A NONCOMMUTATIVE CLOSED FRIEDMAN WORLD MODEL

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A NONCOMMUTATIVE CLOSED FRIEDMAN WORLD MODEL
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A NONCOMMUTATIVE CLOSED FRIEDMAN WORLD MODEL

• Introduction
• Structure of the model
• Closed Friedman universe Geometry and matter
• Singularities
• Concluding remarks

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1. INTRODUCTION
GEOMETRY
MATTER
Machs Principle (MP) geometry from matter
Wheelers Geometrodynamics (WG) matter from
(pre)geometry
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• MP is only partially implemented in Genaral
  Relativity matter modifies the space-time
  structure (Lense-Thirring effect), but
• it does not determine it fully ("empty" de Sitter
  solution),
• in other words,
• SPACE-TIME IS NOT GENERATED BY MATTER

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For Wheeler pregeometry was "a combination of
hope and need, of philosophy and physics and
mathematics and logic''. Wheeler made several
proposals to make it more concrete. Among others,
he explored the idea of propositional logic or
elementary bits of information as fundamental
building blocks of physical reality.A new
possibility PREGEOEMTRY
NONCOMMUTATIVE GEOMETRY
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References
Model

• Int. J. Theor. Phys. 44, 2005, 619.
• J. Math. Phys. 46, 2005, 122501.
• Friedman model
• Gen. Relativ. Gravit. DOI 10.107/s10714-
• 008-0740-3.
• Singularities
• Gen. Relativ. Gravit. 31, 1999, 555
• Int. J. Theor. Phys. 42, 2003, 427

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2. STRUCTURE OF THE MODEL
Transformation groupoid
?E?G
pg
???
E
p
? (p, g)
M
p2
Pair groupod
?1E?E
? (p1, p2)
p1
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? i ?1 are isomorphic
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The algebra
with convolution as multiplication
Z(A) 0
"Outer center"
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We construct differential geometry in terms of
(A, DerA) DerA ? V V1 V2 V3 V1 horizontal
derivations, lifted from M with the help of
connection V2 vertical derivations, projecting
to zero on M V3 InnA ad a a ?A
- gravitational sector
- quantum sector
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Metric
- lifting of the metric g from M
Vertical derivations can be identified with
functions on E with values in the Lie algebra of
G. Natural choice for k is a Killing metric.
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3. CLOSED FRIEDMAN UNIVERSE GEOMETRY AND MATTER
Metric
Total space of the frame bundle
Structural group
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Groupoid
Algebra
"Outer center"
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Metric on V V1?V2
Einstein operator G V ? V
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Einstein equation
G(u)? u, u?V
- generalized eigenvalues of G
?i ? Z
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?i
We find
by solving the equation
Solutions
Generalized eigenvalues
Eigenspaces
WB 1-dimensional
Wh 3-dimensional
Wq 1-dimensional
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By comparing ?B and ?h with the components of
the perfect fluid energy-momentum tenor for the
Friedman model, we find
c 1
We denote
In this way, we obtain components of the
energy-momentum tensor as generalized
eigenvalues of Einstein operator.
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What about ?q?
This equation encodes equation of state
- dust
- radiation
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If we add the cosmological constant ? to the
Einstein operator, its eigenvalue equation
remains the same provided we replace
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Comment
Einstein operator acts on the module of
derivations
and selects submodules
which are identical with the energy-momentum tenso
r components and constraints for eqs of state
to which correspond generalized eigenvalues
Duality in Einsteins eqs is liquidated.
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Quantum sector of the model
- regular representation
Every a ? A generates a random operator ra on
(Hp)p?E
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Random operator is a family of operators r
(rp)p?E, i.e. a function
such that (1) the function r is measurable if
then the function
is measurable with respect to the manifold
measure on E.
(2) r is bounded with respect to the norm r
ess sup r(p) where ess sup means "supremum
modulo zer measure sets".
In our case, both these conditions are satisfied.
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N0 the algebra of equivqlence classes (modulo
equality everywhere) of bounded random operators
ra, a ? A. N N0'' von Neumann algebra,
called von Neumann algebra of the groupoid ?.
In the case of the closed Friedman model
Normal states on N (restricted to N0) are

• - density function which is integrable,
  positive, normalized
• to be faithful it must satisfy the condition ?gt0.

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We are considering the model
Let ? ? 0 or ? ? 0. Since ? is integrable, ?(A)
is well defined for every a on the domain i.e.
the functional ?(A) does not feel singularities.
Tomita-Takesaki theorem ? there exists the
1-parameter group of automotphisms of the algebra
N
A. Connes, C. Rovelli, Class. Quantum Grav.11,
1994, 2899.
which describes the (state dependent) evolution
of random opertors with the Hamiltonian
This dynamics does not feel singularities.
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(M, f), where M von Neumann algebra, f normal
state on M, is a noncommutative probabilistic
space.
f is normal if
f(S Pn) S f(Pn) for any countable family of
mutually orthogonal projections Pn in M.
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The same conclusion can be proved in a
more general way
algebra of random operators before sigularity has
been attached
algebra of random operators after singularity has
been attached
We have
von Neumann algebra
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5. CONCLUDING REMARKS

• Our noncommutative closed Friedman world model
  is a toy model. It is intended to show how
  concepts can interact with each other in the
  framework of noncommutative geometry rather than
  to study the real world. Two such interactions of
  concepts have been elucidated
• Interaction between (pre)geoemtry and matter
  components of the energy-momentum tensor can be
  obtained as generalized eigenvalues of the
  Einsten operator.
• Interaction between singular and nonsingular.

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Usually, two possibilities are considered either
the future quantum gravity theory will remove
singularities, or not. Here we have the third
possibility
Quantum sector of our model (which we have not
explored in this talk) has strong probabilistic
properties all quantum operators are random
operators (and the corresponding algebra is a von
Neumann algebra). Because of this, on
the fundamental level singularities are
irrelevant.
Singularities appear (together with space, time
and multiplicity) when one goes from the
noncommutative regime to the usual space-time
geometry.
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EMERGENCE OF SPACE-TIME
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Therefore, on the fundamental level the concept
of the beginning and end is meaningeless. Only
from the point of view of the macroscopic
observer can one say that the universe had
an initial singularity in its finite past, and
possibly will have a final singularity in its
finite future.
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?
THE END