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Dirac equation

From Wikipedia, the free encyclopedia
Relativistic quantum mechanical wave equation
This article is about a quantum physics equation. For the mathematical function, seeDirac delta function.

Part of a series of articles about
Quantum mechanics
iℏddt|Ψ⟩=H^|Ψ⟩{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }

Inparticle physics, theDirac equation is arelativistic wave equation derived by British physicistPaul Dirac in 1928.[1] In itsfree form, or including electromagnetic interactions, it describes allspin-⁠1/2⁠massive particles, called "Dirac particles", such aselectrons andquarks for whichparity is asymmetry. It is consistent with both the principles ofquantum mechanics and the theory ofspecial relativity,[2] and was the first theory to fully account for special relativity in the context of quantum mechanics. The equation is validated by its rigorous accounting of the observedfine structure of thehydrogen spectrum and has become vital in the building of theStandard Model.[3]

The equation also implied the existence of a new form of matter,antimatter, previously unsuspected and unobserved. The existence of antimatter was experimentally confirmed several years later. It also provided atheoretical justification for the introduction of several component wave functions inPauli'sphenomenological theory ofspin. The wave functions in the Dirac theory are vectors of fourcomplex numbers (known asbispinors), two of which resemblethe Pauli wavefunction in the non-relativistic limit, in contrast to theSchrödinger equation, which described wave functions of only one complex value. Moreover, in the limit of zero mass, the Dirac equation reduces to theWeyl equation. In the context ofquantum field theory, the Dirac equation is reinterpreted to describe quantum fields corresponding to spin-⁠1/2⁠ particles.

Dirac did not fully appreciate the importance of his results; however, the entailed explanation of spin as a consequence of the union of quantum mechanics and relativity—and the eventual discovery of thepositron—represents one of the great triumphs oftheoretical physics. This accomplishment has been described as fully on par with the works ofIsaac Newton,James Clerk Maxwell, andAlbert Einstein before him.[4] The equation has been deemed by some physicists to be "the real seed of modern physics".[5] The Dirac equation has been described as the "centerpiece of relativistic quantum mechanics", with it also stated that "the equation is perhaps the most important one in all of quantum mechanics".[6]

History

[edit]

Early attempts at a relativistic formulation

[edit]

The first phase in the development ofquantum mechanics, lasting between 1900 and 1925, focused on explaining individual phenomena that could not be explained throughclassical mechanics.[7] The second phase, starting in the mid-1920s, saw the development of two systematic frameworks governing quantum mechanics. The first, known asmatrix mechanics, usesmatrices to describephysical observables; it was developed in 1925 byWerner Heisenberg,Max Born, andPascual Jordan.[8]: 51  The second, known as wave mechanics, uses awave equation known as theSchrödinger equation to describe thestate of asystem; it was developed the next year byErwin Schrödinger. While these two frameworks were initially seen as competing approaches, they would later be shown to be equivalent.[9]: 21 

Both these frameworks only formulated quantum mechanics in anon-relativistic setting.[9]: 22  This was seen as a deficiency right from the start, with Schrödinger originally attempting to formulate arelativistic version of Schrödinger equation, in the process discovering theKlein–Gordon equation.[8]: 679  However, after showing that thisequation did not correctly reproduce the relativistic corrections to thehydrogen atomspectrum for which an exact form was known due toArnold Sommerfeld, he abandoned hisrelativistic formulation.[10]: 1025  The Klein–Gordon equation was also found by at least six other authors in the same year.[11]: 6 

During 1926 and 1927 there was a widespread effort to incorporate relativity into quantum mechanics, largely through two approaches. The first was to consider the Klein–Gordon as the correct relativistic generalization of the Schrödinger equation.[11]: 7  Such an approach was viewed unfavourably by many leadingtheorists since it failed to correctly predict numerousexperimental results, and more importantly it appeared difficult to reconcile with the principles of quantum mechanics as understood at the time.[12]: 35  These conceptual issues primarily arose due to the presence of asecond temporal derivative.[10]: 1031 

The second approach introduced relativistic effects as corrections to the known non-relativistic formulas.[12]: 36  This provided many provisional answers that were expected to eventually be supplanted by some yet-unknown relativistic formulation of quantum mechanics. One notable result by Heisenberg and Jordan was the introduction of two terms forspin and relativity into the hydrogenHamiltonian, allowing them to derive thefirst-order approximation of the Sommerfeldfine structure formula.[13]

A parallel development during this time was the concept of spin, first introduced in 1925 bySamuel Goudsmit andGeorge Uhlenbeck.[14] Shortly after, it was conjectured by Schrödinger to be the missing link in acquiring the correct Sommerfeld formula.[12]: 44  In 1927Wolfgang Pauli used the ideas of spin to find aneffective theory for a nonrelativisticspin-⁠1/2⁠particle, thePauli equation.[15] He did this by taking the Schrödinger equation and rather than just assuming that the wave function depends on thephysical coordinate, he also assume that it depends on a spin coordinate that can take only two values±ℏ2{\displaystyle \pm {\tfrac {\hbar }{2}}}. While this was still a non-relativistic formulation, he believed that a fully relativistic formulation possibly required a more complicated model for theelectron, one that moved beyond apoint particle.[12]: 47 

Dirac's relativistic quantum mechanics

[edit]

By 1927, many physicists no longer considered the fine structure of hydrogen as a crucial puzzle that called for a completely new relativistic formulation since it could effectively be solved using the Pauli equation or by introducing a spin-⁠1/2⁠angular momentumquantum number in the Klein–Gordon equation.[12]: 51  At the fifthSolvay Conference held that year,Paul Dirac was primarily concerned with the logical development of quantum mechanics. However, he realized that many other physicists complacently accepted the Klein–Gordon equation as a satisfactory relativistic formulation, which demanded abandoning basic principles of quantum mechanics as understood at the time, to which Dirac strongly objected.[12]: 52  After his return fromBrussels, Dirac focused on finding a relativistic theory for electrons. Within two months he solved the problem and published his results on January 2, 1928.[1]

In hispaper, Dirac was guided by two principles fromtransformation theory, the first being that the equation should beinvariant under transformations of special relativity, and the second that it should transform under the transformation theory of quantum mechanics.[11]: 9  The latter demanded that the equation would have to belinear intemporal derivatives, so that it would admit aprobabilistic interpretation. His argument begins with the Klein–Gordon equation[16]: 32 

