Applying the Linblad equation to quantum dissipative systems

For quantum systems with linear dissipation, we obtain the representation of the Linblad equation in the canonical form via Hermitian operators. Based on this representation, we derive equations for the entropy density and for the statistical projection operator. We consider the quantum harmonic oscillator with linear dissipation as an example. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 2, pp. 288–294, August, 2006. An erratum to this article is available at .     

Asymptotic analysis of the quantum Liouville equation

We present an asymptotic analysis of the quantum Liouville equation with respect to the Planck's constant, which models the temporal evolution of the (quasi)distribution of an ensemble of electrons under the action of a potential. We consider two cases: firstly a smooth potential, and secondly a potential modelled by a δ-distribution. In both cases the zeroth-order term behaves classically. In the smooth case the classical Liouville equation is satisfied and in the case for the δ-potential an interface condition is derived, so that everything is reflected at the potential barrier.     

Quantum Heisenberg group and reflection equation. I

The quantum group of upper triangular matrices and the Hopf algebra associated with the reflection equation are constructed for the R-matrix corresponding to the quantum Heisenberg group. Bibliography: 12 titles. Dedicated to P. P. Kulish on the occasion of his 50th birthday Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 209, 1994, pp. 28–36. Translated by E. V. Damanskinsky.     

Liouville quantum gravity and KPZ

Consider a bounded planar domain D, an instance h of the Gaussian free field on D, with Dirichlet energy (2π)⁻¹∫ _D ∇h(z)⋅∇h(z)dz, and a constant 0≤γ<2. The Liouville quantum gravity measure on D is the weak limit as ε→0 of the measures

Quantum dissipative systems. II. String in a curved affine—Metric spacetime

A closed bosonic string in a curved affine—metric spacetime is considered as an example of dissipative quantum field theory. The conformal anomaly of the trace of the energy—momentum tensor for the string on the affine—metric manifold is investigated. The two-loop metric beta function for the two-dimensional nonlinear dissipative sigma model is calculated. Examples of nonflat manifolds that lead to ultraviolet-finite sigma models are found.Institute of Nuclear Physics, Moscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 101, No. 1, pp. 38–46, October, 1994.     

Quantum groupoids and deformation quantization

The purpose of this Note is to unify quantum groups and star-products under a general umbrella: quantum groupoids. It is shown that a quantum groupoid naturally gives rise to a Lie bialgebroid as a classical limit. The converse question, i.e.. the quantization problem, is posed. In particular, any regular triangular Lie bialgebroid is shown quantizable. For the Lie bialgebroid of a Poisson manifold, its quantization is equivalent to a star-product.     

Up-to-constants comparison of Liouville first passage percolation and Liouville quantum gravity

Liouville first passage percolation(LFPP) with the parameter ξ > 0 is the family of random distance functions ■ on the plane obtained by integrating ■along paths, where ■ is a smooth mollification of the planar Gaussian free field. Recent works have shown that for all ξ > 0, the LFPP metrics,appropriately re-scaled, admit non-trivial subsequential limiting metrics. In the case ξ < ξcrit≈ 0.41, it has been shown that the subsequential limit is unique and defines a metric on γ-Liouville q...     

Canonical analysis and quantization of threedimensional topologically massive gravity

A canonical formulation with constraints for three-dimensional topologically massive gravity is constructed. All the constraints and their algebra are found explicitly. The case of pure Chern—Simons gravity is considered. The local measure in the functional integral is calculated and the role it plays in the cancellation of volume divergences is traced.State Pedagogical Institute, Tomsk. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 100, No. 3, pp. 444–457, September, 1994.     

Canonical quantization for classical Neumann-type dynamical systems within the framework of the Moser spectral approach

The classical Neumann-type dynamical systems describe the motion of particles constrained to remain on an N-sphere S ^N in (N + 1)-dimensional space ^N+1 and subject to quasi-harmonic forces. Following the Moser spectral approach to the connection of Lax finite-zoned infinite-dimensional dynamical systems with Neumann-type finite-dimensional dynamical systems on a sphere in ^N+1 a regular procedure of appropriate quantization of these systems is proposed. Quantum expressions of commutative laws of conservation for Neumann-type quantum dynamical systems are determined in the general case of a canonical Dirac quantization.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 7, pp. 913–922, July, 1992.     

Moser-Trudinger inequalities and Liouville systems

In this Note, we generalize the Moser-Trudinger inequality for one function to a system of functions and give a sufficient condition under which the corresponding functional admits a minimizer, which is a solution of a system of Liouville type.     

Using the Renyi entropy to describe quantum dissipative systems in statistical mechanics

We develop a formalism for describing quantum dissipative systems in statistical mechanics based on the quantum Renyi entropy. We derive the quantum Renyi distribution from the principle of maximum quantum Renyi entropy and differentiate this distribution (the temperature density matrix) with respect to the inverse temperature to obtain the Bloch equation. We then use the Feynman path integral with a modified Mensky functional to obtain a Lindblad-type equation. From this equation using projection operators, we derive the integro-differential equation for the reduced temperature statistical operator, an analogue of the Zwanzig equation in statistical mechanics, and find its formal solution in the form of a series in the class of summable functions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 3, pp. 444–453, September, 2008.     

Dispersion and Strichartz estimates for the Liouville equation

We consider the Liouville equation associated with a metric g of class C² and we prove dispersion and Strichartz estimates for the solution of this equation in terms of geodesics associated with g. We introduce the notion of focusing and dispersive metric to characterize metrics such that the same dispersion estimate as in the Euclidean case holds. To deal with the case of non-trapped long range perturbation of the Euclidean metric, we prove a global velocity moments effect on the solution. In particular, we obtain global in time Strichartz estimates for metrics such that the dispersion estimate is not satisfied.     

Dispersion and Strichartz estimates for the Liouville equation

We consider the Liouville equation associated to a metric g and we prove dispersion and Strichartz estimates for the solution of this equation in terms of the geometry of the trajectories associated to g. In particular, we obtain global Strichartz estimates in time for metrics where dispersion estimate is false even locally in time. We also study the analogy between Strichartz estimates obtained for the Liouville equation and the Schrödinger equation with variable coefficients. To cite this article: D. Salort, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Quantum Twist Maps and Dual Canonical Bases

In this paper, we show that quantum twist maps, introduced by Lenagan-Yakimov, induce bijections between dual canonical bases of quantum nilpotent subalgebras. As a corollary, we show the unitriangular property between dual canonical bases and Poincaré-Birkhoff-Witt type bases under the “reverse” lexicographic order. We also show that quantum twist maps induce bijections between certain unipotent quantum minors.