# Questions about the equivalent forms of Wick's theorem?

NOTE: The problems have been editing with more details.

I have met Wick's theorem first in this book fundamentals of many-body physics when talking about the perturbation expansion of zero temperature Green's function. Later in the perturbation expansion of finite temperature Green's function, I would meet this theorem again. In both expansions, the Wick's theorem is only considered as an operator identity;

1. Zero temperature:

$$\langle 0| T \prod_{i=1}^{2n} \hat{a}_i |0\rangle= \sum (-1)^P \prod \langle 0|T[\hat{a}_j \hat{a}_k]|0\rangle$$

in which $$\hat{a}_i$$ is a creation or annihilation operator, and the sum is over all possible ways picking $$n$$ pairs from among the $$2n$$ operators. And note that the quantum mechanical average is taken to the noninteracting ground state $$|0\rangle$$.

2. Finite temperature:

$$\langle T \prod_{i=1}^{2n} \hat{a}_i \rangle_0 = \sum (-1)^P \prod \langle T[\hat{a}_j \hat{a}_k] \rangle_0$$

in which $$\hat{a}_i$$ is a creation or annihilation operator, and the sum is over all possible ways picking $$n$$ pairs from among the $$2n$$ operators. And note that the ensemble average is taken to the noninteracting density matrix $$\rho_0$$. . There you can see the theorem has nothing to do with the system Hamiltonian that you are caring, no matter the Hamiltonian is quartic (interacting) or quadratic (noninteracting).

Then in this paper Expansion of nonequilibrium Green's functions (by Mathias Wagner ), the author told a general Wick's theorem proved by Danielewicz (Danielewicz, Ann, Phys.152,239(1984)). The general Wick's theorem guarantees the equivalence of the two statements:

1. Wick's theorem holds exactly (in the form of operator identity);
2. The operators to be averaged are noninteracting and the initial density matrix is a one-particle density matrix.

I have struggled with the Danielewicz's paper a few days but I still cannot figure out the deep connection between these two statements, can anybody help me to work out the proof of Danielewicz? Any supporting materials to his proof will also be appreciated. Furthermore, what's the connection between Danielewicz's version and zero/finite temperature version?

Danielewicz's version:

$$\langle \hat{A} \hat{B} \cdots \hat{Y}\hat{Z} \rangle =\hat{A}^{\bullet}\hat{B}^{\bullet}\cdots \hat{Y}^{\bullet\bullet}\hat{Z}^{\bullet\bullet}+\hat{A}^{\bullet}\hat{B}^{\bullet\bullet}\cdots \hat{Y}^{\bullet}\hat{Z}^{\bullet\bullet}+\cdots=\text{sum over all possible contracted pairs}$$ in which the contraction is: $$\hat{A}^\bullet \hat{B}^\bullet= \langle \hat{A} \hat{B}\rangle_0.$$

The following is Danielewicz's proof:

• Not sure you're allowed to reproduce that many pages of the paper here...