SUPPORT THE WORK

GetWiki

Dirac equation

ARTICLE SUBJECTS
aesthetics  →
being  →
complexity  →
database  →
enterprise  →
ethics  →
fiction  →
history  →
internet  →
knowledge  →
language  →
licensing  →
linux  →
logic  →
method  →
news  →
perception  →
philosophy  →
policy  →
purpose  →
religion  →
science  →
sociology  →
software  →
truth  →
unix  →
wiki  →
ARTICLE TYPES
essay  →
feed  →
help  →
system  →
wiki  →
ARTICLE ORIGINS
critical  →
discussion  →
forked  →
imported  →
original  →
Dirac equation
[ temporary import ]
please note:
- the content below is remote from Wikipedia
- it has been imported raw for GetWiki
{{Short description|Relativistic quantum mechanical wave equation}}{{Distinguish|Dirac delta function}}{{Use American English|date=January 2019}}{{Use dmy dates|date=July 2019}}{{Quantum mechanics|cTopic=Equations}}In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-{{1/2}} massive particles, called “Dirac particles”, such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity,BOOK, Quanta: A handbook of concepts, P.W. Atkins, Oxford University Press, 52, 1974, 978-0-19-855493-6, 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 in a completely rigorous way.The equation also implied the existence of a new form of matter, antimatter, 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. The wave functions in the Dirac theory are vectors of four complex numbers (known as bispinors), two of which resemble the Pauli wavefunction in the non-relativistic limit, in contrast to the Schrödinger equation which described wave functions of only one complex value. Moreover, in the limit of zero mass, the Dirac equation reduces to the Weyl equation.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—represents one of the great triumphs of theoretical physics. This accomplishment has been described as fully on a par with the works of Newton, Maxwell, and Einstein before him.BOOK, The New Quantum Universe, T.Hey, P.Walters, Cambridge University Press, 2009, 228, 978-0-521-56457-1, It has been deemed by some physicists to be the “real seed of modern physics”.WEB, Zichichi, Antonino, Antonino Zichichi, 2000-03-02, Dirac, Einstein and physics,physicsworld.com/a/dirac-einstein-and-physics/, 2023-10-22, Physics World, en-GB, In the context of quantum field theory, the Dirac equation is reinterpreted to describe quantum fields corresponding to spin-{{1/2}} particles.The Dirac equation is inscribed upon a plaque on the floor of Westminster Abbey. Unveiled on 13 November 1995, the plaque commemorates Dirac’s life.WEB, Gisela Dirac-Wahrenburg,www.dirac.ch/PaulDirac.html, Paul Dirac, Dirac.ch, 2013-07-12,

