Dirac equation

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 \pm \tfrac{\hbar}{2}. 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. : \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), describing a particle using the wave function \phi(t,x). Here \boldsymbol p^2 = p_1^2+p_2^2+p_3^2 is the square of the momentum, m is the rest mass of the particle, c is the speed of light, and \hbar 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 \boldsymbol p^2+m^2c^2 with \sqrt{\boldsymbol p^2+m^2c^2}. 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 \psi(x). This is a field that assigns a complex vector from \mathbb C^4 to each point in spacetime, In natural units where \hbar = c = 1, 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 : \bar \psi(x)(-i \gamma^\mu \overleftarrow{\partial}_\mu -m)=0. The adjoint spinor is useful in forming Lorentz invariant quantities. For example, the bilinear \psi^\dagger \psi is not Lorentz invariant, but \bar \psi \psi 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 : (i{{\partial}\!\!\!/} - m)\hat \psi(x) = 0. In the path integral formulation of quantum field theory, the spinor field \psi(x) is an anti-commuting Grassmann-valued field that only acts as an integration variable. so particles belong to projective representations rather than regular representations, with the projective representations of the Lorentz group being equivalent to regular representations of \text{SL}(2,\mathbb C). Classical spinor fields do not arise in our universe because the Pauli exclusion principle prevents populating the field with a sufficient number of particles to reach the classical limit. == Properties ==

Lorentz transformations The Lorentz group \text{SO}(1,3), {{refn|group=nb|Strictly speaking the \text{SO}^+(1,3), which is the part of the Lorentz group connected to the identity. It excludes parity and time-reversal transformations.}} describing the transformation between inertial reference frames, can admit many different representations. A representation is a particular choice of matrices that faithfully{{refn|group=nb|A representation, mapping group elements g \in G to matrices \rho(g) \in \text{GL}(V), is faithful if the mapping between group elements and matrices is injective.}} represent the action of the group on some vector space, where the dimensionality of the matrices can differ between representations. For example, the Lorentz group can be represented by 4\times 4 real matrices \Lambda acting on the vector space \mathbb R^{1,3}, corresponding to how Lorentz transformations act on vectors or on spacetime. A smaller representation is a set of 2\times 2 complex matrices acting on Weyl spinors in the \mathbb C^2 vector space. Lie group elements can be generated using the corresponding Lie algebra, which together with a Lie bracket, describes the tangent space of the group manifold around its identity element. The basis elements of this vector space are known as generators of the group. A particular group element is then acquired by exponentiating a corresponding tangent space vector. The generators of the Lorentz Lie algebra must satisfy certain anticommutation relations, known as a Lie bracket. The six vectors can be packaged into an antisymmetric object X^{\mu\nu} indexed by \{\mu,\nu\}, with the bracket for the Lorentz algebra given by : \psi(x) \rightarrow e^{i\beta \gamma^5}\psi(x), \qquad \bar \psi(x) \rightarrow e^{i\beta \gamma^5}\bar \psi(x), where \gamma_5 is the chiral matrix. This arises because in the massless limit the Dirac equation reduces to a pair of Weyl equations. : (i{{\partial}\!\!\!/} +m)(i{{\partial}\!\!\!/} -m)\psi(x) = \left(\partial_\mu \partial^\mu + m^2\right)\psi(x) = 0. As a result, any solution to the Dirac equation is also automatically a solution to the Klein–Gordon equation. The Dirac equation admits positive frequency plane wave solutions : \psi(x) = u(\boldsymbol p)e^{ip_\mu x^\mu} with a positive energy given by p_0 = \sqrt{\boldsymbol p^2+m^2}>0. It also admits negative frequency solutions taking the same form except with p_0 = -\sqrt{\boldsymbol p^2+m^2}. It is more convenient to rewrite these negative frequency solutions by flipping the sign of the momentum to ensure that they have a positive energy p_0>0 and so take the form : \psi(x) = v(\boldsymbol p)e^{-ip_\mu x^\mu}. At the classical level these are positive and negative frequency solutions to a classical wave equation, but in the quantum theory they correspond to operators creating particles with spinor polarization u(\boldsymbol p) or annihilating antiparticles with spinor polarization v(\boldsymbol p). Both these spinor polarizations satisfy the momentum space Dirac equation : ({{p}\!\!\!/} + m)u(\boldsymbol p) = 0, : ({{p}\!\!\!/} - m)v(\boldsymbol p) = 0. Since these are simple matrix equations, they can be solved directly once an explicit representation for the gamma matrices is chosen. In the chiral representation the general solution is given by : u(\boldsymbol p) = {\sqrt{p\cdot \sigma}\xi_s \choose \sqrt{p\cdot \bar \sigma}\xi_s}, \qquad v(\boldsymbol p) = {\sqrt{p\cdot \sigma}\eta_s \choose - \sqrt{p\cdot \bar \sigma}\eta_s}, where \xi_s and \eta_s are arbitrary complex 2-vectors, describing the two spin degrees of freedom for the particle and two for the antiparticle. In the massless limit, these spin states correspond to the possible helicity states that the massless fermions can have, either being left-handed or right-handed. == Related equations ==

Related Dirac equations While the standard Dirac equation was originally derived in a 3+1 dimensional spacetime, it can be directly generalized to arbitrary dimension and metric signatures, where it takes the same covariant form. The crucial difference is that the gamma matrices must be changed to gamma matrices of the Clifford algebra appropriate in those dimensions and metric signature, with the size of the Dirac spinor corresponding to the dimensionality of the gamma matrices. While the Dirac equation always exists, since every dimension admits Dirac spinors, the properties of these spinors and their relation to other spinor representations differs significantly across dimensions. : (i\gamma^\mu D_\mu -m)\psi(x) = 0. Adding self-interaction terms to the Dirac action gives rise to the nonlinear Dirac equation, which allows for the fermions to interact with themselves, such as in the Thirring model. Interactions between fermions can also be introduced through electromagnetic effects. In particular, the Breit equation describes multi-electron systems interacting electromagnetically to first order in perturbation theory. The two-body Dirac equation is a similar multi-body equation. A geometric reformulation of the Dirac equation is known as the Dirac–Hestenes equation. In this formulation all the components of the Dirac equation have an explicit geometric interpretation. Another related geometric equation is the Dirac–Kähler equation, which is a geometric analogue of the Dirac equation that can be defined on any general pseudo-Riemannian manifold and which acts on differential forms. In the case of a flat manifold, it reduces to four copies of the Dirac equation. However, on curved manifolds this decomposition breaks down and the equation fundamentally differs. This equation is used in lattice field theory to describe the continuum limit of staggered fermions. Weyl and Majorana equations The Dirac spinor can be decomposed into a pair of Weyl spinors of opposite chirality \psi^T = (\psi_L,\psi_R). Here the role of the mass is not to make the velocity less than the speed of light, but instead controls the average rate at which these reversals occur; specifically, the reversals can be modelled as a Poisson process. A closely related equation is the Majorana equation, with this formally taking the same form as the Dirac equation except that it acts on Majorana spinors. These are spinors that satisfy a reality condition \psi = C \bar \psi^T, where C is the charge-conjugation operator. In higher dimensions, the Dirac equation has similar relations to the equations describing other spinor representations that arise in those dimensions. Pauli equation In the non-relativistic limit, the Dirac equation reduces to the Pauli equation, which when coupled to electromagnetism has the form : \left[\frac{1}{2m}\left(\boldsymbol \sigma \cdot (\hat{\boldsymbol{p}}-q \boldsymbol A)\right)^2+q\phi \right]|\psi\rangle = 0. Here \boldsymbol \sigma is the vector of Pauli matrices and \hat{\boldsymbol p} = -i\nabla is the momentum operator. The equation describes a fermion of charge q coupled to the electromagnetic field through a magnetic vector potential \boldsymbol A and an electric scalar potential \phi. The fermion is described through the two-component wave function |\psi\rangle, where each component describes one of the two spin states |\psi\rangle = \psi_+|\uparrow\rangle + \psi_-|\downarrow\rangle. The Pauli equation is often used in quantum mechanics to describe phenomena where relativistic effects are negligible but the spin of the fermion is important. It can also be recast in a form which directly shows that the gyromagnetic ratio of the fermion described by the Dirac equation is exactly g=2. In quantum electrodynamics there are additional quantum corrections that modify this value, give rise to a non-zero anomalous magnetic moment. == Gauge symmetry ==

Vector symmetry The vector and axial symmetries of the Dirac action are both global symmetries, in that they act the same everywhere in spacetime. which necessarily transforms in the adjoint representation of the gauge group. The covariant derivative then takes the form The \text{SU}(2) case also plays a role in the Standard Model, describing the electroweak sector. The gauge field in this case is the W-boson, while the Dirac spinors are leptons. == See also ==