Intergratable Quantum Field Theories:New Methods And Applications

可积量子场论:新方法与应用

• 批准号: 1404056
• 负责人: Sergei Lukyanov
• 金额: $ 30万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1404056&HistoricalAwards=false
• 关键词: Intergratable; Quantum; Field; Theories; New

There is large and expanding array of problems in fundamental physics (ranging from sub-nuclear structure of matter through astrophysics and cosmology) which deal with systems whose theoretical description involves continuously many variables with strong interconnection between them. Quantum field theory is a unifying mathematical language for such systems. Appropriately, this language is difficult, and in general its "grammatical rules" are not even yet established. In this situation, any instances, or "models", of quantum field theory whose equations can be worked out all the way through are especially valuable. Such models are known as "integrable quantum field theories". Mainly, they serve as the testing ground for verification and development of ideas and mathematical methods, but remarkably many integrable "models" directly apply to theoretical understanding of properties of complex materials. That is why today the integrable quantum field theories constitute one of the most important and actively developed area of mathematical physics. Moreover, since the discovery of "gauge/string duality" and the role of integrability in this relation, this field of research has found itself at the cutting edge of theoretical high-energy physics. Unfortunately, the integrable models emerging in this context still resist full solution. This research project proposes further development of integrable quantum field theories, with emphasis on the so-called integrable sigma models. This is exactly the general class of models which emerge in the context of the gauge/string duality. Also, similar models are believed to provide theoretical understanding of particularly complicated disordered systems in physics of materials. It is proposed to develop a certain new class of integrable sigma-models, with novel mathematical structure, and find full solutions. This is an intermediate step in the approach to the systems of direct application in the gauge/string dualities. Interestingly, mathematical methods developed for integrable quantum field theories turn out to have broader significance in mathematical physics, and often help to investigate a wider class of non-integrable systems. This area will be explored as well.The activity consists of three projects. (1) Analysis of a new class of integrable sigma models whose target spaces are deformed group manifolds. With the most general known deformations, quantum integrability of such models require significant modifications of standard tools such as the Lax representations, the Yang-Baxter algebras, and Baxter's Q operators and their relations. The project's goal is to develop all such modifications, and use that as the basis for finding full solutions. The key to our approach will be the so-called ODE/IQFT correspondence, the new mathematical tool of integrable quantum field theories ("IQFT side") which allows to "encode" its algebraic structures into a system of ordinary differential equations ("ODE" side). This project lays in the mainstream of the current research in mathematical physics; its significance is in the relation to the gauge/string dualities, and possibly to physics of disordered systems. The other two parts the research is to explore applications of the methods of integrability in non-integrable systems. (2) The relation between the 't Hooft's equation in 1+1 QCD and the Baxter's T-Q equation will be used to analyze the meson decays and scattering amplitudes in the large-N expansion, with the ultimate goal of gaining universal tools for analysis of decays and amplitudes in generic 1+1 systems with confining interactions, which are very common in condensed matter physics. (3) The third part addresses possibility of emergence of near-integrable subsystems in otherwise non-integrable quantum field theories, the possibility which was not previously explored.

在基础物理学（从物质的亚核结构到天体物理学和宇宙学）中有大量和不断扩展的问题，这些问题涉及的系统的理论描述连续涉及许多变量，这些变量之间有很强的相互联系。量子场论是这类系统的统一数学语言。恰当地说，这种语言很难，一般来说，它的“语法规则”甚至还没有建立起来。在这种情况下，量子场论的任何实例或“模型”，其方程都可以一直解出来，这是特别有价值的。这种模型被称为“可积量子场论”。主要是，它们作为验证和发展思想和数学方法的试验场，但值得注意的是，许多可积“模型”直接适用于复杂材料性质的理论理解。这就是为什么今天可积量子场论构成了数学物理学中最重要和最活跃的发展领域之一。此外，自从“规范/弦对偶性”的发现和可积性在这个关系中的作用，这个研究领域已经发现自己处于理论高能物理的前沿。不幸的是，在这种情况下出现的可积模型仍然抵制完全解决方案。本研究计划提出进一步发展可积量子场论，重点是所谓的可积西格玛模型。这正是规范/弦对偶中出现的一般模型。此外，类似的模型被认为提供了对材料物理学中特别复杂的无序系统的理论理解。提出了发展一类新的可积σ模型，具有新的数学结构，并找到完整的解决方案。 这是直接应用于规范/弦对偶系统的方法的中间步骤。有趣的是，为可积量子场论开发的数学方法在数学物理中具有更广泛的意义，并且通常有助于研究更广泛的不可积系统。该活动包括三个项目。(1)目标空间为变形群流形的一类新的可积sigma模型的分析。对于最普遍的已知变形，这种模型的量子可积性需要对标准工具进行重大修改，例如Lax表示，杨-巴克斯特代数和巴克斯特的Q算子及其关系。该项目的目标是开发所有此类修改，并将其作为找到完整解决方案的基础。我们方法的关键是所谓的ODE/IQFT对应，这是可积量子场论的新数学工具（“IQFT侧”），它允许将其代数结构“编码”成常微分方程系统（“ODE”侧）。该项目奠定了当前数学物理研究的主流;其意义在于规范/弦对偶的关系，并可能与无序系统的物理学有关。另外两部分的研究是探索可积性方法在不可积系统中的应用。(2)利用1+1 QCD中的t Hooft方程和巴克斯特的T-Q方程之间的关系，分析大N展开下的介子衰变和散射振幅，最终获得分析凝聚态物理中常见的具有约束相互作用的一般1+1系统的衰变和振幅的通用工具。(3)第三部分讨论了在不可积量子场论中出现近可积子系统的可能性，这种可能性以前没有探讨过。