Dirac field

A Dirac field is a quantum field transforming as a Dirac spinor. Its free Lagrangian density is $\mathcal{L}=i\bar{\psi }{\gamma }^{\mu }{\partial }_{\mu }\psi -m\bar{\psi }\psi$, giving the Dirac equation. Unlike a single-particle wave function, a quantized Dirac field can create and remove particles and antiparticles.

Its mode expansion of a Dirac field contains fermionic annihilation operators for particles and fermionic creation operators for antiparticles, with canonical anticommutation relations enforcing Fermi-Dirac statistics. For the electron field, these excitations are electrons and Positrons. A global phase symmetry gives the conserved Dirac current; coupling that current to the electromagnetic field produces the matter interaction in quantum electrodynamics.

For Minkowski metric $\mathrm{diag}(1,-1)$, the displayed gamma matrices satisfy the Clifford algebra and ${\gamma }^5={\gamma }^0{\gamma }^1$ has chirality eigenvalues $+1,-1$. Ordering the spinor as $({\psi }_R,{\psi }_L{)}^{\mathsf{T}}$ gives $i({\partial }_t+{\partial }_x){\psi }_R=m{\psi }_L$ and $i({\partial }_t-{\partial }_x){\psi }_L=m{\psi }_R$. A plane wave has the Dirac Hamiltonian $p{\sigma }_3+m{\sigma }_1$, whose square is $(p^2+m^2)1$, proving $E^2=p^2+m^2$.

For the coordinate Lorentz boost $t^{\prime }=t\cosh \alpha +x\sinh \alpha$, $x^{\prime }=t\sinh \alpha +x\cosh \alpha$, the spinor convention ${\psi }^{\prime }(x^{\prime })=S\psi (x)$ gives $S=\exp (\alpha {\gamma }^5/2)$. In the Dirac field in two spacetime dimensions basis, this yields the displayed weights. The derivatives ${\partial }_t+{\partial }_x$ and ${\partial }_t-{\partial }_x$ scale by $e^{-\alpha }$ and $e^{\alpha }$, respectively, so the two kinetic terms are invariant. Opposite weights cancel in the Dirac mass term. Reversing the coordinate boost convention reverses both exponents.

The massless Dirac equation in one spatial dimension separates into opposite chirality sectors. Its initial profiles evolve as ${\psi }_R(t,x)=f(x-t)$ and ${\psi }_L(t,x)=h(x+t)$, by the chain rule along the straight characteristics. Thus wave packets translate without distortion at the speed of light. A nonzero Dirac mass term couples the two sectors, so these separate rigid-translation formulas do not apply to a massive field.

For gamma matrices ${\gamma }^0={\sigma }_3$, ${\gamma }^1=i{\sigma }_1$, ${\gamma }^2=i{\sigma }_2$ and Minkowski metric $\mathrm{diag}(1,-1,-1)$, a one-axis spatial reflection acts with ${\gamma }^1$ on the two-component Dirac field. It preserves the kinetic term and reverses the Dirac mass term. Time reversal acts with ${\gamma }^2$ and includes conjugation of the operator coefficients; it has the same mass reversal. Their composition preserves a real mass. Charge conjugation can use ${\psi }_C={\gamma }^1{\psi }^{\ast }$, because ${\gamma }^{\mu }{\gamma }^1=-{\gamma }^1{\gamma }^{\mu \ast }$. The mass bilinear is charge-conjugation even when Grassmann field exchange signs are included, and the kinetic action is invariant up to a boundary term. These statements concern the classical free action.

For a Dirac field with a global phase symmetry, fermion number is the vacuum-subtracted charge of the Dirac current. In a mode expansion it counts particles minus antiparticles, with the overall sign fixed by the chosen generator convention. It is distinct from the total number of fermionic excitations and from fermion parity, which can survive even when interactions violate the continuous charge.

A Dirac field theory invariant under $\psi \mapsto e^{i\alpha }\psi$, $\bar{\psi }\mapsto e^{-i\alpha }\bar{\psi }$ has a conserved global charge, subject to the quantum theory preserving the symmetry. In particle language this charge counts particles minus antiparticles. When the S-matrix commutes with it, a scattering amplitude between states of different charge vanishes. A neutral scalar and a Yukawa interaction preserve the charge, so a process with one incoming Dirac particle and only one outgoing Dirac antiparticle plus neutral particles is forbidden. This net charge is distinct from a sum of positive occupation numbers for all particle and antiparticle modes.

