- Grassmann number
mathematical physics, a Grassmann number (also called an anticommuting number or anticommuting c-number) is a mathematical construction which allows a path integral representation for Fermionic fields. They were discovered by David John Candlinin 1956 [cite journal|journal= Nuovo Cimento | author= D.J. Candlin | volume =4 | title = On Sums over Trajectories for Systems With Fermi Statistics|pages=224 | year=1956] . A collection of Grassman variable are independent elements of an algebra which contains the real numbers that anticommutes 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 grassman integration needs to have the following properties:
* partial integrations 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.
path integral formulationof quantum field theorythe following Gaussian integralof Grassmann quantities is needed for fermionic anticommuting fields:
with being a 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 2"n". These concepts are all named for Hermann Grassmann.
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).
The Grassmann algebra is the
exterior algebraof the vector spacespanned by the generators. The exterior algebra is defined independent of a choice of basis.
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 2"n" × 2"n" square matrices. Physically, these matrices can be thought of as
raising operators acting on a Hilbert spaceof "n" identical fermions in the occupation number basis. Since the occupation number for each fermion is 0 or 1, there are 2"n" possible basis states. Mathematically, these matrices can be interpreted as the linear operators corresponding to left exterior multiplication on the Grassmann algebra itself.
quantum field theory, Grassmann numbers are the "classical analogues" of anticommutingoperators. 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.
Grassmann numbers are also important for the definition of
supermanifolds (or superspace) where they serve as "anticommuting coordinates".
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