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

<br /> \theta \theta = \pm 2\theta_1 \theta_2<br />

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

<br /> (\theta \theta) ( \theta \theta) =0<br />

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 :)