History

The Dirac equation in the form originally proposed by Dirac is:BOOK, Dirac, Paul A.M., Principles of Quantum Mechanics, 4th, 255, Oxford University Press, International Series of Monographs on Physics, 1958, 1982, 978-0-19-852011-5, left(beta mc^2 + c sum_{n = 1}^{3}alpha_n p_nright) psi (x,t) = i hbar frac{partialpsi(x,t) }{partial t} where {{math|ψ(x, t)}} is the wave function for the electron of rest mass {{math|m}} with spacetime coordinates {{math|x, t}}. The {{math|p1, p2, p3}} are the components of the momentum, understood to be the momentum operator in the Schrödinger equation. Also, {{math|c}} is the speed of light, and {{math|ħ}} is the reduced Planck constant. These fundamental physical constants 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.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 stored in the electron’s possibly non-circular orbit of the atomic nucleus, had failed – and the new quantum mechanics of Heisenberg, Pauli, Jordan, 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 {{nowrap|4 × 4}} matrices {{math|α1}}, {{math|α2}}, {{math|α3}} and {{math|β}}, and the four-component wave function {{math|ψ}}. There are four components in {{math|ψ}} because the evaluation of it at any given point in configuration space is a bispinor. It is interpreted as a superposition of a spin-up electron, a spin-down electron, a spin-up positron, and a spin-down positron.The {{nowrap|4 × 4}} matrices {{math|αk}} and {{math|β}} are all Hermitian and are involutory:alpha_i^2 = beta^2 = I_4and they all mutually anticommute:begin{align}
alpha_ialpha_j + alpha_jalpha_i &= 0quad(i neq j)
alpha_ibeta + betaalpha_i &= 0
end{align}These matrices and the form of the wave function have a deep mathematical significance. The algebraic structure represented by the gamma matrices 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 in his Lineare Ausdehnungslehre (Theory of Linear Expansion). 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.{{citation needed |reason=Historical perspective and author editorial |date=October 2018}} (Even more so, a validation of the exquisite insight displayed by the mathematicians Grassmann and Clifford.){{clear}}The single symbolic equation thus unravels into four coupled linear first-order partial differential equations for the four quantities that make up the wave function. The equation can be written more explicitly in Planck units as:BOOK, The Dirac Equation in Curved Spacetime: A Guide for Calculations, Peter, Collas, David, Klein, Springer, 2019, 978-3-030-14825-6, 7,books.google.com/books?id=YymODwAAQBAJ, Extract of page 7i partial_x begin{bmatrix} +psi_4 +psi_3 -psi_2 -psi_1 end{bmatrix}+ partial_y begin{bmatrix} +psi_4 -psi_3 -psi_2 +psi_1 end{bmatrix}+ i partial_z begin{bmatrix} +psi_3 -psi_4 -psi_1 +psi_2 end{bmatrix}- m begin{bmatrix} +psi_1 +psi_2 +psi_3 +psi_4 end{bmatrix} = i partial_t begin{bmatrix} -psi_1 -psi_2 +psi_3 +psi_4 end{bmatrix}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:-frac{hbar^2}{2m}nabla^2phi = ihbarfrac{partial}{partial t}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, and they are related by the relativistically invariant relationE^2 = m^2c^4 + p^2c^2 which says that the length of this four-vector is proportional to the rest mass {{math|m}}. Substituting the operator equivalents of the energy and momentum from the Schrödinger theory produces the Klein–Gordon equation describing the propagation of waves, constructed from relativistically invariant objects,left(-frac{1}{c^2}frac{partial^2}{partial t^2} + nabla^2right)phi = frac{m^2c^2}{hbar^2}phi with the wave function {{math|ϕ}} 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 of finding the electron in a given state of motion. In the Schrödinger theory, the probability density is given by the positive definite expressionrho = phi^*phi and this density is convected according to the probability current vectorJ = -frac{ihbar}{2m}(phi^*nablaphi - phinablaphi^*) with the conservation of probability current and density following from the continuity equation:nablacdot J + frac{partialrho}{partial t} = 0~.The fact that the density is positive definite 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. 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{{explain|reason=Why?|date=November 2021}}rho = frac{ihbar}{2mc^2} left(psi^*partial_tpsi - psipartial_tpsi^* right) .which now becomes the 4th component of a spacetime vector, and the entire probability 4-current density has the relativistically covariant expressionJ^mu = frac{ihbar}{2m} left(psi^*partial^mupsi - psipartial^mupsi^* 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 {{math|ψ}} and {{math|∂tψ}} 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, where it is known as the Klein–Gordon equation, and describes a spinless particle field (e.g. pi meson or Higgs boson). 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. He postulated an equation of the formEpsi = (vec{alpha} cdot vec{p} + beta m) psiwhere the operators (vec{alpha}, beta) must be independent of (vec{p}, t) for linearity and independent of (vec{x}, t) for space-time homogeneity. These constraints implied additional dynamical variables that the (vec{alpha}, beta) operators will depend upon; from this requirement Dirac concluded that the operators would depend upon 4x4 matrices, related to the Pauli matrices.BOOK, Duck, Ian,www.worldscientific.com/worldscibooks/10.1142/3457, Pauli and the Spin-Statistics Theorem, Sudarshan, E C G, 1998, WORLD SCIENTIFIC, 978-981-02-3114-9, en, 10.1142/3457, {{rp|205}}One could, for example, formally (i.e. by abuse of notation) take the relativistic expression for the energyE = c sqrt{p^2 + m^2c^2} ~,replace {{math|p}} by its operator equivalent, expand the square root in an infinite series 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 (see also half derivative) thus:nabla^2 - frac{1}{c^2}frac{partial^2}{partial t^2} = left(A partial_x + B partial_y + C partial_z + frac{i}{c}D partial_tright)left(A partial_x + B partial_y + C partial_z + frac{i}{c}D partial_tright)~.On multiplying out the right side it is apparent that, in order to get all the cross-terms such as {{math|∂x∂y}} to vanish, one must assumeAB + BA = 0, ~ ldots ~withA^2 = B^2 = dots = 1~.Dirac, who had just then been intensely involved with working out the foundations of Heisenberg’s matrix mechanics, immediately understood that these conditions could be met if {{math|A}}, {{math|B}}, {{math|C}} and {{math|D}} 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, something that up until then had been regarded as mysterious, even to Pauli himself. However, one needs at least {{nowrap|4 × 4}} 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 equationleft(Apartial_x + Bpartial_y + Cpartial_z + frac{i}{c}Dpartial_tright)psi = kappapsi with kappa to be determined. Applying again the matrix operator on both sides yieldsleft(nabla^2 - frac{1}{c^2}partial_t^2right)psi = kappa^2psi ~.Taking kappa = tfrac{mc}{hbar} 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 isleft(Apartial_x + Bpartial_y + Cpartial_z + frac{i}{c}Dpartial_t - frac{mc}{hbar}right)psi = 0 ~.SettingA = 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 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{align}
D &= gamma^0,
A &= i gamma^1,quad B = i gamma^2,quad C = i gamma^3,
end{align}and the equation takes the form (remembering the definition of the covariant components of the 4-gradient and especially that {{math|1=∂0 = {{sfrac|1|c}}∂t}}){{Equation box 1|title=Dirac equation|indent=:|equation = (i hbar gamma^mu partial_mu - m c) psi = 0|border|border colour =#50C878|background colour = #ECFCF4}}where there is an implied summation over the values of the twice-repeated index {{math|μ {{=}} 0, 1, 2, 3}}, and {{math|∂μ}} is the 4-gradient. In practice one often writes the gamma matrices in terms of 2 × 2 sub-matrices taken from the Pauli matrices and the 2 × 2 identity matrix. Explicitly the standard representation isgamma^0 = begin{pmatrix} I_2 & 0 0 & -I_2 end{pmatrix},quadgamma^1 = begin{pmatrix} 0 & sigma_x -sigma_x & 0 end{pmatrix},quadgamma^2 = begin{pmatrix} 0 & sigma_y -sigma_y & 0 end{pmatrix},quadgamma^3 = begin{pmatrix} 0 & sigma_z -sigma_z & 0 end{pmatrix}.The complete system is summarized using the Minkowski metric on spacetime in the formleft{gamma^mu, gamma^nuright} = 2 eta^{munu} I_4where the bracket expression{a, b} = ab + badenotes the anticommutator. These are the defining relations of a Clifford algebra over a pseudo-orthogonal 4-dimensional space with metric signature {{math|(+ − − −)}}. The specific Clifford algebra employed in the Dirac equation is known today as the Dirac algebra. Although not recognized as such by Dirac at the time the equation was formulated, in hindsight the introduction of this geometric algebra represents an enormous stride forward in the development of quantum theory.The Dirac equation may now be interpreted as an eigenvalue equation, where the rest mass is proportional to an eigenvalue of the 4-momentum operator, the proportionality constant being the speed of light:P_text{op}psi = mcpsi ,.Using {partial!!!/} mathrel{stackrel{mathrm{def}}{=}} gamma^mu partial_mu ({partial!!!big /} is pronounced “d-slash“),BOOK, Pendleton, Brian,www2.ph.ed.ac.uk/~bjp/qt/rqt.pdf,www2.ph.ed.ac.uk/~bjp/qt/rqt.pdf," title="ghostarchive.org/archive/20221009www2.ph.ed.ac.uk/~bjp/qt/rqt.pdf,">ghostarchive.org/archive/20221009www2.ph.ed.ac.uk/~bjp/qt/rqt.pdf, 2022-10-09, live, Quantum Theory, 2012–2013, section 4.3 “The Dirac Equation”, according to Feynman slash notation, the Dirac equation becomes:i hbar {partial!!!big /} psi - m c psi = 0 ,.In practice, physicists often use units of measure such that {{math|ħ {{=}} c {{=}} 1}}, known as natural units. The equation then takes the simple form{{Equation box 1|title=Dirac equation (natural units)|indent=:|equation = (i{partial!!!big /} - m) psi = 0|border|border colour = #50C878|background colour = #ECFCF4}}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^{muprime} = S^{-1} gamma^mu S ,.If in addition the matrices are all unitary, as are the Dirac set, then {{math|S}} itself is unitary;gamma^{muprime} = U^dagger gamma^mu U ,.The transformation {{math|U}} is unique up to a multiplicative factor of absolute value 1. Let us now imagine a Lorentz transformation to have been performed on the space and time coordinates, and on the derivative operators, which form a covariant vector. For the operator {{math|γμ∂μ}} 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 formbegin{align}
left(iU^dagger gamma^mu Upartial_mu^prime - mright)psileft(x^prime, t^primeright) &= 0
U^dagger(igamma^mupartial_mu^prime - m)U psileft(x^prime, t^primeright) &= 0 ,.
end{align}If the transformed spinor is defined aspsi^prime = Upsithen the transformed Dirac equation is produced in a way that demonstrates manifest relativistic invariance:left(igamma^mupartial_mu^prime - mright)psi^primeleft(x^prime, t^primeright) = 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 {{math|γ’μ’γν}} represent oriented surface 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 isV = frac{1}{4!}epsilon_{munualphabeta}gamma^mugamma^nugamma^alphagamma^beta .For this to be an invariant, the epsilon symbol must be a tensor, and so must contain a factor of {{math|{{sqrt|g}}}}, where {{math|g}} is the determinant of the metric tensor. Since this is negative, that factor is imaginary. ThusV = i gamma^0gamma^1gamma^2gamma^3 .This matrix is given the special symbol {{math|γ5}}, 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 isgamma_5 = begin{pmatrix} 0 & I_{2} I_{2} & 0 end{pmatrix}.This matrix will also be found to anticommute with the other four Dirac matrices:gamma^5 gamma^mu + gamma^mu gamma^5 = 0It 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

