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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.

  1. 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?

  2. 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.

  3. 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]?

  4. 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...

  5. 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.

  6. 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].

  7. In thermal field theory, the definition of normal ordering changes.
    This seems to not be a problem with [1] either.

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    $\begingroup$ I think the lack of answers, despite the bounty, is coming from the fact that the question is kind of hard to understand. The numbered list, for example, doesn't obviously name one thing per item. Perhaps if you can tighten up the question, then it will be more likely for you to get the answer you're looking for. $\endgroup$ – DanielSank Apr 20 '18 at 7:51
  • $\begingroup$ Thanks for the comment. I'll try to explain it better later! $\endgroup$ – Noiralef Apr 20 '18 at 8:04
  • $\begingroup$ A possible generalisation is to consider $q$-statistics $[A,B]_q:=AB-qBA$. The Bose and Fermi cases correspond to $q=\pm1$. Wick's theorem works for any $q\in[-1,1]$. See e.g. journals.aps.org/prd/abstract/10.1103/PhysRevD.43.4111 $\endgroup$ – AccidentalFourierTransform Apr 20 '18 at 13:59
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Various comments to the post (v3):

  1. One may speculate that seemingly unordered operators are in practice always ordered wrt. some order.

  2. -

  3. As long as the fields $\phi_i=\phi_i^{(+)}+\phi_i^{(-)}$ are linear in creation and annihilation operators, this should not be a problem.

  4. -

  5. Isserlis' theorem is related to the path-integral formulation of Wick's theorem, cf. e.g. this Phys.SE post.

  6. -

  7. -

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:

  1. J. Fuchs, Affine Lie Algebras and Quantum Groups, (1992); eq. (3.1.35).
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I will give an answer to explain why there are too many Wick's theorems in condensed matter physics or many-body physics.

Actually, the importance of Wick's theorem is closely related to the calculation of Green's function. Green's function techniques in condensed matter physics or in many-body physics usually rely on expansion of the Green's function in question (generally contains quartic terms in Hamiltonian) in an infinite series of higher Green's functions for a noninteracting solvable system and a subsequent contraction into products of one-particle Green's function. This decomposition is greatly simplified by the use of suggestive diagrammatic representations. The rigorous foundation of this procedure is known as Wick's theorem.

  • The first meet

We first meet Wick's theorem is to formulate the many-body perturbation expansion of zero-temperature Green's function in which the problem can be described by Hamiltonian: $$H=H_0+H_i$$ where $H_i$ is the complex many-body interaction.

  • The second meet

We will meet Wick's theorem again when we perform the many-body expansion of the finite temeprature Green's function in which the problem can also be described by Hamiltonian $H=H_0+H_i$. The big difference compared to zero temperature Green's function is that the system is no longer in a ground state instead of a mixed state by the density matrix $$\rho = \dfrac{e^{-\beta H}}{Tr[e^{-\beta H}]}.$$ One can see the equilibrium many-body density matrix also contains many-body interactions. To formulate the simultaneous expansion on both density matrix and the time evolution operator: $$U(t)=e^{-i H t/\hbar}$$ Matsubara's strategy: replace $\tau=it$ and treat $\tau$ as a real number. As a result of this replacement, many-body perturbation expansion becomes possible.

  • The third meet

Keldysh formalism: which is suitable for the investigation of nonequilibrium many-body problem. (Here the Wick's theorem is much like zero temperature one.)

  • $\cdots \cdots$

The following links are the recommended literature to prove Wick's theorem and discuss the interrelations between different versions of Wick's theorem.

1.Wick theorem for general initial states;

2.Equilibrium and nonequilibrium many-body perturbation theory;

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  • $\begingroup$ Thank you for your reply! I am not really what your actual answer is though - Are you saying that the Wick's theorems appearing in these situations are too different from each other to understand them as special cases of the same thing? $\endgroup$ – Noiralef Apr 22 '18 at 15:03
  • $\begingroup$ To formulate many-body perturbation expansion. $\endgroup$ – Jack Apr 23 '18 at 0:21

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