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. General description and properties Grassmann numbers are individual elements or points of the exterior algebra generated by a set of Grassmann variables or Grassmann directions or supercharges , with possibly being infinite. The usage of the term "Grassmann variables" is historic; they are not variables, per se; they are better understood as the basis elements of a unital algebra. The terminology comes from the fact that a primary use is to define integrals, and that the variable of integration is Grassmann-valued, and thus, by abuse of language, is called a Grassmann variable. Similarly, the notion of direction comes from the notion of superspace, where ordinary Euclidean space is extended with additional Grassmann-valued "directions". The appellation of charge comes from the notion of charges in physics, which correspond to the generators of physical symmetries (via Noether's theorem). The perceived symmetry is that multiplication by a single Grassmann variable swaps the grading between fermions and bosons; this is discussed in greater detail below. The Grassmann variables are the basis vectors of a vector space (of dimension ). They form an algebra over a field, with the field usually being taken to be the complex numbers, although one could contemplate other fields, such as the reals. The algebra is a unital algebra, and the generators are anti-commuting: Since the are elements of a vector space over the complex numbers, they, by definition, commute with complex numbers. That is, for complex , one has The squares of the generators vanish: