Herbert Sydney Green 1920-1999

Written by Angas Hurst.


Bert Green's influence on the development of theoretical science in Australia during the nearly fifty years he lived here cannot be overestimated. From the time he arrived in Adelaide in July 1951 until his death on 16 February 1999 he produced articles and books covering topics as diverse as particle physics, environmental science and neurophysiology. In each of the areas in which he worked, his contributions were always marked by erudition and originality. It is not surprising, therefore, that everyone who came in contact with him, from undergraduate students to international colleagues, wrote in warm and admiring tones about their contact with him.

So, although his official appointment was to the inaugural chair of Mathematical Physics in the University of Adelaide, physics was only one of the fields in which he employed his superb mathematical talent. As an example of some of the outer reaches of his interests, he once used some statistical analyses, prepared by the Dutch composer Henk Badings, of the compositions of some of the great composers to produce by computer (then the very rudimentary IBM1620) lines in the style of Beethoven and Mozart. The then Elder Professor of Music, John Bishop, was highly intrigued.

The Chair of Mathematical Physics in Adelaide was the first chair in theoretical physics in Australia, for, although the Professorial Board at the University of Melbourne in June 1949 had recommended, as a matter of urgency, that a chair of theoretical physics be created, it was not approved until December 1951. That chair was not filled until Associate Professor C.B.O. Mohr was appointed Professor in 1961. In Adelaide, the decision of the then Vice-Chancellor, A.P. Rowe, in consultation with Professor L.G.H. Huxley, Elder Professor of Physics, to create a chair in Mathematical Physics was based on the view that mathematics research in Adelaide had been languishing since the breakdown of Professor J.R. Wilton, and both of these men knew of the chair of mathematical physics in Birmingham held by the distinguished physicist R.E. Peierls. At that time the terms 'theoretical physics' and 'mathematical physics' were used more or less interchangeably. The name 'theoretical physics' was coined by Rudolf Clausius, while J.Willard Gibbs was appointed to a chair of 'mathematical physics' at Yale University in 1871. In recent years the names have begun to acquire slightly different connnotations, with mathematical physics tending to be associated with research having more emphasis on the mathematical structure of the theories, and theoretical physics with attempts to create new theories or to compare theory with experiment. Sadly, these distinctions, minute to an outsider, have begun to show signs of entrenchment, quasi-religious fervour and even vituperative comments. Certainly Bert Green never made distinctions in his choice of research fields, as will be seen from the account of his research work presented in this memoir.

School, university and the war

Bert Green was born on 17 December 1920 at Ipswich, England, the only child of Sydney and Violet Green. Sydney Green had been a mathematics teacher, but increasing deafness meant he had to give up that profession and turn to coach-building. After a period at West Winch, near King's Lynn, Norfolk, the family moved to Felixstowe in 1930. There Bert was a pupil at Langer Road Elementary School until 1932, and even at that early age his outstanding mathematical ability was evident. So it was no surprise when he was awarded a scholarship for Felixstowe County School, where he stayed until 1939. In that year he obtained scholarships to University Colleges at Hull and Nottingham, and passed his Higher School Certificate examination. His results in that examination were so good that he was awarded a Royal Scholarship at the Imperial College of Science and Technology, London. His teachers there included W.G. Penney, later well known for his leadership in the British nuclear bomb program, and Sydney Chapman, of Chapman and Cowling fame. He graduated with a BSc in mathematics with first class honours and a Diploma of Associateship of the Royal College of Science (ARCS) in 1941, a achievement which gave him great pleasure.

By then the war had started and in 1940 he took a summer job building coastal defences, and learnt to ride a bicycle. He also did firewatching on the roof of Imperial College, and filled in the many hours of inactivity by studying text books, with some consequent damage to his eyesight.

After graduation his first position was in the government Scientific Service, but he found the work so dull that he changed over to the Meteorological Office in the Air Ministry, and from there to the RAF in 1941 as a Meteorological Officer with the rank of Flying Officer. The rest of the war was spent on the Isle of Man in the Training Flying Control Centre, advising the RAF on flying hazards such as wing icing. This was clearly very responsible work, because many of the night operations over Europe, particularly in the northern latitudes, were very prone to extreme weather conditions, and the lives of many airmen depended on accurate forecasting. Bert's experience there remained with him throughout his career and gave him special insights into environmental questions and, more remotely, into problems in cosmic ray physics that were out of the normal run. For example, he pointed out that the statistics behind the Adelaide claims for the discovery of tachyons suffered from a defect due to truncating data. Such truncation was known to be a source of error in attempting long-range weather forecasting, because it could lead to spurious peaks. Tachyons are hypothetical particles that travel faster than light, and their discovery would greatly change our view of the universe. There was some evidence from cosmic ray showers being studied in Adelaide that some signals being received appeared to be tachyonic in character, and although the experimentalists could find no flaw in their acquisition of the data, the interpretation had to be reassessed. Bert suggested that a peak in the data, presumed to represent the arrival of tachyonic particles, was actually a truncation error. With this interpretation it was concluded that no evidence for tachyons had been found.

As would be expected by anyone who knew Bert, he was quite unlike the usual run of air force officers. He was not interested in social pastimes such as drinking in the Officer's Mess, finding pleasure more in solitary pursuits such as swimming off the not very attractive rock-strewn beaches.

As the war approached its end, he made plans for resuming his studies, and wrote to Sydney Chapman in March 1945 asking his advice on where he could go, and suggesting London, Cambridge or Edinburgh. For various reasons Chapman, in his reply, agreed that Edinburgh would be the best choice.

The exchange of letters which followed throws some interesting light on Bert's very considerable self-confidence, a trait which was very important for his decision to move to Australia. Enclosed in his letter to Chapman was a small calculation which Bert suggested might be suitable for publication. In his reply, Chapman enclosed a comment from George Temple, a very well-known theoretical physicist. Temple was quite negative in his assessment, saying that first of all the work had been done before, and secondly it was wrong! Instead of being crushed by such a reply, Bert wrote a two-page letter defending his work, and respectfully disagreeing with Temple's arguments. In response Temple acknowledged that he was wrong and Bert was right, and that the note should be published. Bert then wrote to Max Born at Edinburgh, asking whether he could work with him and enclosing this paper for possible publication (it never appeared). He left the RAF in September 1945, and started immediately in Edinburgh.

With Max Born in Edinburgh

Bert's time with Max Born was extraordinarily productive. He obtained his PhD in 1947, and DSc in 1949. Together they published six papers and a book summarising their work on a general kinetic theory of liquids. There were usually from six to eight research students working with Born over that period in rather primitive surroundings in the basement of the Old Infirmary in Drummond Street. Born's practice was to visit the research students' room every day, discussing each student's progress and offering constructive suggestions. He left talking to Bert till the last as their discussions were always very long and lively. In fact Born was once heard to say 'I can't stand Green – he is always right'. They would go to conferences together, and during talks Born would often interrupt and then say 'Green', and Bert would go out to the blackboard and expound the points being made. In a letter to Einstein,1 Born wrote: 'My collaborator Green is hard at work on elementary particles; he is a brilliant man, the best I have had since Pryce'.

Bert submitted his work on kinetic theory for his PhD thesis, and it was considered to be so outstanding that Born proposed that it should be awarded a DSc. However this suggestion was regarded as too radical by some of the conservative members of the examining board, and Bert had to be content with a PhD. However his DSc was not long coming because he submitted another thesis soon afterwards and received his DSc two years later.

Bert was no athlete in the usual sense, lacking the required co-ordination, but he was always very fit. On a working holiday in Europe he cycled over the Swiss Alps. Right up till his last illness he was a great walker, and walking companions often complained of the tremendous pace he used to set. He could be rather idiosyncratic, though, and could pose problems for those with him. Two of my correspondents, Professor L.G. Bowden and Professor A.G. McLellan, tell of a memorable walk in the Lake District when Bert insisted on wearing wellingtons instead of the conventional boots, even though they could at times be dangerous. He also attempted to carry back to Edinburgh a ram's skull as big as a bucket, in the belief that it would have anthropological interest, but was persuaded, much against his wishes, to leave it behind. At the memorial service held in Adelaide after his death, Harry Messel told of another excursion up Snowdon in poor weather, when Bert nearly slid to his death because of the unsuitability of his wellingtons for the terrain. After taking a couple of photographs, Harry pulled him to safety.

The Borns were very hospitable and made their students, who came from all over the world, very much at home. At that time they had with them an attractive au-pair girl from Holland, Marlies Friedheim, and she and Bert soon became very good friends. That friendship culminated in marriage in Dublin in 1951.

