At the De Havilland Aircraft Company, Joe was placed in the Vibrations Department under its Director, R.N. Hadwin, and for the following five years his wartime research centred on electronic instrumentation and different continuous systems and their electrical analogues. In this, his investigation of the mathematical character of complex systems, including air-screw engine combinations, vibration, propeller flutter, and mechanical impedance functions in continuous systems, yielded apparatus and methods of measurement which were then designed and developed by Departmental staff. It was sustained and demanding research that also involved lengthy experimentation and his presence on test flights to check the delicate accuracy of his measurements. Happily, he survived the single occasion when the plane plummeted suddenly to the ground and Joe and his pilot emerged, a little shocked and battered, but with the precious equipment intact.

He would publish his non-confidential statistical and mathematical
engineering research results as the work evolved in a range of scientific
journals: ‘Approximate probability distribution function for the sum of two
independent variates’, in *Journal of the Royal Statistical
Society* in 1942; ‘Rubber as an engineering material’ in the *Journal of the Institution of Production Engineers,* 1944; and his famous ‘Deformation of rubber-like materials’ in *Nature* that year. His paper, ‘Some practical
applications of rubber dampers for the suppression of torsional vibrations
in engine systems’, produced in association with his colleague R. Zdanowich
in 1945, was awarded the Hubert Ackroyd Prize of the Institute of Mechanical
Engineers and published in its *Proceedings*.

The range of his findings and their applications stretched broadly
and, with another departmental colleague, W.P. Fletcher, he published ‘Free
and forced vibrations in the measurement of dynamic properties of rubber’ in
the *Journal of Scientific Instruments*, 1945.
In this succession of papers, Joe made a crucial contribution to the
understanding of rubber-like materials and, active across the spectrum, rose
to become Assistant Director of De Havilland’s wartime Vibrations
Department. With Hadwin, he also produced an overview paper, ‘The
Measurement of Mechanical Impedances’, which brought together their wartime
research on the vibration characteristics of complex systems of air-screw
engines, combinations where individual impedances were known in advance.
Presented at the Sixth International Congress of Applied Mechanics in Paris,
it was published in the *Proceedings* in 1946.

Involved as he was on immediate questions and wartime imperatives, Joe’s reflective mind was also ruminating on larger questions of statistical mathematics, probability, and applications of probability to quantum theory, ideas that arose from his comprehensive pre-war Paris studies and from his evolving work on the theory and practice of vibrations and waves. The intellectual mode of a highly original research scientist struck root. From well outside the ivory tower of physics research, self-impelled and self-reliant, he turned his mind to research at the very forefront of physics, the challenging arena of the subatomic quantum world.

In the last years of the 19th century, the physics of atoms and
particles had entered a radically new phase with the discovery of X-rays and
radioactivity, together with J.J. Thomson’s experimental proof that the
electron found in the outer part of atoms was a particle. These discoveries
had revealed that atoms were not the smallest particles in the universe, and
transformed the way scientists thought about the little known micro-world.
By the century’s end, Max Planck had made the fundamental discovery that the
energy of the atom could not be given off continuously but was emitted in
discrete packets he named ‘quanta’. ‘Planck’s constant’ became a parameter
that signalled a constant quantum that controlled the quantity of all energy
exchanges of atomic systems.^{[1]} Such discoveries fostered a brilliant outburst of Nobel
Laureates in the 20th century and led to the emergence of quantum
mechanics.

In 1905, Albert Einstein, as well as releasing his special theory of
relativity, suggested that light should be regarded as a stream of
particles. Within seven years, Ernest Rutherford determined that atoms had a
nucleus with a positive charge, a hard kernel that was the ‘other partner’
of the negatively-charged electron. On the eve of World War I, Niels Bohr,
working at both Manchester and Copenhagen, had fashioned the Rutherford-Bohr
model which gave the world the iconic image of the atom with electrons in
orbit around the tiny central nucleus.^{[2]}

The Rutherford-Bohr model of the atom combined pieces of classical theory (the idea of orbiting electrons) and pieces of quantum theory (the idea that energy is emitted or absorbed only in discrete quanta) and offered a new approach to probing the little known quantum world. Vital new discoveries were embraced as they emerged — complementarity in de Broglie’s wave-particle duality in the mid-1920s and, critically, Werner Heisenberg’s uncertainty principle, which established that a quantum entity could not have a precise momentum and a precise position at the same time.

In that vibrant period of the mid-1920s, Erwin Schrödinger,^{[3]} addressing aspects of mathematical and quantum statistics and
statistical thermodynamics at Zurich University, used the mathematics of
waves and wave states in wave mechanics to calculate the atomic energy
levels of electrons in orbit around the atomic nucleus and advanced the
importance of the mathematics of waves as a new ingredient of quantum
mechanics.

