q-Deformed superspace andq-extended supersymmetric hamiltonian with arbitrary superpotential

A q-deformation of an extended supersymmetry is described, and a q-extended supersymmetric Hamiltonian is constructed. The procedure of the canonical quantization is developed. Bibliography15 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 235, 1900, pp. 260–272.     

Quantization of model of antisymmetric tensor field in the framework of extended BRST symmetry

The model of an antisymmetric second-rank tensor field is quantized in both the Lagrangian and the Hamiltonian forms of extended BRST quantization. It is shown that the Lagrangian quantization leads to a unitaryS matrix.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 3, pp. 420–428, September, 1995.     

Superfield Quantization in the Sp(2)-Covariant Formalism

We generalize the rules for the superfield Sp(2)-covariant quantization of arbitrary gauge theories to the case of gauge fixing by the generating equations for the gauge functional. We consider possible realizations of the extended antibrackets and show that only one of the realizations is consistent with the extended BRST symmetry transformations in the form of the supertranslations along the Grassmann coordinates of a superspace.     

Obstruction results in quantization theory

Summary Quantization is not a straightforward proposition, as demonstrated by Groenewold's and Van Hove's discovery, exactly fifty years ago, of an “obstruction” to quantization. Their “no-go theorems” assert that it is in principle impossible to consistently quantize every classical observable on the phase spaceR ²ⁿ in a physically meaningful way. A similar obstruction was recently found forS ², buttressing the common belief that no-go theoremss should hold in some generality. Surprisingly, this is not so—it has also just been proven that there is no obstruction to quantizing a torus. In this paper we take first steps towards delineating the circumstances under which such obstructions will appear and understanding the mechanisms which produce them. Our objectives are to conjecture a generalized Groenewold-Van Hove theorem and to determine the maximal subalgebras of observables which can be consistently quantized. This requires a study of the structure of Poisson algebras of classical systems and their representations. To these ends we include an exposition of both prequantization (in an extended sense) and quantization theory—formulated in terms of “basic sets of observables”—and review in detail the known results forR ²ⁿ,S ², andT ². Our discussion is independent of any particular method of quantization; we concentrate on the structural aspects of quantization theory which are common to all Hilbert space-based quantization techniques. This paper is dedicated to the memory of Juan C. Simo Supported in part by NSF Grants DMS 92-22241 and 96-23083 (M.J.G.). This paper was solicited by the editors to be part of a volume dedicated to the memory of Juan C. Simo.     

Noncommutative differential geometry,quantization, and smooth symmetries of the C*-algebras associated to topological dynamics

Noncommutative differential geometric structures are considered for a class of simple C^*-algebras. This structure is defined in terms of smooth Lie group actions on the C^*-algebra in question together with a certain quantization mapping motivated directly by the known cohomological obstructions for the quantum mechanical quantization correspondence. We show that such a quantization mapping may be constructed for the C^*-algebras associated to antisymmetric bi-characters and for the Cuntz/Cuntz-Krieger C^*-algebras associated to topological dynamics. A certain curvature obstruction is defined in terms of the quantization mapping. It is shown that existence of smooth Lie group actions is determined by the curvature obstruction.     

Loop quantum mechanics and the fractal structure of quantum spacetime

We discuss the relation between string quantization based on the Schild path integral and the Nambu-Goto path integral. The equivalence between the two approaches at the classical level is extended to the quantum level by a saddle-point evaluation of the corresponding path integrals. A possible relationship between M-Theory and the quantum mechanics of string loops is pointed out. Then, within the framework of “loop quantum mechanics”, we confront the difficult question as to what exactly gives rise to the structure of spacetime. We argue that the large scale properties of the string condensate are responsible for the effective Riemannian geometry of classical spacetime. On the other hand, near the Planck scale the condensate “evaporates”, and what is left behind is a “vacuum” characterized by an effective fractal geometry.     

