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Anticommutating number
In mathematical physics, a Grassmann number, named after Hermann Grassmann (also called an anticommuting number or supernumber), is an element of the exterior algebra over the complex numbers. The special case of a 1 dimensional algebra is known as a dual number. 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.

Grassmann numbers are generated by anti commuting elements or objects. The idea of anti commuting objects arises in multiple areas of mathematics: they are typically seen in differential geometry, where the differential forms are anti commuting. 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 pre supposed properties. Such objects form an algebra, and specifically the Grassmann algebra or exterior algebra. The Grassmann numbers are elements of that algebra. The appellation of "number" is justified by the fact that they behave not unlike "ordinary" numbers: they can be added, multiplied and divided: they behave almost like a field. More can be done: one can consider polynomials of Grassmann numbers, leading to the idea of holomorphic functions. One can take derivatives of such functions, and then consider the anti derivatives as well. Each of these ideas can be carefully defined, and correspond reasonably well to the equivalent concepts from ordinary mathematics. The analogy does not stop there: one has an entire branch of supermathematics, where the analog of Euclidean space is superspace, the analog of a manifold is a supermanifold, the analog of a Lie algebra is a Lie superalgebra and so on. The Grassmann numbers are the underlying construct that make this all possible. 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. However, supermathematics takes on a special significance in physics, because the anti commuting behavior can be strongly identified with the quantum mechanical behavior of fermions: the anti commutation is that of the Pauli exclusion principle. Thus, the study of Grassmann numbers, and of supermathematics, in general, is strongly driven by their utility in physics. Specifically, in quantum field theory, or more narrowly, second quantization, one works with ladder operators that create multi particle quantum states. 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. In this situation, a Grassmann number corresponds immediately and directly to a wave function that contains some (typically indeterminate) number of fermions. 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. In essence, the Grassmann numbers can be thought of as discarding the relationships arising from spin, and keeping only the relationships due to anti commutation.

Two distinct kinds of supernumbers commonly appear in the literature: those with a finite number of generators, typically 1=n = 1, 2, 3 or 4, and those with a countably infinite number of generators. These two situations are not as unrelated as they may seem at first. First, in the definition of a supermanifold, one variant uses a countably infinite number of generators, but then employs a topology that effectively reduces the dimension to a small finite number. In the other case, one may start with a finite number of generators, but in the course of second quantization, a need for an infinite number of generators arises: one each for every possible momentum that a fermion might carry.

Products of an odd number of Grassmann variables anti commute with each other; such a product is often called an a number. Products of an even number of Grassmann variables commute (with all Grassman numbers); they are often called c numbers. By abuse of terminology, an a number is sometimes called an anticommuting c number. This decomposition into even and odd subspaces provides a grading on the algebra; thus Grassmann algebras are the prototypical examples of supercommutative algebras. Note that the c numbers form a subalgebra of , but the a numbers do not (they are a subspace, not a subalgebra). The definition of Grassmann numbers allows mathematical analysis to be performed, in analogy to analysis on complex numbers. That is, one may define superholomorphic functions, define derivatives, as well as defining integrals. 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 It can be shown that this is essentially equivalent to an algebra with m generators, but this requires work.

The spinor space is defined as the Grassmann or exterior algebra of the space of Weyl spinors (and anti spinors ), such that the wave functions of n fermions belong in .

Grassmann numbers can 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 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.