A local expression containing two fermion fields and a matrix acting on Dirac spinor indices. The choices $\Gamma ={\gamma }^{\mu }$ and $\Gamma ={\gamma }^{\mu }{\gamma }^5$ give the Dirac current and axial current.

A fermion bilinear transforming as a Lorentz four-vector. For a single Dirac field this is the Dirac current; distinct flavours can give a charged or flavour-changing current.

The mode expansion of a free Dirac field decomposes it into positive-frequency particle modes and negative-frequency antiparticle modes:The operators $b^s$ and $d^s$ obey fermionic anticommutation relations.

The positive occupation operators are $N_e={\sum }_s\int d^{3}p{a}_{\mathbf{p}}^{s†}{a}_{\mathbf{p}}^s/(2\pi {)}^3$ and $N_{\bar{e}}={\sum }_s\int d^{3}p{b}_{\mathbf{p}}^{s†}{b}_{\mathbf{p}}^s/(2\pi {)}^3$. Both commute with the free Hamiltonian. The normal-ordered local density integral $\int :{\psi }^{†}\psi :d^{3}x$ is their difference, not their sum. It generates the global phase symmetry. For electron charge $-e$, electric charge is $-eQ$. Pair creation can change total occupation while preserving this net charge.

An antiparticle has the same mass and spin as its particle and opposite additive internal charges. In a relativistic quantum field, antiparticles are created by the negative-frequency part of the field expansion.

The antiparticle of a fermion. In a Dirac field, its external wavefunctions are the negative-frequency $v$ spinors, while its physical energy is positive. Fermion-number arrows on Feynman diagrams run opposite an antifermion's physical momentum.

The antiproton is the antiparticle of the proton: it has the same rest mass and spin $1/2$, with opposite electric charge and baryon number. A negative electric charge alone cannot identify an antiproton, because a negative pion also bends in that direction in a magnetic field. Combining magnetic rigidity with velocity determines a rest mass; agreement with the proton rest mass then makes the identification much more discriminating.

The mode expansion of a Dirac field explains how independent particle and antiparticle excitations enter a relativistic field description. It supplies a theoretical language for the antiproton, while the Chamberlain-Segrè antiproton identification supplies an experimental test. The proton is a composite baryon, so a free Dirac field is not a complete account of its internal dynamics or of antiproton production in a target.

In 1955, Owen Chamberlain, Emilio Segrè, Clyde Wiegand, and Thomas Ypsilantis used Bevatron protons on copper, selecting negative secondaries. Magnetic fields selected $p=1.19\mathrm{GeV}/c$; scintillation counters S1 and S2 measured time of flight over $40\mathrm{ft}$. Cherenkov detector C1 flagged $\beta >0.79$; C2 accepted $0.75<\beta <0.78$ after material losses. S3 checked traversal. The cuts rejected negative pions.^[ref]

Chamberlain's 1959 account describes C2's quartz radiator, mirrors, and photomultiplier readout. The trigger combined delayed scintillation counter signals with C2; photographed oscilloscope traces exposed fast particles and accidental coincidences through C1 pulses or extra pulses. Reversing the magnetic fields provided a positive-proton control and revealed an alignment error. Changing selected momentum at fixed velocity acceptance tested for a proton-mass peak.^[ref]

Terminal-event studies remained unfinished; the 1955 identification did not rely on annihilation.^[ref]

The logic is useful far beyond this experiment. A magnetic field separates electric charge signs and magnetic rigidity, but particles of different rest masses can share both. A time of flight signal adds velocity, while a Cherenkov detector independently checks whether that velocity is plausible. An accidental coincidence can imitate the delay of a slow particle without being one particle traversing the apparatus. Agreement between independent measurements is therefore part of the inference, rather than a redundant decoration.

For a singly charged track perpendicular to a uniform magnetic field, its curvature gives magnetic rigidity $p/|q|=Br$, and the bend direction gives the sign of electric charge. The mass shell relation and four-momentum relation between energy and velocity giveFor a known path length $L$ and constant velocity, time of flight $t=L/(\beta c)$ therefore givesThis measures rest mass under an assigned electric charge magnitude. Curvature alone measures $p/|q|$, so independent evidence for $|q|=e$ matters. In material, the measured delay is instead $t=\int dl/v(l)$; matching a local Cherenkov detector velocity to an upstream momentum requires an energy-loss correction.

As an original illustrative calculation, choose $p=1.20\mathrm{GeV}/c$ and $L=12.0\mathrm{m}$, with rounded input rest energies $m_p{c}^2=0.93827\mathrm{GeV}$ and $m_{\pi }{c}^2=0.13957\mathrm{GeV}$. Constant-velocity propagation givesBoth particles have the same selected momentum and negative electric charge, yet their time of flight separation is about $10.51\mathrm{ns}$. These are calculated values for a deliberately simplified example, not measured discovery events.

For example, the toy energy loss changes the central antiproton velocity from $0.7878c$ to $0.7745c$. Directly substituting the slower downstream value together with the upstream momentum would overestimate the rest mass. The two panels thus describe different stages of a deliberately specified model; their numerical consistency depends on propagating energy between stages. Likewise, the scattered-pion markers represent an erroneous C2 response, not actual slow pions; an upstream C1 signal still rejects them.

The fixed-target production threshold concerns the incident accelerator beam, whereas antiproton mass from momentum and velocity concerns an outgoing secondary beam. To see the distinction, consider a free proton hitting a stationary free proton, producingThe final electric charge is $+2e$ and final baryon number is $3-1=2$, as initially. Approximate all four final rest masses by the same proton rest mass $m$. With metric signature $(+---)$, define the squared centre-of-mass energy by $W^2=c^2(P_1+P_2{)}^2$. The incident proton has total energy $E=m{c}^2+K$, and the target four-momentum is $(mc,0)$. Four-momentum conservation and the mass shell relation giveAt threshold the four final particles have zero relative momentum in their centre-of-momentum frame, so $W_{\mathrm{thr}}=4m{c}^2$. HenceOnly $2m{c}^2$ of new rest energy is created. The larger incident kinetic energy is required because the final system must also carry the initial laboratory momentum. The incident total energy at threshold is $7m{c}^2$; neither this nor $K_{\mathrm{thr}}$ is the selected secondary momentum.

The 1955 authors explicitly qualified the stationary-nucleon threshold: target nucleon motion inside an atomic nucleus can lower the required incident energy.^[ref]

The kinematic reason is visible for two on-shell incoming nucleons:A target momentum directed against the beam increases $W^2$. Applying this illustration to a bound nucleon also requires binding energy and residual-nucleus recoil; the free stationary-target $6m{c}^2$ result is therefore not a universal sharp threshold for a copper target. A low observed secondary momentum is entirely compatible with a much larger accelerator kinetic energy.

Particle-antiparticle annihilation converts a particle and its antiparticle into other particles, conserving energy and applicable quantum numbers. Baryon-antibaryon annihilation commonly produces mesons whose subsequent decays can generate radiation.

Electron-positron annihilation converts an electron and a Positron into other particles while conserving total energy, momentum and electric charge. A common low-energy channel produces two photons. In the centre-of-momentum frame, negligible initial kinetic energy gives two oppositely directed photons, each with energy $m_e{c}^2$.

The back-to-back momentum balance makes paired photon detections a useful signature of annihilation. Motion of the initial pair changes the laboratory photon directions and energies. At larger collision energy, other final states become accessible: an intermediate virtual photon can produce, for example, a muon-antimuon pair. Thus annihilation converts the initial pair into allowed final excitations rather than destroying conserved energy.

For two-photon positron annihilation with negligible initial total momentum, each photon has energy $m_e{c}^2$ and the two propagate oppositely. Their common energy differs from the positron's continuous emission-energy spectrum before slowing in matter.

For an initial Dirac field of mass $m$ annihilating into a distinct Dirac field of mass $M$ through a vector current, put $\Sigma =m^2+M^2$. With Mandelstam variables satisfying $s+t+u=2\Sigma$, the spin average of the two contracted vector currents is $2F$, where $F=(\Sigma -t{)}^2+(\Sigma -u{)}^2+2s\Sigma =t^2+u^2+4s\Sigma -2{\Sigma }^2$. The gamma-matrix trace proof uses $\mathrm{tr}[(/p_2-m){\gamma }^{\mu }(/p_1+m){\gamma }^{\nu }]=4[p_2^{\mu }{p}_1^{\nu }+p_2^{\nu }{p}_1^{\mu }-(s/2)g^{\mu \nu }]$ and the analogous final tensor. Couplings, the mediator denominator and color charge factors multiply this common kinematic expression.

The leading Standard Model gauge amplitudes are s-channel Z boson exchange and t-channel W boson exchange. They interfere. At collision energies well below the weak scale the cross section scales as $G_F^{2}s$. In an inclusive high-energy Born calculation, the forward W-exchange region gives a nonzero limiting scale of order $g^4/m_W^2$, unlike a fixed-angle estimate. Angular cuts and beam polarization can change comparisons with other annihilation channels.

