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%keywords: auto-parametric, charge, resonance, whispering, autonomous, gallery, pulson, modes, nonabelian, nonlinear, non-pertubative, supratransmission , supraconductivity, breather, nonpertubative, pseudosphere, phase, berry, Gordon, sine-Gordon, Baecklund, Thirring, Skyrme, Rayleigh, fine structure, iteration, iterative
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\begin{document}
\title{Higher-Dimensional Solitons Stabilized by Opposite Charge}
\author{B{\sc ernd} ~B{\sc inder}}
\date{30.11.2002}%, updates: \href{http://www.quanics.com}{http://www.quanics.com}}
\affiliation{\it 88682 Salem, Germany,
\href {mailto:binder@quanics.com}{binder@quanics.com} \copyright
2002}
%#############################################################
\begin{abstract}
In this paper it is shown how higher-dimensional solitons can be
stabilized by a topological phase gradient, a field-induced shift
in effective dimensionality. As a prototype, two instable
2-dimensional radial symmetric Sine--Gordon extensions (pulsons)
are coupled by a sink/source term such, that one becomes a stable
1d and the other a 3d wave equation. The corresponding physical
process is identified as a polarization that fits perfectly to
preliminary considerations regarding the nature of electric charge
and background of $1/137$. The coupling is iterative with
convergence limit and bifurcation at high charge. It is driven by
the topological phase gradient or non-local Gauge potential that
can be mapped to a local oscillator potential under $PSL(2,{\Bbb
R})$.
\end{abstract} \maketitle
%#############################################################
%#############################################################
\paragraph*{Introduction.}
%#############################################################
Solitary waves were discovered in the first half of the nineteenth
century by Rusell, the word soliton was invented by Kruskal, the
sine--Gordon (SG) model by Skyrme \cite{Skyrme}. Solitons retain
their identity after collisions, can annihilate with
anti--solitons, many--soliton solutions obey Pauli's exclusion
principle. In 1+1--dim. space--time there are two non--trivial
minimal quantum field theories which describe non--perturbative
phenomena: the SG model and the massive Thirring model
\cite{Thirring} (a self-coupled Dirac field, see the Lagrangians
\cite{Remoissenet}), both are intimately related \cite{Coleman}.
For nonlinear field theory models in 1+1--dim. space--time the
equations of motion admit finite energy and finite width solutions
called solitons \cite{AbS}. In the previous paper fundamental
three--dim. (3d) baryon particles have been assigned to soliton
properties \cite{rayleigh07,beta10}. But how can 3d soliton
properties emerge, and how could a baryon-type 3d SG soliton
subject to distortions be balanced and stabilized?
\\
%#############################################################
\paragraph{Time independent field equations.}
%#############################################################
The low-dim. (bosonic) hermitian scalar field $\theta$ with
Lagrangian density ${\cal L}= {\mu\over 2}
\partial_{\nu}\theta\partial^{\nu}\theta~-V(\theta)$ is a function
of one space dimension and time (1+1--dim.). The time independent
field equations reads $\mu {\partial_r^2 \theta} =
{\partial_\theta V }$ which can also be written as
\begin{equation}
\partial_t V=0,\quad V(\theta) = {\mu \over 2} (\partial_r \theta)^2 \,
\labeqn{potenergy},
\end{equation}
where the sine--Gordon equation (SG)
\begin{equation}
\partial_{rr} \theta
- V_0 \sin\theta = 0 \, \labeqn{sg00},
\end{equation}
has a potential given by
\begin{eqnarray}
{V(\theta)} = V_0(1-\cos\theta) \, \labeqn{pot01}.
\end{eqnarray} In \cite{rayleigh07} it has been shown, that Rayleigh-type
self--excited auto-parametric systems \cite{trvn} can stimulate in
a 3d-situation "whispering gallery modes" (that have been measured
in \cite{Ustinov98, Ustinov00}) and model Coulomb interaction
between sine-Gordon solitons. In \cite{beta10} the same model has
been be applied to determine the most likely Compton mass of the
soliton. In this paper these results will be connected to the
Skyrme baryon model and to the dissipative models including
sink/source term representing the soliton charge.
