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:
- 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.
with A being an N × N matrix.
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).
Grassmann numbers can always be represented by matrices. Consider, for example, the Grassmann algebra generated by two Grassmann numbers and . These Grassmann numbers can be represented by 4×4 matrices:
In general, a Grassmann algebra on n generators can be represented by 2n × 2n square matrices. Physically, these matrices can be thought of as raising operators acting on a Hilbert space of n identical fermions in the occupation number basis. Since the occupation number for each fermion is 0 or 1, there are 2n possible basis states. Mathematically, these matrices can be interpreted as the linear operators corresponding to left exterior multiplication on the Grassmann algebra itself.
In quantum field theory, Grassmann numbers are the "classical analogues" of anticommuting operators. They are used to define the path integrals of fermionic fields. To this end it is necessary to define integrals over Grassmann variables, known as Berezin integrals.