[p2+m2c2]ϕ(t,x)=−ℏ2c2∂2∂t2ϕ(t,x),{\displaystyle \left[{\boldsymbol {p}}^{2}+m^{2}c^{2}\right]\phi (t,x)=-{\frac {\hbar ^{2}}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\phi (t,x),}

describing a particle using the wave functionϕ(t,x){\displaystyle \phi (t,x)}. Herep2=p12+p22+p32{\displaystyle {\boldsymbol {p}}^{2}=p_{1}^{2}+p_{2}^{2}+p_{3}^{2}} is the square of themomentum,m{\displaystyle m} is therest mass of the particle,c{\displaystyle c} is thespeed of light, andℏ{\displaystyle \hbar } is thereduced Planck constant. The naive way to get an equation linear in the time derivative is to essentially consider thesquare root of both sides. This replacesp2+m2c2{\displaystyle {\boldsymbol {p}}^{2}+m^{2}c^{2}} withp2+m2c2{\displaystyle {\sqrt {{\boldsymbol {p}}^{2}+m^{2}c^{2}}}}. However, such asquare root is mathematically problematic for the resulting theory, making it unfeasible.[9]: 379 

Dirac's first insight was the concept of linearization. He looked for some sort ofvariablesαi{\displaystyle \alpha _{i}} that are independent of momentum and spacetime coordinates for which the square root could be rewritten in a linear form[9]: 379 

p12+p22+p32+(mc)2=α1p1+α2p2+α3p3+α4mc.{\displaystyle {\sqrt {p_{1}^{2}+p_{2}^{2}+p_{3}^{2}+(mc)^{2}}}=\alpha _{1}p_{1}+\alpha _{2}p_{2}+\alpha _{3}p_{3}+\alpha _{4}mc.}

By squaring thisoperator and demanding that it reduces to the Klein–Gordon equation, Dirac found that the variables must satisfyαi2=1{\displaystyle \alpha _{i}^{2}=1} andαiαj+αjαi=0{\displaystyle \alpha _{i}\alpha _{j}+\alpha _{j}\alpha _{i}=0} ifi≠j{\displaystyle i\neq j}. Dirac initially considered the2×2{\displaystyle 2\times 2}Pauli matrices as a candidate, but then showed these would not work since it is impossible to find a set of four2×2{\displaystyle 2\times 2} matrices that allanticommute with each other.[8]: 687  His second insight was to instead considerfour-dimensional matrices. In that case the equation would be acting on afour-component wavefunctionψ=(ψ1,ψ2,ψ3,ψ4){\displaystyle \psi =(\psi _{1},\psi _{2},\psi _{3},\psi _{4})}.[17]: 235  Such a proposal was much more bold than Pauli's original generalization to a two-component wavefunction in the Pauli equation.[12]: 57  This is because in Pauli's case, this was motivated by the demand to encode the two spin states of the particle. In contrast, Dirac had no physical argument for a four-component wavefunction, but instead introduced it as a matter of mathematical necessity. He thus arrived at the Dirac equation[1]

(α1p1+α2p2+α3p3+α4mc)ψ(t,x)=iℏc∂∂tψ(t,x).{\displaystyle (\alpha _{1}p_{1}+\alpha _{2}p_{2}+\alpha _{3}p_{3}+\alpha _{4}mc)\psi (t,x)=i{\frac {\hbar }{c}}{\frac {\partial }{\partial t}}\psi (t,x).}

Dirac constructed the correct matricesαi{\displaystyle \alpha _{i}} without realizing that they form a mathematical structure known of since the early 1880s, theClifford algebra.[18]: 320  By recasting the equation in aLorentz invariant form, he also showed that it correctly combines special relativity with his principle of quantum mechanical transformation theory, making it a viable candidate for a relativistic theory of the electron.[8]: 685 

To investigate the equation further, he examined how it behaves in the presence of anelectromagnetic field.[12]: 58  To his surprise, this showed that it described a particle with amagnetic moment arising due to the particle having spin⁠1/2⁠. Spin directly emerged from the equation, without Dirac having added it in by hand. Additionally, he focused on showing that the equation successfully reproduces the fine structure of the hydrogen atom, at least to first order. The equation therefore succeeds where all previous attempts have failed, in correctly describing relativistic phenomena of electrons fromfirst principles rather than through the ad hoc modification of existing formulas.[12]: 60 

Consequences

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Except for his followup paper[19] deriving theZeeman effect and Paschen–Back effect from the equation in the presences of amagnetic fields, Dirac left the work of examining the consequences of his equation to others, and only came back to the subject in 1930.[12]: 65  Once the equation was published, it was recognized as the correct solution to the problem of spin, relativity, and quantum mechanics. At first the Dirac equation was considered the only valid relativistic equation for a particle withmass. Then in 1934 Pauli andVictor Weisskopf reinterpreted the Klein–Gordon equation as the equation for a relativisticspinless particle.[8]: 685 

One of the first calculations was to reproduce the Sommerfeld fine structure formula exactly, which was performed independently byCharles Galton Darwin andWalter Gordon in 1928.[20][21][9]: 377  This is the first time that the full formula has been derived from first principles. Further work on the mathematics of the equation was undertaken byHermann Weyl in 1929.[22] In this work he showed that themassless Dirac equation can be decomposed into a pair ofWeyl equations.

The Dirac equation was also used to study various scattering processes. In particular, theKlein–Nishina formula, looking atphoton-electronscattering, was also derived in 1928.[23]Mott scattering, the scattering of electrons off a heavy target such asatomic nuclei, followed the next year.[24] Over the following years it was further used to derive other standard scattering processes such asMoller scattering in 1932[25] andBhabha scattering in 1936.[26]

A problem that gained more focus with time was the presence ofnegative energystates in the Dirac equation, which led to many efforts to try to eliminate such states. Dirac initially simply rejected the negative energy states as unphysical,[12]: 63  but the problem was made more clear when in 1929Oskar Klein showed that in static fields there exists inevitable mixing between the negative and positive energy states.[9]: 378  Dirac's initial response was to believe that his equation must have some sort of defect, and that it was only the first approximation of a future theory that would not have this problem.[9]: 378  However, he then suggested a solution to the problem in the form of theDirac sea. This is the idea that the universe is filled with an infinite sea of negative energy electrons states. Positive energy electron states then live in this sea and are prevented from decaying to the negative energy states through thePauli exclusion principle.[8]: 715 

Additionally, Dirac postulated the existence of positively chargedholes in the Dirac sea, which he initially suggested could be theproton. However, Oppenheimer showed that in this case stableatoms could not exist[27] and Weyl further showed that the holes would have to have the same mass as the electrons.[28] Persuaded by Oppenheimer's and Weyl's argument, Dirac published a paper in 1931 that predicted the existence of an as-yet-unobserved particle that he called an "anti-electron" that would have the same mass and the opposite charge as an electron and that would mutually annihilate upon contact with an electron. He suggested that every particle may have an oppositely charged partner, a concept now calledantimatter[29][30]: 47 

