Abstract: We consider here a generalisation of
the Majid's mirror product when one of the components of the product
is replaced by a twist. This leads to a new "twisted mirror
product" construction for cocycle bicrossproduct Hopf algebra
and as an application we obtain a canonical cocycle bicrossproduct
for any Hopf algebra H.
We suggest a
new solution of the initial spacetime singularity. In this approach
the initial singularity of spacetime corresponds to a zero-size
singular gravitational instanton characterized by a Riemannian
metric configuration ( + + + + ) in dimension D = 4. Connected
with some unexpected topological data corresponding to the zero
scale of spacetime, the initial singularity is thus not considered
in terms of divergences of physical fields but can be resolved
within the framework of topological field theory. Then it is suggested
that the `zero-scale singularity' can be understood in terms of
topological invariants (in particular, the first Donaldson invariant
∑i(-1)ni).
With this perspective, here we introduce a new topological index,
connected with zero scale, of the form {Z}β =
0 = Tr (-1)s, which we call the `singularity
invariant'. Interestingly, this invariant also corresponds to
the invariant topological current yield by the hyperfinite II∞
von Neumann algebra describing the zero scale of spacetime.
Then we suggest that the (pre-)spacetime is in thermodynamical
equilibrium at the Planck-scale and is therefore subject to the
KMS condition. This might correspond to a unification phase between
the `physical state' (Planck scale) and the `topological state'
(zero scale). Then we conjecture that the transition from the
topological phase of the spacetime (around the zero scale) to
the physical phase observed beyond the Planck scale should be
deeply connected to the supersymmetry breaking of the N
= 2 supergravity.
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We
explore here the consequences of the expected thermal equilibrium
of the spacetime system at the Planck scale. In the context of string
theory, it is supposed that at high temperature (Planck temperature)
this system is subject to a 4D
3D dimensional reduction, the Chern-Simon term becoming dominant
in the action. We then show that the spacetime must be considered
as being subject to the Kubo-Martin-Schwinger (KMS) condition at
the Planck scale. Therefore, in the interior of the KMS strip, i.e.,
from the scale
= 0 to the scale Planck,
the time-like direction should be viewed as complex, the two real
poles being
= 0 and Planck.
This means that, within the limits of the KMS strip, the spacetime
metric should be considered as subject to quantum fluctuations between
the Lorentzian (physical) state and the Euclidean (topological)
state.
The purpose of the present
article, following "Mach's principle" (the main elements
of which have contributed to the foundations of general relativity)
is to propose a new (non-local) interpretation of the inertial
interaction. We then suggest that the inertial interaction can
be correctly described by the topological field theory proposed
by Witten in 1988. In such a context, the instantaneous propagation
and the infinite range of the inertial interaction might be
explained in terms of the topological amplitude connected with
the singular zero size gravitational instanton corresponding
to the Initial Singularity of space-time.
Considering
the expected thermal equilibrium characterizing the physics at
the Planck scale, it is here stated, for the first time, that,
as a system, the space-time at the Planck scale must be considered
as subject to the Kubo-Martin-Schwinger (KMS) condition. Consequently,
in the interior of the KMS strip, i.e. from the scale
= 0 to the scale
= ℓplanck, the fourth coordinate g44
must be considered as complex, the two real poles being
= 0 and
= ℓplanck. This means that 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.
3D dimensional reduction, the Chern-Simon term becoming dominant
in the action. We then show that the spacetime must be considered
as being subject to the Kubo-Martin-Schwinger (KMS) condition at
the Planck scale. Therefore, in the interior of the KMS strip, i.e.,
from the scale 
= 0 to the scale 
Planck,
the time-like direction should be viewed as complex, the two real
poles being