Institute of Advanced Study, Princeton

Bert spent one year, 1949-50, at the Institute of Advanced Study in Princeton, working on problems generated whilst in Edinburgh. His published papers from that time refer to models of quantum mechanics and quantum field theory initiated with Born. From reports, he did not like Princeton much; the only reference I heard him make to it was that he often used to walk in with Gödel and talk to him about mathematical logic. A possible relic of these encounters could be an interesting unpublished manuscript of Bert's on the foundations of mathematical logic. His decision to go to Adelaide rather than accept one of several offers he had received of appointments in the United States reflected his preference to be free to follow his own paths rather than those dictated by passing waves of fashion.

One lasting friendship he made there was with Roy Leipnik, who found his quiet manner and self-suffiency in strong contrast to the other members of the Institute. Roy Leipnik came to Adelaide in 1954 on a Fulbright scholarship, and there they started a collaboration on plasma physics that led to the writing and publication of their book Sources of Plasma Physics, and regular visits by Bert to the United States to work for the United States Navy on plasma physics problems associated with rocket research.

Institute of Advanced Study, Dublin 1950-51

After Princeton Bert went to the Institute of Advanced Study in Dublin and found the atmosphere there much more congenial. He particularly liked the way in which Schrödinger looked at physics, although he was not so impressed by the extent of religious bigotry that was still evident there. It was in Dublin that he met Harry Messel, beginning a life-long friendship and a frenetic period of research productivity into cosmic ray showers which continued on into Adelaide.

He also renewed his friendship with L.G. ('Dook') Bowden, whom he had met in the RAF on the Isle of Man, and, as already mentioned, he married Marlies in 1951 with Harry and Dook as attendants.

When the University of Adelaide advertised for a professor, a reader and a lecturer in mathematical physics, Bert applied from Dublin for the chair and was appointed. This was a great boost for Adelaide's mathematical research, for at a time when employment in the sciences was starting to rise following the wartime achievements of physicists, it would be expected that it would not be easy to attract an outstanding candidate to Australia. Bert Green was just the sort of person who could make such a 'courageous' decision work. He already had a proven research record over a wide range of mathematical physics and, as already mentioned, he was not attracted to the hot-house American research atmosphere. He also preferred to live away from large cities, while his strongly socialist leanings would find even the conservatism of postwar Australia more congenial than the rabid anti-communism leading into McCarthyism of the United States.

Adelaide 1951-99

It was no problem to find someone to fill one of the other positions in mathematical physics. Harry Messel had already started to collaborate with Bert in cosmic ray showers, and so he stepped immediately into the position of senior lecturer. He and Ren Potts, who was going to a lectureship in mathematics, travelled together to Australia by ship, arriving in September 1951, Bert having arrived in August. Bert and Harry were an oddly assorted couple, Bert being reserved and not very interested in socialising while Harry was extremely extroverted and tended to dominate any social gathering. But they got on famously, with their abilities complementing each other's very well. Their common interest was an intense absorption in their research, and one might say that Adelaide did not know what had hit it when they arrived. Harry Messel only stayed nine months before leaving to take up the long-vacant chair of physics in Sydney, but in that time they produced thirteen papers on cosmic rays, not to mention other papers that Bert wrote on other parts of mathematical physics.

During the North Sea Floods in 1953, the Greens' family home in Felixstowe was in danger of being flooded. Sydney Green moved the furniture upstairs to save it from damage, but sadly the effort was too much and he died soon afterwards from a heart attack brought on by his exertions. Bert and his family went to England by ship to look after the family's affairs and spent six months there. Whilst there, Bert made arrangements for his mother to come to Adelaide, which she eventually did.

During Bert's absence Otto Bergmann was acting chairman of the department, with the main responsibility of looking after Ian McCarthy, who had embarked on a PhD with Bert. This was a bold decision of Ian's because the PhD degree was still very new to Australia, there had been no regular teaching programme established in the department, and Adelaide was alone in doing research in mathematical physics. It was also not easy for Ian, as a beginning student, to be in close contact with such a formidable intellect as Bert's. Ian found that Bert was a kind and understanding supervisor, although the presumption that Ian could follow all of Bert's ideas was at times a bit too much, particularly as even the greatest of scientists have their off moments. Out of it came a very interesting thesis on the then new ideas of parastatistics, with Ian working out some of the detailed consequences of the assumption that particles could behave differently from the standard bosonic and fermionic types.

After Harry Messel left, his position was advertised, and it is interesting to see the quality of people who made enquiries about the position. Amongst them were S.F. Edwards, later Sir Sam Edwards, Chief Scientific Advisor to the British Government, and W. Israel and W. Güttinger, both of whom were to become well-known mathematical physicists. J. (John) C. Ward, widely considered as unlucky not to have been awarded a Nobel prize for his later work with Salam, was appointed in October 1953.

Bert's academic life in Adelaide can be divided into four sections, the first being from 1951 until 1959 when he ran what was essentially a research institute with first Harry Messel and then Otto Bergmann and John Ward. It is interesting to read two very contrasting opinions about this period in letters to Bert. They reflect clearly the personalities of the writers.

The first, from Harry Messel, written when he was touring the United States recruiting people for his new institute in Sydney, said: 'Everyone agrees that we have cracked cosmic ray theory and are most pleased with our work. Some of the big experiments now being planned over here (especially by Rossi) depend completely on our work.' The second, from John Ward, written just as he was preparing to leave Adelaide and while Bert was still overseas, paints a very different picture: 'There is no doubt that the state of affairs in theoretical physics is extremely bad (not in Adelaide, I mean everywhere)...One cannot therefore, with a good conscience, recruit students, especially in a place like Adelaide, where the possibilities of jobs are remote, and where the instruction is likely to be poor, and chances of decent research very slight.'

Both of these people only stayed nine months, and the University came to believe that the position was a short-term contract post rather than tenured. This led to a problem for me when I arrived in Adelaide, because the University Council decided that the post did not warrant long-term support such as housing loans and superannuation. The Registrar put the question as to whether it was my intention to stay longer than a year. I actually stayed until my retirement thirty-one years later!

In 1959 a formal Honours course in Mathematical Physics was started (there had been some sporadic teaching before that) with always at least three students each year, and in 1960 third-year courses were introduced. With my arrival in 1957, and Ian McCarthy's in 1961, the department started to be a regular university department with the regular administrative tasks that went with it. Bert was Head of the Department until 1964, and then, following my promotion to a personal chair, the duties of Head alternated until 1973, when departmental government, requiring election of the departmental chairman, was instituted in the University. From 1973 until his retirement in 1985, with departmental government in full swing, Bert shifted much more into the background, although continuing to be very active teaching, doing research and assuming various administrative duties such as Faculty Dean and President of the Australian Mathematical Society.

The final period, from 1985 until his death in 1999, saw no diminution in his research, and continuing activity in teaching and research supervision.

Scientific Work

The main body of Bert's scientific work will be dealt with under the following headings:

  1. Kinetic theory and plasma physics,
  2. Statistical mechanics,
  3. Quantum field theory and particle physics,
  4. Cosmic rays,
  5. Quantum mechanics,
  6. General relativity and gravitation,
  7. Mathematical methods,
  8. Environmental physics,
  9. Biophysics and neurophysiological models.

In addition, Bert published a number of far from trivial papers in nuclear physics [43], [61], [68], [69]; in biophysics with Casley-Smith, Vaccaro and Bass [88], [96], [110], [132], [133], [134]; and in electromagnetic propagation with Wolf [34], [38]. These, however, will not be discussed here.

(a) Kinetic theory and plasma physics

The theory of dilute gases, for which the molecules could be regarded as moving more or less independently of each other, was developed from slightly different standpoints by Maxwell in 1866 and Boltzmann in 1872, and remained the only dynamical theory of complex systems until after the Second World War when, independently, five people developed what came to be known as the BBGKY system of equations for the dynamical behaviour of fluids, with the important extension to dense liquids. These people were Bogoliubov (2) in Moscow, Born and Green [1]-[8] in Edinburgh, Kirkwood (3) at Cornell and Yvon (4) in Paris. Of these, Bert Green was exceptional for whereas the other four were distinguished physicists, he was a beginning research student. Moreover he was not just working under guidance from his supervisor, for Born took the unusual step of adding a footnote to paper [4] on the quantum mechanics of fluids in which he said 'I have signed this paper, as it is a part of the programme with which we started this series. My contribution consists of some general suggestions...The work itself is due to Mr Green.' There were six papers under the general title, 'A General Kinetic Theory of Liquids'. In the first paper their version of the BBGKY equations for classical systems was given, and in the second and third papers the equilibrium and dynamical properties of these equations were discussed (Bert was sole author of the second). The fifth paper gave a kinetic basis for thermodynamics and the last, again with Bert the sole author, was on the difficult question of how to explain the anomalous behaviour of liquid helium II. These papers, the publication of which spread over the years 1946-48, clearly tackled questions of fundamental importance, and the following year Cambridge University Press published them all as a single collection, A General Kinetic Theory of Liquids.

