Towards the Stochastic Quantisation of Interacting Systems of Fermions and Bosons

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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.