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Hugh Montgomery

Mathematical Physics is an exacting discipline. When someone spends the whole of his life working in this field he usually does so out of a deep sense of commitment, and a feeling that the choice was inevitable. However for the Irish scholar William Hamilton the exact opposite seems to be the case; as a young man his talents might have taken him in any of a number of directions, and it was only in his mature years that his commitment to science was complete.

William Rowan Hamilton was born in Dublin in 1805. He was the only son of Archibald Hamilton, a solicitor who was himself born in Scotland, although he had lived in Dublin for most of his life. When his son was three years old Archibald ran into political and financial difficulties, and William was sent to live with his uncle James Hamilton at Trim in County Meath, which lies about 30 miles to the north-west of Dublin. James was a curate in the Church of Ireland, and headmaster of the diocesan school at Trim; in this capacity he was responsible for the whole of William’s schooling. His educational methods seem to have been authoritarian but effective, and under his tutoring William soon developed a prodigious talent for languages. At the age of seven he was reading Hebrew, and this was soon followed by fluency in Latin and Greek; this laid the foundations for a life-long interest in classical literature. When he was 13 William claimed to have mastered one language for each year of his life; these included the major European languages, but he also had some grasp of Persian, Chinese and Hindi. (There was some idea that he might take up a post in the East India Company.) In 1823 he entered Trinity College, where he twice gained the Vice Chancellor’s Prize for English verse. In later life he developed a strong friendship with Wordsworth and to a lesser extent with Coleridge, and it was the latter who introduced him to the philosophy of Kant.

William had acquired an early taste for mathematics during his schoolboy days at Trim, and at the age of ten he devoured a Latin copy of Euclid’s Elements. James Hamilton did not discourage his nephew in this area, but he tried to limit his mathematical studies to books which were thought to be useful for the Entrance Examination at Trinity. Fortunately William escaped from his uncle for a time in the summer of 1822 when he visited Dublin, and he obtained much better guidance from Charles Boyton, the recently appointed Fellow in Mathematics at Trinity College. Ten years earlier the teaching of mathematics at Trinity had been radically reformed by Bartholomew Lloyd, who introduced continental methods at a time when Cambridge was still toiling in the Newtonian straitjacket. Boyton lent William a copy of the Mécanique Céleste of Laplace, and the 17 year-old found an error in one of the propositions. He also solved a problem in Analytical Geometry which was baffling Boyton at the time, and when William entered Trinity a year later his mathematical education was in good hands and advanced rapidly.

However at this stage in his career, it was still not clear to William that he was destined to be a mathematician. The usual aesthetic pursuit for a scientist is music, but this meant nothing to him at all. On the other hand he did think about a career as a classical scholar, and more seriously he considered devoting himself to English poetry. In 1825 he sent some of his verses to Wordsworth, whose advice was harsh but valuable. The writing of poetry required far more than poetic instincts, and young Hamilton would never succeed in that area. William also sent some of his poems to Arabella Lawrence, a friend of Coleridge, and she found them “obscure and unrevealing”. These criticisms were well taken on William’s part, and his reply to Arabella reveals clearly his youthful dilemma:

Despite his misgivings, Hamilton’s powers as a mathematician flowered brilliantly during his undergraduate years. He was extremely successful in examinations, but this did not prevent him from pursuing his own original thoughts. In fact all the work which later made him famous, particularly in geometrical optics and mechanics, germinated during this period. Shortly before he graduated he was made Professor of Astronomy at Trinity, which carried with it the position of Astronomer Royal and Director of the Observatory at Dunsink near Dublin. By modern standards this appointment is astonishing, and it was controversial even at the time; Hamilton had a clear potential in mathematical research, but he had no experience or skill in handling astronomical equipment. However the position suited him in a curious way, because it provided a steady income and a house at Dunsink, and it gave him time and opportunity to develop his mathematical ideas. Hamilton was in essence a lonely thinker, and the relative isolation of Dunsink was more congenial to him than the company of other mathematicians. For six years he shared the house at the observatory with his sisters, and in 1833 he married Helen Bayly and they brought up a family of two sons and one daughter.

