Group theory and nonlinear phenomena in physics

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。SUPERINTEGRABILITY。***超可积系统是哈密顿系统，它允许更多的运动积分，而不是自由度。最著名的也是唯一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。运动的积分形成了有趣的非阿贝尔代数。在经典力学中，极大超积系统中的所有有界轨迹都是闭合的，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，超可积系统还被认为是极其罕见的。最近我们证明了存在无限族的极大超积系统，它们的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和Painleve超越理论的联系。我未来的工作还将包括磁场系统和非零自旋粒子。* * * * * * 2。离散方程的对称性。***这是一个雄心勃勃的计划，其目的是将李群理论转化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。一种是解析法，即利用李论得到这些离散方程的精确解析解。另一种是数值的，与几何积分理论有交叉。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。我们证明了对于常微分方程，保持李点对称可以提高数值性，特别是接近解的奇异性。在接下来的5年里，我计划从这个角度专注于偏微分方程。该计划的一部分是在保持洛伦兹、伽利莱或共形不变性的同时发展晶格上的量子理论。**

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. **