Supervisor: Prof Shahn Majid
Project description:
Quantum Riemannian geometry is a generalisation of geometry that allows coordinates to be noncommutative. The project will focus on a particular approach to this [1] in which differential forms are constructed in an algebraic manner generated by a coordinate algebra A and an exterior differential d. A metric is then a 2-tensor over the algebra A and one has a notion of quantum Levi-Civita connection, curvature and so on. The formalism has already been applied to baby quantum gravity models where A is finite-dimensional as an algebra[2] but there are many more models to which it can be applied, many new tools which are available, such as quantum geodesics, and many more calculations that can be done even in existing models.
The project will be on the one hand to further develop the mathematics needed for applications to mathematical physics, such as the correct notion of variational calculus and conserved quantities and/or on the other hand to further develop applications and models. For example, the algebra A could be a noncommutative algebra of coordinates for `quantum spacetime’, where the noncommutativity is due to Planck scale corrections, or it could be the algebra of observables of a quantum system in quantum mechanics[3], or recent work has included Kaluza-Klein theory with noncommutative `extra dimensions’. There are also potential links with quantum computing and quantum information that could be explored. The applicant can come from either mathematics with an interest in mathematical physics or from mathematical physics with an aptitude for or willing to learn algebra.
References
[1] E.J. Beggs and S. Majid, Quantum Riemannian Geometry, Grundlehren der mathematischen Wissenschaften, vol. 355, Springer (2020) 809pp.
[2] J. Argota-Quiroz and S. Majid, Quantum gravity on finite spacetimes and dynamical mass, PoS (2022) 210 (41pp)
[3] E.J. Beggs and S. Majid, Quantum geodesics in quantum mechanics, J. Math. Phys. 65 (2024) 012101 (37pp)
Please see the supervisor’s website https://webspace.maths.qmul.ac.uk/s.majid/Welcome.html for other recent publications.
Further information:How to apply Entry requirements Fees and funding