**The English physicist Paul Adrien Maurice Dirac (1902-1984) formulated a most general type of quantum mechanics and a relativistic wave equation for the electron which led to the prediction of positive electrons, the first known forms of antimatter.**

Paul Adrien Maurice Dirac was born on Aug. 8, 1902, at Monk Royal in Bristol, England, the son of Charles Adrien Ladislas Dirac and Florence Hannah Holten Dirac. Paul received his secondary education at the old Merchant Venturers' College and, at the age of 16, entered Bristol University. He graduated 3 years later in electrical engineering. Unable to find employment, he studied mathematics for 2 years before moving to Cambridge as a research student and recipient of an 1851 Exhibition scholarship award. His student years (1923-1926) at Cambridge saw the emergence of the mathematical formulation of modern atomic physics in the hands of Louis de Broglie, Werner Heisenberg, Erwin Schrödinger, and Max Born. It was therefore natural that Dirac's attention should turn to a cultivation of mathematics most directly concerned with atomic physics.

## Negative Kinetic Energy

Dirac's first remarkable contribution along these lines came before he earned his doctorate in 1926. In his paper "The Fundamental Equations of Quantum Mechanics" (1925), Dirac decided to extricate the fundamental point in Heisenberg's now famous paper. Before Heisenberg, computation of energy levels of optical and x-ray spectra consisted in a somewhat empirical extension of rules provided by Niels Bohr's theory of the atom. Heisenberg succeeded in grouping terms connected with energy levels in columns forming large squares and also indicated the marvelously simple ways in which any desired energy level could be readily calculated. Dirac found that what Heisenberg really wanted to achieve consisted in a most general type of operation on a "quantum variable" x which was done by "taking the difference of its Heisenberg products with some other quantum variable."

At that time neither Heisenberg nor Dirac had realized that the "Heisenberg products" corresponded to operations in matrix calculus, a fact which was meanwhile being proved by Born and Pascual Jordan in Göttingen. They showed that the noncommutative multiplication of the "Heisenberg quantities" could be summed up in the formula (p X q) (q X p) h/(21), where h is Planck's constant and p and q some canonically conjugate variables. Independently of them, Dirac also obtained the same formula, but through a more fundamental approach to the problem. Dirac's crucial insight consisted in finding that a very simple operation formed the basis of the formula in question. What had to be done was to calculate the value of the classical Poisson bracket [p, q] for p and q and multiply it by a modified form of Planck's constant.

That such a procedure yielded the proper values to be assigned to the difference of p X q and q X p was only one aspect of the success. The procedure also provided an outstanding justification of the principle of correspondence, tying into one logical whole the classical and modern aspects of physics. Dirac once remarked that the moment of that insight represented perhaps the most enthralling experience in his life.

But the most startling result of Dirac's equation for the electron was the recognition of the possibility of negative kinetic energy. In other words, his equations implied for the electron an entirely novel type of motion whereby energy had to be put into the electron in order to bring it to rest. The novelty was both conceptual and experimental and received a remarkably quick elucidation.

The experimental clarification came when C. D. Anderson, doing cosmic-ray research in R. A. Millikan's laboratory in Pasadena, Calif., obtained on Aug. 2, 1932, the photograph of an electron path, the curvature of which could be accounted for only if the electron had a positive charge. The positively charged electron, or positron, was, however, still unconnected with the negative energy states implied in Dirac's theory of the electron. The work needed in this respect was largely done by Dirac, though not without some promptings from others. A most lucid summary of the results was given by Dirac in the lecture which he delivered on Dec. 12, 1933, in Stockholm, when he received the Nobel Prize in physics jointly with Schrödinger.

## World of Antimatter

The most startling consequences of Dirac's theory of the electron consisted in the opening up of the world of antimatter. Clearly, if negative electrons had their counterparts in positrons, it was natural to assume that protons had their counterparts as well. Here Dirac argued on the basis of the perfect symmetry that according to him had to prevail in nature. As a matter of fact, it was a lack of symmetry in Schrödinger's equation for the electron that Dirac tried to remedy by giving it a form satisfactory from the viewpoint of relativity.

All this should forcefully indicate that Dirac was a thinker of most powerful penetration who reached the most tangible conclusions from carrying to their logical extremes some utterly abstract principles and postulates. Thus by postulating the identity of all electrons, he was able to show that they had to obey one specific statistics. This fact in turn provided the long-sought clue for the particular features of the conduction of electricity in metals, a problem with which late classical physics and early quantum theory grappled in vain. This attainment of Dirac paralleled a similar, though less fundamental, work by Enrico Fermi, so that the statistics is now known as the Fermi-Dirac statistics.

This contribution of Dirac came during a marvelously creative period in his life, from 1925 to 1930. Its crowning conclusion was the publication of his Principles of Quantum Mechanics, a work still unsurpassed for its logical compactness and boldness. The latter quality is clearly motivated by Dirac's unlimited faith in the mathematical structuring of nature. The book is indeed a monument to his confidence that future developments will provide the exact physical counterparts that some of his mathematical symbols still lack.

A telling measure of Dirac's main achievements in physics was the recognition that greeted his work immediately. In 1932 he was elected a fellow of the Royal Society and given the most prestigious post in British science, the Lucasian chair of mathematics at Cambridge. He received the Royal Society's Royal Medal in 1939 and its Copley Medal in 1952. He was a member of many academies, held numerous honorary degrees, and was a guest lecturer in universities all over the world. He married Margaret Wigner, sister of Nobel laureate Eugene P. Wigner, in 1937.

The second half of Dirac's working life was occupied mainly with cosmology and the subject of "large numbers," or numbers with cosmic significance. In the 1972, he accepted a post as professor of physics at Florida State University, and he continued there until his death in Tallahassee on October 20, 1984.

## Further Reading on Paul Adrien Maurice Dirac

Humorous details on Dirac's life can be found in George Gamow, Biography of Physics (1961), together with a not too technical discussion of Dirac's theory of holes. See also Niels H. de V. Heathcote, Nobel Prize Winners in Physics, 1901-1950 (1954). For a rigorous account of Dirac's role in quantum mechanics, the standard work is Max Jammer, The Conceptual Development of Quantum Mechanics (1966). Background works which discuss Dirac include James Jeans, Physics and Philosophy (1942), and Barbara Lovett Cline, The Questioners: Physicists and the Quantum Theory (1965).

## Additional Biography Sources

Dirac, Paul, The Principals of Quantum Mechanics, Clarendon Press, 1930.

Dirac, Paul, Spinors in Hilbert Space, University of Miami Center for Theoretical Studies, 1974.

Dirac, Paul, General Theory of Relativity, Wiley, 1975.

Kursunolgu, Behram N., and Eugene P. Wigner, eds., Reminiscences About a Great Physicist: Paul Adrien Maurice Dirac, Cambridge University Press, 1987.