\\
%#############################################################
\paragraph{Polarization and radiative coupling of pulsons.}
%#############################################################
The isolated 1+1--dim. topology of a SG soliton is stable,
integrable, and interaction-free. A single 2-dim. radial symmetric
Sine--Gordon extension (pulsons \cite{Christiansen81}) is not
stable. The dissipative property can be found by regarding the
1+2--dim. pulson solution with dissipative term ${\partial_r
\Theta / r}$ and external coupling term ${\pi M_g / M}$
\begin{equation}
\partial_{rr}\Theta
- V_0 \sin\Theta + {\partial_r \Theta \over r} = {\pi M_g \over M} \,
\labeqn{sg01}.
\end{equation}
Without coupling, the neutral or source free pulson ($M_g=0$) is a
breather-like slowly dying solution \cite{Christiansen81}. But if
we choose the strength of the energy source/sink such, that $M_g
\ne 0$ can compensate the dissipative term, the 2d - pulson can be
either reduced to a pseudo--1d case $\partial_{rr}\Theta - V_0
\sin\Theta = 0$ by compensating the first--order term, or be
promoted to a 3d radial wave equation $\partial_{rr}\Theta - V_0
\sin\Theta + 2{\partial_r \Theta/r}=0$ depending on the sign in
\begin{equation}
{\partial_r \Theta \over r} = \mp {\pi M_g \over M}
%= - \mp M_g {q\over 4\pi}%= \theta_1
\, \labeqn{sg02}.
\end{equation}
This means simply, that a dimensional shift (an additional phase
gradient proportional to the radial distance) is induced by the
Gauge potential of a sink or source. If there is a permanent
external source of stochastic nature (thermal background radiation
or fluctuations), it increases dimensionality and provides for a
basic $r$--independent part in the potential $W(\Theta=0)=V_0$. If
two neutral pulsons become permanently polarized by adding a
source term to one and a sink term with opposite sign to the
other, there will be a radial coupling that can be assigned to a
opposite signed "charge" $\pm M_g$ defined in \eqn{sg02}. The
positive source $M_g>0$ with positive charge will be assigned to
the state that has increased dimensionality driven by a permanent
external stochastic radiation source providing for $V_0>0$. In
this case \eqn{sg02} directly provides for
\begin{equation} W(\Theta) = {\mu \over2} (\partial_r \Theta)^2 + V_0
= {\mu (1+r^2)\over 2} \({\pi M_g \over M}\)^2 \, \labeqn{sg04}.
\end{equation}
As already shown in \cite{sepp,phasetospace,Mapping} \eqn{sg02} is
the projective condition necessary to adjust the topological phase
fields on pseudospheres (constant negative curvature) that
provides for the optimum feedback resonance, coupling strength,
and corresponding fine structure of spin precession. Integrating
\eqn{sg02}
\begin{eqnarray}
{\Theta}(r)-{\Theta}(0) = \int_0^{r}{\partial_r \Theta dr' } = \mp
{r^2\over2 }{\pi M_g \over M} \, \labeqn{sg05},
\end{eqnarray}
relates the potential linearly to the scalar function $\Theta$
\begin{equation}
W(\Theta)=\mp \Theta\mu {\pi M_g \over M} + V_0 , \quad
\Theta(0)=0
\, \labeqn{sg06},
\end{equation}
that gives also the basic potential
\begin{equation}
W(0) = V_0 = {\mu \over 2} \({\pi M_g \over M} \)^2 \,
\labeqn{sg07}.
\end{equation}
The external source or thermal pool provides for the same scalar
angle--dependent energy in both, the oscillator energy responsible
for the dimensional change and polarization in \eqn{sg06}, and for
the soliton coupling energy in \eqn{pot01}
\begin{equation}
W(\theta_M) = {V(\theta_M})
\, \labeqn{sg08}.
\end{equation}
Combining \eqn{pot01} with \eqn{sg06} via \eqn{sg08} allows to
determine the optimum phase shift of resonant coupling $\theta_M$
iteratively from
\begin{eqnarray} M \theta_M = \pm \pi M_g \cos{\theta_M}
,\quad \theta_M = \pi \alpha_M \labeqn{iteration},
\end{eqnarray}
see results in table I.\\
%#############################################################
\paragraph{Topological and geometric phase interpretation.}
%#############################################################
Two neutral pulsons can be mutually stabilized by assigning a
positive "charge" ($M_g>0$) to one and a "negative" charge
($M_g<0$) to the other. This excites $M_g>0$ to a pseudo 1+3--dim.
soliton and reduces $M_g<0$ to the pseudo 1+1--dim. topology of a
SG--soliton. It is known that stereographic mapping can map a
Coulomb potential to an oscillator potential and vice versa. With
\eqn{sg05}, \eqn{iteration}, $\cos\theta=1-2\sin^2(\theta/2)$, the
projective relation is given by
\begin{eqnarray} {r} = 2 \sin(\theta/2) \labeqn{pot05}.
\end{eqnarray} Interpretation: In the resonant case a topological phase
scattering pattern is coupling back to the scatterer. The
resulting non-linear scalar coupling field is a deficit angle
field that obviously can be described by the sine-Gordon equation
on pseudospheres with a local potential given by the square of the
phase gradient. \Eqn{pot05} is a projective scattering condition,
where the isotropic radial coupling connects the pulsons by
projective resonance. It is a projection under $PSL(2,{\Bbb R})$
that maps a local oscillator potential to a global Coulomb
sink/source on the sphere and pseudosphere \cite{NerPogo}. The
mapping is directly controlled by the iterative solution
\eqn{iteration} providing for the generalized fine structure
constants. The coupling or scattering condition \eqn{pot05} maps
the deficit angle to the proper spatial distance by relating the
phase gradient potential to an oscillator potential while changing
the dimension.
% ----------------------------------------------
\begin{table} [h]
\begin{caption}
\newline \small{Convergent fine structure
(re)generation constants\\ $\alpha_M $ for $M_g=1$ and variable
$M>2$ \cite{alpha137MN},\\ check also the simulation at
\cite{online1}. }
\end{caption}\\
\begin{tabular}{p{40pt}p{120pt}}\\
\textit{$M$}& \textit{$1/\alpha_M$}\\
\hline
3 & 4.13669 \\
4 & 4.96178 \\
5 & 5.82662 \\
6 & 6.72097 \\
7 & 7.6371 \\
8 & 8.56944 \\
9 & 9.51399 \\
10 & 10.46789 \\
11 & 11.42906 \\
12 & 12.39597 \\
13 & 13.36747 \\
137 & 137.03600941164\\
\hline \labtab{fsc1}
\end{tabular}
\end{table}
% ----------------------------------------------
%#############################################################
\paragraph{Coupling strength and energy.}
%#############################################################
Compton scattering can model the quantum interaction of a linear
wave and a particle. To define a coupling strength $q$ between SG
kink or antikink induced by the phase fluctuations generated by
background radiation, it will be necessary to define some
potential and energy relations in 1d and 3d. The 1d coupling
energy can be defined by by a temporal average or mean unit energy
$E_{1d}$
\begin{equation}
{E_{1d}} = q^2\overline{(\partial_r \theta)^2}\,=1\mu c^2 = 2 q^2
\overline{V}\, \labeqn{josephson6},
\end{equation} where $\mu$ is a unit mass, $c$ the light velocity,
and $\overline{V} = V_0$ the mean background radiation energy. To
compare our theoretical soliton coupling model to real existing
couplings, mass/energy has to be quantified and geometrized. The
mutual 1-d coupling to photons with amplitude/wavelenght
fluctuation $\lambda_{\mu}$ can be regarded as a permanent Compton
scattering process with mass--energy value related to
$\lambda_{\mu}$ via Compton relation
\begin{equation}
{E_{1d} } = 2 q^2 \overline{V } = {h c \over \lambda_{\mu}} \,
\labeqn{couple1}.
\end{equation}
Let's connect pulsons at distance $2R$ by defining the
3d-potential in accordance with \eqn{sg02}
\begin{equation} \phi_{3d}= {q \over
4\pi R} = {\partial_r \Theta \over 4 \pi R^2} =\mp {M_g \over 2M
R} \, \labeqn{sg02x}.
\end{equation}
Generally, the Gauss relation can connect the 1-d coupling
strength to a 3-d coupling strength with a spherical symmetric
potential $\phi_{3d}(r)$ such, that the radial coupling energy is
defined by
\begin{equation}
E_{3d}(r) = { q \over \epsilon_0} \, \phi_{3d}(r)
\, \labeqn{couple3}.
\end{equation}
The coupling strength $q$ is the charge defining the interaction
\begin{equation}
E_{3d}(r)= - {1 \over \epsilon_0} \int_{\infty}^r \phi_{3d}^2 4\pi
dr' = {q^2 \over 4\pi \epsilon_0 r}, \quad \phi_{3d}= {q \over
4\pi r} \, \labeqn{josephson9}.
\end{equation}
The fine structure constant is defined by
\begin{equation}
\alpha = {q^2 \over 4 \pi \epsilon_0 \hbar c } =
{{E_{3d}(\lambda_{\mu})} \over {E_{1d}}}\, \labeqn{couple3},
\end{equation}
where the relations at the special reference distance
$\lambda_{\mu}$ given by dimensionless Planck units
$h=c=\lambda_{\mu}=1$ must obey the unit condition
\begin{equation} {E_{3d}(\alpha
\lambda_{\mu})} = {E_{1d}} = \phi_{3d}(\alpha \lambda_{\mu}) =
{\phi_{1d}}\equiv 1 \, \labeqn{couple3},
\end{equation}
that provides for
\begin{equation} \alpha = {q \over 4\pi},\quad M =\[ 1 \over \alpha \] \, \labeqn{couple4}.
\end{equation}
where $[\,]$ means next higher integral value. Why integral?
Because of single--valuedness the round-trip path fits integer
numbers, similar to "whispering gallery modes" \cite{rayleigh07}.\\
%#############################################################
\paragraph{Concluding Remarks.}
%#############################################################
It is an interesting question how stationary solitons (like
breather) get their absolute mass/energy. To approach a 3d
scattering we can compare to an ansatz for Skyrme fields in 3d
\cite{MantonPiette01} which uses rational maps between Riemann
spheres under $PSL(2,{\Bbb R})$ and an $SU(2)$ valued Skyrme
field. The lowest energy $E_{1d}$ of Skyrmions which applies to
the rational map ansatz, is more than a factor $12 \pi^2$ lower
than the energy given by the Lagrangian and called
Fadeev-Bogomolny bound, see also
\cite{Campbell86,Houghton98,Kudrya98}. In our case this coupling
ratio is given by $q^{-2}=12 \pi^2$ in \eqn{couple1}. This could
be found by treating the wave--soliton coupling as a
Rayleigh--type auto--parametric system, see
\cite{rayleigh07,beta10}. \Eqn{couple4} provides for $M=137$ and
for a plausible baryon energy limit: $\lambda_{\mu} \approx
1,31777... \cdot 10^{-15}$m or $E_{1d}= \mu c^2 = 940.86369...$
MeV extrapolated to Planck units $\lambda_{\mu}=c=\hbar=1$ emerges
as a system-invariant soliton mass scale that is 1.001382 times
the neutron and 1.002762 times the proton scale \cite{beta10}. The
two oppositely charged 2-dim. radial symmetric pulsons stabilized
and balanced by a topological phase gradient could be assigned to
a proton--electron combination: one pulson is promoted to a
positive charge with $M_g>0$ as a pseudo 1+3--dim. soliton (the
proton) while reducing the second pulson to a negative charge
$M_g<0$ and pseudo 1+1--dim. topology of a SG--soliton (the
electron). The broken symmetry could be supported by the coupling
field that could act as a flexible shield against external
distortion or fluctuations in dimensionality. Generalizations to
higher dimensions and source terms could be assigned to multiple
charged nuclei, where the charge quantity $M_g q$ with $M_g \neq
0$ corresponds to a field--induced shift in effective
dimensionality with $d-2=M_g$, stability and convergence criteria
regarding \eqn{iteration} for $1 \le |M_g|\le M$ can be found in
\cite{alpha137MN}. To account for the (half) spin property of
pulsons with cylindrical symmetry, it should be possible to
characterize the polarized d=1/d=3 soliton system by two coupled
two-spinors under $SU(2)$ including electromagnetic interactions
defined by vector and scalar potentials responsible for the
dimensional shift.
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\end{thebibliography}
\end{document}
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