24.12.2014 Views

pdf 2.1MB

pdf 2.1MB

pdf 2.1MB

SHOW MORE
SHOW LESS

You also want an ePaper? Increase the reach of your titles

YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.

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

Hooray! Your file is uploaded and ready to be published.

Saved successfully !

Ooh no, something went wrong !