{{See also|Pauli equation|Lévy-Leblond equation}}The necessity of introducing half-integer spin goes back experimentally to the results of the Stern–Gerlach experiment. A beam of atoms is run through a strong inhomogeneous magnetic field, which then splits into {{math|N}} parts depending on the intrinsic angular momentum of the atoms. It was found that for silver atoms, the beam was split in two; the ground state therefore could not be integer, 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 {{math|Lz {{=}} −1, 0, +1}}. The conclusion is that silver atoms have net intrinsic angular momentum of {{frac|1|2}}. Pauli set up a theory which explained this splitting by introducing a two-component wave function and a corresponding correction term in the Hamiltonian, representing a semi-classical coupling of this wave function to an applied magnetic field, as so in SI units: (Note that bold faced characters imply Euclidean vectors in 3 dimensions, whereas the Minkowski four-vector {{math|Aμ}} can be defined as A_mu = (phi/c,-mathbf A).)H = frac{1}{2m}left( boldsymbol{sigma}cdotleft(mathbf{p} - e mathbf{A}right)right)^2 + ephi ~.Here {{math|A}} and phi represent the components of the electromagnetic four-potential in their standard SI units, and the three sigmas are the Pauli matrices. 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:H = frac{1}{2m}left(mathbf{p} - e mathbf{A}right)^2 + ephi - frac{ehbar}{2m} boldsymbol{sigma} cdot mathbf{B} ~.This Hamiltonian is now a {{nowrap|2 × 2}} 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, it takes the form:left(gamma^mu(ihbarpartial_mu - eA_mu) - mcright) 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 {{math|i}}, have the same squaring and commutation properties as the Pauli matrices. What is more, the value of the gyromagnetic ratio 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{pmatrix}
mc^2 - E + e phi &
cboldsymbol{sigma}cdot left(mathbf{p} - e mathbf{A}right)
-cboldsymbol{sigma}cdot left(mathbf{p} - e mathbf{A}right) &
mc^2 + E - e phi
end{pmatrix} begin{pmatrix} psi_{+} psi_{-} end{pmatrix} =
begin{pmatrix} 0 0 end{pmatrix} ~.
sobegin{align}
(E - ephi) psi_{+} - cboldsymbol{sigma}cdot left(mathbf{p} - e mathbf{A}right) psi_{-} &= mc^2 psi_{+}
cboldsymbol{sigma}cdot left(mathbf{p} - e mathbf{A}right) psi_{+}-(E - ephi) psi_{-} &= mc^2 psi_{-} end{align}
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, and the momentum going over to the classical value,begin{align}
E - ephi &approx mc^2
mathbf{p} &approx m mathbf{v}
end{align}and so the second equation may be writtenpsi_- approx frac{1}{2mc} boldsymbol{sigma}cdot left(mathbf{p} - e mathbf{A}right) psi_{+} which is of order {{math|{{sfrac|v|c}}}} – 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^2right) psi_{+} = frac{1}{2m} left[boldsymbol{sigma}cdot left(mathbf{p} - e mathbf{A}right)right]^2 psi_{+} + ephi 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 {{math|i}} 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, 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 and the idea of creation and annihilation of particles.

