'time and tide' was exemplified by a coincidence. The day after I circulated my paper, Alan Common returned from a conference and told me that a research student at Birmingham, Mr Lutterodt, had spoken on two-variable Pade approximants. I heard from Birmingham, and sent them a copy of my paper, detailing the coincidence of timing. In fact, the two pieces of work were quite different: whereas I had prescribed firm defining principles leading to a unique form of approximant, Lutterodt had described a broad framework for defining approximants. He came to speak at our seminar, and we acknowledged each other's work in our papers.
I was delighted that several members of the department (Peter Graves-Morris, John McEwan, Dick Hughes-Jones and Gordon Makinson) took up my invitation, and, with David Roberts, they contributed very substantially to the development of multivariate approximants. My initial paper was published in 1973 (ref 27), and by1978, the group had published 25 research and review papers on generalisations of my approximants. Several of these papers were published in the Proceedings of the Royal Society: the first three in the PRS were in collaboration with departmental colleagues (refs 28,29,30). In the first of these papers, John McEwan and I described a region of n-space corresponding to the parameter values of a set of linear equations. We drew a diagram of this region in 3-space: it was formed by cutting a cube down through a face diagonal, and then chopping bits off one half. During this work, I asked my daughter Carol whether she would make a model of this 3-space region, commenting that its volume was a quarter of that of the cube. Next morning, I found four copies of the model awaiting me. When I looked puzzled, Carol picked up the four copies and fitted them together to form the cube. Neither John nor I had had this acute geometric insight. I believe that John still has the four models.
At the third meeting organised by Tony Visconti in 1973, and at a subsequent Euromech meeting in Toulon in 1975, organised by Professor H.Cabannes, I surveyed the work being done by the group (refs 31,32); in these talks, I was happy to use the very elegant diagrams illustrating 2- variable and 3-variable regions drawn by Dick Hughes Jones.
Around 1968, Daniel Bessis and Dino Pusterla had visited Canterbury, because of their interest in Feynman graph calculations. They and their colleagues had found the Pade method useful provided that the calculation did not closely involve branch cuts of matrix elements. Even when methods were proposed to represent branch cuts in the form of the approximants, there was still the difficult problem that scattering amplitudes were to be calculated on the branch cut. I emphasise these words, since they imply (wrongly) that the branch cuts are fixed to the real axis in the invariant energy complex plane, as physicists habitually represent them.
I visited back and forward with Daniel in Saclay and Dino in Padova. On one of Dino's visits, he and I realised that, by expanding the denominator of a scattering amplitude about a point in the upper half of the energy plane, it was very likely that the branch cuts would be placed in the 'shadows' of the branch point - that is, in the lower half plane: this 'shadow property'was a known feature of Pade and continued fractions. This conjecture turned out to be true in practice, and we were able to simply calculate on the real energy axis directly, away from the singularities. I had worked with Alan Genz on numerical integration, and the three of us published this novel idea (ref 33). This was a solution to a major difficulty in computing Feynman amplitudes.
At this time, we had recruited an excellent group of research students. One of these, Leslie Short, took over the study of this method of computation. At the same time, Alan Common worked with a student, the late Terry Stacey, on the similar calculations, but using generalisations of Pade approximants, in particular 'differential approximants'. By now, Shafer had defined 'quadratic approximants'on two Reimann sheets, which were essentially 'Pade plus solving a quadratic equation'. These naturally generalised to 'cubic' and higher order polynomial approximants, defined on several Reimann sheets. 'Differential approximants' were 'Pade plus solving a linear differential equation', essentially the idea of Gammel and Baker. Once they had mastered these various techniques, Leslie and Terry began mixing them: the four of us more or less had a computational factory. My work with Leslie Short was published in a conference proceedings (ref 35). Meanwhile, I had contributed with David Roberts to the series of papers on multivariate approximants (ref 34). In a further two papers published by the Royal Society (refs 36,37) I put together the ideas of multivariate approximation and the representation of branch cuts; these were accompanied by a paper by Leslie Short on the techniques of evaluation of this more general type of multivariate approximant; as I indicated in the last of my two papers, the scope for generalising Pade had become so wide, that it was now a matter of looking at a particular problem, studying the analytic properties of the solution, and defining an approximant with the matching properties. For this reason, I decided to take the multivariate work no further at that time. I reviewed the whole field in several conference reports (refs 38,39,40). As it happened, another student and a strange occurrence led me into a quite different field just at that time.
I had not quite finished with Pade, however - it kept on cropping up over the years. Alan Common and I had worked on another generalisation, to Fourier, Laurent and Chebyshev series. We presented our results to a conference in Antwerp (ref 41). Also, I had been asked to contribute to the volume celebrating Christoffel's 150th birthday, so we prepared a full paper for this book (ref 44). Unfortunately, neither of us were able to attend the celebratory Christoffel Symposium.
