Past exam of the groups and representations course of the University of Oxford / 2025 / 3

Adopt the Minkowski metric $\eta =\mathrm{diag}(-1,+1)$ and the Clifford algebra convention $\{{\gamma }_{\mu },{\gamma }_{\nu }\}=2{\eta }_{\mu \nu }I$. The source's Pauli matrices giveThey satisfy ${\gamma }_0^2=-I$, ${\gamma }_1^2=I$ and ${\gamma }_0{\gamma }_1=-{\gamma }_1{\gamma }_0=-{\sigma }_3$, which verifies the required Clifford algebra. Raising the indices gives ${\gamma }^0=-{\gamma }_0$ and ${\gamma }^1={\gamma }_1$.

Fix the sign of the rapidity $\theta$ by taking the Lorentz boostThe corresponding Spinor representation of the Lorentz group isFor example $S^{-1}{\gamma }^{0}S={\gamma }^0\cosh \theta -{\gamma }^1\sinh \theta$ and $S^{-1}{\gamma }^{1}S={\gamma }^1\cosh \theta -{\gamma }^0\sinh \theta$, establishing Lorentz covariance by $S^{-1}{\gamma }^{\mu }S={\Lambda }^{\mu }{}_{\nu }{\gamma }^{\nu }$.

Thus the Dirac spinor transforms as ${\psi }^{\prime }(x^{\prime })=S(\theta )\psi (x)$, with infinitesimal lawThese variations compare the fields at corresponding spacetime points. At a fixed coordinate one adds the usual orbital term $-\delta x^{\mu }{\partial }_{\mu }\psi$. Reversing the rapidity convention reverses both boost signs consistently. Infinitesimal transformations describe the proper orthochronous Lorentz group.

Choose the chirality operatorA Weyl spinor is a Dirac spinor restricted to one eigenspace of $\Gamma$. The chiral projectors are $P_R=(I+\Gamma )/2$ and $P_L=(I-\Gamma )/2$. ConsequentlyThe two independent Weyl spinors each contain one complex component. Since $[S(\theta ),\Gamma ]=0$, each chirality is preserved by every connected Lorentz transformation, with ${\psi }_R\mapsto e^{\theta /2}{\psi }_R$ and ${\psi }_L\mapsto e^{-\theta /2}{\psi }_L$. A spatial reflection exchanges the two chiralities.

The left/right names depend on the sign convention for $\Gamma$. Choosing ${\gamma }_0{\gamma }_1=-{\sigma }_3$ as the chirality operator exchanges the labels $L$ and $R$; the two invariant one-component representations are the same.

All gamma matrices in this basis are real, and the Lorentz transformation $S(\theta )$ is real. Therefore complex conjugation commutes with the connected Spinor representation of the Lorentz group. With a suitable phase convention, charge conjugation is simply ${\psi }^c={\psi }^{\ast }$. The Majorana spinor condition ${\psi }^c=\psi$ becomesThe reality constraint is preserved by Lorentz transformations because $(S{\psi }_M{)}^{\ast }=S{\psi }_M$. For fermion fields, real components mean self-conjugate Grassmann variables or self-adjoint field components, rather than ordinary commuting real numbers.

Moreover, $\Gamma ={\sigma }_3$ is real, so charge conjugation commutes with the chiral projectors. The Majorana spinor and Weyl spinor conditions can therefore be imposed simultaneously. The Majorana-Weyl spinors areThe independent real component counts areThese are component counts before imposing an equation of motion. In particular the simultaneous reality and chirality constraints are compatible in this spacetime dimension, unlike the familiar four-dimensional case.

Because the present ${\gamma }_0$ is anti-Hermitian, choose the Dirac adjoint with Hermitian intertwining matrix $A=i{\gamma }_0$:This convention makes the Dirac scalar bilinear $\bar{\psi }\psi$ Hermitian. For real $m$, a Lorentz invariant mass term isTo check infinitesimal Lorentz invariance, put $K={\sigma }_3/2$. Then $K^{†}=K$ and $K^{†}A+AK=\{K,A\}=0$. Using a,Equivalently, $\delta ({\psi }_{+}^{\ast }{\psi }_{-})=\theta {\psi }_{+}^{\ast }{\psi }_{-}/2-\theta {\psi }_{+}^{\ast }{\psi }_{-}/2=0$, and similarly for its complex conjugate. Exponentiation gives $S^{†}AS=A$ and therefore ${\bar{\psi }}^{\prime }(x^{\prime })=\bar{\psi }(x)S^{-1}$.

At fixed coordinates ${\mathcal{L}}_m$ has only the orbital variation of a Lorentz scalar; the spacetime measure is invariant under the connected Lorentz group, so the integrated action is invariant. The mass term pairs the two opposite chiralities; retaining only one Weyl spinor removes this Dirac mass. An equivalent spinor phase convention gives a real multiple of ${\psi }_{+}^{\ast }{\psi }_{-}+{\psi }_{-}^{\ast }{\psi }_{+}$ instead.