Paul Dirac

How to use this course

Paul Dirac wrote little and said less, yet much of the language of quantum physics is his: commutators as quantum Poisson brackets, the delta function, bras and kets, creation and annihilation operators, Fermi–Dirac statistics, the relativistic electron equation and the positron. He also posed questions that are still open, such as the magnetic monopole, and planted ideas that others harvested, such as the path integral and the quantization of constrained systems. This course follows that work in order, alongside the life of a man whose shyness became legendary.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations in modern notation and exercises with folded solutions. We use SI units unless stated, \(\hbar=h/2\pi\), and the metric signature \((+,-,-,-)\). Readers who want only the story can read Parts I and III and the opening paragraph of each lesson in Part II.

Timeline

Bristol, 1902–1921

Paul Adrien Maurice Dirac was born in Bristol on 8 August 1902. His father, Charles Adrien Ladislas Dirac, came from the canton of Valais in Switzerland and taught French at the Merchant Venturers’ Technical College. His mother, Florence Hannah Holten, was the daughter of a ship’s captain from Cornwall and worked as a librarian. Paul had an elder brother, Felix, and a younger sister, Betty. The family was legally Swiss until 22 October 1919, when Charles and his children were naturalized.

The household was unhappy. Charles was a strict disciplinarian who required Paul to speak to him only in French. Paul later said that, unable to express himself well in French, he found it easier to stay silent, and he traced his lifelong reticence to these years. He grew up close to neither parent and resented his father for the rest of his life.

School and engineering

He attended Bishop Road primary school and from 1914 the Merchant Venturers’ Technical College, a school with a practical, scientific curriculum and no Latin or Greek. The war had emptied the classes of older boys, and he was allowed to move ahead quickly. In 1918 he entered the University of Bristol to study electrical engineering, following his brother. He graduated with first-class honours in 1921.

Engineering left a mark on his physics. He learned to accept approximations that work, to draw diagrams in projective geometry, and to value methods that give answers. In 1919 the eclipse expedition confirmed Albert Einstein’s general relativity, and relativity became a public sensation. Dirac attended lectures on relativity by the philosopher C. D. Broad in Bristol, and the subject became his first love in theoretical physics.

Mathematics and Cambridge, 1921–1925

In 1921 Britain was in a postwar recession, and Dirac could not find work as an engineer. He won a place at St John’s College, Cambridge, but the scholarship was not enough to live on. Bristol’s mathematics department let him attend its courses without fees, and in two years he took a first-class degree in mathematics, in 1923. With a studentship from the Department of Scientific and Industrial Research he then went to Cambridge as a research student.

He had hoped to work on relativity with Ebenezer Cunningham, who had no room for a student. Instead he was assigned to Ralph Fowler, the leading British authority on statistical mechanics and the new atomic theory, and one of the few people in Cambridge in touch with Copenhagen and Göttingen. Through Fowler, Dirac learned the Bohr–Sommerfeld quantum theory and the methods of Hamiltonian mechanics, the action-angle variables in which it was written. His first papers, from 1924, dealt with statistical mechanics, the Compton effect and adiabatic invariants.

He lived a solitary life, working in his room during the week and walking long distances on Sundays. In March 1925 his brother Felix, unhappy in his work and estranged from their father, killed himself. Dirac rarely spoke of it.

The quantum revolution, 1925–1927

In July 1925 Werner Heisenberg lectured in Cambridge. A few weeks later Fowler received the proofs of Heisenberg’s paper founding matrix mechanics and passed them to Dirac. Dirac was struck by the fact that Heisenberg’s quantities do not commute. He later recalled that during a Sunday walk in October he realized that the commutator resembles the Poisson bracket of classical mechanics, and that he had to wait for the library to open on Monday to check the definition in Whittaker’s Analytical Dynamics. His paper The fundamental equations of quantum mechanics was received by the Royal Society on 7 November 1925 (Lesson 04).