Weyl theory

In the massless case m = 0, the Dirac equation reduces to the Weyl equation, which describes relativistic massless spin-{{frac|2}} particles.BOOK, Tommy, Ohlsson, Tommy Ohlsson, 22 September 2011, Relativistic Quantum Physics: From advanced quantum mechanics to introductory quantum field theory, 86, Cambridge University Press, 978-1-139-50432-4,books.google.com/books?id=hRavtAW5EFcC&pg=PA86, The theory acquires a second text{U}(1) symmetry: see below.

Physical interpretation

Identification of observables

The critical physical question in a quantum theory is this: what are the physically observable quantities defined by the theory? According to the postulates of quantum mechanics, such quantities are defined by Hermitian operators that act on the Hilbert space of possible states of a system. The eigenvalues of these operators are then the possible results of measuring 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 beH = gamma^0 left[mc^2 + c gamma^k left(p_k - q A_kright) right] + c q A^0.where, as always, there is an implied summation over the twice-repeated index {{math|k {{=}} 1, 2, 3}}. This looks promising, because one can see by inspection the rest energy of the particle and, in the case of {{math|A {{=}} 0}}, the energy of a charge placed in an electric potential {{math|cqA0}}. What about the term involving the vector potential? In classical electrodynamics, the energy of a charge moving in an applied potential isH = csqrt{left(mathbf{p} - qmathbf{A}right)^2 + m^2c^2} + 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.{{Citation needed|date=January 2020}}

Hole theory

The negative {{math|E}} 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 photons.To cope with this problem, Dirac introduced the hypothesis, known as hole theory, that the vacuum 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. Since the Pauli exclusion principle 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.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 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, experimentally discovered by Carl Anderson in 1932.BOOK, Penrose, Roger, The Road to Reality, 2004, Jonathan Cape, 0-224-04447-8, 625, 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” background so that the net electric charge density of the vacuum is zero. In quantum field theory, a Bogoliubov transformation on the creation and annihilation operators (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, however, the underlying concepts of “hole theory” are valid. The sea of conduction electrons in an electrical conductor, called a Fermi sea, contains electrons with energies up to the chemical potential of the system. An unfilled state in the Fermi sea behaves like a positively charged electron, and although it too is referred to as an “electron hole”, it is distinct from a positron. The negative charge of the Fermi sea is balanced by the positively charged ionic lattice of the material.

In quantum field theory

{{See also|Fermionic field}}In quantum field theories such as quantum electrodynamics, the Dirac field is subject to a process of second quantization, which resolves some of the paradoxical features of the equation.

Mathematical formulation

In its modern formulation for field theory, the Dirac equation is written in terms of a Dirac spinor field psi taking values in a complex vector space described concretely as mathbb{C}^4, defined on flat spacetime (Minkowski space) mathbb{R}^{1,3}. Its expression also contains gamma matrices and a parameter m > 0 interpreted as the mass, as well as other physical constants. Dirac first obtained his equation through a factorization of Einstein’s energy-momentum-mass equivalence relation assuming a scalar product of momentum vectors determined by the metric tensor and quantized the resulting relation by associating momenta to their respective operators.In terms of a field psi: mathbb{R}^{1,3}rightarrow mathbb{C}^4, the Dirac equation is then{{Equation box 1|title=Dirac equation|indent=:|equation = (i hbar gamma^mu partial_mu - m c) psi(x) = 0|border|border colour =#50C878|background colour = #ECFCF4}}and in natural units, with Feynman slash notation,{{Equation box 1|title=Dirac equation (natural units)|indent=:|equation = (ipartial !!!/ - m) psi(x) = 0|border|border colour =#50C878|background colour = #ECFCF4}}The gamma matrices are a set of four 4 times 4 complex matrices (elements of text{Mat}_{4times 4}(mathbb{C})) which satisfy the defining anti-commutation relations:{gamma^mu, gamma^nu} = 2eta^{munu}I_4where eta^{munu} is the Minkowski metric element, and the indices mu, nu run over 0,1,2 and 3. These matrices can be realized explicitly under a choice of representation. Two common choices are the Dirac representationgamma^0 = begin{pmatrix} I_2 & 0 0 & -I_2 end{pmatrix},quadgamma^i = begin{pmatrix} 0 & sigma^i -sigma^i & 0 end{pmatrix},where sigma^i are the Pauli matrices, and the chiral representation: the gamma^i are the same, but gamma^0 = begin{pmatrix} 0 & I_2 I_2 & 0 end{pmatrix}.The slash notation is a compact notation forA!!!/ := gamma^mu A_muwhere 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 asbarpsi(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^0gamma^mugamma^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:barpsi (x)( - igamma^mu overleftarrow{partial}_mu - m) = 0where the partial derivative overleftarrow{partial}_mu acts from the right on barpsi(x): written in the usual way in terms of a left action of the derivative, we have- ipartial_mubarpsi (x)gamma^mu - mbarpsi (x) = 0.

Klein–Gordon equation

Applying ipartial!!!/ + m to the Dirac equation gives(partial_mupartial^mu + m^2)psi(x) = 0.That is, each component of the Dirac spinor field satisfies the Klein–Gordon equation.

