Minisuperspace Models in Quantum Cosmology

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

2. 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) 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,F} (the cosmic scale factor and the scalar field) ISC2008, Nis, Serbia, August 26 - 31, 2008

4. 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 3-dimensional 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

5. Minisuperspace – anisotropic model • 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 • 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

6. 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 qa(t1)=qa’, qa(t2)=qa’’, in the gauge, N=const, we have • where • ordinary (euclidean) QM propagator between fixed minisuperspace coordinates (qa’, qa’’ ) in a fixed “time” N • S (I) is the action of the minisuperspace model ISC2008, Nis, Serbia, August 26 - 31, 2008

7. Minisuperspace propagator • ordinary QM propagator between fixed minisuperspace coordinates (qa’, qa’’ ) in a fixed time N • S is the action of the minisuperspace model • fab is a minisuperspace metric • with an indefinite signature (-+++…) ISC2008, Nis, Serbia, August 26 - 31, 2008

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

10. 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 for his research on the planet Jupiter. • A. Einstein, A.S. Eddington, P. Ehrenfest, H.A. Lorentz, W. de Sitter in Leiden (1920) ISC2008, Nis, Serbia, August 26 - 31, 2008

11. 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

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

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

15. 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 |.| • {1, 1.4, 1.41, 1.414, 1.4142, 1.41421, 1.414213, …} 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 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 … 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<|x|<|y| then there is some positive integer n such that |nx|>|y|. • There is a quantum gravity uncertainty Dx while measuring distances around the Planck legth ISC2008, Nis, Serbia, August 26 - 31, 2008

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

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

21. Conclusion and перспективе(s) • noncommutative QC • accelerating universe with dynamical compactification of extra dimensions • (4+D)-Kaluza-Klein model • Lagrangian • p-adic ground state WF 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 • 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