It was during these early years at Dunsink that possibly his best work was carried out. He developed a new mathematical approach to geometrical optics, which led among other things to the prediction in 1832 of the phenomenon of conical refraction in biaxial crystals. When Humphrey Lloyd managed to demonstrate this effect two months later the scientific world was deeply impressed, partly because Hamilton’s work was purely abstract and made no appeal to empirical data. In 1835 this achievement won for him the Royal Medal of the Royal Society, and also a knighthood from the Lord Lieutenant of Ireland.

Hamilton then began to develop a formal analogy between the wave theory of optics and Newton’s theory of particle mechanics, which led to a radical reformulation of both subjects. In the nineteenth century this was regarded as a brilliant mental construct without much practical application; as Hertz pointed out Hamilton did not unite optics and mechanics in any physical sense, and the two subjects continued to develop independently. However this situation was completely transformed by the quantum revolution, and Hamilton’s formalism has become the foundation stone of the formal theory of quantum mechanics.

From the point of view of a modern physicist, Hamilton’s work is remarkable for its complete lack of an empirical base. He did not have, nor claim to have, the deep intimate knowledge of the physical world which one finds in a Faraday or a Maxwell. He was a metaphysical thinker, constructing purely abstract mental systems, and as a Kantian Idealist he believed hat these systems must clarify and illuminate the empirical world. Nor did he believe - and here Faraday and Maxwell would agree with him - that the primary aim of Science was to improve the material well-being of humanity. As the years passed the absurdity of his position as Astronomer Royal became increasingly apparent, and his work at the observatory more and more perfunctory. His conscience was smitten by critical reports from the Board of Fellows at Trinity, and in 1843 his friends made the eminently sensible suggestion that he should apply for the vacant Chair of Mathematics. However, this application was opposed by the Board of Fellows, on the grounds that his work at the observatory was not satisfactory. This surely is a perfect example of the Peter Principle, by which a man is always promoted up to the limits of his incompetence.

There were in fact deeper ironies in Hamilton’s position. He was devoting his life to the creation of abstract and some would say useless ideas, at a time when his country was passing through the most bitter and tragic period of its history. Safe in the Nirvana of Dunsink, Hamilton and his family were protected from the political riots in the city, and the Famine did not touch them except in a very peripheral way. Hamilton was a Tory and an Anglican, but his sensitivity would not allow him to try to justify the situation which was unfolding. In 1846 he wrote to his friend Aubrey de Vere, who was doing valiant but hopeless work in the soup kitchens and fever sheds of Limerick:

One feels that by this time Hamilton was completely in thrall to his mathematical daemon, and could not have abandoned it whatever the circumstances. Ever since 1830 he had been struggling with the problem of expressing geometry entirely in algebraic terms, a process which had begun 200 years earlier by Descartes. In a two-dimensional space, directed quantities or vectors can be added and multiplied easily enough, but there was no clear way to do the same for vectors in three-dimensions, or triples as Hamilton called them. It is said that each morning his elder son would ask him with the persistence of youth, “Can you multiply triples yet, Papa?”, at which he would shake his head sadly, and reply, “No, my boy; I can only add and subtract them.”

Hamilton was to wrestle with this problem for 13 years. On the 6^th October 1843 he was walking from the observatory into Dublin, where he was to preside at a meeting of the Royal Irish Academy. Somewhere on that walk inspiration struck. As he later wrote to his son:

i²  =  j²  =  k²  =  ijk  =  -1”

This passage has been quoted so frequently that it has acquired the status of an established myth. Unfortunately it was written 22 years after the event, in the year of Hamilton’s death; memory can play strange tricks when so much is at stake. An earlier account which he sent to P.G. Tait is similar to the above, but differs in a number of details. Hence it seems likely that Hamilton did have a sudden inspiration at about this time, and it might well have been associated with a walk from Dunsink into Dublin.

Hamilton spent much of the rest of his life developing and promoting his new discovery, to which he gave the name quaternions. Their mathematical structure was transparent, although they invoked the revolutionary new concept of factors which fail to commute - i multiplied by j was not the same as j multiplied by i. Hamilton’s great hope was that quaternions would be a vital key to the understanding of physics, but in this he was largely disappointed. One reason for this was that his book “Lectures on Quaternions”, which was published in 1853, was almost unreadable because of its obscurity and verbosity. The only physicist who actively supported the use of quaternions was Peter Guthrie Tait, Professor of Mathematics at Queen’s College Belfast and later of Natural Philosophy at Edinburgh. Nearly every other physicist felt that Hamilton (who was admired profoundly) had taken a wrong turning, and that quaternions were of no importance in physics.

Some of them were probably influenced by the fact that they did not share Hamilton’s metaphysical beliefs. Hamilton seems to have taken this response philosophically, and towards the end of his life he wrote in a letter to Tait:

Hamilton’s personal life was not a happy one. His wife Helen was a semi-invalid, and she was unable to give him the sympathy and understanding he needed so badly. Their relationship was far removed from his ideal of romantic love, and it must have caused a great deal of pain to them both. Even at the practical level Helen was unable to make his life comfortable, and she exerted no control over the band of slatternly Irish servants who ran their household. His study was said to resemble a pigsty. Hamilton died in 1865; he had never complained about his wife’s shortcomings, although for a number of years he had been sinking into serious alcoholism. This certainly undermined his health, but he maintained a prodigious intellectual output to the last. The parallels between Hamilton and Coleridge are considerable.

Sympathy for an individual should play no part in one’s appraisal of a scientific theory. It is however gratifying to find that more than a hundred years after Hamilton’s death, opinion is changing; some physicists (by no means all) feel that quaternion theory is a valuable tool in physics, and can be justified on strictly practical grounds. Many of Hamilton’s ideas have been transformed almost beyond recognition, but his contributions to physics are secure.

Bibliography

1. Thomas L. Hankins, Sir William Rowan Hamilton Johns Hopkins University Press. London 1980
2. Stuart Hollingdale, Makers of Mathematics Penguin Books. London 1989

Dr. John Roche

John Roche introduced his lecture by arguing that modern concepts of the electric and magnetic field contain layer after layer put down during the past three centuries. Almost all of the older concepts in physics are like this. Many of these older levels are still active to some extent but are buried deep in the intuition of the physicist. He maintained that history is the only way of systematically uncovering all of the invisible levels of meaning in a concept of present day physics.

He went on to describe briefly Kelvin and Maxwell’s mechanical ether theory; the theory of physical lines and tubes of force of Faraday, Poynting and J J Thomson; and the non-mechanical ether theory of Hendrik Lorentz. With the collapse of all ether theories in the 1920s, both Maxwell’s and Lorentz’s, ether theories of the field became untenable for the vast majority of physicists. Lorentz’s theory was now the most successful and highly regarded theory in advanced electromagnetism and physicists sought for a way of preserving its main insights while dispensing with his ether theory. The history of the emergence of a new post-ether theory has yet to be written and what follows is necessarily a very rough outline and aspects of this interpretation may be flawed. A compromise theory seems to have emerged which combined elements of the Faraday/Thomson field theory with Lorentz theory. It is difficult to establish that this merging and modification of the two parent theories was developed by any single individual. It seems almost to have been a collective endeavour by the physics community negotiated semi-intuitively in countless publications, conferences, classrooms, laboratories and workshops. This very slowly became the dominant interpretation. It is important to recognise that older theories, that of Maxwell, that of Faraday and that of Poynting and J J Thomson continued to live on side by side with the newly emergent theory, but fading slowly. Furthermore, fragments of older theories are still scattered about in odd corners of present-day electromagnetism and have never been incorporated into the new standard theory. Even at the end of the 20th century classical electromagnetism is not entirely finished in an interpretative sense and retains considerable elements of incoherence.

