Dirac equation

• 1. Introduction

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  On the 13th of November 1995, a plaque was unveiled in the floor of Westminster Abbey. It commemorates the life of the physicist Paul Dirac, and it carries a single equation. The Dirac equation is a relativistic wave equation that the British physicist derived in 1928. It describes massive particles with spin one-half, such as electrons and quarks. Some physicists have called it "the real seed of modern physics." Others have called it "perhaps the most important equation in all of quantum mechanics." The equation also appears in an auditorium in Erice, Sicily, in a lecture hall that bears Dirac's name. Why does one equation earn a place in an abbey? What problem had stumped the leading theorists of the 1920s? And what did the equation reveal that nobody had gone looking for?

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• 2. The Relativity Gap

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  Between 1900 and 1925, quantum mechanics was a series of patches, each explaining one phenomenon that classical mechanics could not. In 1925 Werner Heisenberg, Max Born and Pascual Jordan produced matrix mechanics, which describes physical observables with matrices. The next year Erwin Schrödinger offered wave mechanics, built on his wave equation. The two frameworks first looked like rivals, but they were later shown to be equivalent.

  Both frameworks worked only in a non-relativistic setting, and Schrödinger saw this as a flaw from the start. He tried to build a relativistic version himself and, along the way, found what is now called the Klein-Gordon equation. He tested it on the hydrogen spectrum, for which Arnold Sommerfeld had already supplied an exact form with relativistic corrections. The equation failed that test, and Schrödinger dropped the attempt. At least six other authors found the same equation in the same year.

  During 1926 and 1927, physicists chased relativity along two routes. The first treated the Klein-Gordon equation as the correct relativistic generalization. Many leading theorists disliked this route, because it missed numerous experimental results and seemed hard to square with the principles of quantum mechanics. The conceptual trouble came from a second derivative with respect to time. The second route added relativistic effects as corrections to known formulas. Heisenberg and Jordan inserted two terms, one for spin and one for relativity, into the hydrogen Hamiltonian. That let them derive a first-order approximation of Sommerfeld's fine structure formula.

  Spin ran alongside all of this. Samuel Goudsmit and George Uhlenbeck introduced the idea in 1925, and Schrödinger soon guessed that it was the missing link to the correct Sommerfeld formula. In 1927 Wolfgang Pauli used spin to build an effective theory for a non-relativistic spin-1/2 particle. His wave function depended on a spin coordinate that could take only two values, plus or minus half the reduced Planck constant. Pauli suspected that a fully relativistic version might need an electron that was more than a point particle. A conference in Brussels that same year would put a different objection on the table.

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• 3. Two Months in 1927

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  At the fifth Solvay Conference in 1927, Paul Dirac was mainly occupied with the logical development of quantum mechanics. He noticed that many colleagues had comfortably accepted the Klein-Gordon equation as a satisfactory relativistic answer. Accepting it meant giving up basic principles of quantum mechanics as then understood, and Dirac strongly objected. After he returned from Brussels, he turned to finding a relativistic theory of electrons. Within two months he had solved the problem, and he published the result on the 2nd of January 1928.

  Two principles from transformation theory guided him. The equation had to be invariant under the transformations of special relativity, and it had to obey quantum mechanics' own transformation theory. The second demand meant the equation must be linear in time derivatives, so that it could carry a probabilistic interpretation. The Klein-Gordon equation has a second time derivative. The obvious fix is to take a square root of both sides, but that square root made the resulting theory unworkable.

  His first insight was linearization. Dirac searched for quantities, independent of momentum and spacetime coordinates, that would let the square root be rewritten as a straight sum. Squaring that sum and demanding the Klein-Gordon equation showed what they must do. Each had to square to one, and any two had to anticommute. He first considered the two-by-two Pauli matrices, then showed that four such matrices that all anticommute cannot exist.

  His second insight was to use four-dimensional matrices, which meant a wave function with four components. That was a far bolder step than Pauli's. Pauli had given his wave function two components because a particle has two spin states. Dirac had no physical argument for four components. He introduced them purely as a mathematical necessity. Without knowing it, he had also built a structure that mathematicians had known since the early 1880s, the Clifford algebra.

  Then came the test in an electromagnetic field. To his surprise, the equation described a particle with a magnetic moment, arising because the particle has spin one-half. Spin had emerged from the equation without Dirac putting it in by hand. He also showed that the equation reproduces the fine structure of hydrogen, at least to first order. Earlier attempts had needed ad hoc modifications of existing formulas, while this one worked from first principles. Those four components, two of which resemble Pauli's wave function in the non-relativistic limit, would turn out to hold more than spin.

