Bogolyubov

In the case of finite temperature a similar approach can be used based on the boundary integral method, where instead of the zero temperature Green s function, finite-temperature Green s function derived within TFD formalism is used. Introducing finite-temperature within the thermofield dynamics formalism is based on two steps, doubling of the Hilbert space and Bogolyubov transformations (Takahashi et.ah, 1996 Ademir, 2005). 

The main idea of TFD is the following (Santana, 2004) for a given Hamiltonian which is written in terms of annihilation and creation operators, one applies a doubling procedure which implies extending the Fock space, formally written as Ht = H H. The physical variables are described by the non-tilde operators. In a second step, a Bogolyubov transformation is applied which introduces a rotation of the tilde and non-tilde variables and transforms the non-thermal variables into temperature-dependent form. This formalism can be applied to quite a large class of systems whose Hamiltonian operators can be represented in terms of annihilation and creation operators. 

First we need to rewrite the non-tilde part of the Hamiltonian in the temperature-dependent form using the Bogolyubov transformations which are given by... 

MSN. 173. I. Prigogine, Non-locality and superluminosity, in Proceedings Bogolyubov Conference, Moscow, Sept. 1999. 

E, Bogolyubov Institute for Theoretical Physics, Ukrainian National Academy of Sciences, Kiev, 1994. 

A. Problem of Bound States—Generalization of the Bogolyubov Condition... 

Previously we have considered the reduced density matrices and the equations of motion. These quantities are the representation of the reduced density operators (Bogolyubov,6 Gurov7) defined by... 

In order to get a close equation we must determine the collision integral, that is, we must determine Fn as a function of F (or/j, respectively). For this purpose we can use the formal solution of the Bogolyubov hierarchy, especially for Fn ... 

To use this solution it is necessary to solve two problems. The first problem is the determination of F12(/0). Usually, for deriving a closed equation for F,(f), Bogolyubov s condition of the complete weakening of the initial correlation is used. This is given by8... 

We solve these equations using the Bogolyubov condition ... 

Clearly, it is necessary to generalize the Bogolyubov condition (2.4). It is obvious in generalizing (2.4) to assume that in systems with two-particle bound states, the pair 1 and 2 moved independently of all other particles as t—> -< . Then it is possible to write... 

F c2 is the contribution to F12(f) for pairs in scattering states, and Fb2 is the part for pairs in bound states. Now, in the sense of Bogolyubov, we use for the scattering states the condition of the total weakening of initial correlation ... 

Let us now consider the Bogolyubov condition for the three-particle density operator. In generalizing (2.4), the Bogolyubov condition now takes the form (if three-particle bound states are absent)... 

As in Section II, the initial values F12(t0) and F123(/0) for the binary and the three-particle density operators must be determined. For this purpose we have to generalize the Bogolyubov condition of the weakening of initial correlations given by (2.4) for systems that do not support the formation of bound states. 

The next problem is the determination of the initial value of the three-particle density operator. Using the generalized Bogolyubov asymptotic condition in the approximation (3.39), we obtain finally... 

N. N. Bogolyubov, Problems of the Dynamical Theory in Statistical Mechanics. Gostechisdat, Moscow, 1946. See also Ref. 6, Vol. 2. 

The theory of two band superconductivity, including the configuration interaction of pairs of oppositive spin and momentum in the a-band and b-band, was developed on basis of the Bogolyubov transformations [9-13] where the many body wave function [14] is given by... 

N.N. Bogolyubov, Selected Works on Statistical Physics, Moscow University, Moscow, 1979 (in Russian). 

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