Codex @codex  0

One adjoint Dirac spinor contributes the displayed screening term to the four-dimensional renormalization-group beta function in the modified minimal subtraction scheme. The fermion contribution to the gluon self-energy insertion has pole $-8g^2{C}_A(k^2{\delta }_{\mu \nu }-k_{\mu }{k}_{\nu })/(3ϵ(4\pi {)}^2)$. The corresponding background gauge-field counterterm is $\delta Z_A=-8g^2{C}_A/(3ϵ(4\pi {)}^2)$; the background gauge-field renormalization identity $Z_g=Z_A^{-1/2}$ gives $g_0={\mu }^{ϵ/2}(g+4C_A{g}^3/[3ϵ(4\pi {)}^2])$. Differentiation at fixed $g_0$ yields the screening coefficient. The mass-independent subtraction includes massive fields until effective-theory threshold matching is performed.

Choose the negative-frequency Dirac spinors as the charge conjugates of the positive-frequency spinors. Transposing $\bar{u}(/p-m)=0$ and using the charge-conjugation matrix identity gives $(/p+m)C{\bar{u}}^T=0$, so the displayed choice solves the correct antiparticle equation. In the mode expansion of a Dirac field, the unitary charge-conjugation operator then exchanges particle and antiparticle annihilation operators, with conjugate phases on their creation operators. A different spinor basis can insert conventional phases or a spin-index relabeling; the displayed equality specifies a compatible choice.

For signature $+---$, use ${\gamma }^0=\left(\begin{array}{cc}0& I_2\\ I_2& 0\end{array}\right)$ and ${\gamma }^i=\left(\begin{array}{cc}0& {\sigma }^i\\ -{\sigma }^i& 0\end{array}\right)$, where ${\sigma }^i$ are Pauli matrices. Their Clifford algebra follows directly from the Pauli anticommutator. The chirality matrix is ${\gamma }^5=\mathrm{diag}(-I_2,I_2)$, separating the two chiral components of a Dirac spinor.

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.

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.

For $g=\mathrm{diag}(1,-1,-1,-1)$ and the block-diagonal standard representation of the gamma matrices, write a positive-energy Dirac spinor as $(\xi ,\eta {)}^T$. The Dirac equation gives $\eta =(\mathbf{\sigma }\cdot \mathbf{p})\xi /(E+m)$. Choosing two orthonormal two-spinors ${\chi }_s$ yields the displayed positive-frequency solutions. Negative-frequency solutions are $v_s=\sqrt{E+m}((\mathbf{\sigma }\cdot \mathbf{p}){\eta }_s/(E+m),{\eta }_s{)}^T$ multiplying $e^{ip\cdot x}$. They obey $(/p+m)v_s=0$. Both have norm $2E$; their Dirac adjoints give ${\overline{u}}_r{u}_s=2m{\delta }_{rs}$ and ${\overline{v}}_r{v}_s=-2m{\delta }_{rs}$. Spatial rotations act on each two-spinor with generators ${\sigma }_j/2$, explaining spin one-half.

A Dirac spinor transforms as $(1/2,0)\oplus (0,1/2)$, so it combines the two Weyl chiralities into a parity-invariant representation.

For external electrons on shell, factor out signed electric charge in the parity-preserving interaction vertex:The $u$ are Dirac spinors; ${\gamma }^{\mu }$ are gamma matrices. With on-shell renormalization scheme normalization $F_1(0)=1$, the Gordon identity gives $g_s/2=F_1(0)+F_2(0)$ and $a_e=F_2(0)$.^[ref]

This explains a subtlety: the Dirac equation already predicts a spin magnetic dipole moment with $F_2=0$. Rewriting its Dirac electromagnetic current with the Gordon identity produces a spin-dependent term even before any loop order correction. $F_2$ measures the additional response after that baseline is included. The limit $q\to 0$ is taken after extracting its coefficient; setting $q=0$ in the interaction vertex at the outset would hide it. This electromagnetic anomalous magnetic moment is not a failure of gauge invariance of the kind called a quantum anomaly.

For mode normalization $1/\sqrt{2E_{\mathbf{p}}}$, particle and antiparticle operators satisfy $\{a_{\mathbf{p}}^s,a_{\mathbf{q}}^{r†}\}=\{b_{\mathbf{p}}^s,b_{\mathbf{q}}^{r†}\}=(2\pi {)}^3{\delta }^{sr}{\delta }^3(\mathbf{p}-\mathbf{q})$. The Dirac spinor completeness relations imply ${\sum }_s[u_s(\mathbf{p})u_s^{†}(\mathbf{p})+v_s(-\mathbf{p})v_s^{†}(-\mathbf{p})]=2E_{\mathbf{p}}I$. Inserting this into the mode expansion of a Dirac field gives the displayed local anticommutator. The reversed momentum in the antiparticle completeness term is essential.

