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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