Grassmann Number

In mathematical physics, a Grassmann number, named after Hermann Grassmann, (also called an anticommuting number or anticommuting c-number) is a mathematical construction which allows a path integral representation for Fermionic fields. A collection of Grassmann variables are independent elements of an algebra which contains the real numbers that anticommute with each other but commute with ordinary numbers :

In particular, the square of the generators vanish:

, since

In order to reproduce the path integral for a Fermi field, the definition of Grassmann integration needs to have the following properties:

• linearity

• partial integration formula

This results in the following rules for the integration of a Grassmann quantity:

Thus we conclude that the operations of integration and differentiation of a Grassmann number are identical.

In the path integral formulation of quantum field theory the following Gaussian integral of Grassmann quantities is needed for fermionic anticommuting fields:

with A being an N × N matrix.

The algebra generated by a set of Grassmann numbers is known as a Grassmann algebra. The Grassmann algebra generated by n linearly independent Grassmann numbers has dimension 2n.

Grassmann algebras are the prototypical examples of supercommutative algebras. These are algebras with a decomposition into even and odd variables which satisfy a graded version of commutativity (in particular, odd elements anticommute).