The Fierz–Kofink Identities and the Classification of Spinors

Introduction

A Dirac spinor carries, besides its two complex components, a collection of tensor densities built from it and its adjoint: a scalar, a pseudoscalar, a vector, an axial vector and an antisymmetric tensor. These are the bilinear covariants, sixteen real densities in all. They are not independent. They satisfy a system of quadratic relations — the Fierz–Kofink identities, also called the Fierz–Penrose–Kofink or FPK identities — which are the four-dimensional completion of the internal Fierz identity of Bilinear Operators on the Biquaternion Algebra with Hermitian Adjoint. That article proves the internal, two-dimensional statement and records explicitly that the ambient four-dimensional rearrangement "is a statement about the ambient algebra and is not proved here". This article supplies it.

The identities make possible a classification of spinors by their covariants, due to Lounesto. Lounesto's result is that there are exactly six possible configurations, gathered into three classes of "regular" spinors (the Dirac electron and its relatives) and three of "singular" spinors, with the singular ones carrying a light-like vector. The classification is finite because the FPK identities constrain the covariants so tightly that only six configurations satisfy them. The device that turns the identities into the classification, and whose name the literature borrows, is the Fierz aggregate or boomerang: a single Clifford number built from the covariants, which satisfies a short system of identities and from which the original spinor can be recovered.

This article is pure Clifford-algebra mathematics in the corpus's signature, and it is the one article of the pair drawn from the Fauser paper's aside on Lounesto's programme rather than from its main argument. Its place in the corpus is fixed by the gap the internal Fierz article names: the corpus has the two-dimensional Fierz identity and the four bilinear covariants of the biquaternion spinor module; it does not have the sixteen Dirac covariants in four dimensions, the FPK identities they satisfy, or the classification those identities force. The article also records, with the corpus's usual discipline, which statements are recomputed here and which are the source's.

The Bilinear Covariants

Let the spinor space be $\mathbb C^4$ with the Clifford representation of $\mathrm{Cl}_{1,3}$, $\gamma_\mu\gamma_\nu+\gamma_\nu\gamma_\mu=2\eta_{\mu\nu}$, $\eta=\mathrm{diag}(+1,-1,-1,-1)$, in the Dirac representation

$$ \gamma_0=\begin{pmatrix} I_2 & 0 \\ 0 & -I_2 \end{pmatrix},\qquad \gamma_k=\begin{pmatrix} 0 & \sigma_k \\ -\sigma_k & 0 \end{pmatrix}\ (k=1,2,3), $$

with $\sigma_k$ the Pauli matrices, and pseudoscalar

$$ \gamma_5 = i\,\gamma_0\gamma_1\gamma_2\gamma_3 = \begin{pmatrix} 0 & I_2 \\ I_2 & 0 \end{pmatrix}, \qquad \gamma_5^2=\mathbb 1,\qquad \gamma_5^{\dagger}=\gamma_5 . $$

The corpus's Clifford metric is the same mostly-minus form (Conventions in the Biquaternion Universe), so the conventions agree up to the usual basis freedom, which changes no identity below.

Definition. For a Dirac spinor $\psi\in\mathbb C^4$, with the Dirac adjoint $\bar\psi=\psi^{\dagger}\gamma_0$, the bilinear covariants are

$$ \sigma=\bar\psi\psi,\qquad \omega=i\,\bar\psi\gamma_5\psi,\qquad J_\mu=\bar\psi\gamma_\mu\psi,\qquad K_\mu=-\bar\psi\gamma_5\gamma_\mu\psi,\qquad S_{\mu\nu}=\bar\psi\,i\gamma_\mu\gamma_\nu\,\psi . $$

The names are the standard ones: $\sigma$ is the scalar density, $\omega$ the pseudoscalar, $J$ the (vector) current, $K$ the axial (pseudo-vector) current, and $S$ the antisymmetric tensor (the spin density). The five objects comprise $1+1+4+4+6=16$ real densities, a basis of the space of sesquilinear forms on $\mathbb C^4$: the sixteen Clifford blades $\mathbb 1,\gamma_\mu,\gamma_{\mu\nu},\gamma_{\mu\nu\rho},\gamma_5\gamma_\mu,\gamma_5$ are a basis of $M_4(\mathbb C)$, so every sesquilinear form $\psi\mapsto\phi^{\dagger}M\psi$ is a combination of the covariants with coefficients the entries of $M$. This is the completeness of the action, the ambient form of the corpus's internal Fierz identity: the map $\mathbb B\to\operatorname{End}_{\mathbb C}(S)$ is an isomorphism in the internal case, and in the ambient case the sixteen blades are an isomorphism of $M_4(\mathbb C)$ onto the forms.

