Misunderstanding Wick Ordering

Question

In M. Salmhofer's "Renormalization, An Introduction" Wick ordering is defined as follows:

Let $C = C_\Gamma$ be a nonnegative symmetric operator on $\mathbb{C}^\Gamma$. For $J: \Gamma \to \mathbb{C}$, let

$$\mathcal{W}_\Gamma (J, \phi) = e^{i(J,\phi)_\Gamma + \frac{1}{2} (J, CJ)_\Gamma},$$

where $\Gamma = \Gamma_{\varepsilon,L} = \varepsilon \mathbb{Z}^d / L \mathbb{Z}^d$. Let $\mathcal{P}_\Gamma$ be the algebra of polynomials in $(\phi(x))_{x \in \Gamma}$. Wick ordering is the $\mathbb{C}$-linear map $\Omega_C: \mathcal{P}_\Gamma \to \mathcal{P}_\Gamma$, that takes the following values on the monomials:

$$\Omega_C(1) = 1,$$

and for $n \geq 1$ and $x_1, \dots, x_n \in \Gamma$ (not necessarily distinct)

$$\Omega_C(\phi(x_1) \dots \phi(x_n)) = \left[ \prod_{k=1}^n \frac{1}{i} \frac{\delta}{\delta J(x_k)} \mathcal{W}_\Gamma (J, \phi) \right]_{J=0},$$

where $\frac{\delta}{\delta \phi(x)} = \varepsilon^{-d} \frac{\partial}{\partial \phi(x)}$.

Question 1: What exactly does the $J=0$ part in the last equation do? Also, I don't quite understand the part $\frac{\delta}{\delta J(x_k)}$. We multiply over $k$ and $J$ is a map $\Gamma \to \mathbb{C}$, so $J(x_k) \in \mathbb{C}$ for all $k$. How does it make sense to derive with respect to a constant complex number?

Later, the following theorem is proven (I will only copy the relevant part):

For all $n \geq 1$:

$$\int \, \mathrm d \mu_C(\phi) \; \Omega_C(\phi(x_1) \dots \phi(x_n)) = 0,$$

where $\mathrm d \mu_C(\phi) = (\det 2 \pi C)^{-1/2} e^{-\frac{1}{2} (\phi, C^{-1} \phi)} \mathrm d^N \phi$.

Question 2: Does this imply that the integral of any Wick ordered polynomial vanishes since Wick ordering is $\mathbb{C}$-linear?

Thank you in advance for any help.

Answer 1

Regarding your first question: what is written down in Salmhofer's book is a very technical approach to the generating functional method. It allows one to express correlation functions as functional derivatives with respect to some auxiliary field $J$, which is set to zero before arriving at the final result. A less mathematically rigorous explanation is given in chapters 6-8 of Srednicki and chapter 9 of Peskin and Schroeder.

Regarding your second question and Wick ordering in general: it implies that the integral of any Wick ordered polynomial vanishes under the given measure of integration, i.e. a gaussian measure for a given correlation function $C$. For a discussion of the formalism, I would refer to these notes.