MATHEMATICAL CENTER OF RIEMANNIAN COSMOLOGY

The standard "big-bang theory" sets the origin of the Universe at an infinitesimal limit called the Planck scale (10 power - 33 cm). However, even if it is "very small", this limit is not zero. Therefore, the question is the following : does it exist "something" under the Planck scale? In other words, does it exist a "zero instant" in the remote history of the Universe?

The objective of the Mathematical Center of Riemannian Cosmosology (MCRC) is to demonstrate that there exists a "zero point" corresponding to the origin. Such a zero point is not a physical object but has a pure mathematical content and meaning. From this view, one can say that this singular zero point, could be heuristically compared to a sort of "cosmological code" of the Universe. MCRC currently works on the following domains :

KMS THEORY

TOPOLOGICAL FIELD THEORY

CONTACT riemannian-cosmology@phys-maths.edu.lv

THE QUANTUM DEFORMATION METHOD

One of the main objectives of our work was to establish, in terms of quantum groups, the existence of a natural link between "q_deformation", quantisation of spacetime and "deformation" of the signature of the metric. We showed that in dimension D = 4, the Lorentzian and the Euclidean structures are related by twisting. The only natural signatures at the Planck scale are then the deformations of the Lorentzian (+ + + - ) and Euclidean (+ + + +) signatures. Our results then suggest that, in order to be compatible with noncommutative geometry, only the case of superposition of signatures (+ + + ±) should be considered at the scale of quantum gravity, whereas the ultra-hyperbolic case (+ + - - ) should be excluded. We show this from the viewpoints both of q-Lorentzian symmetry and q-spacetime. Our main mathematical result is then the construction of the new cocycle bicrossproduct where H is a quantum group and c a 2-cocycle of "twisting" type. Such a construction has allowed us to construct the unification of the Lorentzian and Euclidean signatures within a unique structure of quantum group, of the form Uq(so(4)) //Uq(so(3, 1)). We also suggest that the Majid "semidualisation" describes the q-Euclidean to q-Lorentzian transition. In the same way, we study the structures of the q-PoincarÈ group and relate it to the k -PoincarÈ group P k as well as to their different deformations by twisting. We discuss then the conjecture according to which it might exist a general link between deformation cocycle c , curvature (generally of the space of the phases of the system but here curvature of pre-spacetime) and anomalies of the theory. All this work is developed in the paper Construction of Cocycle Bicrossproducts by Twisting;

THE KMS EQUILIBRIUM AT THE PLANCK SCALE

After having established the superposition of qSO(3,1) and qSO(4) within a unique quantum group structure, we have applied this situation within a physical context. The recent literature contains many descriptions regarding the physical state of the universe at the vicinity of the Planck scale. Non commutative geometry, string theory, supergravity or quantum gravity have contributed, independently of each others, to establish on solid basis the data of a "transition phase" in the physical content and the geometric structures of the (pre)spacetime at such a scale. But what is the nature of this dramatic change? We have proposed a novel approach, based on one of the most natural and realistic physical condition predicted by the Standard Model for the (pre)universe. In factn we have shown for the first time that the spacetime -consideredc as a global system- should be expected to reach a thermal equilibrium at the Planck scale. Then as a direct consequence, the spacetime system must be considered as subject to the Kubo-Martin-Schwinger (KMS) condition. Therefore, in the interior of the KMS strip, i.e. from the scale ﬂ = 0 to the scale ﬂ = Planck , the fourth coordinate g 44 must be considered as complex, the two real poles being ﬂ = 0 and ﬂ = Planck . As a result, within the limits of the KMS strip, the Lorentzian and the Euclidean metric are in a "quantum superposition state" (or coupled), this entailing a "unification" (or coupling) between the topological (Euclidean) and the physical (Lorentzian) states of space-time. This KMS state of spacetime at the Planck scale has been developed in several of our papers, amongst which Thermal Equilibrium and KMS Condition at the Planck Scale

