Supersymmetric solvable and integrable systems in quantum and classical physics

量子和经典物理中的超对称可解和可积系统

• 批准号: 41969-2011
• 负责人: Hussin, Véronique
• 金额: $ 1.6万
• 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: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=568878
• 关键词: Supersymmetric; solvable; integrable; systems; quantum

In this research program, I am proposing two work themes which are all related to the study of solvable and integrable supersymmetric systems appearing in quantum and classical physics using group theoretical methods. In particle physics, they are describing in a unified way bosons and fermions. Supersymmetry is also used to include gravitation in the unified scheme involving the different forces of nature. In the first theme, I intend to work on generating new solutions for supersymmetric non-linear differential equations, such as the Korteweg de Vries equation. These solutions are called multi-solitons (or multi solitary waves) as they are well-localised waves that maintain their shape even after interactions with other such waves. In the supersymmetric context, there is a large body of equations to be solved. I hope to extend symmetry reduction and bilinearisation methods to an extensive range of new systems of equations. I will also work with supersymmetric extensions of models used in gauge theories that have been very successful in elementary particle physics. In this work, important properties of supersymmetric sigma models will be exhibited with the aim to better understand the geometry of these systems. In the second theme, I plan to relate known multidimensional integrable quantum systems through supersymmetry and thus determine explicitly the solutions. I also plan the establish relations between supercharges and accidental degeneracy in the spectrum. Finally, from my work on coherent and squeezed states for the one-dimensional Morse potential, I plan to establish rigorously connections between quantum and classical states. Such states are important because they are, in particular, related to the quantum treatment of optical coherence and behave in a similar fashion to classical states for oscillating systems. Nowadays, they are seen as an important instrument for studying quantum information processes and the question of how our different constructions will fit within this scheme is relevant for future developments in the field.

在这个研究项目中，我提出了两个工作主题，它们都与使用群论方法研究量子和经典物理中出现的可解和可积超对称系统有关。 在粒子物理学中，他们以统一的方式描述玻色子和费米子。 超对称性也被用来将引力包含在涉及不同自然力的统一方案中。 在第一个主题中，我打算为超对称非线性微分方程（如Korteweg de弗里斯方程）生成新的解。 这些解被称为多孤子（或多孤波），因为它们是局部化良好的波，即使在与其他波相互作用后也能保持其形状。 在超对称的背景下，有大量的方程需要求解。我希望扩展对称性减少和bilinearisation方法，以广泛的新系统的方程。我也将与规范理论中使用的模型的超对称扩展，在基本粒子物理学中已经非常成功。 在这项工作中，超对称西格玛模型的重要性质将被展示，目的是更好地理解这些系统的几何形状。 在第二个主题中，我计划通过超对称性联系已知的多维可积量子系统，从而明确地确定解决方案。我还计划建立超荷与谱中偶然简并的关系。最后，从我对一维莫尔斯势的相干态和压缩态的研究中，我计划在量子态和经典态之间建立严格的联系。这种状态是重要的，因为它们特别与光学相干性的量子处理有关，并且以类似于振荡系统的经典状态的方式表现。如今，它们被视为研究量子信息过程的重要工具，我们的不同结构如何适应这个方案的问题与该领域的未来发展有关。