Spinors, Conjugations, Bilinears, Chirality, and Fierz Identities

A spinor is not determined by the word “spinor” alone. One must specify the scalar field, dimension, signature, Clifford sign, Spin group or component, and module. Chirality is then an even-dimensional grading. A Majorana or symplectic Majorana condition is an equivariant real structure. An invariant bilinear is an intertwiner from a spinor module to its dual. A Fierz identity is the completeness relation for a trace-dual basis of endomorphisms, with a sign supplied by the order and Grassmann parity of the spinors.

The reliable construction is therefore

$$\begin{gathered} (V,g),\ \operatorname{Cl}(V,g),\ S \longrightarrow \Gamma_*,\ A,\ \mathcal C,\ J,\ \beta \\ \longrightarrow \beta\!\left(\psi,\gamma_{\mu_1\ldots\mu_k}\chi\right) \longrightarrow \text{trace completeness and Fierz rearrangement}. \end{gathered}$$

Every arrow is a calculation or an intertwiner test. None is licensed by a four-dimensional mnemonic. This page develops that reusable method, gives a fully fixed four-dimensional Lorentzian convention card, and ends with a controlled application to Dirac, Majorana, and Weyl bilinears.

Required background. Clifford Algebras and Pin and Spin Groups supplies the Clifford algebra, double cover, and module language; Representations, Intertwiners, Invariants, and Tensor Decomposition supplies intertwiners, invariant forms, and tensor decompositions.

Spinor modules and invariant pairings

Let $(V,g)$ be a nondegenerate real quadratic space of dimension $d=p+q$, where $p$ directions have positive square and $q$ have negative square. The Clifford convention is

$$\gamma(v)\gamma(w)+\gamma(w)\gamma(v) = 2g(v,w)\mathbf1_S.$$

Unless a real structure is under discussion, $S$ is a finite-dimensional complex module. The Spin action is denoted $\rho$, and $\gamma_{\mu_1\ldots\mu_k}$ means the unit-weight antisymmetrized product

$$\gamma_{\mu_1\ldots\mu_k} = \gamma_{[\mu_1}\cdots\gamma_{\mu_k]}.$$

“Unit weight” means that the factor $1/k!$ is included. A source that omits it has different Fierz coefficients.

Minimum data for a spinor statement

Statement	Data that must remain visible
Complex spinor module	Dimension, complex Clifford algebra or Spin representation, and irreducibility notion
Reality condition	Real signature, group or component, antilinear intertwiner, and its square
Chirality condition	Even dimension, orientation, normalized volume element, and eigenvalue convention
Bilinear identity	Pairing, transpose or adjoint convention, spinor order, and Grassmann parity
Fierz coefficient	All preceding data plus gamma normalization, independent basis, and trace dual

The previous Clifford page constructs modules and their Spin action. The representation page identifies invariant bilinear forms with intertwiners $S\to S^*$. Here those two inputs are combined. Geometric spin structures, Dirac operators, fermion dynamics, spin sums, and scattering applications remain outside this page.

Complex modules come before reality conditions

Complexification removes the distinction between positive and negative basis squares. It does not remove the distinction between even and odd dimension. For $d=2m$ and $d=2m+1$,

$$\begin{aligned} \operatorname{Cl}_{2m}^{\mathbb C} &\cong \operatorname{Mat}_{2^m}(\mathbb C),\\ \operatorname{Cl}_{2m+1}^{\mathbb C} &\cong \operatorname{Mat}_{2^m}(\mathbb C) \oplus \operatorname{Mat}_{2^m}(\mathbb C). \end{aligned}$$

Consequently:

• in even dimension there is one irreducible complex module for the full Clifford algebra, of dimension $2^m$; its restriction to the even algebra and the Spin group splits into two half-spin modules;
• in odd dimension there are two inequivalent irreducible modules for the full Clifford algebra, distinguished by the action of the central volume element, but their restrictions give equivalent complex Spin representations of dimension $2^m$.

Thus “irreducible Clifford module” and “irreducible Spin representation” are not interchangeable phrases. The acting algebra or group must be named. This classification follows from the complex matrix-algebra structure, not from a preferred list of gamma matrices; see Figueroa-O’Farrill 2015, §§3–4, PDF and Van Proeyen 1999, §3.1.

The real question is subsequent. A complex Spin representation may be of real, complex, or quaternionic type. That type depends on the real form $\operatorname{Spin}(p,q)$ and can differ from the type of the full real Clifford module. Complexifying first keeps these two questions separate.

Chirality is an even-dimensional grading

Choose an orientation and an oriented orthonormal basis $e_1,\ldots,e_d$. The unnormalized volume operator is

$$\omega = \gamma(e_1)\cdots\gamma(e_d),$$

with

$$\omega^2 = (-1)^{d(d-1)/2} \prod_{a=1}^d g(e_a,e_a)\,\mathbf1_S.$$

In even dimension choose the convention-dependent phase $\kappa$ for which

$$\Gamma_* = \kappa\omega, \qquad \Gamma_*^2=\mathbf1_S.$$

For the declared $v^2=g(v,v)$ convention, one explicit choice is

$$\kappa = i^{d(d-1)/2+q}.$$

The remaining overall sign labels which eigenspace is called positive chirality.

