MATHEMATICAL CENTER OF
RIEMANNIAN COSMOLOGY

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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?

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** 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 : **

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CONTACT US mcrc@phys-maths.edu.lv

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

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-Poincaré group and relate it to
the k -Poincaré 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*;

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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 ß = 0 to the scale ß = Planck , the fourth
coordinate g 44 must be
considered as complex, the two real poles being ß = 0 and ß = 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 - ßH, with ß =
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 ß = 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 ß = 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 (ß = 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 ß = 0 in the partition function Z = Tr
(-1) s e - ßH .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 ß (M0,1/Top) = e - ß H ( M0,1/Top) e ß 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)

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 MŽtrique ˆ l'Echelle de Planck (CNRS Server/ Thses de
MathŽmatique) http://tel.ccsd.cnrs.fr/documents/archives0/00/00/15/02/

Etat Topologique
de l'Espace-Temps ˆ l'Echelle ZŽro (CNRS Server/ Thses de Physique ThŽorique
et MathŽmatique) 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/

Here is our
response to FabienBesnard :

http://igor.bogdanov.free.fr/R%e9ponse%20%e0%20Besnar.htm

Here are the
Thesis Reports

http://igor.bogdanov.free.fr/Thesis
Reports-I.G.Bogdanoff.pdf