# grassmannians – A specific integration with Grassmann variables Answer Hello dear visitor to our network We will offer you a solution to this question grassmannians – A specific integration with Grassmann variables ,and the answer will be typical through documented information sources, We welcome you and offer you new questions and answers, Many visitor are wondering about the answer to this question.

## grassmannians – A specific integration with Grassmann variables

I have recently read (for example, here) that this relation below is true

$$\int dz \: e^{\frac{1}{2} \sum_{ij} z_i A_{ij} z_j} = Pf(\mathbf{A}),$$
where $$Pf(\mathbf{A})$$ is the Pfaffian of an even dimensional skew-(or anti-)symmetric matrix $$\mathbf{A}$$ other $$\{ z_i \}$$ are Grassman variables.

Could anyone point out if there is a similar formula for the form below:

$$\int dz \: \big( \frac{1}{2} \sum_{ij} z_i A_{ij} z_j \big)^k,$$
where $$k > 0$$ is an integer?

My guess is yes since

$$e^{\mathbf{A}} = \sum_{j=0}^\infty \frac{\mathbf{A}^j}{j!},$$
but I do not have enough experience with these types of integrals.