Grassmann number

Anticommuting numbers that satisfy (for Grassmann numbers $\theta_i\,$ and ordinary complex numbers $x_i\,$ )

$\theta_i \theta_j = - \theta_j \theta_i\,$ .

$\theta_i x_j = x_j \theta_i\,$ ,

i.e.,

$\left\{ \theta_i, \theta_j \right\} = 0\,$ ,

$\left[ \theta_i, x_j \right] = 0\,$ .

It follows that

$\theta_i \theta_i = (\theta_i)^2 = 0\,$ .

[edit] Calculus

Calculus over Grassmann numbers is defined using Berezin integrals. Briefly,

In higher dimensions,

$\frac{\partial}{\partial \theta_i} \theta_j + \theta_j \frac{\partial}{\partial \theta_i} = \delta_{ij}\,$ .