Maxwell equations in complex form of Majorana–Oppenheimer,solutions with cylindric symmetry in Riemann <Emphasis Type="Italic">S</Emphasis><Subscript>3</Subscript> and Lobachevsky <Emphasis Type="Italic">H</Emphasis><Subscript>3</Subscript> spaces

Complex formalism of Riemann–Silberstein–Majorana–Oppenheimer in Maxwell electrodynamics is extended to the case of arbitrary pseudo-Riemannian space-time in accordance with the tetrad recipe of Tetrode–Weyl–Fock–Ivanenko. In this approach, the Maxwell equations are solved exactly on the background of static cosmological Einstein model, parameterized by special cylindrical coordinates and realized as a Riemann space of constant positive curvature. A discrete frequency spectrum for electromagnetic modes depending on the curvature radius of space and three parameters is found, and corresponding basis electromagnetic solutions have been constructed explicitly. In the case of elliptical model a part of the constructed solutions should be rejected by continuity considerations. Similar treatment is given for the Maxwell equations in hyperbolic Lobachevsky model, the complete basis of electromagnetic solutions in corresponding cylindrical coordinates has been constructed as well, no quantization of frequencies of electromagnetic modes arises.     

Asymptotics of the quantization errors for in-homogeneous self-similar measures supported on self-similar sets

We study the quantization for in-homogeneous self-similar measures μ supported on self-similar sets.Assuming the open set condition for the corresponding iterated function system, we prove the existence of the quantization dimension for μ of order r ∈(0, ∞) and determine its exact value ξ_r. Furthermore, we show that,the ξ_r-dimensional lower quantization coefficient for μ is always positive and the upper one can be infinite. A sufficient condition is given to ensure the finiteness of the upper quantization coefficient.     

One‐bit sigma‐delta quantization with exponential accuracy

One‐bit quantization is a method of representing bandlimited signals by ±1 sequences that are computed from regularly spaced samples of these signals; as the sampling density λ → ∞, convolving these one‐bit sequences with appropriately chosen filters produces increasingly close approximations of the original signals. This method is widely used for analog‐to‐digital and digital‐to‐analog conversion, because it is less expensive and simpler to implement than the more familiar critical sampling followed by fine‐resolution quantization. However, unlike fine‐resolution quantization, the accuracy of one‐bit quantization is not well‐understood. A natural error lower bound that decreases like 2^?λ can easily be given using information theoretic arguments. Yet, no one‐bit quantization algorithm was known with an error decay estimate even close to exponential decay. In this paper, we construct an infinite family of one‐bit sigma‐delta quantization schemes that achieves this goal. In particular, using this family, we prove that the error signal for π‐bandlimited signals is at most O(2^?.07λ). © 2003 Wiley Periodicals, Inc.     

Conversion of second-class constraints and resolving the zero-curvature conditions in the geometric quantization theory

In the approach to geometric quantization based on the conversion of second-class constraints, we resolve the corresponding nonlinear zero-curvature conditions for the extended symplectic potential. From the zero-curvature conditions, we deduce new linear equations for the extended symplectic potential. We show that solutions of the new linear equations also satisfy the zero-curvature condition. We present a functional solution of these new linear equations and obtain the corresponding path integral representation. We investigate the general case of a phase superspace where boson and fermion coordinates are present on an equal basis.     

广义微分二次量子化算子

该文对任一从 E_c 到 E_c^* 的连续线性算子定义了 其广义微分二次量子化算子, 由Schwartz 核定理得到其Fock 展开,并用张量积的缩合给出复合算子的微分二次量子化算子.     

A Class of Potentials for Which Exact Semiclassical Quantization Can Be Achieved

We consider a class of potentials for which the exact semiclassical quantization is achieved by a certain modification of the quantization condition. A list of potentials for which the new quantization condition is exact coincides with the list of potentials for which the spectrum is determined by the factorization method. We construct a one-parameter family of quantization conditions including the supersymmetric WKB condition as a special case. The new condition allows considering the interrelations between different modifications of the leading approximation and their validity ranges and also allows developing new approximate methods for calculating spectra.     

