Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 307 / 1 / b / Solution

An internal generator $T_a$ is a Lorentz scalar. The Coleman–Mandula theorem and the graded extension allow it to act nontrivially on supercharges only as an R-symmetry. Since R-symmetries are excluded,

Translation invariance of a conserved global supercharge and the graded Jacobi identities giveLorentz covariance requires the supercharges to transform as Weyl spinors,with the complex-conjugate dotted-spinor relation for ${\bar{Q}}_{\dot{\alpha }}$, up to the sign convention used for the action of generators.

The anticommutator $\{Q_{\alpha },{\bar{Q}}_{\dot{\beta }}\}$ transforms as a Lorentz vector because $(1/2,0)\otimes (0,1/2)=(1/2,1/2)$. The only translation generator with that transformation law is $P_{\mu }$, and a normalization of $Q$ fixesAn equal-chirality anticommutator could only contain an antisymmetric spinor contraction times a central charge, but the anticommutator is symmetric under exchange of the complete supercharges. For one supercharge and no central extension this forcesTogether with the stated Poincare brackets, these are the four-dimensional $\mathcal{N}=1$ Super-Poincare relations.