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I know the relationship between normal and correlated two-particle Green Functions for fermions: $$G_c(1,2,3,4)=\Gamma(1,2,3,4)=G(1,2,3,4)+G(1,3)G(2,4)-G(1,4)G(2,3)$$ Also known as irreducible n-particle Green function or n-particle vertex (there are many definitions so I'm confused).

Definitions for G also vary, I use this one: $G(1..2n) = (-1)^{n}\langle T\ c_1\dots c_n c_{n+1}^\dagger\dots c_{2n}^\dagger \rangle$

I need a similar relationship for three-particle functions.

Now I have this formula: $$ \Gamma(1,2,3,4,5,6) = G(1, 2, 3, 4, 5, 6) -2 G(1, 4) G(2, 5) G(3, 6) +2 G(1, 4) G(2, 6) G(3, 5) -2 G(1, 5) G(2, 6) G(3, 4) +2 G(1, 5) G(2, 4) G(3, 6) -2 G(1, 6) G(2, 4) G(3, 5) +2 G(1, 6) G(2, 5) G(3, 4) -G(1, 4) G(2, 3, 5, 6) +G(1, 5) G(2, 3, 4, 6) -G(1, 6) G(2, 3, 4, 5) +G(2, 4) G(1, 3, 5, 6) -G(2, 5) G(1, 3, 4, 6) +G(2, 6) G(1, 3, 4, 5) -G(3, 4) G(1, 2, 5, 6) +G(3, 5) G(1, 2, 4, 6) -G(3, 6) G(1, 2, 4, 5)$$ But it doesn't always satisfy Wick's theorem for the Hamiltonian I'm working with. Is this formula correct?

Also it would be great to get an explanation of how these formulas appear in many-body theory, from logarithm of time-ordered exponential.

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Why do you lack $-G(1,3) G(2,4)$ is your fist equation? General rules for cumulant expansion/Wick pairing contain all possible pairings, unless there are problem-specific reasons for cancellation. –  Slaviks May 3 '13 at 17:14
    
@Slaviks It's annihilation-creation operator pairs, so there is just two of them. –  swish May 3 '13 at 17:39

1 Answer 1

up vote 1 down vote accepted

I was able to confirm both of your formulas, assuming that the operators are fermionic and normal ordered.

Here is the general algorithm for generating a cumulant average of $n$ variables (taken from page page 34 of G.W.Gardiner, Stochastic Methods: A Handbook for the Natural and Social Sciences, attributed to van Kampen):

$$G_c(X_1,X_2,\ldots, X_n)=\sum_{p=0}^{n-1}(-1)^p C_p(X_1,X_2,\ldots,X_n)$$ where $$C_p(X_1,X_2,\ldots,X_n)=G(X_1) G(X_2,X_3,\ldots,X_n)+ \ldots$$ is the sum over all distinct partitions of $n$ variables into $p$ subsets, with the averaging function $G$ applied to each subset. If the operators are fermionic, then each term in $C_p$ has to be multiplied by the sign of the required permutation.

Here are references to the original papers:

  1. E. Meeron, J. Chem. Phys. 27, 1238 (1957); DOI:10.1063/1.1743985
  2. N.G. van Kampen, Physica 74, 215 (1973); 74, 239 (1973.)

Now the computer algebra: using this answer, it is easy to write a Mathematica code computing $C_p$ and thus $G_c$ for the general case.

Assuming that all your variables up to 3 are creation operators, and from 4 to 6 are are annihilation operators (or vice versa), and that the particle number is well-defined, we have to keep averages only of an equal number of creation and annihilation operators. In this way I can confirm the validity of each term in both formulas quoted.

