# Commutation relations in SUSY

ZhenyaT
I hope I put this in the correct section of this forum, I apologize if I didn't.

## Homework Statement

:[/B]

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

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

I am asked to explicitly verify those anti-commutation relations, say for example
$$\{Q_\alpha, Q_\beta\} = 0$$

see above

## The Attempt at a Solution

:[/B]
However, I'm unable to reproduce that result. I might get as far as follows, by simply expanding the anti-commutator, provided I did not make a mistake (I've never had to deal with Grassmann variables before, so that is a real possibility).

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

Also, I think the last term should vanish, due to the anti-commutativity of the $$\bar{\theta}$$
Is this correct so far?
Unfortunately, I'm unable to make further progress.
Any help would be greatly appreciated. Thanks in advance.