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ISC2008, Nis, Serbia, August 26 - 31, 2008 1 Minisuperspace Models in Quantum Cosmology Ljubisa Nesic Department of Physics, University of Nis, Serbia

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ISC2008, Nis, Serbia, August 26 - 31, 2008 2 Minisuperspace Superspace – infinite-dimensional space, with finite number degrees of freedom ( h ij (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)

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ISC2008, Nis, Serbia, August 26 - 31, 2008 3 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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ISC2008, Nis, Serbia, August 26 - 31, 2008 4 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 S 1 xS 2 ) 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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ISC2008, Nis, Serbia, August 26 - 31, 2008 5 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 R 3 ( 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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ISC2008, Nis, Serbia, August 26 - 31, 2008 6 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 ( t 1 )= q ’, q ( t 2 )= q ’’, in the gauge, =const, we have where

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ISC2008, Nis, Serbia, August 26 - 31, 2008 7 Minisuperspace propagator with an indefinite signature (-+++…) 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

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ISC2008, Nis, Serbia, August 26 - 31, 2008 8 Minisuperspace propagator Minisuperspace propagator is for the quadratic action path integral is euclidean classical action for the solution of classical equation of motion for the q

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ISC2008, Nis, Serbia, August 26 - 31, 2008 9 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

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ISC2008, Nis, Serbia, August 26 - 31, 2008 10 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 astronomerMay 61872November 201934Dutch De Sitter made major contributions to the field of physical cosmology.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. 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.de Sitter universecosmological 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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ISC2008, Nis, Serbia, August 26 - 31, 2008 11 Metric and action (Euclidean) Action – for this metric Metric FRW type but… Hamiltonian is not qaudratic “new” metric

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ISC2008, Nis, Serbia, August 26 - 31, 2008 12 Hamiltonian and equation of motion Hamiltonian Equation of motion

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ISC2008, Nis, Serbia, August 26 - 31, 2008 13 Lagrangian and equation of motion Classical action Action and Lagrangian The field equation and constraint Boundary condition

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ISC2008, Nis, Serbia, August 26 - 31, 2008 14 Wheeler DeWitt equation equation de Sitter model ~ particle in constant field Solutions are Airy functions (why is WF “timeless”?)

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ISC2008, Nis, Serbia, August 26 - 31, 2008 15 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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ISC2008, Nis, Serbia, August 26 - 31, 2008 16 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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ISC2008, Nis, Serbia, August 26 - 31, 2008 17 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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ISC2008, Nis, Serbia, August 26 - 31, 2008 18 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 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 |y|. There is a quantum gravity uncertainty x while measuring distances around the Planck legth

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ISC2008, Nis, Serbia, August 26 - 31, 2008 19 p-adic de Sitter model groundstate WF Metric Action Propagator

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ISC2008, Nis, Serbia, August 26 - 31, 2008 20 real and p-adic (adelic) de Sitter model DDiscretization of minisuperspace coordinates adelic ground state WF probability interpretation of the WF at the rational points q

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ISC2008, Nis, Serbia, August 26 - 31, 2008 21 Conclusion and перспективе(s) p-adic ground state WF (4+D)-Kaluza-Klein model accelerating universe with dynamical compactification of extra dimensions Lagrangian noncommutative QC

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ISC2008, Nis, Serbia, August 26 - 31, 2008 22 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 archive 1. 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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