Nikolai N. Bogolubov

The Relativistic Electrodynamics Least Action Principles Revisited: New Charged Point Particle and Hadronic String Models Analysis

The classical relativistic least action principle is revisited from the vacuum field theory approach. New physically motivated versions of relativistic Lorentz type forces are derived, a new relativistic hadronic string model is proposed and analyzed in detail. The reasonings of R. Feynman, who argued that the relativistic dynamical expressions obtain true physical sense only with respect to the proper rest reference frames, are supported by analyzing the dynamical stability of a relativistic charged string model.

The electromagnetic Lorentz condition problem and symplectic properties of Maxwell-and Yang-Mills-type dynamical systems

Symplectic structures associated to connection forms on certain types of principal fiber bundles are constructed via analysis of reduced geometric structures on fibered manifolds invariant under naturally related symmetry groups. This approach is then applied to nonstandard Hamiltonian analysis of of dynamical systems of Maxwell and Yang-Mills type. A symplectic reduction theory of the classical Maxwell equations is formulated so as to naturally include the Lorentz condition (ensuring the existence of electromagnetic waves), thereby solving the well known Dirac -Fock - Podolsky problem. Symplectically reduced Poissonian structures and the related classical minimal interaction principle for the Yang-Mills equations are also considered. 1.

On the Complete Integrability of Nonlinear Dynamical Systems on Functional Manifolds Within the Gradient-Holonomic Approach

A gradient-holonomic approach for the Lax-type integrability analysis of differential-discrete dynamical systems is described. The asymptotic solutions to the related Lax equation are studied, the related gradient identity subject to its relationship to a suitable Lax-type spectral problem is analyzed in detail. The integrability of the discrete nonlinear Schrodinger, Ragnisco–Tu and Burgers–Riemann type dynamical systems is treated, in particular, their conservation laws, compatible Poissonian structures and discrete Lax-type spectral problems are obtained within the gradient-holonomic approach.

FIELD-THEORETIC FOUNDATION OF NO-FIELD QUANTUM HALL EFFECT

Recently, the authors discussed the possibility of the macroscopic quantum state similar to the Quantum Hall Effect in a semi-localized 2D electron system with a toroidal electron-wave amplitude in the absence of any magnetic field. In order to give the concrete statistical foundation of the study, the fermion-boson statistical transformation of the 2D electron system is made using a Chern–Simons gauge potential. Based on the solution of the resultant boson-type Hamiltonian, we construct the fermion-type solution via a unitary transformation. It is shown that the solution in the form of Laughlin function is stable when electrons form pairs. In the presence of hole doping, the pair Laughlin function leads to a representation of a superconducting state when the phase-coherence length λΘ exceeds the incompressibility length λQ, but when λΘ< λQ, it leads to a macroscopic quantum state characterized by particle-number definiteness.

THE MARSDEN–WEINSTEIN REDUCTION STRUCTURE OF INTEGRABLE DYNAMICAL SYSTEMS AND A GENERALIZED EXACTLY SOLVABLE QUANTUM SUPERRADIANCE MODEL

An approach to describing nonlinear Lax type integrable dynamical systems of modern mathematical and theoretical physics, based on the Marsden–Weinstein reduction method on canonically symplectic manifolds with group symmetry, is proposed. Its natural relationship with the well-known Adler–Kostant–Souriau–Berezin–Kirillov method and the associated R-matrix approach is analyzed. A new generalized exactly solvable spatially one-dimensional quantum superradiance model, describing a charged fermionic medium interacting with external electromagnetic field, is suggested. The Lax type operator spectral problem is presented, the related R-structure is calculated. The Hamilton operator renormalization procedure subject to a physically stable vacuum is described, the quantum excitations and quantum solitons, related with the thermodynamical equilibrity of the model, are discussed.

The Bogolubov representation of the polaron model and its completely integrable RPA-approximation

The polaron model in ionic crystal is studied in the Bogolubov representation using a special RPA-approximation. A new exactly solvable approximated polaron model is derived and described in detail. Its free energy at nite temperature is calculated analytically. The polaron free energy in the constant magnetic eld at nite temperature is also discussed. Based on the structure of the Bogolubov unitary transformed polaron Hamiltonian there is stated a very important new result: the full polaron model is exactly solvable.

THEORY OF NO-FIELD QUANTUM HALL EFFECT BASED ON PHASE-CHARGE BOSON WAVE FUNCTION

The derivation of the non-magnetic Laughlin state and other macroscopic quantum states in the semi-localized 2D electron system in the network of circular molecular orbits is made by the study of zero-point plasma oscillation. In the imaginary time representation, the electric field is transformed to the vector potential. After the cancellation of the mean-field component of the inter-electron repulsive field with the ion-lattice field, the boson Hamiltonian with respect to the phase-charge fluctuation is obtained using a Chern–Simons gauge field. Based on the resultant boson wave function, the macroscopic quantum state in hole doping is found to lead to a superfluidity that is described by a coherent function when λΘ > λQ, and to the particle-number-definite state described by a Laughlin function when λQ > λΘ, where λΘ is the phase-coherence length and λQ is the incompressibility length.

PAIRING STATE AND HIGH TEMPERATURE SUPERCONDUCTIVITY OF SEMI-LOCALIZED 2D ELECTRON SYSTEM WITH CIRCULAR MOLECULAR ORBITS

Recently, new types of high temperature superconductors have been found which are characterized by the existence of circular molecular orbits in each unit site of 2D s/p electron system. In view of the characteristic, a new model of superconductivity is studied based on the stability of the correlated state of electrons in the 2D interconnection of circular orbits. This model gives an estimation of the upper bound of superfluidity transition temperature: Tc ~ 130-400 K for fcc C60, and Tc ~ 110-340 K for hole-doped MgB2.

Algebraic aspects of the driven dynamics in the density operator and correlation functions calculation for multi-level open quantum systems

Exact and approximate master equations were derived by the projection operator method for the reduced statistical operator of a multi-level quantum system with finite number N of quantum eigenstates interacting with arbitrary external classical fields and dissipative environment simultaneously. It was shown that the structure of these equations can be simplified significantly if the free Hamiltonian driven dynamics of an arbitrary quantum multi-level system under the influence of the external driving fields as well as its Markovian and non-Markovian evolution, stipulated by the interaction with the environment, are described in terms of the SU(N) algebra representation. As a consequence, efficient numerical methods can be developed and employed to analyze these master equations for real problems in various fields of theoretical and applied physics. It was also shown that literally the same master equations hold not only for the reduced density operator but also for arbitrary nonequilibrium multi-time corr...

OPERATOR ANALYSIS OF AN RPA-REDUCED POLARON MODEL WITHIN THE BOGOLUBOV REPRESENTATION IN MAGNETIC FIELD AT FINITE TEMPERATURE. PART 1

The polaron model in ionic crystal is studied in the N. Bogolubov representation using a special RPA-approximation. A new exactly solvable approximated polaron model is derived and described in detail. Its free energy at finite temperature is calculated analytically. The polaron free energy in the constant magnetic field at finite temperature is also discussed.