In 1928, British physicist Paul Dirac wrote down an equation that combined quantum theory and special relativity to describe the behaviour of an electron moving at a relativistic speed. The equation – which won Dirac the Nobel Prize in 1933 – posed a problem: just as the equation x2 = 4 can have two possible solutions (x = 2 or x = −2), so Dirac's equation could have two solutions, one for an electron with positive energy, and one for an electron with negative energy. But classical physics (and common sense) dictated that the energy of a particle must always be a positive number.
Dirac interpreted the equation to mean that for every particle there exists a corresponding antiparticle, exactly matching the particle but with opposite charge. For example, for the electron there should be an "antielectron", or "positron", identical in every way but with a positive electric charge. The insight opened the possibility of entire galaxies and universes made of antimatter.
But when matter and antimatter come into contact, they annihilate – disappearing in a flash of energy. The Big Bang should have created equal amounts of matter and antimatter.
Editor’s note: In earlier WG articles, here and here, we discussed Dirac’s matter-antimatter clashings, particles “continually popping in and out of existence,” annihilating each other. This occurs in so-called empty space, the void, all around us. In this dynamic process, we might very well be witnessing a microcosm of what happened at the moment of The Big Bang.
I talk a lot about Paul Dirac in my book “Now—The Physics of Time.” Dirac was concerned that the equation of quantum mechanics was not compatible with relativity. A different equation, called the Klein-Gordon equation was compatible, but it had negative energy solutions, which Dirac felt were unphysical. So he tried a different approach at math, one that was far more original and brilliant than that of Klein and Gordon. We now call his solution “The Dirac Equation.”
The Dirac equation did some amazing things. It explained the electron in terms of having a “spin” [value] equal to half of the Planck constant. Amazing! And most importantly, with one minor addition (called “minimal electromagnetic interaction”) it correctly gave the size of the magnetism of the electron! Totally astonishing!
One minor problem: just like the Klein-Gordon equation, it had negative energy solutions. Dirac had rejected KG because of that, but how could he reject a theory that got the magnetism of the electron correct?
Even worse, the existence of negative energy states implied that all electrons were unstable, disagreeing with the known facts. They should all lose energy and drop into the negative energy states, with negative mass.
Dirac’s wrote that he was ignoring this problem because the theory was able to explain so much. A future and more advanced version of the theory would undoubtedly get rid of the negative energy states. Yet that was the very reason he had rejected the Klein-Gordon equation!
Two years later, Dirac “solved” this problem by postulating that all the negative energy states were filled, so the electrons could not drop down. This theory was ad hoc, and made predictions that were obvious nonsense, such as an infinite mass density and charge density to empty space; but he thought they salvaged the theory. And sometimes a particle would jump out of the negative energy state, leaving behind a bubble that acted as if it had positive charge! That bubble, Dirac said in his paper, was the proton! In December 1929 he wrote a paper on this called “Theory of Electrons and Protons.” We now know it was completely wrong.
Actually, it was known to be wrong as soon as it was published. Hermann Weyl showed that if Dirac’s theory had been correct, then the electron and the proton would have had to possess the same mass as each other. (That is not obvious from the theory, but Weyl’s argument was correct, and Dirac agreed.) Yet the proton was 1836 times heavier than the electron! Dirac responded that we need to wait for a later theory that would (some day) address this error of 1836. Then Robert Oppenheimer argued that protons and electrons would annihilate each other, very rapidly! Another great flaw in Dirac’s theory. (Oppenheimer was correct; annihilation is indeed observed between electrons and positrons.)
In desperation, Dirac postulated that there would be a new particle, something he called the “anti-electron” (we now call it the “positron”). But he seemed so unsure of this prediction, that he put it in a paper devoted to a totally different topic; the paper was called “Quantised Singularities in the Electromagnetic Field” published in 1931. Of the 32 paragraphs in that paper, only 2 touched on the negative energy conundrum and his postulate of an anti-electron. He said these particles are not observed because they quickly annihilate (as Oppenheimer had pointed out).
Thus Dirac reluctantly predicted the positron in 1931. In 1932, Anderson observed one, and in 1933, Dirac received the Nobel Prize in physics for his prediction. In his Nobel speech, he never mentioned the seminal contributions of Weyl and Oppenheimer, the two physicists who drove him to making the prediction by pointing out the serious mistakes in his earlier papers.
Yet despite this embarrassing beginning, the Dirac equation has proven to be extremely important. You can study it today with awe and wonder; how could anybody come up with that? That may be why someone who studied the Dirac equation, or noted its importance in physics, told you that — or, concluded that — Dirac was the greatest. Forgotten is the history that he created this equation to avoid the very negative energy states for which he is now given credit. Dirac made numerous mistakes along the way, mistakes that had to be pointed out by others. I would place Dirac in the same category as Louis de Broglie, someone who made a great contribution to the foundations of quantum physics. He was equal to Klein and Gordon (whose equation, despite Dirac’s improper rejection, proved to be true for all integral spin particles.) But he is certainly not of the stature of Isaac Newton, James Maxwell, Neils Bohr, Albert Einstein, nor Richard Feynman.
You can find more detail in my book “Now—The Physics of Time”.
Why is Paul Dirac not known to most people (beyond the physics world)?
Because he is almost never mentioned in high school physics textbooks. If you want to learn about what he proposed in the field of quantum mechanics, you would have to go to college and take a physics course. And so, most folks (never having taken a college physics course) would have never heard of him, and so will never recognize the name.
Although Dirac’s work is equally as important as Einstein’s, he was a quiet man and so not prone to be noticed outside of a physics conference. He built on what Einstein laid as a foundation, and provided insights that allowed us to better understand what quantum mechanics and wave vs. particle really meant. But his work did not lead to weapons (well, not directly, that I know of), and his work is rarely as simple (but just as groundbreaking) as Einstein’s theory of relativity. So unless you want to know more about quantum mechanics, you would rarely hear his name. And so he is not known to most people.
EDIT: And let me say something about “is rarely as simple”. Einstein’s work was groundbreaking not just for what it was, but also for what he did to get there. A “gedanken” (thought-experiment) was probably used by many scientists before Einstein, but not to the extent he used them. Einstein was a great thinker, and managed to get Special Relativity down to the equation we all know. The Three Meanings Of E=mc^2, Einstein's Most Famous Equation
His [Einstein] equation describes things we know and (mostly) understand in everyday life - mass and energy. Dirac’s equation, on the other hand, deals with electrons and 1/2 spin particles like quarks, things that you and I cannot directly experience (and no, getting shocked is not like holding an electron in your hand) Dirac equation - Wikipedia. So what Einstein did is far more easily understandable on a human level than what Dirac did. Not that it is easy to *fully* understand what Einstein did, but the basic parts can be understood by everyday folks.
The merging of quantum mechanics with special relativity was one of the great triumphs of the 20th century physics.
Paul Dirac developed a quantum theory of the electron which in one fell swoop explained the strange quantum property of spin (which naturally emerged from the theory) as well as predicting the existence of antimatter!
Overall, special relativity and quantum mechanics have shown that our common intuition fails as we enter the extremes of measurement. Our intuition seems like the safe zone where the world works according to rather simple rules. However, when reality is pushed to the limits, we find that truth is significantly and remarkably different. Discovering how the universe really works is one of the joys of physics.
Paul Adrien Maurice Dirac was an outstanding twentieth century theoretical physicist whose work was fundamental to the development of quantum mechanics and quantum electrodynamics.
Dirac was awarded the Nobel Prize in Physics jointly with Erwin Schrödinger in 1933 for his contributions to atomic theory, Dirac’s prediction of the existence of antimatter having been experimentally proven by that time. Though he spent most of his life concerned with theoretical science, he had initially intended to pursue a more practical career, earning his bachelor's degree in electrical engineering...
Dirac was born on August 8, 1902, in Bristol, England, though he was registered at birth as Swiss because his father, Charles Adrien Ladislas Dirac, was a Swiss immigrant. His mother, Florence Hannah Holten, was English, and Dirac would himself gain British citizenship in 1919. The younger Dirac had two siblings and studied in his youth at the Merchant Venturer's Technical College, the same school where his father taught French. Dirac excelled in mathematics and science, but his father apparently wished him to excel in French as well, for Dirac reported later in his life that his father would not respond to him unless he spoke in that language. It has been commonly quipped that this habit of his father’s was the reason that Dirac became a man of few words who generally enjoyed solitude.
