Dirac spinor

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 $g=\mathrm{diag}(1,-1,-1,-1)$ 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.

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 mostly-minus signature and the Weyl representation, take $p^0=E=\sqrt{|\mathbf{p}{|}^2+m^2}>0$. The Hermitian matrices $A=p\cdot \sigma =E-\mathbf{p}\cdot \mathbf{\sigma }$ and $B=p\cdot \bar{\sigma }=E+\mathbf{p}\cdot \mathbf{\sigma }$ commute and obey $AB=m^{2}I$. Their positive square roots of a matrix therefore obey $\sqrt{A}\sqrt{B}=mI$. It follows that $u=(\sqrt{A}\xi ,\sqrt{B}\xi {)}^T$ solves $(/p-m)u=0$ and $v=(\sqrt{A}\zeta ,-\sqrt{B}\zeta {)}^T$ solves $(/p+m)v=0$. They multiply $e^{-ipx}$ and $e^{ipx}$ respectively. For normalized two-spinors, $u^{†}u=v^{†}v=2E$. The formula extends by positive-semidefinite square roots to the massless limit.

Take Hermitian generators with $[T^a,T^b]=i{f}^{abc}{T}^c$, $(T_{\mathrm{ad}}^b{)}_{ac}=-i{f}^{bac}$, and $D_{\mu }={\partial }_{\mu }-ig{A}_{\mu }^b{T}_{\mathrm{ad}}^b$. For $S_{\psi }=\int \bar{\psi }({\gamma }_{\mu }{D}_{\mu }+m)\psi$ and $\{{\gamma }_{\mu },{\gamma }_{\nu }\}=2{\delta }_{\mu \nu }$, the Dirac propagator is ${\delta }^{ac}(-i/p+m)/(p^2+m^2)$. Expansion of $e^{-S_{\psi }}$ gives the fermion-gluon vertex $+ig{\gamma }_{\mu }(T_{\mathrm{ad}}^b{)}_{ac}$. There is no two-gluon seagull vertex in this first-order kinetic action, and a closed fermion loop has a minus sign.

With $S^{\mu \nu }=\frac{1}{4}[{\gamma }^{\mu },{\gamma }^{\nu }]$, the spinor representation isIt obeys $S^{-1}{\gamma }^{\mu }S={\Lambda }^{\mu }{}_{\nu }{\gamma }^{\nu }$.

Define ${\sigma }^{\mu \nu }=\frac{i}{2}[{\gamma }^{\mu },{\gamma }^{\nu }]$ using the gamma matrix Clifford algebra. Moving ${\gamma }^{\rho }$ through the two factors with $\{{\gamma }^{\mu },{\gamma }^{\nu }\}=2g^{\mu \nu }$ proves the displayed operator commutator. For real antisymmetric ${\omega }_{\mu \nu }$ and $S=1-\frac{i}{4}{\sigma }^{\mu \nu }{\omega }_{\mu \nu }$, it gives $S^{-1}{\gamma }^{\rho }S={\gamma }^{\rho }+{\omega }^{\rho }{}_{\nu }{\gamma }^{\nu }+O({\omega }^2)$. This is the infinitesimal Lorentz covariance identity required by the Dirac equation.

After complexification, pair the $D=2n$ Clifford algebra generators into $n$ fermionic creation and annihilation operators. Their occupation basis has $2^n$ states, giving the irreducible complex Dirac module. The chirality operator separates its even and odd occupation subspaces, each with $2^{n-1}$ states. Majorana reality conditions concern real dimensions and depend on spacetime signature; they must not be imposed in every even dimension indiscriminately.

A vector-spinor describes a spin-three-halves field after its lower-spin components are removed by constraints and, in the massless theory, gauge equivalence. The massless equation is invariant under ${\psi }_{\mu }\mapsto {\psi }_{\mu }+{\partial }_{\mu }\eta$. Gamma-trace gauge and the field equation give $\gamma \cdot \psi =\partial \cdot \psi =/\partial {\psi }_{\mu }=0$. Only helicities $\pm 3/2$ remain. A four-dimensional Majorana field has two physical polarizations; a complex field also has independent antiparticle modes. The gravitino is the supergravity realization of this field.

The gamma trace contracts the vector index of a Rarita-Schwinger field with a gamma matrix, producing an ordinary spinor. In the massive free theory it vanishes by the field equations. In the massless theory a gauge transformation changes it by ${\gamma }^{\mu }{\partial }_{\mu }ϵ$, allowing a gamma-traceless gauge for propagating modes. Together with transversality this removes the unwanted spin-one-half sector.

Gauge equivalence ${\psi }_{\mu }\sim {\psi }_{\mu }+{\partial }_{\mu }ϵ$ permits gamma-trace gauge. The Rarita-Schwinger field equation then gives transversality and a massless Dirac equation for each component. For momentum along the third axis, residual gauge removes the temporal and longitudinal components. The remaining constraint is ${\psi }_2={\gamma }^1{\gamma }^2{\psi }_1$, with ${\psi }_1$ in the two-dimensional kernel of $/p$. Spinor helicity $\pm 1/2$ is paired with vector helicity of the same sign, leaving only $\pm 3/2$. A Majorana spinor reality condition identifies antiparticles; a Dirac vector-spinor has separate antiparticle modes.

