Motivation for the axioms in Wick product

Question

Here is a link for the definition of Wick product https://encyclopediaofmath.org/wiki/Wick_product, which defines the Wick product recursively. My question is where do these two equations come from? I mean the equations $$ \left\langle: f_{1}^{k_{1}} \cdots f_{n}^{k_{n}}:\right\rangle=0 $$ and $$ \frac{\partial}{\partial f_{i}}\left(: f_{1}^{k_{1}} \cdots f_{n}^{k_{n}}:\right)=k_{i}: f_{1}^{k_{1}} \cdots f_{i}^{k_{i}-1} \cdots f_{n}^{k_{n}}: $$ What is the motivation or intuition behind two equations? Is there a good reference on it?

Answer 1

The Wick product :$A_1A_2A_3$: is a specific way to order noncommuting operators $A_1,A_2,A_3$. The concept was introduced by Gian-Carlo Wick in 1950 to avoid "infinite expectation values" that arise from the zero-point-motion of harmonic oscillators. Basically you reorder the operators in such a way that the expectation value of the reordered product vanishes – that motivates the first of the two equations in the question. The second equation follows because the order of the operators no longer matters once we have reordered them, so we can take the derivative as if these are ordinary functions – rather than non-commuting operators.

A general prescription for Wick ordering in quantum field theory in a creation/annihilation operator formalism is: “Permute all the creation operators $a^\dagger$ and the annihilation operators $a$, treating them as if they commute, so that in the end all $a^\dagger$ are to the left of all $a$.”

An introduction which I found instructive is Wick calculus by Alexander Wurm and Marcus Berg.

Note: Hida and Ikeda introduced a related but distinct concept in probability theory. This is discussed in The Wick product by H. Gjessing et al.