When he was 16, Dirac began his undergraduate career at Bristol University. Following graduation three years later, he was unable to find employment as an electrical engineer, an understandable problem considering the economic depression experienced by Great Britain after World War I. Consequently, he decided to remain at Bristol, where a scholarship enabled him study in the mathematics department for two years free of charge. During that time Dirac appears to have discovered his true calling, for when his scholarship ended, instead of looking for a job in electrical engineering, he applied to be a research assistant in the mathematics department at St. John’s College, Cambridge. He was awarded the position, and there he came to focus his studies on mathematics as it applied to atomic theory.
Having already developed an interest in Albert Einstein’s theories as an undergraduate, Dirac became very familiar at Cambridge with the works of Niels Bohr, Max Born, Werner Heisenberg, Arnold Summerfeld and other great theoretical minds, some of which he was privileged enough to hear speak at the university. Dirac was awarded his Ph.D. in physics in 1926, but had already made his first notable contributions to the field before he ever received the degree.
Dirac’s doctoral research supervisor, Ralph Fowler, received a preview copy of a paper written by Heisenberg in 1925 and recommended that Dirac examine it carefully. In the course of doing so, Dirac became intrigued by an equation developed by Heisenberg, who was working out a new type of quantum mechanics, and later realized that its form was analogous to that of the Poisson Brackets, an operation used in classical mechanics. This Dirac interpreted as a direct link between quantum and classical physics, and the discovery helped him develop a formulation of quantum mechanics that surpassed all others in its generality and logic. Dirac’s generalized quantum mechanics immediately placed him in the same ranks of other illustrious physicists working in the field, though he was only in his mid-twenties and had not yet received his doctorate. More importantly, the formulation enabled him to make several other key theoretical advances.
By applying quantum mechanics to James Clerk Maxwell’s theory of electromagnetism, Dirac established the groundwork of quantum field theory. He also utilized his version of quantum mechanics to develop the transformation theory, which not only united the opposing views of quantum mechanics proposed by Heisenberg and the Austrian physicist Erwin Schrödinger, but also established that other possible versions of quantum mechanics were possible as well.
This momentous achievement, which allowed a much deeper understanding of quantum mechanics than had been previously possible, was made through the use of mathematics alone. Even when he presented the theory in his seminal text The Principles of Quantum Mechanics, which was published in 1930, he did not provide visual analogies or mental pictures in his explanation because he believed that to do so would “introduce irrelevancies.” Indeed, this philosophy seems to have served Dirac very well, for in 1927 he successfully predicted the existence of new subatomic particles solely utilizing mathematical arguments. The theoretical deduction of the new particles, however, was only a byproduct of another landmark undertaking.
Believing that quantum mechanics should take into account Einstein’s theory of special relativity and being unhappy with the results of other physicists that had attempted to achieve this, Dirac decided to develop an equation of the motion of a quantum particle that would not ignore relativistic effects, as did Schrödinger’s wave equation. For his starting point Dirac used only the most basic information known about the electron, its charge and mass. From this simple beginning, he mathematically arrived at a wave equation for a single electron that respects all the requirements of special relativity and quantum mechanics, and was able to predict the properties of the electron, including its spin and magnetic charge, with incredible precision.
The importance of Dirac’s work in this area was recognized by the Nobel Foundation in 1933, when he was awarded the physics prize jointly with Schrödinger. By the time he received the honor, an initially controversial prediction that Dirac made based on his own equation, that there existed a positive counterpart to the electron, had been experimentally verified.
In analysis of his own mathematical equations, Dirac found that electrons could have either positive or negative kinetic energy. This could only be possible, he deduced, if an unidentified particle with the same mass and charge as the electron but positive rather than negative existed. Suggesting that something exists that no one has ever seen evidence of has always been a risky venture, and Dirac was at first hesitant to draw the conclusion his mathematical arguments seemed to lead him to in case others considered it a sign of an error in his work. His fear appears to have been well grounded, as indicated in a letter from Heisenberg, with whom Dirac was on friendly terms, to Wolfgang Pauli, written after Heisenberg had become acquainted with Dirac’s notions of the anti-electron (or positron as it is now called). Heisenberg wrote, “The saddest chapter of modern physics is and remains the Dirac theory.” Presumably Heisenberg had a change of heart regarding the matter when shortly after, in 1932, Carl Anderson experimentally observed the particle of antimatter that Dirac had theoretically anticipated. Years later the anti-proton and anti-neutron, which could be predicted by extension of Dirac’s work, were also proven to exist...