Into this scene came the remarkable young figure of Englishman, Paul Adrien Maurice Dirac. Born in Bristol in 1902, the offspring of a French-speaking Swiss father and an English mother, Dirac had trained for his first degree in engineering and a second degree in mathematics at Bristol University. In 1924, he moved to Cambridge to undertake a doctoral degree, and, assigned to the supervision of lecturer in applied mathematics, Ralph Fowler, entered the world of quantum physics. It was less an entry than an assault. During a visit to Cambridge in 1925, Heisenberg had presented Fowler with an advance copy of his first paper on his matrix mechanics approach to quantum theory (his indeterminacy or uncertainty principle), which Fowler passed to Dirac. Using his exceptional mathematical aptitude, Dirac swiftly developed his own version of quantum theory based on operator algebra. Extending boldly, he visited the Institute for Theoretical Physics which Bohr had established in Copenhagen and demonstrated that both Heisenberg’s matrix mechanics and Schrödinger’s wave mechanics were strictly equivalent — and were special cases of his own operator theory.

Reproduced with permission from The Emilio Segrè Visual Archives, Niels Bohr Library, American Institute of Physics, New York.

By 1927, Dirac was a Fellow of St John’s College Cambridge and
university lecturer, offering the first university course in quantum theory.
The following year he found a celebrated equation that incorporated quantum
physics and the requirements of Einstein’s special theory of relativity to
give a complete description of the electron.^{[4]} This eponymous equation, regarded as his greatest
contribution, situated Dirac as the most creative physicist of his time. His
book *The Principles of Quantum Mechanics* (1930) was set to become a bible of the field; he was appointed to the
famous Lucasian Chair at Cambridge in 1932 at the age of 30 and, a year
later, shared the Nobel Prize for Physics with Schrödinger. In short, Dirac
had independently developed his own formulation of the standard theory of
quantum mechanics, adopting a ‘quantization scheme’ as an independent way of
relating classical to quantum mechanics.

From the late 1920s, other key international mathematicians and
mathematical physicists emerged to define the fundamental symmetry
structures and principles of quantum mechanics and to make initial
contributions to the development of the formulation of quantum mechanics in
phase space. They were, notably, the German mathematician, Hermann Weyl,
with his correspondence of `Weyl-ordered’ operators, Hungarian
mathematician, John von Neumann with his Fourier transform version of the
*-product, and Eugene Wigner’s introduction of the phase space distribution
function controlling quantum mechanical diffusion flow.^{[5]}

It would fall, however, to Joe Moyal, the ‘outsider’ from the British Mandate of Palestine (in conjunction or, as it would prove, in parallel with Weyl, Wigner and H. J. Groenewold) to make the connection of classical mechanics to quantum mechanics firm and transparent through a reformulation of quantum mechanics in phase space. This he did over several years and in the face of dogged resistance and criticism from Paul Dirac, beginning his attempt in 1940 and publishing his seminal and influential paper finally in 1949. The circumstances of this long odyssey are documented in his remarkable correspondence with Professor Dirac, produced in full in Appendix II.

There is evidence that Joe began his first overture on the topic when,
in 1940, he initiated contact and a discussion of his concept of ‘the
possibilities of a statistical basis for quantum mechanics’ with the highly
revered Dirac, then widely judged to be the greatest theoretical British
physicist of the century.^{[6]}

Writing to the eminent Lucasian Professor in 1944, Joe reminds him of
an early conversation the two had had late in 1940 on ‘a possible
statistical base for quantum mechanics’. It seems that, arriving in England
from France in June that year, he carried with him a draft of his earliest
ideas on the concept. Clearly, his thinking on the subject had expanded
during the ensuing wartime years in discussions with Maurice Bartlett and Dr
Harold Jeffreys,^{[7]} two scientists and probabilists he had gotten to know in
wartime, to the point where the idea had been aired with Professor Fowler at
Cambridge and, through him, conveyed to Dirac.

Sir Ralph Fowler FRS, was an important academic figure.^{[8]} He was a prolific researcher in the domains of statistical
mechanics and atomic physics, the author of *Statistical
Mechanics* and *Statistical Thermodynamics*,
and when the youthful Dirac joined him at Cambridge, he was the only
physicist there who grasped the recent development in quantum theory coming
out of Denmark and Germany. His role in steering Dirac’s first revolutionary
paper, `The Fundamental Equations of Quantum Mechanics’, into rapid print in
the *Proceedings of the Royal Society* in 1925
sited him as a man keenly alert to the changing context of discovery in
theoretical physics.

