Towards the Stochastic Quantisation of Interacting Systems of Fermions and Bosons

在过去的二十年中，首先由Da Prato和Debussche开创，然后由Gubinelli扩展，harer等人已经牢固地建立了随机量子化作为构造量子场论（QFT）的基本工具。该方法涉及求解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，函数是如此的不正常以至于你不能把它们和自己相乘。反过来，这意味着我们不能先验地定义非线性应该是什么意思。因此，方程是不适定的。自QFT诞生以来，Feynman、Schwinger、Weinberg等物理学家不断开发处理此类奇异积的方法。我们现在有一个充满了各种方法的容器，这些方法统称为“重整化”，可供我们使用。将这些方法应用于SPDE是一个关键的突破，它使我们能够开始求解随机量化方程，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，因此利用了许多来自概率论和随机分析的技术。然而，粒子物理和自然整体标准模型的一个主要组成部分是费米子，基本上是非交换的对象，人们被迫将它们视为算子代数或类似复杂结构中的对象。因此，我们必须求解随机的算子值奇异函数，而不是仅仅寻找随机的奇异函数。特别是，重整化过程将有界算子（具有非常严格和良好的拓扑结构）转化为无界算子（变化无常的对象），使得人们必须对每个理所当然的操作进行双重检查。在我们的研究中，我们一直致力于用正则结构的语言清晰地表述奇异非交换偏微分方程，并致力于为描述玻色子和费米子相互作用的特定方程推导解理论，以克服上述问题。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），而且包含构成宇宙中所有常规物质的费米子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带鬼的杨-米尔斯等模型。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，对它们的复杂性有更多的了解可能是我们超越标准模型的少数途径之一。因此，我们希望通过我们的研究，能够在最小尺度上对自然的认识做出贡献。该项目属于EPSRC数学物理研究领域。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.