The period 1972-77 had been exceptionally busy for me. In addition to chairing the Sport and Recreation committee, I was for three years Chairman of the School of Mathematics, and in the 'University Cabinet'. Also, I was writing a book on Vector Algebra and Analysis, based on two first-year courses of lectures I had given over a number of years (Book 3); although the book was favourably reviewed, it did not sell particularly well. Our children were all very active in different schools during this period, and we went abroad skiing and for summer holidays each year. It was in 1977 that Carol, the eldest, went off to Cambridge to study Mathematics.
The collaboration over multivariate approximants had been very successful, but a new and longer collaboration was just beginning. Ruth Farwell came to study Mathematics at UKC in 1972; she specialised in Mathematical Physics in her final year and graduated in 1975 with very high marks. She was awarded an SRC studentship, and during her first two terms as a research student, along with Theoretical Physics students, she followed advanced courses given jointly by the Applied Mathematics and Theoretical Physics staff. In the early summer of 1976, I asked her to discuss possible research topics with members of the Applied staff. Although we had a very fruitful and continuing line of business, Ruth had been reading up on gauge theories and topological monopoles, advised by Dr Lewis Ryder in Physics, and wanted to follow this line of research. I was almost totally ignorant of this area of work, so Lewis kindly agreed to supervise Ruth while I got up to date. In this way, Ruth and I studied together. A year later, Ruth had a possible basis for a PhD thesis, but I felt that some further input of ideas was needed. So I asked her to give a seminar, leaving plenty of time for discussion. 'You never know what might turn up in a discussion', I said. Ruth gave the seminar, and we had started talking about topology and strings, when I was struck by what seemed to be a great revelation. I said 'You don't want to do it this way!', and launched into a spontaneous 10-minute lecture on adding to the usual electromagnetic field what I called a magnetoelectric field, interacting with an electron. When Ruth talked this over with Lewis afterwards, he said 'You know what he (me, that is) was talking about - a paper by Cabibbo and Ferrari in 1962'. Ruth told me this the next day, and immediately I remembered, '1962 to 63: I was at CERN; November 1962, Cabibbo gave a seminar'. I can remember Cabibbo giving his talk, but to this day I have no conscious recollection of what he said. But I had been so taken with the ideas, that my subconscious had regurgitated them 15 years later!
So Ruth and I started trying to define a Dirac-type spinor which acted both as an electric pole and as a magnetic monopole, interacting through both the electromagnetic and the new magnetoelectric field. After several weeks, we concluded that this was impossible with the usual 4-component Dirac spinor. So we doubled the size of the spinor, which meant that we doubled the size of the algebra, to an 8x8 matrix algebra. When Ruth mentioned this to Lewis Ryder, he said 'You know what you are doing: you are re-inventing Clifford algebra. Why don't you look at Hestenes' book on the subject?' I had never heard of Clifford algebra, but we took Lewis' advice, and began our long association with this field, which continues today.
I have told this story partly because it set in train much of Ruth's and my research work for over a quarter of a century, but also to emphasise the element of chance in research, as in the rest of our lives. These strange accidents, starting with my hearing Cabibbo's talk in 1962, led to Alan Common and me running the first International Conference on Clifford Algebras in 1985, and eventually to ICCA7 in Toulouse in 2005, and the prospect of ICCA8 in Brazil in 2008.
Ruth and I did not fully realise that we had taken on work that posed many fundamental problems, and would lead us to develop many new ideas. I presented our early ideas on a generalised electron to conferences in Marseille and Lausanne (refs 43,44); already we had realised that our gauge transformations were in 'spin space', and we began to call our ideas 'Spin Gauge Theory'. At the Marseille meeting, Shelley Glashow gave a talk in which he pictured the lepton and quarks of three colours represented by the vertices of a regular tetrahedron. This picture impressed me at the time, but it was nearly ten years before we produced our Clifford algebraic structure corresponding to this tetrahedron.
The work was so complex at this stage that Ruth needed two 'extra' years to complete her doctorate. Fortunately, the SRC was not yet in the hands of The Accountants (as Sir Michael Atiyah has called them), who do not understand that serious research work is unpredictable. The expert committee understood clearly that we were exploring really new ideas, and awarded Ruth a one-year Associateship for her fifth year of research, allowing her to complete her PhD before taking up a Research Fellowship at Imperial College. With the help of a very patient and considerate referee, we wrote up the 'generalised electron' work for the Royal Society (ref 45). From the 'conclusions' section of the paper, it is clear that we had begun to appreciate how different our ideas were from standard gauge theory. We also understood that, algebraically, we were working in a space of six, rather than four dimensions: this opened the doors to later models set in spaces of higher dimensionality.
