Grassmann number

In mathematical physics, a Grassmann number (also called an anticommuting number) is a quantity $\theta _{i}$ that anticommutes with other Grassmann numbers but commutes with ordinary numbers $x_{j}$:

$\theta _{i}\theta _{j}=-\theta _{j}\theta _{i}\qquad \theta _{i}x_{j}=x_{j}\theta _{i}.$

In particular, the square of any Grassmann number vanishes:

$\theta _{i}\theta _{i}=0.\,$

The integration of Grassmann variables needs to fullfill the following properties:

• linearity

$\int [af(\theta )+bg(\theta )]d\theta =a\int f(\theta )d\theta +b\int g(\theta )d\theta$

• partial integrations formula

$\int \left[{\frac {\partial }{\partial \theta }}f(\theta )\right]d\theta =0$

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

$\int 1d\theta =0$

$\int \theta d\theta =1$

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:

$\int exp\left[\theta ^{T}A\eta \right]d\theta d\eta =detA$

with $A$ being a $N\times N$ matrix.

The algebra generated by a set of Grassmann numbers is known as a Grassmann algebra (or an exterior algebra). The Grassmann algebra generated by n linearly independent Grassmann numbers has dimension 2ⁿ. 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).

Matrix representations

Grassmann numbers can always be represented by matrices. Consider, for example, the Grassmann algebra generated by two Grassmann numbers $\theta _{1}$ and $\theta _{2}$. These Grassmann numbers can be represented by 4×4 matrices:

$\theta _{1}={\begin{bmatrix}0&0&0&0\\1&0&0&0\\0&0&0&0\\0&0&1&0\\\end{bmatrix}}\qquad \theta _{2}={\begin{bmatrix}0&0&0&0\\0&0&0&0\\1&0&0&0\\0&-1&0&0\\\end{bmatrix}}\qquad \theta _{1}\theta _{2}=-\theta _{2}\theta _{1}={\begin{bmatrix}0&0&0&0\\0&0&0&0\\0&0&0&0\\1&0&0&0\\\end{bmatrix}}$

In general, a Grassmann algebra on n generators can be represented by 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.

Applications

In quantum field theory, Grassmann numbers 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".