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• 4. The Sea Below Zero

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  Not every solution of the equation looked like an ordinary electron. Negative energy states had been a nagging problem, and over time they provoked many efforts to eliminate them. Dirac at first dismissed them as unphysical. In 1929 Oskar Klein showed that in static fields the negative and positive energy states inevitably mix. Dirac's reaction was to suspect a defect, and to think his equation was only a first approximation to a better theory without the problem.

  He then proposed the Dirac sea. In this picture the universe is filled with an infinite sea of negative energy electron states. Ordinary positive energy electrons live within that sea. The Pauli exclusion principle stops them from decaying into the filled negative states below.

  Dirac also postulated positively charged holes in the sea. He first suggested that these holes might be the proton. Robert Oppenheimer showed that if so, stable atoms could not exist. Hermann Weyl showed that the holes would have to carry the same mass as the electron. Persuaded, Dirac published a paper in 1931 predicting an unobserved particle he called the "anti-electron." It would share the electron's mass, carry the opposite charge, and annihilate with an electron on contact. He suggested that every particle might have an oppositely charged partner, an idea now called antimatter.

  In 1933 Carl Anderson discovered the "positive electron," now called the positron. It had every property of Dirac's anti-electron. The Dirac sea itself was later superseded by quantum field theory, but its legacy survived in the picture of a dynamical vacuum filled with virtual particles. In 1949 Ernst Stueckelberg suggested, and Richard Feynman showed in detail, that negative energy solutions can be read as particles traveling backwards in proper time. Condensed matter physics gives the sea a more literal home in the Fermi sea, made of filled valence electrons below some chemical potential.

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• 5. Early Verdicts

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  After his paper on the Zeeman effect and the Paschen-Back effect in magnetic fields, Dirac mostly left the consequences of his equation to others. He returned to the subject only in 1930. By his own account, as the source tells it, he did not fully appreciate what he had done. Others saw it clearly. The explanation of spin as a product of joining quantum mechanics with relativity, and the later discovery of the positron, have been called a triumph fully on par with the work of Isaac Newton, James Clerk Maxwell and Albert Einstein.

  For a while the Dirac equation stood as the only valid relativistic equation for a particle with mass. Then in 1934 Pauli and Victor Weisskopf reinterpreted the Klein-Gordon equation, which became the equation for a relativistic particle with no spin.

  The first calculations arrived quickly. In 1928 Charles Galton Darwin and Walter Gordon, working independently, reproduced the Sommerfeld fine structure formula exactly. It was the first time the full formula had been derived from first principles. The same year brought the Klein-Nishina formula for photon-electron scattering. Mott scattering, in which electrons bounce off a heavy target such as an atomic nucleus, followed in 1929. Moller scattering came in 1932 and Bhabha scattering in 1936.

  The equation's predictions held up for years of searching for spectroscopic discrepancies. Only in 1947 was the Lamb shift discovered, an effect the equation does not predict. That led in the 1950s to the development of quantum electrodynamics, with the Dirac equation folded into quantum field theory. From there it took a fundamental role in the Standard Model.

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• 6. One Equation, Many Forms

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  In its modern form, the Dirac equation lives in a four-dimensional Minkowski spacetime and is written in terms of a Dirac field. This field assigns a vector of four complex numbers to every point in spacetime. Its defining property is that it transforms as a Dirac spinor under Lorentz transformations. The equation is built from four gamma matrices, which generate what is called the Dirac algebra. Their choice is not unique, and each choice is a different representation. The one Dirac himself found is known as the Dirac representation.

  A companion called the adjoint spinor helps in forming quantities that stay the same in every reference frame. The combination of a spinor with its own conjugate is not invariant under Lorentz transformations, but the adjoint version is. The equation can also be rewritten to resemble the Schrödinger equation, with a Hamiltonian acting on the spinor. That form is useful in quantum mechanics because the Hamiltonian is easy to modify, for instance by adding a potential or coupling to the electromagnetic field.

  A third form comes from the Dirac action. The equation arises as the Euler-Lagrange equation of that action. The action form makes the symmetries of the equation explicit, and Noether's theorem then yields the currents tied to them. It is also usually the starting point for defining the quantum field theory, for example through the path integral.