TOPOLOGICAL FIELD THEORY AT 0 SCALE : AN EVOLUTION IN IMAGINARY TIME

As we saw in the Quantum group part of our work, the bicrossproduct that we have set up suggests an unexpected kind of "unification" between the Lorentzian and the Euclidean Hopf algebras at the Planck scale and yields the possibility of a "q-deformation" of the signature from the Lorentzian (physical) mode to the Euclidean (topological) mode. We also showed that this situation defines implicitly a (semi)duality transformation between Lorentzian and Euclidean quantum groups. Such a dualisation corresponds (as we saw page 3) to the KMS state of spacetime at the Planck scale. But let's considere now the 0 scale. What theory should then prevail at this scale? In our approach, we consider that the Euclidean situation corresponds to the simplest topological field theory. The topological limit of quantum field theory, described in particular by the Witten invariant Z = Tr (-1) n is then given by the usual quantum statistical partition function taken over the (3+1) Minkowskian space-time Z = Tr (-1) n e - ﬂH, with ﬂ = 1/kt and n being the zero energy states number of the theory, for example the fermion number in supersymmetric theories. Then Z describes all zero energy states for null values of the Hamiltonian H.

Now, we propose in our approach a new topological limit of quantum field theory, non-trivial (i.e. corresponding to the non-trivial minimum of the action). Built from scale ﬂ = 0 and independent of H, this unexpected topological limit (in 4D dimensions) is then given by the temperature limit (Hagedorn temperature) of the physical system (3+1)D.Indeed, when applied to quantum space-time, the KMS properties are such that the time-like direction of the system, within the limits of the "KMS strip" (i.e. between the zero scale and the Planck scale) should be considered as complex : +++±. In this case, on the ﬂ = 0 limit, the theory is projected onto the pure imaginary boundary of the KMS strip. Then the partition function gives the pure topological state connected with the zero mode of the scale : Z (ﬂ = 0) = Tr (-1) s, where s represents the instantonic number. This new "singularity invariant", isomorphic to the Witten index Z = Tr (-1) F , can be connected with the initial singularity of space-time, reached for ﬂ = 0 in the partition function Z = Tr (-1) s e - ﬂH .Then the possible resolution of the initial singularity in the framework of topological theory allows us to envisage the existence, before the Planck scale, of a purely topological first phase of expansion of space-time, parameterized by the growth of the dimension of moduli space dim M and described by the Euclidean "pseudo-dynamic" which corresponds to an evolution in imaginary time : s ﬂ (M0,1/Top) = e - ﬂ H ( M0,1/Top) e ﬂ H

This approach is developped in our paper: Toplogical Field Theory of the Initial Singularity of Spacetime

< Here are some of our preprints :

CERN Server :

Construction of Cocycle Bicrossproducts by Twisting ( CERN)

The KMS State of Space-Time at the Planck Scale ( CERN)

Topological Origin of Inertia ( CERN)

Topological Field Theory of the Initial Singularity of Space-Time ( CERN)

ArXiv :

Construction of Cocycle Bicrossproducts by Twisting (Arxiv)

SOME LINKS TO OUR PUBLICATIONS

Here are some links to our published papers and works in the field of theoretical physics and mathematics :

Topological Field Theory of the Initial Singularity of Spacetime (Classical&Quantum Gravity)http://www.iop.org/EJ/abstract/0264-9381/18/21/301/

Kms State of Spacetime at the Planck Scale (Chinese Journal of Physics)http://psroc.phys.ntu.edu.tw/cjp/v40/149/149.htm

Topological Origin of Inertia (Czechoslovak Journal of Physics)

Spacetime Metric and the KMS Condition at the Planck Scale (Annals of Physics) http://www.kluweronline.com/issn/0011-4626/contents

KMS Spacetime at the Planck Scale (Nuovo Cimento B)http://www.sif.it/cgi-bin/publish/gettoc.1999?section=b&volume=117&issue=04&language=english

Thermal Equilibrium and KMS Condition at the Planck Scale (Chinese Annals of Mathematics) http://www.worldscinet.com/cam/24/2402/S0252959903000256.html

Our doctorate thesis can be found on the CNRS / CCSD server :

Fluctuations Quantiques de la Métrique à l'Echelle de Planck (CNRS Server/ Thèses de Mathématique) http://tel.ccsd.cnrs.fr/documents/archives0/00/00/15/02/

Etat Topologique de l'Espace-Temps à l'Echelle Zéro (CNRS Server/ Thèses de Physique Théorique et Mathématique) http://tel.ccsd.cnrs.fr/documents/archives0/00/00/15/03/index_fr.html

Here is the link to HEP names with other links to papers and citations : http://www.slac.stanford.edu/spires/hepnames/

THE QUANTUM DEFORMATION METHOD

THE KMS EQUILIBRIUM AT THE PLANCK SCALE

SOME LINKS TO OUR PUBLICATIONS

ABOUT OUR THESIS

THESIS REPORTS