Dirac Equation
   HOME

TheInfoList



OR:

In
particle physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
, the Dirac equation is a
relativistic wave equation In physics, specifically relativistic quantum mechanics (RQM) and its applications to particle physics, relativistic wave equations predict the behavior of particles at high energies and velocities comparable to the speed of light. In the con ...
derived by British physicist
Paul Dirac Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
in 1928. In its free form, or including electromagnetic interactions, it describes all spin-
massive particle The physics technical term massive particle refers to a massful particle which has real non-zero rest mass (such as baryonic matter), the counter-part to the term massless particle. According to special relativity, the velocity of a massive particl ...
s, called "Dirac particles", such as
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no kn ...
s and
quark A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly o ...
s for which parity is a
symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
. It is consistent with both the principles of
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
and the theory of
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws o ...
, and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the fine structure of the
hydrogen spectrum The emission spectrum of atomic hydrogen has been divided into a number of spectral series, with wavelengths given by the Rydberg formula. These observed spectral lines are due to the electron making transitions between two energy levels in an ...
in a completely rigorous way. The equation also implied the existence of a new form of matter, ''
antimatter In modern physics, antimatter is defined as matter composed of the antiparticles (or "partners") of the corresponding particles in "ordinary" matter. Antimatter occurs in natural processes like cosmic ray collisions and some types of radioac ...
'', previously unsuspected and unobserved and which 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 Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...
. The wave functions in the Dirac theory are vectors of four
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s (known as
bispinor In physics, and specifically in quantum field theory, a bispinor, is a mathematical construction that is used to describe some of the fundamental particles of nature, including quarks and electrons. It is a specific embodiment of a spinor, speci ...
s), two of which resemble the Pauli wavefunction in the non-relativistic limit, in contrast to the
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
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 physics, particularly in quantum field theory, the Weyl equation is a relativistic wave equation for describing massless spin-1/2 particles called Weyl fermions. The equation is named after Hermann Weyl. The Weyl fermions are one of the three ...
. Although Dirac did not at first fully appreciate the importance of his results, the entailed explanation of spin as a consequence of the union of quantum mechanics and relativity—and the eventual discovery of the
positron The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. It has an electric charge of +1 '' e'', a spin of 1/2 (the same as the electron), and the same mass as an electron. When a positron collides ...
—represents one of the great triumphs of
theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
. This accomplishment has been described as fully on a par with the works of
Newton Newton most commonly refers to: * Isaac Newton (1642–1726/1727), English scientist * Newton (unit), SI unit of force named after Isaac Newton Newton may also refer to: Arts and entertainment * ''Newton'' (film), a 2017 Indian film * Newton ( ...
, Maxwell, and
Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
before him. In the context of
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, the Dirac equation is reinterpreted to describe quantum fields corresponding to spin- particles. The Dirac equation appears on the floor of
Westminster Abbey Westminster Abbey, formally titled the Collegiate Church of Saint Peter at Westminster, is an historic, mainly Gothic church in the City of Westminster, London, England, just to the west of the Palace of Westminster. It is one of the United ...
on the plaque commemorating Paul Dirac's life, which was unveiled on 13 November 1995.


Mathematical formulation

In its modern formulation for field theory, the Dirac equation is written in terms of a
Dirac spinor In quantum field theory, the Dirac spinor is the spinor that describes all known fundamental particles that are fermions, with the possible exception of neutrinos. It appears in the plane-wave solution to the Dirac equation, and is a certain combin ...
field \psi taking values in a complex vector space described concretely as \mathbb^4, defined on flat spacetime (
Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inerti ...
) \mathbb^. Its expression also contains
gamma matrices In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\ma ...
and a parameter m > 0 interpreted as the mass, as well as other physical constants. In terms of a field \psi: \mathbb^\rightarrow \mathbb^4, the Dirac equation is then and in
natural units In physics, natural units are physical units of measurement in which only universal physical constants are used as defining constants, such that each of these constants acts as a Coherence (units of measurement), coherent unit of a quantity. For e ...
, with
Feynman slash notation In the study of Dirac fields in quantum field theory, Richard Feynman invented the convenient Feynman slash notation (less commonly known as the Dirac slash notation). If ''A'' is a covariant vector (i.e., a 1-form), : \ \stackrel\ \gamma^1 A_ ...
, The gamma matrices are a set of four 4 \times 4 complex matrices (elements of \text_(\mathbb)) which satisfy the defining ''anti''-commutation relations: \ = 2\eta^. These matrices can be realized explicitly under a choice of representation. Two common choices are the Dirac representation \gamma^0 = \begin I_2 & 0 \\ 0 & -I_2 \end,\quad \gamma^i = \begin 0 & \sigma^i \\ -\sigma^i & 0 \end, where \sigma^i are the
Pauli matrices In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in ...
, and the chiral representation: the \gamma^i are the same, but \gamma^0 = \begin 0 & I_2 \\ I_2 & 0 \end. The slash notation is a compact notation for A\!\!\!/ := \gamma^\mu A_\mu where A is a four-vector (often it is the four-vector differential operator \partial_\mu). The summation over the index \mu is implied.


Dirac adjoint and the adjoint equation

The Dirac adjoint of the spinor field \psi(x) is defined as \bar\psi(x) = \psi(x)^\dagger \gamma^0. Using the property of gamma matrices (which follows straightforwardly from Hermicity properties of the \gamma^\mu) that (\gamma^\mu)^\dagger = \gamma^0\gamma^\mu\gamma^0, one can derive the adjoint Dirac equation by taking the Hermitian conjugate of the Dirac equation and multiplying on the right by \gamma^0: \bar\psi (x)( - i\gamma^\mu \partial_\mu - m) = 0 where the partial derivative acts from the right on \bar\psi(x): written in the usual way in terms of a left action of the derivative, we have - i\partial_\mu\bar\psi (x)\gamma^\mu - m\bar\psi (x) = 0.


Klein–Gordon equation

Applying i\partial\!\!\!/ + m to the Dirac equation gives (\partial_\mu\partial^\mu + m^2)\psi(x) = 0. That is, each component of the Dirac spinor field satisfies the
Klein–Gordon equation The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic wave equation, related to the Schrödinger equation. It is second-order in space and time and manifestly Lorentz-covariant. ...
.


Conserved current

A conserved current of the theory is J^\mu = \bar\gamma^\mu\psi. Another approach to derive this expression is by variational methods, applying Noether's theorem for the global \text(1) symmetry to derive the conserved current J^\mu.


Solutions

Since the Dirac operator acts on 4-tuples of square-integrable functions, its solutions should be members of the same
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
. The fact that the energies of the solutions do not have a lower bound is unexpected.


Plane-wave solutions

Plane-wave solutions are those arising from an ansatz \psi(x) = u(\mathbf)e^ which models a particle with definite 4-momentum p = (E_\mathbf, \mathbf) where E_\mathbf = \sqrt. For this ansatz, the Dirac equation becomes an equation for u(\mathbf): \left(\gamma^\mu p_\mu - m\right) u(\mathbf) = 0. After picking a representation for the gamma matrices \gamma^\mu, solving this is a matter of solving a system of linear equations. It is a representation-free property of gamma matrices that the solution space is two-dimensional (see
here Here is an adverb that means "in, on, or at this place". It may also refer to: Software * Here Technologies, a mapping company * Here WeGo (formerly Here Maps), a mobile app and map website by Here Technologies, Here Television * Here TV (form ...
). For example, in the chiral representation for \gamma^\mu, the solution space is parametrised by a \mathbb^2 vector \xi, with u(\mathbf) = \begin \sqrt\xi \\ \sqrt\xi \end where \sigma^\mu = (I_2, \sigma^i), \bar\sigma^\mu = (I_2, -\sigma^i) and \sqrt is the Hermitian matrix square-root. These plane-wave solutions provide a starting point for canonical quantization.


Lagrangian formulation

Both the Dirac equation and the Adjoint Dirac equation can be obtained from (varying) the action with a specific Lagrangian density that is given by: \mathcal = i\hbar c\overline\gamma^\partial_\psi - mc^2\overline\psi If one varies this with respect to \psi one gets the adjoint Dirac equation. Meanwhile, if one varies this with respect to \bar\psi one gets the Dirac equation. In natural units and with the slash notation, the action is then For this action, the conserved current J^\mu above arises as the conserved current corresponding to the global \text(1) symmetry through
Noether's theorem Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether in ...
for field theory. Gauging this field theory by changing the symmetry to a local, spacetime point dependent one gives gauge symmetry (really, gauge redundancy). The resultant theory is
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
or QED. See below for a more detailed discussion.


Lorentz invariance

The Dirac equation is invariant under Lorentz transformations, that is, under the action of the Lorentz group \text(1,3) or strictly \text(1,3)^+, the component connected to the identity. For a Dirac spinor viewed concretely as taking values in \mathbb^4, the transformation under a Lorentz transformation \Lambda is given by a 4\times 4 complex matrix S
Lambda Lambda (}, ''lám(b)da'') is the 11th letter of the Greek alphabet, representing the voiced alveolar lateral approximant . In the system of Greek numerals, lambda has a value of 30. Lambda is derived from the Phoenician Lamed . Lambda gave rise ...
/math>. There are some subtleties in defining the corresponding S
Lambda Lambda (}, ''lám(b)da'') is the 11th letter of the Greek alphabet, representing the voiced alveolar lateral approximant . In the system of Greek numerals, lambda has a value of 30. Lambda is derived from the Phoenician Lamed . Lambda gave rise ...
/math>, as well as a standard abuse of notation. Most treatments occur at the
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
level. For a more detailed treatment see
here Here is an adverb that means "in, on, or at this place". It may also refer to: Software * Here Technologies, a mapping company * Here WeGo (formerly Here Maps), a mobile app and map website by Here Technologies, Here Television * Here TV (form ...
. The Lorentz group of 4 \times 4 ''real'' matrices acting on \mathbb^ is generated by a set of six matrices \ with components (M^)^\rho_\sigma = \eta^\delta^\nu_\sigma - \eta^\delta^\mu_\sigma. When both the \rho,\sigma indices are raised or lowered, these are simply the 'standard basis' of antisymmetric matrices. These satisfy the Lorentz algebra commutation relations ^, M^= M^\eta^ - M^\eta^ + M^\eta^ - M^\eta^. In the article on the
Dirac algebra In mathematical physics, the Dirac algebra is the Clifford algebra \text_(\mathbb). This was introduced by the mathematical physicist P. A. M. Dirac in 1928 in developing the Dirac equation for spin-½ particles with a matrix representation of t ...
, it is also found that the spin generators S^ = \frac gamma^\mu,\gamma^\nu/math> satisfy the Lorentz algebra commutation relations. A Lorentz transformation \Lambda can be written as \Lambda = \exp\left(\frac\omega_M^\right) where the components \omega_ are antisymmetric in \mu,\nu. The corresponding transformation on spin space is S
Lambda Lambda (}, ''lám(b)da'') is the 11th letter of the Greek alphabet, representing the voiced alveolar lateral approximant . In the system of Greek numerals, lambda has a value of 30. Lambda is derived from the Phoenician Lamed . Lambda gave rise ...
= \exp\left(\frac\omega_S^\right). This is an abuse of notation, but a standard one. The reason is S
Lambda Lambda (}, ''lám(b)da'') is the 11th letter of the Greek alphabet, representing the voiced alveolar lateral approximant . In the system of Greek numerals, lambda has a value of 30. Lambda is derived from the Phoenician Lamed . Lambda gave rise ...
/math> is not a well-defined function of \Lambda, since there are two different sets of components \omega_ (up to equivalence) which give the same \Lambda but different S
Lambda Lambda (}, ''lám(b)da'') is the 11th letter of the Greek alphabet, representing the voiced alveolar lateral approximant . In the system of Greek numerals, lambda has a value of 30. Lambda is derived from the Phoenician Lamed . Lambda gave rise ...
/math>. In practice we implicitly pick one of these \omega_ and then S
Lambda Lambda (}, ''lám(b)da'') is the 11th letter of the Greek alphabet, representing the voiced alveolar lateral approximant . In the system of Greek numerals, lambda has a value of 30. Lambda is derived from the Phoenician Lamed . Lambda gave rise ...
/math> is well defined in terms of \omega_. Under a Lorentz transformation, the Dirac equation i\gamma^\mu\partial_\mu \psi(x) - m \psi(x) becomes i\gamma^\mu((\Lambda^)_\mu^\nu\partial_\nu)S
Lambda Lambda (}, ''lám(b)da'') is the 11th letter of the Greek alphabet, representing the voiced alveolar lateral approximant . In the system of Greek numerals, lambda has a value of 30. Lambda is derived from the Phoenician Lamed . Lambda gave rise ...
psi(\Lambda^ x) - mS
Lambda Lambda (}, ''lám(b)da'') is the 11th letter of the Greek alphabet, representing the voiced alveolar lateral approximant . In the system of Greek numerals, lambda has a value of 30. Lambda is derived from the Phoenician Lamed . Lambda gave rise ...
psi(\Lambda^ x) = 0. Associated to Lorentz invariance is a conserved Noether current, or rather a tensor of conserved Noether currents (\mathcal^)^\mu. Similarly, since the equation is invariant under translations, there is a tensor of conserved Noether currents T^, which can be identified as the stress-energy tensor of the theory. The Lorentz current (\mathcal^)^\mu can be written in terms of the stress-energy tensor in addition to a tensor representing internal angular momentum.


Historical developments and further mathematical details

The Dirac equation was also used (historically) to define a quantum-mechanical theory where \psi(x) is instead interpreted as a wave-function. The Dirac equation in the form originally proposed by
Dirac Distributed Research using Advanced Computing (DiRAC) is an integrated supercomputing facility used for research in particle physics, astronomy and cosmology in the United Kingdom. DiRAC makes use of multi-core processors and provides a variety o ...
is: \left(\beta mc^2 + c \sum_^\alpha_n p_n\right) \psi (x,t) = i \hbar \frac where is the
wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements mad ...
for the electron of
rest mass The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, i ...
with
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
coordinates . The are the components of the
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
, understood to be the
momentum operator In quantum mechanics, the momentum operator is the operator (physics), operator associated with the momentum (physics), linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case o ...
in the
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
. Also, is the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
, and is the
reduced Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
. These fundamental
physical constant A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that is generally believed to be both universal in nature and have constant value in time. It is contrasted with a mathematical constant, ...
s reflect special relativity and quantum mechanics, respectively. Dirac's purpose in casting this equation was to explain the behavior of the relativistically moving electron, and so to allow the atom to be treated in a manner consistent with relativity. His rather modest hope was that the corrections introduced this way might have a bearing on the problem of
atomic spectra Spectroscopy is the field of study that measures and interprets the electromagnetic spectra that result from the interaction between electromagnetic radiation and matter as a function of the wavelength or frequency of the radiation. Matter wa ...
. Up until that time, attempts to make the old quantum theory of the atom compatible with the theory of relativity, which were based on discretizing the
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
stored in the electron's possibly non-circular orbit of the
atomic nucleus The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford based on the 1909 Geiger–Marsden gold foil experiment. After the discovery of the neutron i ...
, had failed – and the new quantum mechanics of
Heisenberg Werner Karl Heisenberg () (5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the main pioneers of the theory of quantum mechanics. He published his work in 1925 in a breakthrough paper. In the subsequent series ...
, Pauli,
Jordan Jordan ( ar, الأردن; tr. ' ), officially the Hashemite Kingdom of Jordan,; tr. ' is a country in Western Asia. It is situated at the crossroads of Asia, Africa, and Europe, within the Levant region, on the East Bank of the Jordan Rive ...
, Schrödinger, and Dirac himself had not developed sufficiently to treat this problem. Although Dirac's original intentions were satisfied, his equation had far deeper implications for the structure of matter and introduced new mathematical classes of objects that are now essential elements of fundamental physics. The new elements in this equation are the four
matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
, , and , and the four-component
wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements mad ...
. There are four components in because the evaluation of it at any given point in configuration space is a
bispinor In physics, and specifically in quantum field theory, a bispinor, is a mathematical construction that is used to describe some of the fundamental particles of nature, including quarks and electrons. It is a specific embodiment of a spinor, speci ...
. It is interpreted as a superposition of a
spin-up Spin-up refers to the process of a hard disk drive or optical disc drive accelerating its platters or inserted optical disc from a stopped state to an operational speed. The period of time taken by the drive to perform this process is referred to ...
electron, a spin-down electron, a spin-up positron, and a spin-down positron. The matrices and are all
Hermitian {{Short description, none Numerous things are named after the French mathematician Charles Hermite (1822–1901): Hermite * Cubic Hermite spline, a type of third-degree spline * Gauss–Hermite quadrature, an extension of Gaussian quadrature m ...
and are involutory: \alpha_i^2 = \beta^2 = I_4 and they all mutually
anticommute In mathematics, anticommutativity is a specific property of some non-commutative mathematical operations. Swapping the position of two arguments of an antisymmetric operation yields a result which is the ''inverse'' of the result with unswapped ...
: \begin \alpha_i\alpha_j + \alpha_j\alpha_i &= 0\quad(i \neq j) \\ \alpha_i\beta + \beta\alpha_i &= 0 \end These matrices and the form of the wave function have a deep mathematical significance. The algebraic structure represented by the
gamma matrices In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\ma ...
had been created some 50 years earlier by the English mathematician W. K. Clifford. In turn, Clifford's ideas had emerged from the mid-19th-century work of the German mathematician
Hermann Grassmann Hermann Günther Grassmann (german: link=no, Graßmann, ; 15 April 1809 – 26 September 1877) was a German polymath known in his day as a linguist and now also as a mathematician. He was also a physicist, general scholar, and publisher. His mat ...
in his ''Lineare Ausdehnungslehre'' (''Theory of Linear Extensions''). The latter had been regarded as almost incomprehensible by most of his contemporaries. The appearance of something so seemingly abstract, at such a late date, and in such a direct physical manner, is one of the most remarkable chapters in the history of physics. The single symbolic equation thus unravels into four coupled linear first-order
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...
s for the four quantities that make up the wave function. The equation can be written more explicitly in Planck units as: i \partial_x \begin -\psi_4 \\ -\psi_3 \\ -\psi_2 \\ -\psi_1 \end + \partial_y \begin -\psi_4 \\ +\psi_3 \\ -\psi_2 \\ +\psi_1 \end + i \partial_z \begin -\psi_3 \\ +\psi_4 \\ -\psi_1 \\ +\psi_2 \end + m \begin +\psi_1 \\ +\psi_2 \\ -\psi_3 \\ -\psi_4 \end = i \partial_t \begin \psi_1 \\ \psi_2 \\ \psi_3 \\ \psi_4 \end which makes it clearer that it is a set of four partial differential equations with four unknown functions.