Over the next two years he worked at astonishing speed. He derived the hydrogen spectrum with his q-number algebra, found the statistics of particles with antisymmetric wave functions independently of Enrico Fermi, and in June 1926 submitted the first doctoral thesis on the new quantum mechanics, titled simply Quantum mechanics.

Copenhagen and Göttingen

In September 1926 he went to Niels Bohr’s institute in Copenhagen. There he wrote his transformation theory, which unified the matrix and wave pictures and introduced the delta function (Lesson 05), and his quantum theory of radiation, which created quantum field theory (Lesson 06). In February 1927 he moved to Göttingen, where he worked alongside Max Born and met J. Robert Oppenheimer. He was elected a fellow of St John’s College in 1927 and attended the Fifth Solvay Conference in Brussels in October. He was twenty-five.

Poisson brackets and q-numbers

For the general reader. Heisenberg found in 1925 that quantum quantities must be multiplied in a definite order: position times momentum differs from momentum times position. Dirac saw that this difference corresponds exactly to an operation in classical mechanics, the Poisson bracket. The correspondence gives a recipe for turning any classical system into a quantum one, and it remains the standard route.

Poisson brackets

For functions \(u,v\) on phase space with coordinates \(q_1,\dots,q_N,p_1,\dots,p_N\), the Poisson bracket is

It is bilinear and antisymmetric, satisfies the Jacobi identity, and obeys the product rule \(\{u_1u_2,v\}=\{u_1,v\}u_2+u_1\{u_2,v\}\). Hamilton’s equations read \(\dot u=\{u,H\}\), and \(\{q_r,p_s\}=\delta_{rs}\).

Dirac’s argument

Suppose quantum variables form a noncommutative algebra, and that there is a quantum bracket \([u,v]\) with the same properties as the Poisson bracket, the product rule holding with the order of factors preserved. Expanding \([u_1u_2,v_1v_2]\) first in the left argument and then in the right, and again in the opposite order, and comparing, gives

Since this holds for independent pairs \((u_1,v_1)\) and \((u_2,v_2)\), the commutator \(uv-vu\) must be a fixed multiple of the bracket \([u,v]\), the constant commuting with everything.

Dirac called the noncommuting quantities q-numbers and ordinary numbers c-numbers. The constant had to be imaginary for the commutator of two Hermitian quantities to be anti-Hermitian, and its size was fixed by agreement with Heisenberg’s results. The correspondence is exact for \(q,p\) and for quadratic functions; for higher polynomials, ordering ambiguities appear at higher order in \(\hbar\), a fact made precise by the Groenewold–van Hove theorem (1946), which shows that no quantization map can preserve all Poisson brackets.

Transformation theory, δ and bra–ket

For the general reader. In 1926 there were two quantum theories, Heisenberg’s matrices and Schrödinger’s waves. Dirac, and independently Pascual Jordan, showed that both are views of one theory, related the way coordinates in different directions describe the same vector. To make the argument work for position, which takes continuous values, Dirac invented a “function” that is zero everywhere except at one point. Mathematicians found it scandalous; physicists found it indispensable. Laurent Schwartz later made it rigorous as a distribution.

The delta function

In The physical interpretation of the quantum dynamics (1927), Dirac introduced \(\delta(x)\) with the defining property

for every continuous \(f\). It is the continuous analogue of the Kronecker delta: an orthonormality relation for a continuous basis reads \(\langle x|x'\rangle=\delta(x-x')\). Useful identities include \(\delta(ax)=\delta(x)/|a|\), \(x\,\delta(x)=0\), \(x\,\delta'(x)=-\delta(x)\), and the Fourier representation \(\delta(x)=\frac{1}{2\pi}\int e^{ikx}\,dk\).

