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Abstract : We propose hereafter that the signature of the Space-Time metric (+++-) is not anymore frozen at the Planck scale and presents quantum fluctuations (+++±) untill 0 scale where it becomes Euclidean (++++). (i) At the albraic level we suggest an oscillation path (3,1) (4,0) exluding (2,2). We built the quotient topological space describing the superposition of the Lorentzian and the Riemanian metrics. In terms of quantum groups we evidence a relation between q-deformation and deformation of the signature. We have obtained a new algebraic construction ( a new cocycle bicross products by twisting) which allowed us to unify the Lorentzian and the Euclidean signatures within a unique quantum group structure. Moreover the q-deformation of space-time shows that the natural structures of q-Minkowski and q-Riemanian spaces are linked by semiduality. (ii) Regarding the physical motivations we suggest that at the Planck Scale the Space-Time is in KMS state. Within the limits of the KMS holomorph strip, the ß timelike parameter is complex. We propose an extension of relativistic gravity which begins at the Planck Scale with the Lagrangian R + R2 + RR*. Then, the infrared limit ot the theory is given at the Planck Scale by the Einstein term in R and corresponds to the Lorentzian metric while the ultra violet limit is given at ß=0 scale by the topological term RR* and corresponds to the Euclidean metric ( topological sector). We propose a duality betweens instantons and monopoles in 4 dimensions giving a representation of the superposition of the metrics. (iii) On the cosmological plan we suggest to describe the Initial Singularity of Space Time by a topological invariant I(S) = Tr(-1)S which is analog to the first Donaldson invariant. The initial singularity must be considered as a singular 0-size gravitational Instanton. The physical observables are therefore replaced by cycles of homology in the moduli space of gravitational Instantons. We propose a conjecture regarding the existence of a topological amplitude associated to a " topological expansion phase " which preceeds the classical comoslogical expansion. This topological phase is also able to be described by the flow of weights of the II * hyperfinite factor type corresponding to the ß=0 initial singularity.
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Contributor : Bogdanoff Igor <>
Submitted on : Thursday, July 18, 2002 - 7:51:41 PM
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  • HAL Id : tel-00001502, version 1


Grichka Bogdanoff. FLUCTUATIONS QUANTIQUES DE LA SIGNATURE DE LA METRIQUE A L'ECHELLE DE PLANCK. Mathématiques [math]. Université de Bourgogne, 1999. Français. ⟨tel-00001502⟩



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