An electron with kinetic energy $\alpha m_e{c}^2$ annihilating with a stationary positron into two equal-angle photons has $\tan \varphi =\sqrt{2/\alpha }$ for $\alpha >0$. Equal photon energies follow from the specified symmetric laboratory geometry.

At tree level in the Standard Model, the diagrams are s-channel photon, Z boson and Higgs boson exchange, and t-channel electron neutrino exchange. The Higgs contribution couples to the electron through its small Yukawa coupling. In a zero-electron-mass approximation that diagram vanishes, leaving the three gauge diagrams. The gauge contributions interfere; their sum has the high-energy behavior required by the electroweak gauge structure.

Antimatter consists of antiparticles, whose rest masses and spins match those of the corresponding particles while additive internal conserved charges are reversed. A Positron, for example, has the electron's rest mass and opposite electric charge. Both have positive physical energy.

When a particle meets its antiparticle, annihilation can convert the pair into other particles while preserving total energy, momentum and applicable conserved charges. Electron-positron annihilation into two photons is a simple example. Producing a few antiparticles in energetic collisions does not establish a large cosmological antimatter population: the production process and the subsequent abundance are separate questions. The antiparticle description is built into relativistic quantum field theory, rather than requiring negative measured mass or energy.

The historical interpretation fills the negative-energy electron states of the Dirac equation and uses the Pauli exclusion principle to prevent additional electrons from falling into them. Removing an electron of energy $-E$ and charge $-e$ gives a hole of excitation energy $E$ and charge $+e$, interpreted as a positron. A quantum field theory instead treats negative-frequency field coefficients as independent antiparticle creation operators in a Fock space, with positive excitation energies after normal ordering. No literal infinite material sea is needed. Bosonic antiparticles also exist, although a filled-sea exclusion argument cannot stabilize bosonic negative-energy levels.

A fermionic annihilation operator removes one fermion from a specified one-particle mode. Together with its adjoint it satisfies a canonical anticommutation relation such as $\{b_r,b_s^{†}\}={\delta }_{rs}$.

A fermionic creation operator is the adjoint of a fermionic annihilation operator and adds one fermion to a mode. Its square vanishes, implementing the Pauli exclusion principle.

The Dirac equation is the relativistic equation of motion for a spin-one-half Dirac field, $(i{\gamma }^{\mu }{\partial }_{\mu }-m)\psi =0$, in units $\hslash =c=1$. The gamma matrices obey the Clifford algebra, ensuring that each component also obeys the free Klein-Gordon equation.

Being first order in time allows an initial spinor to determine its subsequent free evolution. The components encode spin as well as particle and antiparticle solutions; quantization interprets both kinds of excitation with positive physical energy. Coupling through the gauge covariant derivative describes an electron interacting with an electromagnetic field. The nonrelativistic limit contains the coupling of spin to a magnetic field, explaining why a scalar relativistic wave equation alone cannot describe an electron.

Matrices that factor the relativistic energy-momentum relation. In Hamiltonian form, ${\alpha }_i^2={\beta }^2=I$, $\{{\alpha }_i,{\alpha }_j\}=2{\delta }_{ij}I$ and $\{{\alpha }_i,\beta \}=0$. A physical Hamiltonian uses Hermitian matrices; in three spatial dimensions the smallest complex representation has dimension four.

The Dirac Hamiltonian is the first-order time-evolution operator obtained from the Dirac equation. For a free massive particle,where the Dirac matrices satisfy $\{{\alpha }_i,{\alpha }_j\}=2{\delta }_{ij}I$, $\{{\alpha }_i,\beta \}=0$ and ${\beta }^2=I$. These identities give $H^2=c^2{P}^2+m^2{c}^4$, recovering the relativistic energy-momentum relation.

Minimal electromagnetic coupling replaces the canonical momentum by $\mathbf{P}-q\mathbf{A}$ and adds $q\varphi$, in SI units. The wavefunction has spinor components, and the free operator has positive- and negative-energy branches. The Dirac field interprets particle and antiparticle excitations with a stable vacuum; it is more complete than treating the negative branch as an ordinary one-particle spectrum with no lower bound.

The two-component model $H=c(p_x{\sigma }_x+p_y{\sigma }_y)+V{\sigma }_z$ has energies $E_s=s\sqrt{c^2{p}^2+V^2}$. The sign of $V$ changes eigenstates and spin polarization but not the uniform energy dispersion at fixed $|V|$. The Pauli matrices may represent actual spin or an effective pseudospin, depending on the physical realization.