Conserved current

A conserved current of the theory isJ^mu = bar{psi}gamma^mupsi.{{math proof | title = Proof of conservation from Dirac equation | proof =Adding the Dirac and adjoint Dirac equations givesi((partial_mubarpsi)gamma^mupsi+barpsigamma^mu partial_mupsi) = 0so by Leibniz rule,ipartial_mu(barpsigamma^mupsi) = 0}}Another approach to derive this expression is by variational methods, applying Noether’s theorem for the global text{U}(1) symmetry to derive the conserved current J^mu.{{math proof | title = Proof of conservation from Noether’s theorem | proof =Recall the Lagrangian ismathcal{L} = barpsi(igamma^mu partial_mu - m)psi.Under a U(1) symmetry which sends begin{align}psi &mapsto e^{ialpha}psi, barpsi &mapsto e^{-ialpha}barpsi,end{align}we find the Lagrangian is invariant.Now considering the variation parameter alpha to be infinitesimal, we work at first order in alpha and ignore mathcal{O}{alpha^2} terms. From the previous discussion we immediately see the explicit variation in the Lagrangian due to alpha is vanishing, that is under the variation,mathcal{L}mapsto mathcal{L} + deltamathcal{L},where deltamathcal{L} = 0.As part of Noether’s theorem, we find the implicit variation in the Lagrangian due to variation of fields. If the equation of motion for psi, barpsi are satisfied, then{{NumBlk||deltamathcal{L} = partial_muleft(frac{partial mathcal{L}}{partial (partial_mu psi)}deltapsi + frac{partial mathcal{L}}{partial (partial_mu barpsi)}deltabarpsiright) |{{EquationRef|*}}}}This immediately simplifies as there are no partial derivatives of barpsi in the Lagrangian. deltapsi is the infinitesimal variationdeltapsi(x) = ialphapsi(x).We evaluatefrac{partial mathcal{L}}{partial (partial_mu psi)} = ibarpsigamma^mu.The equation ({{EquationNote|*}}) finally is}}

Solutions

{{Further|Dirac spinor|#Hole theory}}Since the Dirac operator acts on 4-tuples of square-integrable functions, its solutions should be members of the same Hilbert space. 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 ansatzpsi(x) = u(mathbf{p})e^{-i p cdot x}which models a particle with definite 4-momentum p = (E_mathbf{p}, mathbf{p}) where E_mathbf{p} = sqrt{m^2 + |mathbf{p}|^2}.For this ansatz, the Dirac equation becomes an equation for u(mathbf{p}):left(gamma^mu p_mu - mright) u(mathbf{p}) = 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).For example, in the chiral representation for gamma^mu, the solution space is parametrised by a mathbb{C}^2 vector xi, withu(mathbf{p}) = begin{pmatrix} sqrt{sigma^mu p_mu}xi sqrt{barsigma^mu p_mu}xi end{pmatrix}where sigma^mu = (I_2, sigma^i), barsigma^mu = (I_2, -sigma^i) and sqrt{cdot} 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{L} = ihbar coverline{psi}gamma^{mu}partial_{mu}psi - mc^2overline{psi}psiIf one varies this with respect to psi one gets the adjoint Dirac equation. Meanwhile, if one varies this with respect to barpsi one gets the Dirac equation.In natural units and with the slash notation, the action is then{{Equation box 1|title=Dirac Action|indent=:|equation = S = int d^4x,barpsi,(ipartial!!!big / - m),psi|border|border colour =#50C878|background colour = #ECFCF4}}For this action, the conserved current J^mu above arises as the conserved current corresponding to the global text{U}(1) symmetry through Noether’s theorem 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 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{SO}(1,3) or strictly text{SO}(1,3)^+, the component connected to the identity.For a Dirac spinor viewed concretely as taking values in mathbb{C}^4, the transformation under a Lorentz transformation Lambda is given by a 4times 4 complex matrix S[Lambda]. There are some subtleties in defining the corresponding S[Lambda], as well as a standard abuse of notation.Most treatments occur at the Lie algebra level. For a more detailed treatment see here. The Lorentz group of 4 times 4 real matrices acting on mathbb{R}^{1,3} is generated by a set of six matrices {M^{munu}} with components(M^{munu})^rho{}_sigma = eta^{murho}delta^nu{}_sigma - eta^{nurho}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^{munu}, M^{rhosigma}] = M^{musigma}eta^{nurho} - M^{nusigma}eta^{murho} + M^{nurho}eta^{musigma} - M^{murho}eta^{nusigma}.In the article on the Dirac algebra, it is also found that the spin generators S^{munu} = frac{1}{4} [gamma^mu,gamma^nu]satisfy the Lorentz algebra commutation relations.A Lorentz transformation Lambda can be written asLambda = expleft(frac{1}{2}omega_{munu}M^{munu}right)where the components omega_{munu} are antisymmetric in mu,nu.The corresponding transformation on spin space isS[Lambda] = expleft(frac{1}{2}omega_{munu}S^{munu}right).This is an abuse of notation, but a standard one. The reason is S[Lambda] is not a well-defined function of Lambda, since there are two different sets of components omega_{munu} (up to equivalence) which give the same Lambda but different S[Lambda]. In practice we implicitly pick one of these omega_{munu} and then S[Lambda] is well defined in terms of omega_{munu}.Under a Lorentz transformation, the Dirac equationigamma^mupartial_mu psi(x) - m psi(x)=0becomesigamma^mu((Lambda^{-1})_mu{}^nupartial_nu)S[Lambda]psi(Lambda^{-1} x) - mS[Lambda]psi(Lambda^{-1} x) = 0.{{math proof | title = Remainder of proof of Lorentz invariance | proof =Multiplying both sides from the left by S^{-1}[Lambda] and returning the dummy variable to x givesiS[Lambda]^{-1}gamma^mu S[Lambda]((Lambda^{-1})_mu{}^nupartial_nu)psi(x) - mpsi(x) = 0.We’ll have shown invariance ifS[Lambda]^{-1}gamma^mu S[Lambda](Lambda^{-1})^nu{}_mu = gamma^nuor equivalentlyS[Lambda]^{-1}gamma^mu S[Lambda] = Lambda^mu{}_nugamma^nu.This is most easily shown at the algebra level. Supposing the transformations are parametrised by infinitesimal components omega_{munu}, then at first order in omega, on the left-hand side we getfrac{1}{2}omega_{rhosigma}(M^{rhosigma})^mu{}_nu gamma^nuwhile on the right-hand side we getleft[frac{1}{2}omega_{rhosigma}S^{rhosigma},gamma^muright] = frac{1}{2}omega_{rhosigma}left[S^{rhosigma},gamma^muright]It’s a standard exercise to evaluate the commutator on the left-hand side. Writing M^{rhosigma} in terms of components completes the proof.}}Associated to Lorentz invariance is a conserved Noether current, or rather a tensor of conserved Noether currents (mathcal{J}^{rhosigma})^mu. Similarly, since the equation is invariant under translations, there is a tensor of conserved Noether currents T^{munu}, which can be identified as the stress-energy tensor of the theory. The Lorentz current (mathcal{J}^{rhosigma})^mu can be written in terms of the stress-energy tensor in addition to a tensor representing internal angular momentum.