It is interesting to see that, apart from Yvon's, all the initial papers of BBGKY were published in 1946. However, neither Yvon's nor Bogoliubov's work was known to the others until later. Bert tells of going to a conference in France, and after giving his talk being approached by a small older Frenchman who told him of similar work he had done. It was Yvon.

Because these first papers of Born and Green set the pattern for the later work of Bert and his collaborators and students, they will be looked at in some detail. The essential difficulty was to find a way to take into account the much stronger mutual influences that molecules in dense fluids can exert on each other.

The starting point for these equations was Liouville's equation:

∂r/∂r/ t + {r, H} = 0.∂

It states simply that probability is conserved during the motion both in classical and quantum theory, and so is an exact but not very useful statement when there is a very large number of particles involved. In classical theory r is the probability of there being N particles present with given co-ordinates and momenta, written f(q1 ,…,qN ; p1 ,…,pN ; t), whilst in quantum theory it denotes the density matrix.

More useful are the partial probabilities associated with considering only some of the particles at a time, irrespective of what the remainder are doing, and these are written, for the classical case, as f(q1 ,…,qn ; p1 ,…,pn ; t) with n < N. The extreme case is to consider only one particle at a time, and that is as far as the well-known equation, due to Boltzmann, went. It was the object of BBGKY to construct equations in which more than one particle at a time is considered, and to do that Liouville’s equation can be replaced by an equivalent system of equations describing the behaviour of the partial probabilities. As a simplifying step that accords well with physics, it is always assumed that the particles interact in pairs – by pair potentials. With this assumption, the equation for the n-particle probability involves the (n + 1)-probability, but no more. These are the BBGKY equations and are still exact and still not very useful.

The important step is to find out how to proceed from here. The different authors proceeded in slightly different ways. Bogoliubov expanded r in powers of the density, and treated this parameter as a perturbation. Kirkwood used time averaging as a representation of the time spread involved in making a measurement. Born and Green, and also Yvon in a more rudimentary form, used what they called, following Kirkwood, the 'superposition approximation'. This supposed that the higher particle number densities depended explicitly on the lower ones. The simplest choice is to write f2(z1,z2) = f1(z1)f1(z2) where z denotes the pair q,p, and this leads back to the Boltzmann equation. Born and Green went further, assuming only f3(z1,z2,z3) = f2(z2,z3)f2(z3,z1)f2(z1,z2)/f1(z1)f1(z2)f1(z3). All of these approximations have their defects, and have been criticised for their rather ad hoc nature. It is sometimes said that this step signals the appearance of irreversibility, because the complete Liouville equation shows no preference for either direction of time, whereas the Boltzmann equation leads to the very fundamental H-theorem – the statement that entropy increases with time. (Boltzmann's suicide is supposed to have been partially caused by the very strong criticism he received following his proof of the H-theorem and his assertion that it was the source of irreversibility.) In the second paper [2] Bert proved, amongst other things, that the H-theorem followed from the superposition approximation. In a his book Molecular Theory of Fluids, he pointed out that, for irreversibility in gas theory, it was necessary to assume binary collisions together with an assumption about 'final encounters'. This was already well-known and not really fundamental enough, so he returned to it in [45], presenting a slightly different definition of H, for which the H-theorem followed more easily.

In [2] and [3], a very important observation was made concerning the difference between gases and liquids, or, in other words, the phenomenon of condensation. Bert observed that as the density of the fluid increased, a mathematical singularity appeared in the equations, and this was interpreted as heralding the onset of condensation. It is this property which still gives the BBGKY equations their significance, for alternative approaches fail in this regard. In [3], the Chapman-Enskog derivation of the equations of hydrodynamics and the coefficients of viscosity from the kinetic theory was presented from a more fundamental basis.

So well constructed were the first three papers on the classical theory of fluids that the transition to quantum theory went through with little fuss. It was possible to construct a quantum version of the equations of hydrodynamics, as well as expressions for the coefficients of viscosity and thermal conduction. The latter have special significance in the behaviour of liquid helium, which is the distinctive quantum fluid, whose paradoxical behaviour results from the special nature of these coefficients. In [5], [7] and [9], Bert showed that quantum mechanical corrections could substantially account for these strange properties. That work had surprisingly little impact, however, and is barely cited. Here Born's comment to Einstein (1) indicates that it missed out in comparison with Landau:

But the use of (sic) helium, which has a liquid phase that behaves curiously, was not as successful as we had hoped. The theory accepted today originated with the Russian Nobel prizewinner of 1962 L.D. Landau.

However it is quite clear that Bert's telling use of density matrix methods set a pattern that was widely followed in all later work.

Although the BBGKY equations went beyond the Boltzmann equation, there was nevertheless still much of interest to be obtained from the latter, particularly when charged fluids were considered. In 1957 Bert was amused to receive a manuscript by J.R. Cotter, who had devoted a large part of his life to trying to solve the Boltzmann equation when the molecules are rigid spheres, something Bert could see could be done much more simply. However he did not publish his solution, leaving it for ten years until a student, P.I. Brooker, took it over as a PhD project culminating in a number of papers, the first of which was [77]. Several times Bert returned to the question of how the Maxwell-Boltzmann equation could best be obtained from the BBGKY equations, both in the classical [71], [79] and quantum form. In the latter case he went beyond the usual rather crude process of simply inserting the quantum-mechanical cross section in the collision integral, and gave a purely quantum-mechanical derivation [32]. As a consequence he found corrections to the standard treatment by Chapman and Cowling.

In 1957, Roy Leipnik, then at the Michelson Laboratory of the US Navy at China Lake, was working on rocket-ground commmunication involving warm plasmas around rockets, and he quickly saw that Bert would be a valuable collaborator. It was natural that someone with Bert's outstanding expertise in the kinetic theory of neutral fluids should be able to consider the case where the fluid can consist of several charged species – a plasma. Bert first visited China Lake in 1958 and made regular visits until 1968. He very rapidly produced a series of papers that extended the theory from neutral to charged fluids: [54] dealt with small disturbances, such as plasma oscillations, in the neighbourhood of equilibrium, [55] constructed the hydrodynamic equations from the micrscopic theory and [59] the thermodynamics. With a student, T. M.L. Wigley, Bert constructed the Boltzmann equation for a charged fluid, with the Coulomb potential replaced by the much more realistic screened Debye potential. This and other work culminated in the book with Roy Leipnik, Sources of Plasma Physics, which brought order to a previously generally scrappily treated subject.

Apart from papers that might be regarded as written in response to practical questions, Bert retained his interest in deeper questions; this is very evident in [66], in which he derives a thermodynamics of complicated systems from general principles. Particular emphasis is placed on describing irreversible processes away from thermodynamical equilibrium but based on the idea that in sufficiently small regions there is equilibrium. Here he is in illustrious company such as Onsager, de Groot, Caratheodory and Born. Again and again we shall see how Bert, even from the remoteness of Adelaide, was prepared to tackle the deepest and most difficult problems of physics. Perhaps people in the bigger centres regarded this as presumptuous.

His early work on kinetic theory, prior to 1960, was presented in book form in several places. The first was the Cambridge collection already referred to, the second was The Molecular Theory of Fluids, originally published by North-Holland in 1952 and then later by Dover in 1969. This book has been very widely cited and is still regarded as the standard account of the kinetic theory of neutral fluids. In 1960 he contributed a chapter, 'The Structure of Liquids', to the prestigious Handbuch der Physik. There were also extensive articles in the Encyclopaedic Dictionary of Physics [63] and Research Frontiers in Fluid Dynamics [72].

(b) Statistical mechanics

Although the time-independent solutions of the BBGKY equations were shown by Bert and others to be, via the H-theorem, the well-known Maxwell-Gibbs expressions of statistical mechanics, the latter are much more general in their application. One very important case is the Ising model, which was solved, for the two-dimensional case by Onsager in 1944 in what was, and is still regarded as a tour-de-force of mathematical physics. This is a simple model of a magnetic material consisting of a rectangular array of elementary magnets that can point only up or down, and that can exert simple magnetic attractions on nearest neighbours. The value of this solution is that it gives an exact description of an interacting many-particle system and exhibits a phase transition at a particular temperature, the critical temperature. At this temperature the system changes from one in which the elementary magnets point predominantly up, or predominantly down, to one in which neither direction is favoured. So it describes a change in the material from a state of being magnetized to one in which it is not. As such, it provides deep insights into such important physical phenomena as melting, evaporation, magnetization and quark-gluon plasmas.

