ISC2008, Nis, Serbia, August 26 - 3 1, 2008 1 Models in Quantum Cosmology Ljubisa Nesic Department of Physics, University of Nis, Serbia
Dec 15, 2015
ISC2008, Nis, Serbia, August 26 - 31, 2008 1
Minisuperspace Models in Quantum
CosmologyLjubisa Nesic
Department of Physics, University of Nis, Serbia
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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
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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)
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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
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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)
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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
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Minisuperspace propagator
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
])[exp()0,';,"( qIDqqNqK
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Minisuperspace propagator
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
])[exp()0,';,"( qIDqqNqK
))0,';,"(exp('"
det2
1';"
2/12
qNqI
IdNqq
))0,';,"(exp('"
det2
1)0,';,"(
2/12
qNqI
IqNqK
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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
The metric of de Sitter model
||||0 yx
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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.
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Metric and action
(Euclidean) Action – for this metric
Metric FRW type but… Hamiltonian is not qaudratic “new” metric
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Hamiltonian and equation of motion
Hamiltonian
Equation of motion
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Lagrangian and equation of motion
Classical action
Action and Lagrangian
The field equation and constraint
Boundary condition
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Wheeler DeWitt equation
equation
de Sitter model ~ particle in constant field
Solutions are Airy functions (why is WF “timeless”?)
0)(142
12
2
dq
dH
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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 |.|.
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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
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
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p-adic de Sitter model
groundstate WF
Metric
Action
Propagator
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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
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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
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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.