Dirac equation
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac particles", such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity, and was the first theory to fully account for special relativity in the context of quantum mechanics. The equation is validated by its rigorous accounting of the observed fine structure of the hydrogen spectrum and has become vital in the building of the Standard Model.
The equation also implied the existence of a new form of matter, antimatter, previously unsuspected and unobserved.
The existence of antimatter was experimentally confirmed several years later. It also provided a theoretical justification for the introduction of several component wave functions in Pauli's phenomenological theory of spin. The wave functions in the Dirac theory are vectors of four complex numbers, two of which resemble the Pauli wavefunction in the non-relativistic limit, in contrast to the Schrödinger equation, which described wave functions of only one complex value. Moreover, in the limit of zero mass, the Dirac equation reduces to the Weyl equation. In the context of quantum field theory, the Dirac equation is reinterpreted to describe quantum fields corresponding to spin- particles.
Dirac did not fully appreciate the importance of his results; however, the entailed explanation of spin as a consequence of the union of quantum mechanics and relativity—and the eventual discovery of the positron—represents one of the great triumphs of theoretical physics. This accomplishment has been described as fully on par with the works of Isaac Newton, James Clerk Maxwell, and Albert Einstein before him. The equation has been deemed by some physicists to be "the real seed of modern physics". The Dirac equation has been described as the "centerpiece of relativistic quantum mechanics", with it also stated that "the equation is perhaps the most important one in all of quantum mechanics".
History
Early attempts at a relativistic formulation
The first phase in the development of quantum mechanics, lasting between 1900 and 1925, focused on explaining individual phenomena that could not be explained through classical mechanics. The second phase, starting in the mid-1920s, saw the development of two systematic frameworks governing quantum mechanics. The first, known as matrix mechanics, uses matrices to describe physical observables; it was developed in 1925 by Werner Heisenberg, Max Born, and Pascual Jordan. The second, known as wave mechanics, uses a wave equation known as the Schrödinger equation to describe the state of a system; it was developed the next year by Erwin Schrödinger. While these two frameworks were initially seen as competing approaches, they would later be shown to be equivalent.Both these frameworks only formulated quantum mechanics in a non-relativistic setting. This was seen as a deficiency right from the start, with Schrödinger originally attempting to formulate a relativistic version of Schrödinger equation, in the process discovering the Klein–Gordon equation. However, after showing that this equation did not correctly reproduce the relativistic corrections to the hydrogen atom spectrum for which an exact form was known due to Arnold Sommerfeld, he abandoned his relativistic formulation. The Klein–Gordon equation was also found by at least six other authors in the same year.
During 1926 and 1927 there was a widespread effort to incorporate relativity into quantum mechanics, largely through two approaches. The first was to consider the Klein–Gordon as the correct relativistic generalization of the Schrödinger equation. Such an approach was viewed unfavourably by many leading theorists since it failed to correctly predict numerous experimental results, and more importantly it appeared difficult to reconcile with the principles of quantum mechanics as understood at the time. These conceptual issues primarily arose due to the presence of a second temporal derivative.
The second approach introduced relativistic effects as corrections to the known non-relativistic formulas. This provided many provisional answers that were expected to eventually be supplanted by some yet-unknown relativistic formulation of quantum mechanics. One notable result by Heisenberg and Jordan was the introduction of two terms for spin and relativity into the hydrogen Hamiltonian, allowing them to derive the first-order approximation of the Sommerfeld fine structure formula.
A parallel development during this time was the concept of spin, first introduced in 1925 by Samuel Goudsmit and George Uhlenbeck. Shortly after, it was conjectured by Schrödinger to be the missing link in acquiring the correct Sommerfeld formula. In 1927 Wolfgang Pauli used the ideas of spin to find an effective theory for a nonrelativistic spin- particle, the Pauli equation. He did this by taking the Schrödinger equation and rather than just assuming that the wave function depends on the physical coordinate, he also assume that it depends on a spin coordinate that can take only two values. While this was still a non-relativistic formulation, he believed that a fully relativistic formulation possibly required a more complicated model for the electron, one that moved beyond a point particle.
Dirac's relativistic quantum mechanics
By 1927, many physicists no longer considered the fine structure of hydrogen as a crucial puzzle that called for a completely new relativistic formulation since it could effectively be solved using the Pauli equation or by introducing a spin- angular momentum quantum number in the Klein–Gordon equation. At the fifth Solvay Conference held that year, Paul Dirac was primarily concerned with the logical development of quantum mechanics. However, he realized that many other physicists complacently accepted the Klein–Gordon equation as a satisfactory relativistic formulation, which demanded abandoning basic principles of quantum mechanics as understood at the time, to which Dirac strongly objected. After his return from Brussels, Dirac focused on finding a relativistic theory for electrons. Within two months he solved the problem and published his results on January 2, 1928.In his paper, Dirac was guided by two principles from transformation theory, the first being that the equation should be invariant under transformations of special relativity, and the second that it should transform under the transformation theory of quantum mechanics. The latter demanded that the equation would have to be linear in temporal derivatives, so that it would admit a probabilistic interpretation. His argument begins with the Klein–Gordon equation
describing a particle using the wave function. Here is the square of the momentum, is the rest mass of the particle, is the speed of light, and is the reduced Planck constant. The naive way to get an equation linear in the time derivative is to essentially consider the square root of both sides. This replaces with. However, such a square root is mathematically problematic for the resulting theory, making it unfeasible.