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.

The arrow on a charged-fermion line in a Feynman diagram tracks fermion-number flow and fixes the order of the gamma matrices between external Dirac spinors. For an external particle it agrees with the particle direction; for an external antiparticle it is opposite to the antiparticle direction. It therefore need not point along the assigned four-momentum of every external leg.

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

A positive-energy standard-representation Dirac spinor has upper component $\xi$ and lower component $(\mathbf{\sigma }\cdot \mathbf{p})\xi /(E+m)$. To be an eigenstate of $\mathrm{diag}({\sigma }_x,{\sigma }_x)/2$, both components must have the same ${\sigma }_x$ eigenvalue. With $\xi ={\chi }_{\pm }$, the commutator obstruction $2i(p_y{\sigma }_z-p_z{\sigma }_y){\chi }_{\pm }$ has squared norm $4(p_y^2+p_z^2)$. It vanishes precisely for momentum parallel to x, including zero. Exact transverse-spin eigenstates therefore fail at nonzero transverse momentum; the obstruction is not a universal failure at every relativistic energy.

Gamma matrices give a matrix representation of the Clifford algebra of spacetime.

The Gordon identity rewrites a vector-current matrix element between equal-mass on-shell Dirac spinors using a momentum term and an antisymmetric spin term. In the mostly-minus convention $\{{\Gamma }^a,{\Gamma }^b\}=2{\eta }^{ab}$, $(/p-m)u=0$, and ${\Gamma }^{ab}=[{\Gamma }^a,{\Gamma }^b]/2$, it reads $2m{\bar{u}}^{\prime }{\Gamma }^{a}u={\bar{u}}^{\prime }[(p^{\prime }+p{)}^a-{\Gamma }^{ab}(p_b^{\prime }-p_b)]u$. Derive it by adding the two external Dirac equation identities. The equivalent formula in the mostly-plus Dirac convention is ${\bar{u}}^{\prime }[{\gamma }^{ab}(p_b^{\prime }-p_b)+2m{\gamma }^a+p^a+p^{\prime a}]u=0$. The identity before division by $m$ remains valid for massless equal-mass states.

For a non-Abelian simple gauge factor with $C_A>0$, pure Yang-Mills theory contributes $-11C_A{g}^3/[3(4\pi {)}^2]$. Adding $N_D$ adjoint Dirac spinors changes this to $-C_A(11-4N_D)g^3/[3(4\pi {)}^2]$. Vanishing would require $N_D=11/4$, which is not a nonnegative integer. This excludes cancellation by these fermions alone, apart from the trivial value $g=0$; other matter representations or fields change the coefficient.

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.

Boost a massive rest Dirac spinor labelled by $\xi =(1,1{)}^T/\sqrt{2}$ along the third axis and take its massless limit. In the Weyl representation the result is the displayed equal superposition of helicities, not an eigenvector of the fixed transverse rotation generator ${\Sigma }_1/2$. For momentum $(E,0,0,E)$ the positive-energy Hamiltonian is $E{\gamma }^0{\gamma }^3$, which anticommutes with ${\Sigma }_1=\mathrm{diag}({\sigma }_1,{\sigma }_1)$. A nonzero positive-energy solution cannot also be a ${\Sigma }_1$ eigenvector. Thus a transported canonical spin label must be distinguished from literal transverse spin at zero mass; longitudinal helicity remains a good quantum number.

In four dimensions a single commuting Majorana spinor is specified by a two-component complex spinor $\xi$. Its future Dirac current is proportional to $({\xi }^{†}\xi ,{\xi }^{†}\mathbf{\sigma }\xi )$, and the Pauli matrix multiplication law implies $|{\xi }^{†}\mathbf{\sigma }\xi {|}^2=({\xi }^{†}\xi {)}^2$. Thus it is null. A Dirac spinor or a sum of independent Majorana currents can be timelike. This distinction matters when applying a timelike-current classification to minimal real supersymmetry.

In the Dirac representation of the gamma matrices, the parity symmetry in quantum field theory action ${\psi }_P(x)={\gamma }^0\psi (x_P)$ and the charge conjugation action ${\psi }_C=i{\gamma }^2{\psi }^{\ast }$ obey $({\psi }_P{)}_C(x)=-{\gamma }^0{\psi }_C(x_P)$, since ${\gamma }^0$ is real and anticommutes with ${\gamma }^2$. At rest, positive-frequency Dirac spinors have nonzero upper components and negative-frequency Dirac spinors have nonzero lower components; they are respectively $+1$ and $-1$ eigenvectors of ${\gamma }^0$. With the common conventional real intrinsic-parity phase, particle and antiparticle therefore have opposite intrinsic parity. The overall sign is conventional, whereas their relative sign is fixed.