No matter what aspect of physics he focused upon, Dirac seems to have been guided by his belief that “physical laws should have mathematical beauty.” The beauty of his own work has been remarked upon by many. It is corroborated by the fact that, of the countless scientists and mathematicians who have labored through the ages, Dirac is the only one to have an equation appear in Westminster Abbey, carved in stone.
Paul Dirac made contributions to theoretical physics that spanned quantum mechanics, relativity, gravity, and magnetism. Among these, the most important was arguably his connecting relativity and quantum mechanics. The Schrödinger equation is only correct for speeds much less than the speed of light. An equation correct for all speeds must include the energy associated with mass: E = mc². The Dirac equation did just that. It also turned out to naturally include spin, which was an ad hoc addition to the wave functions obtained by the Schrödinger equation. He and Schrödinger shared the 1933 Nobel Prize.
Paul Dirac: the purest soul in physics
01 Feb 1998
Paul Dirac published the first of his papers on “The Quantum Theory of the Electron” seventy years ago this month. The Dirac equation, derived in those papers, is one of the most important equations in physics.
Each day, I walk past the road where Paul Adrien Maurice Dirac lived as a child. It is pleasant to have even this tenuous association with one of the greatest intellects of the 20th century. Paul Dirac was born at 15 Monk Road in Bishopston, Bristol, on 8 August 1902, and educated at the nearby Bishop Road Primary School. The family later moved to Cotham Road, near the University of Bristol, and in 1914 the young Dirac joined Cotham Grammar School, formerly the Merchant Venturers.
Dirac was a student at Bristol University between 1918 and 1923, first in electrical engineering and then in applied mathematics. Much later, he said: “I owe a lot to my engineering training because it [taught] me to tolerate approximations. Previously to that I thought…one should just concentrate on exact equations all the time. Then I got the idea that in the actual world all our equations are only approximate. We must just tend to greater and greater accuracy. In spite of the equations being approximate, they can be beautiful.”
Because Dirac was a quiet man – famously quiet, indeed – he is not well known outside physics, although this is slowly changing. In 1995 a plaque to Dirac was unveiled at Westminster Abbey in London and last year Institute of Physics Publishing, which is based in Bristol, named its new building Dirac House.
It is hard to give the flavour of Dirac’s achievements in a non-technical article, because his work was so mathematical. He once said: “A great deal of my work is just playing with equations and seeing what they give.”
When Dirac went to Cambridge in 1923, the physics of matter on the smallest scales – in those days this was the physics of the atom – was in ferment. It had been known for more than a decade that the old mechanics of Newton – “classical” mechanics, as it came to be called – does not apply in the microscopic world. In particular, evidence from the light coming out of atoms seemed to indicate that some quantities that in classical mechanics can take any values are actually restricted to a set of particular values: they are “quantized”.
One of these quantities is the energy of the electrons in an atom. This was strange and shocking. Imagine being told that when your car accelerates from 0 to 70 miles per hour it does so in a series of jumps from one speed to another (say in steps of one thousandth of a mph), with the intermediate speeds simply not existing. It did not make sense, and yet observations seemed to demand such an interpretation.
In the first attempts at a theoretical understanding, physicists tried to find the general rules for imposing these restrictions on classical mechanics – that is rules for quantization. It seemed that in order to quantize, it was necessary first to identify those quantities that do not change when their environment is slowly altered. If a pendulum is slowly shortened, for example, it swings farther and also faster, in such a way that its energy divided by its frequency stays constant. These rules worked for simple atoms and molecules but failed for complicated ones.
Dirac entered physics at the end of this baroque period. One of his first papers was an attempt at a general theory of these unchanging quantities. This is a delicate problem in classical mechanics, not solved even now. It is amazing today to read that paper. In its mathematics it is quite unlike any of Dirac’s later works (for example, he brings in fine differences between rational and irrational numbers), and “pre-invents” techniques developed by other people only decades later. (I say "pre-invents" because the paper was forgotten until recently.)
