Why Do Superfields Have Quadratic Terms in Theta and Theta Bar?

[earth2]

Hi folks,

I just read some stuff about Susy and encountered superfields and their expansion in terms of the supercoords $$x^\mu, \theta, \bar{\theta}$$. Reading that (e.g. in the script of Lykken), I found general expansions like

$$S(x,\theta,\bar{\theta})=...+ \theta\theta \psi + ...+\theta\theta\bar{\theta}\bar{\theta} D$$

But how can terms quadratic in theta/theta bar appear if theta and theta bar are grassmann numbers? Their square should vanish!

I don't get it and any help would be appreciated :)
Thanks,
earth2

 

[haushofer]

The $\theta \theta$ denotes an inner product. This inner product is antisymmetric, so it only contains cross terms, unlike the inner product from vector analysis or any inner product coming from a diagonal metric! Depending on the convention, one has

$$\theta \theta = \pm 2\theta_1 \theta_2$$

This doesn't vanish; $\theta_1 \neq \theta_2$. However, a term like

$$(\theta \theta) ( \theta \theta) =0$$

because $\theta_i \theta_i = 0$.

 

[haushofer]

Glad I could help.

 

[earth2]

Sorry, i had no internet in the past week. Thank you for your answer :)