Tomographic approach to Quantum Cosmology

Cosimo StornaioloINFN – Sezione di Napoli

Fourth Meeting on Constrained Dynamics and Quantum Gravity

Cala Gonone (Sardegna, Italy)September 12-16, 2005

Papers

V.I Man’ko, G. Marmo and C.S.

1. ‘’Radon Transform of the Wheeler-De Witt equation and tomography of quantum states of the universe’’ Gen. Relativ. Gravit. (2005) 37: 99–114

2. ‘’Cosmological dynamics in tomographic probability representation’’ (gr-qc/0412091) submitted to GRG (see references in this paper for extensive treatment of the tomographic approach)

The Tomographic Approach to Quantum Mechanics

Quantum mechanics without wave function and density matrix.New formulation of Q. M. based on the ‘’probability representation’’ of quantum states.Introduction of the marginal probability functions (tomograms)They contain exactly the same informations of the wave functions (or the density matrix or the Wigner distribution)But in this case we deal with the evolution of a measurable quantitywhose evolution is classical or quantum depending on the initial conditions, that can be classical or quantum

The tomographic map

Density

Tomographic map

or in of the Wigner distribution function

),'( t)(x, t),x'(x, * tx

'iexpt),y'ρ(y,|ν|2π

1t)ν,μ,w(X, )'(

2)'(y 22

dydyi yyXy

dqdppqXtpqWtXw )(),,(),,,( 21

) , , (t p q W

Relation between tomograms and wave function

Tomograms contain the same information of wave functions, they are defined by considering the following trasformation

space phaseon ation transform''coordinate'' a is

νpμq

variable there whe

X

2

2 dyyν

y2ν

iμψ(y)exp

ν2π

1) ν ,μ ,(

iXXw

sinν cos μ ss

Properties of the tomograms

1. They are non negative

2.

0) , ,( Xw

1 dX ) , ,( Xw

The Classical Tomogram

The classical tomogram is obtained by substituting the Wigner function with the solution of the classical Liouville equation.

However classical and quantum tomograms ‘’live’’ in the same space and therefore can be compared.

The Tomogram Equation

012)!12(

12 12

12

12

12

0

w

Xx

V

nww n

n

n

n

n

Alternative to the Schroedinger equation

we find the equation for the tomogram

Hi

Wheeler-de Witt equation in Quantum Cosmology

0ψ(a)1

2

1 42

aa

da

da

da

d

ap

p

Here is an example of a Wheeler-deWitt equation in the space of homogeneous and isotropic metrics

3

Ha cos ψ(a)

3

For large values of the expansion factor a the solution is

a model with cosmological constant and no matter fields is considered, the exponent ‘’p’’ reflects the ambiguity of the theory in fixing the order of operators.

The tomogram equation corresponding to the Wheeler de Witt equation

Analogously to the preceding, we are able to express an equation for tomograms in quantum cosmology (see the preceding example)

0),,(24exp2expIm2

2

12expIm)1(

2

12expIm

11

11

211

XwX

iXX

iX

Xi

XXi

Xp

Xi

XXi

X

ik

ekfxf

X

ikX)(~

)(1

Cosmological metric

Homogeneous and isotropic metric

In conformal time

][1

)( 2222

2222 dzdydx

kr

tadtcds

)(η

ta

dtd

2

2222222

1ηc-η)(

kr

dzdydxdads

1- 0 1

kkk

Classical cosmological equations

Friedmann equations

ρπG

a

k

a

a

3

822

2

)3ρ(3

4PπGa

a

a 0

0ρρ

3/4γ:4

1 γ:3

1)ρ-γ(P

Cosmological models as harmonic oscillators

Let us make in a homogeneous and isotropic model in conformal time the change of variables

The evolution cosmological equation takes the form

χaz 1-γ2

3χ

0χ '' 2 zkz

1- 0 1

kkk

Cosmological models as harmonic oscillators (2)

In a similar way for cosmological models with a fluid and a cosmological constant one obtains (in cosmic time) putting

If k=0 we have again the ‘’harmonic oscillator)

z

kzz

-1 z a 1 2

3

Tomographic equation for a harmonic oscillator

An useful equation is the harmonic oscillator equation for the tomograms, which is the same for classical and quantum tomograms