Ruth spent part of her time at Imperial continuing our collaboration, and we came up with an unrealistic model unifying strong, weak and electromagnetic interactions (refs 47,48,49); we filled a copy of Il Nuovo Cimento with our three papers. I see this as a period in which we explored ideas and techniques which we later applied to realistic models. We hinted at the lack of realism of our new invented interacting boson by calling it a 'Gryphon': in an appendix, we noted that the Gryphon was a mythical creature that guarded the gold of the Scythians against the incursions of the one-eyed Arimaspians.
Meanwhile, I still did a little work on Pade, summarising the scope of the subject at an engineering conference in Leuven (ref 46). Also, Alan Common and I got interested in research published in 1983 by George McVittie, our Honorary Professor, who had done a great deal of teaching and had supervised several research students since his 'retirement' from Illinois in 1972. He had completed work begun in 1933 on relativistic spherical expansion of a compressible gas. 'Mac', as we knew him, had found a solution of a second order non-linear differential equation by relating the equation to a Riccati equation. Playing around with these ideas, I found that it was possible to define a continued fraction solution to the general Riccati equation, defining successive terms of this solution iteratively by differentiation. This was a piece of work which pleased me greatly, and I presented it at a conference in Tampa in 1983 (ref 50), at the end of a 4-month, 12-thousand mile tour of Ontario and twenty of the United States. The tour started in Clemson, where Professor Phil Burt had arranged a sale-and-return deal on a car for Monty and me. We were overwhelmed when we were shown our 3.2 litre Chevrolet Monza, with two very comfortable armchairs as front seats. We had a very happy and successful tour, listening to Crazy Otto as we drove, and were made extremely welcome in the twenty places we visited. When I returned to Canterbury, I began organising a one-day meeting to celebrate the 80th birthday of George McVittie, held in June 1984. Professor Bill McCrea chaired the meeting, and the three speakers were Sir Hermann Bondi, the late Professor Keith Runcorn and Dr (now Professor) Malcolm MacCallum. My daughter Carol acted as secretary on the day. Many of Mac's old friends and colleagues attended, and the proceedings were published by the Royal Astronomical Society (ref 51).
World-wide interest in Clifford Algebra was growing rapidly. In 1984, I was sent two important books to review. The book by David Hestenes and Garret Sobczyk (ref 52) was very broad-ranging, reflected Hestenes' thinking over two decades, and indicated that Clifford Algebra had important links with many other fields of mathematics; these and other links have been developed appreciably since that time. I met David Hestenes and Garret Sobczyk for the first time in 1984 when I visited the USA to give the first Sobczyk Lecture, in honour of Garret's father. The book by Richard Delanghe, Fred Brackx and Frank Sommen (ref 53) set out the foundations of Clifford Analysis, which was substantially their own work, in clear and careful detail. I spent a lot of time studying these two texts. I also studied the 1964 paper by Atiyah, Bott and Shapiro linking Topology with Clifford algebra; for a long time, I puzzled over the fact that topology is metric-free, but Clifford algebras are fundamentally related to metric spaces - then I discovered, hidden near the end of the ABS paper, the brief phrase 'and imposing a metric'!
It was evident that, around the world, small groups of researchers were working on Clifford algebra and analysis, and spinors, so I set about contacting these groups. After two rounds of letters, I had contacted about 70 people who were interested in a conference on Clifford Algebra and its Applications. Alan Common and John McEwan had developed an interest in the subject, so Alan and I set about organising one. With ample funds from NATO and (reluctantly) the SERC, we were able to invite and support all invitees, including those from the USA and Eastern Europe. As in 1972, we had a lot of help from John and our three wives, and, over two weeks in 1985, almost all of those working in the field were able to attend talks, get to know each other, discuss at leisure, and enjoy their stay in Canterbury. Alan and I edited the Proceedings (Book 4), and Ruth gave an account of our joint work (ref 54). The meeting was a great success; a spontaneous decision was made to make it the first of a series, and Professor Artibano Micali agreed to organise the second ICCA in Montpellier: it took place in 1989.
Alan and I also carried forward our interest in the McVittie non-linear equation. We were able to find a new class of solutions to a more general type of equation, and to show that a sub-class of these equations reduce to a system of coupled Riccati-type equations (ref 55). This work persuaded Alan to continue to study non-linear differential equations: eventually this led, when I retired, to the appointment of Peter Clarkson to a Chair, and to the development at UKC of a major centre for non-linear differential equations.