Making the Schrödinger equation relativistic

The Dirac equation is superficially similar to the Schrödinger equation for a massive
free particle In physics, a free particle is a particle that, in some sense, is not bound by an external force, or equivalently not in a region where its potential energy varies. In classical physics, this means the particle is present in a "field-free" space. I ...
: -\frac\nabla^2\phi = i\hbar\frac\phi ~. The left side represents the square of the momentum operator divided by twice the mass, which is the non-relativistic kinetic energy. Because relativity treats space and time as a whole, a relativistic generalization of this equation requires that space and time derivatives must enter symmetrically as they do in the Maxwell equations that govern the behavior of light — the equations must be differentially of the ''same order'' in space and time. In relativity, the momentum and the energies are the space and time parts of a spacetime vector, the
four-momentum In special relativity, four-momentum (also called momentum-energy or momenergy ) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is ...
, and they are related by the relativistically invariant relation E^2 = m^2c^4 + p^2c^2 which says that the length of this four-vector is proportional to the rest mass . Substituting the operator equivalents of the energy and momentum from the Schrödinger theory produces the
Klein–Gordon equation The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic wave equation, related to the Schrödinger equation. It is second-order in space and time and manifestly Lorentz-covariant. ...
describing the propagation of waves, constructed from relativistically invariant objects, \left(-\frac\frac + \nabla^2\right)\phi = \frac\phi with the wave function being a relativistic scalar: a complex number which has the same numerical value in all frames of reference. Space and time derivatives both enter to second order. This has a telling consequence for the interpretation of the equation. Because the equation is second order in the time derivative, one must specify initial values both of the wave function itself and of its first time-derivative in order to solve definite problems. Since both may be specified more or less arbitrarily, the wave function cannot maintain its former role of determining the
probability density In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
of finding the electron in a given state of motion. In the Schrödinger theory, the probability density is given by the positive definite expression \rho = \phi^*\phi and this density is convected according to the probability current vector J = -\frac(\phi^*\nabla\phi - \phi\nabla\phi^*) with the conservation of probability current and density following from the continuity equation: \nabla\cdot J + \frac = 0~. The fact that the density is
positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: * Positive-definite bilinear form * Positive-definite f ...
and convected according to this continuity equation implies that one may integrate the density over a certain domain and set the total to 1, and this condition will be maintained by the
conservation law In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, c ...
. A proper relativistic theory with a probability density current must also share this feature. To maintain the notion of a convected density, one must generalize the Schrödinger expression of the density and current so that space and time derivatives again enter symmetrically in relation to the scalar wave function. The Schrödinger expression can be kept for the current, but the probability density must be replaced by the symmetrically formed expression \rho = \frac \left(\psi^*\partial_t\psi - \psi\partial_t\psi^* \right) . which now becomes the 4th component of a spacetime vector, and the entire probability 4-current density has the relativistically covariant expression J^\mu = \frac \left(\psi^*\partial^\mu\psi - \psi\partial^\mu\psi^* \right) . The continuity equation is as before. Everything is compatible with relativity now, but the expression for the density is no longer positive definite; the initial values of both and may be freely chosen, and the density may thus become negative, something that is impossible for a legitimate probability density. Thus, one cannot get a simple generalization of the Schrödinger equation under the naive assumption that the wave function is a relativistic scalar, and the equation it satisfies, second order in time. Although it is not a successful relativistic generalization of the Schrödinger equation, this equation is resurrected in the context of
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, where it is known as the
Klein–Gordon equation The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic wave equation, related to the Schrödinger equation. It is second-order in space and time and manifestly Lorentz-covariant. ...
, and describes a spinless particle field (e.g.
pi meson In particle physics, a pion (or a pi meson, denoted with the Greek letter pi: ) is any of three subatomic particles: , , and . Each pion consists of a quark and an antiquark and is therefore a meson. Pions are the lightest mesons and, more gene ...
or
Higgs boson The Higgs boson, sometimes called the Higgs particle, is an elementary particle in the Standard Model of particle physics produced by the quantum excitation of the Higgs field, one of the fields in particle physics theory. In the Stand ...
). Historically, Schrödinger himself arrived at this equation before the one that bears his name but soon discarded it. In the context of quantum field theory, the indefinite density is understood to correspond to the ''charge'' density, which can be positive or negative, and not the probability density.


Dirac's coup

Dirac thus thought to try an equation that was ''first order'' in both space and time. One could, for example, formally (i.e. by
abuse of notation In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not entirely formally correct, but which might help simplify the exposition or suggest the correct intuition (while possibly minimizing errors an ...
) take the relativistic expression for the energy E = c \sqrt ~, replace by its operator equivalent, expand the square root in an
infinite series In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
of derivative operators, set up an eigenvalue problem, then solve the equation formally by iterations. Most physicists had little faith in such a process, even if it were technically possible. As the story goes, Dirac was staring into the fireplace at Cambridge, pondering this problem, when he hit upon the idea of taking the square root of the wave operator thus: \nabla^2 - \frac\frac = \left(A \partial_x + B \partial_y + C \partial_z + \fracD \partial_t\right)\left(A \partial_x + B \partial_y + C \partial_z + \fracD \partial_t\right)~. On multiplying out the right side it is apparent that, in order to get all the cross-terms such as to vanish, one must assume AB + BA = 0, ~ \ldots ~ with A^2 = B^2 = \dots = 1~. Dirac, who had just then been intensely involved with working out the foundations of Heisenberg's
matrix mechanics Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. It was the first conceptually autonomous and logically consistent formulation of quantum mechanics. Its account of quantum j ...
, immediately understood that these conditions could be met if , , and are ''matrices'', with the implication that the wave function has ''multiple components''. This immediately explained the appearance of two-component wave functions in Pauli's phenomenological theory of
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...
, something that up until then had been regarded as mysterious, even to Pauli himself. However, one needs at least matrices to set up a system with the properties required — so the wave function had ''four'' components, not two, as in the Pauli theory, or one, as in the bare Schrödinger theory. The four-component wave function represents a new class of mathematical object in physical theories that makes its first appearance here. Given the factorization in terms of these matrices, one can now write down immediately an equation \left(A\partial_x + B\partial_y + C\partial_z + \fracD\partial_t\right)\psi = \kappa\psi with \kappa to be determined. Applying again the matrix operator on both sides yields \left(\nabla^2 - \frac\partial_t^2\right)\psi = \kappa^2\psi ~. Taking \kappa = \tfrac shows that all the components of the wave function ''individually'' satisfy the relativistic energy–momentum relation. Thus the sought-for equation that is first-order in both space and time is \left(A\partial_x + B\partial_y + C\partial_z + \fracD\partial_t - \frac\right)\psi = 0 ~. Setting A = i \beta \alpha_1 \, , \, B = i \beta \alpha_2 \, , \, C = i \beta \alpha_3 \, , \, D = \beta ~, and because D^2 = \beta^2 = I_4 , the Dirac equation is produced as written above.