At this time the situation in atomic physics resembled that at the end of the 16th century, when the old Earth-centred astronomy had to be made ever more elaborate in the face of more accurate observations. The difficulties of the 16th and 20th centuries were resolved in the same way: by a complete shift of thought. In atomic physics this happened suddenly, in 1925, with the discovery by Heisenberg of quantum mechanics. This seemed to throw out classical mechanics completely, though it was built in as a limiting case to ensure that, on larger scales, the new mechanics agreed with more familiar experience. The quantum rules emerged automatically, but from a mathematical framework that was peculiar. For example, it involved multiplication where the result depends on the order in which the multiplication is done. It is as though 2 multiplied by 3 is different from 3 multiplied by 2. Heisenberg found this ugly and unsatisfactory. Dirac disagreed, and just a few months after Heisenberg he published the first of a series of papers in which quantum mechanics took the definitive form we still use today.
The main idea is that the multiplied objects – objects that represent variables we can measure in experiments – should be thought of as operations. An experiment is an operation, of course, even though its result is a number. With this interpretation, it is not surprising that the order matters: we all know that putting on our socks and then our shoes gives a result different from putting on our shoes and then our socks. Dirac found the one simple rule by which a multiplied by b differed from b multiplied by a, and from which the whole of quantum mechanics follows.
The same unification was soon found to include Schrödinger’s way of doing quantum mechanics, where the state of a system is represented by a wave whose strength gives the probabilities of the different possible results of measurements on it. For a while this seemed completely different from the framework that Heisenberg had used, but it quickly emerged that in fact each represents Dirac’s operators in a different way. It seemed miraculous.
Although brilliant – in Einstein’s words, “the most logically perfect presentation of quantum mechanics” – this was a reformulation of physics that had, admittedly only just, been discovered. Dirac’s main contribution came several years later, when (still in his mid-twenties) he made his most spectacular discovery.
Before quantum mechanics, there had been another revolution in physics, with Einstein’s discovery in 1905 that Newton’s mechanics fails for matter moving at speeds approaching that of light. To get things right, time had to be regarded as no longer absolute: before-and-after had to be incorporated as a fourth co-ordinate like the familiar three spatial co-ordinates that describe side-to-side, forward-and-backward and up-and-down. Just as what is side-to-side and what is forward-and-backward change when you turn, so time gets mixed in with the other three co-ordinates when you move fast. Now, in the 1920s, came quantum mechanics, showing how Newton’s mechanics failed in a different way: on microscopic scales. The question arose: what is the physics of particles that are at the same time small and moving fast?
This was a practical question: the electrons in atoms are small, and they move fast enough for the new quantum mechanics to be slightly inaccurate, since it had been constructed to have as its large-scale limit Newton’s mechanics rather than Einstein’s. From the start people tried to construct a quantum theory concordant with relativity, but failed to overcome technical obstructions: in particular, their attempts gave probabilities that were negative numbers – something that is nonsense, at least in the usual meaning of probability. The question boiled down to this: what are the right sort of quantum waves describing electrons? And what is the wave equation that governs the dynamics of these waves, while satisfying the requirements of relativity and giving sensible physical predictions?
Dirac’s construction of his wave equation for the electron – published in two papers in the Proceedings of the Royal Society (London) in February and March 1928 – contained one of those outrageous leaps of imagination shared by all great advances in thought. He showed that the simplest wave satisfying the requirements was not a simple number but had four components. This seemed like a complication, especially to minds still reeling from the unfamiliarity of the “ordinary” quantum mechanics. Four components! Why should anybody take Dirac’s theory seriously?
First, and above all for Dirac, the logic that led to the theory was, although deeply sophisticated, in a sense beautifully simple. Much later, when someone asked him (as many must have done before) “How did you find the Dirac equation?” he is said to have replied: “I found it beautiful.” Second, it agreed with precise measurements of the energies of light emitted from atoms, in particularly where these differed from ordinary (non-relativistic) quantum mechanics.
There are two more reasons why the Dirac equation was compelling as the correct description of electrons. To understand them, you should realize that any great physical theory gives back more than is put into it, in the sense that as well as solving the problem that inspired its construction, it explains more and predicts new things. Before the Dirac equation, it was known that the electron spins. The spin is tiny on the scale of everyday but is always the same and plays a central part in the explanation through quantum mechanics of the rules of chemistry and the structure of matter. This spin was a property of the electron, like its mass and its electric charge, whose existence simply had to be assumed before quantum mechanics could be applied. In Dirac’s equation, spin did not have to be imported: it emerged – along with the magnetism of the electron – as an inevitable property of an electron that was both a quantum particle and a relativistic one.
