Commutation relations in SUSY

  • #1
ZhenyaT
1
0
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
$$ \{Q_\alpha, Q_\beta\} = 0 $$

Homework Equations


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.
 

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