Kepler problem in deformed (quantum) four-dimensional space in non relativistic limit with Galilei group of motion
A. N. Leznov
证明了变形（量子）四维空间中的Kepler问题在非相对论极限下的积分可积性。在非相对论量子空间中，运动极限群与伽利略极限群重合。
It is shown that Kepler problem in deformed (quantum) four-dimensional space in non relativistic limit is integrable in quadratures. In non relativistic limit group of motion of quantum space coincide with Galilei one.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Canonical and Lie-algebraic twist deformations of $κ$-Poincare and contractions to $κ$-Galilei algebras
我们提出了$κ$ -变形的Poincare Hopf代数的正则和李代数扭曲变形，从而得到了广义的$κ$ -Minkowski时空关系.计算了相应的形变$κ$ -Poincare量子群.最后，我们对相应的扭曲伽利略代数和对偶伽利略量子群进行了非相对论压缩极限的研究。
The three quantum groups dual to the generalized twist deformed Poincare Hopf algebras are provided with use of FRT procedure. Their Galilean counterparts are obtained by nonrelativistic contraction scheme.
Generalized twist deformations of Poincare and Galilei Hopf algebras
利用twist过程构造了三个新的变形Poincare Hopf代数.相应的相对论时空提供的标准和李代数型的非交换性的总和提出。最后，对相应的Galilei-Hopf代数进行了非相对论压缩极限的研究.
Local representations of the (1+1) quantum extended Galilei algebra
我们提出了（1+1）扩展Galilei群的局部表示的q-变形对应。这些表象作用于定义在时空中的波函数空间。与经典情形一样，q-局部表象是可约的，不可约的条件由q-卡西米尔方程给出，在变形参数趋于零的极限下，该方程变成自由粒子的薛定谔方程。
This work addresses the Hamiltonian dynamics of the Kepler problem in a deformed phase space, by considering the equatorial orbit. The recursion operators are constructed and used to compute the integrals of motion. The same investigation is performed with the introduction of the Laplace-Runge-Lenz vector. The existence of quasi-bi-Hamiltonian structures is also elucidated. Related properties are studied.
(Anti)de Sitter/Poincare symmetries and representations from Poincare/Galilei through a classical deformation approach
一个经典的变形过程，基于通用包络代数，卡西米尔和曲率的对称均匀空间，适用于几种情况下的物理相关性。从（3+1）D Galilei代数，我们描述的水平表示的过程，导致其两个物理上有意义的变形邻居。庞加莱代数是通过在平坦的伽利略相空间（或世界线空间）中引入负曲率而得到的，同时保持平坦的时空。准确地说，从伽利略代数的一个表示开始，两个卡西米尔都不为零，我们得到了庞加莱代数的一个表示，两个卡西米尔都必须等于零。将庞加莱角动量、Pauli-Lubanski分量、位置和速度算符等用“伽利略”算符表示，通过对正常伽利略算符进行变形的表达式来表示。类似地，牛顿-胡克代数的出现是通过赋予时空非零曲率，同时保持平坦的相空间。同样的方法，从（3+1）D庞加莱代数开始，将（反）德西特表示为庞加莱变形。
The quantum oscillator and Kepler-Coulomb problems in $d$-dimensional spaces with constant curvature are analyzed from several viewpoints. In a deformed supersymmetric framework, the corresponding nonlinear potentials are shown to exhibit a deformed shape invariance property. By using the point canonical transformation method, the two deformed Schrödinger equations are mapped onto conventional ones corresponding to some shape-invariant potentials, whose rational extensions are well known. The inverse point canonical transformations then provide some rational extensions of the oscillator and Ke…
We present the quadratic algebra of the generalized MICZ-Kepler system in three-dimensional Euclidean space $E_{3}$ and its dual the four dimensional singular oscillator in four-dimensional Euclidean space $E_{4}$. We present their realization in terms of a deformed oscillator algebra using the Daskaloyannis construction. The structure constants are in these cases function not only of the Hamiltonian but also of other integrals commuting with all generators of the quadratic algebra. We also present a new algebraic derivation of the energy spectrum of the MICZ-Kepler system on the three sphere …
Quantum symmetries in 2+1 dimensions: Carroll, (a)dS-Carroll, Galilei and (a)dS-Galilei
本文利用代数同构与量子收缩，系统分类2+1维中Carroll、Galilei及其(anti-)de Sitter推广代数的量子形变。
We provide fifteen twist-deformed doubly enlarged Newton-Hooke quantum space-times. In $τ$ approaching infinity limit the twisted doubly enlarged Galilei spaces are obtained as well.
