Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 45 / 1 / b

A supercharge is an odd Weyl spinor generator. The mixed anticommutator has one undotted and one dotted index, hence transforms as a Lorentz four-vector. In the minimal Super-Poincaré algebra, its only spacetime generator is the four-momentum, so covariance fixes it to $c{\sigma }_{\alpha \dot{\beta }}^{\mu }{P}_{\mu }$. Normalizing the charges canonically fixes $c=2$:The number two is a normalization, not something Lorentz covariance alone could fix under $Q\mapsto aQ$.

For an explicit closure calculation, use left Grassmann derivatives in superspace and the differential translation convention $P_{\mu }=i{\partial }_{\mu }$:Each of the two derivative-coordinate cross terms contributes $i{\sigma }^{\mu }{\partial }_{\mu }$; the other terms cancel by anticommutativity. This proves the displayed coefficient. The same calculation gives $\{Q,Q\}=\{\bar{Q},\bar{Q}\}=0$ and $[P_{\mu },Q]=[P_{\mu },\bar{Q}]=0$. There is no scalar central charge for a single supersymmetry label: symmetry of the anticommutator is incompatible with a term proportional only to the antisymmetric ${ϵ}_{\alpha \beta }$.

In the massive rest frame $P_{\mu }=(m,0,0,0)$, the mixed relation is $\{Q_{\alpha },Q_{\beta }^{†}\}=2m{\delta }_{\alpha \beta }$. These positive canonical anticommutation relations supply the input for the normalization and finite fermionic Fock space in the question-level continuation.