Normal product contraction

We now introduce a notation involving a normal product with one or more contracted pairs of factors. If U, V, denote a set of free-field creation and annihilation operators, we define the mixed product by... 

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 ... 

A mixed T product can be decomposed into a unique sum of normal products according to the same rule as an ordinary T product, but with the omission of contractions between factors already in normal product form. 

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Next, we have to define the normal product (n-product) and contraction (pairing). The normal product is defined in the following way ... 

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 ... 

Its expression is similar to Eq. (35) but omits contractions between operators within the same normal product. An important consequence of Eqs. (15) and (18) should be emphasized, that... 

For Y operators, as with X operators, we can now define a normal product(which we shall designate N [ ]), a contraction, as well as a normal product with contrac-... 

According to Eq. (37) we know that in expression (164) all the operators must be contracted. Therefore, if we substitute the normal product form (48) for H, we have to consider only the two-electron part of the Hamiltonian, i.e. we can assume v instead of H in the expression (164). Hence, we have... 

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. 

The wave operator is similar to that used in the single reference theory with the normal product imposed to eliminate contractions among the cluster operators themselves (which are not possible in the single reference case),... 

In eq.(28), we have first rewritten qi s in terms of Q,-, and then reordered the product QiQjQkQi as a sum of normal products, using as the vacuum. The traditional Wick s theorem applied to the products of Q, s will lead to pair contractions in the traditional sense between groups of creation-annihilation operators in one Q, with one or more Qy s. If these contractions completely exhaust all the operators present in the composites Q,-, Qj- etc. involved in the contraction, we denote them by bars and centred or filled circles. Joining by some operators will lead to terms with carets and open circles. Thus, the second and the third term in the braces involve incomplete contractions. The second term has connections between operators of Q and Qj and between Qk and Qi. The third term involves connections between Q,- and Q and between Qj and Q(. The fourth term involves complete connections between all the operators of Qi and Qj. The fifth term involves contraction of all the operators of Q,- with those of Qj and of some between Qk and Q . The sixth term involves complete contractions between operators of Q,- and Qj and of Qk and Qi. The seventh term QiQjQkQi indicates that all the operators of Qii Qji Qk and Qi are contracted among themselves which cannot be factored out to pairs such as QiQj QkQi etc. 

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Next, we have to define the normal product or n-product, and contraction or pairing. The simplification of matrix elements requires that we move creation operators to the left of annihilation operators. A normal product is a reordered operator string which satisfies this requirement. A contraction or pairing of creation and/or annihilation operators is their vacuum expectation value. 

Having defined both a normal product of creation and annihilation operators and the contraction of a pair of these operators, we are now in a position to define a normal product with contractions. Consider, for example, the following simple case ... 

In general, a normal product with contraction is given by... 

This can be expressed in a manner similar to eq. (3.109), but with contractions between operators within the same normal product omitted. 

Note that this contraction is a number and that the creation operator is positioned on the right of the annihilation operator. All other contractions are zero. We can define a normal product with contraction in the particle-hole formulation as ... 

It may be shown that any product can be expressed in terms of normal products. For this purpose two more definitions are needed. First we introduce the term contraction . 

Secondly, we must define the contraction of a normal product. 

A contracted normal product is one in which one or more operator pairs is replaced by the scalar factor obtained by contraction and is withdrawn from the normal product. With each contraction is associated a factor (—1) , where v is the number of interchanges needed to bring the paired operators adjacent. 

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. 

Any product of creation and annihilation operators can be written as the sum of the normal product plus all possible singly, multiply and fully contracted normal products. 

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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