Covariant form and relativistic invariance

To demonstrate the
relativistic invariance Relativity may refer to: Physics * Galilean relativity, Galileo's conception of relativity * Numerical relativity, a subfield of computational physics that aims to establish numerical solutions to Einstein's field equations in general relativity ...
of the equation, it is advantageous to cast it into a form in which the space and time derivatives appear on an equal footing. New matrices are introduced as follows: \begin D &= \gamma^0, \\ A &= i \gamma^1,\quad B = i \gamma^2,\quad C = i \gamma^3, \end and the equation takes the form (remembering the definition of the covariant components of the
4-gradient In differential geometry, the four-gradient (or 4-gradient) \boldsymbol is the four-vector analogue of the gradient \vec from vector calculus. In special relativity and in quantum mechanics, the four-gradient is used to define the properties and re ...
and especially that ) where there is an implied summation over the values of the twice-repeated index , and is the 4-gradient. In practice one often writes the
gamma matrices In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\ma ...
in terms of 2 × 2 sub-matrices taken from the
Pauli matrices In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in ...
and the 2 × 2
identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...
. Explicitly the
standard representation In mathematics, the classical groups are defined as the special linear groups over the reals , the complex numbers and the quaternions together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or ske ...
is \gamma^0 = \begin I_2 & 0 \\ 0 & -I_2 \end,\quad \gamma^1 = \begin 0 & \sigma_x \\ -\sigma_x & 0 \end,\quad \gamma^2 = \begin 0 & \sigma_y \\ -\sigma_y & 0 \end,\quad \gamma^3 = \begin 0 & \sigma_z \\ -\sigma_z & 0 \end. The complete system is summarized using the
Minkowski metric In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of Three-dimensional space, three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two Event (rel ...
on spacetime in the form \left\ = 2 \eta^ I_4 where the bracket expression \ = ab + ba denotes the
anticommutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, a ...
. These are the defining relations of a
Clifford algebra In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperc ...
over a pseudo-orthogonal 4-dimensional space with
metric signature In mathematics, the signature of a metric tensor ''g'' (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative an ...
. The specific Clifford algebra employed in the Dirac equation is known today as the
Dirac algebra In mathematical physics, the Dirac algebra is the Clifford algebra \text_(\mathbb). This was introduced by the mathematical physicist P. A. M. Dirac in 1928 in developing the Dirac equation for spin-½ particles with a matrix representation of t ...
. Although not recognized as such by Dirac at the time the equation was formulated, in hindsight the introduction of this ''
geometric algebra In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the ge ...
'' represents an enormous stride forward in the development of quantum theory. The Dirac equation may now be interpreted as an
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
equation, where the rest mass is proportional to an eigenvalue of the
4-momentum operator In quantum mechanics, the momentum operator is the operator associated with the linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case of one particle in one spatial dimensio ...
, the
proportionality constant In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio, which is called the coefficient of proportionality or proportionality constan ...
being the speed of light: P_\text\psi = mc\psi \,. Using \mathrel \gamma^\mu \partial_\mu ( is pronounced "d-slash"), according to Feynman slash notation, the Dirac equation becomes: i \hbar \psi - m c \psi = 0 \,. In practice, physicists often use units of measure such that , known as
natural units In physics, natural units are physical units of measurement in which only universal physical constants are used as defining constants, such that each of these constants acts as a Coherence (units of measurement), coherent unit of a quantity. For e ...
. The equation then takes the simple form A fundamental theorem states that if two distinct sets of matrices are given that both satisfy the Clifford relations, then they are connected to each other by a similarity transformation: \gamma^ = S^ \gamma^\mu S \,. If in addition the matrices are all
unitary Unitary may refer to: Mathematics * Unitary divisor * Unitary element * Unitary group * Unitary matrix * Unitary morphism * Unitary operator * Unitary transformation * Unitary representation * Unitarity (physics) * ''E''-unitary inverse semigrou ...
, as are the Dirac set, then itself is
unitary Unitary may refer to: Mathematics * Unitary divisor * Unitary element * Unitary group * Unitary matrix * Unitary morphism * Unitary operator * Unitary transformation * Unitary representation * Unitarity (physics) * ''E''-unitary inverse semigrou ...
; \gamma^ = U^\dagger \gamma^\mu U \,. The transformation is unique up to a multiplicative factor of absolute value 1. Let us now imagine a
Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation i ...
to have been performed on the space and time coordinates, and on the derivative operators, which form a covariant vector. For the operator to remain invariant, the gammas must transform among themselves as a contravariant vector with respect to their spacetime index. These new gammas will themselves satisfy the Clifford relations, because of the orthogonality of the Lorentz transformation. By the fundamental theorem, one may replace the new set by the old set subject to a unitary transformation. In the new frame, remembering that the rest mass is a relativistic scalar, the Dirac equation will then take the form \begin \left(iU^\dagger \gamma^\mu U\partial_\mu^\prime - m\right)\psi\left(x^\prime, t^\prime\right) &= 0 \\ U^\dagger(i\gamma^\mu\partial_\mu^\prime - m)U \psi\left(x^\prime, t^\prime\right) &= 0 \,. \end If the transformed spinor is defined as \psi^\prime = U\psi then the transformed Dirac equation is produced in a way that demonstrates manifest relativistic invariance: \left(i\gamma^\mu\partial_\mu^\prime - m\right)\psi^\prime\left(x^\prime, t^\prime\right) = 0 \,. Thus, settling on any unitary representation of the gammas is final, provided the spinor is transformed according to the unitary transformation that corresponds to the given Lorentz transformation. The various representations of the Dirac matrices employed will bring into focus particular aspects of the physical content in the Dirac wave function. The representation shown here is known as the ''standard'' representation – in it, the wave function's upper two components go over into Pauli's 2 spinor wave function in the limit of low energies and small velocities in comparison to light. The considerations above reveal the origin of the gammas in ''geometry'', hearkening back to Grassmann's original motivation; they represent a fixed basis of unit vectors in spacetime. Similarly, products of the gammas such as represent ''
oriented surface In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space i ...
elements'', and so on. With this in mind, one can find the form of the unit volume element on spacetime in terms of the gammas as follows. By definition, it is V = \frac\epsilon_\gamma^\mu\gamma^\nu\gamma^\alpha\gamma^\beta . For this to be an invariant, the epsilon symbol must be a
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
, and so must contain a factor of , where is the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
of the
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
. Since this is negative, that factor is ''imaginary''. Thus V = i \gamma^0\gamma^1\gamma^2\gamma^3 . This matrix is given the special symbol , owing to its importance when one is considering improper transformations of space-time, that is, those that change the orientation of the basis vectors. In the standard representation, it is \gamma_5 = \begin 0 & I_ \\ I_ & 0 \end. This matrix will also be found to anticommute with the other four Dirac matrices: \gamma^5 \gamma^\mu + \gamma^\mu \gamma^5 = 0 It takes a leading role when questions of '' parity'' arise because the volume element as a directed magnitude changes sign under a space-time reflection. Taking the positive square root above thus amounts to choosing a handedness convention on spacetime.


