2

Unknown to Dirac, in the summer of 1925 Werner Heisenberg had figured out the skeleton of a new and abstract “quantum mechanics” that promised to be fundamental, logically consistent, and not plagued by the difficulties of the existing quantum theory of atomic structure. Only eight months older than Dirac, Heisenberg was a graduate student of Max Born in Göttingen. Like Dirac, he had not yet obtained his Ph.D. degree.

The young German reasoned that a truly fundamental theory had to contain observable quantities only. By this criterion, electron orbits were no longer legitimate, whereas the frequencies of light emitted by atoms were. After all, who had ever observed an electron orbiting around a nucleus? Transforming this general idea into an abstract mathematical scheme, Heisenberg expressed physical quantities by arrays of symbols soon recognized to be matrices. One of the arrays might represent the position of an electron (Q) and another array its momentum (P, which equals mass times velocity). Heisenberg’s theory led to a mysterious law of multiplication according to which QP differed from PQ, that is, QP ≠ PQ. It was as if 3 × 2 was not always equal to 2 × 3, or that the two numbers do not “commute.” Noting the puzzling fact that in general physical quantities did not commute, Heisenberg, at first, thought it was a flaw in his theory that might disappear when it was further developed.

Heisenberg’s seminal paper was published in a German journal on September 18, 1925, but Dirac knew of it ahead of publication. In August, Heisenberg had sent proofs of his forthcoming paper to Ralph Fowler, who sent them on to Dirac with a note saying, “What do you think of this? I shall be glad to hear.” Dirac first thought that it was of no interest, but a closer study told him a different story. He now realized that far from being a flaw, the strange appearance of non-commuting physical variables was the key element in the new mechanics and, consequently, had to be understood. Some versions of classical mechanics, he remembered, operated with non-commuting variables, which indicated that Heisenberg’s idea might be expressed in formal analogy with classical theory. The crucial insight came to Dirac “in a flash” (as he recalled) one afternoon at the beginning of October. A couple of weeks later he had a paper ready with the ambitious title “The Fundamental Equations of Quantum Mechanics.” This was the first paper that mentioned the term “quantum mechanics” in its title and with a meaning recognizable by modern physicists.

Among the fundamental equations was a significant sharpening of Heisenberg’s non-commuting variables. PQ and QP did not only differ, but, according to Dirac, they differed by a precise amount given by the tiny constant of nature Max Planck had introduced in 1900: PQ – QP ~ h. Planck’s constant is tiny indeed (h = 6.6 × 10^-34 in units of joule × second), which explains why position and momentum commute for a canon-ball but not for an electron. Dirac’s paper also contained a quantum analogue of the classical equation of motion, that is, an expression of how a physical quantity varies in time. Without an equation of motion, quantum mechanics would not be of much value, just as classical mechanics would be of little use without Newton’s second law of motion.

Even back in the 1920s, physics was a very competitive field (as it remains to this day). Dirac was aware that he was in competition with the German physicists, but the question of priority of the laws of quantum mechanics was not of great concern to him. Yet, he must have been disappointed when Heisenberg informed him in a letter of November 20 that most of the results in Dirac’s “extraordinarily beautiful paper” had already been derived by Born and another talented young Göttingen physicist, Pascual Jordan. The two Germans had extended and clarified Heisenberg’s theory, recognizing for the first time that it could be formulated in the language of matrix calculus. Consequently, the Göttingen approach to quantum mechanics was often referred to as “matrix mechanics.” In a lengthy follow-up paper known as the Dreimännerarbeit (“three-man paper”) Born, Heisenberg, and Jordan further developed matrix mechanics into what today is considered to be the first full exposition of quantum mechanics.

Dirac followed the development in Germany with interest, but decided to pursue his own ideas. On their side, the German physicists were quick to appreciate the surprising progress made in Cambridge by the “Englishman working with Fowler”—this is how Heisenberg referred to Dirac in a letter of November 24 to his Austrian colleague Wolfgang Pauli. And this is what Born recalled: “The name Dirac was completely unknown to me. The author appeared to be a youngster, yet everything was perfect in its way and admirable.” The German-speaking physicists were not used to competition and especially not from their British counterparts.

