Service Communautaire d'Information sur la Recherche et le Développement - CORDIS

Final Activity and Management Report Summary - NONCOMMGEOMETRY (Noncommutative geometry from spacetime curvature: Symplectic geometry of quantised geodesics on Riemann manifolds)

The goal of the project was to investigate the transition from classical mechanics to quantum mechanics in the presence of an external field such as a magnetic field or a gravitational field. It has been conjectured but never shown that in the presence of field singularities the quantisation process leads to noncommutative space-time.

It was shown how quantisation by, first, the local linearisation of canonical quantisation by a Lie algebroid, then integrating the Lie algebroid to a Lie groupoid, and finally constructing the convolution algebra of the Lie groupoid can, in principle, produce a noncommutative geometry. In cases where the Lie algebroid is not integrable, it leads to the construction of groupoids internal to differentiable stacks. For gravitational fields, it sheds new light on the groupoid symmetry of the initial value formulation of general relativity.

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