What exactly are "Grassmann-valued fields"?

Question

Peskin & Schroeder define a Grassmann field $\psi(x)$ as a function whose values are anticommuting numbers, that can be written as : [p.301 eq. 9.71]

$$\psi(x) = \sum\psi_i \phi_i(x),\tag{9.71}$$

where $\psi_i$ are said to be "grassmann numbers" and $\phi_i(x)$ are ordinary functions. This seems however inconsistent with the definition of Grassmann numbers as given e.g. in Wikipedia, where a Grassmann number is an element of the $2^n$-dimensional algebra generated by a set of $n$ generators. For example, it is the generators who are anti-commuting and not (necessarily) the numbers. As a simple example if we have a Grassmann algebra with two anticommuting generators $\theta_1,\theta_2$ then a a general Grassmann number is $a+b\theta_1 + c\theta_2 + d\theta_1\theta_2$ where $a,b,c,d$ are complex numbers.

So is it correct to say that when P&S say "Grassmann numbers" they actually mean the generators of the Grassmann Algebra, and that the value the field can have at any point is a linear combination of those generators?

My main question is how to understand the concept of a particular configuration of such a field. At first sight it seems that different configurations correspond to different linear combinations of the generators. But then we are supposed to integrate over all configurations in the path integral, so doesn't that mean we should be integrating over the $\phi_i(x)$'s and not the $\psi_i$'s?

Answer 1

1. On one hand, Wikipedia is talking about the underlying basis $(\theta_i)_{i=1,\ldots, n}$ of Grassmann-odd generators for supernumbers. Usually in physics, $n=\infty$. [These basis-elements $\theta_i$ are not needed for practical calculations, and should not be confused with the Grassmann-odd parameters $\theta$ of a superfield $\Phi(x,\theta)$.]

2. On the other hand, a Grassmann-valued field $x\mapsto \eta(x)$ maps a spacetime point $x$ into a Grassmann-odd supernumber. In particular, a Grassmann-valued Dirac field $x\mapsto \psi(x)$ maps a spacetime point $x$ into a 4-tuple of Grassmann-odd supernumbers, which form a Dirac spinor.

3. P&S eq. (9.71) is at best a very deceptive formula. It's incorrect to view the elements $\psi_i$ as the basis elements $\theta_i$. Rather the $\psi_i$ should be an odd-order monomial of the basis elements, and the sum on the right-hand side eq. (9.71) should be over all such odd-order monomials.

4. Concerning the Berezin integral $$\int_{\mathbb{F}^{0|1}} \!d\eta ~f(\eta)~=~\frac{df(\eta)}{d\eta},\tag{1}$$ the integration variable $\eta$ is not a basis elements $\theta_i$, but rather a Grassmann-odd supernumber $\eta$. In particular, the odd-order monomials of basis elements $\theta_i$ inside the integration variable $\eta$ have no saying in assigning the value of the integral (1). [In the integral (1) the underlying field $\mathbb{F}$ is either $\mathbb{R}$ or $\mathbb{C}$.]

5. See also this, this & this related Phys.SE posts.