Quantization of lie Bialgebras via the Formality of the operad of Little Disks

We give a proof of the Etingof–Kazhdan theorem on quantization of Lie bialgebras based on the formality of the chain operad of little disks and show that the Grothendieck–Teichmüller group acts non-trivially on the corresponding quantization functors. Partially supported by an NSF Grant and A. Sloan Research Fellowship Received: November 2004 Revision: April 2006 Accepted: May 2006     

组合Bose场的分数自旋和分数统计性

对组合Bose子场，采用FS (Faddeev-Senjanovic) 路径积分量子化方法进行量子化.从量子Noether定理出发，给出量子分数自旋和分数统计性质. 关键词： 路径积分量子化 分数自旋 分数统计     

An algebraic index theorem for orbifolds

Using the concept of a twisted trace density on a cyclic groupoid, a trace is constructed on a formal deformation quantization of a symplectic orbifold. An algebraic index theorem for orbifolds follows as a consequence of a local Riemann-Roch theorem for such densities. In the case of a reduced orbifold, this proves a conjecture by Fedosov, Schulze, and Tarkhanov. Finally, it is shown how the Kawasaki index theorem for elliptic operators on orbifolds follows from this algebraic index theorem.     

Velocity Quantization Approach of the One-Dimensional Dissipative Harmonic Oscillator

Given a constant of motion for the one-dimensional harmonic oscillator with linear dissipation in the velocity, the problem to get the Hamiltonian for this system is pointed out, and the quantization up to second order in the perturbation approach is used to determine the modification on the eigenvalues when dissipation is taken into consideration. This quantization is realized using the constant of motion instead of the Hamiltonian. PACS: 03.20.+i, 03.30.+p, 03.65.−w,03.65.Ca     

Action Principle and Modification of the Faddeev–Popov Factor in Gauge Theories

The quantum action (dynamical) principle is exploited to investigate the nature and origin of the Faddeev–Popov (FP) factor in gauge theories without recourse to path integrals. Gauge invariant as well as gauge non-invariant interactions are considered to show that the FP factor needs to be modified in more general cases and expressions for these modifications are derived. In particular we show that a gauge invariant theory does not necessarily imply the familiar FP factor for proper quantization. PACS numbers: 11.15.-q; 12.10.-g; 12.15.-y; 12.38.-t     

Quantized charge transport in adiabatic pumping by a driven double δ-barrier

We investigate the condition that the charge carried by quantum parametric pumping per cycle is quantized in units of the electron charge e and the role of adiabaticity in charge quantization. Using a driven double-δ-potential model and a Floquet scattering approach, it is found that the quantization phenomenon occurs only at very low frequencies of the oscillating potential and adiabaticity of the potential modulation is crucial for quantization.     

Functional Quantization and Small Ball Probabilities for Gaussian Processes

Quantization consists in studying the L ^r -error induced by the approximation of a random vector X by a vector (quantized version) taking a finite number n of values. We investigate this problem for Gaussian random vectors in an infinite dimensional Banach space and in particular, for Gaussian processes. A precise link proved by Fehringer⁽⁴⁾ and Dereich et al. ⁽³⁾ relates lower and upper bounds for small ball probabilities with upper and lower bounds for the quantization error, respectively. We establish a complete relationship by showing that the same holds for the direction from the quantization error to small ball probabilities. This allows us to compute the exact rate of convergence to zero of the minimal L ^r -quantization error from logarithmic small ball asymptotics and vice versa.     

A Generalization of a Two-Parameter Quantization for the Group GL 2(k)

We generalize a well-known two-parameter quantization for the group GL ₂(k) (over an arbitrary field k). Specifically, a certain class of Hopf algebras is constructed containing that quantization. The algebras are constructed given an arbitrary coalgebra and an arbitrary pair of its commuting anti-isomorphisms, and are defined by quadratic relations. They are densely linked to the compact quantum groups introduced by Woronowicz. We give examples of Hopf algebras that can be rowed up to the two-parameter quantization for GL ₂(k).     

约束系统量子化中Dirac方法与Faddeev-Jackiw方法的等价性

对约束系统量子化中Dirac方法和 Faddeev-Jackiw方法进行了讨论，并对它们的运动方程、正则量子化的等价性进行证明.找出了两种方法中约束的对应关系. 关键词： Faddeev-Jackiw方法 Dirac方法 约束系统 正则量子化     

Convergent Star Product Algebras on ‘ax+b’

We define nontempered (exponential growth) function spaces on the Lie group ax+b which are stable under some left-invariant (convergent) star product. The techniques used to achieved the latter come from symmetric spaces geometry and star representation theory.