Towards the Stochastic Quantisation of Interacting Systems of Fermions and Bosons

在过去的二十年里，首先由Da普拉托和Debussche开创，然后由Gubinelli，Hairer等人扩展的工作已经牢固地建立了随机量子化作为建设性量子场论（QFT）的基本工具。这种方法涉及到解决所谓的非线性奇异随机偏微分方程（SPDE）。然而，这样一个方程的任何可能的解必须是固有的奇异的，也就是说，函数是如此的病态，以至于你不能将它们与它们自己相乘。反过来，这意味着我们不能先验地定义非线性意味着什么。因此，方程是不适定的。自量子场论诞生以来，Feynman、Schwinger、温伯格等物理学家不断发展处理这种奇异乘积的方法。我们现在有一个爆炸性的方法，统称为“重整”，在我们的处置。使这些方法适用于SPDE是使我们开始求解随机量子化方程的关键突破，至少对于玻色子是这样。然而，这些新增强的方法依赖于玻色子的交换性质，因此利用了概率论和随机分析的许多技术。然而，粒子物理学标准模型和整个自然界的一个主要组成部分是费米子，基本上是非交换对象，人们被迫将它们实现为算子代数或类似复杂结构中的对象。因此，而不是仅仅找到随机奇异函数，我们必须解决随机算子值奇异函数。特别地，重整化的过程将具有非常刚性和良好拓扑结构的有界算子转变为无界算子，即需要反复检查每个操作的变化无常的对象。我们一直致力于一个单一的非-交换PDE的正则性结构的语言，以及工作推导出一个特定的方程描述玻色子和费米子的相互作用，可以克服上述问题的解决方案理论。如果这项研究成功，将为研究不仅包含玻色子（自然界的力载体），而且包含费米子（构成宇宙中所有常规物质）的物理系统开辟一条全新的途径。其中包括量子电动力学和杨-米尔斯模型。QFT的严格数学基础，特别是标准模型，是基础物理学中最不容易理解的部分之一，深入了解它们的复杂性可能是我们超越标准模型的少数几条道路之一。因此，我们希望通过我们的研究，能够对自然界最小尺度的知识做出贡献。这个项目属于EPSRC数学物理研究领域的福尔斯。该项目由Ajay Chandra博士和Martin Hairer教授监督。

Over the past two decades, work first pioneered by Da Prato & Debussche and then expanded upon by Gubinelli, Hairer et al. has firmly established Stochastic Quantisation as a fundamental tool of Constructive Quantum Field Theory (QFT). This method involves solving so-called non-linear singular stochastic partial differential equations (SPDE). However, any potential solution to such an equation must be inherently singular, that is the functions are so ill-behaved that you cannot multiply them with themselves. This, in turn, means that we cannot a priori define what the nonlinearities are supposed to mean. Hence, the equations are ill-posed. Since the inception of QFT physicists such as Feynman, Schwinger, Weinberg et al. have continuously developed methods to deal with such singular products. We now have a bursting quiver full of methods, collectively known as "Renormalisation", at our disposal. Adapting these methods to SPDE's was the key breakthrough that allowed us to begin solving the stochastic quantisation equations, at least for Bosons. These newly enhanced methods rely, however, on the commutative nature of Bosons and thus on leveraging many techniques from probability theory and stochastic analysis.However, a major ingredient of the Standard Model of Particle Physics and Nature as a whole are Fermions, fundamentally non-commutative objects, and one is forced to realise them as objects in algebras of operators or similarly complicated structures. Thus, instead of just finding random singular functions we have to solve for random operator-valued singular functions. In particular, the procedure of renormalisation turns bounded operators, which have a very rigid and well-behaved topological structure, into unbounded operators, fickle objects which necessitate one to double-check every operation one takes for granted.In our research, we have been working towards a clean formulation of the singular non-commutative PDE's in the language of regularity structures as well as working on deriving a solution theory for specific equations describing the interaction of Bosons and Fermions that can overcome the problems outlined above. If successful this research will open up a whole new avenue for investigating physical systems that do not only contain Bosons, the force carriers of nature, but also Fermions which make up all the conventional matter in the universe. Amongst these are for example models such as Quantum Electrodynamics and Yang-Mills with ghosts.The rigorous mathematical underpinnings of QFT's, and specifically the standard model, are one of the least well-understood parts of fundamental physics and gaining more insight into their intricacies might be one of the few paths open to us to go beyond the Standard Model. Therefore, we hope that we can contribute to the knowledge of nature at its smallest scales with our research.This project falls within the EPSRC Mathematical Physics research area. The project is supervised by Dr. Ajay Chandra and Prof. Martin Hairer.