Feynman Motives

费曼动机

• 批准号: 0901221
• 负责人: Matilde Marcolli
• 金额: $ 22.38万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2014-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901221&HistoricalAwards=false
• 关键词: Feynman; Motives

Quantum field theory is the most sophisticated technique for predictive computations in high-energy and particle physics. The use of Feynman diagrams as computational devices makes it possible to obtain high precision computations of physical processes involving elementary particles and quantum fields. Despite its long history of successful applications to the world of particle physics, the mathematics of quantum field theory is still mysterious and full of beautiful challenges and open problems. Numerical evidence suggests that the procedure of extracting finite values from divergent Feynman integrals gives rise to a class of numbers, multiple zeta values, that are of great significance to number theory and algebraic geometry. This suggests a mysterious relation between quantum field theory and an important research topic of current interest in pure mathematics: Grothendieck's theory of motives of algebraic varieties. The purpose of this research proposal is to understand the nature of this relation and investigate what results can be derived from it, both in terms of gaining some better understanding of the very difficult multi-loop computations of Feynman integrals using tools from algebraic geometry, and conversely of understanding how we can extend our current knowledge of motives using quantum field theory.One of the main questions under investigation is when, possibly after a subtraction of divergences, the computation of a Feynman integral for a scalar quantum field theory results in a period of a mixed Tate motive. Using the Feynman parametric form, this question reflects the motivic nature of a relative cohomology of an affine hypersurface constructed out of the data of the Feynman integral. The subtraction of divergences is encoded in a Hopf algebra structure, which is itself related to Hopf algebras and dual groups that appear naturally in the theory of motives.One of the main steps that are needed to further understand the relation between quantum field theory and motives is combining the more concrete approach via the algebraic geometry of hypersurfaces of Feynman graphs with the more abstract approach via Tannakian categories and Hopf algebras.

量子场论是高能物理和粒子物理中最复杂的预测计算技术。使用费曼图作为计算设备使得有可能获得涉及基本粒子和量子场的物理过程的高精度计算。尽管量子场论的数学在粒子物理领域有着悠久的成功应用历史，但它仍然是神秘的，充满了美丽的挑战和开放的问题。数值证据表明，从发散费曼积分中提取有限值的过程产生了一类数，多重zeta值，这对数论和代数几何具有重要意义。这表明了量子场论和当前纯数学中的一个重要研究课题之间的神秘关系：格罗滕迪克的代数簇动机理论。这项研究计划的目的是了解这种关系的性质，并研究从中可以得出什么结果，无论是在使用代数几何工具更好地理解费曼积分的非常困难的多循环计算方面，还是相反地理解我们如何使用量子场论扩展我们目前对动机的知识。正在研究的主要问题之一是，可能在减去发散之后，标量量子场论的费曼积分的计算导致混合Tate动机的周期。利用Feynman参数形式，该问题反映了由Feynman积分数据构造的仿射超曲面的相对上同调的动机性质。发散的减法被编码在霍普夫代数结构中，进一步理解量子场论与量子动力学之间的关系的一个主要步骤是将通过Feynman图的超曲面的代数几何的更具体的方法与通过Tannakian范畴和Hopf的更抽象的方法相结合代数