In 1933Carl Anderson discovered the "positive electron", now called apositron, which had all the properties of Dirac's anti-electron.[31][9]: 378  While the Dirac sea was later superseded byquantum field theory, its conceptual legacy survived in the idea of adynamical vacuum filled withvirtual particles.[8]: 717  In 1949Ernst Stueckelberg suggested andRichard Feynman showed in detail that the negative energy solutions can be interpreted as particles traveling backwards inproper time.[32]: 61–63 [9]: 379 [33][34] The concept of the Dirac sea is also realized more explicitly in somecondensed matter systems in the form of theFermi sea, which consists of a sea of filledvalence electrons below somechemical potential.[35]: 534 

Significant work was done over the following decades to try to findspectroscopic discrepancies compared to the predictions made by the Dirac equation, however it was not until 1947 thatLamb shift was discovered, which the equation does not predict.[8]: 710  This led to the development ofquantum electrodynamics in 1950s, with the Dirac equation then being incorporated within the context of quantum field theory.[36]: 35  Since it describes the dynamics of Dirac spinors, it went on to play a fundamental role in theStandard Model as well as many other areas of physics. For example, within condensed matter physics, systems whose fermions have a near lineardispersion relation are described by the Dirac equation. Such systems are known asDirac matter and they includegraphene andtopological insulators, which have become a major area of research since the start of the 21st century.[35]: 522 

The Dirac equation is inscribed upon a plaque on the floor ofWestminster Abbey. Unveiled on 13 November 1995, the plaque commemorates Dirac's life.[37] The equation, in itsnatural units formulation, is also prominently displayed in the auditorium at the ‘Paul A.M. Dirac’ Lecture Hall at the Patrick M.S. Blackett Institute (formerly The San Domenico Monastery) of theEttore Majorana Foundation and Centre for Scientific Culture inErice,Sicily.

Formulation

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Covariant formulation

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In its modernfield theoretic formulation, the Dirac equation in 3+1 dimensionalMinkowski spacetime is written in terms of aDirac fieldψ(x){\displaystyle \psi (x)}. This is afield that assigns acomplex vector fromC4{\displaystyle \mathbb {C} ^{4}} to each point inspacetime,[18]: 136  where the key property of the field is that it transforms as aDirac spinor underLorentz transformations.[38]: 28  Innatural units whereℏ=c=1{\displaystyle \hbar =c=1}, theLorentz covariant formulation of the Dirac equation is given by[16]: 168 

Dirac equation

(i∂/−m)ψ(x)=0,{\displaystyle (i{{\partial }\!\!\!/}-m)\psi (x)=0,}

where∂/=γμ∂μ{\displaystyle {{\partial }\!\!\!/}=\gamma ^{\mu }\partial _{\mu }} is acontraction between thefour-gradient∂μ{\displaystyle \partial _{\mu }} and thegamma matricesγμ{\displaystyle \gamma ^{\mu }}.[nb 1] These are a set of four matricesgenerating theDirac algebra, which requires them to satisfy[36]: 40 

{γμ,γν}=2ημνI4,{\displaystyle \{\gamma ^{\mu },\gamma ^{\nu }\}=2\eta ^{\mu \nu }I_{4},}

where{a,b}=ab+ba{\displaystyle \{a,b\}=ab+ba} is theanticommutator,ημν{\displaystyle \eta ^{\mu \nu }} is the Minkowskimetric in a mainlynegative signature,[nb 2] andI4{\displaystyle I_{4}} is the4×4{\displaystyle 4\times 4}identity matrix. The Dirac algebra is a special case of the more general mathematical structure known as aClifford algebra.[16]: 169  The Dirac algebra can also be seen as the real part of thespacetime algebra. There is no unique choice of matrices for the gamma matrices, with different choices known as differentrepresentations of the algebra.[32]: 323  One common choice, originally discovered by Dirac, is known as the Dirac representation. Here the matrices are given by[35]: 531 

γ0=(I200−I2),γi=(0σi−σi0),{\displaystyle \gamma ^{0}={\begin{pmatrix}I_{2}&0\\0&-I_{2}\end{pmatrix}},\qquad \gamma ^{i}={\begin{pmatrix}0&\sigma _{i}\\-\sigma _{i}&0\end{pmatrix}},}

whereσi{\displaystyle \sigma _{i}} are the threePauli matrices fori∈{1,2,3}{\displaystyle i\in \{1,2,3\}}. There are two other common representations for the gamma matrices. The first is thechiral representation, which is useful when decomposing the Dirac equation into a pair ofWeyl equations.[36]: 41  The second is the Majorana representation, for which all gamma matrices areimaginary, so theDirac operator ispurely real.[16]: 344  This representation is useful for studyingMajorana spinors, which are purely real four-componentspinor solutions of the Dirac equation.[35]: 536 

By taking thehermitian conjugate of the Dirac equation and multiplying it byγ0{\displaystyle \gamma ^{0}} from the right, the adjoint Dirac equation can be found, with this being theequation of motion for theDirac adjointψ¯=ψ†γ0{\displaystyle {\bar {\psi }}=\psi ^{\dagger }\gamma ^{0}}. It is given by[39]

ψ¯(x)(−iγμ∂←μ−m)=0.{\displaystyle {\bar {\psi }}(x)(-i\gamma ^{\mu }{\overleftarrow {\partial }}_{\mu }-m)=0.}

The adjoint spinor is useful in forming Lorentz invariant quantities. For example, thebilinearψ†ψ{\displaystyle \psi ^{\dagger }\psi } is not Lorentz invariant, butψ¯ψ{\displaystyle {\bar {\psi }}\psi } is.[35]: 532  Here∂←μ{\displaystyle {\overleftarrow {\partial }}_{\mu }} is shorthand notation for apartial derivative acting on the left. In regular notation, the adjoint Dirac equation is equivalent to[36]: 43 

−i∂μψ¯(x)γμ−mψ¯(x)=0.{\displaystyle -i\partial _{\mu }{\bar {\psi }}(x)\gamma ^{\mu }-m{\bar {\psi }}(x)=0.}

The Dirac equation can be rewritten in a non-covariant form similar to that of theSchrödinger equation[8]: 686 

iℏ∂∂tψ(t,x)=−iℏcα⋅∇ψ(t,x)+βmc2ψ(t,x).{\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi (t,{\boldsymbol {x}})=-i\hbar c{\boldsymbol {\alpha }}\cdot \nabla \psi (t,{\boldsymbol {x}})+\beta mc^{2}\psi (t,{\boldsymbol {x}}).}

