‚ÄčRoy Chisholm (1926 - 2015)

Professor of Applied Mathematics, University of Kent (1965-1994)
Settling in (1966-1970)

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. Memories

A Series of Conferences (1970-72)

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) Memories

Multivariate Approximants (1972-1978)

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. Memories

Approximation of Functions with Branch Points

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. Memories

Clifford Algebra and Spin Gauge Theories: How it Began

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. Memories

Clifford Algebras and Spin Gauge Theories - Second Stage

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. Memories

Clifford Algebras and Spin Gauge Theories - Third Stage

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 new formula 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!

Up to this point, we had treated spinors as Dirac spinors. After 1993, we explored the use of two-sided spin gauge transformations on algebraic spinors, developing Algebraic Spin Gauge Theory. This led to different, but unambiguous results for the Top Mass. We were very pleased that the Top mass came out as 182 Gev, close to the experimental measurement. This result was announced by Ruth Farwell at a conference in London about six years ago, but we have not yet published the calculation.

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.