The gamma matrix Clifford algebra and commuting partial derivatives give $(/\partial {)}^2=\square$. Thus every free Dirac equation solution also solves the Klein-Gordon equation componentwise. The converse is false without the first-order spinor constraint: the second-order equation alone allows twice as much independent initial data.

With $\{{\gamma }^{\mu },{\gamma }^{\nu }\}=2g^{\mu \nu }$ and spin curvature $[{\nabla }_{\mu },{\nabla }_{\nu }]=R_{\mu \nu ab}{\gamma }^{ab}/4$, the Dirac equation operator squares to the rough Laplacian minus $R/4$. In the curvature contraction the four-gamma term vanishes by the first Bianchi identity, the two-gamma terms vanish by Ricci symmetry, and the scalar term is $-2R$. In Riemannian signature $i/\nabla$ is the self-adjoint convention and its square is $-{\nabla }^2+R/4$.

The four-gamma product decomposes into fully antisymmetric, two-gamma and scalar parts. The first part vanishes on contraction by the first Bianchi identity, and the two-gamma parts vanish because they contract symmetric Ricci curvature against antisymmetric gamma matrices. The scalar contractions are $-g^{ac}{g}^{bd}{R}_{abcd}+g^{ad}{g}^{bc}{R}_{abcd}=-2R$. Here $R=g^{ac}{g}^{bd}{R}_{abcd}$ fixes the curvature-slot convention.

// Target: second-fundamental-form.bigb

For a metric-compatible connection with torsion tensor $T^{\rho }{}_{\mu \nu }={\Gamma }_{\mu \nu }^{\rho }-{\Gamma }_{\nu \mu }^{\rho }$, expand the square using Clifford multiplication and the full affine second derivative. Its antisymmetric part is spin curvature minus $T^{\rho }{}_{\mu \nu }{\nabla }_{\rho }$. This gives the displayed formula. The curvature need not have the torsion-free pair symmetry or first Bianchi identity, so its Clifford contraction need not reduce to a scalar $-R/4$. General torsion produces additional first-order and nonscalar curvature couplings.

On a compact boundaryless Riemannian spin manifold with $R\ge 0$, integrating the Lichnerowicz spinor-square formula against a harmonic spinor gives $\int (|\nabla \psi {|}^2+R|\psi {|}^2/4)=0$. Positivity forces the spinor to be parallel. Its norm is constant on each connected component, so a nontrivial solution is nowhere zero there. Strictly positive scalar curvature at any point on such a component excludes nonzero harmonic spinors.

A parallel spinor has zero spin curvature. Contracting the spinorial Ricci identity with a gamma matrix gives $R_{\mu \nu }{\gamma }^{\nu }\psi =0$. For each fixed $\mu$, squaring Clifford multiplication by that Ricci row gives $|R_{\mu \cdot }{|}^2\psi =0$. Riemannian positivity and a nonzero spinor force every row to vanish. This implication is local and does not itself require compactness; the compact hypothesis is used to obtain parallelism from harmonicity.

A spinor Lorentz matrix satisfies $S^{-1}{\gamma }^{\mu }S={\Lambda }^{\mu }{}_{\nu }{\gamma }^{\nu }$. Combining this identity with the transformed derivative shows that $(i{\gamma }^{\mu }{\partial }_{\mu }^{\prime }-m){\Psi }^{\prime }=S(i{\gamma }^{\mu }{\partial }_{\mu }-m)\Psi$. Thus a Lorentz transformation carries every solution of the Dirac equation to another.

Varying the Dirac action with respect to the spinor, rather than its Dirac adjoint, gives the left-acting field equation after integration by parts. Together with the Dirac equation it proves conservation of the vector current.

The massless Dirac equation is ${\gamma }^m{\partial }_m\Psi =0$. The Clifford algebra makes its square the massless wave equation. Quantization of a worldline spin variable as ${\hat{\psi }}^m={\gamma }^m/\sqrt{2}$ turns its fermionic constraint into this equation.

One consistent Dirac equation convention uses $\eta =\mathrm{diag}(-1,1,1,1)$, $\{{\gamma }^a,{\gamma }^b\}=-2{\eta }^{ab}$, and $(i{\gamma }^a{\partial }_a+m)\psi =0$. Taking the negatives of the standard mostly-minus gamma matrices, and defining the Dirac adjoint with the new ${\gamma }^0$, gives the same physical Dirac action. The positive-frequency plane wave is $e^{ip\cdot x}u(p)$ with $p^0>0$, and its equation is $(/p-m)u=0$. The mass shell is $p^2=-m^2$. These signs must be translated together when using a formula stated in another convention.

