Investigations in Quantum Field Theory

量子场论可能是物理学家用来描述物理世界的最成功的工具。例如，标准模型精确地描述了所有已知粒子的行为，是一种量子场论。其他量子场论也被用来描述非常有实际意义的系统，包括超导体和其他奇异材料，它们位于现代和未来电子设备的核心，如量子计算机。量子场论在理论物理中无处不在，这意味着量子场论中的每一个新想法都会在许多不同的学科和科学界产生反响，而不是最初由哪个社区发展起来的。 量子场论的研究经常涉及到非常复杂的数学思想，这导致了对重要数学学科的全新视角，以及以前不相关领域之间的新联系。在物理和数学的交叉点上的这种强有力的跨学科活动对这两个学科都有很大的好处。 尽管量子场论是一个久负盛名的工具，但它的许多方面仍然是神秘的或鲜为人知的。根据具体情况，量子场论可能是“弱耦合”的，也可能是“强耦合”的。弱耦合场论允许我们进行精确的计算，其结果与详细的实验结果相吻合。它们通常被认为是被很好地理解的，尽管许多重要的微妙之处至今仍在被发现。人们对强耦合场论知之甚少，也很难用于精确的计算。许多这样的理论可能还不为人所知，我们也没有系统的方法来寻找它们。 我们对量子场论理解的这些局限性阻碍了我们在各种学科上的进步，从关于宇宙本质的深刻问题到材料科学中非常具体的问题，例如能够在室温下工作的超导体的发展。 这项建议的主要目的是开发新的计算工具和数学思想来研究量子场论，并将这些工具应用于解决高能物理、凝聚态物理和数学中长期存在的理论问题。该计划中的研究活动还将为几名学生和年轻研究人员提供独特的培训机会，他们将在科学界和其他地方传播这些工具和想法。 从长远来看，我们对量子场论的理论理解的这种进步将影响我们为实际应用量身定做奇异量子系统的能力，推进对宇宙基本定律的理解，并导致深刻的数学进步。 除了它的具体应用之外，对量子场论的充分理解将是人类伟大的智力成就之一，它本身就值得追求。

Quantum Field Theory is perhaps the most successful tool physicists use to describe the physical world. For example, the Standard Model, which describes in precise detail the behaviour of all known particles, is a quantum field theory. Other quantum field theories are used to describe systems of very practical interest, including superconductors and other exotic materials which lie at the core of modern and future electronic devices, such as quantum computers. The ubiquity of Quantum Field Theory in theoretical physics means that every novel idea in Quantum Field Theory reverberates through many different subjects and scientific communities, irrespectively of which community originally developed it. The study of Quantum Field Theory often involves ideas of great mathematical sophistication, which have lead to whole new perspectives on important mathematical subjects and novel connections between previously disconnected areas. This strong interdisciplinary activity at the intersection of physics and mathematics greatly benefits both subjects. Despite being such a well-established tool, many aspects of Quantum Field Theory are still mysterious or poorly understood. Depending on the specific situation, a quantum field theory may be "weakly coupled" or "strongly coupled". Weakly coupled field theories allow us precise calculations which match in detail experimental results. They are usually considered to be well understood, though many important subtleties are still being discovered today. Strongly coupled field theories are poorly understood, and very hard to use for precise calculations. Many such theories may be still unknown and we do not have a systematic way to search for them. These limitations in our understanding of Quantum Field Theory obstruct our progress in a variety of subjects, ranging from deep questions about the nature of the Universe to very concrete problems in material science, such as the development of superconductors able to function at room temperature. The main objective of this proposal is to develop new computational tools and mathematical ideas to study Quantum Field Theory and to apply these tools to the solution of long standing theoretical problems in High Energy Physics, Condensed Matter Physics and Mathematics. The research activities in this program will also offer unique training opportunities to several students and young researchers, which will disseminate these tools and idea through the scientific community and beyond. In the long term, this progress in our theoretical understanding of quantum field theories will impact our ability to tailor exotic quantum systems to practical applications, advance out understanding of the fundamental laws of the universe, and lead to deep mathematical advances. Beyond its concrete applications, reaching a full understanding of Quantum Field Theory will be one of humanity great intellectual achievements and it is worth pursuing for its own sake.