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.

在这个研究计划中，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学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。 超对称性还用于将引力纳入涉及不同自然力的统一方案中。 在第一个主题中，我打算致力于为超对称非线性微分方程（例如 Korteweg de Vries 方程）生成新的解。 这些解决方案被称为多孤子（或多孤立波），因为它们是局域化良好的波，即使在与其他此类波相互作用后也能保持其形状。 在超对称背景下，有大量方程需要求解。我希望将对称性约简和双线性化方法扩展到广泛的新方程组。我还将研究规范理论中使用的模型的超对称扩展，这些模型在基本粒子物理学中非常成功。 在这项工作中，将展示超对称西格玛模型的重要属性，旨在更好地理解这些系统的几何结构。 在第二个主题中，我计划通过超对称性将已知的多维可积量子系统联系起来，从而明确地确定解决方案。我还计划在光谱中建立增压和意外简并之间的关系。最后，根据我对一维莫尔斯势的相干态和压缩态的研究，我计划在量子态和经典态之间建立严格的联系。这些状态很重要，因为它们特别与光学相干性的量子处理有关，并且其行为方式与振荡系统的经典状态类似。如今，它们被视为研究量子信息过程的重要工具，而我们的不同结构如何适应该方案的问题与该领域的未来发展相关。