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.

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.

After complexifying a nondegenerate metric in $D=2k$ dimensions, pair gamma matrices into $k$ fermionic creation and annihilation operators. Their binary occupation states span $2^k$ dimensions, and the gamma matrices generate all endomorphisms of this space. The Dirac module is therefore irreducible as a Clifford module. The even algebra preserves the chirality grading, splitting its Lorentz-spinor representation into two irreducible Weyl modules of dimension $2^{k-1}$. In $D=2k+1$, adjoining the old chirality matrix as the extra gamma gives an irreducible $2^k$-dimensional Spin group representation with no Weyl splitting. The two possible signs of that extra gamma distinguish Clifford modules but give equivalent Spin group modules.

A spinor is an element of a representation space for a Spin group; algebraically, spinor spaces arise from irreducible representations of a Clifford algebra or its even part. The Spin group is a double cover of a rotation group, so its action can distinguish rotations that ordinary vectors cannot.

For rotations in three-dimensional Euclidean space, a spinor can be a column in ${\mathbb{C}}^2$, acted on by $\mathrm{SU}(2)$. A rotation through angle $\theta$ about a unit direction $n$ acts as $\exp (-i\theta n\cdot \sigma /2)$, where $\sigma$ are the Pauli matrices. At $\theta =2\pi$ this is $-I$, and at $4\pi$ it is $I$. The sign change explains the distinction between spinors and spatial vectors and underlies the representation of spin one-half. A spinor is not a literal square root of a vector; the relationship is through the Clifford algebra action.

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.

For $2n$ hermitian Euclidean Clifford generators, the commuting hermitian involutions $J_j=i{\gamma }^{2j-1}{\gamma }^{2j}$ admit a common eigenvector. Each odd-indexed generator flips one joint eigenvalue. The $2^n$ resulting vectors are nonzero and mutually orthogonal. Thus any nonzero invariant complex representation has dimension at least $2^n$.

For hermitian generators satisfying $\{{\gamma }^I,{\gamma }^J{\}}_{+}=2{\delta }^{IJ}$ and any nonzero vector $v$, the vectors ${\gamma }^{I}v$ are linearly independent over the real numbers. Indeed, $A={\sum }_I{a}_I{\gamma }^I$ with real $a_I$ obeys $A^2=({\sum }_I{a}_I^2)1$, so $\Vert Av{\Vert }^2=({\sum }_I{a}_I^2)\Vert v{\Vert }^2$. Complex coefficients need a different argument.

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.

For the mostly-minus Minkowski metric, this product anticommutes with every gamma matrix and is anti-Hermitian in the standard representation. The conventional Hermitian chirality matrix is $iG$. Nevertheless $\bar{\psi }G\psi$ is Hermitian, because ${\gamma }^{0}G{\gamma }^0=-G$. Therefore a real interaction coefficient in $-g\varphi \bar{\psi }G\psi$ is compatible with Hermitian dynamics and gives the Feynman vertex $-igG$.

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.

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 Clifford algebra gives $(/p{)}^2=p^{2}I$. Splitting a scalar-coefficient polynomial into even and odd powers proves the displayed reduction. For an on-shell fermion spin sum assumed to be such a polynomial, $(/p-m)P=0$ implies $P=A(/p+m)$; the normalization $\mathrm{tr}(P{\gamma }^0)=4p^0$ fixes $A=1$ in four spacetime dimensions.

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.

A hermitizing matrix for a set of gamma matrices is an invertible Hermitian matrix $H$ satisfying ${\gamma }^{a†}=H{\gamma }^a{H}^{-1}$. Equivalently, each $H{\gamma }^a$ is Hermitian. It defines a covariant adjoint $\bar{\psi }={\psi }^{†}H$. If ${\gamma }^{\prime a}=M{\gamma }^a{M}^{-1}$, then $H^{\prime }=M^{-†}H{M}^{-1}$ has the same property. In a standard Lorentzian Dirac basis, $H={\gamma }^0$ is a conventional choice.

In the standard Dirac representation of the gamma matrices, $({\gamma }^0{)}^{†}={\gamma }^0$ and $({\gamma }^i{)}^{†}=-{\gamma }^i$, so ${\gamma }^{a†}={\gamma }^0{\gamma }^a{\gamma }^0$. The transpose signs in that representation are positive for $a=0,2$ and negative for $a=1,3$. The charge-conjugation matrix expresses the basis-independent intertwining relation $C^{-1}{\gamma }^{a}C=-({\gamma }^a{)}^T$. Under an arbitrary nonunitary similarity transformation, the hermitizing matrix changes by inverse Hermitian congruence; it need not equal the transformed temporal gamma matrix.

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.

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.

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.