Remark (the corpus's internal version). Bilinear Operators on the Biquaternion Algebra with Hermitian Adjoint proves the internal statement: the map $L\colon\mathbb B\to\operatorname{End}_{\mathbb C}(S)$ given by left multiplication on the minimal left ideal $S=\mathbb B\tilde\Pi_1$ is an isomorphism, the four blades being a basis of the four-dimensional endomorphism space, and the four bilinear covariants $b_\mu=(s,e_\mu t)$ a basis of the forms. The present article is the four-dimensional completion: sixteen blades, sixteen covariants. The two are the same statement in dimensions two and four, and the internal one is the case the corpus has already verified.

The Fierz–Kofink Identities

Theorem (the scalar FPK identities). For every Dirac spinor $\psi$, with $J^2=J_\mu J^\mu=\eta^{\mu\nu}J_\mu J_\nu$ and similarly for $K$ and $J\cdot K$,

$$ J^2=\sigma^2+\omega^2,\qquad J^2=-K^2,\qquad J\cdot K=0 . $$

Proof. Each is a quadratic identity in the components of $\psi$, obtained by expanding the left-hand side with the Clifford relations and collecting terms; the three were verified numerically on $100$ random complex spinors, in the representation and with the covariants displayed above, with residuals at machine precision. The identity expresses two facts at once: the square of the current equals the sum of the squares of the two scalars, and the axial current is spacelike (or zero) exactly when the current is timelike, their squares being opposite.

Remark (the rest of the FPK system). The three scalar identities above are not the whole system. The full FPK relations also include tensor identities, which express the antisymmetric combination $J_\mu K_\nu-J_\nu K_\mu$ in terms of the tensor $S$ and its Hodge dual, with coefficients built from $\sigma$ and $\omega$. These are quoted in the modern literature (see Further Reading) with normalisations that depend on the index positions, on the signature and on the sign convention for $\gamma_5$. The corpus verified the three scalar identities numerically and did not reproduce the tensor ones in its own conventions; the article therefore states their existence and does not display a formula it has not checked. Fixing the tensor identities in the corpus's conventions is recorded as an open question.

Remark (the identities constrain the covariants, not the spinor). The three scalar identities involve only the five covariants, not $\psi$ directly. They are the first sign that the covariants cannot be chosen freely: whatever their origin in a spinor, a quadruple $(\sigma,\omega,J,K)$ that violates $J^2=\sigma^2+\omega^2$ cannot come from any spinor. The classification of the next section is the complete exploitation of this constraint.

The Fierz Aggregate and the Boomerang

Definition (the aggregate). The Fierz aggregate of $\psi$ is the Clifford number

$$ Z = \sigma\,\mathbb 1 + J + iS + K\gamma_5 - i\omega\gamma_5, $$

where $J=J_\mu\gamma^\mu$, $K=K_\mu\gamma^\mu$ and $S=S_{\mu\nu}\gamma^{\mu\nu}$ are read as Clifford elements with the covariant coefficients.

Statement (the aggregate form of the FPK identities). The aggregate satisfies

$$ Z^2=4\sigma Z,\qquad Z\gamma_\mu Z=4J_\mu Z,\qquad Z i\gamma_5 Z=4\omega Z,\qquad Z\gamma_5\gamma_\mu Z=4K_\mu Z,\qquad Z i\gamma_\mu\gamma_\nu Z=4S_{\mu\nu}Z . $$

Statement (the inversion theorem). If $\xi$ is a fixed spinor with $\xi^{\dagger}\gamma_0\psi\neq0$, then $\psi$ is recovered from its aggregate by $\psi=Z\xi$. The aggregate therefore returns from the tensor densities to the spinor; this is why Lounesto named it the boomerang.

The two statements are quoted from the literature (see Further Reading) and are not recomputed here; the normalisation of $Z$ is fixed by the factor $4$ in $Z^2=4\sigma Z$, and the corpus did not succeed in matching the literature's sign conventions quickly enough to assert the identity. The boomerang is the conceptual device that motivates the classification and it is recorded for that reason, with the corpus's uncertainty marked.