The operator $\Gamma_*$ anticommutes with every $\gamma(v)$ and commutes with every even Clifford element. It therefore commutes with the Spin action. The projectors

$$P_\pm = \frac12(\mathbf1_S\pm\Gamma_*)$$

split the complex Dirac module into the two invariant half-spin modules $S_\pm=P_\pm S$. Clifford multiplication by a vector reverses chirality:

$$\gamma(v)P_\pm = P_\mp\gamma(v).$$

Orientation reversal or a change in the phase defining $\Gamma_*$ can exchange the names of the two eigenspaces. The invariant content is the splitting and the fact that odd Clifford multiplication exchanges its two parts.

In odd dimension the volume element commutes with every gamma matrix. On an irreducible complex Clifford module it acts as a scalar and distinguishes the two full-algebra modules. It does not furnish a nontrivial projector inside one irreducible Spin module. There is therefore no Weyl chirality in odd dimension.

For the site’s four-dimensional Lorentzian convention,

$$\eta_{\mu\nu} = \operatorname{diag}(1,-1,-1,-1), \qquad \gamma_5 = i\gamma^0\gamma^1\gamma^2\gamma^3,$$

and the labels are fixed by

$$P_L = \frac12(1-\gamma_5), \qquad P_R = \frac12(1+\gamma_5).$$

These labels cannot be imported from a source that uses the opposite sign for $\gamma_5$ without interchanging $L$ and $R$.

Conjugation means an intertwiner, not componentwise conjugation

There are several operations that notation can obscure:

$$S \longrightarrow \overline S, \qquad S \longrightarrow S^*, \qquad S \longrightarrow \overline S^{\,*}.$$

Complex conjugation changes a representation to its conjugate. Transpose changes it to a dual matrix action. Hermitian adjoint combines the two. None is automatically an endomorphism of the original Spin module.

Hermitian adjoints and Dirac adjoints

A sesquilinear pairing can be represented in a basis as

$$h(\psi,\chi) = \psi^\dagger A\chi.$$

Spin invariance requires

$$\rho(g)^\dagger A\rho(g)=A.$$

The matrix $A$ is often called a hermitizing matrix. For a noncompact Lorentz group, $A$ need not be positive definite. Indeed, $\psi^\dagger\chi$ is not generally a Lorentz scalar in a finite-dimensional spinor representation.

Linear charge-conjugation intertwiners

A linear map from a dual or transpose representation back to a spinor representation is encoded here by a matrix $\mathcal C$. A commonly useful Clifford compatibility condition is

$$\mathcal C^{-1}\gamma(v)\mathcal C = -\gamma(v)^{\mathsf T}.$$

The sign on the right and the symmetry of $\mathcal C$ are not universal; both depend on dimension, signature, and which admissible pairing has been chosen. The equation, not the letter $\mathcal C$, identifies the convention.

Antilinear real and quaternionic structures

Let $K$ denote componentwise conjugation in a chosen basis. An antilinear Spin intertwiner has the form

$$J=BK, \qquad J\rho(g)=\rho(g)J.$$

Its square is basis independent:

$$J^2 = B B^*.$$

After normalization, three cases matter:

Representation type detected by an antilinear intertwiner.

Test	Type	Consequence
$J^2=+1$	Real	The fixed subspace $S^J$ is a real Spin module and $S^J\otimes_{\mathbb R}\mathbb C\cong S$
$J^2=-1$	Quaternionic	No nonzero vector obeys $J\psi=\psi$
No such $J:S\to S$	Complex	$S$ and $\overline S$ are inequivalent; a reality condition needs a larger module

A Majorana condition is the fixed-point condition

$$J\psi=\psi$$

when $J^2=+1$. If $J_S^2=-1$, an internal doublet with a quaternionic structure $j^2=-1$ makes

$$(J_S\otimes j)^2=+1.$$

Its fixed-point condition is a symplectic Majorana condition. It is a condition on a doubled representation, not a relabeling of an ordinary Majorana spinor.

In even dimension one must also test

$$J\Gamma_* = \epsilon_\chi\Gamma_*J, \qquad \epsilon_\chi\in\{+1,-1\}.$$

If $\epsilon_\chi=+1$, the reality structure preserves each chiral module and a Majorana–Weyl condition can be possible. If $\epsilon_\chi=-1$, it exchanges the two chiralities, so no nonzero spinor can obey both that Majorana condition and a Weyl condition. This two-sign test—$J^2$ and $\epsilon_\chi$—is safer than remembering a name from a dimension/signature table.

Choosing spinor structures by dimension and signature

Real Clifford algebras and their modules have an eightfold pattern, but a table is only useful after its conventions are matched. The following workflow survives changes of notation.

