NCDIFFGEO · Models of noncommutative differential geometries

Models of noncommutative differential geometries

A standard technique in physics and engineering is to replace continuous geometric backgrounds by discrete approximations (lattice or graph), so that systems become more calculable. It has become clear that this can be done systematically as a case of Noncommutative geometry (NCG), namely one where differentials and functions do not commute. NCG, or `quantum' geometry, is also expected to arise from Quantum Gravity (QG), the physical regime at which gravitational and quantum interactions are equally strong; this occurs at either very high energies, or at very small distances, commonly referred to as the Planck scale. The fundamental assumption is that QG effects at the Planck scale modify the structure of space-time itself, leading to noncommutativity of spacetime coordinates, which are `quantised' in a similar manner as position and momenta in quantum mechanics. Such modification of spacetime introduces uncertainty or `fuzziness' at the Planck scale and also introduces other gravitational and cosmological effects that could ultimately be empirically verified. NCG allows us to model these quantum gravitational corrections in an effective description without full knowledge of QG itself. As such it has its own internal structure as a more general notion of classical geometry and could be used in many other situations where classical geometry breaks down; it amounts to as a general quantisation scheme for geometry itself. The project builds on a specific body of knowledge accumulated over the last 25 years related in part to the notion of `quantum groups' or Hopf algebras. These fully emerged in the 1980s out of quantum integrable systems as a new more general notion of symmetry but they can also feature as `quantum symmetries' of quantum spacetimes. The mathematical theory of `noncommutative Riemannian geometry' was also developed recently. This was originally motivated by the wish to include the NCG of quantum groups themselves (just as classical geometry was driven on part by the geometry of Lie groups) but, as a general scheme, it potentially applies much more widely. The aim of the project was to obtain and study a new generation of quantum spacetime models and their noncommutative differential geometries in this context.

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