Automorphisms of the Weyl Algebra

We discuss a conjecture which says that the automorphism group of the Weyl algebra in characteristic zero is canonically isomorphic to the automorphism group of the corresponding Poisson algebra of classical polynomial symbols. Several arguments in favor of this conjecture are presented, all based on the consideration of the reduction of the Weyl algebra to positive characteristic Mathematics Subject Classification (2000) 13N10, 16S32, 16H05     

On Drinfeld''s realization of quantum affine algebras

We show how to obtain Drinfeld's realization of quantum nontwisted affine algebras from the quantized Cartan-Weyl basis. The formulae for comultiplications in this realization are discussed.     

Aligned Electric and Magnetic Weyl Fields

The results on the non-existence of purely magnetic solutions are extended to the wider class of spacetimes which have homothetic electric and magnetic Weyl fields. This class is a particularization of the spacetimes admitting a direction for which the relative electric and magnetic Weyl fields are aligned. We give an invariant characterization of these metrics and study the properties of their Debever null vectors. The directions observing aligned electric and magnetic Weyl fields are obtained for every Petrov-Bel type.     

Three-dimensional quantum Hall effect in Weyl and double-Weyl semimetals

The well-known quantum Hall effect (QHE) was usually studied in 2D systems. In this work, we investigate the integer QHE in 3D Weyl and double-Weyl semimetals. Based on the lattice models of Weyl and double-Weyl semimetals subjected to a uniform magnetic field, we derive the generalized 3D spinfull Hofstadter Hamiltonians and Harper equations for the two systems, and obtain their corresponding energy spectra. Furthermore, we show that for proper hopping parameters and rational magnetic fluxes, both systems exhibit the 3D QHE when the Fermi level lies in some band gaps. The 3D QHE is topologically characterized by three Chern numbers with one or two nonzero Chern values which are respectively defined for three crystal planes. The possible experimental realization and detection of the 3D QHE are also discussed.     

Re-explanation of Weyl Quantization Scheme via Weyl Ordering Approach

We re-explain the Weyl quantization scheme by virtue of the technique of integration within Weyl ordered product of operators, i.e., the Weyl correspondence rule can be reconstructed by classical functions＇ Fourier transformation followed by an inverse Fourier transformation within Weyl ordering of operators. As an application of this reconstruction, we derive the quantum operator coresponding to the angular spectrum amplitude of a spherical wave.     

Weyl quantization and star products

Many non-linear classical mechanical systems arise as the symplectic reductions of linear systems. The star products on the corresponding quantized algebras can be derived from the Weyl-Moyal product on the algebras of the linear systems. An algebraic approach to Berezin quantization is sketched.     

经典函数与量子算符的Weyl对应

提出了通过定义一种算符编序来处理含有不对易因子的积分的思想,研究了经典函数与量子算符的Weyl对应问题,得到了计算简捷的Weyl对应规则之数学显式,并给出了一种证明量子力学表象完备性关系的新方法.以实例支持了Weyl对应规则的正确性,说明了Weyl对应规则理论是自洽的.     

Supertrace and Superquadratic Lie Structure on the Weyl Algebra,and Applications to Formal Inverse Weyl Transform

Using the Moyal *-product and orthosymplectic supersymmetry, we construct a natural nontrivial supertrace and an associated nondegenerate invariant supersymmetric bilinear form for the Lie superalgebra structure of the Weyl algebra W. We decompose adjoint and twisted adjoint actions. We define a renormalized supertrace and a formal inverse Weyl transform in a deformation quantization framework and develop some examples Mathematics Subject Classification: 53D55, 17B05, 17B10, 17B20, 17B60, 17B65     

Generalized Weyl quantization on the cylinder and the quantum phase

Generalized Weyl quantization formalism for the cylindrical phase space S¹×R¹$S^1\times {\mathbb{R}}^1$ is developed. It is shown that the quantum observables relevant to the phase of the linear harmonic oscillator or electromagnetic field can be represented within this formalism by the self-adjoint operators on the Hilbert space L²(S¹)$L^2\left(S^1\right)$.     

Generalized Weyl Ordering and Nonlinear Coherent States

In the context of the nonlinear coherent state (NLCS) theory we introduce the generalized Weyl orderingoperator formulation. The corresponding generalized Wigner operator turns out to be the Weyl ordered Dirac δ-operatorfunctions. The completeness relation of NLCS is recast into generalized Weyl ordering form. The relationship betweennormal ordering, antinormal ordering and the generalized Weyl ordering is established which constitute a self-consistenttheory for NLCS.     

Construction of a Weyl Representation from a Weak Weyl Representation of the Canonical Commutation Relation

Weak Weyl representations of the canonical commutation relation (CCR) with one degree of freedom are considered in relation to the theory of time operator in quantum mechanics. It is proven that there exists a general structure through which a weak Weyl representation can be constructed from a given weak Weyl representation. As a corollary, it is shown that a Weyl representation of the CCR can be constructed from a weak Weyl representation which satisfies some additional property. Some examples are given. The work is supported by the Grant-in-Aid No.17340032 for Scientific Research from Japan Society for the Promotion of Science (JSPS).     

