Download A Noncommutative Friedman Cosmological Model

A NONCOMMUTATIVE
FRIEDMAN COSMOLOGICAL
MODEL
A NONCOMMUTATIVE FRIEDMAN
COSMOLOGICAL MODEL
1. Introduction
2. Structure of the model
3. Closed Friedman universe – Geometry
and matter
4. Singularities
5. Concluding remarks
1. INTRODUCTION
Rik  2 Rgik  gik  Tik
1
GEOMETRY
MATTER
Mach’s Principle (MP):
geometry from matter
Wheeler’s Geometrodynamics (WG):
matter from (pre)geometry
2
•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
3
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
GEOMETRY
NONCOMMUTATIVE
4
References
• Mathematical structure: J. Math. Phys. 46, 2005,
122501.
• Physical Interpretation: Int. J. Theor. Phys. 46,
2007, 2494.
• Singularities: J. Math. Phys. 36, 1995, 3644.
• Friedman model: Gen. Relativ. Gravit. 41, 2009,
1625.
• Earlier references therein.
2. STRUCTURE OF THE MODEL
Transformation groupoid:
=EG
Lorentz group
pg

E
M
frame bundle
p
 = (p, g)
space-time
p2
Pair groupod:
1=EE
 = (p1, p2)
 i 1 are isomorphic
p1
6
The algebra:

A  Cc (, C)
with convolution as multiplication:
( f1  f 2 )( )  
d (  )
1
1
f1 ( 1 ) f 2 (  )d 1
Z(A) = {0}
"Outer center":
Z   (C (M )),  M : E  M
*
M

 :ZA A
 ( f , a)( p, g )  f ( p)a( p, g )
7
Basic idea: Information about unified GR and
QM is contained in the differential algebra
(A, DerA)
DerA  V = V1 + V2 + V3
V1 – horizontal derivations, lifted from M with the
help of connection
V
3
V2 – vertical derivations, projecting to zero on M
V3 – InnA = {ad a: a A}
V1  V2 - gravitational sector
V3
- quantum sector
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Gravitational sector:
Metric
G(u, v)  g (u1 , v1 )  k (u2 , v2 )
g
- lifting of the metric g from M
k : V2  V2  Z
assumed to be
of the Killing type
9
3. CLOSED FRIEDMAN UNIVERSE – GEOMETRY AND
MATTER

M  ( ,  , ,  ) :  (0, T ),  , ,   S 3
Metric:
  (0, T )  S 3
ds 2  R 2 ( )(d 2  d 2  sin 2  (d 2  sin 2 d 2 ))
Total space of the frame bundle:
E  (,  , ,,  ) :,  , ,  M ,   R  M  R
Structural group:
cosh t sinh t
 sinh t cosh t
G
 0
0

0
 0
0
0
0
0
0
0
, t R
0

0
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Groupoid:
  (,  , ,, 1, 2 ) : 1, 2  R
Algebra:

A  Cc (, C)
(a  b)(,  , ,, 1, 2 )   a(,  , ,, 1,  )b(,  , ,,  , 2 )d
R
"Outer center":
Z  a(,  ) :,   M 
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Metric on V = V1V2:
ds 2   R 2 ( )d 2  R 2 ( )d 2  R 2 ( ) sin 2 d 2
R ( ) sin  sin d  d
2
2
2
2
Einstein operator G: V  V
1
R'2 ( )
B  3( 2
 4
)
R ( ) R ( )
2
B
0

c
Gd   0

0
 0
0 0 0 0
h 0 0 0 
0 h 0 0

0 0 h 0
0 0 0 q 
1
R'2 ( )
R' ' ( )
h 2
 4
2 3
R ( ) R ( )
R ( )
1
R' ' ( )
q  3( 2
 3
)
R ( )
R ( )
12
Einstein equation:
G(u)= u, uV
B
0

0

0
 0
0 0 0 0   u1 
 u1 





u
u
h 0 0 0  2 
 2
0 h 0 0  u3    u3 
 
 
0 0 h 0  u4 
u4 
u5 
0 0 0 q  u5 
  ( 1 ,..., 5 )
- generalized eigenvalues of G
i  Z
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We find i by solving the equation
det(G    I )  0
Solutions:
Generalized eigenvalues:
Eigenspaces:
1
R'2 (t )
 B  3( 2  4 )
R (t ) R (t )
WB – 1-dimensional
1
R'2 (t )
R' ' (t )
h   2  4  2 3
R (t ) R (t )
R (t )
Wh – 3-dimensional
 q  3(
1
R' ' (t )

)
2
3
R (t )
R (t )
Wq – 1-dimensional
14
By comparing B and h with the components of the
perfect fluid enery-momentum tenor for the Friedman
model, we find
 B  8G ( )
 h  8 Gp( )
c=1
We denote
T00   B / 8 G
Tki  ( h / 8 G ) p( ) ki
i, k  1,2,3
In this way, we obtain components of the energy-momentum
tensor as generalized eigenvalues of Einstein operator.
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What about q?
1
3
q  B  h
2
2
 4  G (  (t )  3 p(t ))
This equation encodes equation of state:
 q  4  G
- dust
q  0
- radiation
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INTERPRETATION
• When the Einstein operator is acting on the module of
derivations, it selects the submodule to which there
correspond generalized eigenvalues
• These eigenvalues turn out to be identical with the
components of the energy-momentum tensor and the
equation representing a constraint on admissible equations
of state.
• The source term is no longer made, by our decree, equal
to the purely geometric Einstein tensor, but is produced
by the Einstein operator as its (generalized) eigenvalues.
• In this sense, we can say that in this model ‘pregeometry’
generates matter.
4. SINGULARITIES
Schmidt's b-boundary
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Quantum sector of the model:
 p : A  Bound ( H p ) - regular representation
by
 p (a)( )( )   a( 1 ) ( 1  )d 1
1
p
p  E,   ,   H p  L2 ( p )
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
r : E   Bound ( H p )
pe
such that
(1) the function r is measurable: if  p , p  H p
then the function
E  p  (r ( p) p , p )  C
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
N  L (M , Bound ( L2 ( R))
Normal states on N (restricted to N0) are
 ( A) 
 a(,  ,  ,  ) (,  ,  ,  )d, d , d , d
1
2
1
2
1
2
M  R R
A  ( p (a)) pE
 - density function which is integrable, positive, normalized;
to be faithful it must satisfy the condition >0.
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We are considering the model M  [0, T ]  S 1
Let   0 or   0.
Since  is integrable, (A) is well defined for every a
on the domain M  R  R
i.e. the functional (A) does not feel singularities.
Tomita-Takesaki theorem  there exists the 1-parameter
group of automotphisms of the algebra N

 t (ra ( p))  e
itH p
ra ( p)e
 itH p
A. Connes, C. Rovelli, Class. Quantum
Grav.11, 1994, 2899.
which describes the (state dependent) evolution of
random opertors with the Hamiltonian H p  Log ( p)
This dynamics does not feel singularities.
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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:
1. Interaction between (pre)geoemtry and matter:
components of the energy-momentum tensor can be
obtained as generalized eigenvalues of the Einsten
operator.
2. Interaction between singular and nonsingular.
22
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.
23
Singularities appear (together with space, time and
multiplicity) when one goes from the noncommutative
regime to the usual space-time geometry.
By using Schmidt's b-boundary procedure singularities
appear as the result of taking ratio M  E / G
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