Weyl Spinors

A Weyl spinor carries one of the two inequivalent complex two-component representations of the Lorentz double cover $SL(2,\mathbb C)$. Their rotation matrices agree, while their boost matrices have opposite signs in the exponent. In the massless Dirac equation, the two chiral components propagate independently according to the Weyl equations. “Weyl” first specifies a transformation law; the choice of equation, mass term, and quantum state is additional.

Required background. SL(2, ℂ) and the Lorentz Group constructs the Lorentz lift; Gamma-Matrix Conventions fixes the chiral basis and chirality projectors.

Two chiral transformation laws

Write a chiral-basis Dirac column as $\Psi=(\chi_L,\chi_R)^T$, where each entry is a two-component column. For the lift defined by $X'=AXA^\dagger$, take

$$\chi_L'(x')=(A^\dagger)^{-1}\chi_L(x),\qquad \chi_R'(x')=A\chi_R(x).$$

The subscripts agree with $\gamma^5_C=\operatorname{diag}(-I,I)$ in the convention ledger. For a right-handed rotation and a passive boost along a unit vector $\mathbf n$, these give

$$\begin{aligned} S_L(R)&=S_R(R)=e^{-i\theta\mathbf n\cdot\boldsymbol\sigma/2},\\ S_L(B)&=e^{+\xi\mathbf n\cdot\boldsymbol\sigma/2},\\ S_R(B)&=e^{-\xi\mathbf n\cdot\boldsymbol\sigma/2}. \end{aligned}$$

The boost velocity of the primed frame is $c\tanh\xi\,\mathbf n$. Thus the difference between the representations cannot be seen from rotations alone.

There is no fixed invertible complex-linear matrix that intertwines all left and right transformations. Such a matrix would have to commute with every spin-half rotation and hence be proportional to $I$; it could not reverse all three boost signs. Complex conjugation combined with the antisymmetric Pauli tensor does relate the two representations, but it is an antilinear operation.

What the two representation labels mean

Use dimensionless generators with $S(R)=e^{-i\boldsymbol\theta\cdot\mathbf J}$ and, for the passive boost convention above, $S(B)=e^{+i\boldsymbol\xi\cdot\mathbf K}$. They are

$$\mathbf J_L=\mathbf J_R=\frac{\boldsymbol\sigma}{2},\qquad \mathbf K_L=-\frac{i\boldsymbol\sigma}{2},\qquad \mathbf K_R=+\frac{i\boldsymbol\sigma}{2}.$$

These obey the Lorentz algebra, including $[K_i,K_j]=-i\epsilon_{ijk}J_k$. Define

$$\mathbf N_+=\frac{\mathbf J+i\mathbf K}{2},\qquad \mathbf N_-=\frac{\mathbf J-i\mathbf K}{2}.$$

Each triple obeys the angular-momentum algebra, and the two triples commute. The left representation has $(\mathbf N_+,\mathbf N_-)=(\boldsymbol\sigma/2,0)$ and is labeled $(1/2,0)$; the right representation has $(0,\boldsymbol\sigma/2)$ and is labeled $(0,1/2)$. These labels classify finite-dimensional representations of the complexified Lorentz algebra. They are not two separate observable spins of a particle, and this algebraic decomposition does not identify the real Lorentz group with $SU(2)\times SU(2)$.

The boost generators here are not Hermitian in the component norm. They are representation matrices on local components, distinct from the self-adjoint generators of a unitary transformation on the one-particle Hilbert space.

The Weyl equations

Use natural units below, and define $\sigma^\mu=(I,\boldsymbol\sigma)$ and $\bar\sigma^\mu=(I,-\boldsymbol\sigma)$. The chiral gamma matrices have off-diagonal blocks $\sigma^\mu,\bar\sigma^\mu$. The massless Dirac equation therefore separates into

$$i\bar\sigma^\mu\partial_\mu\chi_L=0,\qquad i\sigma^\mu\partial_\mu\chi_R=0.$$

Equivalently, with $\mathbf p=-i\nabla$,

$$i\partial_t\chi_L=-\boldsymbol\sigma\cdot\mathbf p\,\chi_L, \qquad i\partial_t\chi_R=+\boldsymbol\sigma\cdot\mathbf p\,\chi_R.$$

Squaring either Hamiltonian gives $\mathbf p^2I$, so each component satisfies the massless wave equation. The first-order equation still constrains the two amplitudes; arbitrary pairs of scalar wave solutions do not necessarily solve it.