Remark (why the classification is finite). The aggregate form of the identities shows why there are only finitely many types. The single Clifford number $Z$ must satisfy the polynomial relation $Z^2=4\sigma Z$, which is a very strong condition on a sixteen-component object; the other relations then determine the covariants from $Z$. The solutions fall into finitely many families, and those families are the classes.

The Classification of Spinors

Statement (Lounesto's classification). Subject to the FPK identities, the covariants of a nonzero spinor with $J\neq0$ fall into exactly six classes:

Class	Conditions	Name
1	$\sigma\neq0$, $\omega\neq0$	Dirac (regular, both scalars)
2	$\sigma\neq0$, $\omega=0$	regular, purely scalar
3	$\sigma=0$, $\omega\neq0$	regular, purely pseudoscalar
4	$\sigma=0$, $\omega=0$, $K\neq0$, $S\neq0$	flag-dipole
5	$\sigma=0$, $\omega=0$, $K=0$, $S\neq0$	flagpole
6	$\sigma=0$, $\omega=0$, $K\neq0$, $S=0$	dipole

Classes $1$–$3$ are the regular spinors, those with at least one of $\sigma,\omega$ nonzero; classes $4$–$6$ are the singular spinors, with $\sigma=\omega=0$. The vector $J$ is timelike in the regular classes and light-like in the singular ones, by the identity $J^2=\sigma^2+\omega^2$. The classification is due to Lounesto and is quoted here; the corpus verified that each class is non-empty by exhibiting a representative. The names in the last column are Lounesto's for the three singular classes; the two regular classes $2$ and $3$ carry no standard physical name, and the physical labels that are sometimes attached to them (Weyl, Majorana) depend on the convention for $\gamma_5$ and on which scalar is set to zero, so the article keeps the algebraic description. The singular classes are likewise permuted between sources: the flag-dipole with both $K$ and $S$ nonzero is unambiguous, but the names flagpole and dipole are exchanged in some of the literature, and the algebraic conditions are the safe identifier.

Proposition (all six classes are non-empty). In the Dirac representation and with the covariants above, the following spinors are representatives:

$$ \begin{aligned} \text{class 1:}&\quad \psi=\frac{1}{\sqrt3}\,(1,\,i,\,0,\,1)^{\mathsf T} &&(\sigma=\tfrac13,\ \omega=\tfrac23),\\ \text{class 2:}&\quad \psi=(1,\,0,\,1/\sqrt2,\,0)^{\mathsf T} &&(\sigma=\tfrac12,\ \omega=0),\\ \text{class 3:}&\quad \psi=\frac{1}{\sqrt2}\,(1,\,0,\,i,\,0)^{\mathsf T} &&(\sigma=0,\ \omega=-1),\\ \text{class 4:}&\quad \psi=\frac{1}{\sqrt6}\,(1,\,1+i,\,1,\,-1-i)^{\mathsf T} &&(\sigma=\omega=0,\ K\neq0,\ S\neq0),\\ \text{class 5:}&\quad \psi=\frac{1}{\sqrt2}\,(1,\,0,\,0,\,1)^{\mathsf T} &&(\sigma=\omega=0,\ K=0,\ S\neq0),\\ \text{class 6:}&\quad \psi=\frac{1}{\sqrt2}\,(1,\,0,\,1,\,0)^{\mathsf T} &&(\sigma=\omega=0,\ K\neq0,\ S=0). \end{aligned} $$

Proof. Each representative was computed numerically and its covariants evaluated. The two scalars and the class membership are as displayed; for the singular classes the distinction is between the vanishing of the vector $K$ and the vanishing of the tensor $S$, not of their norms. Class 5 is $(1,0,0,1)^{\mathsf T}/\sqrt2$, the diagonal spinor with equal and opposite space components; there $\sigma=\lvert\chi\rvert^2-\lvert\phi\rvert^2=0$ and $\omega=-2\operatorname{Im}(\chi^{\dagger}\phi)=0$ because $\chi^{\dagger}\phi=0$ is real, while $K$ vanishes identically and $S$ does not. Class 6 is $(1,0,1,0)^{\mathsf T}/\sqrt2$, the spinor with equal upper and lower two-spinors; there $\sigma=0$ and $\omega=0$, $S$ vanishes identically, and $K$ does not — it is light-like, $K^2=0$, which is why it is invisible to the norm $K^2$ and visible only to the vector itself. Class 4 is the least obvious: $(1,1+i,1,-1-i)^{\mathsf T}/\sqrt6$ has $\sigma=\omega=0$ with both $K$ and $S$ nonzero, and it is the type that the special choices $\phi=\pm\chi$ and $\phi\perp\chi$ do not reach.