1. Fix $(p,q)$, the Clifford sign, the scalar field, orientation, and the exact Spin group or connected component.
2. Classify the complex Clifford module and distinguish restriction to the even algebra from a module of the full algebra.
3. In even dimension normalize the volume element and record the eigenvalue convention for $S_\pm$.
4. Solve the intertwiner equations for $A$, $\mathcal C$, or $J$ instead of assuming that componentwise conjugation returns to the same module.
5. Compute $J^2$ and, when chirality exists, the sign $J\Gamma_*J^{-1}\Gamma_*^{-1}$.
6. Construct invariant bilinear forms and determine their symmetry and Clifford-transpose signs.
7. Build an independent trace-dual gamma basis before deriving a Fierz rearrangement.

Stop if the dimension, signature, module, conjugation, gamma normalization, or spinor parity is still undeclared. Under those conditions a remembered reality label or Fierz coefficient is not a well-typed result.

The following cases illustrate the outputs; they are not a substitute for a complete periodicity table.

For a convention-translated summary, set

$$r=p-q\pmod 8.$$

The next table classifies the type of irreducible complex Spin representations for the real form $\operatorname{Spin}(p,q)$. It does not classify irreducible modules of the full real Clifford algebra. In even dimension, $S_\pm$ are the two irreducible half-spin representations; in odd dimension, $S$ is the irreducible complex Spin representation.

Real, complex, and quaternionic types of irreducible complex Spin representations.

$r=p-q$ modulo 8	Spin-representation type	Reality consequence
0	$S_+$ and $S_-$ are real	An ordinary Majorana–Weyl condition is possible
1	$S$ is real	An ordinary Majorana condition is possible
2	$S_+$ and $S_-$ are complex and mutually conjugate	A real structure exists on $S_+\oplus S_-$ and exchanges chirality
3	$S$ is quaternionic	A symplectic Majorana condition is possible after doubling
4	$S_+$ and $S_-$ are quaternionic	A symplectic Majorana–Weyl condition is possible after doubling
5	$S$ is quaternionic	A symplectic Majorana condition is possible after doubling
6	$S_+$ and $S_-$ are complex and mutually conjugate	A real structure exists on $S_+\oplus S_-$ and exchanges chirality
7	$S$ is real	An ordinary Majorana condition is possible

For even $d$, the chirality action is equivalently checked by

$$J\Gamma_*J^{-1} = (-1)^{(p-q)/2}\Gamma_*.$$

The sign is $+$ for residues $0$ and $4$, so the real or quaternionic structure preserves each half-spin representation. It is $-$ for residues $2$ and $6$, so the structure exchanges them. The classification and its domain are treated in Deligne 1999, Table 1.4.1, printed p. 103, PDF; Deligne’s $p$ counts positive and $q$ negative directions, exactly as on this page.

Three convention-fixed examples of the workflow.

Quadratic space	Complex spinors	Reality–chirality result
Lorentzian $(1,3)$ with $v^2=g(v,v)$	A four-complex-dimensional Dirac module splits into two two-complex-dimensional Weyl modules	An ordinary Majorana structure exists on the Dirac module but exchanges the Weyl modules; Majorana and Weyl cannot be imposed simultaneously
Euclidean $(4,0)$ with $v^2=g(v,v)$	The two Weyl modules transform separately under the two $SU(2)$ factors of $\operatorname{Spin}(4)$	Each Weyl module is quaternionic; an ordinary fixed-point Majorana condition fails, while a symplectic condition can be imposed after suitable doubling
Lorentzian $(1,9)$ with $v^2=g(v,v)$	A complex Dirac module has dimension $32$ and each Weyl module has dimension $16$	A real structure can preserve chirality, giving a 16-real-dimensional Majorana–Weyl module

These examples also show why “Wick rotate the Majorana condition” is not a valid rule. The real form has changed, so the intertwiner and its square must be recomputed. Van Proeyen’s Van Proeyen 1999, §§3.2–3.3 and Tables 1–2 give a systematic dimension/signature treatment. That source writes $\eta_{\mathrm{VP}}=\operatorname{diag}(-^t,+^s)$ and organizes reality by $s-t$. For the corresponding site signature, $p_{\mathrm{site}}=t$, $q_{\mathrm{site}}=s$, and therefore

$$r_{\mathrm{site}} = p_{\mathrm{site}}-q_{\mathrm{site}} = -(s-t)_{\mathrm{VP}}.$$

The complex map $\gamma_{\mathrm{site}}=i\Gamma_{\mathrm{VP}}$ matches the Clifford anticommutator but is not a real-Clifford-algebra isomorphism; the reality table still requires the residue conversion above. Van Proeyen also calls $(1+\Gamma_*)/2$ “left,” whereas this page fixes $P_L=(1-\gamma_5)/2$, so projector eigenvalues—not the words left and right—must be translated. Dreiner, Haber, and Martin’s Appendix G.3 instead writes $\eta=(+^t,-^s)$ directly, but its reality tables use $s-t=q-p=-r_{\mathrm{site}}$.

There is one further four-dimensional trap. A Majorana structure on the $\operatorname{Spin}^+(1,3)$ representation does not imply that its real fixed subspace is a module of the full real algebra $\operatorname{Cl}_{1,3}$. Spin-module type and full real Clifford-module type are different classifications.

Invariant bilinears and form-valued covariants

A bilinear form

$$\beta:S\times S\longrightarrow\mathbb C$$

is Spin invariant when

$$\beta\!\left(\rho(g)\psi,\rho(g)\chi\right) = \beta(\psi,\chi).$$

Equivalently, the map

$$\psi \longmapsto \beta(\psi,-)$$

is an intertwiner $S\to S^*$. Infinitesimally, if $\Sigma_{\mu\nu}$ denotes the spin generator,

$$\beta(\Sigma_{\mu\nu}\psi,\chi) + \beta(\psi,\Sigma_{\mu\nu}\chi) = 0.$$

An admissible pairing is further characterized by signs $\sigma,\tau\in\{\pm1\}$ such that

$$\begin{aligned} \beta(\chi,\psi) &= \sigma\,\beta(\psi,\chi),\\ \beta(\gamma(v)\psi,\chi) &= \tau\,\beta(\psi,\gamma(v)\chi). \end{aligned}$$

The signs are properties of the selected pairing, not of spinors in the abstract. They determine the exchange symmetry of the form-valued bilinears

$$\beta_k(\psi,\chi)_{\mu_1\ldots\mu_k} = \beta\!\left( \psi, \gamma_{\mu_1\ldots\mu_k}\chi \right).$$

For homogeneous spinors of Grassmann parities $|\psi|$ and $|\chi|$,

$$\begin{aligned} \beta_k(\chi,\psi) = {}& (-1)^{|\psi||\chi|} \sigma\tau^k (-1)^{k(k-1)/2} \beta_k(\psi,\chi). \end{aligned}$$

This formula separates three sources of signs: the matrix symmetry of the pairing, moving $k$ gamma matrices through the pairing, and exchanging Grassmann coefficients. It is the efficient way to decide whether an identical-spinor bilinear vanishes.

The Clifford covariance identity

$$\rho(g)\gamma(v)\rho(g)^{-1} = \gamma(\Lambda(g)v)$$

then shows that $\beta_k$ transforms as an antisymmetric rank-$k$ tensor. Calling it a scalar, vector, or tensor is a representation statement. Calling it a pseudoscalar or axial vector additionally records behavior under orientation-reversing transformations, which lie outside the connected Spin group.

Although a spinor representation does not descend through the kernel of $\operatorname{Spin}(p,q)\to SO(p,q)$, a bilinear does not see the central element:

$$\beta(\rho(-1)\psi,\rho(-1)\chi) = \beta(-\psi,-\chi) = \beta(\psi,\chi).$$

This cancellation is why tensor bilinears can transform under the determinant-one orthogonal image (or its connected component) even when each individual spinor detects the double cover. Extending the statement to orientation-reversing transformations requires compatible Pin data.

Four-dimensional Lorentzian convention card

Now specialize to

$$d=4, \qquad \eta=\operatorname{diag}(1,-1,-1,-1), \qquad \{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}.$$

Define

$$\gamma_5=i\gamma^0\gamma^1\gamma^2\gamma^3, \qquad \sigma^{\mu\nu} = \frac{i}{2}[\gamma^\mu,\gamma^\nu].$$

No gamma-matrix basis is preferred. Choose the standard hermitizing normalization

$$A=\gamma^0,$$

for which

$$(\gamma^\mu)^\dagger = \gamma^0\gamma^\mu\gamma^0.$$

If $S(\Lambda)$ is the spinor matrix covering $\Lambda\in SO^+(1,3)$, then

$$S(\Lambda)^\dagger\gamma^0 = \gamma^0S(\Lambda)^{-1}.$$

Therefore the Dirac adjoint

$$\overline\psi = \psi^\dagger\gamma^0$$

transforms as

$$\overline\psi \longmapsto \overline\psi\,S(\Lambda)^{-1}.$$

It follows immediately that $\overline\psi\chi$ is a Lorentz scalar and $\overline\psi\gamma^\mu\chi$ is a Lorentz vector. This is the role of $\gamma^0$ in the adjoint; the positive-definite expression $\psi^\dagger\chi$ does not have the required covariance.

Choose the four-dimensional charge-conjugation map $\mathcal C:S^*\to S$ by

$$\mathcal C^{-1}\gamma^\mu\mathcal C = -(\gamma^\mu)^{\mathsf T}, \qquad \mathcal C^{\mathsf T}=-\mathcal C.$$

Define the charge-conjugate spinor by

$$\psi^c = \mathcal C\,\overline\psi^{\,\mathsf T}.$$

Then $\psi^c$ transforms with the same $S(\Lambda)$ as $\psi$. The associated antilinear map is

$$\psi^c = B\psi^*, \qquad B=\mathcal C A^{\mathsf T}.$$

In four-dimensional $(1,3)$ signature it obeys

$$B B^*=\mathbf1, \qquad (\psi^c)^c=\psi.$$

This positive square is the real-type invariant; multiplying the antilinear map by a phase does not change it.

