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%keywords: abelian, local, non-abelian, non-local, non-pertubative, nonabelian, nonlocal, nonpertubative, quantum, computing, geometric, phase, berry, sine-Gordon, Aharonov, Bohm, Thirring, fine structure, memory, curvature, spacetime, iteration, iterative
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\begin{document}
Curvature for the Sake of Single-Valuedness
\title{Iterative Interplay between Aharonov-Bohm Deficit Angle and Berry Phase}
\author{B{\sc ernd}~B{\sc inder} }\affiliation{\footnote{\it
\href{mailto:binder@quanics.com}{binder@quanics.com}, {\it 88682
Salem, Germany \copyright 2002}}}

\date{14.9.2002}
%====================================================================
\begin{abstract}
Geometric phases can be observed by interference as preferred
scattering directions in the Aharonov-Bohm (AB) effect or as Berry
phase shifts leading to precession on cyclic paths. Without
curvature single-valuedness is lost in both case. It is shown how
the deficit angle of the AB conic metric and the geometric
precession cone vertex angle of the Berry phase can be adjusted to
restore single-valuedness. The resulting interplay between both
phases confirms the non--linear iterative system providing for
generalized fine structure constants obtained in the preliminary
work. Topological solitons of the scalar coupling field emerge as
localized, non-dispersive and non-singular solutions of the
(complex) sine-Gordon equation with a relation to the Thirring
coupling constant and non-linear optics.
\end{abstract}
%====================================================================
\pacs {PACS 02.10.De, 03.65.Bz, 03.65.Vf, 03.67.Lx, 05.45.Yv,
06.20.Jr, 12.20.-m, 31.30.Jv, 42.65.T }
%\small All pictures can be found in \cite{online1}
\maketitle

%#############################################################
\section*{Introduction}
%#############################################################

Local confinement requires proper phase relationships defined by
topology and geometry, i.e. to guarantee single-valuedness of the
wave-function. 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
orbit 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 $R^2$ with conic metric
provide for phase evolutions and deficit angles that can be
combined. This paper provides for an exact solution to the most
simple and central case, where the AB deficit angle exactly
compensates the Berry phase shift. The purpose of this paper is
not to outrage conventional wisdom concerning geometric phases,
rather to discover hidden connections on a purely topological
ground.

%#############################################################
\section*{Berry's Topological Phase}
%#############################################################

