Introduction

The differential operator introduced by Dirac in his study of the quantum theory of the electron has turned out to be of fundamental importance both for physics and for mathematics. Essentially the operator is a formal square-root of the wave operator or, with a different signature, of the Laplacian. In this lecture I will attempt to survey its role in mathematics. I will begin with the algebraic underpinnings, which go back to Hamilton and Clifford, and I will then go on to the role of the Dirac operator in Riemannian geometry. In particular I shall discuss various aspects of the index theorem. Finally I will briefly allude to the very recent results on four-dimensional manifolds, arising from new physical ideas of Seiberg and Witten.

Algebraic background

Let us begin by recalling Hamilton's quaternions. These are generated, over the real numbers, by three ‘imaginary” quantities i, j, k, satisfying the relation:

For a general quaternion

We define its conjugate by

Then the norm-squared of x is given by

The quaternions of unit norm form a group (the 3-sphere S3) with the inverse given by

This group acts by left and right multiplication on R4 = C2, giving the two ‘spin representations’ of S3. The group also acts by conjugation

on R3 (the imaginary quaternions, where x0= 0). This gives a double covering

with kernel ± 1.

For three and four dimensions the quaternions provide all one needs to understand spinors (and in due course the corresponding Dirac operator).

Paul Adrien Maurice Dirac was one of the founders of quantum theory and the author of many of its most important subsequent developments. He is numbered alongside Newton, Maxwell, Einstein and Rutherford as one of the greatest physicists of all time. He was born in Bristol on 8 August 1902 and died on 20 October 1984 in Tallahassee, Florida. On Monday 13 November 1995, after evensong, a plaque was dedicated in Westminster Abbey commemorating Paul Dirac. The simplicity and almost austere beauty of the plaque's design reflected in some ways the qualities of Dirac's unique intellect.

After graduating from Bristol University with a first class degree in engineering, Dirac stayed on to study mathematics there before obtaining a studentship in 1923 to enable him to undertake research at St John's College, Cambridge. In 1925, he became a Fellow of St John's College. In 1932, he was elected Lucasian Professor of Mathematics in the University. The Lucasian Professorship was once held by Sir Isaac Newton, and the present holder, Stephen Hawking, was present in the Abbey to give an address at the service of commemoration and the text of this address is included in this volume.

Dirac shared the 1933 Nobel Prize for Physics with Erwin Schrödinger. After retirement from the Lucasian chair in 1969, he accepted a research professorship at the Florida State University in Tallahassee. There he continued to work on fundamental physics, frequently returning to St John's College for summer visits, until shortly before his death.

In the year 1902, the literary world witnessed the death of Zola, the birth of John Steinbeck, and the first publications of The Hound of the Baskervilles, The Immoralist, Three Sisters, and The Varieties of Religious Experience. Monet painted Waterloo Bridge, and Elgar composed Pomp and Circumstance, Caruso made his first phonograph recording and the Irish Channel was crossed for the first time by balloon. In the world of science, Heaviside postulated the Heaviside layer, Rutherford and Soddy published their transformation theory of radioactive elements, Einstein started working as a clerk in the patent office in Berne, and, on August 8, Paul Adrien Maurice Dirac was born in Bristol, one of the children of Charles Dirac (1866-1936), a native of Monthey in the Swiss canton of Valais, and Florence Holten (1878-1941), daughter of a British sea captain. There was also a brother two years older, Reginald, whose life ended in suicide, in 1924, and Beatrice, a sister four years younger. About his father Dirac has recalled:

It is a great privilege to speak to you on this occasion in which we commemorate Paul Dirac. In common with many of my friends in the audience, I enjoyed the good fortune of hearing the lectures on quantum mechanics delivered by him in Cambridge. Actually I was doubly fortunate as, in my year, 1959–60, he added a second course, extending beyond the material in his famous book.

Not only did we learn quantum mechanics as never before, but, very gently, we were shown a standard of logical presentation and clarity, indeed an aesthetic of logic, that was unforgettable. I believe I can say that this experience has affected many of us deeply and provided us with an ideal to which we struggle to aspire in our own research and teaching.