Further discussion of Lorentz covariance of the Dirac equation

The Dirac equation is Lorentz covariant. Articulating this helps illuminate not only the Dirac equation, but also the Majorana spinor and Elko spinor, which although closely related, have subtle and important differences.Understanding Lorentz covariance is simplified by keeping in mind the geometric character of the process.Jurgen Jost, (2002) “Riemannian Geometry and Geometric Analysis (3rd Edition)” Springer Universitext. (See chapter 1 for spin structures and chapter 3 for connections on spin structures) Let a be a single, fixed point in the spacetime manifold. Its location can be expressed in multiple coordinate systems. 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 over a small extended patch of spacetime). One can imagine a as having a fiber of different coordinate frames above it. In geometric terms, one says that spacetime can be characterized as a fiber bundle, and specifically, the frame bundle. The difference between two points x and x’ in the same fiber is a combination of rotations and Lorentz boosts. A choice of coordinate frame is a (local) section through that bundle.Coupled to the frame bundle is a second bundle, the spinor bundle. 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; it is associated to a principal bundle, 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 and the intrinsic angular momentum. 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.James D. Bjorken, Sidney D. Drell (1964) “Relativistic Quantum Mechanics”, McGraw-Hill. (See Chapter 2) 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 x mapsto x’, the Dirac spinor to transform aspsi’(x’) = S psi(x)It can be shown that an explicit expression for S is given byS = expleft(frac{-i}{4} omega^{munu} sigma_{munu}right)where omega^{munu} parameterizes the Lorentz transformation, and sigma_{munu} are the six 4×4 matrices satisfying:sigma^{munu} = frac{i}{2} [gamma^mu,gamma^nu]~.This matrix can be interpreted as the intrinsic angular momentum of the Dirac field. That it deserves this interpretation arises by contrasting it to the generator J_{munu} of Lorentz transformations, having the formJ_{munu} = frac{1}{2} sigma_{munu} + i (x_mupartial_nu - x_nupartial_mu)This can be interpreted as the total angular momentum. It acts on the spinor field aspsi^prime(x) = expleft(frac{-i}{2} omega^{munu} J_{munu}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, having no preferred origin. The generator J_{munu} generates the symmetries of this space: it provides a relabelling of a fixed point x~. The generator sigma_{munu} 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 isigamma^mu frac{partial}{partial x^mu} psi(x) -mpsi(x)=0If the Dirac equation is to be covariant, then it should have exactly the same form in all Lorentz frames:igamma^mu frac{partial}{partial x^{primemu}} psi^prime(x^prime) -mpsi^prime(x^prime)=0The 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. Thus, one may presume that the relation between the two frames can be written aspsi^prime(x^prime) = S(Lambda) psi(x)Inserting this into the transformed equation, the result isigamma^mu frac{partial x^nu}{partial x^{primemu}} frac{partial}{partial x^{nu}} S(Lambda)psi(x) -mS(Lambda)psi(x) = 0The coordinates related by Lorentz transformation satisfy: frac{partial x^nu}{partial x^{primemu}} = {left(Lambda^{-1}right)^nu}_muThe original Dirac equation is then regained ifS(Lambda) gamma^mu S^{-1}(Lambda) = {left(Lambda^{-1}right)^mu}_nu gamma^nuAn explicit expression for S(Lambda) (equal to the expression given above) can be obtained by considering a Lorentz transformation of infinitesimal rotation near the identity transformation:{Lambda^mu}_nu = {g^mu}_nu + {omega^mu}_nu , {(Lambda^{-1})^mu}_nu = {g^mu}_nu - {omega^mu}_nu where {g^mu}_{nu} is the metric tensor : {g^mu}_{nu}=g^{munu’}g_{nu’nu}={delta^mu}_{nu} and is symmetric while omega_{munu}={omega^{alpha}}_{nu} g_{alphamu} is antisymmetric. After plugging and chugging, one obtainsS(Lambda) = I + frac{-i}{4} omega^{munu} sigma_{munu} + mathcal{O}left(Lambda^2right)which is the (infinitesimal) form for S above and yields the relation sigma^{munu} = frac{i}{2} [gamma^mu,gamma^nu] . To obtain the affine relabelling, write
begin{align}
psi’(x’) &= left(I + frac{-i}{4} omega^{munu} sigma_{munu} right) psi(x)
&= left(I + frac{-i}{4} omega^{munu} sigma_{munu} right) psi(x’ + {omega^mu}_nu ,x^{prime,nu})
&= left(I + frac{-i}{4} omega^{munu} sigma_{munu} - x^prime_mu omega^{munu} partial_nuright) psi(x’)
&= left(I + frac{-i}{2} omega^{munu} J_{munu} right) psi(x’)
end{align}After properly antisymmetrizing, one obtains the generator of symmetries J_{munu} given earlier. Thus, both J_{munu} and sigma_{munu} 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, which forces a translation along the fiber of the spinor on the spin bundle, 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.