Joe, in his first letter to Dirac of 18 February, 1944 from Wigston in Leicester where he was stationed that year in connection with his work, wrote accordingly:

Professor Fowler has sent me a copy of his letter to Dr Bartlett in which he writes of his discussion with you and Dr Jeffreys regarding the possibilities of a statistical basis for quantum mechanics.

He suggests I should discuss the matter with you sometime and I should be glad to do so if you can spare the time. I can always manage to come down to Cambridge over a weekend if you will fix the date.

You will remember no doubt we talked about this in December 1940, when I was beginning to consider these ideas.

Yours sincerely

J.E. Moyal

‘Dear Moyal,’ Dirac wrote on February 21, ‘I should be glad to meet you any weekend. So choose any weekend you like.’

Their meeting on 11 March, 1944, at Dirac’s house in Cavendish Avenue,
Cambridge, reopened their discussion. But Dirac’s response to the thrust of
Joe’s presentation and his draft paper was apparently not enthusiastic. As
his biographer, Helge Kragh, points out, Dirac ‘did not consider the
probabilistic interpretation as something inherent in the quantum mechanical
formalism’, a point he stressed in the conclusion to his 1927 paper on ‘The
physical interpretation of the quantum dynamics’. There, he enunciated, ‘The
notion of probabilities does not enter into the ultimate description of
mechanical processes; only when one is given some information that involves
a probability … can one deduce results that involve
probabilities.’^{[9]}

But, reflecting further, and alert to Dirac’s critique, Joe returned to the task, communicating again with him on 26 June, 1944:

On thinking over the objections you raised when I last saw you to my statistical treatment of quantum mechanics, it has occurred to me that the difficulties are chiefly a question of interpretation … One of the difficulties of the theory is that the probability distributions obtained for p and q from single eigenfunctions, can take negative values except perhaps for the ground state. Only linear superpositions of eigenfunctions lead to defined positive probability distributions in phase-space. Now, as I explained in my paper, I consider the form I obtained for the phase-space distribution F (p,q) as in a way an extension, or rather, an exact form of Heisenberg’s principle of uncertainty, in the sense that it imposed not only the well known inequality from the dispersions of p and q, but a special form for their whole probability distribution. Perhaps, then, the fact that phase-space distributors corresponding to single eigenstates can take negative values may be interpreted as meaning that an isolated conservative atomic or molecular system in a single eigenstate is a thing that cannot be generally observed without contradicting this generalised principle of uncertainty. If this can be conceded, and no doubt physical arguments could be brought forward to support such a view, only statistical assemblies and distributions corresponding to linear superpositions of eigenfunctions such as F(p,q,t) is always greater than zero would be observable, and would have an objective reality.

In fact, I regard such dynamical problems as one case where the methods I have outlined may have an advantage over the usual methods. Furthermore, as the theory leads to the distribution of phase-space, and also to correlations at two instants of time, there is a possibility it may lead to experimental verification in the field of electron and molecular beams. Another field where I think the theory may be of some value is in the study of statistical assemblies, since it leads to phase-space distributions for p and q, (equivalent to the Maxwell-Boltzmann distribution) for Fermi-Dirac and Bose-Einstein assemblies. This may be of value in the kinetic theory of non-uniform fluid.

Dirac remained silent, and there is no reply from him in the
Dirac-Moyal Correspondence. At this point, evidently, neither of the two
correspondents was aware that this very distribution in phase-space,
F(p,q,t), had been independently invented by Wigner in a paper, published in
1932, in which he comments on the negative values as a genuine quantum
mechanical peculiarity.^{[10]}

Nine months later, however, (for Dirac was also heavily engaged in war work for the government on uranium separation relating to the construction of atomic bombs) he re-opened communication.

‘Dear Moyal’, he wrote, on 19 March, 1945:

Some work I have been doing lately is connected with your work on a joint probability distribution F (p q t) and has led me to think that there may be a limited region of validity for the use of a joint probability distribution. However, I have rather forgotten the details of your work and would be glad if you could let me see again the part of it dealing with F (p q t). I may get a more favourable opinion of it this time. Have you done any more work on it since our previous correspondence?

In his response of 22 March, Joe, noting that the paper on his work was with Professor Chapman at Imperial College, referred Dirac meanwhile to Maurice Bartlett, ‘back at Queen’s’ and ‘familiar with my work’, who, having ‘worked out a new and improved method of obtaining a joint distribution’, should be able, if desired, to furnish Dirac with any explanations. For his part, though very busy with his engineering research, Joe added, ‘In collaboration with M S Bartlett, I have also carried further the treatment of the harmonic oscillator in phase space.’ ‘I have also,’ he continued, ‘been considering applications to statistical mechanics which, since they require distribution in phase-space, would seem to offer an obvious field for the theory. But apart from equilibrium distributions, I rather hope that the application of the theory of random functions will also lead to methods generally suitable for non-uniform states and fluctuation problems.’

