Quantization in terms of Bogolyubov Group Variables
Oleg Antonovich Khrustalev, S. Yu. Vernov, Маргарита Владимировна Чичикина
The purpose of the present paper is to consider the problem of quantization close to a nontrivial classical field. It is well-known that careless introduction of the classical fields leads to violation of the conservation laws and to the zero-mode problem, which reflects incompatibility between the adopted perturbation theory and the symmetry properties of the system. Bogolyubov group variables are some operators, which can be treated as parameters of the system symmetry group. For example, let’s consider a system with translational symmetry. After Bogolyubov transformation a static classical solution σ(x) transforms in σ(x− â), where â is an operator, canonical conjugate to an operator of the total linear momentum. Group variables are cyclic due to Hamiltonian symmetry, so the conservation laws are accomplished. For a system with adiabatic coupling the problem of combining of the conservation laws with perturbation theory has been solved by N.N. Bogolyubov in 1950, when he studied the polaron problem [1]. The further investigations have shown that Bogolyubov group variables have a wide range of application. These variables have been used in the strong coupling theory [2–8], the relativistic theory of gravitation [9] and other fields of the theoretical physics [10–22]. The Bogolyubov method has been independently rediscovered in the middle of 1970s and named collective coordinate method [23–26]. Connection between various variants of this method has been discussed in [10]. It was revealed however that consideration of systems with a nonstationary classical components is a rather nontrivial problem, because explicit Hamiltonian structure as the time-translation generator becomes clear only after solving the motion equations. So we have to construct such Bogolyubov’s scheme that group variables are defined in
•Jan 1, 2000•0 citations