Bras and kets

In the Principles of Quantum Mechanics (1930) states are vectors, and in a short paper of 1939, adopted in the third edition of the book (1947), Dirac named them: a ket \(|\psi\rangle\) is a vector, a bra \(\langle\phi|\) is a linear functional, and their bracket \(\langle\phi|\psi\rangle\) is a number. An observable \(A\) with eigenvectors \(|a\rangle\) gives a representation in which \(|\psi\rangle\) is described by the components \(\langle a|\psi\rangle\).

Schrödinger’s wave function is \(\psi(x)=\langle x|\psi\rangle\), the components in the position basis. Heisenberg’s matrices are \(\langle E_m|A|E_n\rangle\), the components of an operator in the energy basis. The two theories are one theory in two bases. The momentum relation follows from \([\hat x,\hat p]=i\hbar\): the operator \(e^{-i\hat pa/\hbar}\) translates position eigenstates, \(|x\rangle\mapsto|x+a\rangle\), and differentiating in \(a\) gives the derivative. John von Neumann gave the rigorous version in 1932, using Hilbert space and spectral measures in place of delta functions.

Statistics, radiation, second quantization

For the general reader. Electrons are perfectly identical. Dirac showed in 1926 that this forces their joint wave function to change sign when two of them are exchanged, and worked out how a gas of such particles shares its energy, as Enrico Fermi had done shortly before by another route. A year later he treated light as a collection of oscillators that can be raised or lowered one quantum at a time. This explained how atoms emit light spontaneously, and it was the beginning of quantum field theory.

Antisymmetry and statistics

For two identical particles, the exchange operator \(P_{12}\) commutes with the Hamiltonian, and \(P_{12}^2=1\). Dirac argued in On the theory of quantum mechanics (1926) that the wave function must be either symmetric or antisymmetric, and that electrons, which obey Pauli’s exclusion principle, need the antisymmetric choice. An antisymmetric state of independent particles is a determinant of one-particle states, and it vanishes if two particles occupy the same state.

Each state is either empty or filled, so in the grand canonical ensemble its partition function is \(1+e^{-(\varepsilon-\mu)/kT}\), and the mean occupation follows at once. Dirac later named the two kinds of particles fermions and bosons. In 1926 Fowler applied the new statistics to the electrons in white dwarf stars, the start of the theory of degenerate stars.

Transition rates

In the same 1926 paper Dirac developed time-dependent perturbation theory by the method of variation of constants. For \(H=H_0+V(t)\) and \(|\psi(t)\rangle=\sum_nc_n(t)e^{-iE_nt/\hbar}|n\rangle\), to first order

From the formula above, \(|c_f|^2=|V_{fi}|^2\sin^2(\omega_{fi}t/2)/(\hbar^2(\omega_{fi}/2)^2)\). As \(t\to\infty\) this tends to \(2\pi t\,|V_{fi}|^2\delta(\omega_{fi})/\hbar^2\), linear in \(t\); summing over final states with \(\rho\) gives the rate.

The quantum theory of radiation (1927)

Dirac expanded the electromagnetic field in modes, each a harmonic oscillator, and quantized each oscillator. With \(a_k,a_k^\dagger\) satisfying \([a_k,a_{k'}^\dagger]=\delta_{kk'}\), the mode Hamiltonian is \(\hbar\omega_k(a_k^\dagger a_k+\tfrac12)\), and the number operator \(N_k=a_k^\dagger a_k\) counts photons. Emission into mode \(k\) involves \(a_k^\dagger\), whose matrix element \(\langle n+1|a^\dagger|n\rangle=\sqrt{n+1}\) gives a rate proportional to \(n+1\): the \(n\) is stimulated emission and the \(1\) is spontaneous emission. Dirac thus derived Einstein’s A and B coefficients of 1917 from first principles. In 1928 Jordan and Eugene Wigner gave the fermion version, with anticommutators \(\{b,b^\dagger\}=1\).

