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Book Review

Paul Dirac: The Man and His Work Reviewed by Clifford H. Taubes

Review of Paul Dirac: The Man and His Work d D = i Abraham Pais, Maurice Jacob, David I. Olive, and dt Michael F. Atiyah on complex valued functions. In higher dimen- Cambridge University Press, 1998 sions the analogous operator D acts on certain vec- ISBN 0-521-58382-9 tors of complex valued functions, the ever myste- Hardcover, $19.95 rious spinors. Here are D’s salient features: This operator is first order, it is symmetric, and its square is minus the standard Laplacian. In a This review concerns the book Paul Dirac: The Lorentz signature metric, the square is the wave Man and His Work, which consists of a collection operator. of four essays on the life and work of Paul Dirac. In physics the operator D was used by Dirac to The book also contains a short address by Stephen predict the behavior of relativistic electrons, and Hawking on the occasion of the dedication of a it led directly to Dirac’s prediction of the existence plaque at Westminster Abbey in honor of Dirac. The of antiparticles. (I will explain this below.) These essayists are Abraham Pais (a physicist and histo- days the Dirac operator and the notion of an an- rian of physics), Maurice Jacob (a physicist), David tiparticle are fundamental to physics’s descrip- Olive (a mathematical physicist), and Sir Michael tion of all elementary, spin 1/2 particles. Thus, Atiyah. Peter Goddard served as the book’s editor. Dirac’s equation profoundly changed humanity’s The book is short and wonderful. view of physical reality. Jacob’s essay describes the The essays describe Dirac’s influence in both influence in modern physics of Dirac’s equation and physics and mathematics. The list of Dirac’s fun- his ideas about antiparticles. damental ideas is simply stunning. One might ask In mathematics Dirac’s operator lies at the very which physicists of the twentieth century will be heart of a great deal of late twentieth-century remembered in the thirtieth? Surely Dirac and Ein- geometry; moreover, it still regularly delivers pro- stein and perhaps a few more. In any event, Dirac found surprises (as this writer can avidly attest). is truly one of the heroes of twentieth-century And the Dirac operator promises to deliver sur- physics. The essay by Pais consists of a concise bi- prises into the twenty-first century. The Dirac op- ography of Dirac, and so describes Dirac’s most erator bewitches more than just analysts, because profound scientific achievements. Pais’s essay is some version can be written down on any Rie- full of charming quotes from and about Dirac, who mannian (in fact, conformal) manifold. In this con- was an iconoclast, to say the least. text it is completely natural yet still mysterious. The Dirac is most famous for his celebrated differ- Dirac operator sees something very deep about ential equation, which revolutionized both physics manifolds, and the nature of its vision is still be- and mathematics. This equation is simplest in the yond our ken. Atiyah’s essay sketches the math- case where the space in question is the circle pa- ematics behind Dirac’s operator while describing rameterized by an angle t between 0 and π. In this some of its mathematical incarnations and appli- cations. context the Dirac equation is defined by the oper- Less well known to mathematicians is Dirac’s ator work on magnetic monopoles. These are hypo- Clifford H. Taubes is professor of mathematics at Harvard thetical particles which consist of a single magnetic University. His e-mail address is chtaubes@math. pole, solely “north” or solely “south”. Their exis- harvard.edu. tence restores to Maxwell’s equations for electric-

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ity and magnetism a fundamental symmetry which Of course, Dirac was aware of the is lost in the presence of charges and currents. negative energy problem. But still, his (Modulo a sign, this symmetry interchanges the D explained various major paradoxes electric and magnetic fields.) In spite of some se- about the behavior of electrons, and so rious searching (in laboratories, on earth, and in he was not quick to abandon it. And space), no monopoles have been found to date. In after a time Dirac resolved the nega- any event, Dirac pointed out that the fact that all tive energy problem, and this resolu- electric charges are integer multiples of a single tion leads unavoidably to the anti- constant is a direct consequence of the existence electron. (This particle is now called in this universe of a single monopole. There are the “positron”.) To explain, understand those who speculate that Dirac’s work on magnetic first that the resolution invokes an monopoles will prove as central to twenty-first extra feature of electrons that was well century geometry as the Dirac equation is to geom- known to Dirac but not yet introduced etry of today. The essay by Olive describes Dirac’s here. This feature is the “exclusion work on monopoles and its current remarkable in- principle”, which was first enunciated carnations in certain supersymmetric field theo- by Pauli and which leads to the expla- ries. nation of chemistry. The exclusion The essays of both Olive and Atiyah sparkle to principle asserts that no two electrons can occupy the eyes, and I finished each wishing for more. the same quantum state. (Chemistry appears here Meanwhile, Pais’s essay was entertaining and very for the following reason: The chemical properties touching, for I still have heroes and Dirac is one of an atom are essentially determined by the states of them. Finally, Jacob’s lecture was fascinating, but of its orbital electrons. Meanwhile, the exclusion probably dry without prior knowledge of the principle prevents all but one of the electrons in physics or, at minimum, the mathematics of an- an atom from occupying any given orbital state. tiparticles. For those lacking this knowledge, what Thus, atoms with different numbers of electrons follows is a brief introduction to the subject. must have different chemical properties.) Roughly the antiparticle story is as follows. In With the exclusion principle understood, Dirac the absence of interparticle forces, the dynamics made the audacious proposal that essentially all of a single electron moving in space is controlled of the negative energy states in the universe are by the Dirac operator in the following sense: The already filled with electrons. Then, Pauli’s exclu- possible states of the electron at a given time t are sion principle makes these states inaccessible to described by a spinor (which is to say a vector of the remaining electrons, and the negative energy functions on which the Dirac operator can act) instability disappears. Of course, in making such whose absolute value squared integrates to one. a proposal Dirac was forced to consider the ob- Then the function which is the square of the ab- servational consequences of the occupation of es- solute value of the spinor is meant to be the prob- sentially all of the negative energy states. What ability distribution for finding the electron at points Dirac realized was that this assumption is essen- in space. tially unobservable when all negative energy states Here is how dynamics enters the story: Having are filled. Moreover, an empty negative energy described the electron state at time t by a certain state is essentially indistinguishable from a par- vector of functions whose norm square integrates ticular positive energy state which is filled by the to one, one can obtain the state at any later time antiparticle. The idea here is that a missing nega- from the original via a certain unitary action of the tive charge is indistinguishable from a present group of translations on the line. The key point here positive charge. (Since interparticle interactions is that the generator of this unitary action is sup- would unavoidably knock electrons from filled posed to be the operator D. In particular, the negative energy states into positive energy states, eigenvalues of D are to be interpreted as the al- Dirac was forced to consider the observational lowed energies of electrons whose probability dis- consequences of a relatively small number of neg- tributions are time independent. ative energy states.) So far so good, except that D’s spectrum is not The filling of the negative energy states led bounded from below. For example, the eigenval- Dirac to the antielectron. Remarkably enough, the ues of D for the case where space is the 1-dimen- first observations of the particle occurred only a sional unit radius circle have the form n where n few years later. can be any integer. The fact that D has accessible As I remarked at the outset, this book is quite states which have arbitrarily negative eigenvalues short; it is not a text to learn a great deal about ei- renders D useless for the description of our uni- ther Dirac’s physics or math, or even about his life. verse. Indeed, such a universe would not last long, Rather, this book is a small gem of an introduc- since the slightest generic perturbation would send tion to all three of these subjects. all the electrons pathologically cascading ever deeper into the negative energy states.

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