Group theory and nonlinear phenomena in physics

物理学中的群论和非线性现象

• 批准号: RGPIN-2016-04025
• 负责人: Winternitz, Pavel
• 金额: $ 2.91万
• 依托单位: Université de Montréal
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=687751
• 关键词: Group; theory; nonlinear; phenomena; physics

I plan to concentrate on two related areas namely Superintegrability in Classical and Quantum Theories and Continuous Symmetries of Discrete Equations.The connection between them and the expected impact of both is that I intend to find and study new superintegrable and exactly solvable classical and quantum systems in both continuous and discrete space-time. In the discrete case such systems are are governed by difference equations, rather than differential ones. The essential tool is Lie algebra theory and its generalizations.***1. SUPERINTEGRABILITY.***Superintegrable systems are Hamiltonian systems that allow more integrals of motion than they have degrees of freedom. The best known ones and only rotationally invariant ones are the harmonic oscillator and the Kepler-Coulomb system. They are of physical interest for several reasons. They are exactly solvable . The integrals of motion form interesting non-Abelian algebras. In classical mechanics all bounded trajectories in maximally superintegrable systems are closed and the motion is periodic. In quantum mechanics they exhibit "accidental" degeneracy of energy levels. Until recently superintegrable systems were considered to be extremely rare. Recently we have shown that infinite families of maximally superintegrable systems exist with integrals of motion that are polynomials of arbitrary order in the momenta. Superintegrable systems with their non-Abelian algebras of integrals are finite-dimensional analogs of infinite dimensional soliton systems.***In the next 5 years I plan to systematically study superintegrable scalar systems with integrals of motion that are polynomials of order N in the momenta and to investigate their connection with supersymmetry in quantum mechanics and with the theory of Painleve transcendents. My future work will also include systems with magnetic fields and particles with nonzero spin.*** ***2.SYMMETRIES OF DISCRETE EQUATIONS.***This is a general and ambitious program the aim of which is to turn Lie group theory into an efficient tool for solving equations describing discrete linear and non-linear physical phenomena. The program has two aspects. One is analytic, namely to use Lie theory to obtain exact analytic solutions of these discrete equations. The other is numeric and intersects with the theory of geometric integration. Here the idea is to use symmetry adapted lattices to discretize differential equations while preserving their entire Lie point symmetry groups, or at least a large physical subgroup of the entire symmetry group. We have shown that for ordinary differential equations preserving Lie point symmetries improves the numerics, specially close to singularities of solutions. In the next 5 years I plan to concentrate on partial differential equations from this point of view. Part of the program is to develop quantum theory on lattices while preserving Lorentz, Galilei, or conformal invariance. **

我计划集中研究两个相关领域，即经典和量子理论中的超可积性以及离散方程的连续对称性。它们之间的联系以及两者的预期影响是，我打算在连续和离散时空中寻找和研究新的超可积且可精确解的经典和量子系统。在离散情况下，此类系统由差分方程而不是微分方程控制。基本工具是李代数理论及其推广。***1。超积分。***超可积系统是哈密顿系统，允许比其自由度更多的运动积分。最著名且唯一旋转不变的系统是谐振子和开普勒-库仑系统。 由于多种原因，它们具有物理意义。它们是完全可以解决的。运动积分形成有趣的非阿贝尔代数。在经典力学中，最大超可积系统中的所有有界轨迹都是封闭的，并且运动是周期性的。在量子力学中，它们表现出“偶然”的能级简并。直到最近，超可积系统还被认为是极其罕见的。最近我们已经证明，存在无限族最大超可积系统，其运动积分是动量中任意阶的多项式。具有非阿贝尔积分代数的超可积系统是无限维孤子系统的有限维类似物。***在接下来的 5 年里，我计划系统地研究具有运动积分（动量中的 N 阶多项式）的超可积标量系统，并研究它们与量子力学中的超对称性以及 Painleve 超越理论的联系。我未来的工作还将包括具有磁场和非零自旋粒子的系统。*** ***2.离散方程的对称性。***这是一个通用且雄心勃勃的计划，其目的是将李群理论转变为求解描述离散线性和非线性物理现象的方程的有效工具。该计划有两个方面。一类是解析式的，即利用李理论求得这些离散方程的精确解析解。另一个是数值的，与几何积分理论相交叉。这里的想法是使用对称适应晶格来离散微分方程，同时保留其整个李点对称群，或至少整个对称群的一个大物理子群。我们已经证明，对于常微分方程，保留李点对称性可以改善数值，特别是接近解的奇异性。从这个角度来看，我计划在接下来的5年里专注于偏微分方程。该计划的一部分是发展晶格量子理论，同时保留洛伦兹、伽利略或共形不变性。 **