Quantum Variational Principle and Discrete Integrable Systems

Supervisor(s)

Contact Professor Frank Nijhoff to discuss this project further informally.

Project description

The project deals with a novel quantum theory that is based on the mathematical structure behind so-called integrable systems, i.e., models that on the classical level are described by certain (mostly nonlinear) differential or difference equations and that exhibit the aspect of `exact solvability' - the fact that they are amenable to exact (rather than approximate or numerical) methods for their solution. Such systems possess a rich mathematical structure (e.g. connections to infinite-dimensional Lie algebras and quantum groups). A key feature is the notion of `multidimensional consistency', the property that such models can be extended to compatible systems of equations in spaces of arbitrary dimension. In 2009 Lobb & Nijhoff proposed a novel variational (i.e., least-action) principle that describes this remarkable feature within a Lagrangian formalism.

The new principle forms potentially a paradigm for a new type of fundamental physics.

Recently, the ideas behind this new approach were extended to the quantum realm, and the first steps were taken to formulate a quantum version of this variational principle. The project for a PhD student is to continue this work into the quantum variational principle, elaborating various (integrable) model systems aimed at building a coherent framework together with the development of some new mathematical methodologies.

The project is embedded in the activities of a wider research group in Integrable Systems within the School of Mathematics, comprising several permanent staff, postdocs and postgraduate students. The group runs its own weekly seminar, and entertains close connections with other research groups in the School, e.g. in Algebra, Geometry and Analysis, as well as with the Quantum Information group in Physics.

Entry requirements

Applicants should have, or expect to obtain, a minimum of a UK upper second class honours degree in Mathematics or a related discipline, or equivalent. Familiarity with Quantum Mechanics and Lagrangians would be beneficial. The project will employ techniques from a variety of subjects in mathematics (such as differential geometry, analysis and algebra) but most of those techniques can be mastered by students with a solid background in `standard' pure and applied mathematics.

If English is not your first language, you must provide evidence that you meet the University's minimum English Language requirements.

How to apply

Formal applications for research degree study should be made online through the university's website. Please state clearly in the research information section that the PhD you wish to be considered for is 'Quantum Variational Principle and Discrete Integrable Systems' as well as Professor Frank Nijhoff as your proposed supervisor.

We welcome scholarship applications from all suitably-qualified candidates, but UK black and minority ethnic (BME) researchers are currently under-represented in our Postgraduate Research community, and we would therefore particularly encourage applications from UK BME candidates. All scholarships will be awarded on the basis of merit.

If you require any further information please contact the Graduate School Office, e: maps.pgr.admissions@leeds.ac.uk