As "proof of work" I can offer you the connected function of 8 variables (hope it does not break MathJax in your browser): $$G_c(1,2,3,4,5,6,7,8)=-6 G(1,8) G(2,7) G(3,6) G(4,5)+6 G(1,7) G(2,8) G(3,6) G(4,5)+6 G(1,8) G(2,6) G(3,7) G(4,5)-6 G(1,6) G(2,8) G(3,7) G(4,5)-6 G(1,7) G(2,6) G(3,8) G(4,5)+6 G(1,6) G(2,7) G(3,8) G(4,5)+2 G(3,8) G(1,2,6,7) G(4,5)-2 G(3,7) G(1,2,6,8) G(4,5)+2 G(3,6) G(1,2,7,8) G(4,5)-2 G(2,8) G(1,3,6,7) G(4,5)+2 G(2,7) G(1,3,6,8) G(4,5)-2 G(2,6) G(1,3,7,8) G(4,5)+2 G(1,8) G(2,3,6,7) G(4,5)-2 G(1,7) G(2,3,6,8) G(4,5)+2 G(1,6) G(2,3,7,8) G(4,5)-G(1,2,3,6,7,8) G(4,5)+6 G(1,8) G(2,7) G(3,5) G(4,6)-6 G(1,7) G(2,8) G(3,5) G(4,6)-6 G(1,8) G(2,5) G(3,7) G(4,6)+6 G(1,5) G(2,8) G(3,7) G(4,6)+6 G(1,7) G(2,5) G(3,8) G(4,6)-6 G(1,5) G(2,7) G(3,8) G(4,6)-6 G(1,8) G(2,6) G(3,5) G(4,7)+6 G(1,6) G(2,8) G(3,5) G(4,7)+6 G(1,8) G(2,5) G(3,6) G(4,7)-6 G(1,5) G(2,8) G(3,6) G(4,7)-6 G(1,6) G(2,5) G(3,8) G(4,7)+6 G(1,5) G(2,6) G(3,8) G(4,7)+6 G(1,7) G(2,6) G(3,5) G(4,8)-6 G(1,6) G(2,7) G(3,5) G(4,8)-6 G(1,7) G(2,5) G(3,6) G(4,8)+6 G(1,5) G(2,7) G(3,6) G(4,8)+6 G(1,6) G(2,5) G(3,7) G(4,8)-6 G(1,5) G(2,6) G(3,7) G(4,8)+2 G(3,8) G(4,7) G(1,2,5,6)-2 G(3,7) G(4,8) G(1,2,5,6)-2 G(3,8) G(4,6) G(1,2,5,7)+2 G(3,6) G(4,8) G(1,2,5,7)+2 G(3,7) G(4,6) G(1,2,5,8)-2 G(3,6) G(4,7) G(1,2,5,8)-2 G(3,5) G(4,8) G(1,2,6,7)+2 G(3,5) G(4,7) G(1,2,6,8)-2 G(3,5) G(4,6) G(1,2,7,8)-2 G(2,8) G(4,7) G(1,3,5,6)+2 G(2,7) G(4,8) G(1,3,5,6)+2 G(2,8) G(4,6) G(1,3,5,7)-2 G(2,6) G(4,8) G(1,3,5,7)-2 G(2,7) G(4,6) G(1,3,5,8)+2 G(2,6) G(4,7) G(1,3,5,8)+2 G(2,5) G(4,8) G(1,3,6,7)-2 G(2,5) G(4,7) G(1,3,6,8)+2 G(2,5) G(4,6) G(1,3,7,8)+2 G(2,8) G(3,7) G(1,4,5,6)-2 G(2,7) G(3,8) G(1,4,5,6)-2 G(2,8) G(3,6) G(1,4,5,7)+2 G(2,6) G(3,8) G(1,4,5,7)+2 G(2,7) G(3,6) G(1,4,5,8)-2 G(2,6) G(3,7) G(1,4,5,8)+2 G(2,8) G(3,5) G(1,4,6,7)-2 G(2,5) G(3,8) G(1,4,6,7)-2 G(2,7) G(3,5) G(1,4,6,8)+2 G(2,5) G(3,7) G(1,4,6,8)+2 G(2,6) G(3,5) G(1,4,7,8)-2 G(2,5) G(3,6) G(1,4,7,8)+2 G(1,8) G(4,7) G(2,3,5,6)-2 G(1,7) G(4,8) G(2,3,5,6)-G(1,4,7,8) G(2,3,5,6)-2 G(1,8) G(4,6) G(2,3,5,7)+2 G(1,6) G(4,8) G(2,3,5,7)+G(1,4,6,8) G(2,3,5,7)+2 G(1,7) G(4,6) G(2,3,5,8)-2 G(1,6) G(4,7) G(2,3,5,8)-G(1,4,6,7) G(2,3,5,8)-2 G(1,5) G(4,8) G(2,3,6,7)-G(1,4,5,8) G(2,3,6,7)+2 G(1,5) G(4,7) G(2,3,6,8)+G(1,4,5,7) G(2,3,6,8)-2 G(1,5) G(4,6) G(2,3,7,8)-G(1,4,5,6) G(2,3,7,8)-2 G(1,8) G(3,7) G(2,4,5,6)+2 G(1,7) G(3,8) G(2,4,5,6)+G(1,3,7,8) G(2,4,5,6)+2 G(1,8) G(3,6) G(2,4,5,7)-2 G(1,6) G(3,8) G(2,4,5,7)-G(1,3,6,8) G(2,4,5,7)-2 G(1,7) G(3,6) G(2,4,5,8)+2 G(1,6) G(3,7) G(2,4,5,8)+G(1,3,6,7) G(2,4,5,8)-2 G(1,8) G(3,5) G(2,4,6,7)+2 G(1,5) G(3,8) G(2,4,6,7)+G(1,3,5,8) G(2,4,6,7)+2 G(1,7) G(3,5) G(2,4,6,8)-2 G(1,5) G(3,7) G(2,4,6,8)-G(1,3,5,7) G(2,4,6,8)-2 G(1,6) G(3,5) G(2,4,7,8)+2 G(1,5) G(3,6) G(2,4,7,8)+G(1,3,5,6) G(2,4,7,8)+2 G(1,8) G(2,7) G(3,4,5,6)-2 G(1,7) G(2,8) G(3,4,5,6)-G(1,2,7,8) G(3,4,5,6)-2 G(1,8) G(2,6) G(3,4,5,7)+2 G(1,6) G(2,8) G(3,4,5,7)+G(1,2,6,8) G(3,4,5,7)+2 G(1,7) G(2,6) G(3,4,5,8)-2 G(1,6) G(2,7) G(3,4,5,8)-G(1,2,6,7) G(3,4,5,8)+2 G(1,8) G(2,5) G(3,4,6,7)-2 G(1,5) G(2,8) G(3,4,6,7)-G(1,2,5,8) G(3,4,6,7)-2 G(1,7) G(2,5) G(3,4,6,8)+2 G(1,5) G(2,7) G(3,4,6,8)+G(1,2,5,7) G(3,4,6,8)+2 G(1,6) G(2,5) G(3,4,7,8)-2 G(1,5) G(2,6) G(3,4,7,8)-G(1,2,5,6) G(3,4,7,8)+G(4,8) G(1,2,3,5,6,7)-G(4,7) G(1,2,3,5,6,8)+G(4,6) G(1,2,3,5,7,8)-G(3,8) G(1,2,4,5,6,7)+G(3,7) G(1,2,4,5,6,8)-G(3,6) G(1,2,4,5,7,8)+G(3,5) G(1,2,4,6,7,8)+G(2,8) G(1,3,4,5,6,7)-G(2,7) G(1,3,4,5,6,8)+G(2,6) G(1,3,4,5,7,8)-G(2,5) G(1,3,4,6,7,8)-G(1,8) G(2,3,4,5,6,7)+G(1,7) G(2,3,4,5,6,8)-G(1,6) G(2,3,4,5,7,8)+G(1,5) G(2,3,4,6,7,8)+G(1,2,3,4,5,6,7,8)$$

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