Spinors, covers, and reality conditions

“Weyl,” “Dirac,” and “Majorana” specify different structures. Weyl selects a chiral representation; Dirac combines the two chiralities; Majorana imposes a compatible antilinear reality condition. Whether that condition exists depends on spacetime signature and on every additional representation carried by the field. An invariant antisymmetric tensor alone does not make a representation real.

This page treats four-dimensional spacetime and ordinary finite-dimensional complex field representations. Unless a Euclidean comparison is stated, the metric is $\eta=\operatorname{diag}(+1,-1,-1,-1)$ and the spacetime symmetry is the connected spin cover. Spinors appearing as local fields are Grassmann odd. Commuting external wave functions obey different exchange signs.

The Lorentzian spin cover and chirality

The connected Lorentz group has universal double cover

$$\operatorname{Spin}^{+}(1,3)\cong SL(2,\mathbb C) \longrightarrow SO^{+}(1,3), \qquad \ker=\{I,-I\}.$$

The source is a real six-dimensional group. Its finite-dimensional complex irreducibles have labels $(j_L,j_R)$ with $2j_L,2j_R\in\mathbb Z_{\geq0}$ and dimension $(2j_L+1)(2j_R+1)$. The labels refer to the two factors of the complexified Lie algebra, not two compact subgroups inside the Lorentz group. See real forms and the Lorentz spin map.

Let $S_L=(\tfrac12,0)$ and $S_R=(0,\tfrac12)$. Complex conjugation exchanges these inequivalent representations. We use undotted indices for $S_L$ and dotted indices for $S_R$; index heights are moved by the corresponding antisymmetric tensors. This is the two-component setup of Dreiner–Haber–Martin, §2.

Field representation	Lorentz labels	Independent components before equations of motion
Left Weyl	$(\tfrac12,0)$	$2$ complex
Right Weyl	$(0,\tfrac12)$	$2$ complex
Dirac	$(\tfrac12,0)\oplus(0,\tfrac12)$	$4$ complex
Majorana	A real form of the Dirac representation	$4$ real

A Majorana spinor has both chiral components, related by conjugation; it is not an extra complex irreducible representation. In particular, a nonzero four-dimensional Lorentzian spinor cannot be both Weyl and Majorana in this ordinary pointwise sense. These off-shell component counts are not counts of one-particle states.

The central $-I$ acts on $(j_L,j_R)$ by $(-1)^{2j_L+2j_R}$. Neither Weyl representation descends to an ordinary linear representation of $SO^{+}(1,3)$, whereas the vector $(\tfrac12,\tfrac12)$ does. Field indices use the spin cover. Unitary particle-state representations use the Poincaré group and momentum orbits; see spacetime representations.

A concrete Majorana convention

Use Pauli matrices $\sigma_i$, $\sigma^\mu=(I,\sigma_i)$ and $\bar\sigma^\mu=(I,-\sigma_i)$. In the chiral basis,

$$\gamma^\mu= \begin{pmatrix}0&\sigma^\mu\\\bar\sigma^\mu&0\end{pmatrix}, \qquad \gamma^5=i\gamma^0\gamma^1\gamma^2\gamma^3 =\begin{pmatrix}-I&0\\0&I\end{pmatrix}.$$

Then $\{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}I$. Set

$$C=i\gamma^2\gamma^0, \qquad \bar\Psi=\Psi^\dagger\gamma^0, \qquad \Psi^c=C\bar\Psi^T=i\gamma^2\Psi^*.$$

Here star means componentwise conjugation. The antilinear map $J\Psi=i\gamma^2\Psi^*$ commutes with the connected spin action and satisfies $J^2=1$. Thus the constraint $\Psi^c=\Psi$ is consistent and selects a real vector space of half the Dirac dimension. For example, with $\varepsilon=i\sigma_2=\begin{pmatrix}0&1\\-1&0\end{pmatrix}$ and two-component column $\chi$,

$$\Psi_M=\begin{pmatrix}\chi\\-\varepsilon\chi^*\end{pmatrix} \quad\text{satisfies}\quad\Psi_M^c=\Psi_M.$$

The epsilon implements the intertwiner from a conjugate column to the right-handed module; the displayed sign fixes our charge-conjugation phase. The gamma identities and Majorana definition are given in Dreiner–Haber–Martin, Appendix G.1, equations (G.1.1)–(G.1.2) and (G.1.11)–(G.1.22). Our $C$ is the negative of their numerical convention stated after (G.1.22). Such an overall phase can be absorbed by rephasing the Majorana condition; it must still be kept consistent in explicit component formulas.

An unconstrained Dirac field can instead be assembled from two independent left Weyl fields $\chi,\eta$ as $(\chi,-\varepsilon\eta^*)^T$. If $\chi$ transforms in a gauge representation $R$, taking $\eta$ in $\overline R$ makes both four-component blocks transform in $R$. Setting $\eta=\chi$ must also respect those gauge transformations; a Lorentzian reality condition does not remove this additional requirement.

Derive the identical-fermion sign

For the following contraction take $\epsilon_{12}=+1$ and write the two undotted components with upper indices. Define

$$B_{ab}=\epsilon_{\alpha\beta}\psi_a^\alpha\psi_b^\beta, \qquad \alpha,\beta\in\{1,2\},$$

where $a,b$ distinguish flavors and the fields are evaluated at the same point. Grassmann exchange and epsilon antisymmetry give

$$\begin{aligned} B_{ba} &=\epsilon_{\alpha\beta}\psi_b^\alpha\psi_a^\beta\\ &=-\epsilon_{\alpha\beta}\psi_a^\beta\psi_b^\alpha =-\epsilon_{\beta\alpha}\psi_a^\alpha\psi_b^\beta =B_{ab}. \end{aligned}$$

For one gauge-singlet Weyl field,

$$B_{aa}=\psi_a^1\psi_a^2-\psi_a^2\psi_a^1 =2\psi_a^1\psi_a^2,$$

which need not vanish. A square of one Grassmann component is zero; the product of two different components is not. Consequently

$$\mathcal L_m=-\frac12\left(M_{ab}B_{ab}+\mathrm{h.c.}\right)$$

uses a symmetric complex matrix $M_{ab}=M_{ba}$. The factor $1/2$ removes the double counting in the symmetric sum. With $f$ sterile Weyl flavors there are $f(f+1)/2$ independent complex bilinears before field redefinitions. This is the anticommuting specialization of Dreiner–Haber–Martin, equation (2.60), and the mass matrix in (3.2.1).

For commuting spinor columns, the Grassmann minus sign is absent and the same epsilon contraction is antisymmetric. Substituting a commuting wave-function identity into a local-operator calculation can therefore reverse the conclusion.

Gauge reality is a separate representation problem

For an irreducible unitary representation $R$ of a compact internal group, a group-equivariant antilinear map $J_R$ can square to $+1$ (real type) or $-1$ (quaternionic, also called pseudoreal, type). For complex type, $R$ is inequivalent to $\overline R$ and no such map exists. These are statements about the gauge representation, independent of spacetime chirality.

The fundamental of an internal $SU(2)$ is pseudoreal: $J_Ru=\varepsilon u^*$ has $J_R^2=-1$. It has no nonzero fixed real subspace, since $J_Ru=u$ would imply $-u=u$. A real adjoint representation and a pseudoreal doublet therefore cannot be interchanged when imposing an ordinary Majorana condition on a single irreducible gauge multiplet.

The same distinction appears directly in the mass bilinear. Let $K_{ij}$ be a gauge-invariant bilinear tensor with $K_{ji}=sK_{ij}$, $s=\pm1$, and define

$$\mathcal B_{ab}=K_{ij}\epsilon_{\alpha\beta} \psi_a^{\alpha i}\psi_b^{\beta j}.$$

The preceding exchange calculation now yields $\mathcal B_{ba}=s\mathcal B_{ab}$. A symmetric gauge tensor allows symmetric flavor coefficients. An antisymmetric gauge tensor requires antisymmetric flavor coefficients. In particular, one Weyl doublet of internal $SU(2)$ with no other charges has no such mass bilinear: its unique gauge epsilon has $s=-1$, so $\mathcal B_{aa}=0$. Two independent doublets can pair. This is a kinematic statement; the illustrative field content is not asserted to be anomaly free. See Dreiner–Haber–Martin, §3.2, equations (3.2.35)–(3.2.40).