Ruth and I now set about creating a Clifford algebra model of the electroweak interactions: we introduced several radically new ideas in this work (ref 56). First, space-time and 'higher dimensions' formed a single manifold, based on a particular Clifford algebra. So the Lorentz group and the gauge groups were all set within the same algebra. Second, we used a 4-component spinor for the neutrino, necessitating the introduction of both vector and axial vector interactions. Third, we dispensed with the Higgs field and particle: by factorising the unit operator in the mass term of the Dirac equation, we interpreted the fermion mass as a coupling constant with the Frame Field (the set of Dirac matrices on a manifold, discussed as long ago as 1929 by Fock and Ivanenko). From the beginning, we have viewed the Frame Field as a physical field - a thought that echoes the views of Parmenides and other Greek philosophers about 'space' being a substance of some sort. The frame field interaction term does not commute with the axial vector interaction terms, and this gives rise to precisely the boson mass terms needed to describe the photon and W and Z particles, and, when strong interactions are included, gluons..
The electroweak paper was set in a flat space, spanned by the basis vectors of the (2,6) algebra. We then realised that the Frame Field would contribute extra boson Lagrangian terms in a gravitational theory based on the Lorentz gauge group. In standard theories, the Lorentz group gives terms which are quadratic in curvature. Our additional frame field terms contained one linear in curvature, proportional to the Einstein-Hilbert term, plus a cosmological constant term. The basis for this paper was a manifold whose tangent spaces were spanned by the (1,3) Clifford algebra (ref 57).
George McVittie was very interested in this paper, and he advised us in 1987 to submit it to General Relativity and Gravitation. He was also very critical, as I was, of some modern accounts of differential geometry. When I explained to him how we described manifold structure he was very enthusiastic, and asked to take a part in our work, in particular the study of the fourth order differential equations arising from our Lagrangian. I had arranged to spend the last four months of the year at the University of Adelaide, at the invitation of Professor Angas Hurst, who was Deputy V-C there; we had been research students together. So Ruth and Mac kept in touch, and he corresponded with me about his study. On my return in January 1988, I found Mac in hospital, suffering from cancer of the throat. I visited him regularly for a month or so, and always he had all the work laid out before him, ready for an hour of discussion. I learnt a lot from these visits, but on the last occasion, Mac told me that he would be unable to finish the work. He died a few days later aged 83, a great loss to the University, and certainly to Ruth and me. I still have his work in my files, and I was able to add my tribute to others at his memorial meeting of the Royal Astronomical Society (ref 62). One of my regrets is that I was unable to answer some of Mac's questions about our approach to manifolds:16 years later, I could have given him some answers.
My work with Ruth continued in 1988, and we gave a talk at RAL on our results (ref 58); Ruth also spoke at a small conference arranged in Gent by Professor Richard Delanghe. I gave an introduction to Clifford algebra to sixth-form students visiting UKC (ref 59). John McEwan and I worked with a research student, Eiman el Dahab, setting Hamiltonian Theory within a complex Clifford algebra framework, and I spoke on this at a meeting in Belgium (ref 60). Eiman was then a Colonel in the Egyptian Army; he rose to the rank of General and is now retired.
By this time Ruth and I had combined the two 'frame field' papers, producing a unified electroweak and gravitational spin gauge theory (ref 61). We still had to incorporate quarks and the strong interactions into our models. First, we simplified our electroweak models by using the (1,6) algebra instead of the (2,6) algebra; this change was amply justified later on. Second, the late Ken Greider sent me a preprint in which quarks of three colours were represented by idempotents within a Clifford algebra, with cyclic symmetry. We realised that the fourth idempotent could represent the lepton, and, by changing the group operation, the 3-fold cyclic symmetry could be extended to the tetrahedral group. So, nine years after I saw Glashow's tetrahedron at Marseille, we were able to link it to the idempotents and group symmetries within the (3,1) algebra. Within our 4x4 representation of the algebra, we identified the 3x3 Gell-Mann representation of SU(3), and so were able to introduce the standard gluon interactions. The model unifying electroweak, strong and gravitational interactions, based on the (4,7) Clifford algebra, was presented to a conference in Oxford in September, 1988 (ref 63). We also reported on this work at the second ICCA meeting in Montpellier in 1989, but the proceedings of this meeting took some time to appear (refs 65,66).
At the meeting he organised in Gent in 1988, Richard Delanghe remarked that we had a novel concept of a manifold, and urged us to spell out more details. At the end of 1989, I was invited to the Winter School in Srni by Professors Jarolim Bures and Vladimir Soucek, and I presented a paper (ref 64) spelling out in more detail, but intuitively, how we defined what we came to call 'Clifford manifolds'. We did not claim novelty for our approach, but only that it was a simple and direct way - a 'physicist's way', if you like - of defining the basic structure. This contrasts with the 'mathematician's way', of starting with a very general concept and then refining the definition by putting on bundles of this, bundles of that, and then bundles of the other. Our definition starts with the Clifford bundle, metric, tangent space and connection all 'built in': I spelt out our definition more carefully in a later paper. I also enjoyed the next Winter School at Srni, and the hospitality of the Bures family. The two visits, one just before and one just after the fall of the Berlin Wall, were of great interest politically as well as scientifically.