A tentative description of the new standard theory which emerged from the 1920s:
From Lorentz the theory retained the view that charges were primary and sources of the fields, both bound and free fields. The charges maintained the bound fields. Fields were agents of the charges in their actions upon other charges. Stationary charges produced the electric fields and moving charges produced magnetic and electric fields together. Charges and currents transmit the fields at the speed of light. All electromagnetic radiation fields are produced by accelerating charges, which produce both the electric and the magnetic components of these fields together. From Faraday the fields were thought of as pure forces existing in space, not as a material substance or as a process occurring in a material substance. From Lorentz the principle of superposition was adopted according to which fields do not act upon each other. They act only on charges. Lines of force do not repel each other, nor is there any tension along them.

Aspects of Lorentz theory were not adopted into the standard theory. According to Lorentz theory the induced electric field is directly caused by accelerating charges. According to Faraday theory the induced electric field is caused by a changing magnetic field. In the standard theory, therefore, some electric fields are caused by charges and other electric fields are caused by changing magnetic fields. Only as we approach the end of the century is the Lorentz theory beginning to be accepted, according to which all fields are caused by charges and fields do not cause other fields.

Again, in Lorentz theory there is no such thing as a displacement current but in the standard theory Maxwell’s displacement current is widely accepted. In Lorentz theory the lines of force are not independent physical entities, they are a mapping of force directions, only. The Faraday theory of independent physical lines of force seems to have survived until the 1960s when it gradually disappeared from physics. Even to-day, however, it can still be found occasionally - in the concept of the catapult field, for example, in elementary physics literature.

What is the relationship between electric and magnetic fields? Are the electric and magnetic fields independent entities or are they closely related? Einstein’s analysis has emphasised that they are closely related. Last century André Marie Ampère argued that there is no magnetic field, that there are electrostatic and electrodynamic fields, both electric fields. Special relativity encourages a similar view. For special relativity the magnetic field is a relativistic perturbation of the electrostatic field. Both fields are produced by electric charges. This seems to suggest that the magnetic field is not qualitatively different from the electric field but is only structurally and behaviourally different. This view has not gained universal acceptance in macroscopic physics. It also seems incompatible with a belief in magnetic poles.

Modern definitions of the field
William Thomson in 1851 defined the magnetic field as follows: Any space at every point of which there is a finite magnetic force is called a field of magnetic force. According to A.F Abbott O Level Physics, 1978, the space surrounding a magnet in which magnetic force is exerted is called a magnetic field. The beauty of these definitions is that they say almost nothing about the nature of the field and are not, therefore, controversial. It is perplexing, however, to identify space itself with the field, rather than stating that there is a field in space. Indeed, some modern authors are more careful and state that at every point of space where there is, or could be, a magnetic force there is a magnetic field. This states where the field is but does not explain our concept of it. There is a remarkable reluctance in physics to state openly what our concept of the field is. It is buried deep within physical intuition.

A tentative description of the working concept of the field actually used by physicists was then offered as follows:

• Fields are transparent.
• Fields pass through one another without mutual interference. They are rather like sound waves and light waves in this respect.
• Fields are not material substances.
• Fields are pure forces or the potentiality for forces in space. They act directly on test charges as agencies of distant source charges.
• They are propagated at the speed of light from source charges.

Radiation fields, on the other hand, are commonly thought of as empirical entities, since we can see light and feel infrared radiation. Summing up, Dr Roche argued that for many physicists the working concept of the electric or magnetic bound field is of a hypothetical nonmaterial force condition which actually exists in space ready to act on any test charge which is placed there. Radiation fields are generally thought of as real. He also pointed out that some physicists think of the bound field as a mathematical artifact.

page last updated 24 November2012

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