The Dirac equation

For the general reader. Schrödinger’s equation ignores relativity. The obvious relativistic version, the Klein–Gordon equation, gave nonsense such as negative probabilities. In January 1928 Dirac found a different equation, first order in time like Schrödinger’s and consistent with relativity. To make it work he had to give the electron four components instead of one. The equation then explained, without being asked, why the electron spins and why its magnet is twice as strong as classical theory predicts.

Finding the equation

Dirac wanted an equation \(i\hbar\,\partial_t\psi=H\psi\), first order in time so that \(|\psi|^2\) could be a probability density, with \(H\) linear in momentum:

Requiring \(H^2=\mathbf p^2c^2+m^2c^4\), so that every component obeys the Klein–Gordon equation, forces

No numbers satisfy these, and no 2×2 matrices do either: the three Pauli matrices anticommute, but a fourth anticommuting matrix does not exist in two dimensions. The smallest solution is 4×4. In the Dirac representation, with \(\sigma_i\) the Pauli matrices,

The anticommutation relations define a Clifford algebra. The Klein–Gordon density \(\frac{i\hbar}{2mc^2}(\phi^*\partial_t\phi-\phi\,\partial_t\phi^*)\) can be negative because the equation is second order in time; Dirac’s \(\psi^\dagger\psi\) cannot. Plane-wave solutions \(\psi=u\,e^{-i(Et-\mathbf p\cdot\mathbf x)/\hbar}\) exist for \(E=\pm\sqrt{p^2c^2+m^2c^4}\), each with two spin states. The negative energies are the subject of Lesson 08.

Spin

The orbital angular momentum \(\mathbf L=\mathbf x\times\mathbf p\) does not commute with \(H\). With \(\boldsymbol\Sigma=\begin{pmatrix}\boldsymbol\sigma&0\\0&\boldsymbol\sigma\end{pmatrix}\), one finds

so \(\mathbf J=\mathbf L+\frac{\hbar}{2}\boldsymbol\Sigma\) is conserved. The electron carries spin ½ as a consequence of relativity and the linear form of the equation, with no separate assumption.

The magnetic moment

Couple to an electromagnetic field by \(\mathbf p\to\boldsymbol\pi=\mathbf p-q\mathbf A\) and \(E\to E-q\phi\), with \(q=-e\) for the electron. Write \(\psi=(\varphi,\chi)\) and \(E=mc^2+E'\) with \(E'\ll mc^2\). The lower equation gives \(\chi\approx\frac{\boldsymbol\sigma\cdot\boldsymbol\pi}{2mc}\varphi\), and the upper becomes

The value \(g=2\) had been inferred from spectra and put in by hand by Pauli and others. The small measured excess, \(g-2\approx0.00232\), was explained by Julian Schwinger in 1948 as the radiative correction \(\alpha/\pi\) of quantum electrodynamics.

Fine structure, holes and antimatter

For the general reader. The Dirac equation gave the hydrogen spectrum, including its fine splittings, more accurately than any earlier theory. But it also allowed electrons to have negative energy, and nothing seemed to stop an ordinary electron from falling into such states. Dirac proposed that all the negative-energy states are already full. A missing electron in this “sea” would look like a particle of positive charge. After first identifying it with the proton, Dirac predicted in 1931 a new particle, the anti-electron. Carl Anderson found it in cosmic rays in 1932.

Hydrogen

For an electron in the Coulomb field of a nucleus of charge \(Ze\), the Dirac equation can be solved exactly (Charles Galton Darwin and Walter Gordon, 1928). With \(\alpha=e^2/4\pi\varepsilon_0\hbar c\approx1/137.036\):

The levels depend only on \(n\) and \(j\), so \(2s_{1/2}\) and \(2p_{1/2}\) coincide. The formula is identical to Sommerfeld’s 1916 result from the old quantum theory, with the quantum numbers reinterpreted. In 1947 Willis Lamb and Robert Retherford measured a splitting of about 1058 MHz between \(2s_{1/2}\) and \(2p_{1/2}\), the Lamb shift, whose explanation launched renormalized quantum electrodynamics.

The negative-energy problem

The free equation has solutions with \(E\le-mc^2\). An electron in an atom could radiate and drop into them, releasing unlimited energy. In A theory of electrons and protons (1930), Dirac proposed that in the vacuum all negative-energy states are occupied, one electron per state by the exclusion principle. An unoccupied state, a hole, behaves as a particle with positive energy and positive charge.

Dirac first identified the hole with the proton, hoping to explain all matter with one particle. Hermann Weyl showed that the hole must have the same mass as the electron, and Oppenheimer and Igor Tamm showed that an electron and a proton would then annihilate far too fast for atoms to be stable. In Quantised singularities in the electromagnetic field (1931), Dirac accepted the consequence: the hole is a new particle, an anti-electron, with the electron’s mass and opposite charge. He also suggested an anti-proton.

Anderson photographed a positron track in a cloud chamber in August 1932, without knowing Dirac’s prediction. In 1933 Patrick Blackett and Giuseppe Occhialini, in Cambridge, observed electron–positron pairs created in showers and connected them with Dirac’s theory. In quantum field theory the sea is replaced by a field with creation operators for both particles and antiparticles, but the predictions are the same.

Magnetic monopoles

For the general reader. Every electric charge ever measured is a whole multiple of the electron’s charge. Nobody knew why. In 1931 Dirac showed that if a single isolated magnetic pole exists anywhere in the universe, quantum mechanics forces electric charge to come in whole units. No monopole has been found, but the argument made topology a part of physics, and grand unified theories predict monopoles.

The string argument

Let a monopole of magnetic charge \(g\) (in webers, so that the total flux out of a closed surface is \(g\)) sit at the origin: \(\mathbf B=\frac{g}{4\pi}\frac{\hat{\mathbf r}}{r^2}\). Since \(\nabla\cdot\mathbf B\ne0\) at the origin, no single smooth \(\mathbf A\) exists. One can write \(\mathbf B=\nabla\times\mathbf A\) everywhere except on a line, the Dirac string, running from the monopole to infinity and carrying the return flux \(g\). A particle of charge \(q\) taken around a small loop encircling the string acquires the phase

The string is an artefact of the choice of \(\mathbf A\) and must be undetectable, so this phase must equal 1.

The angular-momentum argument

A static charge \(q\) and monopole \(g\) separated by \(\mathbf r\) have field angular momentum \(\mathbf L_{\text{em}}=\varepsilon_0\int\mathbf x\times(\mathbf E\times\mathbf B)\,d^3x\), which is directed along the line joining them and has magnitude \(qg/4\pi\), independently of the distance (J. J. Thomson, 1904). Requiring this to be a half-integer multiple of \(\hbar\) gives \(qg/4\pi=n\hbar/2\), the same condition. In 1975 Tai Tsun Wu and Chen Ning Yang removed the string by using two patches with different \(\mathbf A\), related by a gauge transformation; the condition becomes the requirement that the transition function be single-valued, an integer Chern number.

The Lagrangian and constrained systems

For the general reader. Quantum mechanics was built on the Hamiltonian, the energy. Dirac wanted a version based on the Lagrangian, which treats space and time more symmetrically. In 1933 he noticed that the quantum amplitude to go from one point to another in a short time behaves like the exponential of the classical action. Richard Feynman read the paper in 1941 and turned the remark into the path integral. Later Dirac solved another problem of the same kind: how to quantize theories such as electromagnetism and gravity, whose variables are not all independent.

The Lagrangian in quantum mechanics

In The Lagrangian in quantum mechanics, published in the Physikalische Zeitschrift der Sowjetunion in 1933, Dirac argued that the transformation function between position eigenstates at nearby times “corresponds to” \(\exp(iL\,\delta t/\hbar)\), and that for a finite interval the amplitude is a product of such factors integrated over intermediate positions.

