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Dirac Centennial |

## Paul Dirac: predictor of antimatter

*Professor Peter Goddard, Master of St John's College, Cambridge*

The 8th of August this year sees the centenary of the birth of one of the founders of quantum theory. Paul Dirac was Lucasian Professor of Mathematics in Cambridge from 1932 to 1969 and is numbered alongside Newton, Maxwell, Einstein and Rutherford, as one of the greatest physicists of all time.

Dirac: by permission of the Master and Fellows of St John's College, Cambridge |

Dirac's father, who was Swiss by birth, was the Head of Modern Languages at the Merchant Venturers' School, where Paul Dirac received his secondary education from 1914 to 1918. The young Dirac was required always to converse with his father in French, and he would remain silent unless he could express himself well in the language. This was probably the origin of his legendary taciturnity.

Dirac took his first degree from Bristol University, graduating with First Class honours in electrical engineering at the age of 19, in the midst of the Depression. Unable to get a job, he remained at Bristol and completed the degree course in mathematics as well. Two years later, in 1923, he succeeded in getting a grant to begin research at St John's College, Cambridge. He had hoped to do research in relativity under Ebenezer Cunningham. But, sensing he was near the end of his research career, Cunningham passed Dirac on to R.H. Fowler. Instead of beginning research on geometrical aspects of relativity, Dirac was introduced by Fowler to atomic physics. Very quickly he was plunged into the most profound problems of atomic physics, which was then in an incomplete and paradoxical state.

On 28 July 1925 Dirac attended a seminar by Werner Heisenberg of Göttingen, who at the end of his talk described some new ideas of his, which later turned out to be the origins of his formulation of quantum mechanics. Fowler, appreciating the potential importance of Heisenberg's remarks, encouraged Dirac to try to understand them.

Heisenberg's approach had been to build up a theory entirely in terms of observable quantities, and the observable quantities in atomic theory were mostly concerned with two states of an atom. In this way, Heisenberg was led to associate two-dimensional arrays with observable quantities, and to develop an algebra of such arrays based on physical motivation, without realising that he was re-inventing the algebra of matrices already known to pure mathematicians. The most striking feature of this algebra is that it is non-commutative, that is u times v is not equal to v times u. Out walking on the Gog Magog hills one Sunday in October 1925, Dirac was thinking about the non-commutativity in Heisenberg's algebra, the difference uv - vu, when he suddenly realised the connection between this and a quantity in classical mechanics called the Poisson bracket. Excited, but unable to decide whether there was anything in the idea, he hurried back to his lodgings. His notes and textbooks contained nothing on Poisson brackets and, being Sunday, the libraries were shut. He spent an anxious night waiting, but with his confidence gradually growing, until the libraries opened and he was able to verify that the analogy was perfect.

In the resulting paper, Dirac solved the central problems of atomic theory which had been baffling theoretical physicists for the previous decade. He said in later years that nothing had ever given him as much satisfaction as this first major discovery.

Then a paper by Erwin Schrödinger appeared giving an apparently completely different version of quantum mechanics, "wave mechanics", couched in mathematics more familiar to many physicists. At first Dirac's reaction was hostile but by August he had mastered Schrödinger's formalism and, in arguably his greatest paper, he established the general mathematical framework in which quantum mechanics is now formulated. Within this framework, the distinctions between the Heisenberg and Schrödinger approaches disappear; they are just different choices of systems of coordinates.

Next Dirac applied his general formalism to the electromagnetic field, showing it to be described in quantum-mechanical terms by an assembly of particles (photons), just as had been conjectured by Planck and Einstein in the work which had motivated much of the development of quantum mechanics. In so doing, he brought together the various strands in the development of the subject into a coherent whole, removing once and for all the dichotomy between waves and particles, and simultaneously he created the subject of quantum electrodynamics.

At the end of 1927, the major outstanding problem was how to reconcile quantum mechanics with the other revolutions in physics that had been made at the beginning of the twentieth century, relativity. Many thought that this problem had already been solved but Dirac saw clearly that the supposed solution was unsatisfactory. In 1928, in two papers that are probably his most famous, he produced his relativistic quantum theory of the electron by constructing what came to be known universally as the Dirac equation. It made the previous candidate theory look thoroughly anaemic. Dirac's theory required the electron to have very definite properties (spin and magnetic moment), in agreement with experiment. Moreover, as Dirac pointed out in 1930, it necessitated the existence of another particle with the opposite electric charge and the same mass as the electron. In this way, it predicted the existence of the positron, the anti-particle of the electron, which was confirmed by experiment in 1932. Dirac's prediction of antimatter was described by Heisenberg as "the most decisive discovery in connection with the properties or the nature of elementary particles ... [It] changed our whole outlook on atomic physics completely".

With these developments, quantum mechanics was in an essentially complete form. Dirac's enormous contributions to it were acknowledged by his election to the Royal Society in 1930 and to the Lucasian Professorship in 1932, and by the award of the Nobel Prize for Physics for 1933, which he shared with Schrödinger.

Dirac continued to lecture on quantum theory in Cambridge until his retirement from the Lucasian Chair in 1969. He supervised comparatively few research students, taking the view that the fundamental problems on which he worked were not suited for most students. For many years, his was the first course in quantum theory that a Cambridge student would take. His presentation followed very closely the treatment in his book but, even so, many would attend the course more than once. His delivery conveyed an integrity and coherence of viewpoint that made the line of argument seem inevitable.

If Dirac had done nothing after the early 1930s, he would still be ranked amongst the greatest names in physics, but his work continued unabated in his later years. The first years of his research career were in a golden age in physics, which he played a major part in creating. His later achievements were not on the same scale but neither were those of other physicists. In the following years he worked on a number of topics, writing many papers of great originality. His work on the possible existence of magnetic monopoles contained the seed of the topological ideas that now play a major role in theoretical physics. The significance of this and much of his other work, such as his approach to constraints in classical mechanics, has grown with the years, and his influence is now as great as ever.

His work did not stop on his retirement from the Lucasian Chair. In 1971 he accepted an appointment as Professor of Physics at Florida State University in Tallahassee. There, having Dirac in the Physics department seemed comparable with having Shakespeare in the English department. He continued to work on theoretical physics, until shortly before his death on 20 October 1984.

*Professor Michael B Green, Department of Applied Mathematics and Theoretical Physics*

Geometry and ArithmeticProfessor Charles B Thomas, Department of Pure Mathematics and Mathematical Statistics |

An American donut |

"Calabi-Yau" manifolds of dimensions 2 and 4 |

Folding, wrinkling and crumplingProfessor L. Mahadevan, Department of Applied Mathematics and Theoretical Physics |