Bogoliubov transformation

Concepts in quantum field approach have been usually implemented as a matter of fundamental ingredients a quantum formalism is strongly founded on the basis of algebraic representation (vector space) theory. This suggests that a T / 0 field theory needs a real-time operator structure. Such a theory was presented by Takahashi and Umezawa 30 years ago and they labelled it Thermofield Dynamics (TFD) (Y. Takahashi et.al., 1975). As a consequence of the real-time requirement, a doubling is defined in the original Hilbert space of the system, such that the temperature is introduced by a Bogoliubov transformation. 

It should be noticed that a((3) and a((3) satisfy the same algebraic relation as those given in Eq. (3), and also that a(/3) 0(/ )) = a(j3) 0(/ )) = 0. Then the thermal state 0(/3)) is a vacuum for a((3) and a(/3) (otherwise, 0,0) is the vacuum for the operators a and a). As a result, the thermal vacuum average of a non-thermal operator is equivalent to the Gibbs canonical average in statistical physics. As a consequence, the thermal problem can be treated by a Bogoliubov transformation, such that the thermal state describes a condensate with the mathematical characteristics of a pure state. 

Generalized Bogoliubov transformations and applications to Casimir effect... 

Abstract. Within the context of the Thermofield Dynamics, we introduce generalized Bogoliubov transformations which accounts simultaneously for spatial com-pactification and thermal effects. As a specific application of such a formalism, we consider the Casimir effect for Maxwell and Dirac fields at finite temperature. Particularly, we determine the temperature at which the Casimir pressure for a massless fermionic field in a cubic box changes its nature from attractive to repulsive. This critical temperature is approximately 100 MeV when the edge of the cube is of the order of the confining length ( 1 fm) for baryons. 

Keywords Thermofield dynamics, Bogoliubov transformation, Compactification,... 

In this talk, we consider the TFD approach for free fields aiming to extend the Bogoliubov transformation to account also for spatial compactification effects. The main application of our general discussion is the Casimir effect for cartesian confining geometries at finite temperature. 

The main goal of this talk is to show that the Bogoliubov transformation of TFD can be generalized to account for spatial compactification and thermal effects simultaneously. These ideas are then applied to the Casimir effect in various cases. 

These results demonstrate explicitly the usefulness of the Bogoliubov transformation to treat confined fields in the context of TFD. From the above considerations, a question emerges naturally what should be the appropriate generalization of the Bogoliubov transformation to account for simultaneously space compactification and thermal effects ... 

We have shown that generalizations of the TFD Bogoliubov transformation allow a calculation, in a very direct way, of the Casimir effect at finite temperature for cartesian confining geometries. This approach is applied to both bosonic and fermionic fields, making very clear the... 

Defining the time- and temperature-dependent annihilation and creation operators through the Bogoliubov transformation... 

Using TFD we are able to find the thermal expectation values of operators. In general, through the Bogoliubov transformation from ai (t), aa(t) to al(P,t),aa(P,t), we find the formula... 

As seen, the SR is to express a matrix element of an operator, between states which are not (usually) of HS nature, by a sum of matrix elements of related operators taken between HS states. The SR formula of (1) is well known see, e.g., Lawson [14]. We gave it, however, in a form which is different from what is usually employed. Namely we expressed the coefficients on the rhs of (la) in terms of the U and V factors. The reason we chose this form is that we want to emphasize the fact that (1) is very similar to what we get when we perform the usual Bogoliubov transformation, i.e., use BCS. In fact, we obtain... 

The mean-fi eld solution of the above model consists of (i) making the Fourier transform of even a-operators and odd a-operators separately, which gives, respectively, ae(k) and a0(k) and (ii) introducing the following Bogoliubov transformations ... 

Uncoupled auxiliary fields are considered at the zeroth order, where the spinon field is diagonalized by applying the Bogoliubov transformation for bosons [4] ... 

The narrow stripon band splits in the SC state, through the Bogoliubov transformation, into the EL(ft) and +(k) bands, given in Eq. (20). The states in these bands are created, respectively, by pl(k) and p+(k), which are expressed in terms of creation and annihilation operators of stripons of the two pairing subsets [see Eq. (16)] through equations of the form ... 

This Bogoliubov transformation is equivalent to a real rotation by an angle cos 1(Ak) it yields... 

Ground state of a weakly interacting Bose gas at zero temperature the Bogoliubov transformation and two-mode squeezing... 

We then ask whether //can be diagonalized by a Bogoliubov transformation, i.e., whether there exist y p) (fermi creation and annihilation operators) and E p) such that... 

Recently Sokolov and Chan published a paper where another quasiparticle-based framework was outlined [40], This approach has the enviable feature of Fermi-vacnnm independence. In that paper the concept of a non-particle-number-conserving canonical transformation was introduced and, as a low-order approximation, the application of a Bogoliubov transformation in a second-order perturbation theory was investigated to describe MR situations. The presented results, as well as the intruder problem on the PES in the BeH2 model, indicate that the applied approximation needs further improvements. 

The situation can be gready simplified by translating into a basis where we replace a system of strongly interacting particles by one of non-interacting quasiparticles by means of the Bogoliubov transformation. The details shall be left to the reader to find in one of the many undergraduate textbooks available (e.g., [8-10]), here we shall concentrate on the interpretation. 

Next, one may evaluate the average interparticle interaction in Eq. (25) by unfolding it through the Bogoliubov transformations ... 

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