In addition to what Joshua said in his nice answer, may favorite (simplified) point of view is looking at a SUSY transformation as a coordinate transformation (translation) in superspace

$$ x' = x + a + \frac{i}{2}\zeta\sigma^{\mu}\bar{\theta} - \frac{i}{2}\theta\sigma^{\mu}\bar{\zeta}$$

$$ \theta'= \theta + \zeta $$

$$ \bar{\theta}'= \bar{\theta} + \bar{\zeta} $$

with $\theta$ and $\bar{\theta}$ denoting the additional Grassmanian coordinates.

The supersymmetry generators or supercharges, when written down as differential operators, contain momentum operators in both, the "usual" even spacetime coordinates and the odd Grassmann coordinates

$$ Q_a = i\partial_a -\frac{1}{2}(\sigma^{\mu})_{a\dot{b}}\bar{\theta}^{\dot{b}}\partial_{\mu}$$

$$ \bar{Q}^{\dot{a}} = i\bar{\partial}^{\dot{a}} -\frac{1}{2}(\bar{\sigma}^{\mu})^{\dot{a}b}\theta_b\partial_{\mu}$$

Where $\partial_a = \frac{\partial}{\partial\theta^a}$ and $\bar{\partial}^{\dot{a}} = \frac{\partial}{\partial\bar{\theta_{\dot{a}}}}$ are the derivatives along the Grassmanian coordinates.

A nice and very readable introduction to the superspace formalism can for example be found in Ch 11 of this book.

This post imported from StackExchange Physics at 2014-03-07 16:41 (UCT), posted by SE-user Dilaton