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 uses a Clifford algebra over the real numbers, a type of geometric algebra.

Coupled Weyl Spinors

As mentioned above, the massless Dirac equation immediately reduces to the homogeneous Weyl equation. By using the chiral representation of the gamma matrices, the nonzero-mass equation can also be decomposed into a pair of coupled inhomogeneous Weyl equations acting on the first and last pairs of indices of the original four-component spinor, i.e. psi = begin{pmatrix} psi_L psi_R end{pmatrix}, where psi_L and psi_R are each two-component Weyl spinors. This is because the skew block form of the chiral gamma matrices means that they swap the psi_L and psi_R and apply the two-by-two Pauli matrices to each:gamma^mu begin{pmatrix}psi_L psi_R end{pmatrix} = begin{pmatrix}sigma^mu psi_R overline{sigma}^mu psi_L end{pmatrix}.So the Dirac equation(igamma^mu partial_mu - m)begin{pmatrix} psi_L psi_R end{pmatrix} = 0becomesibegin{pmatrix} sigma^mu partial_mu psi_R overline{sigma}^mu partial_mu psi_L end{pmatrix} = mbegin{pmatrix} psi_L psi_R end{pmatrix}which in turn is equivalent to a pair of inhomogeneous Weyl equations for massless left- and right-helicity spinors, where the coupling strength is proportional to the mass:isigma^mu partial_mu psi_R = m psi_Lioverline{sigma}^mu partial_mu psi_L = m psi_R.{{clarify|reason=In the Penrose source the RHS is divided by sqrt{2} and there is no imaginary unit on the LHS, but he does not go into the derivation. Other sources -- and the Axial symmetry section above -- seem to agree with the form given here.|date=June 2023}}This has been proposed as an intuitive explanation of Zitterbewegung, as these massless components would propagate at the speed of light and move in opposite directions, since the helicity is the projection of the spin onto the direction of motion.BOOK, Penrose, Roger, The Road to Reality, 2004, Alfred A. Knopf, 0-224-04447-8, 628–632, Sixth Printing, Here the role of the “mass” m is not to make the velocity less than the speed of light, but instead controls the average rate at which these reversals occur; specifically, the reversals can be modeled as a Poisson process.JOURNAL, Gaveau, B., Jacobson, T., Kac, M., Schulman, L. S., Relativistic Extension of the Analogy between Quantum Mechanics and Brownian Motion, Physical Review Letters, 30 July 1984, 53, 5, 419–422, 10.1103/PhysRevLett.53.419,

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{U}(1) symmetry where the fields psi, barpsi transform asbegin{align}psi(x) &mapsto e^{ialpha}psi(x), barpsi(x) &mapsto e^{-ialpha}barpsi(x).end{align}This is a global symmetry, known as the text{U}(1) vector symmetry (as opposed to the text{U}(1) axial symmetry: see below). By Noether’s theorem there is a corresponding conserved current: this has been mentioned previously asJ^mu(x) = barpsi(x)gamma^mupsi(x).

Gauging the symmetry

{{See also|Quantum electrodynamics}}If we ‘promote’ the global symmetry, parametrised by the constant alpha, to a local symmetry, parametrised by a function alpha:mathbb{R}^{1,3} to mathbb{R}, or equivalently e^{ialpha}: mathbb{R}^{1,3} to text{U}(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_muD_mu psi = partial_mu psi + i e A_mupsi,D_mu barpsi = partial_mu barpsi - i e A_mubarpsi.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{U}(1) gauge field, or a text{U}(1) connection.The transformation law under gauge transformations for A_mu is then the usualA_mu(x) mapsto A_mu(x) + frac{1}{e}partial_mualpha(x)but can also be derived by asking that covariant derivatives transform under a gauge transformation asD_mupsi(x) mapsto e^{ialpha(x)}D_mupsi(x),D_mubarpsi(x) mapsto e^{-ialpha(x)}D_mubarpsi(x).We then obtain a gauge-invariant Dirac action by promoting the partial derivative to a covariant one:S = int d^4x,barpsi,(iD!!!!big / - m),psi = int d^4x,barpsi,(igamma^mu D_mu - m),psi.The final step needed to write down a gauge-invariant Lagrangian is to add a Maxwell Lagrangian term,S_{text{Maxwell}} = int d^4x,left[-frac{1}{4}F^{munu}F_{munu}right].Putting these together gives{{Equation box 1|title=QED Action|indent=:|equation = S_{text{QED}} = int d^4x,left[-frac{1}{4}F^{munu}F_{munu} + barpsi,(iD!!!!big / - m),psiright]|border|border colour =#50C878|background colour = #ECFCF4}}Expanding out the covariant derivative allows the action to be written in a second useful form:S_{text{QED}} = int d^4x,left[-frac{1}{4}F^{munu}F_{munu} + barpsi,(ipartial!!!big / - m),psi - eJ^mu A_muright]

Axial symmetry

Massless Dirac fermions, that is, fields psi(x) satisfying the Dirac equation with m = 0, admit a second, inequivalent text{U}(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{pmatrix}psi_1(x)psi_2(x)end{pmatrix},and adopting the chiral representation for the gamma matrices, so that igamma^mupartial_mu may be writtenigamma^mupartial_mu = begin{pmatrix}ibarsigma^mu partial_mu & 0end{pmatrix}where sigma^mu has components (I_2, sigma^i) and barsigma^mu has components (I_2, -sigma^i).The Dirac action then takes the formS = int d^4x, psi_1^dagger(isigma^mupartial_mu)psi_1 + psi_2^dagger(ibarsigma^mupartial_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{U}(1) symmetry manifest:begin{align}psi_1(x) &mapsto e^{ibeta} psi_1(x), psi_2(x) &mapsto e^{-ibeta}psi_2(x).end{align}This can also be expressed at the level of the Dirac fermion as psi(x) mapsto exp(ibetagamma^5) psi(x)where exp is the exponential map for matrices.This isn’t the only text{U}(1) symmetry possible, but it is conventional. Any ‘linear combination’ of the vector and axial symmetries is also a text{U}(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

