shown that this is essentially equivalent to an algebra with generators, but this requires work. Spinor space 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 . Integration Integrals over Grassmann numbers are known as Berezin integrals (sometimes called Grassmann integrals). In order to reproduce the path integral for a Fermi field, the definition of Grassmann integration needs to have the following properties: linearity partial integration formula Moreover, the Taylor expansion of any function terminates after two terms because , 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 , so Berezin defined 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. In the path integral formulation of quantum field theory the following Gaussian integral of Grassmann quantities is needed for fermionic anticommuting fields, with A being an N × N matrix: . Conventions and complex integration An ambiguity arises when integrating over multiple Grassmann numbers. The convention that performs the innermost integral first yields Some authors also define complex conjugation similar to Hermitian conjugation of operators, With the additional convention we can treat and as independent Grassmann numbers, and adopt Thus a Gaussian integral evaluates to and an extra factor of effectively introduces a factor of , just like an ordinary Gaussian, After proving unitarity, we can evaluate a general Gaussian integral involving a Hermitian matrix with eigenvalues , Matrix representations 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. Generalisations There are some generalisations to Grassmann numbers. These require rules