The version of quantum mechanics that Dirac developed in 1926 was known as “q-number algebra,” indicating that it was essentially a mathematical theory that could be applied to problems of physics. Measurable properties are given by numbers read on an instrument—for instance, the weight of a body as read on a balance. They are ordinary or classical numbers that satisfy the law of commutation (3 × 2 is equal to 2 × 3). Quantum variables, on the other hand, do not satisfy the law. They are members of a new class of q-numbers with mathematical properties of its own. Q-numbers are queer. On this basis, Dirac established his abstract q-number algebraic theory. Given the theory’s thoroughly algebraic and unvisualizable character, it is remarkable that it may have been inspired by geometric considerations. Dirac later said that while he published in the algebraic style, he thought in terms of pictures and diagrams. “I prefer the relationships which I can visualize in geometric terms,” he noted.

Despite its apparent remoteness from physical reality, Dirac’s theory turned out to be equivalent to the matrix mechanics of the German physicists, and, as such, described the quantum world of atoms, molecules, and light. For example, Dirac succeeded in explaining the spectrum of hydrogen in accordance with Bohr’s old atomic theory. Symptomatic of the competitiveness of the period, Pauli did the same, using more cumbersome methods of matrix mechanics.

In early 1926, there were basically two versions of quantum mechanics, the Göttingen matrix mechanics, and Dirac’s q-number theory. The latter was very much Dirac’s own and rarely used by other physicists. When the Austrian physicist Erwin Schrödinger published his new “wave mechanics” a few months later, the number of versions grew to three. For a while, confusion increased as Schrödinger’s theory was quite different from the abstract schemes developed in Göttingen and Cambridge. It described particles and atomic phenomena in terms of a continuous wave function (ψ), which satisfied a classical-looking equation of motion soon known as the Schrödinger equation. The physical meaning of the wave function was a matter of some dispute until Born suggested the currently accepted interpretation, namely, that ψ is a measure of the probability of something happening. Dirac, at first, resented the new wave mechanics, which he thought was formulated in a too classical framework. “I definitely had a hostility to Schrödinger’s ideas,” he recalled. However, the hostility did not last long. When it was proved that wave mechanics was mathematically equivalent to matrix mechanics (and also to q-number algebra), Dirac adopted a more pragmatic attitude, often mixing elements of his own theory with those of Schrödinger’s.

As a result of his study of Schrödinger’s theory, in August of 1926 Dirac published an important paper in which he clarified the physical meaning of the mysterious ψ-function and used a generalized version of the Schrödinger equation to solve problems of physics. The most important result was a fundamental distinction between two classes of particles on the basis of their different wave functions. Some particles, such as electrons and protons, were characterized by “anti-symmetrical” wave functions, while other particles, including the light quantum or photon, could be described by “symmetrical” wave functions. The quantum behavior of the first class of particles was initially examined by the Italian physicist Enrico Fermi; they became known as Fermi-Dirac particles (contrary to the photon, which is a Bose-Einstein particle). Later, Dirac invented the more convenient names fermion and boson for the two kinds of particles—designations that have long been part of physicists’ general vocabulary. The latter name is a reference to the Indian physicist Satyendra Nath Bose whose idea was improved by Einstein. As Pauli later proved, all fermions have half-integral spin (½ for an electron) and all bosons feature integral spin (0 for a photon).

Dirac’s paper of August 1926 was widely recognized to be an important contribution to quantum mechanics, but also one that was very hard to understand. It had to be deciphered, line by line. Einstein was not the only one who had troubles with Dirac’s abstract and condensed style of writing. In an October 1926 letter to Bohr, Schrödinger expressed his admiration for Dirac’s paper, adding, however, that there was much in it he found difficult to comprehend: “Dirac has a completely original and unique method of thinking, which—precisely for this reason—will yield the most valuable results, hidden to the rest of us,” he wrote. “But he has no idea how difficult his papers are for the normal human being.”

At the time when Bohr read the letter, Dirac stayed with him in Copenhagen, the first of his many journeys abroad. In May of 1926, he had completed his Ph.D. in Cambridge. The title of his doctoral dissertation was as simple as it was concise—just “Quantum Mechanics,” the first one ever written on the subject. Even before he was granted the doctoral degree, Dirac gave the first course in quantum mechanics taught at a British university. Among the few students who attended was J. Robert Oppenheimer, who was on his way to a distinguished career in physics. During and after World War II, he became a public figure, a result of his appointment as scientific director of the Manhattan Project established to construct the atomic bomb. While Dirac and Oppenheimer hardly knew each other at Cambridge, when they met again in Göttingen the following year, they developed a lasting friendship.

The stay in Copenhagen was immensely fruitful for Dirac, as was his subsequent visit to Göttingen. In the fall of 1926, quantum mechanics had made huge progress. However, a general and unified formalism associated with a conceptual clarification was still missing. It was this excessively difficult task that Dirac addressed in Copenhagen. The result of his intellectual efforts, a fundamental paper on “The Physical Interpretation of the Quantum Dynamics,” was completed in early December. Unknown to Dirac, Jordan was working on the same problem in Göttingen and arrived independently at a theory of the same generality and grandiose scope. Dirac’s paper was a mathematical and conceptual tour de force based on strictly logical deductions, a procedure which appealed to his rational mind. “This work,” he recalled, “gave me more pleasure in carrying it through than any of the other papers which I have written on quantum mechanics either before or after.”

At the time, Born had shown that Schrödinger’s ψ–function could not itself be ascribed a definite physical meaning, in the sense that it was not measurable, but that the square of the function determined a probability density. For example, there is a certain probability that an electron at a time will be located in a volume element dV and this probability is given by ψ²dV. Dirac’s general theory comprised Born’s probabilistic interpretation of quantum mechanics without introducing the interpretation as an axiom. Another characteristic feature of quantum mechanics is the impossibility of knowing the exact location of a particle and the exact value of its momentum simultaneously. This surprising result, so foreign to the determinism inherent in classical mechanics, was first announced by Heisenberg in a classic paper published in May 1927. According to Heisenberg’s principle of uncertainty or indeterminacy, in any measurement, there is a minimum indeterminacy Δq in the position of a particle, and similarly a minimum indeterminacy Δp in its momentum. Each of the uncertainties can be zero, but not both at the same time. They are bound together by the relation ΔqΔp ∼ h, which can be seen as an instance of the lack of commutativity of q and p.

In his Copenhagen paper, Dirac came close to formulating Heisenberg’s deep insight and its consequence that determinism is ruled out in the quantum world: “One can suppose that the initial state of a system determines definitely the state of the system at any subsequent time. If, however, one describes the state of the system at an arbitrary time by giving numerical values to the co-ordinates and momenta, then one cannot actually set up a one-one correspondence between the values of these co-ordinates and momenta initially and their values at a subsequent time.” Although close to the uncertainty relations in December 1926, Dirac felt no need to formulate his insight into a general principle concerning the limitation of measurements.

Another result of Dirac’s paper, a mathematical innovation, merits attention. To facilitate some of his calculations and state them in a more condensed way, he introduced a function δ(x) with the strange property that it is equal to zero for all values of x except at x = 0, where δ(0) equals infinity. The area covered by this function is neither zero nor infinity, as one might expect, but exactly one. Dirac was not really the first to make use of a wildly discontinuous function of this kind, but it was only with his work that it became a powerful tool in quantum mechanics and other areas of theoretical physics. It is not, strictly speaking, a “function,” but what mathematicians call a “distribution.” Dirac was not obsessed by mathematical rigor, and he introduced his delta-function without caring much about whether it was justified by the mathematicians’ standards. What mattered to him was that it was useful from the point of view of physics.

The strange delta-function was eventually justified in the sense that it became part of the general and mathematically rigorous theory of distributions. It is common to see Dirac’s delta-function as a gift from the physicists to the mathematicians, but the actual connection between Dirac’s innovation and the theory of distributions was tenuous. The mathematical distribution theory was invented in 1944 by the French mathematician Laurent Schwartz, who earlier had come across Dirac’s formulae involving the delta-function. However, he found them “so crazy from a mathematical point of view that there was simply no question of accepting them.” Schwartz’s invention was not motivated by physics but by problems in pure mathematics. As he later commented: “It’s a good thing that theoretical physicists do not wait for mathematical justification before going ahead with their theories!” Schwartz and Dirac came to know each other in 1949 when both lectured at a mathematical seminar in Vancouver.

Many people would have rested on their laurels, at least for a while, after having completed work so demanding as the paper on the physical interpretation of quantum mechanics. But not Dirac. On the contrary, he continued straight ahead with another important work, in which he applied quantum mechanics to the electromagnetic field making up the waves of light. Although attempts at a quantum electrodynamics (as the theory is called) had previously been made in 1925 by Jordan in the Dreimännerarbeit, it was only with Dirac’s new paper on the interaction of matter and radiation that a proper foundation of quantum electrodynamics was laid. His theory dealt with emission and absorption of light waves or photons, which to Dirac were just two names for the same thing—the one depicting light as waves and the other as particles. He was pleased to note that his theory contained “a complete harmony between the wave and light-quantum descriptions.”

