%filename="Mapping6.tex", "projection5.ps", "projcoup1.ps"
%keywords: nonlocal, nonabelian, nonlinear, discrete, non-pertubative, supratransmission , supraconductivity, transparency, breather, nonabelian, nonlocal, nonpertubative, computing, pseudosphere, phase, berry, Gordon, sine-Gordon, Baecklund, Aharonov, Bohm, Thirring, Lobachevski, Chebyshev, Kaehler, stereographic, projection, fine structure, iteration, iterative
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\begin{document}
\title{Charge as the Stereographic Projection of Geometric Precession on Pseudospheres}
\author{B{\sc ernd} ~B{\sc inder}}
\date{30.9.2002}
\affiliation{\it 88682 Salem, Germany, \href
{mailto:binder@quanics.com}{binder@quanics.com} \copyright 2002}
\begin{abstract}
In this paper geometric phases (Berry and Aharonov-Bohm) are generalized
to nonlinear topological phase fields on pseudospheres, where the
coordinate vector field is parallel transported along the
signal/soliton vector field with Levi--Civita connection.
Projective $PSL(2,{\Bbb R})$ symmetry describes the relativistic
self-interacting bosonic sine-Gordon field. A Coulomb potential
can be induced as the stereographic projection of a harmonic
oscillator potential mapping angles or phases to distances and
vice versa resulting in mutual coupling with a generalized
coupling constant given by a nonlinear iteration. With
single-valuedness requirement in 137-gonal symmetry it fits within
a few ppb uncertainty to the Sommerfeld fine structure constant.
\end{abstract}
\pacs {02.10.De, 03.65.Bz, 03.65.Vf, 03.65-w, 03.67.Lx, 05.45.Yv,
06.20.Jr, 12.20.-m, 42.65.T } \maketitle
\paragraph*{Introduction.}
The geometric origin of the sine-Gordon equation SG can be
assigned to the study of Riemannian geometry on surfaces of
constant negative scalar curvature, also known as pseudospherical
surfaces that can be considered as varieties embedded (with the
induced natural Riemannian metric) into the three dimensional
Euclidean space $\Bbb R^3$ \cite{AbS, BCP} (very similar to the
Liouville equation). The curvature and coupling parameter are
given by the Levi--Civita connection, where the SG appears by
fixing special local coordinate frames on pseudospherical surfaces
with a Riemann tensor that has only one independent component
$R_{1212}$. The SG has soliton solutions which describe elastic
collision of localizable waves, that can be interpreted as
particles of non--perturbative nature. This is relevant for
spatial and spectral localization of energy, intrinsic nonlinear
modes, self-induced transparency, and supra effects (energy
propagation in the forbidden band gap by means of nonlinear modes)
\cite{GenietLeon}.
Topological and/or geometric phases are subject of concepts in
differential-geometry and topology associated with abelian and
non--abelian groups \cite{berry, AharonovBohm, WIZE}. Generally,
phase factors or phases representing the `holonomy' provide for
important boundary conditions while reducing the degree of
redundancy in variables. This is one of the reasons why phases and
gauge theories are not unimportant in quantum mechanics, despite
of the central role of amplitude densities. Berry showed that the
geometric phase has the same mathematical (gauge) structure as the
Aharonov-Bohm (AB) phase \cite{AharonovBohm} and is the integral
of an effective vector potential along a closed path. Both phases
even combine \cite{AharonovBohmBerry} especially if a charged
particle is in an spatial extended quantum state, i.e. if the
orbital loop includes spatially extended sub-loops. Both, the
local non--abelian Berry phase evolution on $S^2=SU(2)/U(1)$ and
the nonlocal abelian AB scattering effect on $\Bbb R^2$ with conic
metric provide for phase evolutions and deficit angles that can be
combined. In a previous work it was shown how the deficit angle of
the AB conic metric and the geometric precession cone vertex angle
of the Berry phase can be mutually adjusted to restore
single-valuedness. The resulting interplay between both phases
provides a non--linear iterative system providing for generalized
fine structure constants \cite{BerryAB}. Here we start to
generalize the Berry and Aharonov-Bohm phases to nonlinear
topological phase fields on pseudospheres, where projective
$PSL(2,{\Bbb R})$ symmetry describes the relativistic
self-interacting bosonic sine-Gordon field, and where the
iterative interplay can be assigned to a harmonic oscillator
potential.
The central news of this paper is, that oscillators on the sphere
and the pseudosphere are related by the Bohlin--Levi--Civita
transformation with the Coulomb system on the pseudosphere
\cite{ladder, NerPogo}. Consequently, we can introduce the Coulomb
potential and the corresponding coupling constant as the
stereographic projection of the harmonic oscillator potential
mapping angles or phases to distances. With this relations the
previous work \cite {BerryAB} can be fundamentally confirmed and
generalized, SG relativistic
soliton dynamics can be related to electrodynamics.\\
\paragraph*{Strategy.}
This paper outlines the 4 essential steps. Step 1: starting with a
Riemannian geometry of surfaces of constant scalar curvature
(pseudospherical and spherical) embedded into the three
dimensional Euclidean space with symmetry group $SU(2)$ or
$SU(1,1)$. Step 2: fixing local coordinates that have only one
independent component given by the curvature scalar (sine-Gordon
and Liouville). Step 3: introducing feedback coupling by the
stereographic projection onto (conformal-flat) coordinates of the
two-dimensional oscillator with symmetry group $PSL(2,{\Bbb R})$
and identifying the dual potentials with $su(d)$ and $so(d+1)$
symmetry (for $d=2$).
Step 4: fixing the coupling constant by a nonlinear iteration. \\
\paragraph*{B\"acklund transformations.}
The nonlinear SG phase field evolves with a pseudospherical
curvature constraint. This property is found with generalized
Chebyshev coordinates on a plane $\cal S$ embedded in $\Bbb R^3$
\begin{equation}
ds^2=M^{-2} (dx)^2 + M^2(d{y})^2 + 2\cos\theta dx d{y}
\labeqn{Chebyshev}
\end{equation} with
scalar curvature $R=2 R_{1212} /\det(g_{ij})$ \cite{BCP} of the
generalized Chebyshev metric, where
\begin{equation}
\partial_x\partial_{y} \theta= (\pi M_g)^2=-R\sin \theta/2
\labeqn{SineGordon}
\end{equation} is the SG. As a generator of the SG \eqn{SineGordon} and
manifestation of integrability, the B\"acklund transformations
(BT) maps the SG into itself and enables to build new surfaces of
constant negative curvature from old \cite{AbS}. From a given
solution of the SG \cite{AbS, BCP} we can construct new solutions
by solving the ordinary differential equations for a family of
elementary BT $\theta\mapsto\widetilde{\theta} $
\begin{eqnarray}
(\partial_x\widetilde{\theta} + \partial_x\theta)/M =
2\pi M_g \sin[({\widetilde{\theta} - \theta} )/{2}],\nonumber\\
(\partial_{y} \widetilde{\theta}-\partial_{y} \theta)M=
{2 \pi M_g} \sin [({\widetilde{\theta} + \theta} )/{2}]
\labeqn{SGBT}.
\end{eqnarray} The second order equation \eqn{SineGordon} arises
as the integrability conditions of a pair of first order equations
\eqn{SGBT}, i.e. $\partial_{y} (\partial_x\widetilde{\theta})=
\partial_x(\partial_{y} \widetilde{\theta})$. Provided $\theta$ is a solution of
the SG, then $\widetilde{\theta} $ is also a solution. For
simplicity, $\widetilde{\theta} $ will serve as the special
reference field of constant phase given by the rather trivial case
$\widetilde{\theta} =4\pi({1\over 2} + n )$, with quantum gauge
(or spin) dependent winding number $n = 0,1,2, ...$ . This
provides for a simplification and the dimensional reduction
$\partial_x = M^2\partial_{y} $ in \eqn{SGBT} with
\begin{eqnarray}
\partial_{x} \theta /M = M\partial_{y} \theta = 2 \pi M_g \sin(\theta/2) \labeqn{1d}.
\end{eqnarray}
This form corresponds to travelling waves-like solutions with $\xi
= \pi M_g(ax + by)$, where the SG can be reduced to the ordinary
differential equation $ab\partial_{\xi}^2 \psi(\xi)= \sin
\psi(\xi)$, in our case $b=1/a=M$. The stationary SG soliton
solutions are expressed by elliptic functions, the generalized
pseudosphere solution follows immediately from integrating \eqn
{1d} $\psi(\xi) = 4 \arctan \exp(\xi/\sqrt{ab})$. With $\cos
\theta = 1-2\sin^2( \theta/2)$ and introducing $r^2 = x^2 + y^2 =
(1+1/M^4)x^2 =(M^4 + 1)y^2$ with $\partial_{r}^2 = \partial_{x}^2
+ \partial_{y}^2$ the potential is usually given by
\begin{eqnarray}
2 V({\theta}) = ( \partial_{x} \theta / M)^2=(M
\partial_{y} \theta)^2 \nonumber \\ = (M^2+1/M^2)(\partial_{r} \theta )^2
= 2\pi^2 M_g^{2} (1-\cos \theta)
\labeqn{pot01},
\end{eqnarray}
From \eqn{pot01} the self-energy term can be identified as a
constant $\theta$-independent Riemann curvature scalar $R
=-2/\rho^{2} $, with \eqn{SineGordon} $\pi M_g \rho = 1$.