Optimal vector quantization in terms of Wasserstein distance

The optimal quantizer in memory-size constrained vector quantization induces a quantization error which is equal to a Wasserstein distortion. However, for the optimal (Shannon-)entropy constrained quantization error a proof for a similar identity is still missing. Relying on principal results of the optimal mass transportation theory, we will prove that the optimal quantization error is equal to a Wasserstein distance. Since we will state the quantization problem in a very general setting, our approach includes the Rényi-α-entropy as a complexity constraint, which includes the special case of (Shannon-)entropy constrained (α=1) and memory-size constrained (α=0) quantization. Additionally, we will derive for certain distance functions codecell convexity for quantizers with a finite codebook. Using other methods, this regularity in codecell geometry has already been proved earlier by György and Linder (2002, 2003) [11] and [12].     

Error bounds for high-resolution quantization with Rényi-α-entropy constraints

We consider the problem of optimal quantization with norm exponent r > 0 for Borel probability measures on ? ^d under constrained Rényi-α-entropy of the quantizers. If the bound on the entropy becomes large, then sharp asymptotics for the optimal quantization error are well-known in the special cases α = 0 (memory-constrained quantization) and α = 1 (Shannon-entropy-constrained quantization). In this paper we determine sharp asymptotics for the optimal quantization error under large entropy bound with entropy parameter α ∈ [1+r/d,∞]. For α ∈ [0,1 + r/d] we specify the asymptotical order of the optimal quantization error under large entropy bound. The optimal quantization error is decreasing exponentially fast with the entropy bound and the exact rate is determined for all α ∈ [0, ∞].     

Optimal Quantization for Uniform Distributions on Cantor-Like Sets

In this paper, the problem of optimal quantization is solved for uniform distributions on some higher dimensional, not necessarily self-similar N-adic Cantor-like sets. The optimal codebooks are determined and the optimal quantization error is calculated. The existence of the quantization dimension is characterized and it is shown that the quantization coefficient does not exist. The special case of self-similarity is also discussed. The conditions imposed are a separation property of the distribution and strict monotonicity of the first N quantization error differences. Criteria for these conditions are proved and as special examples modified versions of classical fractal distributions are discussed. This work contains and generalizes some parts of the authors doctoral thesis (cf. 16).     

Stability of quantization dimension and quantization for homogeneous Cantor measures

We effect a stabilization formalism for dimensions of measures and discuss the stability of upper and lower quantization dimension. For instance, we show for a Borel probability measure with compact support that its stabilized upper quantization dimension coincides with its packing dimension and that the upper quantization dimension is finitely stable but not countably stable. Also, under suitable conditions explicit dimension formulae for the quantization dimension of homogeneous Cantor measures are provided. This allows us to construct examples showing that the lower quantization dimension is not even finitely stable. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Quantization dimension for infinite self-similar probabilities

The quantization dimension function for a probability measure induced by a set of infinite contractive similarity mappings and a given probability vector is determined. A relationship between the quantization dimension function and the temperature function of the thermodynamic formalism arising in multifractal analysis is established. The result in this paper is an infinite extension of Graf and Luschgy [S. Graf, H. Luschgy, The quantization dimension of self-similar probabilities, Math. Nachr. 241 (2002) 103-109].     

A Local Refinement Strategy for Constructive Quantization of Scalar SDEs

We present a fully constructive method for quantization of the solution X of a scalar SDE in the path space L _p [0,1] or C[0,1]. The construction relies on a refinement strategy which takes into account the local regularity of X and uses Brownian motion (bridge) quantization as a building block. Our algorithm is easy to implement, its computational cost is close to the size of the quantization, and it achieves strong asymptotic optimality provided this property holds for the Brownian motion (bridge) quantization.