Dirac’s reply a month later, on 20 April, 1945, was again far from encouraging:

Dear Moyal,

Thanks for sending me your manuscript again. The situation with regard to joint probability distributions is as follows, as I understand it.

A joint distribution function

F(p,q)should enable one to calculate the mean value of any functionf(p,q)in accordance with the formula(1)

I think it is obvious that there cannot be any distribution function

F(p,q)which would give correctly the mean value of anyf(p,q), since formula (1) would always give the same mean value forpqand forqpand we want their means to differ byiħ. However one can set up a d.f.F(p,q)which gives the correct mean values for a certain class of functionsf(p,q). The d.f. that you propose gives the correct mean value for , forτandθany numbers, but would not give the correct mean value for other quantities, e.g. it would give the same mean value for , whereas we want this second quantity to be times the first. In some work of my own I was led to consider a d f [distribution function] which gives correctly the mean value of any quantities of the form , i.e. all thep’s to the left of all theq’s in every product. My d.f. is not a real number in general, so it is worse than yours, which is real but not always positive, but mine is connected with a general theory of functions of non-commuting observables.

Dirac’s position was firm. From contemporary analysis, however, his
reply indicated that he neither understood nor believed a phase space
approach to be a possibility. Dirac was confusing commuting p and q
variables with noncommuting operators, P and Q as Joe explains in his
subsequent rebuttal.^{[11]} Moreover, Dirac did not appreciate the mathematical
implications of Weyl’s correspondence, namely that it gave a formula for any
quantum mechanical observable (in more mathematical terms) an expression for
any hermitian operator.^{[12]}

Satisfied with this dismissal, and committed to his own interpretation
of non-commuting observables in the paper he was preparing for *Reviews of Modern Physics*, ’On the analogy between
quantum and classical mechanics’,^{[13]} Dirac proposed that he refer to Joe’s work somewhat in these
terms:

The possibility of setting up a probability for non-commuting observable in quantum mechanics to have specified values has been previously considered by J.E. Moyal, who obtained a probability for a coordinate

qand a momentumpat any time to have specified values, which probability gives correctly the averages of any quantity of the form , whereτandθare real numbers. Moyal’s probability is always real, though not always positive, and in this respect is more physical than the probability of the present paper, but its region of applicability is rather restricted and it does not seem to be connected with a general theory of functions like the present one.

‘Do you think', Dirac asked, ‘this reference would correctly describe your work and do you object to such a reference?’

Joe’s reply of 29 April 1945,^{[14]} built on rising frustration, was robust:

If I understand correctly your remarks concerning joint probability distributions, you consider them as functions of the non-commuting variables P, Q which will give correct averages for certain classes of functions of the latter … Such functions may of course prove extremely useful mathematically, but they can hardly be called probability distributions in the ordinary sense.

My approach to this problem has been entirely different. I have looked for a probability distributions in the ordinary sense, which will be a function of the ordinary, commuting variables p, q. Its connection with functions of the corresponding non-commuting operators P, Q of quantum mechanics, is that it should give correct means for such of these functions (i.e. Hermitian operators) as are formed to represent physical quantities. If a physical quantity is given in classical mechanics by a function M (p,q), (i.e. a Hamiltonian, or an angular momentum) a Hermitian operator M (P,Q) is formed to represent it according to certain rules. I have looked for an F (p,q) such that it will always give …

It is obvious that such a function F (p,q) should be connected with a unique method of forming the quantum mechanics operators from the corresponding classical mechanics functions of p and q (I am speaking of course, of the classical quantum mechanics for particles without spin). A first test for the correctness of such an F (p q), will therefore be that the corresponding method of forming operators should give correctly at least all the known Hermitian operators of the theory since a general method for forming these operators is not generally agreed upon in the standard theory.

The F (p, q) which I propose in my paper fulfils these conditions … It is consequently incorrect in my view to say that the F (p, q) in my paper will give correct averages only for functions of the form . Actually, it will give the right averages for all the Hermitian operators considered in the classical quantum mechanics of particles without spin, e.g. Hamiltonian, angular momentum, total angular momentum, radial momentum, etc.

