Supersymmetry in Elementary Particle Physics · Problem 2.7

Formalism of Supersymmetry

Physics — Particle Physics · superspace-operator-commutation

Problem

Show that the superspace operators Qξ=(−∂∂θ−iθˉσˉm∂m)ξ+ξ†(∂∂θˉ+iσmθ∂m)\mathcal{Q}_\xi = \left(-\frac{\partial}{\partial\theta} - i\bar{\theta}\bar{\sigma}^m\partial_m\right)\xi + \xi^\dagger\left(\frac{\partial}{\partial\bar{\theta}} + i\sigma^m\theta\partial_m\right) satisfy the commutation relation [Qξ,Qη]=−2i(ξ†σˉmη−η†σˉmξ)∂m.[\mathcal{Q}_\xi, \mathcal{Q}_\eta] = -2i\left(\xi^\dagger \bar{\sigma}^m\eta - \eta^\dagger \bar{\sigma}^m\xi\right)\partial_m .

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