Generalized normal ordering Wick theorem

For the formulation of the generalized Wick theorem corresponding to the generalized normal ordering, we need the matrix element rf, Eq. (65), of the one-hole density matrix and the cumulants kjc, Eqs. (39)-(47), of the fc-particle density matrices. 

Generalized normal ordering is intimately linked to the cumulants Xk of the k-particle density matrices (for short, density cumulants). The contractions in the sense of the generalized Wick theorem involve the... 

Tensors, from the same or different fields, can be combined by outer multiplication, denoted by juxtaposing indices with order preserved on the resultant tensor.33 It is possible that an index is present both in the covariant and contravariant index sets then with the repeated index summation convention, both are eliminated and a tensor of lower rank results. The elimination of pairs of indices is known as contraction, and outer multiplication followed by contraction is inner multiplication.33 In multiplication between tensors, contractions cannot take place entirely within one normal product (i.e., the generalized time-independent Wick theorem see Section IV) hence such tensors are called irreducible. 

A somewhat more general version of Wick s theorem may be developed which involves products of operator strings, some or all of which may be normal-ordered. The original form of Wick s theorem is only slightly modified in that the contractions need be evaluated only between normal-ordered strings and not within them. For example, for a product of two normal-ordered strings, the generalized Wick s theorem says that... 

The generalized form of Wick s theorem (see Eq. [91]) says that this product of normal-ordered operator strings may be written using only contractions between the two strings. That is,... 

Since each term of the operators Yr, H and exp(T) o is in normal order, using the generalization of our extended Wick s theorem- eq.(36) we may write each of the expectation values on the left and right side of eq.(60) as a sum over completely contracted quantities. This should exclude contractions within each group of Yr, H and exp(T) o. 

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