# Commutation relations of symmetry generators in SUSY

It is well known that the generators $$Q_\alpha = \frac{\partial}{\partial \theta^\alpha} - i \sigma^\mu_{\alpha \dot \beta} \bar{\theta}^\dot{\beta} \partial_\mu$$ and $$\bar{Q}_\dot{\alpha} = -\frac{\partial}{\partial \bar{\theta}^\dot{\alpha}} + i \theta^\beta\sigma^\mu_{\beta \dot \alpha} \partial_\mu$$

where $$\theta^\alpha$$, $$\bar{\theta}^\dot{\beta}$$ are Grassmann variables, obey the anti-commutation relations

$$\{Q_\alpha, \bar{Q}_\dot{\alpha}\} = 2i \sigma^\mu_{\alpha \dot \alpha} \partial_\mu$$ $$\{Q_\alpha, Q_\beta\} = \{\bar{Q}_\dot{\alpha}, \bar{Q}_\dot{\beta}\} = 0$$

Question: I want to explicitly verify those anti-commutation relations, say for example $$\{Q_\alpha, Q_\beta\} = 0$$.

However, I'm unable to reproduce that result. I might get as far as follows:

$$\{Q_\alpha, Q_\beta\} = \{\frac{\partial}{\partial \theta^\alpha}, \frac{\partial}{\partial \theta^\alpha}\} - i \sigma^\mu_{\beta \dot \beta} \bar{\theta}^\dot{\beta} \{\frac{\partial}{\partial \theta^\alpha}, \partial_\mu\} - i \sigma^\mu_{\alpha \dot \beta} \bar{\theta}^\dot{\beta} \{ \partial_\mu, \frac{\partial}{\partial \theta^\beta} \} - \sigma^\mu_{\alpha \dot \beta} \sigma^\nu_{\beta \dot \gamma} \{ \bar{\theta}^\dot{\beta} \partial_\mu, \bar{\theta}^\dot{\gamma} \partial_\nu \}$$

where the last term vanishes due to the anti-commutation of the $$\bar{\theta}$$.

Any help on how to proceed with the calculation towards the desired result is greatly appreciated.

• How do you get anticommutators in the 2 cross-terms? Jan 20, 2019 at 2:34
• I think the two terms in the middle should not be there if you use Leibniz rule for Grassmann variables
– Kosm
Jan 20, 2019 at 6:26
• @Qmechanic I simply pulled out the $\sigma^\mu_{\beta \dot \beta} \bar{\theta}^\dot{\beta}$ factor out of the cross term of the commutator, am I not allowed to do that? Jan 20, 2019 at 11:45
• Only if you account for sign factors when supercommuting objects. Jan 20, 2019 at 12:29

1. $$Q_{\alpha}$$ and $$Q_{\beta}$$ supercommute because they only consist of objects that manifestly supercommute with each other.
2. In contrast, $$Q_{\alpha}$$ and $$\overline{Q}_{\dot{\alpha}}$$ only fail to supercommute because the Grassmann-variables and their corresponding derivatives don't supercommute $$\{\frac{\partial}{\partial \theta^{\alpha}},~ \theta^{\beta}\}_+ ~=~\delta_{\alpha}^{\beta},\qquad \{\frac{\partial}{\partial \overline{\theta}^{\dot{\alpha}}},~ \overline{\theta}^{\dot{\beta}}\}_+ ~=~\delta_{\dot{\alpha}}^{\dot{\beta}}.$$