Comparison with related theories


Pauli theory

The necessity of introducing half-integer
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...
goes back experimentally to the results of the
Stern–Gerlach experiment The Stern–Gerlach experiment demonstrated that the spatial orientation of angular momentum is quantized. Thus an atomic-scale system was shown to have intrinsically quantum properties. In the original experiment, silver atoms were sent throug ...
. A beam of atoms is run through a strong
inhomogeneous Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
, which then splits into parts depending on the
intrinsic angular momentum Spin is a conserved quantity carried by elementary particles, and thus by composite particles (hadrons) and atomic nuclei. Spin is one of two types of angular momentum in quantum mechanics, the other being ''orbital angular momentum''. The orbit ...
of the atoms. It was found that for
silver Silver is a chemical element with the Symbol (chemistry), symbol Ag (from the Latin ', derived from the Proto-Indo-European wikt:Reconstruction:Proto-Indo-European/h₂erǵ-, ''h₂erǵ'': "shiny" or "white") and atomic number 47. A soft, whi ...
atoms, the beam was split in two; the
ground state The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state. ...
therefore could not be
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
, because even if the intrinsic angular momentum of the atoms were as small as possible, 1, the beam would be split into three parts, corresponding to atoms with . The conclusion is that silver atoms have net intrinsic angular momentum of . Pauli set up a theory which explained this splitting by introducing a two-component wave function and a corresponding correction term in the
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
, representing a semi-classical coupling of this wave function to an applied magnetic field, as so in
SI units The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...
: (Note that bold faced characters imply
Euclidean vectors In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors ...
in 3 
dimensions In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordina ...
, whereas the Minkowski
four-vector In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a ...
can be defined as A_\mu = (\phi/c,-\mathbf A).) H = \frac\left( \boldsymbol\cdot\left(\mathbf - e \mathbf\right)\right)^2 + e\phi ~. Here and \phi represent the components of the
electromagnetic four-potential An electromagnetic four-potential is a relativistic vector function from which the electromagnetic field can be derived. It combines both an electric scalar potential and a magnetic vector potential into a single four-vector.Gravitation, J.A. Whe ...
in their standard SI units, and the three sigmas are the
Pauli matrices In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in ...
. On squaring out the first term, a residual interaction with the magnetic field is found, along with the usual classical Hamiltonian of a charged particle interacting with an applied field in
SI units The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...
: H = \frac\left(\mathbf - e \mathbf\right)^2 + e\phi - \frac \boldsymbol \cdot \mathbf ~. This Hamiltonian is now a matrix, so the Schrödinger equation based on it must use a two-component wave function. On introducing the external electromagnetic 4-vector potential into the Dirac equation in a similar way, known as
minimal coupling In analytical mechanics and quantum field theory, minimal coupling refers to a coupling between field theory (physics), fields which involves only the electric charge, charge distribution and not higher multipole moments of the charge distribution. ...
, it takes the form: \left(\gamma^\mu(i\hbar\partial_\mu - eA_\mu) - mc\right) \psi = 0 ~. A second application of the Dirac operator will now reproduce the Pauli term exactly as before, because the spatial Dirac matrices multiplied by , have the same squaring and commutation properties as the Pauli matrices. What is more, the value of the
gyromagnetic ratio In physics, the gyromagnetic ratio (also sometimes known as the magnetogyric ratio in other disciplines) of a particle or system is the ratio of its magnetic moment to its angular momentum, and it is often denoted by the symbol , gamma. Its SI u ...
of the electron, standing in front of Pauli's new term, is explained from first principles. This was a major achievement of the Dirac equation and gave physicists great faith in its overall correctness. There is more however. The Pauli theory may be seen as the low energy limit of the Dirac theory in the following manner. First the equation is written in the form of coupled equations for 2-spinors with the SI units restored: \begin mc^2 - E + e \phi & c\boldsymbol\cdot \left(\mathbf - e \mathbf\right) \\ -c\boldsymbol\cdot \left(\mathbf - e \mathbf\right) & mc^2 + E - e \phi \end \begin \psi_ \\ \psi_ \end = \begin 0 \\ 0 \end ~. so \begin (E - e\phi) \psi_ - c\boldsymbol\cdot \left(\mathbf - e \mathbf\right) \psi_ &= mc^2 \psi_ \\ -(E - e\phi) \psi_ + c\boldsymbol\cdot \left(\mathbf - e \mathbf\right) \psi_ &= mc^2 \psi_ \end Assuming the field is weak and the motion of the electron non-relativistic, the total energy of the electron is approximately equal to its
rest energy The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, ...
, and the momentum going over to the classical value, \begin E - e\phi &\approx mc^2 \\ \mathbf &\approx m \mathbf \end and so the second equation may be written \psi_- \approx \frac \boldsymbol\cdot \left(\mathbf - e \mathbf\right) \psi_ which is of order – thus at typical energies and velocities, the bottom components of the Dirac spinor in the standard representation are much suppressed in comparison to the top components. Substituting this expression into the first equation gives after some rearrangement \left(E - mc^2\right) \psi_ = \frac \left boldsymbol\cdot \left(\mathbf - e \mathbf\right)\right2 \psi_ + e\phi \psi_ The operator on the left represents the particle energy reduced by its rest energy, which is just the classical energy, so one can recover Pauli's theory upon identifying his 2-spinor with the top components of the Dirac spinor in the non-relativistic approximation. A further approximation gives the Schrödinger equation as the limit of the Pauli theory. Thus, the Schrödinger equation may be seen as the far non-relativistic approximation of the Dirac equation when one may neglect spin and work only at low energies and velocities. This also was a great triumph for the new equation, as it traced the mysterious that appears in it, and the necessity of a complex wave function, back to the geometry of spacetime through the Dirac algebra. It also highlights why the Schrödinger equation, although superficially in the form of a
diffusion equation The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's la ...
, actually represents the propagation of waves. It should be strongly emphasized that this separation of the Dirac spinor into large and small components depends explicitly on a low-energy approximation. The entire Dirac spinor represents an ''irreducible'' whole, and the components just neglected here to arrive at the Pauli theory will bring in new phenomena in the relativistic regime –
antimatter In modern physics, antimatter is defined as matter composed of the antiparticles (or "partners") of the corresponding particles in "ordinary" matter. Antimatter occurs in natural processes like cosmic ray collisions and some types of radioac ...
and the idea of
creation Creation may refer to: Religion *''Creatio ex nihilo'', the concept that matter was created by God out of nothing * Creation myth, a religious story of the origin of the world and how people first came to inhabit it * Creationism, the belief tha ...
and
annihilation In particle physics, annihilation is the process that occurs when a subatomic particle collides with its respective antiparticle to produce other particles, such as an electron colliding with a positron to produce two photons. The total energy a ...
of particles.


Weyl theory

In the massless case m = 0, the Dirac equation reduces to the
Weyl equation In physics, particularly in quantum field theory, the Weyl equation is a relativistic wave equation for describing massless spin-1/2 particles called Weyl fermions. The equation is named after Hermann Weyl. The Weyl fermions are one of the three ...
, which describes relativistic massless spin- particles. The theory acquires a second \text(1) symmetry: see below.


Physical interpretation


Identification of observables

The critical physical question in a quantum theory is this: what are the physically
observable In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum ph ...
quantities defined by the theory? According to the postulates of quantum mechanics, such quantities are defined by
Hermitian operator In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to it ...
s that act on the
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
of possible states of a system. The eigenvalues of these operators are then the possible results of
measuring Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. In other words, measurement is a process of determining how large or small a physical quantity is as compared t ...
the corresponding physical quantity. In the Schrödinger theory, the simplest such object is the overall Hamiltonian, which represents the total energy of the system. To maintain this interpretation on passing to the Dirac theory, the Hamiltonian must be taken to be H = \gamma^0 \left c^2 + c \gamma^k \left(p_k - q A_k\right) \right+ c q A^0. where, as always, there is an implied summation over the twice-repeated index . This looks promising, because one can see by inspection the rest energy of the particle and, in the case of , the energy of a charge placed in an electric potential . What about the term involving the vector potential? In classical electrodynamics, the energy of a charge moving in an applied potential is H = c\sqrt + qA^0. Thus, the Dirac Hamiltonian is fundamentally distinguished from its classical counterpart, and one must take great care to correctly identify what is observable in this theory. Much of the apparently paradoxical behavior implied by the Dirac equation amounts to a misidentification of these observables.


