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).
Read more about Grassmann Number: Exterior Algebra, Matrix Representations, Applications
Famous quotes containing the word number:
“[The] elderly and timid single gentleman in Paris ... never drove down the Champs Elysees without expecting an accident, and commonly witnessing one; or found himself in the neighborhood of an official without calculating the chances of a bomb. So long as the rates of progress held good, these bombs would double in force and number every ten years.”
—Henry Brooks Adams (1838–1918)