Weyl spinor

A left- or right-handed Weyl spinor transforms in the Lorentz representation $(1/2,0)$ or $(0,1/2)$. Parity exchanges the two chiralities.

A spacetime field valued in one irreducible chiral spinor representation. A free massless left-handed Weyl field obeys $i({\partial }_t-\mathbf{\sigma }\cdot \nabla ){\chi }_L=0$. The left-handed massless plane-wave spinor and helicity content of a quantized Weyl field describe its modes and particle states. Field chirality fixes one spinor representation; it does not give the same particle and antiparticle helicity.

For mostly-minus metric signature, the Weyl sigma matrices are ${\sigma }^{\mu }=(I,{\sigma }_1,{\sigma }_2,{\sigma }_3)$ and ${\bar{\sigma }}^{\mu }=(I,-{\sigma }_1,-{\sigma }_2,-{\sigma }_3)$, where ${\sigma }_i$ are the Pauli matrices. The Pauli matrix multiplication law provesA real Lorentz four-vector is equivalently a Hermitian matrix $X=x^{0}I+x^i{\sigma }_i$ with $\det X=(x^0{)}^2-|\mathbf{x}{|}^2$. The transformation $X\mapsto MX{M}^{†}$ for $M\in \mathrm{SL}(2,\mathbb{C})$ preserves this determinant and gives the Lorentz spinor double cover. The indices ${\sigma }_{\alpha \dot{\beta }}^{\mu }$ therefore convert a vector index into one undotted and one dotted Weyl spinor index. They are the invariant tensors in the mixed supercharge anticommutator $\{Q_{\alpha },{\bar{Q}}_{\dot{\beta }}\}=2{\sigma }_{\alpha \dot{\beta }}^{\mu }{P}_{\mu }$.

A left-handed massless Weyl spinor has a positive-energy plane wave transforming by $e^{i\alpha /2}$ under a rotation about its momentum. Its particle has helicity $-1/2$. The negative-frequency term multiplies an antiparticle fermionic creation operator, so its corresponding one-particle state carries the conjugate rotation phase and helicity $+1/2$. Thus a left-handed field has one particle and one antiparticle helicity; the right-handed field interchanges the signs. A full massless Dirac field contains both chiral fields and hence both particle helicities.

In the standard Weyl representation of the gamma matrices and mostly-minus metric signature, the left chirality component obeys $i({\partial }_t-\mathbf{\sigma }\cdot \nabla ){\chi }_L=0$. A plane wave with $p^0>0$ exists precisely on the massless mass shell $p^0=|\mathbf{p}|$. Its coefficient is a negative helicity eigenspinor. For direction angles $\vartheta ,\phi$, one normalized choice isThe nonzero solution is unique up to overall normalization and phase. Different patches are required for a continuous phase choice over all directions; the direction of zero momentum is undefined.

With $K_i=M_{0i}$ and $M_{\mu \nu }=i{\gamma }_{\mu \nu }/2$ in the supplied mostly-minus index convention, the two Weyl spinor blocks have equal rotation generators $J_i={\sigma }_i/2$ and opposite boost generators. A complex-linear intertwiner must commute with every SU(2) rotation and hence be scalar by the Schur lemma, but no nonzero scalar intertwines the opposite boosts. This proves inequivalence of the two complex representations; parity exchanges them.

For odd Grassmann variables, $(\chi \psi )={\chi }^{\alpha }{\psi }_{\alpha }$ is symmetric under exchanging the two spinors: the Grassmann sign cancels the sign of the antisymmetric epsilon contraction. With ${ϵ}^{12}=1$, lower components $\chi =(a,b)$ and $\psi =(c,d)$ give $(\chi \psi )=bc-ad$. Commuting numerical spinors instead give an antisymmetric contraction.

For four independent odd Grassmann variables $a,b,c,d$, $(bc-ad{)}^2=-2abcd$, whereas $(\psi \psi )(\chi \chi )=(-2cd)(-2ab)=4abcd$. This proves the displayed two-component identity and fixes its sign. It is not valid with the same interpretation for commuting spinors, whose self-contractions vanish.

Chirality distinguishes the two irreducible Weyl representations of the proper Lorentz group. For a massless particle it agrees with helicity up to the particle-antiparticle convention, while a mass term couples opposite chiralities.

The projectors $P_L=(1-{\gamma }^5)/2$ and $P_R=(1+{\gamma }^5)/2$ select the two chirality eigenspaces of a Dirac field. They satisfy $P_L+P_R=I$, $P_L{P}_R=0$, and $P_L^2=P_L$, $P_R^2=P_R$.

