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

ikikikik TgRgR 2

1

GEOMETRY MATTER

Mach’s Principle (MP): geometry from matter

Wheeler’s Geometrodynamics (WG): matter from (pre)geometry

1. INTRODUCTION1. INTRODUCTION

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: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.

=EG

E

M = (p, g)

p

pg

1=EE = (p1, p2)p1

Transformation groupoid:

Pair groupod:

i 1 are isomorphic

p2

2. STRUCTURE OF THE MODEL2. STRUCTURE OF THE MODEL

6

space-time

frame bundle

Lorentz group

),( C cCA

with convolution as multiplication:

)(1

1121121 )()())((

d

dffff

The algebra:

Z(A) = {0}

MEMCZ MM :)),((* "Outer center":

),()(),)(,(

:

gpapfgpaf

AAZ

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

V2 – vertical derivations, projecting to zero on M

V3 – InnA = {ad a: a A}

8

21 VV

3V

3V

- gravitational sector

- quantum sector

Metric

),(),(),( 2211 vukvugvuG

g - lifting of the metric g from M

ZVVk 22: assumed to be of the Killing type

9

Gravitational sector:Gravitational sector:

33 ),0(,,),,0(:),,,( STSTM

))sin(sin)(( 22222222 ddddRds

RR MME ,,,,:),,,,(

R

ttt

tt

G ,

0000

0000

00coshsinh

00sinhcosh

3. CLOSED FRIEDMAN UNIVERSE – GEOMETRY AND3. CLOSED FRIEDMAN UNIVERSE – GEOMETRY ANDMATTERMATTER

Metric:

Total space of the frame bundle:

Structural group:

10

R 2121 ,:),,,,,(

Groupoid:

),( C cCA

Algebra:

R

dbaba ),,,,,(),,,,,(),,,,,)(( 2121

MaZ ,:),(

"Outer center":

11

22222

22222222

sinsin)(

sin)()()(

ddR

dRdRdRds

Metric on V = V1V2:

Einstein operator G: V V

q

h

h

h

B

Gcd

0000

0000

0000

0000

0000

))(

)('

)(

1(3

4

2

2

R

R

RB

)(

)(''2

)(

)('

)(

134

2

2

R

R

R

R

Rh

))(

)(''

)(

1(3

32

R

R

Rq

12

Einstein equation: G(u)= u, uV

5

4

3

2

1

5

4

3

2

1

0000

0000

0000

0000

0000

u

u

u

u

u

u

u

u

u

u

q

h

h

h

B

),...,( 51 - generalized eigenvalues of G

i Z13

iWe find by solving the equation

0)det( IG

))(

)('

)(

1(3

4

2

2 tR

tR

tRB

Solutions:Generalized eigenvalues: Eigenspaces:

WB – 1-dimensional

)(

)(''2

)(

)('

)(

134

2

2 tR

tR

tR

tR

tRh Wh – 3-dimensional

))(

)(''

)(

1(3

32 tR

tR

tRq Wq – 1-dimensional

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By comparing B and h with the components of theperfect fluid enery-momentum tenor for the Friedmanmodel, we find

)(8 GB

)(8 Gph c = 1

We denote

GT B 8/00

3,2,1,)()8/( kipGT ikh

ik

In this way, we obtain components of the energy-momentum tensor as generalized eigenvalues of Einstein operator.

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What about q?

hBq 2

3

2

1

This equation encodes equation of state:

))(3)((4 tptG

Gq 4

0q

- dust

- 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 theequation 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. SINGULARITIES4. SINGULARITIES

Schmidt's b-boundary

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Quantum sector of the model:

p

daa

by

HBoundA

p

pp

11

11 )()())()((

)(:

- regular representation

)(,, 2 pp LHEp

Every a A generates a random operator ra on (Hp)pE

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Random operator is a family of operators r = (rp)pE,i.e. a function

ep

pHBoundEr

)(:

such that(1) the function r is measurable: if ppp H ,

then the function CprpE pp ),)((

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 equalityeverywhere) 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

))((,( 2 RLBoundMLN

Normal states on N (restricted to N0) are

RRM

ddddaA 212121 ,,,),,,(),,,()(

Epp aA ))(( - 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 1],0[ STM Let 0 or 0.Since is integrable, (A) is well defined for every aon the domaini.e. the functional (A) does not feel singularities.

RRM

Tomita-Takesaki theorem there exists the 1-parametergroup of automotphisms of the algebra N

pp itHa

itHat eprepr )())((

which describes the (state dependent) evolution ofrandom opertors with the Hamiltonian )( pLogH p

This dynamics does not feel singularities. 21

A. Connes, C. Rovelli, Class. QuantumGrav.11, 1994, 2899.

5. CONCLUDING REMARKSOur 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.

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Quantum sector of our model (which we have not exploredin this talk) has strong probabilistic properties: all quantumoperators are random operators (and the correspondingalgebra is a von Neumann algebra). Because of this, on thefundamental level singularities are irrelevant.

Usually, two possibilities are considered: either the futurequantum gravity theory will remove singularities, or not. Here we have the third possibility:

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By using Schmidt's b-boundary procedure singularitiesappear as the result of taking ratio GEM /

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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Singularities appear (together with space, time and multiplicity) when one goes from the noncommutativeregime to the usual space-time geometry.

?THE END