The Dirac adjoint of a spinor is $\bar{\psi }={\psi }^{†}{\gamma }^0$. It transforms contragrediently, making $\bar{\psi }\psi$ a Lorentz scalar and $\bar{\psi }{\gamma }^{\mu }\psi$ a Lorentz vector.

The Dirac scalar bilinear is invariant under the simultaneous spinor and adjoint transformations because ${\bar{\Psi }}^{\prime }{\Psi }^{\prime }=\bar{\Psi }{S}^{-1}S\Psi$. It is the mass term of the Dirac action. At fixed coordinates it transforms as a scalar field, with an orbital transport term and no spin term.

The free Dirac action isIts Euler-Lagrange equations are the Dirac equation and its adjoint.

For on-shell Dirac spinors in standard relativistic normalization,These completeness identities turn spin-summed squared amplitudes into gamma-matrix traces.

A left- or right-handed Weyl spinor transforms in the Lorentz representation $(1/2,0)$ or $(0,1/2)$. Parity exchanges the two chiralities.

The free massless relativistic field equation for one chiral two-component spinor. With mostly-minus metric, the left-chiral equation has the minus sign and the right-chiral equation the plus sign. The positive-energy particle helicities are respectively $-1/2$ and $+1/2$; antiparticle helicities are opposite.

For a coherent two-component field $\psi =e^{-i{V}_{0}t}(\eta (t-z),\rho (t+z){)}^T$, the physical current is $\mathbf{J}=(2\mathrm{Re}{\eta }^{\ast }\rho ,2\mathrm{Im}{\eta }^{\ast }\rho ,|\eta {|}^2-|\rho {|}^2)$. The density is $|\eta {|}^2+|\rho {|}^2$. Transverse components generally survive interference even though the field has no transverse coordinate dependence. They have zero transverse divergence; together with the longitudinal current they obey local conservation and $|\mathbf{J}{|}^2=({\psi }^{†}\psi {)}^2$.

The isotropic Weyl Hamiltonian acts on a two-component spinor as $H=v\mathbf{\sigma }\cdot \mathbf{p}$. For a plane wave of momentum $\hslash \mathbf{k}$, its two energy eigenvalues are $E_{\pm }=\pm \hslash |v||\mathbf{k}|$. For $v>0$ the corresponding Bloch vectors point along and opposite $\mathbf{k}$, respectively. If the Pauli matrices describe physical spin one-half, the mean spin is $\pm (\hslash /2)\hat{\mathbf{k}}$; in an effective two-band model the same algebra can describe an internal pseudospin instead. The bands meet linearly at the origin. ^[ref]

Adding a spatially constant Hermitian matrix $V=dI+\mathbf{b}\cdot \mathbf{\sigma }$ to an isotropic Weyl Hamiltonian preserves each momentum sector. By the Hermitian two-by-two matrix in the Pauli basis decomposition, the exact energies are $d\pm |\hslash v\mathbf{k}+\mathbf{b}|$. The touching point moves to ${\mathbf{k}}_{\ast }=-\mathbf{b}/(\hslash v)$ at energy $d$, without opening a gap in three-dimensional momentum space.

For a fixed $k_z=0$ slice, however,This slice has minimum branch separation $2|b_z|$ if $b_z\ne 0$, because it misses the shifted node. A one-dimensional cut can likewise look gapped while a crossing exists elsewhere. No nonzero two-by-two matrix anticommutes with all three Pauli matrices, which is another way to see why an independent constant mass matrix is unavailable for this isolated three-dimensional two-component model.

A spacetime field valued in one irreducible chiral spinor representation. A free massless left-handed Weyl field obeys $i({\partial }_t-\mathbf{\sigma }\cdot \nabla ){\chi }_L=0$. The left-handed massless plane-wave spinor and helicity content of a quantized Weyl field describe its modes and particle states. Field chirality fixes one spinor representation; it does not give the same particle and antiparticle helicity.

