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.

For a vector space $V$ over a field of characteristic different from two and a quadratic form $Q$, its Clifford algebra is the quotient of the tensor algebra by the relations $v^2=Q(v)1$. Writing $B(u,v)=(Q(u+v)-Q(u)-Q(v))/2$ gives $uv+vu=2B(u,v)1$.

For a real two-dimensional positive-definite form and an orthonormal basis $e_1,e_2$, the generators satisfy $e_1^2=e_2^2=1$ and $e_1{e}_2=-e_2{e}_1$. The algebra has basis $1,e_1,e_2,e_1{e}_2$. Its Clifford modules provide spinor spaces, while products of suitable unit vectors represent rotations. The sign convention on $Q$ must be stated, especially when comparing geometric and physical conventions.

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.

The group of unitary Clifford gates, defined by normalization of the Pauli group. For a single qubit, $H$ and $S$ generate its usual finite gate group, and modulo global phase its action permutes the six Pauli eigenstates. On multiple qubits, one adds an entangling controlled-NOT gate. This quantum-computing use of the name is distinct from groups attached to a general Clifford algebra.

For a quadratic form $Q$ on a vector space $V$, a Clifford module is a module over the Clifford algebra $\mathrm{Cl}(V,Q)$. Equivalently it is a vector space $S$ with a linear assignment $v\mapsto c(v)\in \mathrm{End}(S)$ satisfyingOver a field of characteristic different from two, expanding $c(v+w{)}^2$ gives $c(v)c(w)+c(w)c(v)=2B(v,w)I$, with $B$ the bilinear form associated to $Q$. The universal property of the Clifford algebra extends these generator actions to an action of the whole algebra over a field.

For $Q(x,y)=x^2+y^2$ over $\mathbb{R}$, the matrices $c(e_1)=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$ and $c(e_2)=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ both square to $I$ and anticommute. They make ${\mathbb{R}}^2$ a Clifford module. Such modules provide the algebraic setting for spinors; changing the sign convention for $Q$ changes the required generator squares.

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.

Choose five Hermitian four-by-four matrices obeying the Euclidean Clifford algebra relations ${\gamma }_a{\gamma }_b+{\gamma }_b{\gamma }_a=2{\delta }_{ab}I$. For $M_{ab}=E_{ab}-E_{ba}$, the displayed anti-Hermitian matrices satisfy the Special orthogonal Lie algebra Lie bracket relations. A direct proof first establishes $[\rho (M_{ab}),{\gamma }_c]={\delta }_{bc}{\gamma }_a-{\delta }_{ac}{\gamma }_b$, then applies the commutator product rule to $[{\gamma }_c,{\gamma }_d]$. One explicit choice is ${\gamma }_1={\sigma }_1\otimes I$, ${\gamma }_2={\sigma }_2\otimes I$, ${\gamma }_3={\sigma }_3\otimes {\sigma }_1$, ${\gamma }_4={\sigma }_3\otimes {\sigma }_2$, ${\gamma }_5={\sigma }_3\otimes {\sigma }_3$, with the Pauli matrices. The Cartan subalgebra operators $-i\rho (M_{12})$ and $-i\rho (M_{34})$ have simultaneous eigenvalues $(\pm 1/2,\pm 1/2)$, identifying the four-dimensional spinor representation. A full $2\pi$ rotation acts as $-I_4$, proving it does not descend to $SO(5)$.

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.

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 $E^2=p^2+m^2$.

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 represent the spacetime Clifford algebra throughFor the Minkowski metric with signature $(+,-,-,-)$, ${\gamma }^0$ squares to $I$ and each spatial gamma matrix squares to $-I$. Distinct orthogonal generators anticommute. A four-dimensional complex representation is used for Dirac spinors in four-dimensional spacetime.

The particular matrix entries depend on the chosen basis; simultaneous similarity transformations preserve the defining relations. Their usefulness in the Dirac equation is that contracting them with derivatives produces a first-order operator whose square reduces to the second-order relativistic wave operator, apart from interactions or curvature contributions.

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.