Wick’s theorem, generalized

Expressed in words, the product of creation and annihilation operators is equal to a normal product of these operators plus the sum of normal products with one contraction plus the sum of normal products with two contractions etc., up to the normal product where all operators are contracted. This theorem can be generalized (to the so-called generalized Wick s theorem) in the way that also a product of the following form can be handled ... 

We already know that in order to calculate expression (74) using the generalized Wick s theorem, we have to perform contractions between the -products on the right hand side of Eq. (74). We also know that according to Eq. (37) all the operators have to be contracted. Here the diagrams can be introduced because these contractions can be represented diagrammatically. 

According to the generalized Wick s theorem, we can form new diagrams ( -diagrams in notation of Ref.48)) on the left hand side of Eq. (126) that connect the H-and T-diagrams. We can distinguish two types of R-diagrams ... 

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 theorem (3.109) can be extended to the so-called generalized Wick s theorem. This extension allows products of the following form to be handled ... 

Since DN(UV-- -XY)Z = N(DUV- - -XY)Z, the theorem is proved for n + 1 factors. This lemma can be generalized by multiplying both sides of Eq. (10-196) by an arbitrary number of contracted factors, and using Eq. (10-195) to bring these factors within the N products. Wick s theorem now states that a T product can be decomposed into a unique sum of normal products as follows ... 

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,... 

As an example, consider the CCSD energy equation derived earlier in Eq. [134] using Wick s theorem. Each term of the general expression... 

Because of the validity of eqs. (29) and (30), the completely contracted 9 s in any o can be brought adjacent to one another with a proper parity factor. We shall make use of this generalized version of our new Wick s theorem in our development of the Coupled Cluster theory in terms of tfa. The hamiltonian for the electronic system may be written as... 

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. 

Needless to say, it was the second quantization formalism of quantum field theory, enabling the exploitation of Wick s theorem together with a representation via Feynman-like graphs or diagrams—the mathematical techniques relied upon by all the above authors [32-34]—that made it possible to carry out the general proof of the extensive nature of RSPT and to unscramble the general structure of MBPT wave functions and energies. The principal results of these efforts are usually referred to as the linked cluster and connected cluster theorems (see below). 

However, for Fermi superoperators life is more complicated. The anticommutator corresponding to only the left or the right Fermi superoperators are numbers but that for the left and right superoperators, in general, is not a number. Thus, the Fermi superoperators are not Gaussian. However, since the left and right superoperators always commute, the following Wick s theorem [49] can be applied to the time-ordered product... 

Apart from the necessity to project onto the physical subspace Fj, the form of the slave boson Hamiltonian H b has the advantage that usual many-body techniques, like the generalization of Wick s theorem to finite temperatures, can be used to produce a systematic perturbation expansion in the mixing term. Coleman performs such an expansion neglecting the restriction on Q and carries out the projection onto the 2 = 1 subspace in the end of the calculation. For that purpose he extended Abrikosov s scheme (Abrikosov 1965, Barnes 1976) of weighting the 2 > 1 states by... 

The general rules for the evaluation of matrix elements can be formulated in a more strict manner. They are contained in the so-called Wick s theorem which we will not use directly in its original formulation (Wick 1950). The algebraic properties of creation and annihilation operators given by Eqs. (2.48-50), or the above rules based on these properties, give sufBcient information to evaluate any matrix element we need. For some further reading, we refer to the more advanced... 

Analogous relations hold for three-fermion terms, which also occur in the transformed Hamiltonian. After complex application of the Wick s theorem on all fermion operators we get the normal form of the Hamiltonian in the general representation. 

This is demonstrated by using the anticommutation rules on all types of pair. In words, any 2-tiictor product can be written as the normal product plus its contraction. Wick s theorem is the generalization of this result to products of n factors, and may be proved by induction, starting from the case n = 2. The theorem is as follows. 

Clearly Wick s theorem achieves in a very general way results that could be obtained step by step, using the anticommutation rules. The great importance of (9.2.8) lies in the fact that, since normal products operating on 0) always give zero, the expectation value of any product arises only from fully contracted products. For example, the product considered above has zero expectation value because it contains no non-zero fully contracted terms. On the other hand, b,.b,bjbj would admi two potentially non-zero contractions, namely b b bjbj and b,bjbJbJ, and hence... 

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