I shall attempt to give you some idea of how a theoretical physicist works — how he sets about trying to get a better understanding of the laws of nature.

One can look back over the work that has been done in the past. In doing so one has the underlying hope at the back of one's mind that one may get some hints or learn some lessons that will be of value in dealing with present-day problems. The problems that we had to deal with in the past had fundamentally much in common with the presentday ones, and reviewing the successful methods of the past may give us some help for the present.

One can distinguish between two main procedures for a theoretical physicist. One of them is to work from the experimental basis. For this, one must keep in close touch with the experimental physicists. One reads about all the results they obtain and tries to fit them into a comprehensive and satisfying scheme.

The other procedure is to work from the mathematical basis. One examines and criticizes the existing theory. One tries to pin-point the faults in it and then tries to remove them. The difficulty here is to remove the faults without destroying the very great successes of the existing theory.

I am honoured to speak about P. A. M. Dirac whom we all loved and whom I so greatly admired. I am also glad to see so many friends in the audience. As an old Johnian myself, I would particularly like to mention Sir Harry Hinsley, the Master of St. John's (Dirac's College). Sir Harry is an eminent historian. To him I shall address my remarks, so as to assure you all that you will be spared as many technical details as possible.

Paul Adrian Maurice Dirac was undoubtedly one of the greatest physicists of this or of any century. In three decisive years, 1925, 1926, and 1927, with three papers, he laid the foundations, first of the Theory of Quantum Mechanics, second of the Quantum Theory of Fields, and third — with his famous equation of the electron — of the Theory of Elementary Particles. (In the course of this lecture, I shall explain the relevant concepts of the Quantum Theory of Fields and the Dirac equation for the electron.) When one met Dirac, one could see the complete and utter dedication of a great scientist. One could feel with him the pleasure of scientific creation at its noblest, and the highest personal integrity.

A few years ago I described a simple device that reveals in a very elementary way the extremely perplexing character the data from the Bohm–Einstein–Podolsky–Rosen experiment assumes in the light of the analysis of J. S. Bell. There is a second, closely related form of that gedanken demonstration, which I would like to examine for several reasons.

1. It is simpler: there are only two (not three) settings for each switch.

2. The gedanken data resemble more closely the data collected in actual realizations of the device.

3. None of the possible switch settings produce the perfect correlations found in the first version of the gedanken demonstration, where the lights always flash the same color when the switches have the same setting. Since absolutely perfect correlations are never found in the imperfect experiments we contend with in the real world, an argument that eliminates this feature of the ideal gedanken data can be applied to real data from real experiments. (If you believe, however, along with virtually all physicists, that the quantum theory gives the correct ideal limiting description of all phenomena to which it can be applied, then this is not so important a consideration.)

4. Because the ideal perfect correlations are absent from this version of the gedanken demonstration, one is no longer impelled to assert the existence of impossible instruction sets. To establish that the new data nevertheless remain peculiar, it is necessary to take a different line of attack, which has again intriguing philosophical implications, but of a rather different character.

The modified demonstration

In the modified gedanken demonstration there are only two switch settings (1 and 2) at each detector.

Over fifty years ago Einstein, Podolsky, and Rosen published a striking argument that quantum theory provides only an incomplete description of physical reality – that is, that some elements of physical reality fail to have a counterpart in quantum-theoretical description. What made this a challenge to the quantum-theoretical Weltanschauung was the very mild character of the sufficient condition for the reality of a physical quantity on which their argument hinged: “If, without in any way disturbing a system, we can predict with certainty the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity.” The argument consists in pointing out that it is possible to construct a quantum-mechanical state ψ with the following properties:

1. The state ψ describes two noninteracting systems (I and II). In almost all discussions the two systems are taken to be two particles, and the absence of relevant interaction is built in by taking ψ to assign negligible probability to finding the particles closer together than some macroscopically large distance.

2. By measuring an observable A of system I, one can predict with certainty the result of a subsequent measurement of a corresponding observable P of system II.