Instead of treating the characteristic quantities of the radiation field as ordinary c-numbers, he now treated them as q-numbers or quantum variables, which is the essence of so-called “second quantization.” This important idea goes back to Dirac’s work published in 1927. It enabled him to calculate the probabilities of emission, absorption, and scattering of light in agreement with experimental knowledge and previous theoretical formulae. In addition, he could explain dispersion, the phenomenon where light changes its wavelength after having passed matter. Yet Dirac was not entirely satisfied, as he had merely justified on the basis of quantum mechanics what was already known. Referring to second quantization, he remarked at the end of his life, “It was a bit of disappointment to find that nothing really new came out of the idea.” Important as Dirac’s radiation theory was, it was restricted to particles satisfying Bose-Einstein statistics (bosons), which in practice meant light in the form of photons. At the time, Dirac did not attempt to extend his theory to cover also fermions such as electrons. It was left to Jordan and other physicists in the German tradition to generalize the method of second quantization in such a way that it was applicable to electrons and other particles carrying mass.

Between February and June of 1927, Dirac stayed in Göttingen, which was the center of theoretical physics even more than Copenhagen. There, he met not only the German quantum theorists, including Born, Jordan, and Heisenberg, but also many visiting foreigners. One of them was the American physicist Raymond Birge, who in a letter to a colleague in the United States reported on the impression Dirac made in the German university town: “When he talks, Born just sits and listens to him open-mouthed. That Dirac thinks of absolutely nothing but physics.” The Russian Igor Tamm, a later Nobel laureate, was another visiting physicist greatly impressed by Dirac. Back in Moscow, Tamm wrote in a letter to a relative about his unforgettable meeting with this “true man of genius,” whom he had had the privilege of befriending. “Do not smile that it sounds high-flown; I really mean it. I know that when I grow old I’ll be telling my grandchildren with pride about that acquaintance of mine.”

On his way back to Cambridge, Dirac visited another of the continental centers of quantum physics, the University of Leiden in the Netherlands. He was invited by Paul Ehrenfest, who was a close friend of Bohr and had done important work in both the old and the new quantum theory. Ehrenfest had recognized Dirac’s genius at an early stage and was eager to know more of his methods. Shortly before Dirac arrived in Leiden, Ehrenfest wrote to him, in much the same vein as Schrödinger and Einstein had previously: “We spent many, many hours going over a few pages of your work before we understood them! And many points are still as dark to us as the most moonless night!”

In the late 1920s, physicists discussed not only the physical meaning of quantum mechanics, but also its broader philosophical implications. Although the two questions were intimately connected, Dirac felt much more at home on the first question than the second one. Basically, he was not very interested in the interpretation debate in so far that it was unrelated to the equations of quantum mechanics. For example, he disliked Bohr’s complementarity principle, a broadly formulated interpretation of the quantum world and its relation to the classical world that Bohr had suggested in 1927. Dirac felt that it involved too much philosophy and too little physics. “It doesn’t provide you with any equations which you didn’t have before,” was his reason for dismissing Bohr’s principle.

But Dirac could not avoid the issue of interpretation such as it came up in the discussions between Bohr, Einstein, Heisenberg, Schrödinger, and others. Although he differed in some respects from the Bohr-Heisenberg view known as the Copenhagen interpretation, on the whole, he accepted that quantum mechanics was not a theory of nature but solely a way of calculating and predicting measurable quantities. Born paraphrased Dirac’s view, or what he called his l’art pour l’art attitude, as follows: “The existence of a mathematically consistent theory is all we want. It represents everything that can be said about the empirical world; we can predict with its help unobserved phenomena, and that is all we wish. What you mean by an objective world we don’t know and don’t care.”

Dirac believed that getting the right equations was more important than discussing questions of interpretation, which, in the end, might be a matter of opinion rather than fact. On the other hand, he was neither uninformed nor uninterested in the interpretation issue. As a participant in the famous 1927 Solvay conference in Brussels, he witnessed the epic debate between Bohr and Einstein, but without taking much part in the discussion. As he explained it, “I listened to their arguments, but I did not join in them, essentially because I was not very much interested. I was more interested in getting the correct equations.” However, he did give a fairly elaborate exposition of his own view, which was critically discussed by Born and Heisenberg. While Heisenberg tended to conceive of nature as the product of the observer’s free will, Dirac suggested the much less subjectivist view that it was nature itself that determined the outcome of an experiment. Of course, he added, only human observers could decide what kind of experiment to make.