The right-hand side of the equation is theHamiltonian acting on the Dirac spinorHψ{\displaystyle H\psi }. Hereα{\displaystyle {\boldsymbol {\alpha }}} andβ{\displaystyle \beta } are a set of fourHermitian4×4{\displaystyle 4\times 4} matrices that all anticommute with each other and square to the identity. They are related to the gamma matrices throughγ0=β{\displaystyle \gamma ^{0}=\beta } andγi=βαi{\displaystyle \gamma ^{i}=\beta \alpha _{i}}.[9]: 375  This form is useful inquantum mechanics, where the Hamiltonian can be easily modified to solve a wide range of problems, such as by introducing apotential[8]: 702  or through aminimal coupling to theelectromagnetic field.[8]: 688 

Dirac action

[edit]

The Dirac equation can also be acquired from aLagrangian formulation of the field theory, where the Diracaction is given by[35]: 535 

S=∫d4xψ¯(i∂/−m)ψ.{\displaystyle S=\int d^{4}x{\bar {\psi }}(i{{\partial }\!\!\!/}-m)\psi .}

The equation then arises as theEuler–Lagrange equation of this action, found by varying the spinorψ¯(x){\displaystyle {\bar {\psi }}(x)}.[36]: 43  Meanwhile, the adjoint Dirac equation is acquired by varying the adjoint spinorψ(x){\displaystyle \psi (x)}. The action formulation of the Dirac equation has the advantage of making thesymmetries of the Dirac equation more explicit, since they leave its action invariant.[16]: 166–168 Noether's theorem then allows for the direct calculation ofcurrents corresponding to these symmetries.[40]: 166  Additionally, the action is usually used to define the associatedquantum field theory, such as through thepath integral formulation.[16]: 251 

Meaning of the Dirac fields

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In quantum mechanics, the Dirac spinorψ(x){\displaystyle \psi (x)} corresponds to a four-component spinorwave function describing thestate of aDirac fermion.[8]: 685  Its positionprobability density, the probability of finding the fermion in a region of space, is described by the zeroth component of its vector current,ψ†ψ{\displaystyle \psi ^{\dagger }\psi }.[8]: 689  In the case of a large number ofparticles, it can also be interpreted as thecharge density.[18]: 137  An appropriatenormalization is required to ensure that the total probability across all of space is equal to one, withprobability conservation following directly from theconservation of the vector current. The Dirac equation is therelativistic analogue of the Schrödinger equation for the Dirac fermion wavefunction.

In thesecond quantization form of quantum field theory the Dirac spinor isquantized to be a operator-valued spinor fieldψ^(x){\displaystyle {\hat {\psi }}(x)}.[41] In contrast to quantum mechanics, it no longer represents the state in theHilbert space, but is rather theoperator that acts on states to create or destroy particles.[16]: 22 [42]: 92 Observables are formed usingexpectation values of these operators.[43]: 41  The Dirac equation then becomes an operator equation describing the state-independent evolution of the operator-valued spinor field[44]: 35 

(i∂/−m)ψ^(x)=0.{\displaystyle (i{{\partial }\!\!\!/}-m){\hat {\psi }}(x)=0.}

In thepath integral formulation of quantum field theory, the spinor fieldψ(x){\displaystyle \psi (x)} is an anti-commutingGrassmann-valued field that only acts as an integration variable.[38]: 29  The Dirac equation then emerges as the classicalsaddle point behaviour of the path integral. It also arises as an equation of the expectation value of the classical field variables[44]: 35 

⟨(i∂/−m)ψ(x)⟩=0,{\displaystyle \langle (i{{\partial }\!\!\!/}-m)\psi (x)\rangle =0,}

in the sense of theSchwinger–Dyson equations.[16]: 80  This version of the equation can also be acquired by taking the expectation value of the operator equation.

The Dirac equation also arises in describing thetime evolution of a spinor field inclassical field theory. Such a field theory would have thespecial linear groupSL(2,C){\displaystyle {\text{SL}}(2,\mathbb {C} )} as itsspacetime symmetry group rather than the Lorentz group, since the latter does not admitspinor representations.[40]: 117  This is in contrast to the quantum theory which does admit spinor representations even when the spacetime symmetries are described by the Lorentz group. This is because the states in a Hilbert space are defined only up to a complex phase,[45]: 81  so particles belong toprojective representations rather than regular representations, with the projective representations of the Lorentz group being equivalent to regular representations ofSL(2,C){\displaystyle {\text{SL}}(2,\mathbb {C} )}.[16]: 176  Classical spinor fields do not arise in our universe because thePauli exclusion principle prevents populating the field with a sufficient number of particles to reach theclassical limit.[40]: 347 

Properties

[edit]

Lorentz transformations

[edit]

TheLorentz groupSO(1,3){\displaystyle {\text{SO}}(1,3)},[nb 3] describing the transformation betweeninertial reference frames, can admit many differentrepresentations.[46]: 38  A representation is a particular choice ofmatrices thatfaithfully[nb 4] represent the action of thegroup on somevector space, where thedimensionality of the matrices can differ between representations.[47]: 54  For example, the Lorentz group can be represented by4×4{\displaystyle 4\times 4} real matricesΛ{\displaystyle \Lambda } acting on the vector spaceR1,3{\displaystyle \mathbb {R} ^{1,3}}, corresponding to howLorentz transformations act on vectors or onspacetime.[16]: 159  Another representation is a set of4×4{\displaystyle 4\times 4} complex matrices acting onDirac spinors in the complex vector spaceC4{\displaystyle \mathbb {C} ^{4}}.[48]: 24  A smaller representation is a set of2×2{\displaystyle 2\times 2} complex matrices acting onWeyl spinors in theC2{\displaystyle \mathbb {C} ^{2}} vector space.[49]: 209 

Lie group elements can be generated using the correspondingLie algebra, which together with aLie bracket, describes thetangent space of the groupmanifold around its identity element.[50]: 64  Thebasis elements of this vector space are known asgenerators of the group. A particular group element is then acquired byexponentiating a corresponding tangent space vector.[51]: 295  The generators of the Lorentz Lie algebra must satisfy certainanticommutation relations, known as a Lie bracket. The six vectors can be packaged into an antisymmetric objectXμν{\displaystyle X^{\mu \nu }} indexed by{μ,ν}{\displaystyle \{\mu ,\nu \}}, with the bracket for the Lorentz algebra given by[36]: 39 