The four-component Majorana condition is

$$\psi^c=\psi.$$

These equations are basis covariant. Under $\gamma'^\mu=U\gamma^\mu U^{-1}$ and $\psi'=U\psi$,

$$\begin{aligned} A' &= (U^\dagger)^{-1}A U^{-1},\\ \mathcal C' &= U\mathcal C U^{\mathsf T}. \end{aligned}$$

Thus the transformed matrix $\gamma'{}^0=U\gamma^0U^{-1}$ and the hermitizing matrix $A'$ need not be the same under an arbitrary nonunitary similarity. With the correctly transformed $A'$, $\overline{\psi'}=\overline\psi U^{-1}$ and $(\psi')^c=U\psi^c$.

The matrix of the associated invariant complex bilinear is a different intertwiner:

$$C_\beta = (\mathcal C^{-1})^{\mathsf T}, \qquad \beta(\psi,\chi) = \psi^{\mathsf T}C_\beta\chi.$$

It transforms as

$$C_\beta' = U^{-\mathsf T}C_\beta U^{-1}.$$

In the common standard normalization $\mathcal C^2=-\mathbf1$ and $\mathcal C^{\mathsf T}=-\mathcal C$, the two matrices happen to have the same numerical entries, $C_\beta=\mathcal C$. Their categorical roles and general basis-transformation laws remain different.

Charge conjugation reverses four-dimensional chirality:

$$\begin{aligned} (P_L\psi)^c &= P_R\psi^c,\\ (P_R\psi)^c &= P_L\psi^c. \end{aligned}$$

Hence a nonzero four-dimensional Lorentzian spinor cannot be both Majorana and Weyl. A Majorana spinor can instead be built from one Weyl spinor and its charge conjugate:

$$\Psi_M = \psi_L+(\psi_L)^c.$$

It obeys $\Psi_M^c=\Psi_M$ but contains both chiralities.

The five four-dimensional bilinear types

The sixteen matrices

$$\mathbf1, \quad \gamma_5, \quad \gamma^\mu, \quad \gamma^\mu\gamma_5, \quad \sigma^{\mu\nu}\quad(\mu<\nu)$$

form a basis of $\operatorname{End}(\mathbb C^4)$. They give the standard Dirac bilinears.

The dimension count is

$$1+1+4+4+6 = 16 = \dim_{\mathbb C}\operatorname{End}(\mathbb C^4).$$

Four-dimensional bilinears in the (+---) convention.

Name	Bilinear	Proper-Lorentz type	Identical Grassmann-odd Majorana field
Scalar	$\overline\psi\psi$	Scalar	May be nonzero
Pseudoscalar	$\overline\psi\gamma_5\psi$	Scalar under the connected group; parity distinguishes it	May be nonzero
Vector	$\overline\psi\gamma^\mu\psi$	Vector	Zero
Axial vector	$\overline\psi\gamma^\mu\gamma_5\psi$	Vector under the connected group; parity distinguishes it	May be nonzero
Tensor	$\overline\psi\sigma^{\mu\nu}\psi$	Antisymmetric rank-two tensor	Zero

For a Majorana field, $\overline\psi=\psi^{\mathsf T}C_\beta$. The matrices $C_\beta$, $C_\beta\gamma_5$, and $C_\beta\gamma^\mu\gamma_5$ are antisymmetric, while $C_\beta\gamma^\mu$ and $C_\beta\sigma^{\mu\nu}$ are symmetric. Grassmann anticommutation therefore allows the first set and kills the second set for identical fields.

For two distinct Grassmann-odd Majorana spinors $\psi$ and $\chi$, the same calculation gives

$$\begin{array}{c|ccccc} \Gamma & \mathbf1 & \gamma_5 & \gamma^\mu & \gamma^\mu\gamma_5 & \sigma^{\mu\nu} \\ \hline \overline\chi\Gamma\psi & +\overline\psi\Gamma\chi & +\overline\psi\Gamma\chi & -\overline\psi\Gamma\chi & +\overline\psi\Gamma\chi & -\overline\psi\Gamma\chi \end{array}$$

These signs reverse in the appropriate places for commuting test spinors. That is why a bilinear table without a Grassmann-parity declaration is incomplete. Factors of $i$ are often inserted into the pseudoscalar or other bilinears to make Hermiticity manifest; those factors do not change the Spin-representation type.

Weyl selection rules

Because $\gamma^0$ anticommutes with $\gamma_5$,

$$\overline{\psi_L} = \overline\psi P_R, \qquad \overline{\psi_R} = \overline\psi P_L.$$

The matrices $\mathbf1$, $\gamma_5$, and $\sigma^{\mu\nu}$ commute with $\gamma_5$, whereas $\gamma^\mu$ and $\gamma^\mu\gamma_5$ anticommute with it. Hence:

• scalar, pseudoscalar, and tensor bilinears pair opposite chiralities;
• vector and axial-vector bilinears pair the same chirality.

