Grassmann number

Grassmann number

In 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 Candlin in 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 heta_i are independent elements of an algebra which contains the real numbers that anticommutes with each other but commute with ordinary numbers x:

: heta_i heta_j = - heta_j heta_iqquad heta_i x = x heta_i.

In particular, the square of the generators vanish:

: ( heta_i)^2 = 0,, since heta_i heta_i = - heta_i heta_i.

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

* linearity: int, [a f( heta) + b g( heta) ] , d heta = a int,f( heta), d heta + b int,g( heta), d heta

* partial integrations formula

: int left [frac{partial}{partial heta}f( heta) ight] , d heta = 0

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

: int, 1, d heta = 0: int, heta, d heta = 1

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:

: int expleft [ heta^TAeta ight] ,d heta,deta = det A

with A being a N imes 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 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).

Exterior algebra

The Grassmann algebra is the exterior algebra of the vector space spanned by the generators. The exterior algebra is defined independent of a choice of basis.

Matrix representations

Grassmann numbers can always be represented by matrices. Consider, for example, the Grassmann algebra generated by two Grassmann numbers heta_1 and heta_2. These Grassmann numbers can be represented by 4×4 matrices:

: heta_1 = egin{bmatrix}0 & 0 & 0 & 0\1 & 0 & 0 & 0\0 & 0 & 0 & 0\0 & 0 & 1 & 0\end{bmatrix}qquad heta_2 = egin{bmatrix}0&0&0&0\0&0&0&0\1&0&0&0\0&-1&0&0\end{bmatrix}qquad heta_1 heta_2 = - heta_2 heta_1 = egin{bmatrix}0&0&0&0\0&0&0&0\0&0&0&0\1&0&0&0\end{bmatrix}

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 space of "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.


In 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".


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Grassmann–Cayley algebra — is a form of modelling algebra for projective geometry, based on work by German mathematician Hermann Grassmann on exterior algebra, and, subsequently, by British mathematician Arthur Cayley s work on matrices and linear algebra. It is also known …   Wikipedia

  • Hermann Grassmann — Infobox Scientist name = Hermann Günther Grassmann box width = image width =150px caption = Hermann Günther Grassmann birth date = April 15, 1809 birth place = Stettin (Szczecin) death date = September 26, 1877 death place = Stettin residence =… …   Wikipedia

  • C-number — The term C number is an old nomenclature used by Paul Dirac to distinguish between real or complex numbers (c numbers or classical numbers) and operators (q numbers or quantum numbers) in quantum mechanics.Although c numbers are commuting, in… …   Wikipedia

  • Marcelo Grassmann — (Marcello Grassmann) Brazilian engraver and draughtsman born in 1925. Initially interested in sculpture, Grassmann became a wood engraver in the 1940s and in the 1950s became famous as a metal engraver and draughtsman. He won several… …   Wikipedia

  • Exterior algebra — In mathematics, the exterior product or wedge product of vectors is an algebraic construction generalizing certain features of the cross product to higher dimensions. Like the cross product, and the scalar triple product, the exterior product of… …   Wikipedia

  • Superspace — has had two meanings in physics. The word was first used by John Wheeler to describe the configuration space of general relativity; for example, this usage may be seen in his famous 1973 textbook Gravitation .The second meaning refers to the… …   Wikipedia

  • An Exceptionally Simple Theory of Everything — is a preprint proposing a basis for a unified field theory, named E8 Theory , which attempts to describe all known fundamental interactions in physics, and to stand as a possible theory of everything. The preprint was posted to the physics arXiv… …   Wikipedia

  • List of mathematics articles (G) — NOTOC G G₂ G delta space G networks Gδ set G structure G test G127 G2 manifold G2 structure Gabor atom Gabor filter Gabor transform Gabor Wigner transform Gabow s algorithm Gabriel graph Gabriel s Horn Gain graph Gain group Galerkin method… …   Wikipedia

  • Representation of a Lie superalgebra — In the mathematical field of representation theory, a representation of a Lie superalgebra is an action of Lie superalgebra L on a Z2 graded vector space V , such that if A and B are any two pure elements of L and X and Y are any two pure… …   Wikipedia

  • Quenched approximation — In particle physics, quenched approximation is an approximation often used in lattice gauge theory in which the quantum loops of fermions in Feynman diagrams are neglected. Equivalently, the corresponding one loop determinants are set to one.… …   Wikipedia