[Xμν,Xρσ]=i(ηνρXμσ−ημρXνσ+ημσXνρ−ηνσXμρ).{\displaystyle [X^{\mu \nu },X^{\rho \sigma }]=i(\eta ^{\nu \rho }X^{\mu \sigma }-\eta ^{\mu \rho }X^{\nu \sigma }+\eta ^{\mu \sigma }X^{\nu \rho }-\eta ^{\nu \sigma }X^{\mu \rho }).}

This algebra admits numerous representation, where each generator is represented by a matrix, with eachalgebra representation generating a correspondingrepresentation of the group.[51]: 662  For example, the representation acting on real vectors is given by the six matricesXμν=Mμν{\displaystyle X^{\mu \nu }=M^{\mu \nu }} where[51]: 664 

(Mμν)ρσ=i(ημρδνσ−ηνρδμσ).{\displaystyle \left(M^{\mu \nu }\right)^{\rho }{}_{\sigma }=i(\eta ^{\mu \rho }\delta ^{\nu }{}_{\sigma }-\eta ^{\nu \rho }\delta ^{\mu }{}_{\sigma }).}

The Lorentz transformation matrix can then be acquired from these generators through an exponentiation[16]: 169 

Λ=exp⁡(i2ωμνMμν),{\displaystyle \Lambda =\exp \left({\tfrac {i}{2}}\omega _{\mu \nu }M^{\mu \nu }\right),}

whereωμν{\displaystyle \omega _{\mu \nu }} is an antisymmetric matrix encoding the sixdegrees of freedom of the Lorentz group used to specify the particular group element. These correspond to the three boosts and threerotations.[36]: 40 

Another representation for the Lorentz algebra is thespinor representation where the generators are given by[16]: 169 

Sμν=i4[γμ,γν].{\displaystyle S^{\mu \nu }={\tfrac {i}{4}}[\gamma ^{\mu },\gamma ^{\nu }].}

In this case the Lorentz group element, specified byωμν{\displaystyle \omega _{\mu \nu }}, is given by

S[Λ]=exp⁡(i2ωμνSμν).{\displaystyle S[\Lambda ]=\exp \left({\tfrac {i}{2}}\omega _{\mu \nu }S^{\mu \nu }\right).}

The mapping ofΛ↦S[Λ]{\displaystyle \Lambda \mapsto S[\Lambda ]} is not one-to-one since there are two consistent choices forωμν{\displaystyle \omega _{\mu \nu }} that give the sameΛ{\displaystyle \Lambda } but a differentS[Λ]{\displaystyle S[\Lambda ]}.[50]: 45  This is a consequence of the spinor representation beingprojective representations of the Lorentz groupSO(1,3){\displaystyle {\text{SO}}(1,3)}.[45]: 81  Equivalently, they are regular representation ofSL(2,C){\displaystyle {\text{SL}}(2,\mathbb {C} )}, which is adouble cover of the Lorentz group.[40]: 106 

Under a Lorentz transformation, spacetime coordinates transform under the vector representationx′μ=Λμνxν{\displaystyle x'^{\mu }=\Lambda ^{\mu }{}_{\nu }x^{\nu }}, while the spinors transform under the spinor representation

ψ′(x′)=S[Λ]ψ(x).{\displaystyle \psi '(x')=S[\Lambda ]\psi (x).}

The Dirac equation is a Lorentz covariant equation, meaning that it takes the form in all inertial reference frames.[48]: 5  That is, it takes the same form when for a spinorψ(x){\displaystyle \psi (x)} withcoordinatesx{\displaystyle x}, as well as for a Lorentz transformed spinorψ′(x′){\displaystyle \psi '(x')} in the Lorentz transformed coordinatesx′=Λx{\displaystyle x'=\Lambda x}

(i∂/′−m)ψ′(x′)=0,{\displaystyle (i{{\partial }\!\!\!/}'-m)\psi '(x')=0,}

where∂μ′{\displaystyle \partial _{\mu }'} is thefour-gradient for the new coordinatesx′{\displaystyle x'}.[nb 5] Meanwhile, the Dirac action is Lorentz invariant, meaning that it is the same in all reference framesS′=S{\displaystyle S'=S}.[16]: 166–168 

Symmetries

[edit]

Dirac's theory is invariant under a globalU(1){\displaystyle {\text{U}}(1)}symmetry acting on thephase of the spinor[16]: 174 

ψ(x)→eiαψ(x),ψ¯(x)→e−iαψ¯(x).{\displaystyle \psi (x)\rightarrow e^{i\alpha }\psi (x),\qquad {\bar {\psi }}(x)\rightarrow e^{-i\alpha }{\bar {\psi }}(x).}

This has a correspondingconserved current that can be derived from the action usingNoether's theorem, given by[35]: 532 

jμ=ψ¯γμψ.{\displaystyle j^{\mu }={\bar {\psi }}\gamma ^{\mu }\psi .}

This symmetry is known as the vector symmetry because its current transforms as avector under Lorentz transformations. Promoting this symmetry to agauge symmetry gives rise toquantum electrodynamics.

In themassless limit, the Dirac equation has a second inequivalentU(1){\displaystyle {\text{U}}(1)} symmetry known as theaxial symmetry, which acts on the spinors as[52]: 99 

ψ(x)→eiβγ5ψ(x),ψ¯(x)→eiβγ5ψ¯(x),{\displaystyle \psi (x)\rightarrow e^{i\beta \gamma ^{5}}\psi (x),\qquad {\bar {\psi }}(x)\rightarrow e^{i\beta \gamma ^{5}}{\bar {\psi }}(x),}

whereγ5{\displaystyle \gamma _{5}} is the chiral matrix. This arises because in the massless limit the Dirac equation reduces to a pair ofWeyl equations.[16]: 168  Each of these is invariant under aU(1){\displaystyle {\text{U}}(1)} phase symmetry. These two symmetries can then be grouped into the vector symmetry where both Weyl spinors transform by the same phase, and the axial symmetry where they transform under the opposite signs of the phase.[16]: 621  The current corresponding to the axial symmetry is given by[35]: 532 

j5μ=ψ¯γμγ5ψ.{\displaystyle j_{5}^{\mu }={\bar {\psi }}\gamma ^{\mu }\gamma ^{5}\psi .}

This transforms as apseudovector, meaning that its spatial part is odd under parity transformations.[49]: 469 Classically, the axial symmetry admits a well-formulated gauge theory, but at thequantum level it has achiral anomaly that provides an obstruction towards gauging.[16]: 616 

The spacetime symmetries of the Dirac action correspond to thePoincaré group, a combination of spacetimetranslations and the Lorentz group.[16]: 109  Invariance under the four spacetime translations yields the Diracstress-energy tensor as itsfour-currents[52]: 98 

Tμν=iψ¯γμ∂νψ−ημνLDirac,{\displaystyle T^{\mu \nu }=i{\bar {\psi }}\gamma ^{\mu }\partial ^{\nu }\psi -\eta ^{\mu \nu }{\mathcal {L}}_{\text{Dirac}},}

where the last term is the Dirac Lagrangian, which vanisheson shell. Invariance under Lorentz transformations meanwhile yields a set of currents indexed byρ{\displaystyle \rho } andσ{\displaystyle \sigma }, given by[52]: 99 

(Jμ)ρσ=xρTμσ−xσTμρ+ψ¯γμSρσψ,{\displaystyle ({\mathcal {J}}^{\mu })^{\rho \sigma }=x^{\rho }T^{\mu \sigma }-x^{\sigma }T^{\mu \rho }+{\bar {\psi }}\gamma ^{\mu }S^{\rho \sigma }\psi ,}

whereSρσ{\displaystyle S^{\rho \sigma }} are the spinor representation generators of the Lorentz Lie algebra, used to define how spinors transform under Lorentz transformations.

Plane wave solutions

[edit]
Main article:Plane-wave solutions to the Dirac equation

Acting on the Dirac equation with the operator(i∂/+m){\displaystyle (i{{\partial }\!\!\!/}+m)} gives rise to theKlein–Gordon equation for each component of the spinor[53]: 93 

(i∂/+m)(i∂/−m)ψ(x)=(∂μ∂μ+m2)ψ(x)=0.{\displaystyle (i{{\partial }\!\!\!/}+m)(i{{\partial }\!\!\!/}-m)\psi (x)=\left(\partial _{\mu }\partial ^{\mu }+m^{2}\right)\psi (x)=0.}

As a result, any solution to the Dirac equation is also automatically a solution to the Klein–Gordon equation.[40]: 349  Its solutions can therefore be written as alinear combination ofplane waves.[54]: 59 

The Dirac equation admits positivefrequency plane wave solutions[54]: 59 

ψ(x)=u(p)eipμxμ{\displaystyle \psi (x)=u({\boldsymbol {p}})e^{ip_{\mu }x^{\mu }}}

with a positive energy given byp0=p2+m2>0{\displaystyle p_{0}={\sqrt {{\boldsymbol {p}}^{2}+m^{2}}}>0}. It also admitsnegative frequency solutions taking the same form except withp0=−p2+m2<0{\displaystyle p_{0}=-{\sqrt {{\boldsymbol {p}}^{2}+m^{2}}}<0}.[16]: 188  It is more convenient to rewritten these negative frequency solutions by flipping the sign of the momentum to ensure that they have a positiveenergyp0>0{\displaystyle p_{0}>0} and so take the form[36]: 48 

ψ(x)=v(p)e−ipμxμ.{\displaystyle \psi (x)=v({\boldsymbol {p}})e^{-ip_{\mu }x^{\mu }}.}

At the classical level these are positive and negative frequency solutions to a classical wave equation, but in the quantum theory they correspond to operators creating particles with spinorpolarizationu(p){\displaystyle u({\boldsymbol {p}})} orannihilatingantiparticles with spinor polarizationv(p){\displaystyle v({\boldsymbol {p}})}.[16]: 188  Both these spinor polarizations satisfy themomentum space Dirac equation[54]: 59 

(p/+m)u(p)=0,{\displaystyle ({{p}\!\!\!/}+m)u({\boldsymbol {p}})=0,}
(p/−m)v(p)=0.{\displaystyle ({{p}\!\!\!/}-m)v({\boldsymbol {p}})=0.}

Since these are simple matrix equations, they can be solved directly once an explicit representation for the gamma matrices is chosen. In the chiral representation the general solution is given by[16]: 190 

u(p)=(p⋅σξsp⋅σ¯ξs),v(p)=(p⋅σηs−p⋅σ¯ηs),{\displaystyle u({\boldsymbol {p}})={{\sqrt {p\cdot \sigma }}\xi _{s} \choose {\sqrt {p\cdot {\bar {\sigma }}}}\xi _{s}},\qquad v({\boldsymbol {p}})={{\sqrt {p\cdot \sigma }}\eta _{s} \choose -{\sqrt {p\cdot {\bar {\sigma }}}}\eta _{s}},}

whereξs{\displaystyle \xi _{s}} andηs{\displaystyle \eta _{s}} are arbitrary complex 2-vectors, describing the two spin degrees of freedom for the particle and two for the antiparticle.[36]: 45  In the massless limit, these spin states correspond to the possiblehelicity states that the massless fermions can have, either beingleft-handed or right-handed.[16]: 190 

Related equations

[edit]

Related Dirac equations

[edit]

While the standard Dirac equation was originally derived in a3+1{\displaystyle 3+1} dimensionalspacetime, it can be directly generalized to arbitrarydimension andmetric signatures, where it takes the same covariant form.[55]: 51  The crucial difference is that thegamma matrices must be changed to gamma matrices of theClifford algebra appropriate in those dimensions and metric signature, with the size of theDirac spinor corresponding to the dimensionality of the gamma matrices. While the Dirac equation always exists, since every dimension admits Dirac spinors, the properties of these spinors and their relation to otherspinor representations differs significantly across dimensions.[38]: 59  Other differences include the absence of achirality matrix in odd dimensions.[35]: 534 

The equation can also be generalized from flatMinkowski spacetime tocurved spacetime through the introduction of a spinorcovariant derivative[38]: 179 

Dμ=∂μ−i4(ωαβ)μγαγβ,{\displaystyle D_{\mu }=\partial _{\mu }-{\tfrac {i}{4}}\left(\omega _{\alpha \beta }\right)_{\mu }\gamma ^{\alpha }\gamma ^{\beta },}

whereωαβ{\displaystyle \omega _{\alpha \beta }} is thespin connection that can be defined using thetetrad formalism. TheDirac equation in curved spacetime then takes the form[56]: 862 

(iγμDμ−m)ψ(x)=0.{\displaystyle (i\gamma ^{\mu }D_{\mu }-m)\psi (x)=0.}

Adding self-interaction terms to the Dirac action gives rise to thenonlinear Dirac equation, which allows for thefermions to interact with themselves, such as in theThirring model.[57]: 81  Interactions between fermions can also be introduced throughelectromagnetic effects. In particular, theBreit equation describes multi-electron systems interacting electromagnetically to first order inperturbation theory.[58] Thetwo-body Dirac equation is a similar multi-body equation.

