Tomographic approach to
Quantum Cosmology
Cosimo Stornaiolo
INFN – 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.
2.
‘’Radon Transform of the Wheeler-De Witt equation
and tomography of quantum states of the universe’’
Gen. Relativ. Gravit. (2005) 37: 99–114
‘’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 quantity
whose evolution is classical or quantum depending on the
initial conditions, that can be classical or quantum
The tomographic map
Density  (x, x' , t)   (x, t) *( x' , t )
Tomographic map

1
w(X, μ, ν, t) 
ρ(y, y' , t)  exp i

2π | ν |
 (y 2  y '2 )
2
i
or in of the Wigner distribution function
w( X ,  ,  , t ) 
1
2
 W ( q, p, t )  ( X
X ( y  y ')

dydy'
W (q, p, t )
  q   p )dqdp
Relation between tomograms
and wave function
Tomograms contain the same information
of wave functions, they are defined by
considering the following trasformation
1
w( X , μ , ν ) 
2π ν
 iμ 2 iX 
 ψ(y)exp 2ν y  ν y dy
2
where the variable
X  μq  νp
is a ' ' coordinate' ' transformationon phasespace
μ  s cos
ν  s sin 
Properties of the tomograms
1.
They are non negative
2.
 w( X ,  ,  ) dX  1
w( X ,  ,  )  0
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
Alternative to the Schroedinger equation
i  H
we find the equation for the tomogram
w  


2 n 1
1
 V
w2 
2 n 1
n  0 (2n  1)! x

  


 2 X 
2 n 1
 12n 1 w  0
Wheeler-de Witt equation
in Quantum Cosmology
Here is an example of a Wheeler-deWitt equation in
the space of homogeneous and isotropic metrics
11 d p d
2
4
a

a


a
 p
ψ(a)  0
2  a da
da

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.
For large values of the expansion factor a the
solution is
Ha 3
ψ(a)  cos
3
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)   
~ ikX
f (k )e
  f ( x)  
ik
 X 
1
2
1
1
  




 1 
 
   
     
 i
 i



 
Im exp2


X



X
2

X

X








   
 
1
     1 
  1 
    
( p  1) Imexp2
 i
 i



 

X  2 X  X   
   X  
1
     1 
 
  
   
 2 Imexp2
 i
 2i
 w( X ,  , )  0


   exp 4

X



X

X



X



  
 

Cosmological metric
Homogeneous and isotropic metric
2
a
(t )
2
2
2
2
2
2
ds  c dt 
[
dx

dy

dz
]
2
1  kr
k  1
k  0
k  -1
In conformal time
dt
dη 
a (t )
2
2
2


dx

dy

dz
2
2
2
2
ds  a (η)- c dη 

2
1

kr


Classical cosmological equations
Friedmann equations
a
k 8πG


ρ
2
2
a
a
3
2
4
a   πG (ρ  3P)
3
P  ( γ - 1)ρ

a 0
ρ  ρ0 
a
 3 :γ  1

4 : γ  4 / 3
Cosmological models as
harmonic oscillators
Let us make in a homogeneous and isotropic model
in conformal time the change of variables
3
χ  γ -1
2
z  aχ
The evolution cosmological equation takes the form
z' 'k χ z  0
2
k  1
k  0
k  -1
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
a z

  32   1
  1- 

k
z 
z 

z
If k=0 we have again the ‘’harmonic oscillator)
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
W
W
2 W
μ
ω ν
0
t
ν
μ
Uncertainty Relations for
tomograms
The uncertainty relation is
 w( X ,1,0) XdX  
 w( X ,0,1) X dX   w( X ,0,1) X dX    1


 4
 w( X ,1,0) X 2 dX 

2
2
2
Propagators in the tomographic
approach
The evolution of a tomogram can be
described by the transition probability
(X, μ, ν, t, X' , μ' , ν' , t 0 )
with the equation
W  X , μ, ν, t     (X, μ, ν, t, X' , μ' , ν' , t 0 ) W  X ' , μ' , ν' , t 0 dX ' d μ' d ν'
Evolution of a tomogram of the
universe
The transition probility Π satisfies the
following equation



2
 μ
ω ν
 δ(μ - μ' )δν - ν'δX - X'δ(t - t 0 )
t
ν
μ
Solutions for the transition
probabilities
In the minisuperspace considered in this talk, the
transition probabilities are
 osc (X, μ, ν, t, X' , μ' , ν' , t 0 )  δX - X' δμ'-μ cosω t - t 0   ν sinωt - t 0 
μ


 δ ν'-ν cosω t - t 0   sinωt - t 0 
ω


 free (X, μ, ν, t, X' , μ' , ν' , t 0 )  δX - X' δ(μ'-μ)δν'-ν  μt - t 0 
 rep (X, μ, ν, t, X' , μ' , ν' , t 0 )  δX - X' δ(μ'-μcoshω t - t 0   ν sinhω t - t 0 
μ


 δ ν'-ν coshω t - t 0   sinhω t - t 0 
ω


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 measurable
In the particular case discussed before the classical
and quantum equations are the same
In future work we shall need to define the
measurement of a cosmological tomogram
This will enable us to study the initial conditions
problem from a phenomenological point of view
Moreover 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
Entropy
Cosmic background radiation fluctuations
Approximate homogeneity and isotropy
Formation 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 view
Moreover
• 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
Cosmology
Why Tomographic approach to Quantum
Cosmology?
Cosmological tomograms vs cosmological wave
functions
Cosmologies as “ harmonic oscillators”
Towards a phenomenological approach to
Quantum Cosmology
Perspectives and conclusions
Wheeler-de Witt equation
in Quantum Gravity
canonical approach
equation in the space of three
dimensional metrics
x  hij

 Gijkl
metrica spaziale
δ2
3R(h)h1 / 2  2h1 / 2 ψh ij   0
δh ijδh kl
Gijkl metricthespaceof threegeometries(superspace)
Quantum Mechanics
Uncertainty principle
Schrödinger Equation
pq   / 2
ψ(x, t)
ˆ
Hψ(x, t)  i
t
Observables and measurements
Physical interpretation
Wheeler-de Witt equation
in Quantum Cosmology
Here is an example of a Wheeler-deWitt equation in
the space of homogeneous and isotropic metrics
11 d p d
2
4
a

a


a
 p
ψ(a)  0
2  a da
da

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.
For large values of the expansion factor a the
solution is
Ha 3
ψ(a)  cos
3
Quantum Cosmology
1.
2.
3.
4.
5.
Minisuperspace: considering only homogeneous
metrics
cosmological models as a point particles
working with a finite number of degrees of
freedom
violates the uncertainty principle fixing
contemporarily a zero infinite variables and their
momenta
Not Copenhagen interpretation of this quantum
theory
Boundary Conditions
Wick 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 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 view
• Moreover
• Formulate a classical theory of fluctuations in G.R.
• Extension of our analysis to Quantum Gravity