1. Amusing properties of Klein- Gordon solutions on manifolds with variable dimensions D.V. Shirkov, P. P. Fiziev BLThPh, JINR, Dubna Talk at the Workshop Bogoliubov Readings Dubna, September 25, 2010 The Goal: To Open the Padlocks of Nature!

2. The main question: Where we can find the KEY ? The Tool

3. The basic problem of the standard approach to quantum gravity is caused by the very classical Einsten-Hilbert action in D = 1 + d : A New Idea: => for dimension D > 2 D. V. Sh., Particles and Nuclei (PEPAN), Lett. No 6 (162), 2010

4. Examples of 2-dim manifolds with variable geometries (surface of buttles)

5. Then we have local solutions: We consider the toy models in which the physical space is a continuous merger : and THE TIME IS GLOBAL ! The Klein Gordon Equation on Manifolds with variable topological dimension Assume (at least locally) With common frequency: x KG Equation:

6. Wave Equation in (1+2)-Dim Spacetime with Cylindrical Symmetry Using proper changes of variables we can transform the Z-equation in the Schrodinger-like form: Shape function: The basic Theorem: Standard anzatz: Simple problems: The only nontrivial problem: Z-equation x

7. Some Explicit Examples Two Cylinders of Constant Radiuses R and r < R, Connected Continuously by a Part of Cone: The shape function:

8. x

9. Exact Solutions and the limit * Continuous spectrum:

10. The Resonance States: The nontrivial dependence on the Klein- Gordon mass M: Z M = 0

11. A simple assymptotic formula for resonances:

12. A simple class of exactly solvable models X X X

13. Vertex angle:

14. The spectra for different values of the vertex angle: REAL frequencies => Two series of real frequencies:

15. CONCLUDING REMARKS 1. A signal, related to degree of freedom specific for the higher-dim part does not penetrate into the smaller-dim part, because of centrifugal force at the junction. 2. Our New THEOREM relates the KG problem on variable geometry to the Schrodinger-type eq with potential, generated by the variation. 3. The specific spectrum of scalar excitations characterizes the junction Geometry. This observation suggest an idea: To explain the observed particles spectra by geometry of the junction between domains of the space-time with different topological dimension. between domains of the space-time with different topological dimension. 4. The parity violation, due to the asymmetry of space geometry could yield the CP-violation. This, in turn, gives a hope to discover a simple natural basis for Explanation of the real situation, concerning C, P, and T properties of the particles.

16. Thank You for your attention We are still looking for the KEY ! ?