Supersymmetry generator ($Q_{\alpha }^A,{\bar{Q}}_{\dot{\alpha }A}$)

A supersymmetry generator is an odd Weyl spinor generator of the Super-Poincaré algebra. Acting on a physical state reverses fermion parity while preserving its four-momentum.

A supersymmetry transformation is the infinitesimal action generated by an odd supercharge, with constant anticommuting spinor parameter in a global theory. In superspace it acts as $\delta \mathcal{S}=(ϵQ+\overline{ϵ}\bar{Q})\mathcal{S}$, with possible conventional overall factors of $i$. On the fermion of a chiral superfield, the transformation includes $\delta \psi =\sqrt{2}ϵF_{\mathrm{aux}}+i\sqrt{2}{\sigma }^{\mu }\overline{ϵ}{\partial }_{\mu }\varphi$. A nonzero vacuum auxiliary field therefore gives an inhomogeneous fermion shift, identifying the Goldstino direction when global supersymmetry is spontaneously broken.

Let $[J_i,q_a]=(R_i{)}_{ab}{q}_b$, with numerical coefficient matrices and $[J_i,J_j]=i{ϵ}_{ijk}{J}_k$. The Jacobi identity then gives $(R_j{R}_i-R_i{R}_j)q=i{ϵ}_{ijk}{R}_{k}q$, so component coefficient matrices obey the opposite bracket to state-representation matrices. Thus $R_i=-{\sigma }_i/2$ is consistent for an upper-column spinor, whereas $+{\sigma }_i/2$ in that same component prescription is not. Inverse adjoint actions and dual row conventions must be compared with their index contractions intact.