Our interpretation of the frame field and fermion mass gave rise to a newformula relating all fermion masses to the known mass of the W boson. Since all the fermion masses were known except that of the then unobserved Top Quark, we could calculate its mass, based on our models. We were strongly urged by John Bell to carry out this calculation. Now, the result could have been almost anything, since it depended crucially on numbers 3 and 32 which were dictated by the structure of our chosen algebra: we were very pleased when the result turned out to be in the right ball-park. The predictions given by the Standard Model had been around 90 Gev, then 110 Gev, sliding up to around 125 Gev, and that model had a lot of flexibility built in. Our figure was precise, 151.7 Gev. We published our result in 1992 (ref 67), and I presented it to the Fermilab conference in the same year (ref 68). Normally, Ruth and I do not try to 'sell the product', but on this occasion, we circulated a clever cartoon representation by my son Dave (now recovered from the bite) of interlocking quarks of three colours, one wearing a Top Hat.
At Fermilab, I was very kindly shown round by Professor Dante Amidei, and I have since then had communication with him about the Fermilab measurement of 174 Gev for the Top mass, now rising to 178 Gev. We are still very interested in the interpretation of the measurement, since it is hard to compare exactly what the relation is between the 'bare mass' that we predict, and the observed mass. I was pretty sceptical when the Standard Model figure from CERN suddenly shot up to around 175 Gev. Although we were encouraged by John Bell, he was told that there was no interest at CERN in a talk on our work. I suspect that a theory which does not depend upon the existence of a Higgs particle is pretty unpalatable to those who have long and deeply committed themselves to finding one!
We had a third very welcome, but surprising, spur to our work. I heard from Professor Li Deming of Shanxi that he was interested in our work, and would find support to visit us. At that time, around 1991, it was by no means easy for him to arrange to come: he had been one of those vilified by the Mao regime and sent away for ten years to work in the fields. Li Deming brought to our attention certain properties of Clifford algebras, in particular the fact that not all algebras possessed a charge conjugation operator. For our four-force model, he pointed out that our (4,7) algebra did not possess the necessary operators on which to base a physical theory. But the (3,8) algebra possessed all of the desirable physical properties, and also retained the mathematical structure of our 'lepton-quark tetrahedron', since the (2,2) sub-algebra is isomorphic to the (3,1) algebra. From then on, we have used the (3,8) algebra to describe our 'family models', and have developed from Li Deming's ideas the concept of 'physical algebras'.
When he returned to China, Deming very kindly arranged for me to be invited to two conferences in 1992, at the Nankai Institute and the University of Shanxi. Monty and I were very happy to visit, and we were extremely well looked after. The Nankai meeting was on differential geometric methods, and was a celebration of the 70th birthday of Nobel Laureate Professor C.N.Yang. I had been trying to understand what an 'Instanton' was, and had studied the book by Nash and Sen. I was very unhappy with their approach, which gave a description mixing different formalisms, quaternionic, topological and Grassmann algebraic, in the same calculation. I could not understand this mathematical tapas, and so developed a simple descripton of the k=1 Instanton based solely on the (4,0) Clifford algebra. I spoke on this at Nankai (ref 69). Shortly before my talk, I learnt that my allocated time had been doubled, so I quickly prepared a short resume of our spin gauge theory work (ref 70). In Shanxi, I gave a plenary lecture on our four-force model, and I was complimented when Professor Yang stayed an extra night so that he could hear my talk (ref 71). Li Deming also spoke at the Shanxi meeting, on the structure of Clifford algebras. At Shanxi, Monty and I were introduced to Tai Chi: we got up at 5 o'clock every morning for our pre-breakfast lesson.
Ruth and I had reached a plateau in our spin gauge work, and we were now just refining our ideas and presentation; the only new idea was to note that there were several ways in which the (3,8) algebra might give rise to a multiplicity three of families of particles, but we came to no firm conclusions about this. At the third ICCA meeting, held in 1993 at Dienze in Belgium and organised by Richard Delanghe and Professor Fred Brackx, we offered only a short talk outlining our theories and stating our results (ref 72). The Dienze meeting, ICCA3 as we now call it, saw the broadening and strengthening of the whole Clifford field, notably through Gent's great area of expertise, Clifford Analysis, but also in growing links with other branches of geometry and with computing.
I was now close to retirement, and Ruth was becoming more and more a senior administrator at the University of Brighton. It was several years before we generated further new ideas. But a totally new enterprise was opening up.
Shortly after Monty and I had returned, late in 1991, from our second autumn visit to Australia, invited by Professor Paul Davies and Angas Hurst, Monty discovered that William Clifford had married Lucy Lane in 1875. The Cliffords had been at the centre of London scientific and literary culture during their brief marriage, and, in her 50 years of widowhood, Lucy had been a friend and confidante of many famous people, notably Henry James, Rudyard Kipling and Thomas Huxley. Monty began researching Lucy Clifford, and collected around two hundred letters written by her: she began to think of writing a book about her research. One of the Cliffords' daughters had married into the well-known Dilke family We contacted her descendents and were eventually invited to Dorset in April 1993 by Alice Dilke, the holder of letters and memorabilia from Lucy Clifford. There, Monty was offered sole access to nearly a thousand letters to Lucy, stored in an old trunk, from a wide variety of distinguished figures, from the Victorian age through to the Bloomsbury set. She knew that it was 'treasure trove', and later gave a talk 'Secrets of the Lost Trunk'in UKC. Monty also spoke at the Wigner Symposium in Christ Church, Oxford, in September 1993; but her talk, like those given by Ruth and me, and by the rest of the speakers, was never published. That same year, a joint paper (ref 73) on generalised classical mechanics was presented at a different Wigner Symposium in Goslar, by John McEwan and Eiman Abou el Dahab.
By a strange coincidence, Professor Marysa Demoor, of the University of Gent, learnt at this time of Richard Delanghe through their University newsletter's report of his Honorary Degree at UKC, and of his association with Clifford's mathematics. Marysa is a literary scholar who was interested in Lucy Clifford, and she contacted Richard. Although they were in the same University, Richard had not come across Marysa before, but he put her in touch with Monty. In May 1993, before the third ICCA meeting in Dienze, Richard arranged a lunch at which he and we met Marysa for the first time. Piling coincidence on coincidence, it emerged at this lunch that Richard and Marysa had regularly taught in the same building, they came from the same locality in Belgium, and Marysa's father had been Richard's school headmaster!
The meeting with Marysa led to the creation of the 'William and Lucy Clifford Research Group', consisting of Ruth, Marysa, Monty and me. With Alice Dilke's agreement, Monty unselfishly offered to share with Marysa her access to Lucy's letters, and we arranged for ourselves, Ruth, and Marysa and her husband Patrick, to visit Dorset to make two copies of all of the letters. At the beginning of July, we were all set to go, but our first two granddaughters were born on July 3rd, 1993: so Monty did not take part in the frantic copying weekend in Dorset.
Monty, Ruth and I began to find out more about the Cliffords. Ruth started enquiries at the Public Records Office and in 1994, Monty and I visited Exeter to discover what we could from the local press of the time of William's childhood there, and were greatly helped in our research by a local historian, Hazel Harvey. Monty started a collection of books written by the Cliffords, and I started studying William's books, in particular his philosophical 'Lectures and Essays'. Monty had plenty to do studying the mass of letters and Lucy's books, and we all began to learn more about the friends and colleagues of William and Lucy, and the age they lived in.
Professor Parra Serra of Barcelona is a 'Clifford man', interested in his mathematics, his philosophy and his life. In June 1994, he invited Monty, Ruth and me to join him for ten days in an official discussion group on William Clifford. For some time, Ruth and I had intended to do something to celebrate William Clifford's 150th birthday on May 6, 1995. The three of us had a day or two on our own in Barcelona: during this time, we decided to organise a meeting celebrating the lives of both William and Lucy Clifford: we would tell the main story ourselves, and arrange for a variety of expert speakers to talk on special themes. We also concocted the idea of a 'Quotes and Profiles' exhibition, for which we would collect images of important people who had spoken or written about the Cliffords: their words would come out of their mouths as 'cartoon bubbles'. When we contacted her, Marysa readily agreed to work with us towards the meeting, but the organisation and creating of the exhibition naturally fell to the three of us. Collecting the sayings and finding the images of 40 'contributers' to Quotes and Profiles took a lot of time, but it was very interesting work, resulting in a most successful display (physically created at the University of Brighton).
The 'Two Lives' meeting was held at UKC on May 4, 1995, and the four of us were delighted that a number of very eminent scholars had agreed to speak, including Sir Roger Penrose and Professor Gillian Beer. We had great support from the Dilke family and from the British Society for the History of Mathematics. The meeting went extremely well, and ended with an evening party at our home on a delightfully warm evening. What we did not then know was the way in which this success would lead to a host of invitations around the world. My retirement in 1994 came just in time for me to be free to accept these.
At the end of May, 1995, Monty gave a talk on the Cliffords at Christ Church College, Canterbury (which has recently become a University), and in July, Ruth shared the platform with Roger Penrose, giving a lunchtime talk on William Clifford at the National Portrait Gallery. We were invited further afield in July that year. Ruth delivered three lectures on the basis of Spin Gauge Theories to the 1995 Banff Summer School, run by Professor Bill Baylis (ref 75). Here, we clearly defined what we called 'physical algebras', which have all the properties needed for describing quantum physical systems; we noted the pleasing fact that the Clifford algebras which we had chosen for our models on phenomenological grounds, the (1,6) and the (3,8) algebras, had turned out to be physical algebras. Ruth also took a portable copy of the Q and P exhibition to Banff, and gave a general talk on William Clifford. At the same time, Monty and I were fulfilling an invitation to Madeira, where William Clifford died, taking the exhibition and giving talks on William and Lucy. While we were there, Monty began to research the last months of William's life: she found out who travelled to Madeira with the Cliffords in January, 1879, and where William died. Our year continued with a visit to Mexico, where Professor Jaime Keller held a Clifford algebra conference. We took the exhibition, and again each gave talks on the Cliffords, in addition to my conference contribution.
