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Amusing properties of Klein-

Gordon solutions on manifolds

with variable dimensions

The

Goal: To

Open the

Padlocks

of

Nature!

D.V. Shirkov, P. P. Fiziev

BLThPh, JINR, Dubna

Talk at the Workshop

Bogoliubov Readings

Dubna, September 25, 2010

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 :

for dimension D > 2

=>

A New Idea:

D. V. Sh., Particles and Nuclei (PEPAN), Lett. No 6 (162), 2010

Examples of 2-dim manifolds with

variable geometries

(surface of buttles)

KG Equation:

x

Assume (at least locally)

The

Klein

Gordon

Equation

on

Manifolds

with

variable

topological

dimension

We consider the toy models in which

the physical space is a continuous

merger :

and THE TIME IS GLOBAL !

Then we have local solutions:

With common frequency:

Wave Equation in (1+2)-Dim Spacetime with Cylindrical Symmetry

Shape function:

Standard anzatz:

x

Simple problems:

The only nontrivial problem:

Z-equation

The basic

Theorem:

Using proper changes of variables we can transform the Z-equation in the

Schrodinger-like form:

Some Explicit Examples

Two Cylinders of Constant Radiuses R and r < R,

Connected Continuously by a Part of Cone:

The shape

function:

Continuous spectrum:

Exact Solutions and the limit *

The Resonance States:

M = 0

Z

The

nontrivial

dependence

on the

Klein-

Gordon

mass M:

A simple assymptotic formula for resonances:

A simple class of exactly solvable models

X

X

X

Vertex

angle:

Two series of real frequencies:

=>

REAL

frequencies

The spectra for different values of the vertex angle:

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.

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.

We

are

still

looking

for

the

KEY !

Thank You

for your attention