3. If, instead, one chooses to measure a different observable B of system I, one can predict with certainty the result of a subsequent measurement of a corresponding observable Q of system II.

4. The observables P and Q are represented in quantum theory by noncommuting operators, which means that they cannot both have definite values.

This delightful volume does for most of one's favorite special functions what arithmetic does for the exponential: it displays them not as solutions to differential equations but as matrix elements of representations of elementary Lie groups.

If you feel deeply that there is nothing more to say after pointing out that ex is the solution of f′ = f(withf(0) = 1) and do not care that it is also 2.71828 … multiplied by itself x times, then you also might feel inclined to ignore this book. This would be a pity, because almost half is devoted to the kind of general exposition of Lie groups, Lie algebras and their representations that few mathematicians are capable of treating us to. It is an exposition that prefers words to symbols, is intuitive and does not require us to scale a mountain when we only want to peek over the garden hedge. Knowing only a little linear algebra, one can read this much with joy and enlightenment without compromising one's devotion to differential equations.

If, however, you have always suspected that there was more to Bessel functions, Gegenbauer polynomials, associated Laguerre polynomials, and Hermite polynomials than Morse and Fesh bach or Watson were letting on, then you can indulge in the feast as well as the cocktails. And there, in the algebraic paté, you will come upon those higher transcendental truffles you long ago snuffled in the garden of analysis. Only a hard man could fail to be moved by this transformation, and you can see what it has done to a soft-hearted esthete like your reviewer.

Greenwood has given a derivation of the relativistic addition law for parallel velocities that makes no explicit use of the Lorentz transformation equations, relying instead on direct applications of length contraction and time dilation. This kind of approach is necessary in a general education physics course, where more abstract derivations can only obscure the physics. Arguing from the Lorentz transformation in such courses is as inappropriate as trying to teach school children elementary geometrical facts about circles and triangles by starting from their algebraic representations in Cartesian coordinates.

In this spirit I would like to describe another derivation of the addition law that dispenses not only with the Lorentz transformation, but also makes no use of length contraction, time dilation, or the relativity of simultaneity. This argument extracts the result as an immediate consequence of the constancy of the velocity of light.

As in Greenwood's gedanken experiment, we consider a sequence of events taking place on a long straight uniformly moving train. I shall describe all the events from the viewpoint of a frame of reference S in which the train moves parallel to its own length with speed u. All references to speeds, distances, lengths, and times refer to their values in the frame S.

At a certain instant a photon (speed c) and a massive particle (speed w less than c, but greater than u) begin a race from the rear of the train to the front. The photon, of course, wins, taking a time T to reach the front.

Stirling's approximation to the factorial function, n!∼(2πn)½(n/e)n (1) plays a central role in any number of investigations of statistical physics, and is invaluable in the kinds of simple probabilistic studies that can convey to students in a general education course the nature of entropy and irreversibility. Unfortunately, the usual derivations of (1) are inaccessible to such students and even to many beginning physics majors. One can, of course, simply verify its remarkably accurate performance, but the better students are bound to find this frustrating: Why is it that Stirling's formula works as well as it does?

I provide here an elementary answer to this question that can be adapted to give convincing explanations at a range of levels of mathematical innocence. For the crudest argument it is only necessary to know the elementary definition of the number e that arises in the theory of compound interest. Students who also know that the natural logarithm has the expansion

can be given a really intimate glimpse into the workings of Stirling's formula, while those who are willing to approximate a few simple sums by integrals can acquire a level of understanding possessed, I suspect, by few professionals.

Stirling's formula begins to yield up its secrets with the observation that n! can evidently be written in the form

This can be written in the equivalent form,

which immediately calls to mind the definition of e as the limiting value for large.

I have to say that the only other time I was asked to talk at a commencement was 1952, when I graduated from high school. So while I suppose I should have spent the last month thinking hard about the great challenges lying ahead for all of you, I was actually more preoccupied with the great challenge lying ahead for me. What can a middle aged theoretical physicist have to say to the graduating class of this unique college?

