Mathematics and Physics have always had a close relationship, though the level of friendliness between the two disciplines has, of course, varied from time to time. This little book focuses on a physicist who was trained as a mathematician and who brought a very mathematical turn of mind to bear on physical problems in a very fruitful way.

As Peter Goddard says in his introduction, "Dirac cited mathematical beauty as the ultimate criterion for selecting the way forward in theoretical physics." This method was sometimes enormously successful, as when Dirac predicted the existence of the positron simply because his mathematics required such particles to exist. This is a feat on the order of the prediction of the existence of Uranus and Neptune on the basis of the gravitational anomalies in the solar system. In both cases, a mathematical theory led physicists to postulate, and then verify, the existence of some objective entity; they both demonstrate the real power of mathematics as a tool to understand the universe.

The story isn't all positive, though, as Abraham Pais makes clear in his survey of Dirac's life and work. After his initial successes in the 1930s, Dirac seems to have been far less fortunate in his research. Though he remained an interesting and powerful voice, he seemed seriously distressed by the mathematical difficulties in quantum theory, most particularly in quantum electrodynamics, of which he said "the resulting theory is an ugly and incomplete one." The "renormalization" procedure which is used to deal with the divergent expansions in the theory did not satisfy him at all, and he spent great energy in trying to find the "right" theory. Pais quotes a letter that expresses his final attitude on this subject:

The rules of renormalization give surprisingly, excessively good agreement with experiments. Most physicists say that these working rules are, therefore, correct. I feel that is not an adequate reason. Just because the results happen to be in agreement with observation does not prove that one's theory is correct.

At this point, perhaps we should see Dirac's mathematical instincts as a roadblock to his success as a physicist. Or perhaps not. Recent developments in mathematical physics have once again brought it close to mathematics, as the final two papers in the book (dealing with magnetic monopoles and with connections between physics and geometry) make clear. We may find, at the end, that Dirac was right to be unsatisfied with QED. At the time, however, it certainly set him on a lonely way, his thinking diverging in many ways from the orthodoxies of the time.

This book, intended as a sort of tribute to Paul Dirac, consists of a short memorial address by Stephen Hawking plus four papers related to Dirac and his work. The memorial address was given in 1995, when a plaque commemorating Dirac's memory was dedicated in Westminster Abbey. The first paper is a historical survey by Abraham Pais, giving us an overview of Dirac's life and work. The following three papers are more technical, dealing with "Antimatter" (Maurice Jacob), "The monopole" (David I. Olive), and "The Dirac equation and geometry" (Michael F. Atiyah). These papers give us modern views of some of the ideas that originated with Dirac. The first, on antimatter, focuses mostly on experimental work, ranging from the original work confirming the existence of the positron to very recent work in particle physics. The last two essays are more mathematical, with Atiyah's (unsurprisingly) requiring the most background.

For a mathematician, the most fascinating aspect of the book are the issues it raises about the interaction of mathematics and physics. As we pointed out above, Dirac's life displays both the power and the limitations of mathematical methods in physics, and both aspects should be pondered by those of us who teach mathematics (particularly if we teach mathematics to students of physics). Also of value is to note the *kind* of mathematics that appears in the last two essays. This goes far beyond the standard differential equations that we associate with applications to physics, reaching the farthest ends of differential and algebraic geometry and other advanced areas that we tend to regard as very "pure" mathematics. One even meets plane lattices and fractional linear transformations, an epsilon away from the modular forms that have been at the center of this reviewers work in number theory.

Overall, this book is certainly worth looking at. While it's probably too hard for our students, it is valuable reading for their professors, who probably know less about what's going on in theoretical physics than we should.

Fernando Q. Gouvêa is associate professor and chair of the Department of Mathematics and Computer Science at Colby College. His research interests are in number theory and the history of mathematics.