NONCOMMGEOMETRY · Non-commutative geometry from spacetime curvature: Symplectic geometry of quantized geodesics on Riemann manifolds
FP6 — Marie Curie Actions (Human Resources and Mobility)
- Duration
- 2005-01-01 → 2007-09-30
- EU contribution
- €250,364
- Participants
- 2
- Scheme
- OIF
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Results in brief
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.
Data: CORDIS, © European Union
Project objective
The development of a mathematical framework for the unified description of all fundamental forces of nature is one of the biggest challenges in mathematical physics and is, therefore, an important objective of European fundamental research.In order to describe gravity together with the other forces, such a framework will have to unite and generalize the differential geometrical methods of general relativity and the operator algebraic methods of quantum theory. One of the promising candidates for this is non-commutative geometry.However, its relation to classical gravity and curvature is still an open problem. The project objective is to construct and study non-commutative geometry as induced by curvature on Riemann manifolds, in collaboration with A. Weinstein (Berkeley) and P. Schupp (IU Bremen). Due to the uncertainty principle, a quantum particle in curved spacetime is always subject to tidal forces even if it moves on a geodesic.Analogous to the motion in a magnetic field, the quantization of the motion in tidal forces yields, in the strong field limit, non-commutative coordinates. The main idea is that, locally, the motion in tidal forces can be reinterpreted as free motion on this non-commutative geometry.For a mathematically rigorous implementation of this idea, the Hamiltonian formulation of geodesic motion is to be linearized, the local symplectic structure to be quantized, leading to a Lie algebroid description of a bundle of local non-commutative geometries.The global non-commutative geometry is described in the integrable case by the convolution algebra of the corresponding Lie groupoid, in the non-integrable case by the star product realization of the corresponding Poisson algebra.These general constructions are to be computed explicitly for the examples of Schwarzschild and Robertson-Walker spacetimes. The representation theory of the resulting algebras will be studied and compared with black hole and cosmological phenomenology.
Original text from CORDIS.
Participants
- JACOBS UNIVERSITY BREMEN GGMBH · BREMENCoordinatorCity levelGermany
- UNIVERSITY OF CALIFORNIA, BERKELEY · BERKELEYCity levelUnited States
Links
Data: CORDIS, © European Union