The answer came to me a few weeks ago, when I read in a pamphlet about St. John's College that the principal goal of a liberal education is to acquire the skills of rational thought, careful analysis, logical choice, imaginative experimentation, and clear communication. Having always regarded these as the primary tools of the physicist, I realized that I could do no better than to call to your attention a few examples of the application of these skills in public affairs, in private life, and on the frontiers of science.

Let's begin with clear communication in public affairs. Several years ago I was half listening to an early speech by a new President who was acquiring a reputation as a clear communicator. Talking about a trillion dollar national debt, he was saying: “A trillion dollars is so much money that it's hard to grasp the idea, so I want to tell you how to make it a little more real.”

Once I tried to teach some quantum mechanics to a class of law students, philosophers, and art historians. As an advertisement for the course I put together the most sensational quotations I could collect from the most authoritative practitioners of the subject. Heisenberg was a goldmine: “The concept of the objective reality of the elementary particles has thus evaporated…”; “the idea of an objective real world whose smallest parts exist objectively in the same sense as stones or trees exist, independently of whether or not we observe them … is impossible …” Feynman did his part too: “I think I can safely say that nobody understands quantum mechanics.” But I failed to turn up anything comparable in the writings of Bohr. Others attributed spectacular remarks to him, but he seemed to take pains to avoid any hint of the dramatic in his own writings. You don't pack them into your classroom with “The indivisibility of quantum phenomena finds its consequent expression in the circumstance that every definable subdivision would require a change of the experimental arrangement with the appearance of new individual phenomena,” or “the wider frame of complementarity directly expresses our position as regards the account of fundamental properties of matter presupposed in classical physical description but outside its scope.”

I was therefore on the lookout for nuggets when I sat down to review these three volumes – a reissue of Bohr's collected essays on the revolutionary epistemological character of the quantum theory and on the implications of that revolution for other scientific and non-scientific areas of endeavor (the originals first appeared in 1934, 1958, and 1963.)

A major impediment to writing physics gracefully comes from the need to embed in the prose many large pieces of raw mathematics. Nothing in freshman composition courses prepares us for the literary problems raised by the use of displayed equations. Our knowledge is acquired implicitly by reading textbooks and articles most of whose authors have also given the problem no thought. When I was a graduate teaching assistant in a physics course for non-scientists, I was struck by the exceptional clumsiness with which extremely literate students, who lacked even the exposure to such dubious examples, treated mathematics in their term papers. The equations stood out like dog turds upon a well manicured lawn. They were invariably introduced by the word “equation” as in “Pondering the problem of motion, Newton came to the realization that the key lay in the equation

To these innocents equations were objects, gingerly to be pointed at or poked, not inseparably integrated into the surrounding prose.

Clearly people are not born knowing how to write mathematics. The implicit tradition that has taught us what we do know contains both good strands and bad. One of my defects of character being a preference for form over substance, I have worried about this over the years, collecting principles that ought to govern the marriage of equations to readable prose. I present a few of them here, emphasizing that the list makes no claim to be complete.

Imagine a New Yorker profile in the literary style of Krokodil, and you have this book. Not a biography of Lev Davidovich Landau, it is an attempt to portray the human and scientific style of this extraordinary man and the remarkable school of theoretical physics he single-handedly created and sustained. Written for the edification of youth, it suffers from that turgid sentimentality – the rapture of an elephant in heat – that seems to accompany all Soviet efforts at morally elevating prose. It should nevertheless be fascinating reading for anyone working along the trails first blazed by Landau himself, that is, for almost all physicists.

After two introductions (why two? why not, I suppose, but this is typical of the literary anarchy that follows) Anna Livanova gives us a tantalizing glimpse of Landau before the age of sixteen, when he entered the university in Leningrad. Aside from a nod to his wife (an engineer in a chocolate factory; love at first sight; a son, Igor, in 1946) the rest is, so to speak, history, and the history is told (and this is the saving grace of the book) primarily through a wonderful series of anecdotes and vignettes.

