Relativistic quantum field

Microcausality requires local bosonic observables at spacelike separation to commute. For a free real scalar field, the field commutator is the difference of its two Wightman functions; it vanishes at equal time and, by Lorentz invariance, at every spacelike separation. Nonzero spacelike vacuum correlations do not violate this condition, since a correlation is different from the response to a local intervention. Odd fermionic fields satisfy a corresponding spacelike anticommutation condition, while physical even local observables commute.

For a free complex scalar field, independent particle annihilators and antiparticle creators giveThe invariant mass shell integral vanishes at spacelike separation: transform to equal time, then reverse the spatial momentum in one term. Retaining only the particle-annihilation term would leave a Wightman function, which is generally nonzero even at spacelike separation. Thus both frequency branches are needed for a local relativistic field. For a real scalar field, the creation term supplies the same cancellation but creates the same self-conjugate particle species.

A free massive spin-two field is a symmetric tensor obeying a massive Klein-Gordon equation, vanishing divergence, and vanishing trace. Its rest-frame polarizations form the symmetric traceless square of the $SO(D-1)$ vector, with $(D-2)(D+1)/2$ degrees of freedom.

When ${\partial }_{-}$ is invertible, divergence determines components with a plus index from $h_{--},h_{-I},h_{IJ}$. The trace constraint fixes the trace of $h_{IJ}$. The remaining transverse group representations are a symmetric traceless square, a vector representation, and a scalar.

In units with $\hslash =1$, the inverse lifetime of an unstable particle. A neutral meson mixing effective Hamiltonian operator incorporates decay through its anti-Hermitian part $-i\Gamma /2$.

A branching fraction is the probability that a decay occurs through a particular mutually exclusive channel, equal to its partial decay width divided by the total decay width. Inclusive channel fractions sum to one. Rounding displayed fractions can make their numerical sum differ from one.

The invariant decay amplitude is the transition matrix element for one incoming particle and a specified on-shell final state, with the overall momentum delta function factored out. For covariantly normalized states, write the connected S-matrix element as $i(2\pi {)}^4{\delta }^4(p_i-p_f)\mathcal{M}(i\to f)$. Its overall phase does not affect the decay width. The rate is $\Gamma =(2M_i{)}^{-1}{\sum }_{\mathrm{spins}}\int d{\Phi }_f|\mathcal{M}{|}^2$, with the initial spin average included when the initial state is unpolarized. For a spin-zero parent and an angle-independent two-body amplitude, the two-body Lorentz-invariant phase space gives $\Gamma =|\mathbf{k}|\sum |\mathcal{M}{|}^2/(8\pi M_i^2)$. In leptonic pion decay, contracting the axial hadronic matrix element with the chiral weak charged current makes this amplitude proportional to the charged-lepton mass, proving helicity suppression.

An absorptive phase arises from physical intermediate states or rescattering in a decay amplitude. For a pair of CP symmetry conjugate reactions with otherwise symmetric dynamics, this phase is the same in both amplitudes, whereas a weak phase changes sign. Their differing transformation laws allow interference to produce a direct CP asymmetry.

An unpolarized decay of a massive spin-one particle averages the squared amplitude over three parent spin states. Its polarization sum is $-{\eta }_{\mu \nu }+p_{\mu }{p}_{\nu }/m^2$. For a conserved external current the longitudinal $p_{\mu }{p}_{\nu }$ term drops out, but this does not replace the spin average by $1/2$: the physical massive parent still has three states.

For a pseudoscalar meson of mass $M$ decaying into a pseudoscalar meson of mass $m$ and two massless leptons through a weak charged current, the transverse massless leptonic current selects $f_{+}(s)$. With hadronic current normalized as $(p+k{)}^{\mu }{f}_{+}(s)+(p-k{)}^{\mu }{f}_{-}(s)$ and flavour coefficient $V$,The Källén function has mass dimension four, and the coefficient has mass dimension $-7$. This decay rate assumes the stated form-factor normalization and no additional isospin factor. It follows by the integrated massless leptonic tensor and the pion momentum change of variable in the parent's centre-of-momentum frame.

