Paul Dirac - Career

Career

Dirac established the most general theory of quantum mechanics and discovered the relativistic equation for the electron, which now bears his name. The remarkable notion of an antiparticle to each particle - i.e. the positron as antiparticle to the electron - stems from his equation. He was the first to develop quantum field theory, which underlies all theoretical work on sub-atomic or "elementary" particles today, work that is fundamental to our understanding of the forces of nature. He proposed and investigated the concept of a magnetic monopole, an object not yet known empirically, as a means of bringing even greater symmetry to Maxwell's equations of electromagnetism.

He quantized the gravitational field, and developed a general theory of quantum field theories with dynamical constraints, which forms the basis of the gauge theories and superstring theories of today. The influence and importance of his work has increased with the decades, and physicists daily use the concepts and equations that he developed. Dirac's first step into a new quantum theory was taken late in September 1925. Ralph Fowler, his research supervisor, had received a proof copy of an exploratory paper by Werner Heisenberg in the framework of the old quantum theory of Bohr and Sommerfeld, which leaned heavily on Bohr's correspondence principle but changed the equations so that they involved directly observable quantities. Fowler sent Heisenberg's paper on to Dirac, who was on vacation in Bristol, asking him to look into this paper carefully.

Dirac's attention was drawn to a mysterious mathematical relationship, at first sight unintelligible, that Heisenberg had reached. Several weeks later, back in Cambridge, Dirac suddenly recognized that this mathematical form had the same structure as the Poisson Brackets that occur in the classical dynamics of particle motion. From this thought he quickly developed a quantum theory that was based on non-commuting dynamical variables. This led him to a more profound and significant general formulation of quantum mechanics than was achieved by any other worker in this field

Dirac noticed an analogy between the Poisson brackets of classical mechanics and the recently proposed quantization rules in Werner Heisenberg's matrix formulation of quantum mechanics. This observation allowed Dirac to obtain the quantization rules in a novel and more illuminating manner. For this work, published in 1926, he received a Ph.D. from Cambridge.

In 1928, building on 2×2 spin matrices which he discovered independently of Wolfgang Pauli's work on non-relativistic spin systems, (Abraham Pais quoted Dirac as saying "I believe I got these (matrices) independently of Pauli and possibly Pauli got these independently of me") he proposed the Dirac equation as a relativistic equation of motion for the wavefunction of the electron. This work led Dirac to predict the existence of the positron, the electron's antiparticle, which he interpreted in terms of what came to be called the Dirac sea. The positron was observed by Carl Anderson in 1932. Dirac's equation also contributed to explaining the origin of quantum spin as a relativistic phenomenon.

The necessity of fermions (matter being created and destroyed in Enrico Fermi's 1934 theory of beta decay), however, led to a reinterpretation of Dirac's equation as a "classical" field equation for any point particle of spin ħ/2, itself subject to quantization conditions involving anti-commutators. Thus reinterpreted, in 1934 by Werner Heisenberg, as a (quantum) field equation accurately describing all elementary matter particles – today quarks and leptons – this Dirac field equation is as central to theoretical physics as the Maxwell, Yang–Mills and Einstein field equations. Dirac is regarded as the founder of quantum electrodynamics, being the first to use that term. He also introduced the idea of vacuum polarization in the early 1930s. This work was key to the development of quantum mechanics by the next generation of theorists, and in particular Schwinger, Feynman, Sin-Itiro Tomonaga and Dyson in their formulation of quantum electrodynamics.

Dirac's Principles of Quantum Mechanics, published in 1930, is a landmark in the history of science. It quickly became one of the standard textbooks on the subject and is still used today. In that book, Dirac incorporated the previous work of Werner Heisenberg on matrix mechanics and of Erwin Schrödinger on wave mechanics into a single mathematical formalism that associates measurable quantities to operators acting on the Hilbert space of vectors that describe the state of a physical system. The book also introduced the delta function. Following his 1939 article, he also included the bra-ket notation in the third edition of his book, thereby contributing to its universal use nowadays.

In 1933, following his 1931 paper on magnetic monopoles, Dirac showed that the existence of a single magnetic monopole in the universe would suffice to explain the observed quantization of electrical charge. In 1975, 1982, and 2009 intriguing results suggested the possible detection of magnetic monopoles, but there is, to date, no direct evidence for their existence.

Dirac was the Lucasian Professor of Mathematics at Cambridge from 1932 to 1969. In 1937, he proposed a speculative cosmological model based on the so-called large numbers hypothesis. During World War II, he conducted important theoretical and experimental research on uranium enrichment by gas centrifuge.