Because of the difficulty of Onsager's solution – which was not helped by the well-known obscurity of his writings – there was a need for a simpler treatment. The first people to provide this were Mark Kac and John Ward (5), using a clever combinatorial method based on the properties of determinants. Whilst trying to understand their paper, I constructed a graphical picture that seemed to describe their construction. However when this was shown to Bert, he first pronounced it a very interesting idea, and then later showed that it would not work in all cases. What followed is an illustration of what working with Bert could be like. Despite several urgings from him that this idea should be written up, I did nothing about it, and eventually Bert produced, without any preliminary discussion, a complete manuscript describing a new solution to the Ising problem. To do this he used the mathematical formalism of Pfaffians, which had been introduced by E.R. Caianiello (6) to describe fermions in quantum field theory and had already been used by me, and then the previous difficulty disappeared. The paper [58] made an immediate impact, leading quickly to the first solution of the complete dimer problem by H.N.V. Temperley and M.E. Fisher and P.W. Kasteleyn (7). (Bert independently also solved this problem, but the publication of his solution was delayed until the appearance in 1964 of the book, Order-Disorder Phenomena.)

This solution, eventually called the free fermion field, simplified the solution of the two-dimensional Ising model so much that it could easily be given in an undergraduate course. Ilya Prigogine asked Bert to prepare a review article, and this he started to do in collaboration with me, at a time when I was on sabbatical leave in Edinburgh. In the course of preparing this article, which soon turned into the book already mentioned, all the Ising models that had been solved up till that time were found to be particular cases of the free fermion model. No further essential progress was made in this field until 1967 when Elliott Lieb (8) solved the six-vertex model, leading on to Rodney Baxter's solution (9) of the eight-vertex model and the enormously significant Yang-Baxter relations.

Despite all these new results and the continued reference made to this book in the literature, there is a sense in which this work did not receive the recognition it perhaps should have. The primary reason for this was that a book is not the best medium for publishing new results. By the time the book appeared, many of the results had appeared elsewhere and priority was lost. Also it appeared that the cursory treatment given to approximate methods – included only as an afterthought in response to a request from Prigogine – did not go down well with those who had invested considerable effort in this important part of the understanding of the properties of cooperative systems. As approximate methods were outside the intended scope of the book, it would have been much more politic to have left them out completely.

(c) Quantum field theory and particle physics

In 1939 Born (10) suggested what he called the principle of reciprocity, which proposed that natural laws are symmetric under the interchange of position and momentum co-ordinates. He based this on the fact that the canonical commutation relations of quantum mechanics and the components of angular momentum display this symmetry. This idea was severely criticised by Peierls, who cited in particular that the principle does not apply to translations, as is very evident from the difference between the two in the actual world. Nevertheless Born remained attracted to this idea, which has still some relevance at the present day in the occurrence of 'duality' in string theory – although it is now quite unlike anything Born and his collaborators contemplated. The main idea as expounded in [8], [10]-[13], [18] and [75] was to use this symmetry to constrain the structure of relativistic wave equations, and consequently the possible values of the masses of the mesons known at the time. It was also hoped that this principle would ameliorate, if not remove, the divergences that still plague elementary particle theory. It is clear from Born's letters to Einstein (1) that he was very optimistic about the value of this program:

Now the divergences in quantum mechanics seem to indicate that an absolute length does exist in the world. I presume that this will have to be included in the general transformation group. We have gone to a great deal of trouble over this. My pupil Green, a highly gifted man (whom I am going to send to you in Princeton next year) may possibly make some progress with it; he has good ideas and great mathematical skill.

It is equally clear from Einstein's replies that he had little interest in what he regarded as the very deficient machinery of quantum mechanics.

Like most physicists who worked on particle physics in the 1950s, Bert had a great interest in trying to formulate the equations so as to be free from unacceptable infinities. At that time the work of Dyson, Feynman and Schwinger had shown that these equations could be expressed as a set of integro-differential equations, the Dyson-Schwinger equations, and if these equations could be made to behave, then one had a fundamental theory. Bert wrote several papers, [33], [39], [106] and [125], in which the divergent quantities could be removed, leaving behind some very complicated but putatively finite equations. In the process of doing this, he derived a generalization of the Ward identities, which are some of the most important constraints on the fundamental equations. However his priority in this has been only occasionally recognised, and they are usually called the Ward-Takahashi or simply the Ward identities. This failure to recognise him was a source of the very rare occasions when Bert showed what could almost be described as anger.

In 1953, Bert wrote a short paper [31] which, more than any other of his writings, has made his name widely known. In this paper he described what is known as parastatistics, which is a symmetry extending the well-known Bose and Fermi statistics. Bose statistics apply in particular to photons and are central to the operation of the laser, while Fermi statistics provide the mathematical foundation for Pauli's exclusion principle and hence the structure of the periodic table of elements. Parastatistics can be regarded as interpolating between these two, and for some time it looked as if they were the appropriate language for describing quarks. Even though they missed out on that score, they are still the subject of immense research at the present time, being re-expressed in terms of more and more exotic mathematical schemes. This theory has gradually assumed importance, more as an elegant framework on which to build other algebraic structures. Even since 1989, there have been well over several hundred citations of this single paper. The 'Green Ansatz' which appeared there for the first time is a standard tool in describing what are also called 'Generalised Statistics'. The 'para-Bose' algebra, in particular, can be seen today as an example of a 'Lie superalgebra', although superalgebras were not introduced formally into physics by others until much later (12). For the people, including Bert, who were working in Adelaide in the late '60s, trying to put internal and space-time symmetries together in a non-trivial way, the failure to spot the potential of superalgebras in this connection, rather than Lie algebras, surely represents what Dyson would have called a 'missed opportunity'. As is so often one of the ironies of life, this paper was written whilst Bert was in the midst of what he regarded as much more important work with Harry Messel on cosmic rays, and it was regarded by him as an amusing sideline. The work on cosmic rays is now part of history.

Bert, in association with A.J. Bracken and later P.D. Jarvis, continued to investigate paraparticles and their generalisations as models for the known collection of particles. In the paper [89], they constructed a generalised parastatistics in which the quarks did not have definite isospin or hypercharge. The advantage of this model was that the quark could be regarded as a simple parafermion, so that an additional colour label is not required. However it was a parastatistic model of order 3, and no consistent way of quantising this could be found that would respect the correspondence principle. This led Bert to a further generalisation called modular statistics, [94], [100] and [119], for which a consistent quantisation could be defined. This appeared to be a more economical model of quarks, without the multiplicity of colours. However as quarks are not observed there needs to be some explanation for this, and Bert made the bizarre suggestion that they are tachyons – particles that can travel faster than light, and that are certainly not observed. More conventionally, in association with Peter Jarvis, he constructed a model whereby all the standard particles, including quarks, were composites of modular particles, with quarks being more complex composites than electrons.

The problem of how to describe bound states in quantum field theory was advanced by the appearance of the Bethe-Salpeter equation and the related Wick equation (11), which however did not conform to the standard form of bound-state equations coming from the Schrödinger equation. Bert and S.N. Biswas obtained solutions [44], [46], [76] and [87] without making the usual instantaneous interaction approximation, and Bert showed in [48] that the Wick equation separated nicely in bipolar coordinates, with the consequent appearance of a separation constant called a relativistic quantum number, which might be interpreted as labelling the so-called 'strange' particles L and q. Later on, Biswas and collaborators showed (13) that this separability was due to the symmetry of the Wick equation under the group O(5).

Bert always had a fondness for using the non-compact de Sitter group SO(4,1) in cosmology and in particle physics. In the latter case it appears as the group generated by the G-matrices of the Bhabha equation, and it is this equation (with a modified mass term) that Bert employed to construct equations describing particles of higher spin [102], and in particular charged particles. This is not straightforward because the usual versions of such particles suffer from defects of non-causality and an indefinite metric. However, Bert was able to construct an electrodynamics of charged particles with higher spin [106] that was free from causality defects.

(d) Cosmic rays

The Green-Messel papers were written and published in the early 1950s when not much was known about the interactions of pions with nuclei, although by then the distinction between pions and muons had been demonstrated by C.F. Powell and co-workers. Strange Particles and Associated Production were under discussion, but the particle physics side of cosmic ray studies was still to settle down to the eight-fold way and the eventual quark model.