Therefore, it is plausible to decompose energy in \eqn{pot01} into
at least two terms: a self-energy term $\pi^2 M_g^{2} $ and a
dynamic coupling term $\pi^2 M_g^{2} \cos \theta$ that accounts
for the field evolution based on the BT.
\paragraph*{Coupling space and phase coordinate by harmonic oscillation.}
Searching for external coupling and synchronization, \eqn{pot01}
allows to force global harmonic oscillations via potential
\begin{eqnarray}
V_o(r ) ={1\over2}
\({r \over \rho} \)^2 ={1\over2} \(\pi M_g{r } \)^2 =
-{1 \over 4} R{r}^2 \labeqn{pot02}.
\end{eqnarray}
Regarding \eqn{1d} and the square roots of \eqn{pot01} and \eqn
{pot02}, space and phase coordinate become directly coupled
\begin{eqnarray}
{r} = \mp \rho \sqrt{M^2 + 1/M^2} \partial_{r} \theta = \mp 2
\sin(\theta/2) \labeqn{pot05},
\end{eqnarray}
which provides for an additional dimensional reduction. The
coupling space and phase coordinate provides for a coupling
constant since integration of \eqn{pot05} provides for adynamic
coupling term $\mp \pi M_g \sqrt{M^2 + 1/M^2} \theta $ that can be
combined with a self-energy term and integration constant to
\begin{eqnarray}
V({\theta}) = V_o(y ) = \pi ^2 M_g^2 \mp \pi M_g \sqrt{M^2 +
1/M^2} \theta
\labeqn{pot04}.
\end{eqnarray}
Comparing the correspondent parts of self-energy and dynamic
coupling in \eqn{pot04} and \eqn{pot01}, we immediately obtain an
iterative equation of cooperative macroscopic phase shift driven
by stereographic feedback
\begin{eqnarray} \sqrt{M^2 + 1/M^2} \theta = \pm \pi M_g \cos{\theta}
\labeqn{iteration},
\end{eqnarray}
where the coupling allows for two possible signs.\\
\paragraph*{Coulomb potential from fractional linear transformations.}
% ------------------------------------------
\begin{figure}
\center\includegraphics[scale=0.4] {projection5.ps}
\caption{\small{Oscillation by precession on the sphere $S^2$ with
amplitude $r = 2|A-B|$ (blue) stereographically projected to the
Coulomb distance $r_c = |O-Q|$ (red) with $r = 2 \sin(\theta/2)$
and $r_c = \cot(\theta/2)$. The horizontal distance $d =
|A-Q|=|A-O|^2/|A-B|=\cos(\theta/2)r_c=(1 + \cos \theta )/r$. }}
\labfig{projection5.ps}
\end{figure}
% ------------------------------------------
The relation between $r$ and $\theta$ in \eqn{pot05} is a
stereographic projection with (pseudo)spherical angle $\theta/2$
onto the conformally-flat ($x,y$)-plane, see
fig.\fig{projection5.ps}. Defining Lobachevskian planes and
constructing a Lie--B\"acklund transformation which relates the
Liouville equation to the SG \cite{BCP}, these systems possess
nonlinear hidden symmetries providing for properties similar to
those of conventional oscillator and Coulomb systems. In the
previous work is was proposed, that the iterative solution
$\alpha(M)= \theta(M)/\pi$ could be interpreted as a generalized
spin-orbit or fine structure constant, since $\alpha$ enters in
\cite{BerryAB} as a Newton-type coupling constant of the conic
metric. Let where $r_c$ and $r$ denote the radial coordinates of
Coulomb and oscillator systems, respectively. Under stereographic
projection the conventional Bohlin transformation $r_c=r^2$ plus
inversion relates the harmonic oscillator potential \eqn{pot02} on
the (pseudo)sphere to the Coulomb system on pseudosphere, as well
as those interacting with specific external magnetic fields
\cite{NerPogo}. Since the group of the isometries of the
Lobachevsky and Kaehler metric coincides with $PSL(2,{\Bbb R})$,
it acts by projective fractional linear (or M\"obius)
transformations \cite{BCP} and allows to obtain the typical
electromagnetic field patterns. In our case, see
fig.\fig{projection5.ps} and \eqn{pot05}, the parameterization is
nothing but the stereographic projection of the two-dimensional
(pseudo)sphere
\begin{equation}
z= r_c{\rm e}^{i\varphi} =\left \{\begin{array}{cc}
\cot\frac{\theta}{2} {\rm e}^{i\varphi}
\;{\rm sphere}; \qquad\\
\coth\frac{\theta}{2}{\rm e}^{i\varphi} \;{\rm pseudosphere},
\end{array} \right.