For example,

$$\overline{\psi_L}\chi_L=0, \qquad \overline{\psi_L}\gamma^\mu\chi_R=0,$$

while $\overline{\psi_L}\chi_R$ and $\overline{\psi_L}\gamma^\mu\chi_L$ are not forced to vanish. These are algebraic chirality rules. Chirality is a Lorentz-representation grading; helicity is a momentum-dependent spin projection, and the two should not be identified outside their appropriate massless on-shell setting.

Fierz identities are completeness relations

Let $N=\dim_{\mathbb C}S$, and let $\{\Gamma_A\}$ be an independent basis of $\operatorname{End}(S)$. Choose its trace-dual basis $\{\Gamma^A\}$ so that

$$\operatorname{tr}(\Gamma_A\Gamma^B) = N\delta_A{}^B.$$

Then every endomorphism has the basis-independent expansion

$$M = \frac1N \Gamma_A\, \operatorname{tr}(\Gamma^A M).$$

For homogeneous spinors, apply this identity to the rank-one endomorphism $u\overline v$. Moving $\overline v$ past $u$ when taking the trace gives

$$u\overline v = \frac{(-1)^{|u||v|}}{N} \Gamma_A \left(\overline v\Gamma^A u\right).$$

This is the master Fierz identity. A Fierz rearrangement is obtained by inserting it between the remaining spinors. The sign is $-$ for two Grassmann-odd spinors and $+$ for two commuting spinors.

In even dimension a basis can be assembled from independent antisymmetrized gamma products. In odd dimension the volume element relates $k$-fold and $(d-k)$-fold products, so including both creates a redundant set. The trace-dual construction detects this problem: a redundant Gram matrix cannot be inverted.

In even dimension, multiplication by $\Gamma_*$ likewise relates $\gamma_{(k)}$ to the orientation-dual $\gamma_{(d-k)}$, with a phase fixed by the chosen volume element and epsilon convention. After restricting to a Weyl module, only the blocks that map the selected source chirality to the selected target chirality remain. A Weyl Fierz identity must therefore be derived in the relevant $\operatorname{Hom}(S_\pm,S_\pm)$ or $\operatorname{Hom}(S_\pm,S_\mp)$ space; it is not obtained by erasing terms from a Dirac identity without rechecking completeness.

Four-dimensional completeness

With the four-dimensional matrices defined above, trace orthogonality gives

$$\begin{aligned} \operatorname{tr}(\gamma^\mu\gamma^\nu) &= 4\eta^{\mu\nu},\\ \operatorname{tr}( \gamma^\mu\gamma_5\gamma^\nu\gamma_5) &= -4\eta^{\mu\nu},\\ \operatorname{tr}( \sigma^{\mu\nu}\sigma^{\rho\sigma}) &= 4\left( \eta^{\mu\rho}\eta^{\nu\sigma} - \eta^{\mu\sigma}\eta^{\nu\rho} \right). \end{aligned}$$

The second line fixes the minus sign in the axial trace-dual element. The last line fixes the tensor normalization. Thus

$$\begin{aligned} M = \frac14\Big[ &\mathbf1\,\operatorname{tr}(M) + \gamma_5\,\operatorname{tr}(\gamma_5M) \\ &+ \gamma^\mu\,\operatorname{tr}(\gamma_\mu M) - \gamma^\mu\gamma_5\, \operatorname{tr}(\gamma_\mu\gamma_5M) \\ &+ \frac12\sigma^{\mu\nu}\, \operatorname{tr}(\sigma_{\mu\nu}M) \Big]. \end{aligned}$$

The last line sums over all ordered $\mu,\nu$; its factor $1/2$ avoids double counting the six independent tensors. If the sum is restricted to $\mu<\nu$, that factor is absent.

For four Grassmann-odd fields in the displayed order, the scalar channel therefore rearranges as

$$\begin{aligned} (\overline\psi\chi)(\overline\lambda\rho) = -\frac14\Big[ & (\overline\psi\rho)(\overline\lambda\chi) + (\overline\psi\gamma_5\rho) (\overline\lambda\gamma_5\chi) \\ &+ (\overline\psi\gamma^\mu\rho) (\overline\lambda\gamma_\mu\chi) \\ &- (\overline\psi\gamma^\mu\gamma_5\rho) (\overline\lambda\gamma_\mu\gamma_5\chi) \\ &+ \frac12 (\overline\psi\sigma^{\mu\nu}\rho) (\overline\lambda\sigma_{\mu\nu}\chi) \Big]. \end{aligned}$$

For commuting spinors the overall minus sign becomes a plus sign. This single derivation explains the coefficient pattern and makes its hypotheses inspectable. The general completeness identity and its four-dimensional specialization are developed in Dreiner, Haber, and Martin 2010, Appendix G.1, especially G.1.100–G.1.104, while the elementary two-component exchange and Fierz signs appear in §2, equations (2.59)–(2.70), pp. 16–17.