So, electron spin was the third reason for believing Dirac’s mathematically inspired equation. The fourth came from a consequence of the equation that was puzzling for a few years at first. Related to its four components was the fact that any solution of the equation where the electron had a positive energy had a counterpart where the energy was negative. It gradually became clear that these counterpart solutions could be interpreted as representing a new particle, similar to the electron but with positive rather than negative charge; Dirac called it an “anti-electron”, but it soon came to be known as the positron. If an electron encounters a positron, Dirac predicted, the two charges cancel and the pair annihilates, with the combined mass transforming into radiation in the most dramatic expression of Einstein’s celebrated equation E = mc2. Thus was antimatter predicted. When the positron was discovered by Anderson in 1932, Dirac’s immortality was assured. Dirac and Schrödinger shared the Nobel Prize for Physics in 1933.
Nowadays, positrons are used every day in medicine, in PET (positron emission tomography) scanners that pinpoint interesting places in the brain (e.g. places where drugs are chemically active). These work by detecting the radiation as the positrons emitted from radioactive nuclei annihilate with ordinary electrons nearby.
The Dirac equation for an electron moving in an arbitrary electromagnetic field can be written in many ways. In Dirac’s original papers it is written as
where p0 = ih∂/c∂t (the energy operator), e is the charge on the electron, A0 is the scalar potential associated with the electromagnetic field, c is the speed of light, α1 are 4×4 matrices derived from the Pauli matrices, p1 = -ih∂/∂x is a momentum operator (p2 = -ih∂/∂y, p3 = -ih∂/∂z). A1 are the three components of the electromagnetic vector potential, m is the mass of the electron and Ψ is the wavefunction of the electron.
The wavefunction Ψ is a 4×1 column vector (also known as a spinor) and each element is a function of space and time, representing the spin state (up or down) of the electron and the associated positron solution. As explained in the main text, the equation was able to explain the results of all of the experiments at the time, to explain the origin of electron spin and to predict the existence of antimatter.
The equation can be written in more compact form. In §67 of The Principles of Quantum Mechanics (4th edn, Oxford University Press) it is written as
{p0+ecA0–ρ1(σ,p+ecA)–ρ3mc}ψ=0
where ρ1 and ρ3 are 4×4 matrices (related to α1 and the Pauli matrices), σ is a three-component vector of 4×4 matrices, and p is a three-component vector of momentum operators. The version of the equation in Westminster Abbey is even more compact and reads iγ⋅∂ψ = mψ where γ is a 4×4 matrix and ∂ is a 4-vector.
Having explained spin, it was natural for Dirac to try to explain electric charge, and in particular the mysterious fact that it is quantized: all charges found in nature are multiples of the charge on the electron. In classical electricity, there is no basis for this: charges can have any value.
In 1931 Dirac gave a solution of this problem in an application of quantum mechanics so original that it still astounds us to read it today. He combined electricity with magnetism, in a return to the 18th-century notion of a magnet being a combination of north and south magnetic poles (magnetic charges), in the same way that a charged body contains positive and negative electric charges. That symmetry was lost in the 19th century with the discoveries of Oersted, Ampère and Faraday, culminating in Maxwell’s synthesis of all electromagnetic and – in another example of getting out more than you put in – optical phenomena. In its place came a greater simplicity: there are only electric charges, whose movement generates magnetism (and now the motive power for much of our civilisation). The absence of isolated magnetic poles – magnetic monopoles – was built into classical electromagnetism, and also the quantum mechanics that grew out of it.
Dirac wondered if there was any way that magnetic monopoles could be brought into quantum physics without spoiling everything that had grown out of assuming that they did not exist. He found that this could be done, but only if the strength of the monopole (the “magnetic charge”) was linked to that of the electric charge, and if both were quantized. This solved the original problem: for consistency with quantum mechanics, the existence of even one monopole anywhere in the universe would suffice to ensure that electric charge must be quantized. The implication is compelling: to account for the quantization of electricity, magnetic poles must exist. After this, Pauli referred to Dirac as “Monopoleon”.
