Grassmann number

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 $\theta_i$ are independent elements of an algebra which contains the real numbers that anticommute with each other but commute with ordinary numbers $x$:

$\theta_i \theta_j = -\theta_j \theta_i\qquad\theta_i x = x \theta_i.$

In particular, the squares of the generators vanish:

$(\theta_i)^2 = 0,\,$ since $\theta_i \theta_i = -\theta_i \theta_i.$

In other words, a Grassmann number is a non-zero square-root of zero.

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

• linearity

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

• partial integration 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:

$\int\, 1\, d\theta = 0$

$\int\, \theta\, d\theta = 1.$

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:

$\int \exp\left[-\theta^TA\eta\right] \,d\theta\,d\eta = \det A$

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).

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 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 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.

Grassmann numbers are also important for the definition of supermanifolds (or superspace) where they serve as "anticommuting coordinates". In the 1970s a completely new type of symmetry was discovered which provided a physical mechanism to cancel infinite ground state fluctuations. Supersymmetry is a trait of contemporary mathematical models that can be described in several ways. We can say first of all that spacetime has dimensions other than those to which we daily experience: they are called the Grassmann dimensions because they are measured with Grassmann numbers rather than ordinary real numbers.

Generalisations

There are some generalisations to Grassmann numbers. These require rules in terms of N variables such that:

$\theta_{i_1} \theta_{i_2}\cdots\theta_{i_N} + \theta_{i_N}\theta_{i_1}\theta_{i_2}\cdots +\cdots = 0$

where the indices are summed over all permutations so that as a consequence:

$(\theta_i)^N = 0\,$

for some N > 2. These are useful for calculating hyperdeterminants of N-tensors where N > 2 and also for calculating discriminants of polynomials for powers larger than 2. There is also the limiting case as N tends to infinity in which case one can define analytic functions on the numbers. For example in the case with N = 3 a single grassmann number can be represented by the matrix:

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

so that $\theta^3=0$. For two grassmann numbers the matrix would be of size 10×10.

For example, the rules for N = 3 with two Grassmann variables imply:

$\theta_1 (\theta_2)^2 + \theta_2 \theta_1 \theta_2 + (\theta_2)^2 \theta_1 = 0$

so that it can be shown that

$\theta_1 (\theta_2)^2 = -\frac{1}{2} \theta_2 \theta_1 \theta_2 = (\theta_2)^2 \theta_1$

and so

$(\theta_1)^2(\theta_2)^2 = (\theta_2)^2(\theta_1)^2 = \theta_1(\theta_2)^2 \theta_1 = \theta_2(\theta_1)^2 \theta_2 = -\frac{1}{2} \theta_1 \theta_2 \theta_1 \theta_2 = -\frac{1}{2} \theta_2 \theta_1 \theta_2 \theta_1,$

which gives a definition for the hyperdeterminant of a 2×2×2 tensor as

$(A^{abc}\theta_a\eta_b\psi_c)^4 = \det(A)(\theta_1)^2(\theta_2)^2(\eta_1)^2(\eta_2)^2(\psi_1)^2(\psi_2)^2.$

See also

• Grassmannian
• Hermann Grassmann (linguist and mathematician)
• Superspace
• Exterior algebra