\end{equation}
where $\theta,\varphi$ are the (pseudo)spherical coordinates
\cite{NerPogo}. The radial dependence of the Coulomb potential is
$V_c \propto 1/r_c$, the oscillator potential $V_o \propto r^2$.
In both, classical and quantum cases, the fractional linear
transformation and successive transformation $r_c=r^2$ converts
the $su(d)$ symmetry algebra of the oscillator to the $so(d+1)$
symmetry of the Coulomb system \cite{NerPogo}.
Fig.\fig{projcoup1.ps} extends the principle to
mutual interaction of geometric precession.\\
% ------------------------------------------
\begin{figure}
\center\includegraphics[scale=0.4] {projcoup1.ps}
\caption{\small{Precessional coupling by stereographic projection:
mutual geometric interaction of oscillators. }}
\labfig{projcoup1.ps}
\end{figure}
% ------------------------------------------
\paragraph*{The coupling strength.}
The coupling strength obtained with the radial distance on the
projective plan $r$ differs from the coupling strength obtained
from a pure one-dimensional definition \cite{BerryAB,alpha137top}
by a factor $\sqrt{1 + 1/M^4}$, for $M=137$ a relative reduction
in coupling strength of about $1.42 \cdot 10^{-9}$. The iteration
\eqn{iteration} is now invariant with respect to the inversion and
duality $M\leftrightarrow 1/M$. Inversion seems to be in our case
the central linear fractional transformation between local and
non--local holonomy relating the Coulomb and oscillator potential.
$M$-type inversion could also characterize the relations between
the electric and magnetic monopole charge $g e=1$ with $g/e=M^2$,
and also between group and phase velocity of a wave packet in the
ground state $v_g v_p = 1$ with $v_p/v_g = M^2$. The coupling
strength is balanced by the orbital degree of degeneracy $M$ or
$1/M$ of the precessional field, the B\"acklund parameter
introduced in \eqn{SGBT}. $M$ as an integral quantum number
describes the phase-locked and single-valued field \cite{BerryAB}
and provides for integrability. The coupling constant and special
$\theta$-value or oscillation range is iteratively obtained in
\eqn{iteration}, where $M=137$ or $M=1/137$ provides with $M_g=1$
for $1/\alpha=137.03600960$ that fits within some ppb's to the
Sommerfeld fine structure constant obtained in neutron
interferometry. The meaning of the number $137$ remains unclear.\\
\paragraph*{Conclusion.}
There is a clear geometrical interpretation: the coordinate vector
field is parallel transported along the signal/soliton vector
field with respect to the Levi--Civita connection. A "privileged"
surface ${\Bbb H} $ of scalar curvature $R=-2$ is given by the
Lobachevskian plane and Poincar\'e disks. The potential
\eqn{pot02} provides for a global harmonic precession balanced by
the topological phase shift $\theta(r)$, where the usual SG
coupled pendulum interpretation is extended to a macroscopically
coupled spin interpretation. The situation becomes stable and
self-consistent if the Coulomb feedback synchronizes to local
soliton oscillations (breather) that generate the Coulomb
potential by stereographic projection. Regarding the recent work
of \cite{GenietLeon}, the nonlinear mechanism behind
$V[{\theta}({r})]$ could have a strong relevance for self-induced
transparency and nonlinear supratransmission. \Eqn{iteration} is
an chaotic algorithm, bifurcation starts above a special values of
$M_g$. In \cite{GenietLeon} the bifurcation of energy transmission
is demonstrated numerically and experimentally on the chain of
coupled pendula (sine-Gordon and nonlinear Klein-Gordon
equations). Energy propagation in the forbidden band gap by means
of nonlinear modes requires a degree of macroscopic coherence
initiated i.e. by \eqn{pot02}. It appears, that both $\alpha(M)$
and SG-solitons could be simultaneously observed in Josephson
ladders \cite{ladder} in the context of supratransmission and
supraconductivity.
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\end{document}