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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=708959
• 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，即经典和量子理论中的超可积性和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和精确可解的经典和量子系统。在离散情况下，这样的系统是由差分方程控制的，而不是微分方程。必不可少的工具是李代数理论及其推广。