Remark (the degenerate choices). The three singular classes are easy to confuse, because the families $\phi=\chi$, $\phi=-\chi$, $\phi\perp\chi$ and $\phi=\pm i\chi$ produce class 6, class 6, class 5 and (with $\sigma\neq0$) class 3 respectively, while class 4 is reached only by a genuinely generic phase. The special spinor $(1,0,1,0)^{\mathsf T}$ has $K\neq0$ but $K^2=0$, so a check that tests $\lvert K\rvert^2$ rather than the vector $K$ will misreport it as class 5; the vector test is the correct one. This is recorded because it is exactly the trap a later editor would fall into.

Remark (the flag-dipole is the type Fauser singles out). The class $4$ spinors — $\sigma=\omega=0$ with both $K$ and $S$ nonzero — are the type that the usual menagerie of Dirac, Majorana and Weyl spinors does not exhaust, and they are the reason the classification is more than a relabelling of the known spinors. In the modern literature they are called type-4 or flag-dipole spinors, and they include the Elko (mass-dimension-one) spinors. This is the class Fauser, citing Lounesto, describes as "a new unknown type of spinor", and the one that the Fierz aggregate was introduced to expose.

Remark (where the corpus's spinors sit). The corpus's Dirac field is a general element of $\mathbb B$ on eight real parameters, and a general element corresponds to a spinor of class 1: for a generic field both $\sigma$ and $\omega$ are nonzero. The corpus's massless Weyl spinors are not in any of the six classes as written, because they are not four-component spinors with the same dual; the massless limit of a massive Dirac spinor is class 1 with $J$ light-like. The corpus's minimal-left-ideal spinors are the two chiral halves, and the classification applies to their four-component recombination. The article records this so that the six classes are not mistaken for six kinds of corpus field.

Relation to the Corpus

The internal and the ambient Fierz identities. The corpus's internal Fierz identity says that the left action $L\colon\mathbb B\to\operatorname{End}_{\mathbb C}(S)$ is an isomorphism, so every endomorphism of the two-dimensional module is a left multiplication and the four blades are a basis. The ambient identity of this article says that the sixteen Clifford blades are a basis of $M_4(\mathbb C)$, so every sesquilinear form on the four-dimensional spinor space is a combination of the sixteen covariants. The two are the same theorem in dimensions two and four; the internal one is the case the corpus has already verified, and the ambient one is the statement that Bilinear Operators on the Biquaternion Algebra with Hermitian Adjoint records as "a statement about the ambient algebra and is not proved here". This article supplies it.

The classification and the corpus's idempotents. The corpus builds spinors as minimal left ideals $\mathbb B\tilde\Pi_1$, generated by primitive idempotents. The Lounesto classification is a classification of four-component spinors by their tensor densities, which is a different slicing of the same space: the idempotent fixes the module and the chirality, while the covariants fix the type. The two are compatible — every Lounesto class contains idempotent-generated spinors — and the relation between the primitive idempotent and the class is not recorded in the corpus and is left to the open questions.

The classification and the corpus's discrete symmetries. The discrete group $D=\Gamma_{1,3}/\Gamma^+_{1,3}$ of Parra's Four Options of the Dirac Equation and the Discrete Symmetries acts on the spinor and therefore on its covariant tuple, each symmetry acting on $(\sigma,\omega,J,K,S)$ by a definite $\pm$ pattern. Which classes those transformations permute, and which they fix, is not recorded in the corpus and is a natural question; the point of this remark is only that the six classes are permuted by the discrete symmetries and are not a fixed labelling of the corpus's fields.

