Operators normal-ordered

Normal ordering operator allowing to pass from function of Bosons to function of complex scalars. 

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 discussed in detail earlier, products of normal-ordered operators can be simplified algebraically using Wick s theorem by evaluating pairs of contractions between the component annihilation and creation operators. Many of these contractions produce mathematically redundant terms that can be combined after complicated manipulation to eventually produce a much simpler expression. Diagrams provide a straightforward scheme by which these redundancies may be eliminated. 

Construct the quantity HN exp(7 ) d>0)c. First we expand exp(T) subject to the limitation on the excitation level in step 1 above. Then we add the correct components of HN to each T vertex or combination of T vertices to retain the correct excitation level and the connected requirement on the diagram. This procedure exploits the fact that the only contractions that need to be considered are those among different normal-ordered operators. 

The total intracavity variances are expressed through the stochastic variables using the relationships between normally-ordered operator averages and stochastic moments with respect to the P-function. Restoring the previous phase structure of intracavity interaction, we obtain that V+ = V- = V and... 

To illustrate the spatial properties of the polarization of multipole radiation, consider the normal-ordered operator polarization matrix (133) in the case of monochromatic electric-type pure y-pole radiation. Assume that the radiation field is in a single-photon state lm) with given m. Then, the average of (133) takes the form... 

U(a,a, t) by using the definition of normal order operator [2], and finally we obtain... 

Evaluation of the operator products as they occur either on the rhs of fhe Bloch Eq. (23) or in fhe definifion of fhe effective operators (15) and (49). The aim of this step is to bring all the creation and annihilation operators (in each term of the expansions) into the extended normal-order form (50). The resulf is a sequence of normal-ordered operator ferms (briefly referred to as Feynman-Goldstone diagrams). 

Arguments normal-ordered operators anti-symmetrized with respect to indices of the same type. Diagram technique can be easily extended to include operator self-contractions, but this is not needed in the CASCC theory. 

Operator representation in general, a normal-ordered operator splits into different terms having different excitation/de-excitation, hole/particle ranks. For example, the normal-ordered Hamiltonian ... 

The above graphical constructs represent individual normal-ordered operators. Using these graphical representations one can derive all unique combinations (operator contractions) which contribute to the considered matrix element of an operator product. The rules for the diagram manipulations are standard and can be found elsewhere [13,51]. General-order coupled cluster equations can be derived from the general-order coupled cluster functional ... 

Finally, in expectation values sequences of annihilation and creation operators stemming from the second-quantized Hamiltonian and from the states in bra and ket of the full bra-ket must be evaluated for which rules such as Wick s theorem, which implements the anticommutation relations of operator pairs to obtain a relation to normal ordered operator products, can be beneficial [65,353]. 

With respect to the old vacuum, the new vacuum is called a dressed vacuum and the states in it are called dressed states. The new particles are called quasiparticles because they are a composite of an old particle and a series of old pairs they are dressed with a series of pairs rather than being bare particles. The new vacuum is also called a polarized vacuum because with respect to the old vacuum there is a charge polarization expressed by the presence of undressed pairs. The concept of vacuum polarization will be discussed more later. Finally, another way of looking at the change in the vacuum is that because the transformation mixes positron creation and electron annihilation operators, a normal-ordered operator in the new basis will definitely not be normal-ordered in the old basis. 

The partition of O Eq. 28.14 is based on the so-called normal ordered operators and, Qn (Cammi 2009). Specifically, Qn(A, T) is the component of the solvent reaction potential due to the correlation CC electronic density, and H(0)n is the normal ordered form of Hamiltonian of the solute in presence of the frozen Hartree-Fock reaction field Qhf-... 

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