Berezin integral

In mathematical physics, the Berezin integral, named after Felix Berezin, (also known as Grassmann integral, after Hermann Grassmann), is a way to define integration for functions of Grassmann variables (elements of the exterior algebra). It is not an integral in the Lebesgue sense; the word "integral" is used because the Berezin integral has properties analogous to the Lebesgue integral and because it extends the path integral in physics, where it is used as a sum over histories for fermions.

Let ${\displaystyle \Lambda ^{n}}$ be the exterior algebra of polynomials in anticommuting elements ${\displaystyle \theta _{1},\dots ,\theta _{n}}$ over the field of complex numbers. (The ordering of the generators ${\displaystyle \theta _{1},\dots ,\theta _{n}}$ is fixed and defines the orientation of the exterior algebra.)

The Berezin integral over the sole Grassmann variable ${\displaystyle \theta =\theta _{1}}$ is defined to be a linear functional

where we define

so that :

These properties define the integral uniquely and imply

Take note that ${\displaystyle f(\theta )=a\theta +b}$ is the most general function of ${\displaystyle \theta }$ because Grassmann variables square to zero, so ${\displaystyle f(\theta )}$ cannot have non-zero terms beyond linear order.

The Berezin integral on ${\displaystyle \Lambda ^{n}}$ is defined to be the unique linear functional ${\displaystyle \int _{\Lambda ^{n}}\cdot {\textrm {d}}\theta }$ with the following properties: