QGNC · Non commutative geometry and quantum gravity
FP7 — People (Marie Curie Actions)
- Duration
- 2009-03-16 → 2012-03-15
- EU contribution
- €45,000
- Participants
- 1
- Scheme
- MC-ERG
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Results in brief
Non-commutative geometry and quantum gravity
Work on this project produced several scientific results, mainly relevant for the study of non-commutative geometry and algebra quantum field theory. Concerning non-commutative geometry the equality, first noticed by Rieffel, between Connes distance (in the commutative case) and the Wasserstein distance in the theory of optimal transport was extended to the locally compact case. And a proposal was made for some novel directions to follow in order to develop a theory of optimal transport in non-commutative geometry. Connes' distance was also applied in deformation quantisation. Recall that the Moyal algebra is the non-commutative deformation - via a star product - of the algebra of Schwartz function on R2n. The project also produced a computation of the distance between a certain class of states of the Moyal algebra (corresponding to the eigenstates of the quantum harmonic oscillator). And general results were obtained on the Moyal plane: the distance between any state of the Moyal algebra and any of its translation is precisely the amplitude of the translation. The project also resulted in a proposal of general framework for comparing Connes distance in the Moyal plane to the quantum length that has been defined in various model of quantum spacetime (like the Doplicher-Fredenhagen-Robert (DFR) model) as the spectrum of a suitable length operator. Applied to the eigenstates of the quantum harmonic oscillator, it was found that the length has the spectrum of an operator and Connes distance correspond to two different ways of integrating the same non-commutative line element on a non-commutative space. Work done during the project also showed that the product of the Moyal space with the spectral triple of C2 - restricted to coherent states - is orthogonal in the sense of Pythagoras theorem. Concerning modular flow in algebraic quantum field theory the project applied the thermal time hypothesis of Connes-Rovelli to double-cone regions in a bi-dimensional conformal field theory with boundary. It was found that the modular flow associated to Longo's ad hoc state was purely geometrical (as for a qft in Minkowski spacetime), while the modular flow associated to the vacuum of the two-dimensional (2D) boundary conformal field theory (CFT) combines the geometrical action with a term that mixes the components of the field on the edge of the double cone. This result is particularly interesting for it gives an explicit illustration of Connes theorem, according to which the modular flow defined by two distinct states are unitarily equivalent. Here, it was found that the action of Connes cocycle was purely non-geometrical. And part of the project was also devoted to exploring the implication of these results the nature of time.
Data: CORDIS, © European Union
Project objective
The project aims at deepening the relationship between various approaches to quantum gravity and noncommutative geometry.Indeed there is a strong theoretical evidence that the usual continuum model of space-time is not correct at very short distances where both gravitational and quantum mechanics effects come into play. Several theories propose a (incomplete) treatment of quantum gravity, but even the most popular of them (string theory, loop quantum gravity) have not been experimentally tested for the energy scale corresponding to the Planck length is far beyond our experimental possibilities. However it has been stressed that quantum gravity theories, by giving a non continuum structure to space-time, may lead to some observational effects. Most of the time this non continuum structure has a mathematical translation in terms of quantum, or non commutative, spacetimes. Namely one posits a noncommutative algebraic structure of the coordinates of spacetime by a deformation of the classical algebra, just as quantum groups are deformation of classical Lie algebras.However the physical status of such space-times is not clear. Naive questions like ""what are the points of a quantum space-time ?"", ""what happens to the notion of distance ?"" have no clear answer.These questions are addressed by the conceptual approach to Noncommutative Geometry (NCG) developed by Connes. NCG provides a formulation of standard geometric and topological concepts (like spin and differential structures) within a purely algebraic framework. Riemannian geometry is encompassed as a particular case, commutative, of a more general theory.The project aims at deepening the relation between the more down-to-earth-approach to quantum space-time developped by theoretical physicist on the one side, and the more fundamental (but difficult) theory of Connes on the other side.It is a continuation of EIF fellowship made by Martinetti in Amelino-Camelia`s group in Rome in 2006/08.""
Original text from CORDIS.
Participants
- UNIVERSITA DEGLI STUDI DI ROMA LA SAPIENZA · RomaCoordinatorItaly
Links
Data: CORDIS, © European Union