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Quantum Field Theory and Operator Algebras

This research project is in competition for funding with one or more projects available across the EPSRC Doctoral Training Partnership (DTP). Usually the projects which receive the best applicants will be awarded the funding. Find out more information about the DTP and how to apply.

Application deadline: 15 March 2019

Start date: 1 October 2019

The research area of this project is an operator-algebraic approach to Quantum Field Theory.


Quantum Field Theory (QFT) is the theoretical framework in which the physics of elementary particles can be formalised in a mathematical language. It is widely recognised for its rich mathematical structure that is a driving force and inspiration behind many developments in pure mathematics, similar to how quantum mechanics initiated the study of Hilbert spaces and linear operators on them.

This project is an operator-algebraic approach to QFT, modelling physical observables as elements of C*- or von Neumann algebras, symmetries by group actions of such algebras and states by positive linear functionals on them. This formulation makes it possible to use advanced tools such as Tomita-Takesaki theory in a QFT context.

Within this general setting, there are various opportunities for PhD projects, for example in connection with the explicit constructing of model theories, or the detailed investigation of the analytic and algebraic properties of certain scattering operators appearing in QFT. Although the background and motivation of these topics is from physics, this PhD project will focus on mathematical aspects.

Project aims and methods

You will learn about mathematical aspects of QFT and operator algebras in an interdisciplinary research area at the interface of mathematics and physics. This will involve in particular the Haag-Kastler approach to QFT, Tomita-Takesaki modular theory of von Neumann algebras in standard form, reproducing kernel Hilbert spaces like Hardy spaces on tubes, and unitary group representations on them.

Other more specialised techniques, developed by the supervisor in collaboration with an international team of colleagues, address constructive aspects of algebraic QFT and will give you direct access to expert knowledge. Depending on the precise direction of the project, there will be the opportunity to interact with colleagues from different universities such as Leipzig, Rome, Sao Paulo, or York. Since the background of this project comes from quantum physics, you will also learn about mathematical modelling in the natural sciences.


Picture of Gandalf Lechner

Dr Gandalf Lechner

Senior Lecturer

+44(0)29 208 75553

Programme information

For programme structure, entry requirements and how to apply, visit the Mathematics programme.

View programme
Join us on 27 March to find out more about postgraduate funding opportunities for September 2019.

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