The theoretical issues of the day were concerned with the development of the cosmic ray cascade in matter in general and the atmosphere in particular. The electromagnetic cascade itself (via bremsstrahlung and pair production) was understood, but the nuclear cascade was still only vaguely understood, as was the way it affected the development of cascades in the atmosphere.

For example, the discovery of the p0 and its decay immediately gave a means to feed energy from the nuclear interactions into the electromagnetic cascade. It was not, however, clear how the primary nuclei distributed energy into the secondary nuclear cascade. A common question was whether the pions were produced by 'multiple' or 'plural' production. Angular distributions gave some clues.

Bert and Harry addressed these issues with an essentially analytical approach in [35] and [36]. Later on, cascade calculations used Monte Carlo methods with heavy computer back-up, such as Harry developed with the new Silliac computer. As a result, these papers, although a tour-de-force of mathematical analysis, have been superseded by less elegant methods and much more powerful computing resources. Unfortunately it therefore seems that this work was a decade or so too early, and most of this intense effort was wasted.

(e) Quantum mechanics

Running through most of Bert's work, and particularly in quantum mechanics, was a preference for using algebraic as opposed to analytic methods. Probably his undergraduate background did not develop a feeling for mathematical rigour, and algebraic methods usually gave a better understanding of what the mathematics meant. His book, Matrix Mechanics, which arose out of his third-year lectures but also reflected many of his research techniques, was translated into German, Russian and Japanese as well as appearing in two English printings. This book continues to provide a useful and remarkably compact introduction to quantum mechanics for the beginning student. In it, Bert solved a wide variety of problems in quantum mechanics by purely algebraic methods. Even the fine structure of the energy levels of the hydrogen atom was obtained by such methods, perhaps for the first time. However it is typical of Bert's approach to problem-solving generally that the solutions presented in the book are often not completely rigorous. On a close examination, many subtle difficulties reveal themselves, usually relating to the precise definitions of the algebraic objects being manipulated. But it is also typical that by examining these subtleties and attempting to find out how and why Bert's methods work, one can obtain new insights into the underlying physics as well as the mathematics of the problem at hand.

In 1958 Bert published one of his best papers [53]. It was entitled 'Observation in Quantum Mechanics' and addressed one of the outstanding problems of modern physics, namely the process by which indeterminate superpositions in quantum mechanics become converted to the determinate, although possibly unknown, alternatives of ordinary macroscopic physics. For many years the prescription of von Neumann, usually called the 'collapse of the wave packet', was the accepted view of how this happened. As it assumed that some processes outside quantum mechanics had to be invoked, even going so far as involving the brain of the human observer, people were not comfortable with it, although it seemed the only possible answer. The best known representation of this difficulty appears in the well-known Schrödinger's cat paradox. Bert, together with a number of others such as Wakita and Ludwig, found a much more satisfying explanation, which is basically still the received description, although nowadays in various forms. The idea was to suppose that a measuring apparatus could be of almost any form so long as it was very complicated, that is, contained a very large number (often for mathematical convenience taken to be infinite) of components such as molecules or electrons. The system being measured could be microscopic. When the two systems interact, any 'interference terms' in the state of the microscopic system become vanishingly small purely as a consequence of the size of the measuring instrument. There are, of course, many processes in nature in which a human observer is not involved – especially before homo sapiens evolved – and the von Neumann description is quite unable to say how these could happen. However with Bert's theory all one has to do is to replace the measuring apparatus by the environment to bring about the necessary disappearance of interferences. The only place where this very satisfactory explanation might run into some difficulty is in the early evolution of the universe, where there is no environment!

From his work with Born, Bert became convinced that quantum mechanics could only properly be discussed when states are described by density matrices rather than wave functions. (Perhaps even that can be a bit restrictive, as for example when dealing with supplementary conditions.) The paper just described emphasised the fact that the state of the measuring apparatus could not be known exactly, and a density matrix must be used. On a more general level, he presented an abstract formulation of quantum mechanics in terms of semi-groups [113] – he also considered semigroups in [108] – with states being given by density matrices. In the spirit of the comments made above, there is no attempt here to discuss the topological requirements of this theory, so it is not clear what sort of algebra of quantum variables is actually being defined.

(f) General relativity and gravitation

Although Bert wrote only three papers that could be clearly classified under this heading, they are so interesting that they deserve a separate classification. Bert had strong views about gravitation and cosmology, insisting right up to his last book that physicists were in error in not confining themselves to the de Sitter universe and de Sitter group.

In the first paper [51], he tackled the problem that occupied Einstein in the latter years of his life, namely to construct a unified theory of gravitation and electromagnetism. His approach was the opposite of that usually followed. Instead of setting up a gravitation theory and then incorporating the Dirac equation, he started with the Dirac matrices, spinors and Lagrangian, and from them constructed the metric tensor and the total Lagrangian including gravitation. The latter step is commonplace now under the heading of local gauge invariance, but it was quite unfamiliar at the time. But not only did this gauge-invariant Lagrangian include gravitation, it also contained the electromagnetic field! In a follow-up paper [52], Bert showed that this theory admits teleparallelism, meaning that vectors can be parallel transferred around the space, even though it is not flat. It also describes mesonic fields in addition to gravitation and electromagnetism. Bert never seemed to have followed up this paper, although several thesis projects came out of it.

(g) Mathematical methods

All the people who worked with Bert speak of his very strong mathematical ability, and there are at least fifteen of his papers that can be classified as mathematics rather than physics. But Bert was in no sense a mathematician, and would not have wished to be regarded as such. As Freeman Dyson in his talk to the Australian Academy of Science at its jubilee meeting in 1979 would describe it, Bert was engaged in what G.H. Hardy called 'schoolboy mathematics'. This is not as pejorative as it sounds, because he was in the company of almost all theoretical physicists. As already remarked in subsection (e), he was not concerned with nice points of mathematical rigour but rather with obtaining results. So long as the problems were essentially finite-dimensional or involved analytic functions, there was little to worry about, and his most elegant work was in this field. Also his collaborators Tony Bracken, Peter Jarvis and Denis O'Brien, having been educated under a more modern syllabus, were able to sound warning bells if necessary.

The schoolboy mathematics that Dyson referred to was mainly centred around the problem of how to classify the very large number of 'elementary' particles that appeared in the 1950s and later. The appropriate mathematical tools for this were finite-dimensional Lie algebras and their representations, and their study required expert and often ingenious algebraic manipulations – right in Bert's field. In collaboration with Tony Bracken, he developed what they called characteristic identities that are analogous to the more elementary Cayley-Hamilton theorems of matrix algebra [83], [84] and [95]. These identities are a very powerful tool for constructing representations of Lie algebras, and have applications to parastatistics [85]. They can be generalised to Lie superalgebras, [109] and [120], which had been found essential for the non-trivial combination of internal and space-time symmetries and particle physics.

(h) Environmental physics

On one of his regular visits to the United States, Bert became involved in studies in environmental physics, and his paper [90] discussed the spreading of pollutants. His extensive experience in kinetic theory and fluid dynamics was important in constructing the correct equations describing this process. The classical diffusion equation needed to be supplemented by the stochastic theory of Brownian motion. This work led to a dramatic confrontation with the South Australian government in 1984, following a decision by that government to construct a large petrochemical complex at Redcliff at the northern end of Spencer Gulf. The principal product of this enterprise would be dichlorethane, a rather nasty substance, and there was a risk that in loading this on to ships there could be spills into the gulf.

At the time Bert was a member of the Board of Environmental Studies at the University, which oversaw the work of the Department of Environmental Studies, under the direction of Dr J.R. Hails. This was a postgraduate department, and students were expected to undertake research in some environmental project. Bert suggested that a study could be made of the Redcliff project because he had been told about its possible hazards by Professor Rainer Radok, Head of the Horace Lamb Institute at Flinders University. Because of his previous work in this area, Bert eventually took over the study and wrote a submission to the Redcliff Environmental Inquiry in which he pointed out that the dangers were so real that the project should be discontinued. This caused such a stir that he appeared on the ABC television program, 'The 7.30 Report'. This report concluded with Bert sitting in a small and unsteady dinghy in the middle of Spencer Gulf with the ABC reporter Pru Goward, and saying in response to her asking what would happen if a spill occurred that 'Whyalla and Port Augusta would have to be evacuated'. He left for the United States shortly after appearing on this program, and the South Australian government felt obliged to reply. The relevant Minister, Roger Goldsworthy, cast doubt on Bert's report and interviewed comments, saying that they were nonsense. I regarded this as professional libel and contacted the ABC. I spoke to a very nervous Pru Goward, who was well aware of the ABC's vulnerability. I then wrote a letter to the Minister, taking strong exception to the tone of his remarks, and eventually received a three-page reply defending the Government's position and which was almost entirely beside the point. The letter was clearly written by a scientist but not an environmental physicist, and it later emerged that it had been written by a pair of chemists from Michigan, USA, who had been contracted by the South Australian Government. The final irony was that the chemists contacted Bert in the United States, asking his advice on the problem! The conclusion of the whole matter was that the petrochemical project at Redcliff was abandoned, and nothing further has been heard of it. This incident is a very good example of how independent academics can provide advice that is free from contractual pressures. It is to be regretted that now such independence is rapidly disappearing. This will be a cost to the community.