For mostly-minus metric signature, the Weyl sigma matrices are ${\sigma }^{\mu }=(I,{\sigma }_1,{\sigma }_2,{\sigma }_3)$ and ${\bar{\sigma }}^{\mu }=(I,-{\sigma }_1,-{\sigma }_2,-{\sigma }_3)$, where ${\sigma }_i$ are the Pauli matrices. The Pauli matrix multiplication law provesA real Lorentz four-vector is equivalently a Hermitian matrix $X=x^{0}I+x^i{\sigma }_i$ with $\det X=(x^0{)}^2-|\mathbf{x}{|}^2$. The transformation $X\mapsto MX{M}^{†}$ for $M\in \mathrm{SL}(2,\mathbb{C})$ preserves this determinant and gives the Lorentz spinor double cover. The indices ${\sigma }_{\alpha \dot{\beta }}^{\mu }$ therefore convert a vector index into one undotted and one dotted Weyl spinor index. They are the invariant tensors in the mixed supercharge anticommutator $\{Q_{\alpha },{\bar{Q}}_{\dot{\beta }}\}=2{\sigma }_{\alpha \dot{\beta }}^{\mu }{P}_{\mu }$.

A left-handed massless Weyl spinor has a positive-energy plane wave transforming by $e^{i\alpha /2}$ under a rotation about its momentum. Its particle has helicity $-1/2$. The negative-frequency term multiplies an antiparticle fermionic creation operator, so its corresponding one-particle state carries the conjugate rotation phase and helicity $+1/2$. Thus a left-handed field has one particle and one antiparticle helicity; the right-handed field interchanges the signs. A full massless Dirac field contains both chiral fields and hence both particle helicities.

In the standard Weyl representation of the gamma matrices and mostly-minus metric signature, the left chirality component obeys $i({\partial }_t-\mathbf{\sigma }\cdot \nabla ){\chi }_L=0$. A plane wave with $p^0>0$ exists precisely on the massless mass shell $p^0=|\mathbf{p}|$. Its coefficient is a negative helicity eigenspinor. For direction angles $\vartheta ,\phi$, one normalized choice isThe nonzero solution is unique up to overall normalization and phase. Different patches are required for a continuous phase choice over all directions; the direction of zero momentum is undefined.

With $K_i=M_{0i}$ and $M_{\mu \nu }=i{\gamma }_{\mu \nu }/2$ in the supplied mostly-minus index convention, the two Weyl spinor blocks have equal rotation generators $J_i={\sigma }_i/2$ and opposite boost generators. A complex-linear intertwiner must commute with every SU(2) rotation and hence be scalar by the Schur lemma, but no nonzero scalar intertwines the opposite boosts. This proves inequivalence of the two complex representations; parity exchanges them.

For odd Grassmann variables, $(\chi \psi )={\chi }^{\alpha }{\psi }_{\alpha }$ is symmetric under exchanging the two spinors: the Grassmann sign cancels the sign of the antisymmetric epsilon contraction. With ${ϵ}^{12}=1$, lower components $\chi =(a,b)$ and $\psi =(c,d)$ give $(\chi \psi )=bc-ad$. Commuting numerical spinors instead give an antisymmetric contraction.

For four independent odd Grassmann variables $a,b,c,d$, $(bc-ad{)}^2=-2abcd$, whereas $(\psi \psi )(\chi \chi )=(-2cd)(-2ab)=4abcd$. This proves the displayed two-component identity and fixes its sign. It is not valid with the same interpretation for commuting spinors, whose self-contractions vanish.

Chirality distinguishes the two irreducible Weyl representations of the proper Lorentz group. For a massless particle it agrees with helicity up to the particle-antiparticle convention, while a mass term couples opposite chiralities.

The chiral projectors $P_{L,R}=(1\mp {\gamma }^5)/2$ separate the components of a Dirac spinor. The weak charged current selects one chirality. A massive fixed-helicity state generally contains both chiral components; the simple helicity-chirality identification requires the massless limit and a specified particle-antiparticle convention.

For a positive-energy Dirac spinor normalized to unit probability and with positive helicity, the left-chiral component has squared weight $(1-\beta )/2$, where $\beta =|\mathbf{p}|/E$ in natural units. Its right-chiral weight is $(1+\beta )/2$; for negative helicity the two are exchanged. In an ultrarelativistic positive-helicity particle, the left-chiral amplitude is suppressed by a mass-to-energy ratio.

The projectors $P_L=(1-{\gamma }^5)/2$ and $P_R=(1+{\gamma }^5)/2$ select the two chirality eigenspaces of a Dirac field. They satisfy $P_L+P_R=I$, $P_L{P}_R=0$, and $P_L^2=P_L$, $P_R^2=P_R$.