Hole theory

The negative solutions to the equation are problematic, for it was assumed that the particle has a positive energy. Mathematically speaking, however, there seems to be no reason for us to reject the negative-energy solutions. Since they exist, they cannot simply be ignored, for once the interaction between the electron and the electromagnetic field is included, any electron placed in a positive-energy eigenstate would decay into negative-energy eigenstates of successively lower energy. Real electrons obviously do not behave in this way, or they would disappear by emitting energy in the form of
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they always ...
s. To cope with this problem, Dirac introduced the hypothesis, known as hole theory, that the
vacuum A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often dis ...
is the many-body quantum state in which all the negative-energy electron eigenstates are occupied. This description of the vacuum as a "sea" of electrons is called the
Dirac sea The Dirac sea is a theoretical model of the vacuum as an infinite sea of particles with negative energy. It was first postulated by the British physicist Paul Dirac in 1930 to explain the anomalous negative-energy quantum states predicted by th ...
. Since the
Pauli exclusion principle In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulated ...
forbids electrons from occupying the same state, any additional electron would be forced to occupy a positive-energy eigenstate, and positive-energy electrons would be forbidden from decaying into negative-energy eigenstates. If an electron is forbidden from simultaneously occupying positive-energy and negative-energy eigenstates, then the feature known as
Zitterbewegung In physics, the zitterbewegung ("jittery motion" in German, ) is the predicted rapid oscillatory motion of elementary particles that obey relativistic wave equations. The existence of such motion was first discussed by Gregory Breit in 1928 and la ...
, which arises from the interference of positive-energy and negative-energy states, would have to be considered to be an unphysical prediction of time-dependent Dirac theory. This conclusion may be inferred from the explanation of hole theory given in the preceding paragraph. Recent results have been published in Nature . Gerritsma, G. Kirchmair, F. Zaehringer, E. Solano, R. Blatt, and C. Roos, Nature 463, 68-71 (2010)in which the Zitterbewegung feature was simulated in a trapped-ion experiment. This experiment impacts the hole interpretation if one infers that the physics-laboratory experiment is not merely a check on the mathematical correctness of a Dirac equation solution but the measurement of a real effect whose detectability in electron physics is still beyond reach. Dirac further reasoned that if the negative-energy eigenstates are incompletely filled, each unoccupied eigenstate – called a hole – would behave like a positively charged particle. The hole possesses a ''positive'' energy because energy is required to create a particle–hole pair from the vacuum. As noted above, Dirac initially thought that the hole might be the proton, but
Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...
pointed out that the hole should behave as if it had the same mass as an electron, whereas the proton is over 1800 times heavier. The hole was eventually identified as the
positron The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. It has an electric charge of +1 '' e'', a spin of 1/2 (the same as the electron), and the same mass as an electron. When a positron collides ...
, experimentally discovered by Carl Anderson in 1932. It is not entirely satisfactory to describe the "vacuum" using an infinite sea of negative-energy electrons. The infinitely negative contributions from the sea of negative-energy electrons have to be canceled by an infinite positive "bare" energy and the contribution to the charge density and current coming from the sea of negative-energy electrons is exactly canceled by an infinite positive "
jellium Jellium, also known as the uniform electron gas (UEG) or homogeneous electron gas (HEG), is a quantum mechanical model of interacting electrons in a solid where the positive charges (i.e. atomic nuclei) are assumed to be uniformly distributed in ...
" background so that the net electric charge density of the vacuum is zero. In
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, a
Bogoliubov transformation In theoretical physics, the Bogoliubov transformation, also known as the Bogoliubov–Valatin transformation, was independently developed in 1958 by Nikolay Bogolyubov and John George Valatin for finding solutions of BCS theory in a homogeneous s ...
on the
creation and annihilation operators Creation operators and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator (usually ...
(turning an occupied negative-energy electron state into an unoccupied positive energy positron state and an unoccupied negative-energy electron state into an occupied positive energy positron state) allows us to bypass the Dirac sea formalism even though, formally, it is equivalent to it. In certain applications of
condensed matter physics Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the sub ...
, however, the underlying concepts of "hole theory" are valid. The sea of
conduction electron In solid-state physics, the valence band and conduction band are the bands closest to the Fermi level, and thus determine the electrical conductivity of the solid. In nonmetals, the valence band is the highest range of electron energies i ...
s in an
electrical conductor In physics and electrical engineering, a conductor is an object or type of material that allows the flow of charge (electric current) in one or more directions. Materials made of metal are common electrical conductors. Electric current is gener ...
, called a
Fermi sea A composite fermion is the topological bound state of an electron and an even number of quantized vortices, sometimes visually pictured as the bound state of an electron and, attached, an even number of magnetic flux quanta. Composite fermions we ...
, contains electrons with energies up to the
chemical potential In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...
of the system. An unfilled state in the Fermi sea behaves like a positively charged electron, though it is referred to as a "hole" rather than a "positron". The negative charge of the Fermi sea is balanced by the positively charged ionic lattice of the material.


In quantum field theory

In
quantum field theories In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles ...
such as
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
, the Dirac field is subject to a process of
second quantization Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quantization, in which the fields (typically as t ...
, which resolves some of the paradoxical features of the equation.


Further discussion of Lorentz covariance of the Dirac equation

The Dirac equation is
Lorentz covariant In relativistic physics, Lorentz symmetry or Lorentz invariance, named after the Dutch physicist Hendrik Lorentz, is an equivalence of observation or observational symmetry due to special relativity implying that the laws of physics stay the same ...
. Articulating this helps illuminate not only the Dirac equation, but also the
Majorana spinor In physics, the Majorana equation is a relativistic wave equation. It is named after the Italian physicist Ettore Majorana, who proposed it in 1937 as a means of describing fermions that are their own antiparticle. Particles corresponding to this e ...
and
Elko spinor Elko may refer to: Place names Canada *Elko, British Columbia United States *Elko, Nevada *Elko County, Nevada * Elko, Georgia * Elko, Minnesota * Elko, Missouri * Elko, New York *Elko Tract in Henrico County, Virginia * Elko, South Carolina *El ...
, which although closely related, have subtle and important differences. Understanding Lorentz covariance is simplified by keeping in mind the geometric character of the process. Let a be a single, fixed point in the
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
. Its location can be expressed in multiple
coordinate systems In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
. In the physics literature, these are written as x and x', with the understanding that both x and x' describe ''the same'' point a, but in different local frames of reference (a
frame of reference In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points― geometric points whose position is identified both mathema ...
over a small extended patch of spacetime). One can imagine a as having a
fiber Fiber or fibre (from la, fibra, links=no) is a natural or artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineering materials often incorporate ...
of different coordinate frames above it. In geometric terms, one says that spacetime can be characterized as a
fiber bundle In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
, and specifically, the
frame bundle In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts natur ...
. The difference between two points x and x' in the same fiber is a combination of
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
s and
Lorentz boost In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation i ...
s. A choice of coordinate frame is a (local)
section Section, Sectioning or Sectioned may refer to: Arts, entertainment and media * Section (music), a complete, but not independent, musical idea * Section (typography), a subdivision, especially of a chapter, in books and documents ** Section sign ...
through that bundle. Coupled to the frame bundle is a second bundle, the
spinor bundle In differential geometry, given a spin structure on an n-dimensional orientable Riemannian manifold (M, g),\, one defines the spinor bundle to be the complex vector bundle \pi_\colon\to M\, associated to the corresponding principal bundle \pi_\c ...
. A section through the spinor bundle is just the particle field (the Dirac spinor, in the present case). Different points in the spinor fiber correspond to the same physical object (the fermion) but expressed in different Lorentz frames. Clearly, the frame bundle and the spinor bundle must be tied together in a consistent fashion to get consistent results; formally, one says that the spinor bundle is the
associated bundle In mathematics, the theory of fiber bundles with a structure group G (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from F_1 to F_2, which are both topological spaces wit ...
; it is associated to a
principal bundle In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equip ...
, which in the present case is the frame bundle. Differences between points on the fiber correspond to the symmetries of the system. The spinor bundle has two distinct generators of its symmetries: the
total angular momentum In quantum mechanics, the total angular momentum quantum number parametrises the total angular momentum of a given particle, by combining its orbital angular momentum and its intrinsic angular momentum (i.e., its spin). If s is the particle's s ...
and the
intrinsic angular momentum Spin is a conserved quantity carried by elementary particles, and thus by composite particles (hadrons) and atomic nuclei. Spin is one of two types of angular momentum in quantum mechanics, the other being ''orbital angular momentum''. The orbit ...
. Both correspond to Lorentz transformations, but in different ways. The presentation here follows that of Itzykson and Zuber.Claude Itzykson and Jean-Bernard Zuber, (1980) "Quantum Field Theory", McGraw-Hill ''(See Chapter 2)'' It is very nearly identical to that of Bjorken and Drell. A similar derivation in a general relativistic setting can be found in Weinberg.Steven Weinberg, (1972) "Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity", Wiley & Sons ''(See chapter 12.5, "Tetrad formalism" pages 367ff.)''. Here we fix our spacetime to be flat, that is, our spacetime is Minkowski space. Under a
Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation i ...
x \mapsto x', the Dirac spinor to transform as \psi'(x') = S \psi(x) It can be shown that an explicit expression for S is given by S = \exp\left(\frac \omega^ \sigma_\right) where \omega^ parameterizes the Lorentz transformation, and \sigma_ are the six 4×4 matrices satisfying: \sigma^ = \frac gamma^\mu,\gamma^\nu. This matrix can be interpreted as the
intrinsic angular momentum Spin is a conserved quantity carried by elementary particles, and thus by composite particles (hadrons) and atomic nuclei. Spin is one of two types of angular momentum in quantum mechanics, the other being ''orbital angular momentum''. The orbit ...
of the Dirac field. That it deserves this interpretation arises by contrasting it to the generator J_ of
Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation i ...
s, having the form J_ = \frac \sigma_ + i (x_\mu\partial_\nu - x_\nu\partial_\mu) This can be interpreted as the
total angular momentum In quantum mechanics, the total angular momentum quantum number parametrises the total angular momentum of a given particle, by combining its orbital angular momentum and its intrinsic angular momentum (i.e., its spin). If s is the particle's s ...
. It acts on the spinor field as \psi^\prime(x) = \exp\left(\frac \omega^ J_\right) \psi(x) Note the x above does ''not'' have a prime on it: the above is obtained by transforming x \mapsto x' obtaining the change to \psi(x)\mapsto \psi'(x') and then returning to the original coordinate system x' \mapsto x. The geometrical interpretation of the above is that the frame field is
affine Affine may describe any of various topics concerned with connections or affinities. It may refer to: * Affine, a relative by marriage in law and anthropology * Affine cipher, a special case of the more general substitution cipher * Affine comb ...
, having no preferred origin. The generator J_ generates the symmetries of this space: it provides a relabelling of a fixed point x~. The generator \sigma_ generates a movement from one point in the fiber to another: a movement from x \mapsto x' with both x and x' still corresponding to the same spacetime point a. These perhaps obtuse remarks can be elucidated with explicit algebra. Let x' = \Lambda x be a Lorentz transformation. The Dirac equation is i\gamma^\mu \frac \psi(x) -m\psi(x)=0 If the Dirac equation is to be covariant, then it should have exactly the same form in all Lorentz frames: i\gamma^\mu \frac \psi^\prime(x^\prime) -m\psi^\prime(x^\prime)=0 The two spinors \psi and \psi^\prime should both describe the same physical field, and so should be related by a transformation that does not change any physical observables (charge, current, mass, ''etc.'') The transformation should encode only the change of coordinate frame. It can be shown that such a transformation is a 4×4
unitary matrix In linear algebra, a complex square matrix is unitary if its conjugate transpose is also its inverse, that is, if U^* U = UU^* = UU^ = I, where is the identity matrix. In physics, especially in quantum mechanics, the conjugate transpose is ...
. Thus, one may presume that the relation between the two frames can be written as \psi^\prime(x^\prime) = S(\Lambda) \psi(x) Inserting this into the transformed equation, the result is i\gamma^\mu \frac \frac S(\Lambda)\psi(x) -mS(\Lambda)\psi(x) = 0 The coordinates related by Lorentz transformation satisfy: \frac = _\mu The original Dirac equation is then regained if S(\Lambda) \gamma^\mu S^(\Lambda) = _\nu \gamma^\nu An explicit expression for S(\Lambda) (equal to the expression given above) can be obtained by considering a Lorentz transformation of infinitesimal rotation near the indentity transfomation: _\nu = _\nu + _\nu\ ,\ _\nu = _\nu - _\nu where _ is the
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
: _=g^g_=_ and is symmteric while \omega_=_ g_ is antisymmetric. After plugging and chugging, one obtains S(\Lambda) = I + \frac \omega^ \sigma_ + \mathcal\left(\Lambda^2\right) which is the (infinitesimal) form for S above and yields the relation \sigma^ = \frac gamma^\mu,\gamma^\nu/math> . To obtain the affine relabelling, write \begin \psi'(x') &= \left(I + \frac \omega^ \sigma_ \right) \psi(x) \\ &= \left(I + \frac \omega^ \sigma_ \right) \psi(x' + _\nu \,x^) \\ &= \left(I + \frac \omega^ \sigma_ - x^\prime_\mu \omega^ \partial_\nu\right) \psi(x') \\ &= \left(I + \frac \omega^ J_ \right) \psi(x') \\ \end After properly antisymmetrizing, one obtains the generator of symmetries J_ given earlier. Thus, both J_ and \sigma_ can be said to be the "generators of Lorentz transformations", but with a subtle distinction: the first corresponds to a relabelling of points on the affine
frame bundle In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts natur ...
, which forces a translation along the fiber of the spinor on the
spin bundle In differential geometry, given a spin structure on an n-dimensional orientable Riemannian manifold (M, g),\, one defines the spinor bundle to be the complex vector bundle \pi_\colon\to M\, associated to the corresponding principal bundle \pi_\col ...
, while the second corresponds to translations along the fiber of the spin bundle (taken as a movement x \mapsto x' along the frame bundle, as well as a movement \psi \mapsto \psi' along the fiber of the spin bundle.) Weinberg provides additional arguments for the physical interpretation of these as total and intrinsic angular momentum.Weinberg, "Gravitation", ''op cit.'' ''(See chapter 2.9 "Spin", pages 46-47.)''


Other formulations

The Dirac equation can be formulated in a number of other ways.


Curved spacetime

This article has developed the Dirac equation in flat spacetime according to special relativity. It is possible to formulate the
Dirac equation in curved spacetime In mathematical physics, the Dirac equation in curved spacetime is a generalization of the Dirac equation from flat spacetime (Minkowski space) to curved spacetime, a general Lorentzian manifold. Mathematical formulation Spacetime In full ...
.


The algebra of physical space

This article developed the Dirac equation using four vectors and Schrödinger operators. The
Dirac equation in the algebra of physical space In physics, the algebra of physical space (APS) is the use of the Clifford algebra, Clifford or geometric algebra Cl3,0(R) of the three-dimensional Euclidean space as a model for (3+1)-dimensional spacetime, representing a point in spacetime via a ...
uses a Clifford algebra over the real numbers, a type of geometric algebra.


U(1) symmetry

Natural units are used in this section. The coupling constant is labelled by convention with e: this parameter can also be viewed as modelling the electron charge.


Vector symmetry

The Dirac equation and action admits a \text(1) symmetry where the fields \psi, \bar\psi transform as \begin \psi(x) &\mapsto e^\psi(x), \\ \bar\psi(x) &\mapsto e^\bar\psi(x). \end This is a global symmetry, known as the \text(1) vector symmetry (as opposed to the \text(1) axial symmetry: see below). By
Noether's theorem Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether in ...
there is a corresponding conserved current: this has been mentioned previously as J^\mu(x) = \bar\psi(x)\gamma^\mu\psi(x).


Gauging the symmetry

If we 'promote' the global symmetry, parametrised by the constant \alpha, to a local symmetry, parametrised by a function \alpha:\mathbb^ \to \mathbb, or equivalently e^: \mathbb^ \to \text(1), the Dirac equation is no longer invariant: there is a residual derivative of \alpha(x). The fix proceeds as in scalar electrodynamics: the partial derivative is promoted to a covariant derivative D_\mu D_\mu \psi = \partial_\mu \psi + i e A_\mu\psi, D_\mu \bar\psi = \partial_\mu \bar\psi - i e A_\mu\bar\psi. The covariant derivative depends on the field being acted on. The newly introduced A_\mu is the 4-vector potential from electrodynamics, but also can be viewed as a \text(1)
gauge field In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups) ...
, or a \text(1) connection. The transformation law under gauge transformations for A_\mu is then the usual A_\mu(x) \mapsto A_\mu(x) + \frac\partial_\mu\alpha(x) but can also be derived by asking that covariant derivatives transform under a gauge transformation as D_\mu\psi(x) \mapsto e^D_\mu\psi(x), D_\mu\bar\psi(x) \mapsto e^D_\mu\bar\psi(x). We then obtain a gauge-invariant Dirac action by promoting the partial derivative to a covariant one: S = \int d^4x\,\bar\psi\,(iD\!\!\!\!\big / - m)\,\psi = \int d^4x\,\bar\psi\,(i\gamma^\mu D_\mu - m)\,\psi. The final step needed to write down a gauge-invariant Lagrangian is to add a Maxwell Lagrangian term, S_ = \int d^4x\,-\fracF^F_. Putting these together gives Expanding out the covariant derivative allows the action to be written in a second useful form: S_ = \int d^4x\,-\fracF^F_ + \bar\psi\,(i\partial\!\!\!\big / - m)\,\psi - eJ^\mu A_\mu


Axial symmetry

Massless Dirac fermions, that is, fields \psi(x) satisfying the Dirac equation with m = 0, admit a second, inequivalent \text(1) symmetry. This is seen most easily by writing the four-component Dirac fermion \psi(x) as a pair of two-component vector fields, \psi(x) = \begin \psi_1(x)\\ \psi_2(x) \end, and adopting the chiral representation for the gamma matrices, so that i\gamma^\mu\partial_\mu may be written i\gamma^\mu\partial_\mu = \begin 0 & i\sigma^\mu \partial_\mu\\ i\bar\sigma^\mu \partial_\mu\ & 0 \end where \sigma^\mu has components (I_2, \sigma^i) and \bar\sigma^\mu has components (I_2, -\sigma^i). The Dirac action then takes the form S = \int d^4x\, \psi_1^\dagger(i\sigma^\mu\partial_\mu)\psi_1 + \psi_2^\dagger(i\bar\sigma^\mu\partial_\mu) \psi_2. That is, it decouples into a theory of two Weyl spinors or Weyl fermions. The earlier vector symmetry is still present, where \psi_1 and \psi_2 rotate identically. This form of the action makes the second inequivalent \text(1) symmetry manifest: \begin \psi_1(x) &\mapsto e^ \psi_1(x), \\ \psi_2(x) &\mapsto e^\psi_2(x). \end This can also be expressed at the level of the Dirac fermion as \psi(x) \mapsto \exp(i\beta\gamma^5) \psi(x) where \exp is the exponential map for matrices. This isn't the only \text(1) symmetry possible, but it is conventional. Any 'linear combination' of the vector and axial symmetries is also a \text(1) symmetry. Classically, the axial symmetry admits a well-formulated gauge theory. But at the quantum level, there is an anomaly, that is, an obstruction to gauging.