(i) Biological and neurophysiological models

Professor Terry Triffet met Bert at a Quantum Chemistry and Biology Workshop at Sanibel Island, Florida, in 1963 and began a friendship and collaboration that lasted until Bert's death. Among many common interests, the most fruitful was their belief in the influence quantum effects had on mental processes. Initially they concentrated on developing the mathematical, physical and chemical tools that they knew would be required [81], [97]-[99]. Here they looked at a model of a 'small interconnected group of neurons' from the point of view of the electrochemical dynamics, using Hamiltonian and matrix-operator notation. As this was unfamiliar to the neuroscience community, it was initially regarded with considerable suspicion. With the development of computing capacity, they were able to provide a sound basis for neural modelling by applying the laws and methods of theoretical and mathematical physics. The 'small interconnected group of neurons' was modified to represent 'unit circuits' known to constitute primary functional units throughout the brain, and these were interconnected to form an extensive network capable of simple image recognition and certain other simple basic brain processes.

As their methodology allowed them to extend the model into the quantum realm, they could bring digitalised information to bear on their linearised macroscopic model, leading to the concept of 'computing the uncomputable'. This led to the treatment of computational brain processes as a 'quantal Turing machine' and finally to a general theory of how the mind operates by gaining and creating information. This was all summarised in their book Sources of Consciousness, published in 1997. While far from a complete model of the mind's operation, the theory remains unique in its mathematically structured incorporation of physical laws and biochemical facts to describe interactions within the brain, extending from the quantally dependent interactions of metastable ions in the membrane channels of neurons, to the interactions of currents flowing in networks of these neurons with the electromagnetic fields they generate.

For some time Bert and Terry were convinced that the processes by which nerve signals were propagated could be explained by a Hamiltonian model of nonlinear oscillators, rather than by the opening and closing of membrane ion channels as proposed by Hodgkin and Huxley. However this part of their theory had to be recast when it became clear from experimental evidence that ion channels did exist and could open and close by way of conformational changes in proteins embedded in the membrane

There is an unpublished paper, 'Formation and Impairment of Sequential Memory: A Contribution from a Case of Transient Global Amnesia', that was written in response to a transient ischemic attack which Bert suffered in 1990 whilst attending a lecture at the University of Arizona. For a period of nine hours he suffered what is commonly called a 'black-out', and when he awoke the next morning, he had no memory from the middle of the lecture until 9 pm. His experience was incorporated into their model of brain processes.

The whole course of their investigations is described in twenty five papers [111], [112], [114], [115], [121]-[123], [126]-[131], [135], [136], [138], [140]-[144] and [149]-[151].

Personal Aspects

It is clear from what has been written here – which covers only a part of Bert's published scientific work – that he was a scientist of extraordinary breadth and depth. There was nothing routine or pot-boiling about any of his publications, so that even in areas such as general relativity, in which he published little, he produced material that could prove fertile for many years. In the preparation of this memoir, I received reports from a wide variety of people who found themselves both baffled and enormously impressed by Bert. The word 'genius' keeps appearing in comments by people who would not be expected to speak extravagantly. There are some reasons which can account for these strong reactions. First of all, Bert had a privileged upbringing in so far as he was a very talented only child, and although he always had a very friendly nature, he was perfectly satisfied with his own company. This self-isolating tendency was exacerbated when he started to go deaf in his twenties, which meant that his social exchanges became much less easy. For students this would mean a remoteness that they would find very hard to bridge, and for colleagues a difficulty in free exchange of ideas. So scientific discussions with him were rarely of the kicking-the-ball-around type, but rather a formulation of the problem, after which Bert would disappear, to return next morning, say, with it all worked out. Sometimes his solution would not be correct, and it could be quite tiring convincing Bert to change because, like most people, he did not take kindly to being corrected, and he could be very stubborn. Tony Bracken puts it admirably:

Sometimes his ideas were wrong, but he had the magical gift that, when he was wrong, it was almost always wrong in an interesting way. Those who worked with Bert sometimes had the feeling that our main task was to keep the locomotive on the rails. It was no good saying to Bert, 'I think this is wrong'. He would just say 'Oh?' and shrug. The only thing that worked was to produce a counterexample; then he would alter track slightly, and by repetitions of this process you could gradually come to a satisfactory conclusion.

Most of the time, though, the solution would be both ingenious and complete, bringing in ideas from unexpected directions. Bert had a very wide range of knowledge and could call up the relevant parts without difficulty, and with great clarity and cogency. He did not appear to search around for the correct path to a solution, so that there was rarely much sign of the usual false starts and well-filled waste paper basket. His use of University of Adelaide examination booklets, 16-page blue paper covered foolscap, in which to write his notes almost without blemish and apparently straight out, was legendary. Beginning research students repeatedly told of having a problem suggested to them, and of being given an examination booklet setting out in impeccable style, and in characteristic semi-neat writing, all the relevant points.

This leads to the most puzzling question of all when considering Bert's career. Despite his great scientific productivity and originality, he did not receive the sort of recognition that would be expected. There were no honorary doctorates, no elections to foreign scientific societies and academies (including the Royal Society of London), no invitations to give plenary addresses or to act as rapporteur at conferences, and no memberships of international committees. His deafness must not be discounted in considering his disinclination to feature at international meetings. It is not generally appreciated how disabling deafness can be in so much of ordinary intercourse. For example, on one occasion Bert gave an important seminar at Princeton that went well until question time, when, because he could not hear the questions, he simply smiled and nodded, giving the impression to the audience that he was not really on top of his subject. He was very reluctant to say that he was deaf, so people who did not know him would think he was very slow.

He was often invited to accept positions in the United States, both before coming to Adelaide and after, but he refused them all, saying that he much preferred the lifestyle of Adelaide, both at the University and elsewhere, and that his scientific work could function perfectly satisfactorily there. There is no doubt from his record that that is true, but there is also no doubt that his occasional visits to the United States and Europe were not sufficient to make a big impact on the international scene. Added to this was his very wide interests, which prevented him from making a strong impact in any one area, especially in such a highly competitive area as particle physics and field theory. His strongest reputation is undoubtedly in kinetic theory, in which he published not only fundamental papers, but also very highly regarded books and monographs. He was not particularly upset or bitter about his lack of recognition, feeling that he was very fortunate to do the things he wanted to do, surrounded by very bright young students, with periodic overseas visits to keep in touch. He certainly had no wish to push himself forward in any administrative role, no matter how prestigious. He took on the Presidency of the Australian Mathematical Society out of a sense of duty, and worked very hard then to assist Soviet refusniks. In the same way he took on the job of Dean of the Mathematical Sciences Faculty at Adelaide, and membership of Academy Sectional Committees, but had no interest in being on the University or Academy Councils.

At a personal level, Bert is remembered, despite his apparent remoteness, with great respect and often affection by those who worked with him, as students or as colleagues and often as both. Like a lot of people with introverted personalities, he found it easier to get on with extroverts as they would make most of the running. Two traits of his character that commanded great respect were his absolute integrity and his moral courage. He had very clear ideas about acknowledgement of other people's work, often to the extent of citing papers that were only marginally related to what he had done, and firm views about serving out a proper time after a job had been taken on. So not only did his students receive fine training in research but they were also given guidance on how to behave in later life. These instructions were given by example rather than precept.

As already mentioned, although Bert was not an athlete, he was a tremendous walker and he knew the Adelaide Hills and Flinders Ranges very well. He also walked extensively around Tucson, climbing the 2885m Mt Wrightson at the age of 67. Despite his deafness he was a very keen and discriminating concert goer, this being recognised by the choice of music at his funeral and memorial services. His favourite hobby was the game of Go, regarded by Japanese as their especial province so that defeat by a non-Japanese was a very serious matter. So it was not surprising that from time to time Bert would have to be regarded as an honorary Japanese in order for his opponent not to lose face. He organised a Go club in Adelaide, which used to meet regularly at his home.

