Grassmann Integral - Substitution Formula

Substitution Formula

Consider now a mixture of even and odd variables, i.e. x_a and θ_i. Again we assume a coordinate transformation as where x_a are even functions and θ_i are odd functions. We assume the functions x_a and θ_i to be defined on an open set U in Rm. The functions x_a map onto the open set U' in Rm.

The change of the integral will depend on the Jacobian

This matrix consists of four blocks:

A and D are even functions due to the derivation properties, B and C are odd functions. A matrix of this block structure is called even matrix.

The transformation factor itself depends on the oriented Berezinian of the Jacobian. This is defined as:

For further details see the article about the Berezinian.

The complete formula now reads as:

$\int_U f(x_a,\theta_i) \, d(x,\theta) = \int_{U'} f(x_a,\theta_i) \operatorname{Ber}_{+-}\, \frac{\partial(x_a,\theta_i)}{\partial(y_b,\xi_j)} \, d(y,\xi).$

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