Towards the Stochastic Quantisation of Interacting Systems of Fermions and Bosons

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.

Probabilistic modelling permeates the Financial services, healthcare, technology and other Service industries crucial to the UK's continuing social and economic prosperity, which are major users of stochastic algorithms for data analysis, simulation, systems design and optimisation. There is a major and growing skills shortage of experts in this area, and the success of the UK in addressing this shortage in cross-disciplinary research and industry expertise in computing, analytics and finance will directly impact the international competitiveness of UK companies and the quality of services delivered by government institutions.
By training highly skilled experts equipped to build, analyse and deploy probabilistic models, the CDT in Mathematics of Random Systems will contribute to
- sharpening the UK's research lead in this area and
- meeting the needs of industry across the technology, finance, government and healthcare sectors

MATHEMATICS, THEORETICAL PHYSICS and MATHEMATICAL BIOLOGY

The explosion of novel research areas in stochastic analysis requires the training of young researchers capable of facing the new scientific challenges and maintaining the UK's lead in this area. The partners are at the forefront of many recent developments and ideally positioned to successfully train the next generation of UK scientists for tackling these exciting challenges.
The theory of regularity structures, pioneered by Hairer (Imperial), has generated a ground-breaking approach to singular stochastic partial differential equations (SPDEs) and opened the way to solve longstanding problems in physics of random interface growth and quantum field theory, spearheaded by Hairer's group at Imperial. The theory of rough paths, initiated by TJ Lyons (Oxford), is undergoing a renewal spurred by applications in Data Science and systems control, led by the Oxford group in conjunction with Cass (Imperial). Pathwise methods and infinite dimensional methods in stochastic analysis with applications to robust modelling in finance and control have been developed by both groups.
Applications of probabilistic modelling in population genetics, mathematical ecology and precision healthcare, are active areas in which our groups have recognized expertise.

FINANCIAL SERVICES and GOVERNMENT

The large-scale computerisation of financial markets and retail finance and the advent of massive financial data sets are radically changing the landscape of financial services, requiring new profiles of experts with strong analytical and computing skills as well as familiarity with Big Data analysis and data-driven modelling, not matched by current MSc and PhD programs. Financial regulators (Bank of England, FCA, ECB) are investing in analytics and modelling to face this challenge. We will develop a novel training and research agenda adapted to these needs by leveraging the considerable expertise of our teams in quantitative modelling in finance and our extensive experience in partnerships with the financial institutions and regulators.

DATA SCIENCE:

Probabilistic algorithms, such as Stochastic gradient descent and Monte Carlo Tree Search, underlie the impressive achievements of Deep Learning methods. Stochastic control provides the theoretical framework for understanding and designing Reinforcement Learning algorithms. Deeper understanding of these algorithms can pave the way to designing improved algorithms with higher predictability and 'explainable' results, crucial for applications.
We will train experts who can blend a deeper understanding of algorithms with knowledge of the application at hand to go beyond pure data analysis and develop data-driven models and decision aid tools
There is a high demand for such expertise in technology, healthcare and finance sectors and great enthusiasm from our industry partners. Knowledge transfer will be enhanced through internships, co-funded studentships and paths to entrepreneurs