Once when he was quite ill with a viral disease, and not able to concentrate on his beloved mathematics, Bert decided to write a detective novel. Instead of composing it as he wrote, he worked out the complete story, complete with racy dialogue, in his mind and then typed it out. He submitted the story to Penguin publishers but they did not accept it, and he did not bother any further. The manuscript is still in existence and it would be interesting to see whether a posthumous spy novel would have a sale now.

With the very strong support of his wife, Marlies, he made a point of regularly entertaining members of the department at their home. This produced a marvellous spirit within the department which had a great influence on the students. Even though the department no longer exists, this influence is still very noticeable in Australia, where so many of its graduates have risen to responsible positions. Bert always considered the existence of the Department of Mathematical Physics to be as great a contribution to the development of Australian science as his own research publications. Because of its special origin and the unique way in which Bert and his colleagues set it in motion, it was always regarded by the rest of the University with a mixture of pride, incomprehension and envy. Henry Basten, former Vice-Chancellor of Adelaide, once referred to it as the jewel in Adelaide's crown. Consequently it was a continuing source of sadness to Bert in his final years that what he had worked so hard to create should have been virtually destroyed, despite being given a resounding commendation by the Review Committee set up in 1984 to make recommendations for its future.

Bert was always a committed socialist of the Fabian variety and so believed in rule by an enlightened minority, sometimes being scornful of the 'tyranny of the majority'. His son, Roy, felt compelled at times to argue strongly that socialism not based on true democracy would be a stunted and dangerous form of political organization. Bert's answer to this was that no decisions of any significance should be left in human hands, and that artificial intelligence, rather than the market place, would be our saviour.

When Bert was in Dublin, he got to know de Valera well, because the latter, though president of the republic, maintained an interest in theoretical physics throughout his life – so much so that he intervened constantly in the seminar program that Bert co-ordinated, and if he had some political objection or score to settle he would strike the offending invitees off Bert's list! Despite their difference in age and religious belief, the two had many political views in common. Although Bert's religious views were agnostic, he believed in toleration and supported his daughter Johanne's marriage taking place in her husband's church, recognising its importance to him. Bert meditated regularly for the last half of his life. This was more for health and relaxation purposes than as any particular religious practice. He did, however, have some respect for certain aspects of Buddhism and at times referred to himself as a Humanist.


I wish to acknowledge the contributions made to this memoir by the twenty-five people who responded to my invitation to write about their association with Bert. They provided me with a wonderful range of impressions of Bert as an outstanding scientist, as a kindly mentor and as a warm and constant friend. I particularly want to thank his wife Marlies and children Roy and Johanne for details of their family life, and their experience of living in Adelaide. Professors Leon Bowden and Alister McLellan and Mr Arthur Birt were very helpful about Bert's life before coming to Adelaide, providing some very enjoyable lighter touches which were very revealing of sides of Bert's character. Professor Terry Triffet gave me a definitive account of their work together on neurophysiology and Professors David Hoffman, Tony Bracken and John Prescott helped to set Bert's work in the context of present knowledge. The detailed accounts of Bert's work from Associate Professors Peter Jarvis and Robin Storer were also much appreciated. Without their help I would have found it almost impossible to assimilate and judge the extraordinarily wide corpus of Bert's scientific writings.

About this memoir

This memoir was originally published in Historical Records of Australian Science, vol.13, no.3, 2001. It was written by Angas Hurst, Department of Physics and Mathematical Physics, University of Adelaide.

Numbers in brackets refer to the references, and numbers in square brackets refer to the bibliography.


  1. The Born-Einstein Letters (Macmillan, 1971).
  2. N.N. Bogoliubov, J.Phys. (U.S.S.R.) 10, 256 (1946). An English translation is given in Studies in Statistical Mechanics, Vol.I (North-Holland, 1962).
  3. J.G. Kirkwood, J.Chemical Physics 14, 180 (1946).
  4. J. Yvon, La Théorie Statistique des Fluides et l'Equation d'État. Actualités Scientifique et Industrielles # 203 (1935).
  5. M. Kac and J.C. Ward, Phys.Rev. 88, 1332 (1952).
  6. E.R. Caianiello, Nuovo Cimento 11, 492 (1954).
  7. H.N.V. Temperley and M.E. Fisher, Phil.Mag. 6, 1061 (1961); P.W. Kasteleyn, Physica 27, 1209 (1961).
  8. E.H. Lieb, Phys.Rev. 162, 162 (1967).
  9. R.J. Baxter, Ann.Phys. 70, 192 (1972).
  10. M.Born, Proc.Roy.Soc.Edin. A59, 219 (1939).
  11. E.E. Salpeter and H.A. Bethe, Phys.Rev. 84, 1232 (1951); G.C. Wick, Phys.Rev. 96, 1124 (1954).
  12. J. Wess and B. Zumino, Nucl.Physics B70, 39 (1974).
  13. D. Basu and S.N. Biswas, J.Math.Phys. 11 (1970).

Curriculum vitae

  • Visiting Professorships, Dublin Institute of Advanced Studies, University of Florida, Michigan State University and University of Arizona.
  • Fellow, Australian Academy of Science 1954–99.
  • Fellow, Australian Institute of Physics.
  • Life Member, Australian Mathematical Society (Vice-President 1973–74 and 1976–77; President 1974–76).
  • Life Member, Royal Zoological Society of South Australia.
  • Editorial Boards: (sometime member and chairman) Australian Journal of Physics; (sometime member) International Journal of Engineering Science; (member) Mathematical and Computer Modelling.



  • M. Born and H.S. Green, A General Kinetic Theory of Liquids (Cambridge University Press, 1949).
  • H.S. Green, Molecular Theory of Fluids, 264pp. (North-Holland Publishing Co., 1952).
  • H.S. Green, 'The Structure of Liquids', Handbuch der Physik, 10, 1–133, (1960).
  • H.S. Green and C. A. Hurst, Order-Disorder Phenomena, 363pp. (Interscience Publishers, London, 1964).
  • H.S. Green, Matrix Mechanics, 118 pp. (P.Noordhoff Ltd., Groningen, 1965).
  • H.S. Green, Research Frontiers in Fluid Dynamics, Ch. 4: Molecular Theory of Fluids, 105–143 (Editors, Seeger and Temple) (Interscience, New York, 1965).
  • H.S. Green, Quantenmechanik in Algebraischer Darstellung, 106pp. (Springer, Berlin, 1966).
  • H.S. Green, Matrichnaya Kvantovaya Mechanika, 163pp. (Ed. A.A. Sokolov) (Izdat. 'Mir', Moscow, 1968).
  • H.S. Green, Matrix Methods in Quantum Mechanics (Barnes and Noble, New York, 1968).
  • H.S. Green and R. B. Leipnik, Sources of Plasma Physics, 630pp. (Wolters-Noordhoff, Groningen, 1970).
  • H.S. Green, Matrix Methods in Quantum Mechanics (Japanese edition, with additional material) (Kodansha, Tokyo, 1980).
  • H.S. Green and T. Triffet, Sources of Consciousness (World Scientific, Singapore, 1997).
  • H.S. Green, Information Theory and Quantum Physics: Physical Foundations for Understanding the Conscious Process (Springer, Berlin, 1999).

Journal Articles

(Letters in parentheses refer to corresponding sections in the above discussion of Green's scientific work.)

