Supersymmetric solvable and integrable systems in quantum and classical physics

在这个研究计划中，我提出了两个工作主题，它们都与用群论方法研究量子和经典物理中出现的可解和可积的超对称系统有关。在粒子物理学中，他们以统一的方式描述玻色子和费米子。超对称性也被用来在涉及不同自然力的统一方案中包括引力。 在第一个主题中，我打算致力于生成超对称非线性微分方程的新解，例如Korteweg de Vries方程。这些解被称为多孤子(或多孤波)，因为它们是局域化的波，即使在与其他此类波相互作用后也保持其形状。在超对称的背景下，有一大堆方程需要求解。我希望将对称约化和双线性化方法扩展到更广泛的新方程组。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性质，目的是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。我还计划在谱中建立超荷和偶然简并之间的关系。最后，通过我对一维Morse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。这种状态很重要，因为它们尤其与光学相干性的量子处理有关，其行为方式与振荡系统的经典状态相似。如今，它们被视为研究量子信息过程的重要工具，我们不同的结构如何适应这个方案的问题与该领域的未来发展有关。

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.