For an unbroken $U(1)$, a single left Weyl field of charge $q$ has a bilinear of charge $2q$, so its Majorana mass requires $q=0$. Two left Weyl fields of opposite charges can instead form a Dirac mass. Additional scalars, symmetry breaking, or flavor structure change the problem; the representation and charges must be stated anew.

What changes in four Euclidean dimensions

For positive-definite four-dimensional geometry,

$$\operatorname{Spin}(4)=SU(2)_L\times SU(2)_R, \qquad SO(4)=\frac{SU(2)_L\times SU(2)_R}{\langle(-I,-I)\rangle}.$$

These are compact groups. The two Euclidean Weyl modules are $(\mathbf2,\mathbf1)$ and $(\mathbf1,\mathbf2)$. Unlike the Lorentzian case, complex conjugation preserves the isomorphism class of each chirality: the invariant epsilon acts within its own $SU(2)$ factor. The Euclidean spin covering fixes the diagonal kernel; Haber, Solution Set 4, pp. 7–8, equation (23) describes this cover. Its diagonal kernel can be checked in the quaternionic action $x\mapsto axb^{-1}$: fixing $x=1$ gives $a=b$, and fixing every quaternion then requires $a=b=\pm1$. See Garrett, §2.3, pp. 4–5.

On either Weyl module the equivariant antilinear map is

$$J_Eu=\varepsilon u^*, \qquad J_E^2=\varepsilon\varepsilon^*=-I.$$

Thus it is a quaternionic structure, not an ordinary reality condition. The two chiral modules are inequivalent, so an equivariant conjugation cannot simply exchange them as in Lorentzian signature. A single Euclidean Dirac module also has no ordinary $\operatorname{Spin}(4)$-equivariant real structure. This is the four-dimensional case of Dreiner–Haber–Martin, Appendix G.3, equations (G.3.7)–(G.3.10) and the discussion following them.

A doubled multiplet can change the answer. If an auxiliary two-dimensional space has its own antilinear map $J_F^2=-1$, then on $S_L\otimes F$ the combined map satisfies

$$(J_E\otimes J_F)^2=(-1)(-1)=+1.$$

Its fixed points give a symplectic reality condition, compatible with the extra-index transformations. This is the representation-theoretic reason symplectic Majorana conditions require additional structure; see Dreiner–Haber–Martin, equations (G.3.16)–(G.3.20). It is not a reality condition on one undoubled Euclidean Weyl spinor.

These statements concern pointwise representations. They do not forbid a Euclidean formulation of a Lorentzian theory containing Majorana fermions. Analytic continuation and the conjugation or reflection operation of the functional integral must be specified separately; one cannot transfer the Lorentzian component constraint unchanged.

Connect the indices to an operator calculation

For a left Weyl field with gauge representation $R$ and flavor space $F$, the complete component space is $S_L\otimes R\otimes F$. Repeating the same Grassmann field twice uses

$$\Lambda^2(S_L\otimes R\otimes F),$$

before projecting to Lorentz and gauge singlets. It does not use separate exterior powers of the spin and gauge factors. That single operation reproduces both the nonzero sterile bilinear and the vanishing one-flavor internal-doublet bilinear.

The fields, Lorentz symmetry, and statistics lesson develops this counting method, and the local-operator workbench implements bounded derivative-free complex invariant counts. Its compact Lorentz projection is an algebraic device for the same complexified invariants, not a change to Euclidean spacetime or an automatic Majorana constraint. The Weinberg operator supplies a worked example with two Weyl leptons, commuting Higgs fields, gauge contractions, and symmetric flavor coefficients. A count of bilinears or invariant operators is not yet a count of physical masses after flavor-basis changes.