Extension to color symmetry

We can extend this discussion from an abelian \text(1) symmetry to a general non-abelian symmetry under a
gauge group In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations ( Lie group ...
G, the group of color symmetries for a theory. For concreteness, we fix G = \text(N), the
special unitary group In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
of matrices acting on \mathbb^N. Before this section, \psi(x) could be viewed as a spinor field on Minkowski space, in other words a function \psi: \mathbb^\mapsto \mathbb^4, and its components in \mathbb^4 are labelled by spin indices, conventionally Greek indices taken from the start of the alphabet \alpha,\beta,\gamma,\cdots. Promoting the theory to a gauge theory, informally \psi acquires a part transforming like \mathbb^N, and these are labelled by color indices, conventionally Latin indices i,j,k,\cdots. In total, \psi(x) has 4N components, given in indices by \psi^(x). The 'spinor' labels only how the field transforms under spacetime transformations. Formally, \psi(x) is valued in a tensor product, that is, it is a function \psi:\mathbb^ \to \mathbb^4 \otimes \mathbb^N. Gauging proceeds similarly to the abelian \text(1) case, with a few differences. Under a gauge transformation U:\mathbb^ \rightarrow \text(N), the spinor fields transform as \psi(x) \mapsto U(x)\psi(x) \bar\psi(x)\mapsto \bar\psi(x)U^\dagger(x). The matrix-valued gauge field A_\mu or \text(N) connection transforms as A_\mu(x) \mapsto U(x)A_\mu(x)U(x)^ + \frac(\partial_\mu U(x))U(x)^, and the covariant derivatives defined D_\mu\psi = \partial_\mu \psi + igA_\mu\psi, D_\mu\bar\psi = \partial_\mu \bar\psi - ig\bar\psi A_\mu^\dagger transform as D_\mu\psi(x) \mapsto U(x)D_\mu\psi(x), D_\mu\bar\psi(x) \mapsto (D_\mu\bar\psi(x))U(x)^\dagger. Writing down a gauge-invariant action proceeds exactly as with the \text(1) case, replacing the Maxwell Lagrangian with the Yang–Mills Lagrangian S_ = \int d^4x \,-\frac\text(F^F_) where the Yang–Mills field strength or curvature is defined here as F_ = \partial_\mu A_\nu - \partial_\nu A_\mu - ig\left _\mu,A_\nu\right/math> and cdot,\cdot/math> is the matrix commutator. The action is then


Physical applications

For physical applications, the case N=3 describes the
quark A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly o ...
sector of the
Standard model The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions - excluding gravity) in the universe and classifying a ...
which models strong interactions. Quarks are modelled as Dirac spinors; the gauge field is the
gluon A gluon ( ) is an elementary particle that acts as the exchange particle (or gauge boson) for the strong force between quarks. It is analogous to the exchange of photons in the electromagnetic force between two charged particles. Gluons bind q ...
field. The case N=2 describes part of the
electroweak In particle physics, the electroweak interaction or electroweak force is the unified field theory, unified description of two of the four known fundamental interactions of nature: electromagnetism and the weak interaction. Although these two force ...
sector of the Standard model. Leptons such as electrons and neutrinos are the Dirac spinors; the gauge field is the W gauge boson.


Generalisations

This expression can be generalised to arbitrary Lie group G with connection A_\mu and a representation (\rho, G, V), where the colour part of \psi is valued in V. Formally, the Dirac field is a function \psi:\mathbb^ \to \mathbb^4\otimes V. Then \psi transforms under a gauge transformation g:\mathbb^ \to G as \psi(x) \mapsto \rho(g(x))\psi(x) and the covariant derivative is defined D_\mu\psi = \partial_\mu\psi + \rho(A_\mu)\psi where here we view \rho as a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
representation of the Lie algebra \mathfrak = \text(G) associated to G. This theory can be generalised to curved spacetime, but there are subtleties which arise in gauge theory on a general spacetime (or more generally still, a manifold) which, on flat spacetime, can be ignored. This is ultimately due to the contractibility of flat spacetime which allows us to view a gauge field and gauge transformations as defined globally on \mathbb^.


See also


Articles on the Dirac equation

* Dirac field *
Dirac spinor In quantum field theory, the Dirac spinor is the spinor that describes all known fundamental particles that are fermions, with the possible exception of neutrinos. It appears in the plane-wave solution to the Dirac equation, and is a certain combin ...
*
Gordon decomposition In mathematical physics, the Gordon decomposition (named after Walter Gordon) of the Dirac current is a splitting of the charge or particle-number current into a part that arises from the motion of the center of mass of the particles and a part tha ...
* Klein paradox *
Nonlinear Dirac equation :''See Ricci calculus and Van der Waerden notation for the notation.'' In quantum field theory, the nonlinear Dirac equation is a model of self-interacting Dirac fermions. This model is widely considered in quantum physics as a toy model of self-i ...


Other equations

*
Breit equation The Breit equation is a relativistic wave equation derived by Gregory Breit in 1929 based on the Dirac equation, which formally describes two or more massive spin-1/2 particles (electrons, for example) interacting electromagnetically to the firs ...
* Dirac–Kähler equation *
Klein–Gordon equation The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic wave equation, related to the Schrödinger equation. It is second-order in space and time and manifestly Lorentz-covariant. ...
*
Rarita–Schwinger equation In theoretical physics, the Rarita–Schwinger equation is the relativistic field equation of spin-3/2 fermions. It is similar to the Dirac equation for spin-1/2 fermions. This equation was first introduced by William Rarita and Julian Schwinge ...
*
Two-body Dirac equations In quantum field theory, and in the significant subfields of quantum electrodynamics (QED) and quantum chromodynamics (QCD), the two-body Dirac equations (TBDE) of constraint dynamics provide a three-dimensional yet manifestly covariant reformula ...
*
Weyl equation In physics, particularly in quantum field theory, the Weyl equation is a relativistic wave equation for describing massless spin-1/2 particles called Weyl fermions. The equation is named after Hermann Weyl. The Weyl fermions are one of the three ...
*
Majorana equation In physics, the Majorana equation is a relativistic wave equation. It is named after the Italian physicist Ettore Majorana, who proposed it in 1937 as a means of describing fermions that are their own antiparticle. Particles corresponding to this ...


Other topics

*
Fermionic field In quantum field theory, a fermionic field is a quantum field whose quanta are fermions; that is, they obey Fermi–Dirac statistics. Fermionic fields obey canonical anticommutation relations rather than the canonical commutation relations of bos ...
*
Feynman checkerboard The Feynman checkerboard, or relativistic chessboard model, was Richard Feynman’s sum-over-paths formulation of the kernel for a free spin-½ particle moving in one spatial dimension. It provides a representation of solutions of the Dirac equ ...
*
Foldy–Wouthuysen transformation The Foldy–Wouthuysen transformation was historically significant and was formulated by Leslie Lawrance Foldy and Siegfried Adolf Wouthuysen in 1949 to understand the nonrelativistic limit of the Dirac equation, the equation for spin-½ partic ...
*
Quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
*
Quantum chromodynamics In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type ...


References


Citations


Selected papers

* * * * *


Textbooks

* * * * * * *


External links


The Dirac Equation
at MathPages
The Nature of the Dirac Equation, its solutions, and Spin



Pedagogic Aids to Quantum Field Theory
click on Chap. 4 for a step-by-small-step introduction to the Dirac equation, spinors, and relativistic spin/helicity operators.
BBC Documentary ''Atom 3 The Illusion of Reality''
{{Authority control 1928 introductions Fermions Partial differential equations
Equation In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
Quantum field theory Spinors