Open Questions

The tensor identities. The corpus did not reproduce the tensor part of the FPK system — the expression of $J_\mu K_\nu-J_\nu K_\mu$ through $S$ and its Hodge dual — in its own conventions; a first fit attempt did not converge, which is why the article states only that the identities exist. Fixing the sign and the index positions in the signature the corpus uses for the Clifford metric is the open task.
The aggregate normalisation. The FPK identities in aggregate form, $Z^2=4\sigma Z$ and the rest, were quoted in the literature's normalisation and not recomputed. Fixing the aggregate $Z=\sigma+J+iS+K\gamma_5-i\omega\gamma_5$ against the corpus's covariants would settle it; the corpus records the uncertainty.
Where the corpus's fields sit. A general biquaternion field is class 1. Do the corpus's plane-wave solutions, its chiral pair, and its retired antilinear-mass equation fall into definite classes, and is the class of a solution invariant under the corpus's discrete symmetries?
Idempotents and classes. Does a primitive idempotent $\tilde\Pi$ of $\mathbb B$ determine a Lounesto class for the spinor $\mathbb B\tilde\Pi$, and if so which? The classification slices the spinor space one way and the ideals another; the map between the two slicings is not written down.
Class 4 and the framework. The flag-dipole class is the "new type" the classification exposes and the locus of the Elko and mass-dimension-one spinors. Does the corpus's construction reach it — that is, is there an element of $\mathbb B$, or a natural idempotent, whose four-component recombination is of class 4?
The fifteen-dimension count. The covariants span the sixteen-dimensional space of sesquilinear forms; the classification is a statement about the image of the spinor under the sixteen covariants modulo the FPK relations. Is there a corpus statement about the dimension of that image — sixteen minus the number of independent FPK relations — that the classification would sharpen?

Summary

A Dirac spinor carries sixteen bilinear covariants — a scalar, a pseudoscalar, a vector, an axial vector and an antisymmetric tensor — which are not independent but satisfy the Fierz–Kofink identities. Three of these are scalar and were verified on $100$ random spinors: $J^2=\sigma^2+\omega^2$, $J^2=-K^2$, $J\cdot K=0$. The system also has tensor components, which the article states but does not display, since their normalisation was not reproduced here. The identities are equivalently expressed through the Fierz aggregate $Z=\sigma+J+iS+K\gamma_5-i\omega\gamma_5$, which satisfies $Z^2=4\sigma Z$ and its companions and from which the spinor is recovered, $=\ Z\xi$ — Lounesto's boomerang. The identities force the covariants into exactly six classes, three regular ($\sigma$ or $\omega$ nonzero) and three singular ($\sigma=\omega=0$, with the possible vanishings of $K$ and $S$): Dirac, the two other regular types, the flag-dipole, the flagpole and the dipole. All six classes were verified to be non-empty by explicit representatives. The flag-dipole is the type the usual Dirac–Weyl–Majorana menagerie does not exhaust and the one the classification was introduced to expose. This article supplies the four-dimensional completion of the corpus's internal Fierz identity, which the bilinear article records as missing, and it states the classification in the corpus's signature, marking clearly which statements are recomputed here and which are Lounesto's. It is pure Clifford-algebra mathematics, with no framework speculation, and it is the ambient companion of the two-dimensional Fierz identity the corpus already carries.

Summary of Notation

Symbol	Meaning
$\gamma_\mu$, $\gamma_\mu\gamma_\nu+\gamma_\nu\gamma_\mu=2\eta_{\mu\nu}$	Clifford generators, $\eta=\mathrm{diag}(+1,-1,-1,-1)$
$\gamma_5=i\gamma_0\gamma_1\gamma_2\gamma_3$	Pseudoscalar
$\bar\psi=\psi^{\dagger}\gamma_0$	Dirac adjoint
$\sigma=\bar\psi\psi$, $\omega=i\bar\psi\gamma_5\psi$	Scalar and pseudoscalar densities
$J_\mu=\bar\psi\gamma_\mu\psi$	Vector current
$K_\mu=-\bar\psi\gamma_5\gamma_\mu\psi$	Axial current
$S_{\mu\nu}=\bar\psi\, i\gamma_\mu\gamma_\nu\,\psi$	Antisymmetric tensor density
$J^2=\sigma^2+\omega^2=-K^2$, $J\cdot K=0$	The scalar FPK identities
$Z=\sigma+J+iS+K\gamma_5-i\omega\gamma_5$	Fierz aggregate (boomerang)
$Z^2=4\sigma Z$	The aggregate form of the identities
$\psi=Z\xi$	The inversion theorem
flag-dipole, flagpole, dipole	The three singular classes $4,5,6$