Believing that Dirac’s proposed reference limited the range of applicability of his work, Joe protested:

I do not … think that your reference to my work gives a correct description of it. It is certainly not correct in my view to say that form (2) for

F(p,q)is limited to giving correctly averages for quantities of the form ; in fact, it will give averages for all observables formed as in (3), and this includes as far as I know, all the observables ordinarily considered in classical quantum theory.^{[15]}

‘This would perhaps not matter a great deal,’ he continued, in a manner that pulled no punches: ‘if my work was already published, since readers could then refer to the original. I have not however been able so far to arrange for its publication, due largely, as you will no doubt remember, to your veto which made the late Professor Fowler hesitate about presenting it to the Royal Society. Your criticism is thus left without an answer. Your objection at the time, if I remember rightly, was chiefly that joint distributions for p and q had no physical meaning and consequently no validity or usefulness. I am glad to notice that you now think they open up an interesting field of research.’

With spirit and courtesy, Joe had, he thought, settled this wave of reservations satisfactorily. Yet, as an academic outsider pinning his hopes of a research career on his research achievement, his frustration was real:

Regarding your query as to whether I shall be able to do further work on this subject,’ he concluded, `my main difficulty is again the fact that my existing work is not yet published … It is also discouraging to accumulate for years unpublished results as I have been doing … The papers you have seen represent my first real effort at research in pure mathematics and theoretical physics; I was hoping that their publication would eventually enable me to transfer my activities entirely from the field of research in engineering and applied physics to that of pure science, and do some serious work on theoretical physics. Failure to obtain publication has forced me to adjourn such plans

sine die, and my present work is leaving me less and less time for pure research.

Joe Moyal had run against a paradigm. Dirac, a man of pre-eminent
reputation, the most esteemed figure of quantum mechanics in Britain, held
an entrenched and dominant position within the discipline. He himself had
always conducted his research at Cambridge on his own — in contrast to his
European colleagues, who had the advantage of both formal and informal
collaboration — and was, from his earliest endeavours, exacting,
introspective and tenacious in his confidence of his own views. With some 64
research papers behind him in 1945 and his foundation book, he appeared, as
one distinguished mathematician has noted, `intellectually incapable of, and
unwilling to give ground’.^{[16]}

In two subsequent letters, on 11 and 18 May, 1945, Dirac again resisted Joe’s position, attempted to show that his argument was trivially wrong, and appeared not to fully appreciate the underlying Weyl correspondence principle and the relation of Joe’s theory to it.

‘In Bartlett’s paper which you just sent me,’ Dirac argued, on May 18,
‘the quantum values for the energy of the harmonic oscillator are assumed
and the correct value for was obtained because of this assumption. You can
always get the right answer by borrowing sufficient results from the
ordinary quantum theory. The true test of a theory is whether it always
gives consistent results whichever way it is applied, and my way of
evaluating , given above shows that your theory does not always
give consistent results.’ However, stirred in part, perhaps, by Joe’s
heartfelt charge over publication, Dirac suggested, ‘I would be willing to
help you publish if you would change it [your presentation] so that it does
not contain any general statements which I think to be wrong.’^{[17]}

In this contest of opinion, the persistence of the two protagonists testifies to the importance of the sustained debate. Fearless as an outsider, Joe defended his position. In his letter of 25 April, 1945, he conveys the essence of his theory and its equivalence to classical wave mechanics:

If, as I think, this equivalence is correct, then the theory should lead to correct results for the various quantities obtained by wave mechanics, such as frequencies and transition probabilities, even when dealing with negative functions F(p, q). The appearance of the latter should then be taken to mean that the situation is such that simultaneous prediction of the value of p and q is impossible, but would not impair the calculation of other experimentally determinable quantities.

‘Summarizing,’ he concluded, on 15 May, 1945, ‘I think it would be fair to say that my paper gives a derivation of classical quantum mechanics on a purely statistical basis (plus Newtonian mechanics) which is equivalent to the standard matrix theory with the addition of Weyl’s postulate for a quantum kinematics [Moyal’s underlining] and furthermore that it shows the consequences such a theory entails with regards to the problems of determinism, probability distributions, fluctuations, quantum statistics etc. Would you agree to this position?’

Joe’s tenacity owed something to his `Israeli’ background: he was not easily intimidated. He also had faith in the rigour of his mathematical formulation. In a further letter to Dirac on 26 May, 1945, he asserted vigorously:

I don’t think your remark on [my] getting the right answer 'by borrowing sufficient results from the ordinary quantum theory' quite fair. In so far as my theory is equivalent to the ordinary theory, it leads to the same eigenvalues for the mean of the energy, as I have shown in my paper. In order to prove an inherent inconsistency in my theory one would have to show that the method you use follows necessarily from my basic postulates, but this is not the case. My method on the other hand is based on a theory for statistical assemblies resulting from these postulates. As such it is quite consistent with the rest of the theory, and also appears to lead to correct results.

