Commutation relations normal

The effective operator A is the state-independent part of the definition AL/3, i = I-III. The operator A can thus be obtained by combining the perturbation expansions of its normalization factors and of A into a single expression [73] or by computing these normalization factors and A separately. These combined and noncombined forms of A[, may differ when computed approximately (see Section VI and paper II). The calculation of with the noncombined form is the same as with A since the model eigenvectors used with A are obtained by multiplying those utilized with A[,p by the above normalization factors. The operators and A are nevertheless different and, thus, do not have necessarily the same properties, for example, the conservation of commutation relations studied in Section IV. 

Therefore, whenever the normal form of the quadrature variance is negative, this component of the field is squeezed or, in other words, the quantum noise in this component is reduced below the vacuum level. For classical fields, there is no unity coming from the boson commutation relation, and the normal form of the quadrature component represents true variance of the classical stochastic variable, which must be positive. 

We now note that the only difference between (153) and commutation relations (23) is the presence of position-dependent factors in the right-hand side of (153). It seems to be quite tempting to introduce the normalized local operators... 

The commutation relations (42) can be used to normalize the A-electron states obtained by acting on (A — 2)-electron states with Kramers pair creation operators [8], Suppose that P) is a properly normalized (N — 2)-electron state which is annihilated by By, i.e., suppose that... 

Below we provide the relation of the pth moment to the pth-order commutator of H and T. In particular, we shall argue that the p = 0 moment, i.e., the normalization of the spectrum corresponds to the limiting short time information on the dynamics. Higher-order moments are seldom sufficient as the Taylor expansion... 

Along this line, in a recent paper [37] we introduced the so-called quasiparticle-based MR CC method (QMRCC). The mathmatical structure of QMRCC is more or less the same as that of the well-known SR CC theory, i.e., the reference function is a determinant, commuting cluster operators are applied, normal-ordering and diagram techniques can be used, the method is extensive, etc. The point where the MR description appears is the application of quasiparticle slates instead of the ordinary molecular orbitals. These quasiparticles are second-quantized many-particle objects introduced by a unitary transformation which allows us to represent the reference CAS function in a determinant-like form. As it is shown in the cited paper, on one hand the QMRCC method has some advantages with respect to the closely related SR-based MR CC theory [22, 31, 34] (more... 

To sum up we note that the present level of formulation does not distinguish between classical- and quantum mechanics. A further characteristic reveals biorthogonality implying that the coefficients c,- are not to be associated with a probability interpretation, since they obey the rule Cj + c = 1. As emphasized, the operators in Eq. 1.63 are principally non-self-adjoint and non-normal and hence they might not commute with each other as well as their own adjoint. The order appearing in the resulting operator relations therefore has to be respected. 

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