Renormalization of Quantum Field Theory in Curved Space-Time and Renormalization Group Equations

The structure of quantum field theory renormalization in curved space-time is investigated. The equations allowing us to investigate the behaviour of vacuum energy and vertex functions in the limit of small distances in the external gravitational field are established. The behaviour of effective charges corresponding to the parameters of nonminimal coupling of the matter with the gravitational field is studied and the conditions under which asymptotically free theories become asymptotically conformally invariant are found. The examples of asymptotically conformally invariant theories are given. On the basis of a direct solution of renormalization group equations the effective potential in the external gravitational field and the effective action in the gravity with the high derivatives are obtained. The expression for the cosmological constant in terms of R²-gravity Lagrangian parameters is given which does not contradict the observable data. Renormalization and renormalization group equations for the theory in curved space-time with torsion are investigated.     

On the Asymptotic Behavior of the Hyperbolic Brownian Motion

The main results in this paper concern large and moderate deviations for the radial component of a $n$ -dimensional hyperbolic Brownian motion (for $n\ge 2$ ) on the Poincaré half-space. We also investigate the asymptotic behavior of the hitting probability $P_\eta (T_{\eta _1}^{(n)}<\infty )$ of a ball of radius $\eta _1$ , as the distance $\eta $ of the starting point of the hyperbolic Brownian motion goes to infinity.     

Asymptotic Behavior for the Liouville Equations

A new proof of the diffusion approximation for ordinary differential equations is given. It is based on an asymptotic expansion of the solution of the corresponding Liouville partial differential equations. In contrast to previous results obtained for the suspension under Holderian mappings of subshift of finite type or Fourier analysis techniques, our proof relies only on symbolic dynamics.     

Renormalization of the Quantum Theory of the Solid State

The perturbative structure of a quantum field theory of the solid state, including allradiative corrections, is applied for the calculation of renormaiization factors and counterterms. After a general outline, approximation schemes for the explicit evaluation of renormalization parameters are developed.     

Summing Logarithms in Quantum Field Theory: The Renormalization Group

The process of renormalization in quantum fieldtheory necessarily involves the introduction of anarbitrary mass scale ² into the theory.The effect of having this parameter appear due toquantum effects can be analyzed from many points of view; thegeneral topic is usually called therenormalization group. In this paper, oneaspect of this feature of quantum field theory isdiscussed in some detail. It is shown how the appearance of thisarbitrary mass scale imposes consistency conditions onquantum-induced corrections to the classical action ofa model. This has the effect of determining higher order corrections in terms of lower ordercorrections in the perturbative expansion of theeffective action, which in turn permits at least partialsummation of all terms in the perturbative expansion. This is illustrated in the context of twosimple, well-understood models; a ⁴model in four dimensions and a ³ modelin six dimensions. The technicalities associated withthe renormalization procedure itself are not discussed.     

Nonlinear Theory of Quantum Brownian Motion

A nonlinear theory of quantum Brownian motion in classical environment is developed based on a thermodynamically enhanced nonlinear Schrödinger equation. The latter is transformed via the Madelung transformation into a nonlinear quantum Smoluchowski-like equation, which is proven to reproduce key results from the quantum and classical physics. The application of the theory to a free quantum Brownian particle results in a nonlinear dependence of the position dispersion on time, being quantum generalization of the Einstein law of Brownian motion. It is shown that the time of decoherence from quantum to classical diffusion is proportional to the square of the thermal de Broglie wavelength divided by the classical Einstein diffusion constant.     

Interacting Quantum and Classical Continuous Systems II. Asymptotic Behavior of the Quantum Subsystem

In the framework of event-enhanced quantum theory the dynamical equation for the reduced density matrix of a quantum system interacting with a continuous classical system is derived. The asymptotic behavior of the corresponding dynamical semigroup is discussed. The example of a quantum–classical coupling on Lobatchevski space is presented.     

Heisenberg Equations of Motion in a Nonabelian Gauge Theory

Heisenberg type equations of motion are established in a nonabelian gauge theory with minimal and nonminimal couplings and various relativistic particle equations of motion are derived from them. These equations for pointlike particles possessing a nonabelian gauge interaction (chosen for definiteness to be of SO(4,1) type) ore obtained in classical limit, ħ → 0, or in a semiclassical limit in which contributions of first order in ħ are retained. As a byproduct of the formalism, which can be applied to an arbitrary gauge group, a simple derivation of the Lorentz equation and the Bargmann-Michel-Telegdi equation from spinor electrodynamics with anomalous (i.e. nonminimal) coupling is given starting from the associated quantum mechanical Heisenberg equations of motion and specializing the gauge group to the electromagnetic U(1) group.     

On the Graded Quantum Yang-Baxter and Reflection Equations

We clarify the extra signs appearing in the graded quantum Yang-Baxter reflection equations, when they are written in a matrix form. We find the boundary K-matrix for the Perk-Schultz six-vertex model, thus give a general solution to the graded reflection equation associated with it.     

Asymptotic Behavior of Compressible Navier-Stokes Equations with Density-Dependent Viscosity and Vacuum

In this paper, we study the one-dimensional Navier-Stokes equations connecting to vacuum state with a jump in density when the viscosity depends on the density. Precisely, when the viscosity coefficient μ(ρ) is proportional to ρ ^θ with θ > 0, where ρ is the density, we give the asymptotic behavior and the decay rate of the density function ρ(x, t). Furthermore, the behavior of the density function ρ(x, t) near the interfaces separating the gas from vacuum and the expanding rate of the interfaces are also studied. The analysis is based on some new mathematical techniques and some new useful estimates. This fills a final gap on studying Navier-Stokes equations with the viscosity coefficient μ(ρ) dependent on the density ρ.     

Oscillatory and Asymptotic Behavior of a Second-Order Nonlinear Functional Differential Equations

This paper is concerned with oscillatory and asymptotic behavior of solutions of a class of second order nonlinear functional differential equations. By using the generalized Riccati transformation and the integral averaging technique, new oscillation criteria and asymptotic behavior are obtained for all solutions of the equation. Our results generalize and improve some known theorems.     

Connections of Coherent Information,Quantum Discord,and Entanglement

For a pure non-markovian dephasing model we derive analytic expressions of coherent information,quantum discord,and entanglement.We find that for the cases of the initial Werner states,the dynamical behavior of coherent information is similar to that of quantum discord but different from that of entanglement.Coherent information,as well as quantum discord,can reveal the quantum correlations in some mixed-states,in which the entanglement is zero.     

The Pullback Asymptotic Behavior of the Solutions for 2D Nonautonomous G-Navier-Stokes Equations

The pullback asymptotic behavior of the solutions for 2D Nonautonomous G-Navier-Stokes equations is studied, and the existence of its $L^2$-pullback attractors on some bounded domains with Dirichlet boundary conditions is investigated by using the measure of noncompactness. Then the estimation of the fractal dimensions for the 2D G-Navier-Stokes equations is given.     

应用级联运动方程方法研究并联双量子点的量子纠缠

Quantum dots comprise a type of quantum impurity system. The entanglement and coherence of quantum states are significantly influenced by the strong electron-electron interactions among impurities and their dissipative coupling with the surrounding environment. Competition between many-body effects and transfer couplings plays an important role in determining the entanglement among localized impurity spins. In this work, we employ the hierarchical-equations-of-motion approach to explore the entanglement of a strongly correlated double quantum dots system. The relation between the total system entropy and those of subsystems is also investigated.     

Renormalization Theory of a Two Dimensional Bose Gas: Quantum Critical Point and Quasi-Condensed State

We present a renormalization group construction of a weakly interacting Bose gas at zero temperature in the two-dimensional continuum, both in the quantum critical regime and in the presence of a condensate fraction. The construction is performed within a rigorous renormalization group scheme, borrowed from the methods of constructive field theory, which allows us to derive explicit bounds on all the orders of renormalized perturbation theory. Our scheme allows us to construct the theory of the quantum critical point completely, both in the ultraviolet and in the infrared regimes, thus extending previous heuristic approaches to this phase. For the condensate phase, we solve completely the ultraviolet problem and we investigate in detail the infrared region, up to length scales of the order \((\lambda ^3\rho _0)^{-1/2}\) (here \(\lambda \) is the interaction strength and \(\rho _0\) the condensate density), which is the largest length scale at which the problem is perturbative in nature. We exhibit violations to the formal Ward Identities, due to the momentum cutoff used to regularize the theory, which suggest that previous proposals about the existence of a non-perturbative non-trivial fixed point for the infrared flow should be reconsidered.     

State Representations for Renormalized Energy Equations in Quantum Field Theory

Quantum fields with interaction do not allow the application of the Fock representation. Rather the algebraic G.N.S. procedure has to be used which leads to nonorthonormal basis states. This raises the problem of explicit state construction with respect to such states. In the present paper this problem is treated for the case of a sufficiently regularized, selfinteracting spinorfield. By some theorems it is demonstrated that a consequent treatment of its field Hamiltonian which respects the general algebraic requirements leads to Dyson's renormalized energy equation. In addition this approach allows explicit state constructions which so far have not been realized in conventional quantum field theory.