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Consider a collection of creation $a^\dagger$and annihilation operators $a$. In physics one defines Wick ordering (also known as normal ordering) as a prescription to place all creation operators before annihilation operators and this is commonly denoted by a set of dots $::$. For example, $$:a^\dagger a a a^\dagger a: = a^\dagger a^\dagger a a a.$$ The motivation for this is to have finite vacuum expectation values for quantum fields that are built out of creation and annihilation operators.

In rigorous approaches to quantum field theory (such as the books Quantum Physics: A Functional Integral Point of View by Glimm & Jaffe or The $P(\Phi)_2$ Euclidean (Quantum) Field Theory by Barry Simon) they also use an operation known as Wick ordering (or Wick powers/products), which I believe originates from probability theory.

I have seen various definitions for what the rigorous/probabilistic version of Wick ordering is. On Wikipedia and in Simon's book it is defined recursively as follows: suppose $X_1 \ldots X_n$ are random variables with finite moments. Then the Wick product $:X_1 \ldots X_n$: is defined so that

  1. The empty Wick product is 1, i.e. $:~: = 1$
  2. $$\frac{\partial}{\partial X_i}:X_1\ldots X_n:~ = ~:X_1\ldots X_{i-1}\hat{X}_{i}X_{i+1}\ldots X_n:$$ where $\hat{X_i}$ means $X_i$ is omitted.
  3. Each Wick product has zero expectation: $$\langle :X_1\ldots X_n:\rangle = 0$$

Another definition is in terms of Hermite polynomials, as presented in this arxiv paper or in Glimm & Jaffe's book. Yet another definition is given in Glimm & Jaffe's book, which is an orthogonal projection onto a subspace of Fock space.

I am getting confused with all these different definitions. The answer to this question gives some great motivation, but from a mathematical point of view I'm still not clear what exactly Wick ordering is doing (especially when viewed as an orthogonal projector). What is its relationship to the physicist's prescription of moving all creation operators to the left and annihilation operators to the right? Are all these definitions equivalent?

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    $\begingroup$ one thing to keep in mind, when considering different definitions of Wick ordering: the ordering is defined relative to a vacuum state (ground state of a Hamiltonian), which is the state that is implicit in the equation $\langle :X_1\ldots X_n:\rangle = 0$; a different vacuum state will produce a different Wick ordering; this applies to the physics context when there is superconducting pairing; then the prescription "move all creation operators to the left and annihilation operators to the right" has to be changed. $\endgroup$
    – Carlo Beenakker
    Commented Apr 26 at 9:22
  • $\begingroup$ @CarloBeenakker Thank you, so in probabilistic terms this would mean the Wick ordering depends on the probability measure (where the probability measure is defined using the ground state of the Hamiltonian)? $\endgroup$
    – CBBAM
    Commented Apr 26 at 15:52
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    $\begingroup$ indeed, that is how I understand it. $\endgroup$
    – Carlo Beenakker
    Commented Apr 26 at 15:56
  • $\begingroup$ @CarloBeenakker Thank you. With this insight what I understand so far is the purpose of Wick ordering is to redefine polynomials so that they are orthogonal with respect to a Hamiltonian dependent probability measure. This is done by either the recursive definition above or equivalently using Hermite polynomials (both of which agree with the physicist's idea of "move all creation operators to the left and annihilation operators to the right"). This orthogonality is important since without it we would not get vanishing vacuum expectation values. $\endgroup$
    – CBBAM
    Commented Apr 26 at 16:30
  • $\begingroup$ @CarloBeenakker Assuming my understanding is correct, then as a follow up how would one define Wick ordering for operators that are not multiplication operators, where integration against the probability measure isn't well-defined? $\endgroup$
    – CBBAM
    Commented Apr 26 at 16:30

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