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. One can say there is a one-to-one correspondence between Green's functions and 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. And the corresponding Wick's theorem has been proved by Matsubara.

• 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;

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;