As \(\hbar\to0\) the phase oscillates rapidly except near paths where \(S\) is stationary, so the integral is dominated by the classical path: Hamilton’s principle emerges from interference, as Dirac had noted. Feynman later recalled telling Dirac that the two are in fact proportional; Dirac only replied, “Are they?”

Constrained Hamiltonian systems

In gauge theories the Legendre transformation is not invertible. In electromagnetism, \(\mathcal L=-\tfrac14F_{\mu\nu}F^{\mu\nu}\) contains no \(\dot A_0\), so its conjugate momentum vanishes identically. In Generalized Hamiltonian dynamics (1950) and his Lectures on Quantum Mechanics (1964), Dirac gave the general procedure.

For electromagnetism, \(\pi^0\approx0\) is primary and Gauss’s law \(\nabla\cdot\mathbf E\approx0\) is secondary; both are first class and generate gauge transformations. Dirac applied the method to general relativity in 1958–59, and Richard Arnowitt, Stanley Deser and Charles Misner developed the related ADM formalism. The method is the foundation of canonical quantum gravity.

Lucasian Professor, 1929–1945

In 1929 Dirac spent five months in the United States, lecturing at Wisconsin and elsewhere, then crossed the Pacific with Heisenberg to lecture in Japan and returned through Siberia. He became a Fellow of the Royal Society in 1930, the year The Principles of Quantum Mechanics appeared. The book set out the theory from a few abstract principles, in Dirac’s spare style, and remains in print.

In 1932 he was elected Lucasian Professor of Mathematics at Cambridge, the chair once held by Isaac Newton, at the age of thirty. In 1933 he shared the Nobel Prize in Physics with Erwin Schrödinger “for the discovery of new productive forms of atomic theory”. He considered refusing it to avoid publicity, until told that a refusal would attract more.

Friends and family

He visited the Soviet Union several times in the 1930s and was a close friend of Pyotr Kapitsa, whose Cambridge discussion group, the Kapitza Club, he attended. During a year at the Institute for Advanced Study in Princeton in 1934–35, Eugene Wigner introduced him to his sister Margit (Manci) Balázs, a divorcée with two children. They married in London in January 1937. Dirac raised her children, Judith and Gabriel, and the couple had two daughters, Mary and Monica. Marriage, by all accounts, made him much happier.

His research in these years ranged widely. In 1937 he proposed the large numbers hypothesis: the ratio of the electric to the gravitational force between a proton and an electron, about \(10^{39}\), is close to the age of the universe in atomic units of time, and he suggested that the gravitational constant decreases as the universe ages. In 1938 he derived the equation of motion of a radiating classical electron. During the war he worked on methods of separating uranium isotopes for the British Tube Alloys project, including the centrifuge, and gave the 1941 Bakerian Lecture of the Royal Society. He received the Royal Medal in 1939.

Cambridge, Florida and legacy, 1945–1984

After the war Dirac spent 1947–48 at the Institute for Advanced Study at Oppenheimer’s invitation. In 1954, in the McCarthy era, he was refused a US visa; it was granted in August, by which time he had arranged to spend the year in India. He returned to Princeton in 1958–59 and worked on the Hamiltonian form of general relativity. He received the Copley Medal and the Max Planck Medal in 1952, and became a member of the Pontifical Academy of Sciences in 1961.

He was deeply dissatisfied with renormalization, the method by which Feynman, Schwinger, Tomonaga and Dyson made quantum electrodynamics finite. He accepted that it worked but regarded subtracting infinities as a sign that the theory was wrong, and he spent his later years looking for a better one. He took few research students. They included Harish-Chandra, Dennis Sciama, Behram Kurşunoğlu, Richard Eden and C. J. Eliezer, and Homi Bhabha and Fred Hoyle also worked under his guidance at Cambridge.