Dirac's quantum electrodynamics made predictions that were – more often than not – infinite and therefore unacceptable. A workaround known as renormalization was developed, but Dirac never accepted this. "I must say that I am very dissatisfied with the situation," he said in 1975, "because this so-called 'good theory' does involve neglecting infinities which appear in its equations, neglecting them in an arbitrary way. This is just not sensible mathematics. Sensible mathematics involves neglecting a quantity when it is small – not neglecting it just because it is infinitely great and you do not want it!" His refusal to accept renormalization resulted in his work on the subject moving increasingly out of the mainstream.

However, from his once rejected notes he managed to work on putting quantum electrodynamics on "logical foundations" based on Hamiltonian formalism that he formulated. He found a rather novel way of deriving the anomalous magnetic moment "Schwinger term" and also the Lamb shift, afresh, using the Heisenberg picture and without using the joining method used by Weisskopf and French, the two pioneers of modern QED, Schwinger and Feynman, in 1963. That was two years before the Tomonaga–Schwinger–Feynman QED was given formal recognition by an award of the Nobel Prize for physics.

Weisskopf and French (FW) were the first to obtain the correct result for the Lamb shift and the anomalous magnetic moment of the electron. At first FW results did not agree with the incorrect but independent results of Feynman and Schwinger (Schweber SS 1994 "QED and the men who made it: Dyson,Feynman,Schwinger and Tomonaga", Princeton :PUP). The 1963–1964 lectures Dirac gave on quantum field theory at Yeshiva University were published in 1966 as the Belfer Graduate School of Science, Monograph Series Number, 3. After having relocated to Florida in order to be near his elder daughter, Mary, Dirac spent his last fourteen years (of both life and physics research) at the University of Miami in Coral Gables, Florida and Florida State University in Tallahassee, Florida.

In the 1950s in his search for a better QED, Paul Dirac developed the Hamiltonian theory of constraints (Canad J Math 1950 vol 2, 129; 1951 vol 3, 1) based on lectures that he delivered at the 1949 International Mathematical Congress in Canada. Dirac (1951 “The Hamiltonian Form of Field Dynamics” Canad Jour Math, vol 3, 1) had also solved the problem of putting the Tomonaga–Schwinger equation into the Schrödinger representation (See Phillips R J N 1987 “Tributes to Dirac” p31 London:Adam Hilger) and given explicit expressions for the scalar meson field (spin zero pion or pseudoscalar meson), the vector meson field (spin one rho meson), and the electromagnetic field (spin one massless boson, photon).

The Hamiltonian of constrained systems is one of Dirac’s many masterpieces. It is a powerful generalization of Hamiltonian theory that remains valid for curved spacetime. The equations for the Hamiltonian involve only six degrees of freedom described by, for each point of the surface on which the state is considered. The (m = 0, 1, 2, 3) appear in the theory only through the variables, which occur as arbitrary coefficients in the equations of motion. There are four constraints or weak equations for each point of the surface = constant. Three of them form the four vector density in the surface. The fourth is a 3-dimensional scalar density in the surface H_L ≈ 0; H_r ≈ 0 (r = 1, 2, 3)

In the late 1950s, he applied the Hamiltonian methods he had developed to cast Einstein’s general relativity in Hamiltonian form (Proc Roy Soc 1958,A vol 246, 333,Phys Rev 1959,vol 114, 924) and to bring to a technical completion the quantization problem of gravitation and bring it also closer to the rest of physics according to Salam and DeWitt. In 1959 also he gave an invited talk on "Energy of the Gravitational Field" at the New York Meeting of the American Physical Society later published in 1959 Phys Rev Lett 2, 368. In 1964 he published his “Lectures on Quantum Mechanics” (London:Academic) which deals with constrained dynamics of nonlinear dynamical systems including quantization of curved spacetime. He also published a paper entitled “Quantization of the Gravitational Field” in 1967 ICTP/IAEA Trieste Symposium on Contemporary Physics.

If one considers waves moving in the direction resolved into the corresponding Fourier components (r, s = 1, 2, 3), the variables in the degrees of freedom 13, 23, 33 are affected by the changes in the coordinate system whereas those in the degrees of freedom 12, (11 − 22) remain invariant under such changes. The expression for the energy splits up into terms each associated with one of these six degrees of freedom without any cross terms associated with two of them. The degrees of freedom 13, 23, 33 do not appear at all in the expression for energy of gravitational waves in the direction . The two degrees of freedom 12, (11 − 22) contribute a positive definite amount of such a form to represent the energy of gravitational waves. These two degrees of freedom correspond in the language of quantum theory, to the gravitational photons (gravitons) with spin +2 or −2 in their direction of motion. The degrees of freedom (11 + 22) gives rise to the Newtonian potential energy term showing the gravitational force between the two positive mass is attractive and the self energy of every mass is negative.

Amongst his many students was John Polkinghorne, who recalls that Dirac "was once asked what was his fundamental belief. He strode to a blackboard and wrote that the laws of nature should be expressed in beautiful equations."