Alas, no magnetic monopole has ever been found. Perhaps they do not exist, or perhaps (and there are hints of this in the theory) positive and negative monopoles are so tightly bound together that they have not been separated. Much later, Dirac referred to this theory as “just a disappointment”. However, the mathematics he invented to study the monopole – combining geometry with analysis – now forms the basis of the modern theories of fundamental particles.
There were two other seminal contributions to physics in those early years. I have space only to mention them. Dirac applied quantum mechanics to the way light and matter interact. This made him realize that it was necessary to quantize not only particles but the electromagnetic field itself, and led him to the first consistent theory of photons (which had been discovered several decades previously in the beginnings of quantum mechanics). This led to the elaborate and thriving quantum field theories of today.
Dirac also showed how quantum waves for many electrons had to be constructed, incorporating the philosophically intriguing fact that any two of these particles are absolutely identical and so cannot be distinguished in any way. This produced the definitive understanding of earlier rules about how quantum mechanics explains the periodic table of the elements, and provided the basis for the theory of metals and the interior of stars.
Like all scientists at the highest level, Dirac was not afraid to descend from the pinnacle and discuss more down-to-earth matters. Here are two examples. Much of our knowledge comes from light scattered by matter; in particular, that is how we see. In a clever stroke of lateral thinking, Dirac realized that the quantum symmetry between waves of light and waves of matter implied that it is also possible for material particles to be scattered by light, a ghostly possibility that could be observed, as he showed in 1933 in a paper with Peter Kapitza. This was observed for the first time about ten years ago and the manipulation of atoms by laser beams is now a thriving area of applied quantum mechanics – a fact recognized with a Nobel prize last year (Physics World November 1997 p51, print version).
The second example is his Second World War work. In the Manhattan Project to develop the first nuclear bombs, it was necessary to separate isotopes of uranium. One class of methods involved the centrifugal effects of fluid streams that were made to bend. Dirac put the theory of these techniques on a firm basis, and indeed his work in this field has been described as seminal.
Dirac stories
It is not my intention to write about what sort of person Dirac was. But I must mention the genre of “Dirac stories”. He was so unusual in the logic and precision of his interaction with the world, both in and out of physics, that tales have become attached to him and have acquired a life of their own. I suppose it matters to a historian whether they are true or apocryphal (or as Norman Mailer says, “factoids”), but to us they have a deeper resonance that transcends fact. Resisting temptation, I retell just two less well known ones.
Like many scientists, Dirac was known to sleep during (other people’s) lectures, and then wake and suddenly make a penetrating remark. Once, a speaker stopped, scratched his head and declared: “Here is a minus where there should be a plus. I seem to have made an error of sign.” Dirac opened one eye and said: “Or an odd number of them.” Another time, Dirac was at a meeting in a castle, when another guest remarked that a certain room was haunted: at midnight, a ghost appeared. In his only reported utterance on matters paranormal, Dirac asked: “Is that midnight Greenwich time, or daylight saving time?”
Dirac’s writing was famous for its clarity and simplicity. Every physicist knows his Principles of Quantum Mechanics – such a perfect and complete summary of his views that in later years his lectures consisted of readings from it. There is the story that he was once present when Niels Bohr was writing a scientific paper – with many hesitations and redraftings, as was his custom. Bohr stopped: “I do not know how to finish this sentence.” Dirac replied: “I was taught at school that you should never start a sentence without knowing the end of it.”
Many physicists have spoken of Dirac with awe. John Wheeler, referring to the sharp light of his intelligence, said “Dirac casts no penumbra.” Niels Bohr said: “Of all physicists, Dirac has the purest soul.” He is also reported as saying (I cannot now find this quotation): “Dirac did not have a trivial bone in his body.”
The mathematician Mark Kac divided geniuses into two classes. There are the ordinary geniuses, whose achievements one imagines other people might emulate, with enormous hard work and a bit of luck. Then there are the magicians, whose inventions are so astounding, so counter to all the intuitions of their colleagues, that it is hard to see how any human could have imagined them. Dirac was a magician.
Sir Michael Berry is Royal Society Research Professor at the H H Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, UK. This is a version of a talk delivered in September 1997 at the official opening of Dirac House, the headquarters of Institute of Physics Publishing.