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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 1 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 ISC2008, Nis, Serbia, August 26 - 31, 2008 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) ISC2008, Nis, Serbia, August 26 - 31, 2008 3 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) ISC2008, Nis, Serbia, August 26 - 31, 2008 4 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) ISC2008, Nis, Serbia, August 26 - 31, 2008 5 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 ISC2008, Nis, Serbia, August 26 - 31, 2008 6 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 (-+++…) ISC2008, Nis, Serbia, August 26 - 31, 2008 7 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)) ISC2008, Nis, Serbia, August 26 - 31, 2008 8 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 ISC2008, Nis, Serbia, August 26 - 31, 2008 9 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) 10 Metric and action Metric FRW type but… Hamiltonian is not qaudratic “new” metric (Euclidean) Action – for this metric ISC2008, Nis, Serbia, August 26 - 31, 2008 11 Hamiltonian and equation of motion Hamiltonian Equation of motion ISC2008, Nis, Serbia, August 26 - 31, 2008 12 Lagrangian and equation of motion Action and Lagrangian The field equation and constraint Boundary condition Classical action ISC2008, Nis, Serbia, August 26 - 31, 2008 13 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”?) ISC2008, Nis, Serbia, August 26 - 31, 2008 14 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, …} 15 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 ISC2008, Nis, Serbia, August 26 - 31, 2008 16 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 … ISC2008, Nis, Serbia, August 26 - 31, 2008 17 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 ISC2008, Nis, Serbia, August 26 - 31, 2008 18 p-adic de Sitter model Metric Action Propagator groundstate WF ISC2008, Nis, Serbia, August 26 - 31, 2008 19 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 xZ x Q \ Z Discretization of minisuperspace coordinates ISC2008, Nis, Serbia, August 26 - 31, 2008 20 Conclusion and перспективе(s) noncommutative QC dri dr i da 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 ISC2008, Nis, Serbia, August 26 - 31, p 2008 21 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. ISC2008, Nis, Serbia, August 26 - 31, 2008 22