0μ

νων

μ 2

WW

t

W

Uncertainty Relations for tomograms

The uncertainty relation is

4

1)1,0,()1,0,(

)0,1,()0,1,(

22

22

dXXXwdXXXw

XdXXwdXXXw

Propagators in the tomographic approach

The evolution of a tomogram can be described by the transition probability

)t,ν',μ',X't,ν,μ,(X, 0

with the equation

ν' dμ' d'dt,ν' ,μ' ,' )t,ν',μ',X't,ν,μ,(X,tν, μ, , 00 XXWXW

Evolution of a tomogram of the universe

The transition probility Π satisfies the following equation

)t-δ(tX'-Xδν'-ν)δμ'-δ(μμ

νων

μ 02

t

Solutions for the transition probabilities

00

000

t-tsinωω

μt-t cosω ν'-νδ

t-tsinω νt-t cosωμ'-μ δ X'-Xδ)t,ν',μ',X't,ν,μ,(X,osc

00 t-tμ ν'-νδ δ(μ'-μ) X'-Xδ)t,ν',μ',X't,ν,μ,(X, free

00

000

t-t sinhωω

μt-t coshω ν'-νδ

t-t sinhω νt-t coshωδ(μ'-μ X'-Xδ)t,ν',μ',X't,ν,μ,(X,rep

In the minisuperspace considered in this talk, the transition probabilities are

The initial condition problem

Quantum cosmology can be considered as the theory of the initial conditions of the universe

Differently from the wave function approach we can deal classical and quantum cosmology with the same variable.

The difference is just in the initial conditions

Do we have to postulate these conditions?

Phenomenological Quantum Cosmology

Our approach appears to be promising, because tomograms are in principle measurableIn the particular case discussed before the classical and quantum equations are the sameIn future work we shall need to define the measurement of a cosmological tomogramThis will enable us to study the initial conditions problem from a phenomenological point of viewMoreover we hope to be able to distinguish the quantum evolution from the classical one.

What we can know from observations?

We can expect that observations put some constraints on the present tomogram (and consequently to the initial conditions) , we must use observations of EntropyCosmic background radiation fluctuationsApproximate homogeneity and isotropyFormation of structures

Conclusions and perspectives

Conclusions•We saw that there are models that have a simple description•This result seems promising to develop a phenomenological study of the initial conditions problem•We have proposed a novel way to deal with Quantum Cosmology

Perspectives• Determine how to measure a cosmological tomogram• Analyze quantum decoherence from our point of viewMoreover• Formulate a classical theory of fluctuations in G.R.• Extension of our analysis to Quantum Gravity

Motivations for this work

The initial conditions problem in Quantum CosmologyWhy Tomographic approach to Quantum Cosmology?Cosmological tomograms vs cosmological wave functionsCosmologies as “ harmonic oscillators” Towards a phenomenological approach to Quantum CosmologyPerspectives and conclusions

Wheeler-de Witt equation in Quantum Gravity

canonical approachequation in the space of three dimensional metrics

0hψ2)(δhδh

δij

2/12/13

klij

2

hhhRGijkl

ijhx spaziale metrica

e)(superspac geometries threeof space themetric ijklG

Quantum Mechanics

Uncertainty principle Schrödinger Equation

Observables and measurementsPhysical interpretation

tiH

t)ψ(x,

t)ψ(x,ˆ

2/ qp

Wheeler-de Witt equation in Quantum Cosmology

0ψ(a)1

2

1 42

aa

da

da

da

d

ap

p

Here is an example of a Wheeler-deWitt equation in the space of homogeneous and isotropic metrics

3

Ha cos ψ(a)

3

For large values of the expansion factor a the solution is

a model with cosmological constant and no matter fields is considered, the exponent ‘’p’’ reflects the ambiguity of the theory in fixing the order of operators.

Quantum Cosmology

1. Minisuperspace: considering only homogeneous metrics

2. cosmological models as a point particles 3. working with a finite number of degrees of

freedom 4. violates the uncertainty principle fixing

contemporarily a zero infinite variables and their momenta

5. Not Copenhagen interpretation of this quantum theory

Boundary ConditionsWick rotation Euclidean 4 space It is important to note that the cosmological evolution is determined by the (initial) boundary conditions

Two proposals:

• Hartle and Hawking, no boundary conditions

• Vilenkin, the universe ‘’tunnels into existence from nothing’’

Need or a fundamental law of the initial condition (Hartle) or to derive it from the phenomenology

Conclusions and perspectivesConclusionsWe saw that there are models that have a simple descriptionThis result seems promising to develop a phenomenological study of

the initial conditions problemWe have proposed a novel way to deal with Quantum Cosmology

Perspectives• Determine how to measure a cosmological tomogram• Analyze quantum decoherence from our point of view• Moreover• Formulate a classical theory of fluctuations in G.R.• Extension of our analysis to Quantum Gravity