Kepler's laws of planetary motion are deduced from those of a harmonic oscillator following Arnold. Conversely, the circular orbits through the Earth's center suggested by Galilei are consistent with an $r^{-5}$ potential as found before by Newton. Both the Kepler/oscillator correspondance and circular orbits are examples of dual potentials.
We construct the integrals of motion for the 5D deformed Kepler system with non-central potentials in $su(2)$ Yang-Coulomb monopole field. We show that these integrals form a higher rank quadratic algebra $Q(3; L^{so(4)}, T^{su(2)})\oplus so(4)$, with structure constants involving the Casimir operators of $so(4)$ and $su(2)$ Lie algebras. We realize the quadratic algebra in terms of the deformed oscillator and construct its finite-dimensional unitary representations. This enable us to derive the energy spectrum of the system algebraically. Furthermore we show that the model is multiseparable a…
We show that the Lorentzian Snyder models, together with their Galilei and Carroll limiting cases, can be rigorously constructed through the projective geometry description of Lorentzian, Galilean and Carrollian spaces with nonvanishing constant curvature. The projective coordinates of such curved spaces take the role of momenta, while translation generators over the same spaces are identified with noncommutative spacetime coordinates. In this way, one obtains a deformed phase space algebra, which fully characterizes the Snyder model and is invariant under boosts and rotations of the relevant …
We introduce a new superintegrable Kepler-Coulomb system with non-central terms in $N$-dimensional Euclidean space. We show this system is multiseparable and allows separation of variables in hyperspherical and hyperparabolic coordinates. We present the wave function in terms of special functions. We give a algebraic derivation of spectrum of the superintegrable system. We show how the $so(N+1)$ symmetry algebra of the $N$-dimensional Kepler-Coulomb system is deformed to a quadratic algebra with only 3 generators and structure constants involving a Casimir operator of $so(N-1)$ Lie algebra. We…
Superintegrable systems with monopole interactions in flat and curved spaces have attracted much attention. For example, models in spaces with a Taub-NUT metric are well-known to admit the Kepler-type symmetries and provide non-trivial generalizations of the usual Kepler problems. In this paper, we overview new families of superintegrable Kepler, MIC-harmonic oscillator and deformed Kepler systems interacting with Yang-Coulomb monopoles in the flat and curved Taub-NUT spaces. We present their higher-order, algebraically independent integrals of motion via the direct and constructive approaches…
The Kepler problem on 3D spaces of variable and constant curvature from quantum algebras
在变曲率和常曲率的三维空间上，利用量子sl（2，R）余代数（带变形参数z）构造了超可积的Kepler势，其中包括经典的球面、双曲和（反）de Sitter空间以及它们的非常曲率类似空间.在这种情况下，非变形极限z = 0与导致适当的欧几里得和闵可夫斯基空间/势的平坦收缩相一致。相应的哈密顿量承认运动的三个常数来自余代数结构。此外，开普勒势在常曲率空间上的最大超可积性是通过找到一个额外的运动常数来明确显示的，该运动常数来自于一个额外的对称性，而这个对称性是不能从量子代数中推导出来的。由此导出了这类空间的Laplace-Runge-Lenz向量，并分析了它的代数性质。
Quasi-exactly solvable extensions of the Kepler-Coulomb potential on the sphere
本文在形变超对称框架下研究了 $d$ 维球面上 Kepler-Coulomb 势的扩展族，通过施加约束和相容性条件构建了具有已知基态和第一激发态的准精确可解势，并利用生成函数法推导了超势与本征态公式，提出了关于常数的猜想。
We study presence of an additional symmetry of a generic central potential in the $κ$-space-time. An explicit construction of Fradkin and Bacry, Ruegg, Souriau (FBRS) for a central potential is carried out and the piece-wise conserved nature of the vector is established. We also extend the study to Kepler systems with a drag term, particularly Gorringe-Leach equation is generalized to the $κ$-deformed space. The possibility of mapping Gorringe-Leach equation to an equation with out drag term is exploited in associating a similar conserved vector to system with a drag term. An extension of dual…