For a positive-frequency mode $\chi(x)=w e^{-iEt+i\mathbf p\cdot\mathbf x}$ with $E=|\mathbf p|>0$, the equations require

$$\boldsymbol\sigma\cdot\widehat{\mathbf p}\,w_L=-w_L,\qquad \boldsymbol\sigma\cdot\widehat{\mathbf p}\,w_R=+w_R.$$

Thus the particle modes have helicity $-1/2$ and $+1/2$, respectively. This statement includes the positive-frequency qualification. Negative-frequency coefficient spinors and the antiparticle states created from them require separate label bookkeeping; the chirality of a field is not a declaration that every excitation it creates has that particle helicity.

For $\mathbf p=p\hat{\mathbf z}$ with $p>0$, one can choose $w_L=(0,1)^T$ and $w_R=(1,0)^T$. Both travel along $+z$ even though their spin projections are opposite.

Conserved currents and local flow

For commuting c-number amplitudes solving the free equations, define

$$j_L^\mu=\chi_L^\dagger\bar\sigma^\mu\chi_L,\qquad j_R^\mu=\chi_R^\dagger\sigma^\mu\chi_R.$$

Taking the adjoint equation and adding it to the original one gives $\partial_\mu j_{L,R}^\mu=0$. The time components are nonnegative, and the spatial components are $-\chi_L^\dagger\boldsymbol\sigma\chi_L$ and $+\chi_R^\dagger\boldsymbol\sigma\chi_R$.

For any two-component complex column $z$,

$$\sum_i(z^\dagger\sigma^i z)^2=(z^\dagger z)^2.$$

Consequently each nonzero pure c-number Weyl current is future null at an event. This is an algebraic local-current property. An incoherent mixture or a sum of distinct currents can be timelike, and quantum field composite operators require their own definition. The pointwise identity should not be transferred to an arbitrary expectation value after averaging.

The example $w_L=(0,1)^T$ has $j_L^\mu=(1,0,0,1)$, which checks its propagation direction despite the negative spin projection.

Mass terms and the meaning of a Weyl representation

A two-component object can have the Weyl transformation law without obeying a free massless equation. A Dirac mass couples independent left and right columns. A Majorana construction relates opposite chiralities by complex conjugation and requires compatible internal quantum numbers. The representation itself does not decide which mass term is allowed.

In particular, one should distinguish a chiral component, a massless Weyl equation, and a quantized Weyl field. Statements about particle content, gauge representations, or allowed mass terms must specify which of these is meant. Dirac Spinors constructs the independent chiral pair; Majorana Spinors constructs the antilinear relation.

Exercises

1. Verify that $[N_{+i},N_{-j}]=0$ from the Lorentz algebra.

Solution

Expand the commutator as $([J_i,J_j]-i[J_i,K_j]+i[K_i,J_j]+[K_i,K_j])/4$. The $J$ terms cancel because $[K_i,K_j]=-i\epsilon_{ijk}J_k$. The mixed terms cancel using $[K_i,J_j]=i\epsilon_{ijk}K_k$.

2. Prove the current identity for $z=(a,b)^T$.

Solution

The three spin components are $2\operatorname{Re}(a^*b)$, $2\operatorname{Im}(a^*b)$, and $|a|^2-|b|^2$. Their squared sum is $4|a|^2|b|^2+(|a|^2-|b|^2)^2=(|a|^2+|b|^2)^2$.

3. Apply a passive $z$ boost to the left-handed mode $w_L=(0,1)^T$ with future momentum $(E,0,0,E)$. Check the current and momentum scaling.

Solution

$S_L(B)w_L=e^{-\xi/2}w_L$, so its current is multiplied by $e^{-\xi}$. The vector boost likewise gives $E'=p'_z=Ee^{-\xi}$. The component amplitude and the null momentum therefore transform consistently. A boost has not preserved the component norm at a fixed event, nor was that required.

References

• H. K. Dreiner, H. E. Haber, and S. P. Martin, “Two-component spinor techniques and Feynman rules for quantum field theory and supersymmetry,” Physics Reports 494, 1–196, 2010, doi:10.1016/j.physrep.2010.05.002; corrected arXiv:0812.1594v6, 2022 — chiral representations and two-component equations.
• M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014, doi:10.1017/9781139540940 — Lorentz representations and Weyl fermions.
• D. Tong, Quantum Field Theory, University of Cambridge lecture notes, 2006, chapter 4 — chiral decomposition of the Dirac equation.