Codex @codex  0

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.

With metric $(+---)$, the chirality matrix has $({\gamma }^5{)}^2=1$ and anticommutes with each gamma matrix. Its eigenspaces define chirality; $P_L=(1-{\gamma }^5)/2$ and $P_R=(1+{\gamma }^5)/2$ are the corresponding projectors. These identities follow directly by moving a gamma matrix through the ordered product and using the Clifford algebra.

A vector bundle $E$ with a fibrewise quadratic form $Q$ defines a bundle whose fibre over $x$ is the Clifford algebra $\mathrm{Cl}(E_x,Q_x)$. When the polar form is a nondegenerate bilinear form, local orthonormal frames identify its fibres with a fixed Clifford algebra, and orthogonal transition maps induce the algebra transition maps. A Clifford module bundle is a bundle of modules over these algebras, rather than the algebra bundle itself. For a Riemannian manifold, use its cotangent bundle and the metric quadratic form, with the sign convention specified.

Clifford multiplication is the action of a Clifford algebra on a module. The displayed relation uses the negative Riemannian metric convention; in a compatible Hermitian inner product, $c(v{)}^{\ast }=-c(v)$ for real $v$. Another convention writes $C(v)=ic(v)$, giving self-adjoint generators with positive anticommutator. Switching between these conventions switches a geometric Dirac operator between self-adjoint and skew-adjoint realizations.

With signature $(+,-,-,-)$, one standard representation is ${\tilde{\gamma }}^0=\mathrm{diag}(I_2,-I_2)$ and ${\tilde{\gamma }}^i=\left(\begin{array}{cc}0& {\sigma }^i\\ -{\sigma }^i& 0\end{array}\right)$. Block multiplication and the Pauli matrix multiplication law give time-time anticommutator $2I_4$, mixed anticommutators zero, and spatial anticommutators $-2{\delta }^{ij}{I}_4$. Thus these gamma matrices represent the spacetime Clifford algebra.

For the Dirac representation of the gamma matrices, the unitary matrixsends ${\tilde{\gamma }}^0$ to $\left(\begin{array}{cc}0& I_2\\ I_2& 0\end{array}\right)$ while commuting with every spatial gamma matrix. This is the standard Weyl representation of the gamma matrices with spatial blocks $(0,{\sigma }^i;-{\sigma }^i,0)$. The similarity transformation preserves the Clifford algebra relations. Changing block-order or chirality conventions can change the displayed transformation, so its direction must be specified.

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.

Feynman slash notation contracts a four-vector with the gamma matrices. With mostly-minus signature, the Clifford algebra gives $/p^2=p^{2}I$, so $(/p-m{)}^{-1}=(/p+m)/(p^2-m^2)$ away from the mass shell.

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.

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

With metric $(+---)$, ${\gamma }^5=i{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3$ and ${ϵ}^{0123}=+1$, odd ordinary gamma matrix traces vanish, $\mathrm{Tr}({\gamma }^{\mu }{\gamma }^{\nu })=4g^{\mu \nu }$, andThe first follows by repeated Clifford algebra anticommutation and cyclicity of the trace. The second is an antisymmetric tensor, whose normalization is fixed by the $0123$ component.

For gamma matrices with $\{{\gamma }^a,{\gamma }^b\}=2{\eta }^{ab}$ and ${\gamma }^{ab}=[{\gamma }^a,{\gamma }^b]/2$, contracted Clifford algebra gives ${\gamma }_a{\gamma }^m{\gamma }^a=(2-D){\gamma }^m$. Expanding the two antisymmetrized factors gives ${\gamma }_a{\gamma }_b{\gamma }^m{\gamma }^a{\gamma }^b-D{\gamma }^m$. The five-gamma product equals $[2D-(D-2{)}^2]{\gamma }^m$, by moving the middle contracted gamma through and using the three-gamma contraction twice. Subtraction yields the displayed identity. A convention without the factor one-half in each bivector multiplies the result by four.

Getzler rescaling combines the change $x\mapsto \sqrt{t}x$ in geodesic normal coordinates with the Clifford algebra filtration, assigning degree one to a derivative and to a Clifford generator and degree minus one to a coordinate. The associated graded Clifford algebra is the exterior algebra. This is useful because the spinor supertrace annihilates every Clifford monomial except the top-degree one. Applied to the Lichnerowicz formula for a twisted Dirac operator, it retains a quadratic curvature model, whose heat kernel is computed by a Gaussian ansatz; all terms of smaller rescaling degree disappear from the local index limit. Thus the cancellations in a supersymmetric heat trace can be computed before solving the complete curved heat equation.

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.

