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