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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=668722
• 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. **

我计划集中在两个相关的领域，即超可积性在经典和量子理论和连续对称性的离散方程。它们之间的连接和预期的影响都是，我打算找到和研究新的超可积和精确可解的经典和量子系统在连续和离散时空。在离散的情况下，这样的系统是由差分方程，而不是微分方程。基本工具是李代数理论及其推广。1.超级集成性。*超可积系统是允许运动积分多于自由度的哈密顿系统。最著名的和唯一的旋转不变的是谐振子和开普勒-库仑系统。 它们具有物理意义有几个原因。它们是完全可解的。运动积分构成有趣的非阿贝尔代数。在经典力学中，极大超可积系统的所有有界轨线都是封闭的，并且运动是周期的。在量子力学中，它们表现出能级的“偶然”简并。直到最近，超可积系统被认为是极其罕见的。最近，我们证明了无穷族的极大超可积系统存在的运动积分是多项式的任意阶的动量。超可积系统及其非阿贝尔积分代数是无限维孤子系统的有限维类似物。在接下来的5年里，我计划系统地研究运动积分为动量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和Painleve超越理论的联系。我未来的工作也将包括具有磁场和非零自旋的粒子的系统。* 2.离散方程的对称性。*这是一个通用的和雄心勃勃的计划，其目的是把李群理论成为一个有效的工具，用于求解方程描述离散的线性和非线性物理现象。该方案有两个方面。一种是解析的，即利用李理论得到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。这里的想法是使用对称适应格来离散微分方程，同时保留它们的整个李点对称群，或者至少是整个对称群的一个大的物理子群。我们已经表明，常微分方程保持李点对称性提高了数值，特别是接近奇异的解决方案。在未来5年内，我计划集中在偏微分方程从这个角度来看。该计划的一部分是发展晶格上的量子理论，同时保持洛伦兹、伽利略或共形不变性。**