Meanwhile, Sir Michael Atiyah (then President of the Royal Society and Master of Trinity College) had invited us to arrange a one-day meeting at the Newton Institute in Cambridge. Ruth did the organising; we took the Q and P exhibition, and Monty and I each gave talks at the meeting; other speakers included Gillian Beer and Richard Delanghe. Monty and I also took the exhibition to Swansea in April 1966, and gave accompanying talks at the 75th University College celebrations; in addition, I gave a seminar on our spin gauge theory work. We again gave talks and took the exhibition to ICCA4 in Aachen that May, where Ruth and I only offered a summary of our work. Immediately after the Aachen conference, Professor Wolfgang Sproessig held a meeting in Freiberg; I gave a summary of the Banff lectures (ref 77), while Monty gave a talk on the Cliffords' life and times.
So, in the year from May 1995 to April 1996, we made ten different biographical and historical presentations about William and Lucy Clifford, in various parts of the world. In addition to having access to the treasure trove of letters through Alice Dilke, Monty and I were invited by Dr Fisher Dilke to study the papers left by William Clifford. William was very careless about correspondence and papers, and his literary executors had seen that almost everything he wrote was published: so there was very little of importance in the residue of his papers. But one sheet of paper stood out, and I published a short historical note on it (ref 80). The paper appeared to be a plan for a publication drawing together a vast range of different strands of physical science. Clifford concluded with an enigmatic statement that:
All of these things must come out of the knowledge of the form of atoms and their relation to the ether. What is pointed to is therefore a connection between kinetic theory and undulatory theory.
I posed the question whether this connection was an anticipation of the wave-particle duality of de Broglie.
Our absorbtion in this historical, philosophical and biographical work, organising the 'Two Lives' meeting and creating the exhibition, must have been part of the reason why Ruth and I did not develop our models much between 1993 and 1996. We spent time studying the basis of our models: prior to preparing the Banff lectures on the basic concepts of Spin GaugeTheories, Ruth and I wrote a substantial review (ref 74) in 1995. This study of the basics led Ruth and I to question the use of 'standard' gauge transformations on algebraic spinors: these spinors are elements of the relevant Clifford algebra, and we could argue that gauge transformations should be 'two-sided' equivalence transformations (the same as for the general element of the algebra), instead of 'left-handed' ones, applicable to column spinors. During the period 1977-79, Ruth was very busy moving house and also moving from the University of Brighton, to take up a very senior administrative post (eventually Pro-Vice-Chancellor) at South Bank University. Nevertheless, we produced a substantial paper (ref 78) studying the effect of applying equivalence gauge transformations to our model of electroweak interactions of leptons, based on the algebra Cl(1,6). The result was very interesting. The electroweak interaction, originally formulated by Glashow in 1961, contains a 'chirality-minus' interaction that is the sum of two very different terms. It turned out that if we used a 'two-sided' gauge transformation containing only one part of the usual 'left-handed generator', the resulting left- and right-handed interaction terms could be identified precisely with the two terms in the Glashow interaction. So we obtained a simpler gauge theory. Another point was forced upon us: there is a choice to be made in defining an algebraic spinor, but whatever choice is made, the spinor necessarily breaks many symmetries of the model. This fact has become important in later work.
After this publication, Ruth could not find time for research - she was running a University with 12,000 students! For the next 5 years, I worked essentially on my own, but kept Ruth informed. Our latest model was set in a flat space, and did not deal with gravity; nor did it include quarks and strong interactions. In order to extend our 'equivalence spin gauge theories' to include gravitation, I started to make a detailed examination of the definition of Clifford manifolds: the structure is not complicated, and I was able to introduce metric, tangent spaces, connections and curvature systematically and coherently. In order to discuss gauge transformations, it was necessary to explain how a bundle of tangent spaces could be uniquely defined throughout a Clifford manifold. This problem was spelt out by a very kind referee of my paper, and after some thought, I was able to give a unique definition of a bundle by parallel transport from an asymptotically flat submanifold, where parallel transport is unambiguous; my colleague Jim Shank helped me to find a suitable theorem to backup my geometric intuition. Using this structure, I was able to introduce gravitational 'equivalence' gauge transformations: as an example, I extended our latest electroweak lepton model. The gauge transformation, as expected, gave rise to a standard 'left-hand' interaction term; the right-hand term was a completely new type of gravitational term at particle level - it broke the T-symmetry between forward and backward time directions, and gave a preferred local direction in physical 3-space. In practice, we do experience a preferential time direction in the universe. I do not know what the (small) breaking of space symmetry at particle level implies.