Controlled QFT example: Majorana currents and four-fermion channels

Let $\psi$ be a classical Grassmann-odd Majorana field in four-dimensional Lorentz signature. From

$$(C_\beta\gamma^\mu)^{\mathsf T} = C_\beta\gamma^\mu$$

and $\overline\psi=\psi^{\mathsf T}C_\beta$,

$$\overline\psi\gamma^\mu\psi = \psi^{\mathsf T}C_\beta\gamma^\mu\psi = 0.$$

The diagonal vector current vanishes algebraically. The axial current $\overline\psi\gamma^\mu\gamma_5\psi$ need not vanish because $C_\beta\gamma^\mu\gamma_5$ is antisymmetric. Thus “a Majorana field has no bilinears” is false; the result is channel dependent.

For four distinct fermion fields, the scalar Fierz identity above rewrites one contraction pattern into scalar, pseudoscalar, vector, axial, and tensor channels with the second and fourth spinors exchanged. This is the algebraic step used when comparing four-fermion operator bases or changing a spinor-contraction channel. It says nothing by itself about equations of motion, operator independence after integration by parts, renormalization, or the value of an amplitude.

Quantum fields are operator-valued distributions. Coincident composite operators require a regulator and a renormalization prescription even when the formal spinor-index identity is exact. The construction of free Dirac, Majorana, and Weyl fields, their wave functions, spin sums, and physical bilinears continues in The Dirac Field and Plane Waves, Spin Sums, and Bilinears.

Convention translation and failure diagnosis

For an independent four-dimensional treatment of Lorentz spinors, the Dirac adjoint, chirality, and charge conjugation, see Tong 2006, §§4.1–4.5.

Do not identify transpose, conjugate, and adjoint

The matrices $\gamma^\mu{}^{\mathsf T}$, $\gamma^\mu{}^*$, and $\gamma^\mu{}^\dagger$ belong to different intertwiner equations. The charge map $\mathcal C$, bilinear matrix $C_\beta$, matrix $B$ in the antilinear map $BK$, and hermitizing matrix $A$ have different source and target spaces. Their matrices may coincide after a special normalization without becoming the same construction.

Do not impose Majorana and Weyl conditions from dimension alone

Even dimension permits chirality. It does not guarantee that a real structure exists or preserves a chiral summand. Compute both $J^2$ and $J\Gamma_*J^{-1}$.

Do not call $\psi^\dagger\chi$ a Lorentz scalar

Finite-dimensional Lorentz spinor representations are nonunitary. The invariant adjoint contains a hermitizing intertwiner, which is $\gamma^0$ in the four-dimensional convention used here.

Do not copy a Fierz table before reconstructing its trace dual

Changing any of the following can change signs or coefficients:

• metric signature or the sign in the Clifford relation;
• the phase of $\Gamma_*$ or $\gamma_5$;
• the definition of $\sigma^{\mu\nu}$;
• unit-weight versus unnormalized antisymmetrization;
• summing all tensor index pairs versus only independent pairs;
• the conjugation used in the bilinear;
• the order and Grassmann parity of the spinors.

The round-trip test is to reconstruct an arbitrary matrix from the proposed gamma basis. If the reconstruction fails, the imported Fierz identity has not been translated correctly.

The same warning applies to dimensional regularization. The sixteen-matrix identity is four-dimensional and must not be imported unchanged into $d=4-2\epsilon$. The dimension-specific failure and evanescent structures belong to Fierz Relations and Dimension-Specific Identities.

Do not confuse an algebraic identity with a dynamical reduction

Fierz completeness changes spinor contractions. Equations of motion, integration by parts, gauge-index identities, flavor symmetry, and renormalized operator relations are separate reductions.

Exercises

1. Distinguish Clifford and Spin irreducibility

For $d=2m$, explain why the unique irreducible complex module of $\operatorname{Cl}_{2m}^{\mathbb C}$ becomes reducible under the Spin action.

Solution

The normalized volume operator $\Gamma_*$ anticommutes with odd Clifford generators but commutes with the even algebra. Its two eigenspaces $S_\pm$ are therefore preserved by the even algebra and by Spin. Odd Clifford multiplication exchanges them, so neither eigenspace is a module for the full Clifford algebra. The full module can be irreducible while its Spin restriction is reducible.

2. Prove that vectors reverse chirality

Starting from $\{\Gamma_*,\gamma(v)\}=0$, show that $\gamma(v)P_\pm=P_\mp\gamma(v)$.

Solution

Using $P_\pm=(1\pm\Gamma_*)/2$,

$$\begin{aligned} \gamma(v)P_\pm &= \frac12\gamma(v)(1\pm\Gamma_*)\\ &= \frac12(1\mp\Gamma_*)\gamma(v)\\ &= P_\mp\gamma(v). \end{aligned}$$

Thus an odd Clifford element maps one half-spin module into the other.

3. Check covariance of the Dirac adjoint

Assume $S^\dagger\gamma^0=\gamma^0S^{-1}$. Derive the transformation law of $\overline\psi$ and prove that $\overline\psi\chi$ is invariant.