Periastron shift in Weyl class spacetimes

The periastron position advance for geodesic motion in axially symmetric solutions of the Einstein field equations belonging to the Weyl class of vacuum solutions is investigated. Explicit examples corresponding to either static solutions (single Chazy-Curzon, Schwarzschild and a pair of them), or stationary solutions (single rotating Chazy-Curzon and Kerr black hole) are discussed. The results are then applied to the case of S2-SgrA^* binary system of which the periastron position advance will be soon measured with a great accuracy.     

Gravitational Potentials of Triaxial Ellipsoids in Weyl Gravity

We present analytic expressions for the gravitational potentials associated with triaxial ellipsoids, spheroids, spheres and disks in Weyl gravity. The gravitational potentials of these configurations in Newtonian gravity, i.e. the potentials derived by integration of the Poisson equation Green's function 1/|r – r| over the volume of the configuration, are well known in the literature. Herein we present the results of the integration of |r – r|, the Green's function associated with the fourth order Laplacian ⁴ of Weyl gravity, over the volume of the configuration to obtain the resulting gravitational potentials within this specific theory. As an application of our calculations, we solve analytically Euler's equations pertaining to incompressible rotating fluids to show that, as in the case of Newtonian gravity, homogeneous prolate configurations are not allowed within Weyl gravity either.     

Localization and the Weyl algebras

Physics of Atomic Nuclei - Let W n (ℝ) be the Weyl algebra of index n. It is well known that so(p, q) Lie algebras can be viewed as quadratic polynomial (Lie) algebras in W n (ℝ) for p...     

A conjecture on the use of quantum algebras in the treatment of discrete systems

The interactions between atomic spin-states, and between them and an external radiation field, can be described in terms of quantum algebras by a trade-off of bosonic and fermionic degrees of freedom and q-deformed schemes. In this Letter we discuss the use of this concept concerning the calculation of a spin observable, like the spin squeezing.     

On the Weyl transverse frames in type I spacetimes

We apply a covariant and generic procedure to obtain explicit expressions of the transverse frames that a type I spacetime admits in terms of an arbitrary initial frame. We also present a simple and general algorithm to obtain the Weyl scalars ₂^T, ₀^T and ₄^T associated with these transverse frames. In both cases it is only necessary to choose a particular root of a cubic expression.     

The Weyl Tensor and Equilibrium Configurations of Self-Gravitating Fluids

It is shown that (except for two well defined cases), the necessary and sufficient condition for any spherically symmetric distribution of fluid to leave the state of equilibrium (or quasi-equilibrium), is that the Weyl tensor changes with respect to its value in the state of equilibrium (or quasi-equilibrium).     

Special tensors in the deformation theory of quadratic algebras for the classical Lie algebras

Using deformation theory, Braverman and Joseph constructed certain primitive ideals in the enveloping algebras of the simple Lie algebras. Except in the case sl(2,C)$\mathfrak{s}\mathfrak{l}(2,\mathbb{C})$, there is a special value of the deformation parameter giving an ideal of infinite codimension. For the classical Lie algebras, the uniqueness of the special value is equivalent to the existence of tensors with very particular properties. The existence of these tensors was concluded abstractly by Braverman and Joseph but here we present explicit formulae. This allows a rather direct computation of the special value of the deformation parameter.     

Holographic p-wave superfluid with Weyl corrections

In this work,we study the effects of the Weyl corrections on the p-wave superfluid phase transition in terms of an EinsteinMaxwell theory coupled to a complex vector field.In the probe limit,it is observed that the phase structure is significantly modified owing to the presence of the higher order Weyl corrections.The latter,in general,facilitates the emergence of the superfluid phase as the condensate increases with the Weyl coupling measured byγ.Moreover,several features about the phase structure of the holographic superfluid are carefully investigated.In a specific region,the phase transition from the normal phase to the superfluid phase is identified to be the first order,instead of being the second order,as in the cases for many holographic superconductors.By carrying out a numerical scan of model parameters,the boundary dividing these two types of transitions is located and shown to be rather sensitive to the strength of Weyl coupling.Also,a feature known as"Cave of Winds",associated with the emergence of a second superfluid phase,is observed for specific choices of model parameters.However,it becomes less prominent and eventually disappears asγincreases.Furthermore,for temperature in the vicinity of the critical one for vanishing superfluid velocity,denoted by T0,the supercurrent is found to be independent of the Weyl coupling.The calculated ratio,of the condensate with vanishing superfluid velocity to that with maximal superfluid velocity,is in good agreement with that predicted by Ginzburg-Landau theory.While compared with the impact on the phase structure owing to the higher curvature corrections,the findings in our present study demonstrate entirely different characteristics.Further implications are discussed.     

Graded quasi-Lie algebras

This paper is concerned with a new class of graded algebras naturally uniting within a single framework various deformations of the Witt, Virasoro and other Lie algebras based on twisted and deformed derivations, as well as color Lie algebras and Lie superalgebras. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005. Supported by the Liegrits network Supported by the Crafoord foundation