To no avail. In further communication, on 6 June, 1945, Dirac returned to the problem of dispersion of energy in a stationary state which he saw as ‘the simplest example which shows the limitation of your theory’.

It was a strenuous contest for an independent scientist on the edge of a research career, fought out firmly, point by point. Yet the explicit persistence of Joe’s challenge, no doubt rare in Dirac’s illustrious career, had some effect. In June 1945, the high priest of physics, was offering — with the limitations of his reservations clearly stated — to help Joe to publish his work, divided into two parts.

‘The quantum theory part of your work,’ he advised, ‘could form a
paper which I could communicate to a scientific journal. With regard to the
remainder, I do not know how much of it represents new research and how much
is an exposition of known results. What did Fowler say?’^{[18]}

Joe’s answer was precise. ‘My work on Random Functions is new,’ he
replied on 17 June, 1945. ‘The late Professor Fowler’s original intention
had been to present the whole work for publication in the Proceedings of the
Royal Society as three separate papers which I then intended to condense to
three papers of 15 or 20 pages each.’ A week later, Dirac renewed his offer.
‘If it does not divide naturally [into two parts],’ he wrote, ‘probably the
Proc. Roy. Soc. is the best journal for them.’^{[19]}

Joe, however, was happy to agree to Dirac’s suggestion to arrange his
material in two parts. ‘I am now rewriting the part of my work on quantum
mechanics as a separate paper,’ he told Dirac in a letter from London of 10
July. ‘As regards the rest, I am rewriting it as a paper in two parts, which
could then appear either separately or together, whichever is more
convenient.’^{[20]}^{}His relief at the outcome of so
stiff a contest emerges in his conciliatory final paragraph:

I enclose some notes in which I have tried to develop a method which could overcome the difficulty about non-zero dispersions for eigenvalues in my theory and also extend in character, and there are several things I still want to clear up, but I should be glad in the meantime to have your opinion on this development. I also enclose some notes comparing the results in your paper with mine.

Dirac’s response was to invite Joe to attend the weekly Colloquium at
Cambridge. `We would be glad,’ wrote the `holy Dirac’ (as Schrödinger dubbed
him), ‘if you could come to any of them.’ In his last letter, six months
later, he pressed Joe to give a talk on his quantum theory work. ‘I think it
would be a good idea to have it discussed,’ he wrote, ‘if you do not mind
possible heavy criticism.’^{[21]}

Characteristically, Dirac showed no concession to Joe’s views in the
paper he published in the April-July issue of *Reviews of Modern Physics* of
1945,^{[22]} ‘On the Analogy between Classical and Quantum Mechanics,’ much
of which lay at the heart of their long discussion. Here, he opened his
argument working with noncommuting variables, which must have underlain his
resistance to the simpler Moyal approach in phase-space.

J.E. Moyal Papers, 45/3, Basser Library, Australian Academy of Science, Canberra.

‘In the case when the non-commuting quantities are observables,’ he
wrote, ‘one can set up a theory of functions of them of almost the same
generality as the usual functions of commuting variables and one can use
this theory to make the analogy between classical and quantum mechanics.’
Here, too, in the body of the text, (despite their subsequent detailed and
contested correspondence), he referred to Joe’s work exactly as he had
specified it in his letter to Joe of 20 April, 1945, a description which Joe
had strenuously rejected. In one of his rare references to a contemporary
researcher (outside the tried and true band of Heisenberg, Jordan, Pauli and
Born), he added: `This work is not yet published. I am indebted to J.E.
Moyal for letting me see the manuscript.’^{[23]}

Paul Dirac had held on to Joe Moyal’s manuscript for many months. In the event, however, Joe was right. Professor Alan McIntosh, former Head of the Centre for Mathematics and its Applications at The Australian National University, noted after reading the correspondence:

Joe had come up with a sound formulation of quantum mechanics, the phase space approach … But Dirac didn’t take Joe’s theory seriously; he didn’t understand it; he didn’t think it possible … and he contradicts himself … Joe is putting forward an entirely different formulation of quantum mechanics [from the Schrödinger and Heisenberg formulations], a formulation which he is claiming is equivalent to the others and more useful in solving evolution equations, how the system evolves from time to time — and this is precisely why his work and his statistical method is being used so widely today.