{{See also | quantum chromodynamics}}We can extend this discussion from an abelian text{U}(1) symmetry to a general non-abelian symmetry under a gauge group G, the group of color symmetries for a theory.For concreteness, we fix G = text{SU}(N), the special unitary group of matrices acting on mathbb{C}^N.Before this section, psi(x) could be viewed as a spinor field on Minkowski space, in other words a function psi: mathbb{R}^{1,3}mapsto mathbb{C}^4, and its components in mathbb{C}^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{C}^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^{i,alpha}(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{R}^{1,3} to mathbb{C}^4 otimes mathbb{C}^N.Gauging proceeds similarly to the abelian text{U}(1) case, with a few differences. Under a gauge transformation U:mathbb{R}^{1,3} rightarrow text{SU}(N), the spinor fields transform aspsi(x) mapsto U(x)psi(x)barpsi(x)mapsto barpsi(x)U^dagger(x).The matrix-valued gauge field A_mu or text{SU}(N) connection transforms asA_mu(x) mapsto U(x)A_mu(x)U(x)^{-1} + frac{1}{g}(partial_mu U(x))U(x)^{-1},and the covariant derivatives defined D_mupsi = partial_mu psi + igA_mupsi,D_mubarpsi = partial_mu barpsi - igbarpsi A_mu^daggertransform as D_mupsi(x) mapsto U(x)D_mupsi(x),D_mubarpsi(x) mapsto (D_mubarpsi(x))U(x)^dagger.Writing down a gauge-invariant action proceeds exactly as with the text{U}(1) case, replacing the Maxwell Lagrangian with the Yang–Mills LagrangianS_{text{Y-M}} = int d^4x ,-frac{1}{4}text{Tr}(F^{munu}F_{munu})where the Yang–Mills field strength or curvature is defined here asF_{munu} = partial_mu A_nu - partial_nu A_mu - igleft[A_mu,A_nuright]and [cdot,cdot] is the matrix commutator.The action is then{{Equation box 1|title=QCD Action|indent=:|equation = S_{text{QCD}} = int d^4x,-frac{1}{4}text{Tr}(F^{munu}F_{munu}) + barpsi,(iD!!!!big / - m),psi|border|border colour =#50C878|background colour = #ECFCF4}}

Physical applications

For physical applications, the case N=3 describes the quark sector of the Standard model which models strong interactions. Quarks are modelled as Dirac spinors; the gauge field is the gluon field. The case N=2 describes part of the electroweak 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{R}^{1,3} to mathbb{C}^4otimes V.Then psi transforms under a gauge transformation g:mathbb{R}^{1,3} to G aspsi(x) mapsto rho(g(x))psi(x)and the covariant derivative is definedD_mupsi = partial_mupsi + rho(A_mu)psiwhere here we view rho as a Lie algebra representation of the Lie algebra mathfrak{g} = text{L}(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{R}^{1,3}.

See also

{{Col-begin}}{{Col-break}}

Articles on the Dirac equation

{{Col-break}}

Other equations

{{Col-break}}

Other topics

{{col-end}}

References

Citations

{{Reflist}}

Selected papers

  • JOURNAL, Anderson, Carl, 1933, 10.1103/PhysRev.43.491, The Positive Electron, Physical Review, 43, 6, 491, 1933PhRv...43..491A, free,
  • JOURNAL, Arminjon, M., F. Reifler


, 2013
, Equivalent forms of Dirac equations in curved spacetimes and generalized de Broglie relations
, Brazilian Journal of Physics
, 43
, 64–77
, 1–2
, 1103.3201, 10.1007/s13538-012-0111-0
, 2013BrJPh..43...64A, 38235437
,

Textbooks

  • BOOK, Bjorken, J D, Drell, S, Relativistic Quantum mechanics, 1964, New York, McGraw-Hill,archive.org/details/relativisticquan0000bjor, registration,
  • BOOK, Halzen, Francis, Francis Halzen, Martin, Alan, Alan Martin (physicist), Quarks & Leptons: An Introductory Course in Modern Particle Physics,archive.org/details/quarksleptonsint0000halz, registration, John Wiley & Sons, 1984, 9780471887416,
  • BOOK, Griffiths, D.J., Introduction to Elementary Particles, 2nd, Wiley-VCH, 2008, 978-3-527-40601-2,
  • BOOK, Rae, Alastair I. M., Jim Napolitano, Quantum Mechanics, 6th, Routledge, 2015, 978-1482299182,
  • BOOK, Schiff, L.I., Quantum Mechanics,archive.org/details/quantummechanics0000schi, registration, 3rd, McGraw-Hill, 1968,
  • BOOK, Shankar, R., Principles of Quantum Mechanics, 2nd, Plenum, 1994,
  • BOOK, Thaller, B., The Dirac Equation, Texts and Monographs in Physics, Springer, 1992,

External links

{{Quantum mechanics topics}}{{Authority control}}

- content above as imported from Wikipedia
- "Dirac equation" does not exist on GetWiki (yet)
- time: 2:26am EDT - Wed, May 22 2024
[ this remote article is provided by Wikipedia ]
LATEST EDITS [ see all ]
GETWIKI 21 MAY 2024
GETWIKI 09 JUL 2019
Eastern Philosophy
History of Philosophy
GETWIKI 09 MAY 2016
GETWIKI 18 OCT 2015
M.R.M. Parrott
Biographies
GETWIKI 20 AUG 2014
CONNECT