Ageometric reformulation of the Dirac equation is known as the Dirac–Hestenes equation.[59] In this formulation all the components of the Dirac equation have an explicit geometric interpretation. Another related geometric equation is theDirac–Kähler equation, which is a geometric analogue of the Dirac equation that can be defined on any generalpseudo-Riemannian manifold and which acts ondifferential forms.[60] In the case of aflat manifold, it reduces to four copies of the Dirac equation. However, on curved manifolds this decomposition breaks down and the equation fundamentally differs.[61] This equation is used inlattice field theory to describe the continuum limit ofstaggered fermions.

Weyl and Majorana equations

[edit]

The Dirac spinor can be decomposed into a pair of Weyl spinors of opposite chiralityψT=(ψL,ψR){\displaystyle \psi ^{T}=(\psi _{L},\psi _{R})}.[40]: 340  UnderLorentz transformations, one transforms as a left-handed Weyl spinorψL{\displaystyle \psi _{L}} and the other as a right-handed Weyl spinorψR{\displaystyle \psi _{R}}. In the chiral representation of the gamma matrices, the Dirac equation reduces to the pair of equations for the Weyl spinors[54]: 57 

iσμ∂μψR(x)=mψL(x),{\displaystyle i\sigma ^{\mu }\partial _{\mu }\psi _{R}(x)=m\psi _{L}(x),}
iσ¯μ∂μψL(x)=mψR(x).{\displaystyle i{\bar {\sigma }}^{\mu }\partial _{\mu }\psi _{L}(x)=m\psi _{R}(x).}

In particular, in themassless limit the Weyl spinorsdecouple and the Dirac equation is equivalent to a pair ofWeyl equations.[16]: 168 .

This decomposition has been proposed as an intuitive explanation ofZitterbewegung, as these massless components would propagate at thespeed of light and move in opposite directions, since thehelicity is the projection of thespin onto the direction ofmotion.[62] Here the role of themass is not to make thevelocity less than the speed of light, but instead controls theaverage rate at which these reversals occur; specifically, the reversals can be modelled as aPoisson process.[63]

A closely related equation is theMajorana equation, with this formally taking the same form as the Dirac equation except that it acts on Majorana spinors.[16]: 193  These are spinors that satisfy a reality conditionψ=Cψ¯T{\displaystyle \psi =C{\bar {\psi }}^{T}}, whereC{\displaystyle C} is thecharge-conjugation operator.[38]: 56  Inhigher dimensions, the Dirac equation has similar relations to the equations describing other spinor representations that arise in those dimensions.

Pauli equation

[edit]
Main article:Pauli equation

In the non-relativistic limit, the Dirac equation reduces to thePauli equation,[32]: 332  which when coupled to electromagnetism has the form

[12m(σ⋅(p^−qA))2+qϕ]|ψ⟩=0.{\displaystyle \left[{\frac {1}{2m}}\left({\boldsymbol {\sigma }}\cdot ({\hat {\boldsymbol {p}}}-q{\boldsymbol {A}})\right)^{2}+q\phi \right]|\psi \rangle =0.}

Hereσ{\displaystyle {\boldsymbol {\sigma }}} is the vector ofPauli matrices andp^=−i∇{\displaystyle {\hat {\boldsymbol {p}}}=-i\nabla } is themomentum operator. The equation describes a fermion ofchargeq{\displaystyle q} coupled to theelectromagnetic field through amagnetic vector potentialA{\displaystyle {\boldsymbol {A}}} and anelectric scalar potentialϕ{\displaystyle \phi }.[17]: 258  The fermion is described through the two-componentwave function|ψ⟩{\displaystyle |\psi \rangle }, where each component describes one of the two spinstates|ψ⟩=ψ+|↑⟩+ψ−|↓⟩{\displaystyle |\psi \rangle =\psi _{+}|\uparrow \rangle +\psi _{-}|\downarrow \rangle }.[8]: 711 

The Pauli equation is often used inquantum mechanics to describe phenomena whererelativistic effects are negligible but the spin of the fermion is important.[8]: 711–715  It can also be recast in a form which directly shows that thegyromagnetic ratio of the fermion described by the Dirac equation is exactlyg=2{\displaystyle g=2}.[54]: 74  Inquantum electrodynamics there are additional quantum corrections that modify this value, give rise to a non-zeroanomalous magnetic moment.[8]: 712 

Gauge symmetry

[edit]

Vector symmetry

[edit]

The vector andaxial symmetries of the Dirac action are both globalsymmetries, in that they act the same everywhere inspacetime.[16]: 34  Inclassical field theory, theLagrangian can always be modified in a way to elevate a global symmetry to a local symmetry, which can act differently at different spacetime locations.[40]: 166–167  In the case of the vector symmetry, which corresponding to a global change of thespinorfield by aphaseψ→eiαψ{\displaystyle \psi \rightarrow e^{i\alpha }\psi },gauging would result in anaction invariant under a local symmetry where the phase can take different values at different pointsψ→eiα(x)ψ{\displaystyle \psi \rightarrow e^{i\alpha (x)}\psi }.[54]: 70  While symmetries can always be gauged in classical field theory, this may not always be possible in the fullquantum theory due to various obstructions such asanomalies, which signal that the full quantum theory is not invariant under the local symmetry despite its classical Lagrangian being invariant.[48]: 177  For example, the axial symmetry in themassless Dirac theory with onefermion is anomalous due to thechiral anomaly and cannot be gauged.[36]: 661 

Elevating the vector symmetry to a local symmetry means that the original action is no longer invariant under the symmetry due to the appearance of a∂μα(x){\displaystyle \partial _{\mu }\alpha (x)} term arising from thekinetic term.[40]: 167  Instead, a new fieldAμ(x){\displaystyle A_{\mu }(x)}, known as a gauge field, must be introduced. It must also transform under the local symmetry as[36]: 78 

Aμ→Aμ+1e∂μα,{\displaystyle A_{\mu }\rightarrow A_{\mu }+{\tfrac {1}{e}}\partial _{\mu }\alpha ,}

wheree{\displaystyle e} plays the role of thecharge of theDirac spinor to the gauge field. The Dirac action can then be made invariant under the local symmetry by replacing the derivative term with a newgauge covariant derivative[16]: 173 

Dμψ=∂μψ+ieAμψ.{\displaystyle D_{\mu }\psi =\partial _{\mu }\psi +ieA_{\mu }\psi .}