In Dirac's hands the beauty of mathematical logic and rational argument was not merely a tool for establishing sound proofs. Rather it could be a weapon of discovery that could lead to the most unexpected yet perfectly valid conclusions, which, once understood and assimilated, were unassailable in their beauty and Tightness. The importance of this is that it is the discovery of the unexpected truth that changes the direction of scientific development, in both theoretical and experimental work. Dirac repeatedly showed how implacable yet beautiful logic could achieve these ends, as the other speakers also demonstrate.

I already have gray hair but I belong to a generation which grew up in physics calculating Feynman graphs and using the CPT invariance of Quantum Field Theory. The world would look very different if we could reverse the flow of time (an operation denoted by T), inverse all directions in space (an operation denoted by P) and change all particles into their antiparticle (an operation denoted by C). Yet the laws of physics would remain the same and all phenomena would occur in the same way. Our present understanding of physics implies the existence of antimatter, and all the properties of antimatter are predictable from the known properties of matter. All this looked so powerful, so beautiful and almost so natural to us, as we were learning modern physics in the late 1950s and early 1960s. The two ways to read the same simple Feynman graph, using it to describe, for instance, either the exchange of a photon between two electrons, or electron–positron annihilation and formation through one photon, looked like an obvious part of the calculation rules. This is shown in Figure 2.1. One can read it horizontally. This is scattering. One can also read it vertically. This is annihilation and pair formation. The same term can be used to describe both processes.

In this and the next two chapters we discuss the conceptual cornerstone of classical physics, namely Newtonian mechanics. We shall see how Newton, employing certain general principles of reasoning as guides, used the regularities of the solar system, in the form of Kepler's laws, to arrive at his own law of gravitation. He then reversed this reasoning process to derive not only Kepler's laws but also to account for the phenomena of the ocean tides. The stereotypical image of Newton represents him as the embodiment and culmination of the Age of Reason, as in the 1689 Godfrey Kneller portrait of him in the robes of a Cambridge don. It may come as a surprise to learn that Newton had serious and long-standing interests in certain unorthodox theological subjects and in alchemy

ISAAC NEWTON

Isaac Newton was born in the small English village of Woolsthorpe, in Lincolnshire, on Christmas day, 1642, eleven months after Galileo's death. (On the Gregorian calendar that was not adopted in England until 1752, Newton's birth date is January 4, 1643.) His father (1606–1642), also named Isaac Newton, died three months before his son's birth. Newton was a sickly baby and was not expected to live long. In 1645 his mother, Hannah Ayscough (d. 1679), married a wealthy minister, Barnabas Smith (1582–1653), and moved to a neighboring village, leaving young Isaac to be raised by his maternal grandmother.

This book has grown out of an elective, one-semester junior/senior level interdisciplinary course I have taught for several years to students in arts and letters, science, and engineering at the University of Notre Dame. It allows one to examine a selection of philosophical issues in the context of specific episodes of the development of physical laws and theories. Many students with science and engineering backgrounds find this exercise informative – for some unsettling, but still rewarding. Although a major goal of the exposition is to impress upon the reader the essential and ineliminable role that philosophical considerations have played in the actual practice of science, more space is devoted to the history and content of science than to philosophy per se. The reason for this is that I believe that meaningful and useful philosophy of science can only be done within the context of the often tortuous historical route to new insights. Another way to put this is that it takes a lot of history of science to anchor even a little philosophy of science.

Some necessary background from the history of ancient and early modern science is presented first, but major emphasis is given to the immediate precursors to and the content of the watersheds of twentieth-century physics: relativity and, especially, quantum mechanics. This is not a systematic exposition of either the history or the philosophy of science, but an individualistic, perhaps to some even an idiosyncratic, selection of topics and episodes from the history and philosophy of physics.

In order to contrast by means of specific examples two different ways of reaching general conclusions about the phenomena of reality, we compare the work of Aristotle with that of Sir Francis Bacon (1561–1626). As we shall see, the difference is one of emphasis and execution rather than of principle. Aristotle favors the discussion of general principles that can be arrived at on the basis of an intuitive grasp of natures, while Bacon stresses the importance of slow and careful induction from specific cases.