My first account of this work was given at a delightful conference on Clifford Analysis in Prague, run by Professors Fred Brackx, Vladimir Soucek and Jarolim Bures. I had not then sorted out the parallel transport problem, so my conference talk [81] is vague at this point. The full and coherent account of gravitational equivalence gauge theory [82] was published in 2002. Although I am not an expert in Clifford analysis, I was invited to act as an editor of the proceedings of the Prague conference [book 5].
Monty and I had thought that we had finished with talks about the lives of the Cliffords, but we were each invited by the Australian Mathematical Society to take the Q&P exhibition and to speak at their annual conference in Canberra in 2001. We were made very welcome by Professor Alan Macintosh and his colleagues, with whom I had very helpful discussions, and had the good fortune to stay with old Canterbury friends, Jon and Rosella Hampshire. We also spoke at Melbourne, invited by Professor Geoff Opat, and went on to spend a delightful fortnight in our old haunts, with old friends in Adelaide. We performed yet again, and I gave research talks supported by the Australian Physical Society.
The sixth conference in the ICCA series was held in 2002 at Cookeville, Tennessee, efficiently organised by Professor Rafal Ablamowicz; here I presented an account of my Clifford manifold work. Yet another conference based on Clifford algebras was held in 2002, in Clifford's home institution, Trinity College, Cambridge; it was run by the Institute of Mathematics and its Applications, and organised by Professor Anthony Lasenby, Dr Joan Lasenby and Dr Chris Doran. Monty and I did not give any talks at this meeting, but we set up the Q&P exhibition. I was asked to give the after-dinner talk at the banquet; it was an appropriate occasion to talk about Clifford's first discovery of Clifford algebras in his famous 1873 paper on biquaternions, where he defined what we now call the Cl(0,4) algebra. This preceded his general definition given in an abstract for the LMS in 1876, published posthumously, and his other paper in 1878. It has been said that Grassmann discovered Clifford algebra in 1877. However, his ideas were not quite the same as Clifford's, and Grassmann's whole motivation was very different from that of Clifford. It is a great pity that this misleading and demeaning argument has been made: the thinking of both Grassmann and Clifford was decades ahead of their own time, and each made a fundamental contribution through the related, but quite distinct, Grassmann and Clifford algebras. History has plenty of space to honour both of these men of genius, and to remember their names. I was very pleased when the IMA invited me to publish my talk (ref 82).
Ruth and I still had not included quarks and the strong interactions in our equivalence gauge theory models, and I worked on this during 2002-2004. Monty and I were involved in moving to a smaller house, and I still found working alone hard. In 2004, I thought that I could write a paper including the strong interactions: but the more I wrote, the more I realised that I had not appreciated the way in which half a dozen of my ideas did not quite fit together: around page 17 of my write-up, I became completely bogged down. It was very fortunate that, a few months earlier, Ruth had arranged that South Bank University should recognise that she needed time to resume research work, and she was able to carve out a little of her time to help sort out which of my ideas should be retained, and to contribute several of her own. It was a great delight to find that the old combination was still very effective.
In the autumn of 2004, Monty and I were again invited to give talks on the Cliffords, this time to a very interested and knowledgable audience at the Ethical Society in London. It was most interesting to meet and talk with members of this society of sceptics, one of the earliest groups of nineteenth century free-thinkers. We were pleased that both of our talks were published by the Ethical Society (ref 84), since they represented the talks we had given in many parts of the world since 1995.
In May 2005, Professor Pierre Angles ran ICCA7 at Toulouse. The number of participants was higher than at previous ICCA conferences, and there was a very wide spread of lectures. By now, Ruth and I had sorted out many totally new ideas about our equivalence theory 'family model', and we each gave a plenary talk; Monty also gave a talk based on her twin biography of William and Lucy Clifford, Such Silver Currents. Ruth and I submitted a 'progress report' for publication in the proceedings, since we had not completed our study. At the end of 2005, Ruth and I still have decisions to make about the details of a very complex piece of work. At the beginning of the meeting, Pierre Angles had arranged a delightful surprise for the company, a visit to the headquarters of the ancient Order de la Dive Bouteille de Gaillac. Those of us at the conference who had organised earlier ICCA meetings were accorded the distinction of being enrolled as Chevalliers of the Order at an ancient ceremony, where we had to demonstrate our ability to consume considerable quantities of Gaillac wine. So I am now a Chevalier, equipped with a heavy medal representing a bottle, and a vintners apron.
©
Roy
Chisholm