Solution

For $\psi'=S\psi$,

$$\overline{\psi'} = \psi^\dagger S^\dagger\gamma^0 = \psi^\dagger\gamma^0S^{-1} = \overline\psi S^{-1}.$$

If $\chi'=S\chi$, then

$$\overline{\psi'}\chi' = \overline\psi S^{-1}S\chi = \overline\psi\chi.$$

4. Diagnose the Majorana vector current

Let $\psi$ be Grassmann odd and suppose $(C_\beta\gamma^\mu)^{\mathsf T}=C_\beta\gamma^\mu$. Show directly that $\psi^{\mathsf T}C_\beta\gamma^\mu\psi=0$. What changes for commuting spinors?

Solution

Write $M=C_\beta\gamma^\mu$. Then

$$\psi_aM_{ab}\psi_b = -\psi_bM_{ab}\psi_a = -\psi_aM_{ba}\psi_b.$$

If $M^{\mathsf T}=M$, the expression equals its negative and vanishes. For commuting components the first minus sign is absent, so a symmetric matrix can contribute; the Grassmann-odd zero cannot be reused.

5. Recover the overall Fierz sign

Use matrix completeness on $u\overline v$ and explain why its coefficient is negative when both spinors are Grassmann odd.

Solution

Completeness gives

$$u\overline v = \frac1N \Gamma_A \operatorname{tr}(\Gamma^A u\overline v).$$

Inside the trace, writing the result as $\overline v\Gamma^A u$ moves $\overline v$ past $u$. The exchange gives $(-1)^{|u||v|}$. It is $-1$ for two Grassmann-odd spinors and $+1$ for two commuting spinors.

Where to continue

• For the first physical application to Dirac, Majorana, and Weyl field bilinears, continue to Plane Waves, Spin Sums, and Bilinears, after its prerequisite, The Dirac Field.
• This page supplies reusable algebraic machinery, not field equations, canonical anticommutators, propagators, external-state normalizations, or a theorem-first classification in every dimension and signature.
• Direct mathematical consumers include Fierz Relations and Dimension-Specific Identities, Supersymmetry Across Dimensions, Signatures, and Reality Conditions, and the component-multiplet closure page. Each destination states its additional prerequisites and physical or regularization scope.

References

• Pierre Deligne, Notes on Spinors, PDF, in Quantum Fields and Strings: A Course for Mathematicians, American Mathematical Society, 1999; official IAS publication record. Exact locators are Table 1.4.1, printed p. 103, for type by $p-q$; Table 1.5.1, p. 103, for complex pairing symmetry; §§4.8–4.9, pp. 121–123, for the $\operatorname{End}(S)$ and exterior-power decomposition; and Theorem 6.1, pp. 129–131, for the $(+,-,\ldots,-)$ specialization. This is a structural source for representation types, invariant pairings, and the decomposition underlying gamma completeness. Its ordinary representation-vector signs are not QFT Grassmann exchange signs. The official PDF is an image-only, untagged scan, so the metadata page and printed locators are supplied for verification and its tables are not reproduced here.
• Herbi K. Dreiner, Howard E. Haber, and Stephen P. Martin (2010), Two-component spinor techniques and Feynman rules for quantum field theory and supersymmetry, arXiv:0812.1594v6, §2, equations (2.59)–(2.70), and Appendices A, B.1, G.1, and G.3. These locations fix the (+---) convention, commuting-versus-anticommuting exchange signs, four-dimensional bilinears, Majorana swap rules, charge conjugation, and Fierz completeness. The authors’ errata and metric-convention page states that version 6 incorporates all known published-version corrections and lists no correction to v6.
• José Figueroa-O’Farrill (2015), Majorana Spinors, PDF, §§1 and 3–5. These notes support the complex Clifford classification and the invariant real/quaternionic-structure method. They describe themselves as preliminary and do not contain the planned full bilinear section, so bilinear and Fierz claims above are cross-checked against the other references rather than resting on these notes alone.
• David Tong (2006), Quantum Field Theory, §4, “The Dirac Equation”, especially §§4.1–4.5, Cambridge Part III lecture notes. This is the teaching source for Lorentz spinors, the Dirac adjoint, chiral decomposition, charge conjugation, and the bounded four-dimensional application. Tong defines $\gamma_5$ with the opposite sign, so his left/right projector labels have been interchanged to match the site’s declared $\gamma_5=+i\gamma^0\gamma^1\gamma^2\gamma^3$.
• Antoine Van Proeyen (1999), Tools for Supersymmetry, arXiv: hep-th/9910030v7, §3, especially §§3.1–3.4, equations (3.1)–(3.44), and Tables 1–2. These locations establish dimension/signature-dependent gamma bases, charge conjugation, reality, chirality, bilinear signs, and general Fierz completeness. Version 7 is cited because version 4 corrected the factor in Fierz equation (3.44) and version 6 corrected the sign in reality equation (3.25). Its timelike/spacelike convention, signature residue, and chirality labels are translated explicitly above.