Transcript Cambridge Paper
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
R ik
1 2
Rg ik
g ik
T ik
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 NONCOMMUTATIVE GEOMETRY 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 E frame bundle = (p, g) M space-time Pair groupod: 1 =E E = (p 1 , p 2 )
i
1 are isomorphic
p 1 p p 2 pg 6
The algebra:
A
C c
( ,
C
) with convolution as multiplication: (
f
1
f
2 )( )
d
( )
f
1 ( 1 )
f
2 ( 1 1 )
d
1
Z(A) = {0}
"Outer center":
Z
*
M
(
C
(
M
)), ( :
Z
A
f A
,
a
)(
p
,
g
)
f
(
p
)
a
(
p
,
g
)
M
:
E
M
7
Basic idea: Information about unified GR and QM is contained in the differential algebra (
A,
Der
A
) Der
A
V = V 1 + V 2 + V 3 V 1
– horizontal derivations, lifted from M with the help of connection
V
3
V 2
– vertical derivations, projecting to zero on M
V 3
– Inn
A
= {ad a: a
A
}
V
1
V
2 - gravitational sector
V
3 - quantum sector 8
Gravitational sector:
Metric
G
(
u
,
v
)
g
(
u
1
,
v
1
)
k
(
u
2
,
v
2
)
g
- lifting of the metric
g
from M
k
:
V
2
V
2
Z
assumed to be of the Killing type 9
3. CLOSED FRIEDMAN UNIVERSE – GEOMETRY AND MATTER
M
( , , , ) : ( 0 ,
T
), Metric: , ,
S
3
ds
2
R
2 ( )(
d
2
d
2 sin 2 (
d
2 sin 2
d
2 )) ( 0 ,
T
)
S
3 Total space of the frame bundle:
E
( , , , , ) : , , ,
M
,
R
M
R
Structural group:
G
cosh sinh 0 0
t t
sinh
t
cosh
t
0 0 0 0 0 0 0 0 0 0 ,
t
R
10
Groupoid: ( , , , , 1 , 2 ) : 1 , 2
R
Algebra:
A
C c
( ,
C
) (
a
b
)( , , , , 1 , 2 )
R
a
( , , , , 1 , )
b
( , , , , , 2 )
d
"Outer center":
Z
a
( , ) : ,
M
11
Metric on V = V 1 V 2 :
ds
2
R
2 ( )
d
2
R
2 ( )
d
2
R
2 ( ) sin 2
d
2
R
2 ( ) sin 2 sin 2
d
2 Einstein operator
G: V
V
B
3 (
R
2 1 ( )
R
' 2
R
4 ( ) ( ) )
d
2
G c d
B
0 0 0 0 0
h
0 0 0 0 0
h
0 0 0 0 0
h
0 0 0 0 0
q
h
1
R
2 ( )
R
' 2 ( )
R
4 ( ) 2
R
'' ( )
R
3 ( )
q
3 ( 1
R
2 ( )
R
'' ( )
R
3 ( ) ) 12
Einstein equation:
G
(
u
)=
u
,
u
V
B
0 0 0 0 0
h
0 0 0 0 0
h
0 0 0 0 0
h
0 0 0 0 0
q
u u
1 2
u
u u
3 4 5
u u u
1 2 3
u u
4 5
(
1
,...,
5
)
i Z - generalized eigenvalues of
G
13
We find i by solving the equation det(
G
I
) 0 Solutions: Generalized eigenvalues: Eigenspaces:
B
3 ( 1
R
2 (
t
)
R
' 2 (
t
)
R
4 (
t
) )
h
1
R
2 (
t
)
R
' 2 (
t
)
R
4 (
t
) 2
R
'' (
t R
3 (
t
) ) W B – 1-dimensional W h – 3-dimensional
q
3 ( 1
R
2 (
t
)
R
''
R
3 (
t
(
t
) ) ) W q – 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
T
0 0
B
/ 8
G T k i
(
h
/ 8
G
)
p
( )
k i 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
?
q
1 2
B
3 2
h
4
G
( (
t
) 3
p
(
t
)) This equation encodes equation of state:
q
4
G
- dust
q
0
- radiation
16
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 17
Quantum sector of the model:
p
:
A
Bound
(
H p
) - regular representation
by
p
(
a
)( )( )
p a
( 1 )
p
E
, , ( 1 1 )
d
1
H p
L
2 (
p
) Every
a
A
generates a random operator
r a on
(
H p
) p E 18
Random operator is a family of operators
r
= (
r p
)
p
E
, i.e. a function
r
:
E
p
e Bound
(
H p
) such that (1) the function
r
is measurable: if then the function
E
p
,
p
H p 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
N 0 –
the algebra of equivqlence classes (modulo equality everywhere) of bounded random operators
r a , a
A
.
N = N 0 '' –
von Neumann algebra, called von Neumann algebra of the groupoid .
In the case of the closed Friedman model
N
L
(
M
,
Bound
(
L
2 (
R
)) Normal states on
N
(restricted to
N 0
) are (
A
)
M
R a
(
R
, , 1 , 2 ) ( , , 1 , 2 )
d
,
d
,
d
1 ,
d
2
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 Since 0 or 0.
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
(
r a
(
p
))
e itH
p r a
(
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
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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?