There are two main sections: an evocation of the school of Landau as it operated under the Master, and an attempt to convey in nontechnical terms the nature of his contributions to the theory of superfluid helium. The former is the better part.

This is the third edition of a little volume, first published in 1952, that traces the uses of the variational principle in physical science all the way from Hero of Alexandria to Schwinger of Massachusetts. The occasion for the new edition is the addition of an essay by Laurence Mittag, Michael Stephen, and Wolfgang Yourgrau on the hydrodynamics of normal fluids and superfluids, emphasizing variational formulations.

The book, however, is worth commenting on as a whole, for it has languished in an undeserved obscurity. It offers concise and elegant formulations of the great variational principles of optics, mechanics, electrodynamics, quantum theory (old and new), and now, hydrodynamics. Although each topic is skillfully presented in analytic terms congenial to the modern reader, the historical background is almost always preserved. This is a luxury almost universally abandoned by authors of scientific books, and though the reasons for this are obvious, the loss is quite possibly greater than many of us think. It is not just that the subject is thereby dehumanized; the accompanying loss of perspective on our own efforts, both as physicists and as people, is equally to be regretted. A trivial example is the discussion of Hamilton–Jacobi theory (the concise presentation of which, by the way, is the best I have seen this side of Landau and Lifshitz) which treats us to the spectacle of Jacobi, unable to resist the needle, in presenting his modest refinement of Hamilton's remarkable edifice (page 58): “I therefore do not know why Hamilton… requires the introduction of a function S of 6n + 1 variables … while, as we have seen, it is completely sufficient to…”

Attitudes toward quantum mechanics differ interestingly from one generation of physicists to the next. The first generation are the Founding Fathers, who struggled through the welter of confusing and self contradictory constructions to emerge with the modern theory of the atomic world and supply it with the “Copenhagen interpretation.” On the whole they seem to have taken the view that while the theory is extraordinarily strange (Bohr is said to have remarked that if it didn't make you dizzy then you didn't really understand it), the strangeness arises out of some deeply ingrained but invalid modes of thought. Once these are recognized and abandoned the theory makes sense in a perfectly straightforward way. The word “irrational”, which appears frequently in Bohr's early writings about the quantum theory, is almost entirely absent from his later essays.

The second generation, those who were students with the Founding Fathers in the early post-revolutionary period, seem firmly – at times even ferociously – committed to the position that there is really nothing peculiar about the quantum world at all. Far from making bon mots about dizziness, or the opposite of deep truths being deep truths, they appear to go out of their way to make quantum mechanics sound as boringly ordinary as possible.

The third generation – mine – were born a decade or so after the revolution and learned about the quantum as kids from popular books like George Gamow's. We seem to be much more relaxed about it than the other two. Few of us brood about what it all means, any more than we worry about how really to define mass or time when we use classical mechanics.

Last October I received a phone call asking if I could survey Landau's contributions to condensed matter physics at a commemorative 80th birthday symposium to be held in Tel Aviv the following June. I said no, I couldn't. I wasn't an historian of science, and wouldn't pretend to be one. And as a physicist I felt that Landau's contributions in condensed matter physics spanned an area broader than my competence to review. Then I hesitated. I never had the opportunity to meet Landau, but there have been times in my life when I have felt his intellectual presence so vividly that, more than a teacher, he appeared to me almost as a scientific muse. How could I turn down an opportunity to pay public homage to the man who, more than any other, built and permeates the edifice, poking around the corners of which has been the largest part of my experience as a physicist?

So I told my caller that although I couldn't talk on the topic he wanted, I could talk about the ways in which my own career has intertwined with the thoughts of Landau; how no matter what new area I turned my attention to, I invariably found myself working with concepts of his devising. Such a talk would not be historical. It would simply illustrate how the work of Landau impinged on a typical run-of-the-mill condensed matter physicist of a later generation, and it might more appropriately be titled “My life with Landau.” “Great,” said my caller, “just what we want.”