The Cliffords: Algebra, Philosophy and Biographical Study

 

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. 
 

A New Idea: Equivalence Gauge Transformations 

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. he 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.