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Minisuperspace Models in
Quantum Cosmology
Ljubisa Nesic
Department of Physics,
University of Nis, Serbia
ISC2008, Nis, Serbia, August 26 - 31,
2008
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Minisuperspace



Superspace – infinite-dimensional space, with finite number
degrees of freedom (hij(x), F(x)) at each point, x in S
In practice to work with inf.dim. is not possible
One useful approximation – to truncate inf. degrees of freedom
to a finite number – minisuperspace model.



Homogeneity
isotropy or anisotropy
Homogeneity and isotropy


instead of having a separate Wheeler-DeWitt equation for each point of the
spatial hypersurface S, we then simply have a SINGLE equation for all of S.
metrics (shift vector is zero)

ds   N (t )dt  hij (q (t ))dx dx ,   1,2,..., n
2
2
2
i
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j
2
Minisuperspace – isotropic model

The standard FRW metric

Model with a single scalar field simply has TWO
minisuperspace coordinates {a, F} (the cosmic scale factor and
the scalar field)
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Minisuperspace – anisotropic model

All anisotropic models



Kantowski-Sachs models
Bianchi
Kantowski-Sachs models, 3-metric


THREE minisuperspace coordinates {a, b, F} (the cosmic scale
factors and the scalar field) (topology is S1xS2)
Bianchi, most general homogeneous 3-metric with a 3dimensional group of isometries (these are in 1-1
correspondence with nine 3-dimensional Lie algebras-there are
nine types of Bianchi cosmology)
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2008
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Minisuperspace – anisotropic model




Bianchi, most general homogeneous 3-metric with a 3-dimensional
group of isometries (these are in 1-1 correspondence with nine 3dimensional Lie algebras-there are nine types of Bianchi cosmology)
The 3-metric of each of these models can be written in the form
wi are the invariant 1-forms associated with a isometry group
The simplest example is Bianchi 1, corresponds to the Lie group R3
(w1=dx, w2=dy, w3=dz)

FOUR minisuperspace coordinates {a, b, c, F} (the cosmic scale
factors and the scalar field)
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Minisuperspace propagator


For the minisuperspace models path (functional) integral is reduced
to path integral over 3-metric and configuration of matter fields, and
to another usual integration over the lapse function N.
For the boundary condition q(t1)=q’, q(t2)=q’’, in the gauge,
N=const, we have
q "; q '   dNK (q " , N ; q ' ,0)

where
K (q " , N ; q ' ,0)   Dq exp(  I [q ])


ordinary (euclidean) QM propagator between fixed minisuperspace
coordinates (q’, q’’ ) in a fixed “time” N
S (I) is the action of the minisuperspace model
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K (q " , N ; q ' ,0)   Dq exp(  I [q ])
Minisuperspace propagator


ordinary QM propagator between fixed minisuperspace coordinates
(q’, q’’ ) in a fixed time N
S is the action of the minisuperspace model
 1

I [q ]   dtN  2 f (q)q  q   U (q)
 2N

0
1

f is a minisuperspace metric
dsm2  f dq dq 

with an indefinite signature (-+++…)
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K (q " , N ; q ' ,0)   Dq exp(  I [q ])
Minisuperspace propagator

for the quadratic action path integral is
1
K (q " , N ; q ' ,0) 
2


I (q " , N ; q ' ,0)

1/ 2

 I 
  det 

 
q " q ' 

2
exp(  I (q " , N ; q ' ,0))
euclidean classical action for the solution of
classical equation of motion for the q
Minisuperspace propagator is
1
q "; q ' 
2


1/ 2

 I 
 dN   det q " q ' 
2
exp(  I (q " , N ; q ' ,0))
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de Sitter minisuperspace model



simple exactly soluble model
model with cosmological constant and without matter field
E-H action with GHY surface term
0| x|| y|

The metric of de Sitter model
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de Sitter?