Florida

Dirac retired from the Lucasian chair in 1969. After a visiting post at the University of Miami, he moved to Florida State University in Tallahassee, where he was a professor from 1971 until his death. He received the Order of Merit in 1973. He kept working, on the large numbers hypothesis and on cosmology, and swam in the lakes near Tallahassee. He died there on 20 October 1984, aged 82, and is buried in Roselawn Cemetery. On 13 November 1995 a memorial stone was unveiled in Westminster Abbey, near Newton’s grave, carrying his equation in the form \(i\gamma\cdot\partial\psi=m\psi\).

Legacy

• Formalism: commutators from Poisson brackets, transformation theory, bra–ket notation and the delta function are the working language of quantum mechanics.
• Quantum field theory: the quantized radiation field, creation and annihilation operators, and Fermi–Dirac statistics; the Dirac equation describes every spin-½ particle of the Standard Model.
• Antimatter: predicted in 1931, observed in 1932, now used daily in positron emission tomography.
• Ideas for the future: the monopole and charge quantization, the Lagrangian behind the path integral, constrained quantization behind gauge theories and canonical gravity, and the Dirac–Born–Infeld action of string theory.
• Honours: Nobel Prize (1933), Royal Medal (1939), Copley Medal (1952), Max Planck Medal (1952), Order of Merit (1973). The Dirac Medal of the International Centre for Theoretical Physics has been awarded annually since 1985.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Paul Dirac, MacTutor History of Mathematics.
2. Paul Dirac, Wikipedia (with extensive references).
3. Paul A. M. Dirac, biographical, NobelPrize.org.
4. St John’s College, Cambridge, Paul Adrien Maurice Dirac (1902–1984).
5. G. Farmelo, Paul Dirac: the Mozart of science, Institute for Advanced Study, 2008.
6. G. Farmelo, The Strangest Man: The Hidden Life of Paul Dirac, Quantum Genius, Faber and Faber, 2009.
7. R. H. Dalitz and R. Peierls, Paul Adrien Maurice Dirac, Biographical Memoirs of Fellows of the Royal Society 32 (1986) 137–185.
8. H. Kragh, Dirac: A Scientific Biography, Cambridge University Press, 1990.

Physics

1. P. A. M. Dirac, The fundamental equations of quantum mechanics, Proc. R. Soc. A 109 (1925) 642–653.
2. P. A. M. Dirac, On the theory of quantum mechanics, Proc. R. Soc. A 112 (1926) 661–677.
3. P. A. M. Dirac, The physical interpretation of the quantum dynamics, Proc. R. Soc. A 113 (1927) 621–641.
4. P. A. M. Dirac, The quantum theory of the emission and absorption of radiation, Proc. R. Soc. A 114 (1927) 243–265.
5. P. A. M. Dirac, The quantum theory of the electron, Proc. R. Soc. A 117 (1928) 610–624.
6. P. A. M. Dirac, A theory of electrons and protons, Proc. R. Soc. A 126 (1930) 360–365.
7. P. A. M. Dirac, Quantised singularities in the electromagnetic field, Proc. R. Soc. A 133 (1931) 60–72.
8. P. A. M. Dirac, The Lagrangian in quantum mechanics, Physikalische Zeitschrift der Sowjetunion 3 (1933) 64–72.
9. P. A. M. Dirac, A new notation for quantum mechanics, Mathematical Proceedings of the Cambridge Philosophical Society 35 (1939) 416–418.
10. P. A. M. Dirac, Generalized Hamiltonian dynamics, Canadian Journal of Mathematics 2 (1950) 129–148; Lectures on Quantum Mechanics, Yeshiva University, 1964.
11. P. A. M. Dirac, The Principles of Quantum Mechanics, Oxford, 1930; 4th edition 1958.
12. P. A. M. Dirac, Theory of electrons and positrons, Nobel Lecture, 1933.
13. C. D. Anderson, The positive electron, Physical Review 43 (1933) 491–494.
14. R. P. Feynman, Space-time approach to non-relativistic quantum mechanics, Reviews of Modern Physics 20 (1948) 367–387.