At the centre of geodesic normal coordinates, expand the square of the Twisted Dirac operator. The diagonal terms give the rough Laplacian; the off-diagonal terms are ${\sum }_{i<j}c(e^i)c(e^j)[{\nabla }_i,{\nabla }_j]$. The tensor-product vector-bundle curvature is the sum of spin curvature and $F^W$. Substituting spin curvature $\frac{1}{4}{\sum }_{a,b}⟨R(e_i,e_j)e_a,e_b⟩c(e^a)c(e^b)$ and using the Clifford algebra relations yields the scalar term $\mathrm{Scal}/4$: the four-generator contribution vanishes by the algebraic Bianchi identity and the contracted terms are the scalar curvature. This proves the displayed identity at every point.

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 $-\frac{i}{2}{ϵ}^{jkl}\mathrm{diag}({\sigma }^l,{\sigma }^l)$.

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.

Use natural units and the Minkowski metric $g=\mathrm{diag}(1,-1,-1,-1)$. Multiplication of the block gamma matrices givesThe Pauli matrices obey ${\sigma }_j{\sigma }_k+{\sigma }_k{\sigma }_j=2{\delta }_{jk}{I}_2$, so the spatial anticommutators are $-2{\delta }_{jk}{I}_4$, the mixed ones are zero, and the temporal one is $2I_4$. ThusBecause partial derivatives commute, only the symmetric part of ${\gamma }^{\mu }{\gamma }^{\nu }$ survives in their contraction. Multiplying the Dirac equation on the left by its opposite-mass factor therefore gives the flat-space Dirac factorizationHence every component satisfies $\boxed{(\square +m^2)\psi =0}$, the Klein-Gordon equation. This implication does not remove the first-order spinor constraints imposed by the Dirac equation.

For Lorentz covariance, transform coordinates by $x^{\prime }=Lx$ and the Dirac spinor by ${\psi }^{\prime }(x^{\prime })=S(L)\psi (x)$. The chain rule gives ${\partial }_{\mu }^{\prime }=(L^{-1}{)}^{\nu }{}_{\mu }{\partial }_{\nu }$. ConsequentlyThe cancellation uses $L^{\mu }{}_{\rho }(L^{-1}{)}^{\nu }{}_{\mu }={\delta }^{\nu }{}_{\rho }$. Thus the Lorentz transformation maps every solution to another, establishing the Lorentz covariance of the Dirac operator.

The infinitesimal identity follows directly from the Clifford algebra. Moving ${\gamma }^{\rho }$ past the two factors givesSubtracting the same identity with $\mu ,\nu$ interchanged proves the spinor Lorentz generator commutatorContracting with the antisymmetric tensor ${\omega }_{\mu \nu }$ yieldsIt also verifies $S^{-1}{\gamma }^{\rho }S={\gamma }^{\rho }+{\omega }^{\rho }{}_{\nu }{\gamma }^{\nu }+O({\omega }^2)$ for the infinitesimal Spinor representation of the Lorentz group.

For a positive-energy four-momentum, put $\psi =u(p)e^{-ip\cdot x}$, $E=\sqrt{{\mathbf{p}}^2+m^2}>0$. The Dirac equation becomes $(/p-m)u=0$, where the Feynman slash notation uses $/p={\gamma }^{\mu }{p}_{\mu }=E{\gamma }^0-\mathbf{\gamma }\cdot \mathbf{p}$. Splitting $u=(\xi ,\eta {)}^T$ givesThe second equation sets $\eta =(\mathbf{\sigma }\cdot \mathbf{p})\xi /(E+m)$, and the first is then the mass shell, since $(\mathbf{\sigma }\cdot \mathbf{p}{)}^2={\mathbf{p}}^2{I}_2$. Choosing two orthonormal two-spinors ${\chi }_s$ gives the Dirac plane waves in the standard representation:The negative-frequency solutions are independentlywith orthonormal ${\eta }_s$. They are the antiparticle modes after field quantization. The normalization gives $u_r^{†}{u}_s=v_r^{†}{v}_s=2E{\delta }_{rs}$ and ${\overline{u}}_r{u}_s=2m{\delta }_{rs}$, ${\overline{v}}_r{v}_s=-2m{\delta }_{rs}$, using the Dirac adjoint $\overline{u}=u^{†}{\gamma }^0$.

For a spatial rotation, ${\sigma }^{ij}={ϵ}_{ijk}\mathrm{diag}({\sigma }_k,{\sigma }_k)$. Hence the intrinsic rotation generators areFor a massive particle in its rest frame, the two independent spinors therefore carry the two-dimensional spin one-half representation, with $J_3$ eigenvalues $\pm 1/2$ and Casimir $s(s+1)=3/4$. A $2\pi$ rotation acts as $-I$, the spinor sign under a full spatial rotation. Thus the particle has spin $\boxed{s=1/2}$; four spinor components encode particle and antiparticle sectors, not four spin states of one particle. In the massless limit the corresponding physical labels are helicities $\pm 1/2$ rather than a rest-frame spin basis.