^{[24]}

Similarly, Dr John Corbett, emeritus quantum physicist at Macquarie
University, notes succinctly that the Dirac/Moyal correspondence reveals
‘not only how new ideas and approaches are only accepted reluctantly; but
how even very good scientists can read their own problems into another’s
work’. For Corbett, Dirac was too concerned with the quantization problem.
He also judged, in respect of Dirac’s criticism of Joe’s quantization method
(deriving as it did from Weyl), that Dirac’s own method did not give a
one-to-one correspondence between a classical quantity and its quantum
counterpart. ‘Dirac,’ he concludes, ‘failed to yield answers and throughout
played his cards close to his chest.’^{[25]}

Professor Curtright at the University of Miami suggests that it is noteworthy that the 'Moyal bracket' is not discussed in the two men’s correspondence.

If it had been, one would have expected Dirac the formalist to pick up the technical sweetness of the construction. But the bracket is certainly there in the final published paper of Moyal, so if Dirac did read the final version he must have seen it. Yet, even if Dirac did not read the necessary part of the paper, it is a key component of Moyal’s work which was later canonized, so Dirac must have been aware of the bracket subsequent to the publication of Moyal’s papers. Yet again, there appears to have been no response on Dirac’s part.

^{[26]}

Importantly, during 1946, Joe had been alerted to Wigner’s paper of
1932, which was unknown to him (and clearly unknown to Dirac to whose
attention he had brought it) when he worked out his theory. At the time he
devised his theory, he sought the distribution that would yield quantum
expectation values most compactly. ‘I tried to look for a more direct
generalization, which was much nearer to the original form of classical
mechanics in Hamiltonian form,’ he said in interview in 1979. ‘I was not aware that Eugene Wigner had already done
something on that in a very brief paper on statistical mechanics in the
early 1930s. It was brought to my attention later. So I developed and worked
out the whole thing by myself. I worked out the whole formalism later with
lots of applications and developed it further and more rigorously.’ However,
given Wigner’s earlier part (which Joe included in his published paper’s
references), the theory became known as the ‘Wigner-Moyal
formalism’.^{[27]}

Equally significantly, during his revisions Joe also had occasion to
communicate with Hilbrand Groenewold^{[28]} in Holland (and with J. Bass) who `had studied the same
subject independently’, and profited from correspondence with them.
Groenewold’s paper, based on his Ph.D. dissertation and published in 1946,
came to the subject with a foreknowledge of Wigner’s earlier work. It
systematically developed the Weyl correspondence and arrived at similar
mathematical constructions to Joe’s — from a different point of view. He
also developed the *-product, whose antisymmetization comprises the Moyal
Brackets.

Joe Moyal’s separate paper ‘Quantum Mechanics as a Statistical Theory’
was submitted to the Cambridge Philosophical Society in November 1947 from
his post at the Department of Mathematics and Physics of The Queen’s
University, Belfast, and was presented for publication in the *Proceedings* by M.S. Bartlett. In his acknowledgements,
the author made public his indebtedness to Professors Dirac, Harold Jeffreys
and Ralph Fowler for their criticisms, suggestions and arrangements and,
most warmly, to Maurice Bartlett for his many invaluable communications and
discussions and results incorporated in the text. He also acknowledged
correspondence with H.J. Groenewold^{[29]} and J. Bass in 1949.

Given the subsequent enormous influence of the paper, the terms that flowed from it and the stimulus it gave to a spread of multi-disciplinary research, it is worth citing its eloquent introduction:

Statistical concepts play an ambiguous role in quantum theory. The critique of acts of observation, leading to Heisenberg’s 'principle of uncertainty' and to the necessity for considering dynamical parameters as statistical variates, not only for large aggregates, as in classical kinetic theory, but also of isolated atomic systems, is quite fundamental in justifying the basic principles of quantum theory; yet, paradoxically, the expression of the latter in terms of operations in an abstract space of 'state' vectors is essentially independent of any statistical ideas. These are only introduced as a post hoc interpretation, the accepted one being that the probability of a state is equal to the square of the modulus of the vector representing it; other and less satisfactory statistical interpretations have also been suggested.

^{[30]}

One is led to wonder whether this formalism does not disguise what is an essentially statistical theory, and whether a reformulation of the principles of quantum mechanics in purely statistical terms would not be worth while in affording us a deeper insight into the meaning of the theory. From this point of view, the fundamental entities would be the statistical variates representing the dynamical parameters of each system; the operators, matrices and wave functions of quantum theory would no longer be considered as having an intrinsic meaning, but would rather appear as aids to the calculation of statistical averages and distributions. Yet there are serious difficulties in effecting such a reformulation. Classical statistical mechanics is a 'crypto-deterministic' theory, where each element of the probability distribution of the dynamical variables specifying a given system evolves with time according to deterministic laws of motion; the whole uncertainty is contained in the form of initial distributions. A theory based on such concepts could not give a satisfactory account of such non-deterministic effects as radioactive decay or spontaneous emission. Classical statistical mechanics is, however, only a special case in the general theory of dynamical statistical (stochastic) processes. In the general case, there is the possibility of `diffusion’ of the probability 'fluid', so that the transformation with time of the probability distribution need not be deterministic in the classical sense. In this paper, we shall attempt to interpret quantum mechanics as a form of such a general statistical dynamics.

