This is a follow-up question to QMechanic's great answer in this question. They give a formulation of Wick's theorem as a purely combinatoric statement relating two total orders $\mathcal T$ and $\colon \cdots \colon$ on an algebra.
I have come across "Wick's theorems" in many contexts. While some of them are special cases of the theorem [1], others are — as far as I can see — not. I am wondering if there is an even more general framework in which Wick's theorem can be presented, showing that all of these theorems are in fact the same combinatoric statement.
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Wick's theorem applies to a string of creation and annihilation operators, as described e.g. on Wikipedia:
$$ ABCD = \mathopen{\colon} ABCD \mathclose{\colon} + \sum_{\text{singles}} \mathopen{\colon} A^\bullet B^\bullet CD \mathclose{\colon} + \cdots \tag{*} $$
Here, the left hand side is "unordered" and it seems to me that [1] is not valid? -
The creation and annihilation operators in (*) can be either bosonic or fermionic.
This technicality is not a problem in [1] since it allows for graded algebras. -
Wick's theorem can also be applied to field operators:
$$ \mathcal T\, \phi_1 \cdots \phi_N = \mathopen{\colon} \phi_1 \cdots \phi_N \mathclose{\colon} + \sum_{\text{singles}} \mathopen{\colon} \phi_1^\bullet \phi_2^\bullet \cdots \phi_N \mathclose{\colon} + \cdots $$
Since the mode expansion of a field operator $\phi_k$ consists of annihilation and creation operators, normal ordering is actually not simply a total order on the algebra of field operators. Once again, we can not apply [1]? -
In a class I am taking right now, we applied Wick's theorem like this to field operators that didn't depend on time:
$$ \phi_1 \cdots \phi_N = \mathopen{\colon} \phi_1 \cdots \phi_N \mathclose{\colon} + \sum_{\text{singles}} \mathopen{\colon} \phi_1^\bullet \phi_2^\bullet \cdots \phi_N \mathclose{\colon} + \cdots $$
This seems to combine the issues of points 1 and 3… -
In probability theory, there is Isserlis' Theorem:
$$ \mathbb E(X_1 \cdots X_{2N}) = \sum_{\text{Wick}} \prod \mathbb E(X_i X_j) $$
This looks like it should also be a consequence from one and the same theorem, but I don't even know what the algebra would be here. -
My string theory lectures were quite a while ago, but I vaguely remember that there we had radial ordering instead of time ordering. Also there seems to be some connection to OPEs.
This seems to not be a problem with [1]. -
In thermal field theory, the definition of normal ordering changes.
This seems to not be a problem with [1] either.
Best Answer
Various comments to the post (v3):
One may speculate that seemingly unordered operators are in practice always ordered wrt. some order.
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As long as the fields $\phi_i=\phi_i^{(+)}+\phi_i^{(-)}$ are linear in creation and annihilation operators, this should not be a problem.
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Isserlis' theorem is related to the path-integral formulation of Wick's theorem, cf. e.g. this Phys.SE post.
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The single-most important generalization of the operator formulation of Wick's theorem (as compared to my Phys.SE answer) is to consider contractions that doesn't belong to the algebra center. This is often used in CFT, see e.g. Ref. 1.
References: