In mathematical physics, a Grassmann number, named after Hermann Grassmann (also called an anticommuting number or supernumber), is an element of the exterior algebra of a complex vector space.
Grassmann numbers saw an early use in physics to express a path integral representation for fermionic fields, although they are now widely used as a foundation for superspace, on which supersymmetry is constructed.
Differential forms are normally defined in terms of derivatives on a manifold; however, one can contemplate the situation where one "forgets" or "ignores" the existence of any underlying manifold, and "forgets" or "ignores" that the forms were defined as derivatives, and instead, simply contemplate a situation where one has objects that anti-commute, and have no other pre-defined or presupposed properties.
More can be done: one can consider polynomials of Grassmann numbers, leading to the idea of holomorphic functions.
Each of these ideas can be carefully defined, and correspond reasonably well to the equivalent concepts from ordinary mathematics.
Of course, one could pursue a similar program for any other field, or even ring, and this is indeed widely and commonly done in mathematics.
Thus, the study of Grassmann numbers, and of supermathematics, in general, is strongly driven by their utility in physics.
The ladder operators for fermions create field quanta that must necessarily have anti-symmetric wave functions, as this is forced by the Pauli exclusion principle.
When the number of fermions is fixed and finite, an explicit relationship between anticommutation relations and spinors is given by means of the spin group.
This group can be defined as the subset of unit-length vectors in the Clifford algebra, and naturally factorizes into anti-commuting Weyl spinors.
Both the anti-commutation and the expression as spinors arises in a natural fashion for the spin group.
That is, for complex x, one has The squares of the generators vanish: In other words, a Grassmann variable is a non-zero square-root of zero.
), and larger finite products, can be seen here to be playing the role of a basis vectors of subspaces of
The Grassmann algebra generated by n linearly independent Grassmann variables has dimension 2n; this follows from the binomial theorem applied to the above sum, and the fact that the (n + 1)-fold product of variables must vanish, by the anti-commutation relations above.
The special case of n = 1 is called a dual number, and was introduced by William Clifford in 1873.
In case V is infinite-dimensional, the above series does not terminate and one defines The general element is now where
Some of the basic concepts are developed in greater detail in the article on dual numbers.
As a general rule, it is usually easier to define the super-symmetric analogs of ordinary mathematical entities by working with Grassmann numbers with an infinite number of generators: most definitions become straightforward, and can be taken over from the corresponding bosonic definitions.
For example, a single Grassmann number can be thought of as generating a one-dimensional space.
A vector space, the m-dimensional superspace, then appears as the m-fold Cartesian product of these one-dimensional
[clarification needed] It can be shown that this is essentially equivalent to an algebra with m generators, but this requires work.
[6][clarification needed] The spinor space is defined as the Grassmann or exterior algebra
, and quantum field theory additionally require invariance under the shift of integration variables
such that The only linear function satisfying this condition is a constant (conventionally 1) times B, so Berezin defined[7] 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.
The convention that performs the innermost integral first yields Some authors also define complex conjugation similar to Hermitian conjugation of operators,[8] With the additional convention we can treat θ and θ* as independent Grassmann numbers, and adopt Thus a Gaussian integral evaluates to and an extra factor of θθ* effectively introduces a factor of (1/b), just like an ordinary Gaussian, After proving unitarity, we can evaluate a general Gaussian integral involving a Hermitian matrix B with eigenvalues bi,[8][9] Grassmann numbers can be represented by 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.
Mathematically, these matrices can be interpreted as the linear operators corresponding to left exterior multiplication on the Grassmann algebra itself.
These require rules in terms of N variables such that: where the indices are summed over all permutations so that as a consequence: for some N > 2.
For example, in the case with N = 3 a single Grassmann number can be represented by the matrix: so that
For example, the rules for N = 3 with two Grassmann variables imply: so that it can be shown that and so which gives a definition for the hyperdeterminant of a 2×2×2 tensor as