We derive the non-relativistic $c\to\infty$ and ultra-relativistic $c\to 0$ limits of the $κ$-deformed symmetries and corresponding spacetime in (3+1) dimensions, with and without a cosmological constant. We apply the theory of Lie bialgebra contractions to the Poisson version of the $κ$-(A)dS quantum algebra, and quantize the resulting contracted Poisson-Hopf algebras, thus giving rise to the $κ$-deformation of the Newtonian (Newton-Hooke and Galilei) and Carrollian (Para-Poincaré, Para-Euclidean and Carroll) quantum symmetries, including their deformed quadratic Casimir operators. The corres…
Superintegrable systems are a class of physical systems which possess more conserved quantities than their degrees of freedom. The study of these systems has a long history and continues to attract significant international attention. This thesis investigates finite dimensional quantum superintegrable systems with scalar potentials as well as vector potentials with monopole type interactions. We introduce new families of $N$-dimensional superintegrable Kepler-Coulomb systems with non-central terms and double singular harmonic oscillators in the Euclidean space, and new families of superintegra…
The formalism of SUSYQM (SUperSYmmetric Quantum Mechanics) is properly modified in such a way to be suitable for the description and the solution of a classical maximally superintegrable Hamiltonian System, the so-called Taub-Nut system, associated with the Hamiltonian: $$ \mathcal{H}_η({\mathbf{q}}, {\mathbf{p}}) = \mathcal{T}_η({\mathbf{q}}, {\mathbf{p}}) + \mathcal{U}_η({\mathbf{q}}) = \frac{|{\mathbf{q}}| {\mathbf{p}}^2}{2m(η+ |{\mathbf{q}}|)} - \frac{k}{η+ |{\mathbf{q}}|} \quad (k>0, η>0) \, .$$ In full agreement with the results recently derived by A. Ballesteros et al. for the quantum c…
Relativistic time-commutative dynamics with $κ$-plane noncommutativity
本文针对时空非对易性相关物理意义存争议的问题，构建首个在非对易时空上两粒子的完全相对论性初量子化模型，刻画了变形对称代数及其极限，构建单粒子模型，展示两粒子系统描述，揭示对称生成元与相互作用定律变形的关联。
Superintegrable potentials on 3D Riemannian and Lorentzian spaces with non-constant curvature
在三维弯曲空间上构造了一个量子sl（2，R）余代数，这类超可积势包括球面空间、双曲空间和（反）de Sitter空间的非常曲率类似物.通过在两个不同的相空间上工作，充分研究了这些“变形”空间的连接和曲率张量。前者直接来自变形余代数的3D辛实现，而后者是通过映射到球型相空间而获得的。在这个框架中，非变形极限与导致欧几里得和闵可夫斯基空间/势的平坦收缩相一致。由此产生的哈密顿总是承认，至少有三个功能独立的运动常数来自余代数结构。此外，确定了这种非常曲率的黎曼和洛伦兹空间上的内禀振子和开普勒势，并明确给出了几个例子。
We analyse the $n$-dimensional superintegrable Kepler-Coulomb system with non-central terms. We find a novel underlying chain structure of quadratic algebras formed by the integrals of motion. We identify the elements for each sub-structure and obtain the algebra relations satisfied by them and the corresponding Casimir operators. These quadratic sub-algebras are realized in terms of a chain of deformed oscillators with factorized structure functions. We construct the finite-dimensional unitary representations of the deformed oscillators, and give an algebraic derivation of the energy spectrum…
By applying the recurrence approach and coupling constant metamorphosis, we construct higher order integrals of motion for the Stackel equivalents of the $N$-dimensional superintegrable Kepler-Coulomb model with non-central terms and the double singular oscillators of type ($n, N-n$). We show how the integrals of motion generate higher rank cubic algebra $C(3)\oplus L_1\oplus L_2$ with structure constants involving Casimir operators of the Lie algebras $L_1$ and $L_2$. The realizations of the cubic algebras in terms of deformed oscillators enable us to construct finite dimensional unitary repr…
Integrable potentials on spaces with curvature from quantum groups
通过定义在sl（2）Poisson余代数的非标准量子变形上的Hamilton算子，构造了一类定义在二维球面、双曲和（反）de Sitter空间变形上的经典可积系统.所有这些空间都具有取决于变形参数z的非恒定曲率。作为特例，提出了谐振子和Kepler-Coulomb势在这种空间上的类似物。另一个变形的哈密顿量也被证明提供超可积系统通常的球，双曲和（反）德西特空间与常曲率，正好符合z。根据每个特定的空间，所得到的势被解释为中心谐振子与两个以上的振子或离心屏障的叠加。所有这些哈密顿量的非变形极限z=0可以被认为是零曲率极限（收缩），它导致平坦欧氏空间和闵可夫斯基空间上相应的（超）可积系统。
Two-dimensional classical superintegrable systems: polynomial algebra of integrals
本文证明二维经典超可积系统的运动积分生成四维多项式代数，提出不依赖可分离性的构造方法，并应用于多个物理势，显式推导经典轨迹。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。