isc2008, nis, serbia, august 26 - 31, 2008 1 minisuperspace models in quantum cosmology ljubisa...

ISC2008, Nis, Serbia, August 26 - 31, 2008 1

Minisuperspace Models in Quantum

CosmologyLjubisa Nesic

Department of Physics, University of Nis, Serbia


Minisuperspace Superspace – infinite-dimensional space, with finite number degrees of freedom (hij(x), (x)) at each point, x

in 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 , we then simply have a SINGLE

equation for all of . metrics (shift vector is zero)

ndxdxtqhdttNds jiij ,...,2,1,))(()( 222


Minisuperspace – isotropic model

The standard FRW metric

Model with a single scalar field simply has TWO minisuperspace coordinates {a, } (the cosmic scale factor and the scalar field)


Minisuperspace – anisotropic model All anisotropic models

Kantowski-Sachs models Bianchi

THREE minisuperspace coordinates {a, b, } (the cosmic scale factors and the scalar field) (topology is S1xS2)

Bianchi, most general homogeneous 3-metric with a 3-dimensional group of isometries (these are in 1-1 correspondence with nine 3-dimensional Lie algebras-there are nine types of Bianchi cosmology)

Kantowski-Sachs models, 3-metric


Minisuperspace – anisotropic model

i are the invariant 1-forms associated with a isometry group The simplest example is Bianchi 1, corresponds to the Lie group R3

(1=dx, 2=dy, 3=dz)

Bianchi, most general homogeneous 3-metric with a 3-dimensional group of isometries (these are in 1-1 correspondence with nine 3-dimensional Lie algebras-there are nine types of Bianchi cosmology)

The 3-metric of each of these models can be written in the form

FOUR minisuperspace coordinates {a, b, c, } (the cosmic scale factors and the scalar field)


Minisuperspace propagator

ordinary (euclidean) QM propagator between fixed minisuperspace coordinates (q’, q’’ ) in a fixed “time” N S (I) is the action of the minisuperspace model

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, =const, we have

)0,';,"(';" qNqdNKqq

where

])[exp()0,';,"( qIDqqNqK



with an indefinite signature (-+++…)

dqdqfdsm

2

1

02

)()(2

1][ qUqqqf

NdtNqI

ordinary QM propagator between fixed minisuperspace coordinates (q’, q’’ ) in a fixed time N

S is the action of the minisuperspace model

f is a minisuperspace metric




Minisuperspace propagator is

)0,';,"( qNqI

for the quadratic action path integral is

euclidean classical action for the solution of classical equation of motion for the q


))0,';,"(exp('"

det2

1';"

2/12

qNqI

qq

IdNqq

))0,';,"(exp('"

det2

1)0,';,"(

2/12

qNqI

qq

IqNqK


de Sitter minisuperspace model simple exactly soluble model model with cosmological constant and without matter field E-H action with GHY surface term

The metric of de Sitter model

||||0 yx


de Sitter?

A. Einstein, A.S. Eddington, P. Ehrenfest, H.A. Lorentz, W. de Sitter in Leiden (1920)

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 for his research on the planet Jupiter.


Metric and action

(Euclidean) Action – for this metric

Metric FRW type but… Hamiltonian is not qaudratic “new” metric


Hamiltonian and equation of motion

Hamiltonian

Equation of motion


Lagrangian and equation of motion

Classical action

Action and Lagrangian

The field equation and constraint

Boundary condition


Wheeler DeWitt equation

equation

de Sitter model ~ particle in constant field

Solutions are Airy functions (why is WF “timeless”?)

0)(142

12

2

qq

dq

dH


Next step…maybe … number theory!?

The field Q is Causchi incomplete with respect to the usual absolute value |.| {1, 1.4, 1.41, 1.414, 1.4142, 1.41421, 1.414213, …}

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 |.|.


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


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 …


Measuring of distances

which restricts priority of archimedean gemetry based on real numbers and gives rise to employment of nonarchimedean geometry based on p-adic numbers

mc

Glx Planck

353

106,1

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


p-adic de Sitter model

groundstate WF

Metric

Action

Propagator


real and p-adic (adelic) de Sitter model

Discretization of minisuperspace coordinates

p

pR q )|(||||| 22

adelic ground state WF p

pR q)(

ZQx

Zxqq R

\0

|)(||)(|

22

probability interpretation of the WF

at the rational points q


Conclusion and перспективе(s)

p-adic ground state WF

(4+D)-Kaluza-Klein model

accelerating universe with dynamical compactification of extra dimensions

Lagrangian

22

22

4

2222

'1)(

1)(

~2

k

ddta

drdrtRdtNds

aa

kr

ii

dtLSdRddtRgS mD 3~

DD

DDD

aRNRaNk

aRaRN

DaaR

N

DDRRa

NL

3

122232

~6

1~2

1

~2

~12

)1(~

2

1

)|(|)|(|),( ppp yxyx

pp DDN |6/)5(1|||

noncommutative QC


Literature 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 archive1. 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.

isc2008, nis, serbia, august 26 - 31, 2008 1 minisuperspace models in quantum cosmology ljubisa...

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