For ${\gamma }^{\mu \nu \rho }{\partial }_{\nu }{\psi }_{\rho }-m{\gamma }^{\mu \rho }{\psi }_{\rho }=0$ with $m\ne 0$, divergence gives ${\gamma }^{\mu \rho }{\partial }_{\mu }{\psi }_{\rho }=0$. Gamma contraction then forces $\gamma \cdot \psi =0$, hence $\partial \cdot \psi =0$. The remaining equation is $(/\partial +m){\psi }_{\mu }=0$ in a mostly-plus convention. In the rest frame the temporal component vanishes, leaving a spatial vector times an on-shell two-component spinor; its gamma trace removes the spin-one-half sector. The remaining spin-three-halves representation has four physical states, including longitudinal helicities $\pm 1/2$.

With $\{{\gamma }_{\mu },{\gamma }_{\nu }\}=2{\eta }_{\mu \nu }$, $\Omega ={\omega }^{\mu \nu }{\gamma }_{\mu \nu }/4$, and $\Lambda =e^{\omega }$, the Clifford algebra gives $[\Omega ,{\gamma }^{\rho }]=-{\omega }^{\rho }{}_{\mu }{\gamma }^{\mu }$. Exponentiation proves the inverse Lorentz action displayed above, equivalently $S^{-1}{\gamma }^{\rho }S={\Lambda }^{\rho }{}_{\mu }{\gamma }^{\mu }$. This second order is what combines with $\psi \mapsto S\psi$ and the Dirac adjoint action $\bar{\psi }\mapsto \bar{\psi }{S}^{-1}$ to make the gamma bilinears transform as ordinary contravariant tensors.

The generators in the Spinor representation of the Lorentz group satisfy $M^{†}{\gamma }^0+{\gamma }^{0}M=0$, so the finite transformation preserves the indefinite form $S^{†}{\gamma }^{0}S={\gamma }^0$. This makes $\bar{\psi }\psi$ a Lorentz scalar through the Dirac adjoint. Boost generators are Hermitian rather than anti-Hermitian: a boost has eigenvalues $e^{\pm \eta /2}$ and is not unitary in the ordinary positive-definite component norm. Rotations are unitary. This component representation is distinct from the unitary action on the physical state space.

In a real-generator convention, $M^{\rho \sigma }=\frac{1}{4}[{\gamma }^{\rho },{\gamma }^{\sigma }]$ obeys $[M^{\rho \sigma },{\gamma }^{\mu }]=g^{\sigma \mu }{\gamma }^{\rho }-g^{\rho \mu }{\gamma }^{\sigma }$. Applying this identity to the commutator of two gamma matrices proves the Lorentz algebra relations. With the matching vector generators and real antisymmetric parameters, $S=\exp (\frac{1}{2}{\Omega }_{\rho \sigma }{M}^{\rho \sigma })$ obeys $S^{-1}{\gamma }^{\mu }S={\Lambda }^{\mu }{}_{\nu }{\gamma }^{\nu }$. A convention using Hermitian Lorentz generators instead must consistently insert factors of $i$.

The spinor Lorentz matrices preserve the indefinite form defined by ${\gamma }^0$: $S^{†}{\gamma }^{0}S={\gamma }^0$. This follows from the gamma-matrix adjoint identities for infinitesimal generators and extends to finite connected transformations. Consequently the Dirac adjoint transforms with $S^{-1}$, even though a boost matrix is not ordinarily unitary.

For $\Lambda =1+\omega$ with antisymmetric lowered parameters, the spin matrix is $S=1-(i/4){\omega }_{\mu \nu }{\sigma }^{\mu \nu }$, where ${\sigma }^{\mu \nu }=(i/2)[{\gamma }^{\mu },{\gamma }^{\nu }]$. At transformed coordinates this is the full component transformation. At fixed coordinates an additional orbital term $-{\omega }^{\mu }{}_{\nu }{x}^{\nu }{\partial }_{\mu }\Psi$ appears.

For a Dirac spinor a spatial rotation acts in each chiral block by a matrix exponential $e^{\pm i\theta \mathbf{n}\cdot \mathbf{\sigma }/2}$, depending on angle orientation. Since $(\mathbf{n}\cdot \mathbf{\sigma }{)}^2=I_2$ for a unit axis, the Pauli matrix multiplication law gives this exponential as $\cos (\theta /2)I_2\pm i\sin (\theta /2)\mathbf{n}\cdot \mathbf{\sigma }$. At $\theta =2\pi$ it is $-I_2$, and at $4\pi$ it is $I_2$. A vector returns already at $2\pi$. This describes the lift of the rotation loop to the Spin group, rather than a single-valued representation of the ordinary rotation group on spinors.

The Clifford algebra gives $S^{\mu \nu }=\frac{1}{2}{\gamma }^{\mu }{\gamma }^{\nu }-\frac{1}{2}{\eta }^{\mu \nu }I$. Reordering one more gamma matrix gives $[S^{\mu \nu },{\gamma }^{\rho }]={\eta }^{\nu \rho }{\gamma }^{\mu }-{\eta }^{\mu \rho }{\gamma }^{\nu }$. The commutator derivation identity then givesso these matrices represent the Lorentz algebra. This convention has no extra factor of $i$; the spatial generators are $-\frac{i}{2}{ϵ}^{jkl}\mathrm{diag}({\sigma }^l,{\sigma }^l)$.

A Dirac mass term pairs left- and right-handed Weyl fields in conjugate Lorentz representations. Gauge invariance requires the two chiral fields to carry matching internal quantum numbers, directly or after multiplication by a scalar field that acquires a vacuum expectation value.