For two outgoing particles the Lorentz-invariant phase space is $d{\Phi }_2=(2\pi {)}^4{\delta }^{(4)}(p-k-k^{\prime })d^{3}k/((2\pi {)}^{3}2E_k)d^3{k}^{\prime }/((2\pi {)}^{3}2E_{k^{\prime }})$. With massless daughters in their center-of-mass frame, $d{\Phi }_2=d\Omega /(32{\pi }^2)$ and $\int d{\Phi }_2=1/(8\pi )$. This fixes normalization factors in decay widths and scattering rates.

For two distinct massless final particles in their centre-of-momentum frame, integrating the energy-momentum delta function gives $d{\Phi }_2=d\Omega /(32{\pi }^2)$. A massless two-particle initial state has flux $2s$, hence $d\sigma /d\cos \theta =\overline{|\mathcal{M}{|}^2}/(32\pi s)$. Average only over the initial spin states physically present.

For two distinct massless final particles, $d{\Phi }_2/d\Omega =1/(32{\pi }^2)$ in the centre-of-mass frame. The invariant initial flux is $4p_1\cdot p_2=2s$. Their ratio gives the displayed differential cross-section. Equivalently the denominator is $128{\pi }^2{p}_1\cdot p_2$. Initial spins are averaged and final spins and colors summed when defining an inclusive unpolarized cross-section; averaging final colors would change the observable.

A Lorentz four-vector coefficient selecting a spin-one polarization in a mode expansion of a free field. Its allowed components depend on the field equations and gauge fixing.

A physical polarization of a massive spin-one gauge boson whose spatial polarization points along its spatial momentum. Together with two transverse polarizations it supplies the three physical spin states. In the Higgs mechanism, a broken-direction Goldstone boson supplies this extra state as the gauge boson acquires mass.

Two orthonormal transverse polarization vectors span the plane perpendicular to a nonzero momentum. Summing their dyadic products gives the spatial transverse projector. It has trace and rank two and acts as the identity on physical photon polarizations.

For $m>0$, the three physical polarization vectors are transverse to $k$ and have norm $-1$ in metric $(+---)$. In the rest frame their sum is $\mathrm{diag}(0,1,1,1)$; Lorentz covariance then gives the displayed expression for $k^2=m^2$. This includes the longitudinal mode. It cannot be specialized to $m=0$ by setting its denominator to zero.

A timelike basis polarization in covariant photon quantization has positive Minkowski metric squared length, conventionally ${ϵ}^{0\mu }=(1,0)$ with signature $(+,-,-,-)$. It is distinct from the spatial longitudinal polarization ${ϵ}^{3\mu }=(0,\hat{\mathbf{p}})$. The extra minus sign in the covariant oscillator commutator makes its one-photon state a negative-norm photon state. Neither this temporal polarization nor an isolated longitudinal photon survives the Gupta-Bleuler null-state quotient.

The Fierz-Pauli equations describe a free massive spin-two field. A symmetric tensor $h_{\mu \nu }$ obeys the Klein-Gordon equation together with ${\partial }^{\mu }{h}_{\mu \nu }=0$ and $h^{\mu }{}_{\mu }=0$, leaving five propagating polarizations in four dimensions.

A free Dirac field has Lagrangian density $\mathcal{L}=i\bar{\psi }{\gamma }^{\mu }{\partial }_{\mu }\psi -m\bar{\psi }\psi$ and obeys the Dirac equation.

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.

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.

Antimatter is an assembly of antiparticles. A cosmological antimatter population is distinct from the small secondary antiparticle populations produced by energetic collisions.

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 spin-one-half field equation $(i{\gamma }^{\mu }{\partial }_{\mu }-m)\psi =0$.

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.

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 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 $P_L^2=P_L$, $P_R^2=P_R$.

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.