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 (known as Dirac spinors), 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", and as "perhaps the most important [equation] 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. 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. These conceptual issues primarily arose due to the presence of a second temporal derivative.
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. He did this by taking the Schrödinger equation and, rather than just assuming that the wave function depends on the physical coordinate, he also assumed that it depends on a spin coordinate that can take only two values <math>\pm \tfrac{\hbar}{2}</math>. 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.
:<math>
\left[\boldsymbol p^2+m^2c^2\right]\phi(t,x) = -\frac{\hbar^2}{c^2}\frac{\partial^2}{\partial t^2} \phi(t,x),
</math>
describing a particle using the wave function <math>\phi(t,x)</math>. Here <math>\boldsymbol p^2 = p_1^2+p_2^2+p_3^2</math> is the square of the momentum, <math>m</math> is the rest mass of the particle, <math>c</math> is the speed of light, and <math>\hbar</math> 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 <math>\boldsymbol p^2+m^2c^2</math> with <math>\sqrt{\boldsymbol p^2+m^2c^2}</math>. However, such a square root is mathematically problematic for the resulting theory, making it unfeasible. Such a proposal was much more bold than Pauli's original generalization to a two-component wavefunction in the Pauli equation. 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. 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. 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, 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. 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. 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 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 (formerly The San Domenico Monastery) of the Ettore Majorana Foundation and Centre for Scientific Culture in Erice, Sicily.
Formulation
Covariant formulation
In its modern field theoretic formulation, the Dirac equation in 3+1 dimensional Minkowski spacetime is written in terms of a Dirac field <math>\psi(x)</math>. This is a field that assigns a complex vector from <math>\mathbb C^4</math> to each point in spacetime, In natural units where <math>\hbar = c = 1</math>, the Lorentz covariant formulation of the Dirac equation is given by One common choice, originally discovered by Dirac, is known as the Dirac representation. Here the matrices are given by
:<math>
\bar \psi(x)(-i \gamma^\mu \overleftarrow{\partial}_\mu -m)=0.
</math>
The adjoint spinor is useful in forming Lorentz invariant quantities. For example, the bilinear <math>\psi^\dagger \psi</math> is not Lorentz invariant, but <math>\bar \psi \psi</math> is. Additionally, the action is usually used to define the associated quantum field theory, such as through the path integral formulation. In contrast to quantum mechanics, it no longer represents the state in the Hilbert space, but is rather the operator that acts on states to create or destroy particles. Observables are formed using expectation values of these operators. The Dirac equation then becomes an operator equation describing the state-independent evolution of the operator-valued spinor field
:<math>
(i