The Dirac action then takes the form[54]: 71 

S=∫d4xψ¯(iD/−m)ψ=∫d4x[ψ¯(i∂/−m)ψ−ejμAμ].{\displaystyle S=\int d^{4}x{\bar {\psi }}(i{{D}\!\!\!\!/}-m)\psi =\int d^{4}x\left[{\bar {\psi }}(i{{\partial }\!\!\!/}-m)\psi -ej^{\mu }A_{\mu }\right].}

This result can also be directly acquired through the Noether procedure, which is the general principle that a global symmetry can be gauged through the introduction of a term coupling the gauge field to the appropriate global symmetrycurrent.[40]: 161  Additionally introducing the kinetic term for the gauge field results in the action forquantum electrodynamics.[36]: 78 

General symmetries

[edit]

The symmetries that can be gauged can be greatly expanded by considering a theory withN{\displaystyle N}identical Dirac spinorsψa{\displaystyle \psi ^{a}} labelled by a newindexa{\displaystyle a}. Together these spinors can be considered as being part of a single objectψ{\displaystyle \psi } with componentsψi,a{\displaystyle \psi ^{i,a}} wherei{\displaystyle i} labels the four spin components, anda{\displaystyle a} the different spinors.[16]: 491  The largest global symmetry of this action is then given by theunitary groupU(N){\displaystyle {\text{U}}(N)}.[36]: 490 

Any continuoussubgroup ofU(N){\displaystyle {\text{U}}(N)} can then be gauged. In particular, if one wishes to gauge the symmetry acting on allN{\displaystyle N} components, then the symmetry being gauged must admit anN{\displaystyle N}-dimensionalunitary representation acting on the spinors.[49]: 420  That is, for everyg∈G⊆U(N){\displaystyle g\in {\text{G}}\subseteq {\text{U}}(N)}, there exists anN×N{\displaystyle N\times N} dimensional matrixrepresentationρ(g){\displaystyle \rho (g)} such that

ψ(x)→ρ(g)ψ(x){\displaystyle \psi (x)\rightarrow \rho (g)\psi (x)}

forms afaithful representation of thegroup. Gauging the largest continuous subgroup,SU(N){\displaystyle {\text{SU}}(N)}, requires the spinors to transform in thefundamental orantifundamental representation.[16]: 484 [nb 6] Gauging elevates the representation from being spacetime independent to being spacetime dependentρ(g(x)){\displaystyle \rho (g(x))}.[36]: 490  It requires introducing a gauge fieldAμa{\displaystyle A_{\mu }^{a}}, formally theconnection on theprincipal bundle,[64] which necessarily transforms in theadjoint representation of thegauge group. The covariant derivative then takes the form[40]: 223 

Dμψ=∂μψ+ρ~(Aμ)ψ.{\displaystyle D_{\mu }\psi =\partial _{\mu }\psi +{\tilde {\rho }}(A_{\mu })\psi .}

One symmetry that frequently gets gauged is thespecial unitary groupSU(N){\displaystyle {\text{SU}}(N)} symmetry. Spinors transforming in the fundamental representation transform as[49]: 416 

ψ(x)→U(x)ψ(x),{\displaystyle \psi (x)\rightarrow U(x)\psi (x),}

whereU(x){\displaystyle U(x)} is aN×N{\displaystyle N\times N}unitary matrix corresponding to a particular group element. The gauge field is a matrix valued gauge fieldAμa{\displaystyle A_{\mu }^{a}} which transforms in the adjoint representation as[16]: 492 

Aμ(x)→U(x)Aμ(x)U−1(x)+1g(∂μU(x))U−1(x).{\displaystyle A_{\mu }(x)\rightarrow U(x)A_{\mu }(x)U^{-1}(x)+{\frac {1}{g}}(\partial _{\mu }U(x))U^{-1}(x).}

The covariant derivative then takes the form

Dμψ=∂μψ+igAμψ.{\displaystyle D_{\mu }\psi =\partial _{\mu }\psi +igA_{\mu }\psi .}

By also introducing the kinetic term for the gauge field, one constructs the action forquantum chromodynamics[36]: 489 

SQCD=∫d4x[−14Tr(FμνFμν)+ψ¯(iD/−m)ψ],{\displaystyle S_{\text{QCD}}=\int d^{4}x\left[-{\tfrac {1}{4}}{\text{Tr}}(F^{\mu \nu }F_{\mu \nu })+{\bar {\psi }}(i{{D}\!\!\!/}-m)\psi \right],}

where in the gauge field kinetic term, theYang–Mills fieldsstrength tensor is defined as

Fμν=∂μAν−∂νAμ−ig[Aμ,Aν].{\displaystyle F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }-ig[A_{\mu },A_{\nu }].}

The case ofSU(3){\displaystyle {\text{SU}}(3)} describes strong interactions of thequark sector of theStandard Model, with the gauge field corresponding togluons and the Dirac spinors to the quarks.[65]: 54  TheSU(2){\displaystyle {\text{SU}}(2)} case also plays a role in the Standard Model, describing theelectroweak sector. The gauge field in this case is theW-boson, while the Dirac spinors areleptons.[16]: 592 

See also

[edit]

Notes

[edit]
  1. ^This article uses theEinstein summation convention, where there is an implicit summation over spacetime coordinates for any pairs of repeated indices.
  2. ^In the mainly negative signatureημν{\displaystyle \eta ^{\mu \nu }} is adiagonal matrix with +1 forμ=ν=0{\displaystyle \mu =\nu =0} and -1 for the remaining components.
  3. ^Strictly speaking theSO+(1,3){\displaystyle {\text{SO}}^{+}(1,3)}, which is the part of the Lorentz group connected to theidentity. It excludesparity andtime-reversal transformations.
  4. ^A representation, mapping group elementsg∈G{\displaystyle g\in G} to matricesρ(g)∈GL(V){\displaystyle \rho (g)\in {\text{GL}}(V)}, is faithful if the mapping between group elements and matrices isinjective.
  5. ^The proof relies on the fact thatgamma matrices satisfy the identityS[λ−1]γμS[Λ]=Λμνγν{\displaystyle S\left[\lambda ^{-1}\right]\gamma ^{\mu }S[\Lambda ]=\Lambda ^{\mu }{}_{\nu }\gamma ^{\nu }}.
  6. ^Gauging a larger dimensional representation ofSU(N){\displaystyle {\text{SU}}(N)} with representation dimensionM{\displaystyle M} requires instead consideringM{\displaystyle M} identical spinors and gauging theSU(N)⊂U(M){\displaystyle {\text{SU}}(N)\subset {\text{U}}(M)} subgroup.

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