Two years after the Solvay Conference, Dirac was approached on behalf of Oxford University Press by James G. Crowther, a science journalist and author, who wanted him to write a textbook on quantum mechanics. Crowther recalled that Dirac “was living in a simply furnished attic in St. John’s College. He had a wooden desk of the kind that is used in schools. He was seated at this apparently writing the great work straight off.” The result of this meeting was Principles of Quantum Mechanics (1930), probably the most important textbook on the subject ever published, and a rich source for understanding Dirac’s view of quantum physics. Dirac clearly had an agenda in writing the book, namely to disseminate what he thought were the basic principles and proper methods of quantum mechanics. He wanted to shape a theory, which had not yet found its final form.

Principles did much to disseminate views of the Copenhagen school to a generation of young physicists. In agreement with Bohr, in the preface, Dirac called attention to “the increasing recognition of the part played by the observer himself in introducing the regularities that appear in his observations.” This he considered “very satisfactory from a philosophical point of view.” Dirac further pointed out that the fundamental laws of nature “do not govern the world as it appears in our mental picture in any very direct way, but instead they control a substratum of which we cannot form a mental picture without introducing irrelevancies.” From the beginning to the end, Principles was based on what he called the “symbolic method.” Dirac wanted to present the general theory of quantum mechanics in a way that was free of physical interpretation. The symbols, he said, were used “in an abstract way, the algebraic axioms that they satisfy and the connection between equations involving them and physical conditions being all that is required.”

Compared with other textbooks in theoretical physics,—or textbooks of any kind—Principles was unusual. With no illustrations, no index, and almost no references, it was not a reader-friendly work. Nor had Dirac found it necessary to add the customary list with suggestions for further reading. The book was also completely ahistorical, presenting its subject as if it were an abstract mathematical discipline coming down from the heavens. Experiments were given very low priority. By ordinary pedagogical standards, it was simply a catastrophe, but pedagogy was obviously not what interested Dirac most. The exposition was highly original. While all other textbooks introduce Planck’s constant early on, it only appeared on page 95 of the book’s 264 pages. Similarly, readers had to wait until page 104 to meet the Schrödinger equation. This was fully intentional, a consequence of the symbolic method.

Despite its obvious pedagogical deficiencies, Principles became a great success. The first English edition sold about 2,000 copies, and the German and Russian translations sold even more. The book quickly established itself as the standard work on quantum mechanics, used not only by students as a textbook, but also by many experienced research physicists. It was widely reviewed in the physics journals, in almost all cases positively and in some even enthusiastically. It was a common feature of the reviews to praise the book for its directness, generality, and completeness. According to Oppenheimer, it was “astonishingly complete” and “unitary and coherent.” He observed that Principles was “clear, with a clarity dangerous for the beginner, deductive, and in its foundation abstract” and also that “the virtual contact with experiment is made quite late in the book.” However, in spite of all its qualities, Oppenheimer added a warning: “The book remains a difficult book, and one suited only to those who come to it with some familiarity with the theory. It should not be the sole text, nor the first text, in quantum theory.” Ehrenfest had more than just some familiarity with quantum theory but nonetheless found Principles a hard read. “A terrible book—you can’t tear it apart!” is how he is said to have reacted.

Philosophically minded reviewers preferred to interpret Dirac’s textbook in accordance with their own philosophical tastes. Eddington, at the time Britain’s best-known scientist and popular writer of science, favored a philosophical view of science that could be described as a hybrid of rationalism and idealism. He was fascinated by Dirac’s approach to physics in general and his wave equation of the electron in particular. Eddington praised Dirac’s symbolic version of quantum mechanics as “highly transcendental, almost mystical” and found that it agreed with his own view of the universe as a dematerialized world of shadows dressed up in mathematical symbols. Dirac was probably surprised, not knowing that he was almost a mystic. The Austrian physicist and philosopher Philipp Frank, a professor in Prague and a leading figure in the school of logical positivism, read Principles as indirect support of positivist philosophy. He was pleased with what he saw as Dirac’s philosophical position, namely, that physical theory can only answer questions that relate to the outcome of experiments, whether real or imagined. Frank was a close friend of Einstein, who was no less impressed by Dirac’s book. Referring to the standard probabilistic interpretation of quantum mechanics (which Einstein did not accept) he wrote that it was to Dirac “to whom, in my opinion, we owe the most logically perfect presentation of this theory.”