Grassmann Delta function

A function of the spinorial coordinates of an n–dimensional anticommuting space defined by the condition [1,2,3,4]

(1)

for any function f. In the simplest case of a one–dimensional anticommuting space, a generic function has the expansion . Inserting in (1) one can easily infer the result

(2)

From this identity it follows . Moreover, the Berezin integral as defined in (1) is invariant under coordinate translations

The procedure which leads to (2) can be implemented in the case of a generic n–dimensional anticommuting space to give

(3)

From the general definition (3) is then easy to establish the transformation properties of the n–dimensional Grassmann delta function under translations and reflection, by simply generalizing the results for the one–dimensional case. As an example, we consider the four–dimensional N– extended superspace . A generic superfield defined on it is a function of four bosonic coordinates x ^μ, and 4N spinorial ones, , . In this case, the Grassmann delta function...