Willem de Sitter (May 6, 1872 – November
20, 1934) was a Dutch mathematician,
physicist and astronomer
De Sitter made major contributions to the
field of physical cosmology.
He co-authored a paper with Albert Einstein
in 1932 in which they argued that there
might be large amounts of matter which do
not emit light, now commonly referred to as
dark matter.
He also came up with the concept of the de
Sitter universe, a solution for Einstein's
general relativity in which there is no matter 
and a positive cosmological constant.
This results in an exponentially expanding,
empty universe. De Sitter was also famous
ISC2008, Nis,
Serbia, August 26 - 31,
for his research on the planet
Jupiter.
2008
A. Einstein, A.S. Eddington,
P. Ehrenfest, H.A. Lorentz,
W. de Sitter in Leiden (1920)
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Metric and action

Metric





FRW type
but…
Hamiltonian is not qaudratic
“new” metric
(Euclidean) Action – for this metric
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Hamiltonian and equation of motion

Hamiltonian

Equation of
motion
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Lagrangian and equation of motion

Action and
Lagrangian

The field equation and
constraint

Boundary
condition

Classical
action
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Wheeler DeWitt equation

1  d2
H   4 2  q  1(q)  0
2  dq


equation

de Sitter model ~
particle in constant
field

Solutions are Airy functions (why is WF “timeless”?)
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Next step…maybe … number theory!?


number sets
The field of real numbers R is the
result of completing the field of
rationals Q with the respect to
the usual absolute value |.|.

The field Q is Causchi incomplete with respect to the usual absolute
value |.|

ISC2008, Nis, Serbia, August 26 - 31,
2008
{1, 1.4, 1.41, 1.414, 1.4142, 1.41421, 1.414213, …}
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Next step…



number sets
Ostrowski theorem describing all
norms on Q. According to this
theorem: any nontrivial norm on
Q is equivalent to either ordinary
absolute value or p-adic norm for
some fixed prime number p.
This norm is nonarchimedean
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Next step…



In computations in everyday life, in scientific experiments and on
computers we are dealing with integers and fractions, that is with
rational numbers and we newer have dealings with irrational
numbers.
Results of any practical action we can express only in terms of
rational numbers which are considered to have been given to us by
God.
But …
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Measuring of distances

Archimedean axiom



“Any given large segment of a straight line can be surpassed
by successive addition of small segments along the same
line.”
A more formal statement of the axiom would be that if
0<|x|<|y| then there is some positive integer n such that
|nx|>|y|.
There is a quantum gravity uncertainty x while
measuring distances around the Planck legth
G
35
x  lPlanck 

1
,
6

10
m
3
c

which restricts priority of archimedean gemetry based
on real numbers and gives rise to employment of
nonarchimedean geometry based on p-adic numbers
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p-adic de Sitter model

Metric

Action

Propagator

groundstate WF
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real and p-adic (adelic) de Sitter model

adelic ground state WF
  R  p (q)
p



probability interpretation
of the WF
at the rational points q
|  |2 | R |2  (| q | p )
p
2

|

(
q
)
|
|  ( q ) |2   R
0

xZ
x Q \ Z
Discretization of minisuperspace coordinates
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Conclusion and перспективе(s)



noncommutative QC
dri dr i
da d a
~2 2
accelerating universe with
2
2
2
ds   N dt  R (t )
 a (t )
dynamical compactification
2 2
2 2
kr
1 k' 
1 4
of extra dimensions




~
S    g R dt d 3R d D   Sm    dtL
(4+D)-Kaluza-Klein
model
1
D( D  1) 3 D  2 2 D 2 D 1 
D 2
Ra
R

a  ~ R a Ra
~
~ R a
2N
12 N
2N
1 ~
1~
 kNRa D  N  R3a D
2
6
L

Lagrangian

p-adic ground state WF
 p ( x, y)  (| x | p )(| y | p )
| N | | 1  D( D  5) / 6 | p
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p
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Literature



1.
B. de Witt, “Quantum Theory of Gravity. I. The
canonical theory”, Phys. Rev. 160, 113 (1967)
C. Mysner, “Feynman quantization of general
relativity”, Rev. Mod. Phys, 29, 497 (1957).
D. Wiltshire, “An introduction to Quantum
Cosmology”, lanl archive
G. S. Djordjevic, B. Dragovich, Lj. Nesic,
I.V.Volovich, p-ADIC AND ADELIC
MINISUPERSPACE QUANTUM COSMOLOGY, Int.
J. Mod. Phys. A 17 (2002) 1413-1433.
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