With Minkowski metric signature $+---$ and $\mathrm{tr}({\gamma }^{\mu }{\gamma }^{\nu }{\gamma }^{\rho }{\gamma }^{\sigma }{\gamma }_5)=4i{ϵ}^{\mu \nu \rho \sigma }$, define $H^{\mu \nu }(a,b)=a^{\mu }{b}^{\nu }+a^{\nu }{b}^{\mu }-g^{\mu \nu }a\cdot b+i{ϵ}^{\mu \nu \rho \sigma }{a}_{\rho }{b}_{\sigma }$. Then $\mathrm{tr}[/a{\gamma }^{\mu }(1-{\gamma }_5)/b{\gamma }^{\nu }(1-{\gamma }_5)]=8H^{\mu \nu }(a,b)$. The symmetric-tensor contraction gives twice the sum of the two possible dot-product pairings; the Levi-Civita symbol contraction, including the two factors of $i$, gives twice their difference. Their sum is the displayed identity, explaining the single pairing in left-chiral weak rates.

A vectorlike gauge spectrum pairs complex gauge group representations $R$ and $\bar{R}$ among independent left-handed Weyl spinors. Such a pair permits a gauge-invariant fermion mass and occurs in a full charged hypermultiplet of four-dimensional $\mathcal{N}=2$ extended supersymmetry.

A chiral gauge spectrum has independent left-handed Weyl spinors in complex gauge group representations without an obligatory conjugate pair of left-handed Weyl spinors. The right-handed antiparticles required by the CPT theorem do not themselves make a spectrum vectorlike.

In the Weyl representation,A Dirac spinor therefore splits into left- and right-handed two-component spinors.

A Majorana spinor equals its charge conjugate. In the Weyl representation it can be written ${\psi }_M=(u_L,i{\sigma }^2{u}_L^{\ast }{)}^T$ from one left-handed Weyl spinor.

For a Majorana spinor $\lambda$ with odd Grassmann variables as components, ${\lambda }^{T}M\lambda$ sees only the antisymmetric part of $M$. With $C^T=-C$ and $({\gamma }^a{)}^T=-C{\gamma }^a{C}^{-1}$, reversing an antisymmetrized gamma product gives the displayed transpose rule. Thus the same-spinor vector and bivector bilinears vanish, whereas scalar, pseudoscalar and three-form bilinears can be nonzero. Four-dimensional Hodge duality gives the same vanishing result for a bivector multiplied by ${\gamma }_5$. Commuting spinors instead see the symmetric part, so their selection rule is different.

For Grassmann-odd four-dimensional Majorana spinors, choose $C^T=-C$ and $({\gamma }^a{)}^T=-C{\gamma }^a{C}^{-1}$. Then $C{\gamma }^a$ is symmetric whereas $C{\gamma }^{abc}$ is antisymmetric. Swapping Grassmann components therefore changes the sign of a vector bilinear but preserves a three-gamma bilinear. Integration by parts and antisymmetry of the vector indices then make the two spinor variations of the quadratic Rarita-Schwinger field action equal. Commuting auxiliary spinors have different interchange signs.

A Majorana-Weyl spinor satisfies both a Majorana spinor reality condition and a Weyl spinor chirality condition when dimension and signature permit them. In ten-dimensional Lorentzian spacetime it has sixteen real components off shell and eight physical massless spin-one-half polarizations.

For Grassmann-odd spinors in two dimensions with $\bar{\psi }={\psi }^{T}C$, $C^T=-C$ and $({\gamma }^{\mu }{)}^{T}C=-C{\gamma }^{\mu }$, moving components past each other gives $\bar{\chi }ϵ=\overline{ϵ}\chi$ and $\bar{\chi }{\gamma }^{\mu }{\gamma }^{\nu }ϵ=\overline{ϵ}{\gamma }^{\nu }{\gamma }^{\mu }\chi$. The transpose of a two-gamma product reverses its order; the Grassmann sign and the antisymmetry of $C$ cancel. For $\delta \psi =B{\gamma }^{\mu }{\partial }_{\mu }Xϵ$, this also gives $\delta \bar{\psi }=-B\overline{ϵ}{\gamma }^{\mu }{\partial }_{\mu }X$. These identities fix the boundary variation in worldsheet supersymmetry; commuting spinors would not obey the same interchange rule.

A Majorana mass term pairs a Weyl field with itself and violates its continuous phase symmetry. Gauge invariance therefore permits it only when the field is neutral under every unbroken gauge charge.

For anticommuting components and an antisymmetric numerical matrix $A$, $\delta ({\psi }^{T}A\psi )=2\delta {\psi }^{T}A\psi$: move the second varied field past the first and use $A^T=-A$. Applying this to $C^{-1}$ and to $C$ cancels the factor $1/2$ in $\mathcal{L}=\bar{\psi }i/\partial \psi +\frac{1}{2}(m{\psi }^T{C}^{-1}\psi -m^{\ast }\bar{\psi }C{\bar{\psi }}^T)$. Varying the Dirac adjoint gives the first displayed equation. Integration by parts in the spinor variation gives $i{\gamma }^{\mu T}{\partial }_{\mu }{\bar{\psi }}^T+m{C}^{-1}\psi =0$; multiplication by $C$ gives the second. Squaring the coupled equations yields $(\square +|m{|}^2)\psi =0$, so the physical Majorana fermion mass is $|m|$.