In concluding the paper, Joe took the opportunity to deal confidently
with Dirac’s obstinate resistance to the introduction of an additional
postulate on the form of the phase space distribution, ‘the equivalent to a
theory of functions of non-commuting observables’. ‘Dirac’, he writes of the
physicist’s *Reviews of Modern Physics* paper of 1945,
‘has given a theory of functions of non-commuting observables which differs
from the one obtained in section 5 of this paper; it has the advantage of
being independent of the basic set of variables, but, as might be expected
from the foregoing discussion, it leads to complex quantities for the
phase-space distributions which can never be interpreted as probabilities.’
(p. 119) With his paper in proof form, he also added to his references
Richard Feynman’s recently published paper in *Reviews of Modern Physics* (1948), 20,
377–87.

The reconstitution of this historical controversy is illuminating for
the light it sheds on a hitherto unknown piece of the history of quantum
theory. J.E. Moyal first came to public attention in the brief allusion to
his unpublished research in Dirac’s paper of 1945. Yet the background to
that allusion marks one of the most extensive correspondences Paul Dirac
engaged in relating to any one of his research contributions.^{[31]}

Operating as he was in a very small, tight, highly competitive
research community in quantum mechanics, Dirac was not given to discursive
overtures. An inveterate self-referencer, he eschewed even the practice of
courtesy referencing and ignored the work of upcoming men. Yet on this
occasion he carried on a protracted correspondence — albeit at times a
stubbornly tendentious one — for some 18 months or more with a researcher
outside academia, from off-field. In it he stamped himself as intellectually
self-protective, reluctant to step outside the intellectual framework he had
devised, a man whom his biographer, Helge Kragh, has characterized as one
who, having developed the celebrated standard theory of quantum mechanics,
was satisfied that the theory was complete and his methodology appropriate
for further development.^{[32]}

The opinion of two American physicists reputed in the field and who
have studied the correspondence, offers an informed scientific
judgment.^{[33]} As Professor Thomas Curtright of the Department of Physics at
the University of Miami sums up: ‘the letters definitely show Dirac to be
wrong about some really basic points in quantum mechanics. That by itself is
most remarkable. But then they also show that Dirac is basically unfair and
incredibly stubborn.’ Indeed, he adds, `it is stunning to a reader
well-versed in quantum mechanics that Dirac — the master formalist — makes
such *silly* mistakes and commits them in
writing for all posterity.’ Concomitantly, Dr Cosmas Zachos of the Division
of High Energy Physics at Argonne National Laboratory contends, ‘Moyal’s
innovations are now seen to be compatible with this methodology, and it is
puzzling why Dirac did not jump at the opportunity to embrace them. Even
after publication of Moyal’s and Groenewold’s papers which established the
Moyal Bracket as the proper generalization of the Poisson bracket (an object
which Dirac himself had analogized to Quantum commutators) he still failed
to acknowledge this essential completion of his own proposal.’ For
Curtright, the correspondence also exposed the point that ‘Moyal deserves
full credit for having the insight to look at quantum mechanics in terms of
distributions on phase-space completely independently of Wigner’.

Joe himself knew he had fought a singular fight and, while averse to
keeping personal correspondence, he preserved this correspondence for
posterity. He would absorb his substantial other material in his `Stochastic
Processes and Statistical Physics’, published in the *Journal of
the Royal Statistical Society* subsequently. In a later interview,
however, he declared, ‘my first paper really contained all the essentials of
the formalism, the version of quantum which is an equivalent of older
mechanics.’^{[34]}

‘Quantum Mechanics as a Statistical Theory’ proved to be far ahead of its time. Slow to move, received as it was initially by a small range of researchers in quantum fields, it gathered expanding range and impact from the 1960s as the international research community grew, until it exploded into high prominence in an evolving series of mathematical and practical applications nearly half a century after its publication.

The paper’s route to publication had proven long and challenging from its embryonic beginnings in the early 1940s. But, as Henri Poincaré, writing on mathematical creativity, once pertinently observed, `Ideas lock into the brain and are stirred but not replaced by interruption’.