Download Cambridge Paper

A NONCOMMUTATIVE
CLOSED FRIEDMAN
WORLD MODEL
A NONCOMMUTATIVE CLOSED
FRIEDMAN WORLD 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
•
•
•
•
•
Model:
Int. J. Theor. Phys. 44, 2005, 619.
J. Math. Phys. 46, 2005, 122501.
Friedman model:
Gen. Relativ. Gravit. DOI 10.107/s10714008-0740-3.
Singularities:
Gen. Relativ. Gravit. 31, 1999, 555
Int. J. Theor. Phys. 42, 2003, 427
5
2. STRUCTURE OF THE MODEL
Transformation groupoid:
=EG
pg

E
p
 = (p, g)
M
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
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
8
Metric
G(u, v)  g (u1, v1 )  k2 (u2 , v2 )
g
- lifting of the metric g from M
k : V2 V2  Z
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.
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
10
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 
11
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
13
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 energy-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.
15
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
16
If we add the cosmological constant  to the Einstein
operator, its eigenvalue equation remains the same
provided we replace:
Comment:
Einstein operator acts
on the module of derivations
to which correspond
generalized eigenvalues
and selects
submodules
which are identical with
the energy-momentum
tensor components and
constraints for eqs of state
Duality in Einstein’s eqs is liquidated.
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
18
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.
19
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.
20
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.
21
(M, φ), where M – von Neumann algebra,
φ – normal state on M, is a noncommutative
probabilistic space.
φ is normal if:
φ(Σ Pn) = Σ φ(Pn) for any countable family
of mutually orthogonal projections Pn in M.
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
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.
Singularities appear (together with space, time and
multiplicity) when one goes from the noncommutative
regime to the usual space-time geometry.
23
EMERGENCE OF SPACE-TIME
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.
24
?
THE END