  • [1] M. Born and H.S. Green, A General Kinetic Theory of Liquids I: The Molecular Distribution Functions. Proc. Roy. Soc. A188, 10–18 (1946).(a)
  • [2] H.S. Green, A General Kinetic Theory of Liquids II: Equilibrium Properties. Proc. Roy. Soc. A189, 103–117 (1947).(a)
  • [3] M. Born and H.S. Green, A General Kinetic Theory of Liquids III: Dynamical Properties. Proc. Roy. Soc. A190, 455–473 (1947).(a)
  • [4] M. Born and H.S. Green, A General Kinetic Theory of Liquids IV:Quantum Mechanics of Fluids. Proc. Roy. Soc. A191 168–181 (1947).(a)
  • [5] M. Born and H.S. Green, Quantum Theory of Liquids. Nature 159, 738–739 (1947).(a)
  • [6] M. Born and H.S. Green, A General Kinetic Theory of Liquids V: The Kinetic Basis of Thermodynamics. Proc. Roy. Soc. A192, 166–180 (1948).(a)
  • [7] H.S. Green, A General Kinetic Theory of Liquids VI: Liquid Helium II. Proc. Roy. Soc. A194, 244–258 (1948).(a)
  • [8] H.S. Green, The Relativistic Quantum Mechanics of the Elementary Particles. Proc. Cambridge Phil. Soc. 45, 263–274 (1948).(c)
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  • [67] H.S. Green, Plasma Dynamics and Thermonuclear Reactions. Atomic Energy 6, 2–6 (1964).(a)
  • [68] H.S. Green, Structure and Energy Levels of Light Nuclei. Nuclear Physics 54, 505–515 (1964).(c)
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  • [96] H.S. Green and J. Casley-Smith et al., The Quantitative Morphology of Skeletal Muscle Capillaries in Relation to Permeability. Microvascular Research 10,43–64 (1975).(i)
  • [97] H.S. Green and T. Triffet, An Electrochemical Model of the Brain: General Theory and the Simple Neuron. J. Biol. Phys. 3, 53–76 (1975).(i)
  • [98] H.S. Green and T. Triffet, An Electrochemical Model of the Brain: Collective Behaviour, Irreversibility and Information. J. Biol. Phys. 3, 77–93 (1975).(i)
  • [99] H.S. Green and T. Triffet, Quantum Mechanics and the Brain. Int.J. Quantum Chem: Quantum Biology Symp. 2, 289–296 (1975).(i)
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  • [104] H.S. Green, Quantum Mechanics of Space and Time. Foundations of Physics 8, 753–591 (1978).(e)
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  • [106] H.S. Green, J. F. Cartier and A. A. Broyles, Electron Propagator without Renormalization. Phys. Rev. D18, 1102–1109 (1978).(c)
  • [107] H.S. Green, Distribution of Arrival Times in Cosmic Ray Showers. Adv. Appl. Prob. 10, 730–735 (1978).(d)
  • [108] H.S. Green, Semigroups in Relativistic Quantum Mechanics. Structures of Time and Space 3, 183–194 (1979).(c)
  • [109] P. D. Jarvis and H.S. Green, Casimir Invariants and Characteristic Identities. J. Math. Phys. 20, 2115–2122 (1979).(g)
  • [110] S. Vaccaro and H.S. Green, Ionic Processes in Excitable Membranes. J. Theor. Biol. 81, 777–802 (1979).(i)
  • [111] H.S. Green and T. Triffet, Mathematical Modelling of Nervous Systems. Math. Modelling 1, 41–61 (1980).(i)
  • [112] T. Triffet and H.S. Green, Information and Energy Flow in a Simple Nervous System. J. Theor. Biol. 86, 3–44 (1980).(i)
  • [113] H.S. Green, Semigroups and the Density Matrix Formulation of Quantum Mechanics. Int. J. Quantum Chem. 17, 121–132 (1981).(e)
  • [114] H.S. Green and T. Triffet, Non-linear Ion Dynamics. Dynamical Systems 2, 80–91 (1981).(i)
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  • [116] H.S. Green, Colour Algebras and Generalized Statistics. Pp.346–350 in Lecture Notes in Physics 180: Group Theoretical Methods in Physics (Springer, Berlin, 1983).(g)
  • [117] H.S. Green, Entropy and Human Activity
    in Environment and Population: Problems of Adaptation
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  • [121] H.S. Green and T. Triffet, Calcium Dynamics at a Plastic Synapse in Aplysia. J. Theor. Biol. 100, 649–674 (1983).(i)
  • [122] T. Triffet and H.S. Green, in Membrane Permeability: Experiments and Models. (Ed. A. H. Bretag), 31–35 (Techsearch, Adelaide, 1983).(i)
  • [123] T. Triffet and H.S. Green, Ionic Currents and Field Effects in Neural Extracellular Spaces. In Nonlinear Electrodynamics In Biological Systems (Eds. W.R. Adey and A.F. Lawrence), (Plenum Pulishing Co., 1984).(i)
  • [124] H.S. Green, Fluid Transport Processes in Upper Spencer Gulf. Marine Geology 61, 181–195 (1984).(h)
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  • [127] H.S. Green and T. Triffet, Extracellular Fields within the Cortex. J. Theor. Biol. 115, 43–64 (1985).(i)
  • [128] H.S. Green and T. Triffet, Electromagnetic Waves in Cortex Layers. Winner of Best Paper Award and Maxwell Prize in Proceedings, 5th International Conference on Mathematical Modelling, Berkeley, August 1985 (Pergamon, 1985).(i)
  • [129] T. Triffet and H.S. Green, Information Transfer by Electromagnetic Waves in Cortex Layers. J. Theor. Biol. 131, 199–221 (1988).(i)
  • [130] H.S. Green and T. Triffet, Information Processing by the Cortex. Comput. Math. Appl. 15, 743–756 (1988).(i)
  • [131] T. Triffet and H.S. Green, Information Transfer in the Cortex. Mathl. Comput. Modelling 11, 832–836 (1988).(i)
  • [132] A.J. Bracken, H.S. Green and L. Bass, Groups Defined on Images in Fluid Diffusion. J. Austral. Math. Soc. B30, 101–119 (1988).(i)
  • [133] L. Bass, A.J. Bracken and H.S. Green, Boundary Layers and Images in Dispersed Flow Reactors: A Green's Function Approach. Chemical Engineering Science 43, 1583–1590 (1988).(i)
  • [134] L. Bass, H.S.Green and H. Boxenbaum, Gompertzian Mortality Derived from Competition Between Cell Types: Congenital, Toxicologic and Biometric Determinants of Longevity. J. Theor. Biol. 140, 263–278 (1988).(i)
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  • [136] T. Triffet and H.S. Green, Unit Circuit Neural Networks of the Cortex. Mathl. Comput. Modelling 12, 673–694 (1989).(i)
  • [137] H.S. Green, A. J. Bracken and L. Bass, Harmonic Functions Satisfying a Radiation Boundary Condition. Computers Math. Applic. 22, 23–38 (1991).(g)
  • [138] H.S. Green and T. Triffet, Quantum Mechanics, Real and Artificial Intelligence. Aust. J. Phys. 44, 323–334 (1991).(i)
  • [139] H.S. Green, A. J. Bracken and L. Bass, Harmonic Functions Satisfying a Radiation Boundary Condition. Computers Math.Applic. 122, 23–38 (1991).(g)
  • [140] T. Triffet and H.S. Green, A Model of an Artificial Electrochemical Synapse. Intelligent Engineering Systems Through Artificial Neural Networks (C.H. Dagli, L.I. Burke and Y.C. Shin, Eds.) 12, 51–60 (ASME Press, New York, 1992).(i)
  • [141] H.S. Green and T. Triffet, Modelling Intelligent Behavior. Journal of Intelligent Material Systems and Structures 4, 35–42 (1993).(i)
  • [142] T. Triffet and H.S. Green, Structured Neurobiological Networks. Mathl.Comput. Modelling 17, 75–88 (1993).(i)
  • [143] T. Triffet and H.S. Green, Development of an Electrochemical Transistor for Use as an Artificial Synapse. Proc. of Third International Conference on Microelectronics for Neural Networks, 195–205 (Univ. of Edinburgh Technologies Ltd., Edinburgh, 1993).(i)
  • [144] H.S. Green and T. Triffet, Artificial Neural Processing. Mathl. Comput. Modelling 18, 1–18 (1993).(i)
  • [145] H.S. Green, A Cyclic Symmetry Principle in Physics, Aust. J. Phys. 47, 25–43 (1994). (e)
  • [146] H.S. Green, Statistical Symmetries in Physics, Aust. J. Phys. 47, 109–122 (1994). (e)
  • [147] H.S. Green, Contiguity and the Quantum Theory of Measurement, Aust. J. Phys. 48, 613–633 (1995).(e)
  • [148] S. N. Biswas and H.S, Green, Symmetry Breaking by Deformations, J. Phys. A28, L339–342 (1995).(c)
  • [149] T. Triffet and H.S. Green, Consciousness: Computing the Uncomputable. Mathl. Comput. Modelling 24, 37–56 (1996) (i)
  • [150] H.S. Green and T. Triffet, The Cortex as a Quantal Turing Machine. The Mathematical Scientist 21, 73–84 (1996).(i)
  • [151] H.S. Green and T. Triffet, The Animal Brain as a Quantal Computer. J.Theor.Biol. 184, 385–403 (1997).(i)
  • [152] H.S. Green, Quantum Theory of Gravitation. Aust. J. Phys. 51, 459–475 (1998).(f)

Unpublished Manuscripts

(In Green Papers, Department of Physics and Mathematical Physics, University of Adelaide.)

  1. H.S. Green, On the Foundations of Mathematical Logic.
  2. H.S. Green, Generation of Small Random Numbers.

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