  What the spinor means depends on the setting. In quantum mechanics it is a four-component wave function describing the state of a Dirac fermion, and a component of its vector current gives the probability density of finding the fermion in a region of space. In second quantization it becomes an operator that creates or destroys particles. In the path integral it is an anti-commuting variable that serves only as something to integrate over.

  Under the hood, the Lorentz group has six degrees of freedom, three boosts and three rotations. Spinors transform under a representation in which each Lorentz transformation corresponds to two possible choices, because the group SL(2,C) is a double cover of the Lorentz group. The equation keeps the same form in every inertial frame. Any solution of it is also a solution of the Klein-Gordon equation, so solutions can be written as sums of plane waves. Positive frequency waves create particles in the quantum theory, while negative frequency ones annihilate antiparticles. Each case has two spin degrees of freedom.

  A purely classical spinor field never appears in our universe. The Pauli exclusion principle prevents the field from being filled with enough particles to reach the classical limit.

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• 7. Symmetry and Gauge Fields

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  The Dirac action is unchanged if the phase of the spinor is shifted by the same amount everywhere. This global U(1) symmetry is called the vector symmetry, and Noether's theorem gives it a conserved current. Promoting it to a local symmetry, where the phase can differ at each point, forces a new ingredient. A gauge field must be introduced, and the ordinary derivative must be replaced by a gauge covariant derivative. Adding a kinetic term for that gauge field produces the action of quantum electrodynamics.

  In the massless limit the equation gains a second, inequivalent U(1) symmetry, called the axial symmetry. It exists because the massless equation splits into two pieces, each with its own phase symmetry. The axial current behaves as a pseudovector, meaning its spatial part is odd under parity transformations. Classically this symmetry could be gauged, but at the quantum level a chiral anomaly blocks the way. In the massless theory with a single fermion, the axial symmetry cannot be gauged for that reason.

  Spacetime symmetries add more. The Dirac action is invariant under the Poincare group, which combines spacetime translations with the Lorentz group. The four translations yield the Dirac stress-energy tensor, and Lorentz transformations yield a further set of conserved currents.

  The real expansion comes from using N identical Dirac spinors at once. The largest global symmetry then becomes the unitary group U(N), and any continuous subgroup can be gauged. Gauging the special unitary group SU(N) and adding a kinetic term for the gauge field gives the action of quantum chromodynamics. The case of SU(3) describes the strong interactions of the Standard Model, with gluons as the gauge field and quarks as the Dirac spinors. The SU(2) case describes the electroweak sector, where the gauge field is the W-boson and the spinors are leptons.

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• 8. Relatives and Descendants

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  In 1929 Hermann Weyl showed that the massless Dirac equation splits into a pair of Weyl equations. The Dirac spinor decomposes into two Weyl spinors of opposite chirality, one left-handed and one right-handed. The mass term is what couples them together. In the massless limit they decouple and move on separately.

  This split has been offered as an intuitive explanation of Zitterbewegung, a rapid trembling motion associated with the electron. The two massless pieces would travel at the speed of light in opposite directions. In that picture the mass does not slow anything down. Instead it sets the average rate at which the motion reverses, and the reversals can be modeled as a Poisson process. A close cousin is the Majorana equation, which takes the same form but acts on spinors that satisfy a reality condition.

  Going the other way, the non-relativistic limit of the Dirac equation is the Pauli equation. It describes phenomena where relativistic effects are negligible but the spin of the fermion matters. Recast in the right form, it shows that the gyromagnetic ratio of a particle obeying the Dirac equation is exactly 2. Quantum electrodynamics adds corrections that give rise to a non-zero anomalous magnetic moment.

  The equation has been stretched in many directions. It generalizes to any dimension and any metric signature, provided the gamma matrices are swapped for those of the appropriate Clifford algebra. It also extends to curved spacetime through a spinor covariant derivative built from the spin connection. Self-interaction terms lead to the nonlinear Dirac equation, as in the Thirring model. The Breit equation handles multi-electron systems interacting electromagnetically, and the two-body Dirac equation is a similar multi-body relative. Geometric rewritings include the Dirac-Hestenes equation and the Dirac-Kahler equation. The latter acts on differential forms and is used in lattice field theory to describe the continuum limit of staggered fermions.

  The most surprising descendants live on a desktop. In condensed matter physics, systems whose fermions have a nearly linear dispersion relation are described by the Dirac equation. These systems are known as Dirac matter, and they include graphene and topological insulators. Since the start of the 21st century they have become a major area of research.

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