ARISTOTLE

Aristotle was born in 384 B.C. at Stagira, a Greek colonial town on the peninsula of Chalcidice in Macedonia on the northern shores of the Aegean. His father Nicomachus (d. c. 374 B.C.) was a physician to Amyntas II of Macedon (d. 370/369 B.C.), grandfather of Alexander the Great (356–323 B.C.). In 367 B.C. Aristotle, at age seventeen, came to Athens and began his studies as a pupil in Plato's Academy. Plato was at that time a man of sixty. Aristotle's life falls rather naturally into three major periods. He spent twenty years at the Academy until Plato's death (348/347 B.C.). The anti-Macedonian mood prevalent in Athens may have been partly responsible for his leaving that city and traveling for the next twelve years. In this second period he went to Macedon in 342 B.C. at the behest of Philip II (382–336 B.C.) to tutor that king's son, Alexander. During his thirteenth through sixteenth years, Alexander was taught by Aristotle.

In the previous two chapters we discussed, largely in terms of modern mathematical notation, Newton's three laws of motion and his deduction of the law of universal gravitation. Here we attempt to give the reader some appreciation of the type of mathematical arguments that Newton actually employed in the Principia. Then we show how his laws of motion and of gravitation can be used to deduce Kepler's three laws as necessary logical consequences of this overarching theory. (Some of the material here is a bit more technically involved that that in previous chapters. The reader whose tastes such details do not suit can simply move onto Chapter 10.) As we have already seen in the last chapter, Newton accepted Kepler's laws and was able to argue from them to the inverse-square nature of the gravitational attraction between any two bodies. He then reversed the line of argument and, assuming the validity of the law of gravitation, derived Kepler's three laws of planetary motion. For example, we find in the Principia:

This contains Kepler's first and second laws together. Of course, we expect to be able to do more than simply recover Kepler's laws that were, after all, used to deduce the law of gravitation itself.

Both empirical considerations and mechanical models of the aether provided the foundation upon which Maxwell built his classical theory of electromagnetism. In this chapter we first discuss some formal aspects of this theory and then turn to observational consequences of it that led to a profound conflict between the principles of classical mechanics and those of electromagnetic theory. This will set the stage for our treatment of relativity in later chapters.

MAXWELL'S EQUATIONS

Prior to Maxwell's great synthesis, the basic laws of the separate fields of electricity and of magnetism were, respectively, Coulomb's law for the electric field E produced by a static point charge q and the Biot–Savart law for the magnetic field B produced by a wire carrying a current i. (See Section 14.A for mathematical statements of these laws and for the mathematical details that support many of the claims made in the present section.) Each of these two laws involves a proportionality constant (say, k1 and k2, respectively) that must be determined by experiment. (These are analogous to the constant G in Newton's law of gravitation, Eq. (8.4).) That is, the constants k1 and k2 are fixed independently and by different types of phenomena (electrostatics and magnetostatics, respectively). It turns out that the ratio k1/k2 has the dimensions of a velocity squared that we denote by c2 for reasons that will become evident shortly.

In the popular mind Albert Einstein is identified as the architect of the theory of relativity and as the embodiment of modern scientific genius. Also, among a somewhat older set, he is probably remembered as a man with great humanitarian concerns, as well as the person whose theoretical work was the foundation for the atomic bomb. For one who was to become a singular legend even in his own lifetime, Einstein's background and early years were inauspicious enough. In fact, in later life his own assessment of his ability was that ‘I have no particular talent, I am merely extremely inquisitive.’ Like Newton, he seems to have had an exceptional power of concentration. We begin with a sketch of Einstein the man and then turn to his arguments for the relativity principle.

ALBERT EINSTEIN

Einstein was born in Ulm, Germany, into a family none of whose members had shown any brilliance. One year after Albert's birth, the family moved to Munich. His father ran a series of small businesses, most of which ended in failure. Young Albert was not a particularly good pupil, largely because he disliked the rigid German school system that placed great emphasis on learning by rote. He attended a Catholic elementary school, where he was the only Jewish student in the class, and then entered the Luitpold Gymnasium in Munich when he was ten.