A Lorentz scalar Dirac bilinear connects opposite chiralities. In the stated on-shell normalization ${\bar{u}}_R{u}_L=m_{\mathrm{spinor}}$, whereas ${\bar{u}}_L{u}_L=0$. Such an overlap verifies the allowed Lorentz structure but does not determine an independent coefficient multiplying a different mass operator.

With Minkowski metric signature $+---$ and $\mathrm{tr}({\gamma }^{\mu }{\gamma }^{\nu }{\gamma }^{\rho }{\gamma }^{\sigma }{\gamma }_5)=4i{ϵ}^{\mu \nu \rho \sigma }$, define $H^{\mu \nu }(a,b)=a^{\mu }{b}^{\nu }+a^{\nu }{b}^{\mu }-g^{\mu \nu }a\cdot b+i{ϵ}^{\mu \nu \rho \sigma }{a}_{\rho }{b}_{\sigma }$. Then $\mathrm{tr}[/a{\gamma }^{\mu }(1-{\gamma }_5)/b{\gamma }^{\nu }(1-{\gamma }_5)]=8H^{\mu \nu }(a,b)$. The symmetric-tensor contraction gives twice the sum of the two possible dot-product pairings; the Levi-Civita symbol contraction, including the two factors of $i$, gives twice their difference. Their sum is the displayed identity, explaining the single pairing in left-chiral weak rates.

A vectorlike gauge spectrum pairs complex gauge group representations $R$ and $\bar{R}$ among independent left-handed Weyl spinors. Such a pair permits a gauge-invariant fermion mass and occurs in a full charged hypermultiplet of four-dimensional $\mathcal{N}=2$ extended supersymmetry.

A chiral gauge spectrum has independent left-handed Weyl spinors in complex gauge group representations without an obligatory conjugate pair of left-handed Weyl spinors. The right-handed antiparticles required by the CPT theorem do not themselves make a spectrum vectorlike.

In the Weyl representation,A Dirac spinor therefore splits into left- and right-handed two-component spinors.

A Majorana spinor equals its charge conjugate. In the Weyl representation it can be written ${\psi }_M=(u_L,i{\sigma }^2{u}_L^{\ast }{)}^T$ from one left-handed Weyl spinor.

For a Majorana spinor $\lambda$ with odd Grassmann variables as components, ${\lambda }^{T}M\lambda$ sees only the antisymmetric part of $M$. With $C^T=-C$ and $({\gamma }^a{)}^T=-C{\gamma }^a{C}^{-1}$, reversing an antisymmetrized gamma product gives the displayed transpose rule. Thus the same-spinor vector and bivector bilinears vanish, whereas scalar, pseudoscalar and three-form bilinears can be nonzero. Four-dimensional Hodge duality gives the same vanishing result for a bivector multiplied by ${\gamma }_5$. Commuting spinors instead see the symmetric part, so their selection rule is different.

For Grassmann-odd four-dimensional Majorana spinors, choose $C^T=-C$ and $({\gamma }^a{)}^T=-C{\gamma }^a{C}^{-1}$. Then $C{\gamma }^a$ is symmetric whereas $C{\gamma }^{abc}$ is antisymmetric. Swapping Grassmann components therefore changes the sign of a vector bilinear but preserves a three-gamma bilinear. Integration by parts and antisymmetry of the vector indices then make the two spinor variations of the quadratic Rarita-Schwinger field action equal. Commuting auxiliary spinors have different interchange signs.

A Majorana-Weyl spinor satisfies both a Majorana spinor reality condition and a Weyl spinor chirality condition when dimension and signature permit them. In ten-dimensional Lorentzian spacetime it has sixteen real components off shell and eight physical massless spin-one-half polarizations.

For Grassmann-odd spinors in two dimensions with $\bar{\psi }={\psi }^{T}C$, $C^T=-C$ and $({\gamma }^{\mu }{)}^{T}C=-C{\gamma }^{\mu }$, moving components past each other gives $\bar{\chi }ϵ=\overline{ϵ}\chi$ and $\bar{\chi }{\gamma }^{\mu }{\gamma }^{\nu }ϵ=\overline{ϵ}{\gamma }^{\nu }{\gamma }^{\mu }\chi$. The transpose of a two-gamma product reverses its order; the Grassmann sign and the antisymmetry of $C$ cancel. For $\delta \psi =B{\gamma }^{\mu }{\partial }_{\mu }Xϵ$, this also gives $\delta \bar{\psi }=-B\overline{ϵ}{\gamma }^{\mu }{\partial }_{\mu }X$. These identities fix the boundary variation in worldsheet supersymmetry; commuting spinors would not obey the same interchange rule.