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 ReadingsDubna, September 25, 2010

The Goal: ToOpen thePadlocks of Nature!

The main question:

Where we can find the KEY

? The Tool

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

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

Then we have local solutions:

We consider the toy models in which the physical space is a continuous merger :and THE TIME IS GLOBAL !

TheKlein

Gordon Equation

on

Manifolds with variable topologicaldimension

Assume (at least locally)

With common frequency:

x

KG Equation:KG Equation:

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

xx

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

The shape function:

x

Exact Solutions and the limit *

Continuous spectrum:

The Resonance States:

The nontrivialdependenceon theKlein-Gordonmass M:

Z

M = 0

A simple assymptotic formula for resonances:

A simple class of exactly solvable models

X

XX

Vertex angle:

The spectra for different values of the vertex angle:

REALfrequencies=>=>

Two series of real frequencies:Two series of real frequencies:

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 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.

Thank You for your attention

Wearestilllookingfor the KEY !

?