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A NONCOMMUTATIVE CLOSED FRIEDMAN WORLD MODEL

A NONCOMMUTATIVE CLOSED FRIEDMAN WORLD MODEL. A NONCOMMUTATIVE CLOSED FRIEDMAN WORLD MODEL. Introduction Structure of the model Closed Friedman universe – Geometry and matter Singularities Concluding remarks. 1. INTRODUCTION. GEOMETRY. MATTER. Mach’s Principle (MP):

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A NONCOMMUTATIVE CLOSED FRIEDMAN WORLD MODEL

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  1. A NONCOMMUTATIVE CLOSED FRIEDMAN WORLD MODEL

  2. A NONCOMMUTATIVE CLOSED FRIEDMAN WORLD MODEL • Introduction • Structure of the model • Closed Friedman universe – Geometry and matter • Singularities • Concluding remarks

  3. 1. INTRODUCTION GEOMETRY MATTER Mach’s Principle (MP): geometry from matter Wheeler’s Geometrodynamics (WG): matter from (pre)geometry 2

  4. MP is only partially implemented in Genaral Relativity: matter modifies the space-time structure (Lense-Thirring effect), but • it does not determine it fully ("empty" de Sitter solution), • in other words, • SPACE-TIME IS NOT GENERATED BY MATTER 3

  5. For Wheeler pregeometry was "a combination of hope and need, of philosophy and physics and mathematics and logic''. Wheeler made several proposals to make it more concrete. Among others, he explored the idea of propositional logic or elementary bits of information as fundamental building blocks of physical reality.A new possibility:PREGEOEMTRY NONCOMMUTATIVE GEOMETRY 4

  6. References Model: • Int. J. Theor. Phys. 44, 2005, 619. • J. Math. Phys. 46, 2005, 122501. Friedman model: • Gen. Relativ. Gravit. DOI 10.107/s10714- 008-0740-3. Singularities: • Gen. Relativ. Gravit. 31, 1999, 555 • Int. J. Theor. Phys. 42, 2003, 427 5

  7. 2. STRUCTURE OF THE MODEL Transformation groupoid: =EG pg  E p  = (p, g) M p2 Pair groupod: 1=EE  = (p1, p2) p1 6  i 1 are isomorphic

  8. The algebra: with convolution as multiplication: Z(A) = {0} "Outer center": 7

  9. We construct differential geometry in terms of (A, DerA) DerA  V = V1 + V2 + V3 V1 – horizontal derivations, lifted from M with the help of connection V2 – vertical derivations, projecting to zero on M V3 – InnA = {ad a: a A} - gravitational sector - quantum sector 8

  10. Metric - lifting of the metric g from M Vertical derivations can be identified with functions on E with values in the Lie algebra of G. Natural choice for k is a Killing metric. 9

  11. 3. CLOSED FRIEDMAN UNIVERSE – GEOMETRY AND MATTER Metric: Total space of the frame bundle: Structural group: 10

  12. Groupoid: Algebra: "Outer center": 11

  13. Metric on V = V1V2: Einstein operator G: V  V 12

  14. Einstein equation: G(u)= u, uV - generalized eigenvalues of G i  Z 13

  15. i We find by solving the equation Solutions: Generalized eigenvalues: Eigenspaces: WB – 1-dimensional Wh – 3-dimensional Wq – 1-dimensional 14

  16. By comparing B and h with the components of the perfect fluid energy-momentum tenor for the Friedman model, we find c = 1 We denote In this way, we obtain components of the energy-momentum tensor as generalized eigenvalues of Einstein operator. 15

  17. What about q? This equation encodes equation of state: - dust - radiation 16

  18. If we add the cosmological constant  to the Einstein operator, its eigenvalue equation remains the same provided we replace:

  19. Comment: Einstein operator acts on the module of derivations and selects submodules which are identical with the energy-momentum tensor components and constraints for eqs of state to which correspond generalized eigenvalues Duality in Einstein’s eqs is liquidated.

  20. Quantum sector of the model: - regular representation Every a  A generates a random operator ra on (Hp)pE 18

  21. Random operator is a family of operators r = (rp)pE, i.e. a function such that (1) the function r is measurable: if then the function is measurable with respect to the manifold measure on E. (2) r is bounded with respect to the norm ||r|| = ess sup ||r(p)|| where ess sup means "supremum modulo zer measure sets". In our case, both these conditions are satisfied. 19

  22. N0 – the algebra of equivqlence classes (modulo equality everywhere) of bounded random operators ra, a A. N = N0'' – von Neumann algebra, called von Neumann algebra of the groupoid . In the case of the closed Friedman model Normal states on N (restricted to N0) are • - density function which is integrable, positive, normalized; to be faithful it must satisfy the condition >0. 20

  23. We are considering the model Let   0 or   0. Since  is integrable, (A) is well defined for every a on the domain i.e. the functional (A) does not feel singularities. Tomita-Takesaki theorem  there exists the 1-parameter group of automotphisms of the algebra N A. Connes, C. Rovelli, Class. Quantum Grav.11, 1994, 2899. which describes the (state dependent) evolution of random opertors with the Hamiltonian This dynamics does not feel singularities. 21

  24. (M, φ), where M – von Neumann algebra, φ – normal state on M, is a noncommutative probabilistic space. φ is normal if: φ(ΣPn) = Σφ(Pn) for any countable family of mutually orthogonal projections Pn in M.

  25. The same conclusion can be proved in a more general way algebra of random operators before sigularity has been attached algebra of random operators after singularity has been attached We have: von Neumann algebra

  26. 5. CONCLUDING REMARKS Our noncommutative closed Friedman world model is a toy model. It is intended to show how concepts can interact with each other in the frameworkof noncommutative geometry rather than to study the real world. Two such interactions of concepts have been elucidated: • Interaction between (pre)geoemtry and matter: components of the energy-momentum tensor can be obtained as generalized eigenvalues of the Einsten operator. • Interaction between singular and nonsingular. 22

  27. Usually, two possibilities are considered: either the future quantum gravity theory will remove singularities, or not. Here we have the third possibility: Quantum sector of our model (which we have not explored in this talk) has strong probabilistic properties: all quantum operators are random operators (and the corresponding algebra is a von Neumann algebra). Because of this, on the fundamental level singularities are irrelevant. Singularities appear (together with space, time and multiplicity) when one goes from the noncommutative regime to the usual space-time geometry. 23

  28. EMERGENCE OF SPACE-TIME

  29. Therefore, on the fundamental level the concept of the beginning and end is meaningeless. Only from the point of view of the macroscopic observer can one say that the universe had an initial singularity in its finite past, and possibly will have a final singularity in its finite future. 24

  30. ? THE END

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