Dirac's first insight was the concept of linearization. He looked for some sort of variables that are independent of momentum and spacetime coordinates for which the square root could be rewritten in a linear form
By squaring this operator and demanding that it reduces to the Klein–Gordon equation, Dirac found that the variables must satisfy and if. Dirac initially considered the Pauli matrices as a candidate, but then showed these would not work since it is impossible to find a set of four matrices that all anticommute with each other. His second insight was to instead consider four-dimensional matrices. In that case the equation would be acting on a four-component wavefunction. Such a proposal was much more bold than Pauli's original generalization to a two-component wavefunction in the Pauli equation. This is because in Pauli's case, this was motivated by the demand to encode the two spin states of the particle. In contrast, Dirac had no physical argument for a four-component wavefunction, but instead introduced it as a matter of mathematical necessity. He thus arrived at the Dirac equation
Dirac constructed the correct matrices without realizing that they form a mathematical structure known of since the early 1880s, the Clifford algebra. By recasting the equation in a Lorentz invariant form, he also showed that it correctly combines special relativity with his principle of quantum mechanical transformation theory, making it a viable candidate for a relativistic theory of the electron.
To investigate the equation further, he examined how it behaves in the presence of an electromagnetic field. To his surprise, this showed that it described a particle with a magnetic moment arising due to the particle having spin. Spin directly emerged from the equation, without Dirac having added it in by hand. Additionally, he focused on showing that the equation successfully reproduces the fine structure of the hydrogen atom, at least to first order. The equation therefore succeeds where all previous attempts have failed, in correctly describing relativistic phenomena of electrons from first principles rather than through the ad hoc modification of existing formulas.
Consequences
Except for his followup paper deriving the Zeeman effect and Paschen–Back effect from the equation in the presences of a magnetic fields, Dirac left the work of examining the consequences of his equation to others, and only came back to the subject in 1930. Once the equation was published, it was recognized as the correct solution to the problem of spin, relativity, and quantum mechanics. At first the Dirac equation was considered the only valid relativistic equation for a particle with mass. Then in 1934 Pauli and Victor Weisskopf reinterpreted the Klein–Gordon equation as the equation for a relativistic spinless particle.One of the first calculations was to reproduce the Sommerfeld fine structure formula exactly, which was performed independently by Charles Galton Darwin and Walter Gordon in 1928. This is the first time that the full formula has been derived from first principles. Further work on the mathematics of the equation was undertaken by Hermann Weyl in 1929. In this work he showed that the massless Dirac equation can be decomposed into a pair of Weyl equations.
The Dirac equation was also used to study various scattering processes. In particular, the Klein–Nishina formula, looking at photon-electron scattering, was also derived in 1928. Mott scattering, the scattering of electrons off a heavy target such as atomic nuclei, followed the next year. Over the following years it was further used to derive other standard scattering processes such as Moller scattering in 1932 and Bhabha scattering in 1936.
A problem that gained more focus with time was the presence of negative energy states in the Dirac equation, which led to many efforts to try to eliminate such states. Dirac initially simply rejected the negative energy states as unphysical, but the problem was made more clear when in 1929 Oskar Klein showed that in static fields there exists inevitable mixing between the negative and positive energy states. Dirac's initial response was to believe that his equation must have some sort of defect, and that it was only the first approximation of a future theory that would not have this problem. However, he then suggested a solution to the problem in the form of the Dirac sea. This is the idea that the universe is filled with an infinite sea of negative energy electrons states. Positive energy electron states then live in this sea and are prevented from decaying to the negative energy states through the Pauli exclusion principle.
Additionally, Dirac postulated the existence of positively charged holes in the Dirac sea, which he initially suggested could be the proton. However, Oppenheimer showed that in this case stable atoms could not exist and Weyl further showed that the holes would have to have the same mass as the electrons. Persuaded by Oppenheimer's and Weyl's argument, Dirac published a paper in 1931 that predicted the existence of an as-yet-unobserved particle that he called an "anti-electron" that would have the same mass and the opposite charge as an electron and that would mutually annihilate upon contact with an electron. He suggested that every particle may have an oppositely charged partner, a concept now called antimatter
In 1933 Carl Anderson discovered the "positive electron", now called a positron, which had all the properties of Dirac's anti-electron. While the Dirac sea was later superseded by quantum field theory, its conceptual legacy survived in the idea of a dynamical vacuum filled with virtual particles. In 1949 Ernst Stueckelberg suggested and Richard Feynman showed in detail that the negative energy solutions can be interpreted as particles traveling backwards in proper time. The concept of the Dirac sea is also realized more explicitly in some condensed matter systems in the form of the Fermi sea, which consists of a sea of filled valence electrons below some chemical potential.
Significant work was done over the following decades to try to find spectroscopic discrepancies compared to the predictions made by the Dirac equation, however it was not until 1947 that Lamb shift was discovered, which the equation does not predict. This led to the development of quantum electrodynamics in 1950s, with the Dirac equation then being incorporated within the context of quantum field theory. Since it describes the dynamics of Dirac spinors, it went on to play a fundamental role in the Standard Model as well as many other areas of physics. For example, within condensed matter physics, systems whose fermions have a near linear dispersion relation are described by the Dirac equation. Such systems are known as Dirac matter and they include graphene and topological insulators, which have become a major area of research since the start of the 21st century.
The Dirac equation is inscribed upon a plaque on the floor of Westminster Abbey. Unveiled on 13 November 1995, the plaque commemorates Dirac's life. The equation, in its natural units formulation, is also prominently displayed in the auditorium at the ‘Paul A.M. Dirac’ Lecture Hall at the Patrick M.S. Blackett Institute of the Ettore Majorana Foundation and Centre for Scientific Culture in Erice, Sicily.