Two time- or lengthscales are natural requirements of geometric
phases, especially in the adiabatic Berry case: the long scale is
given by the path of a vector signal on a curved manifold, i.e.
the orbital coupling period, the short scale defines the vector
signal, i.e. it's spinning period. The non-adiabatic
generalization of \cite{ahanan} defines a geometric phase factor
for any cyclic evolution of a quantum system, where the geometric
phase is an eigenstate of the time evolution operator describing
the holonomy of a natural connection in the Hilbert space of
states. Consider a $T$-periodic cyclic vector $\ket{\psi(\tau)}$
that evolves on a closed path ${\cal C}$ according to
\begin{equation}
\ket{\psi(T)}= e^{i\varphi(T)}\;\ket{\psi(0)} \labeqn{sv02},
\end{equation}
where the total phase $\varphi(T)$ acquired by the cyclic vector
can naturally be decomposed into a geometric $\varphi_{g}(T)$ and
dynamical phase $\varphi_{d}(T)$
\begin{equation}
\varphi(T) = \varphi_{g}(T) + \varphi_{d}(T) \labeqn{sv03}.
\end{equation}
The dynamical phase for one loop $t\in [0;T]$ is with the
Schr\"odinger equation and hamilton operator $H$ given by
\begin{equation}
\varphi_{d}(T) = - {1 \over \hbar}\int\limits_0^T
\bra{{\psi}(\tau)} H(\tau) \ket{{\psi}(\tau)}d\tau
 \labeqn{sv05}.
\end{equation}
The Berry phase or geometric phase depends not on the explicit
time dependence of the trajectory and is for one loop given by
\begin{equation}
\varphi_{g}(T) = i \oint_{\cal C} \bra{{\psi}(\tau)} {d
}\ket{{{\psi}(\tau)}}  \labeqn{sv04}.
\end{equation}
The `parallel transported' spin vector will come back after every
loop with a directional change $\varphi_{g}(T)$ equal to the
curvature enclosed by the path $\cal C$. On the unit sphere
$S^2=SU(2)/U(1)$ the curvature increment is proportional to the
area increment that can be a spherical triangle with area given by
\begin{eqnarray}
  d\Omega := [1-\cos \theta_{B}(\tau)] d\varphi(\tau),
\end{eqnarray}
the area on the Bloch sphere enclosed by the closed evolution loop
of the eigenstate equal to
\begin{eqnarray}
 \Omega = \oint_{\cal C} d\Omega :=
 \int_0^T d\tau [1-\cos \theta_{B}(\tau)] \dot \varphi(\tau).
\end{eqnarray}
The Berry phase $\varphi_{g}(T) = J \Omega $ and the total phase
are proportional to spin $J$. In the standard case of conic
precession on the sphere
\begin{equation}
  \varphi_{g}(T) = 2\pi J (1-\cos\theta_{B}), \quad \varphi(T)= 2\pi J  \labeqn{sv08},
\end{equation}
where $\theta_{B}$ is the vertex cone semiangle, $\varphi_{d}(T)=
2\pi J \cos\theta_{B}$, see fig.\fig{ABconic1}. Introducing a
total phase evolution frequency $ \dot \varphi(\tau)=\omega$, the
phase evolution of the geometric phase is given by the angular
frequency
\begin{equation}
   \omega_{p} = J {d\Omega \over d t} = {d\varphi_{g}\over d t}
   = (1-\cos\theta_{B})\gamma \omega
   \labeqn{freq1},
\end{equation}
where $\gamma = d\tau/ d t$ is a relativistic factor. Usually,
$\omega_{p}$ can be assigned to precession, the difference
$\gamma\omega-\omega_{p}$ is given by the evolution of the
dynamical phase that will be defined by
\begin{equation}
   \omega_{d} = M\omega_{M} = \gamma \omega \cos\theta_{B}
   \labeqn{freq2},
\end{equation}
where $M$ is the quantum number of regular $M$-gonal symmetry that
divides all loops phases in $M$ equal sub-loop parts, i.e.
$\varphi_{d}(T) = M\triangle\varphi_{d}(T)$. With $n$ parameters
$\lambda_\mu(t)$, $\mu=1,2,...,n$ that span a closed curve $\cal
C$ in the $T$-periodic parameter space
$\lambda_\mu(0)=\lambda_\mu(T)$, the Berry phase may be
represented in terms of the `gauge potential' $A$ with connection
matrix
\begin{equation}
(A_{\mu})^{\alpha\beta}:= \langle\psi^\alpha(\lambda)|
\,{\partial}/{\partial\lambda^\mu}\,
|\psi_{}^\beta(\lambda)\rangle \label{conn}
\end{equation}
where $A =\sum_\mu A_{\mu}\,d\lambda_\mu,$ and
\begin{equation}
\varphi_{g}(T)  =  \oint_{\cal C} A  =  \int\limits_{\cal S_{\cal
C}} F , \quad F = dA \labeqn{berry-geometric}.
\end{equation}
$A$ is the non--abelian gauge potential that can be regarded as a
winding number density and allows for parallel transport of
vectors over $\cal S_{\cal C}$, an arbitrary surface in the
parameter space bounded by the contour $\cal C$. For more details
regarding monopoles and Wilson loops on the lattice in
non--abelian
gauge theories, see e.g. \cite{Zakharov}.\\
Generalized spin-orbit coupling strength can be defined as
\begin{equation}
 {\alpha(M) } ={J\triangle\varphi_{d}(T) \over \varphi(T)}= { J\omega_{M} \over \gamma\omega}
\labeqn{phase02d},
\end{equation}
where the coupling is proportional to the evolution of the
dynamical part $\triangle \varphi_{d}(T)/\varphi(T)$ and spin $J$.
With \eqn{freq2} in \eqn{phase02d} the dynamical part of
\eqn{freq1} provides for
\begin{equation}
{M \alpha }  = J \cos{\theta_{B}} \labeqn{spinorbit11}.
\end{equation}

% ------------------------------------------
\begin{figure}
\center\includegraphics[scale=0.6] {ABconic2.ps}
\caption{\small{The radial projection from the conic $z$-offset on
$S^2$ onto the polar plane for $M=7$. }} \labfig{ABconic1}
\end{figure}
% ------------------------------------------

%#############################################################
\section*{The AB topological phase}
%#############################################################

Nonlinear AB effects will arise if the pattern of the scattered
particle/wave couples back to the magnetic flux that generates the
AB phase, see fig.\fig{AB}. In this case the magnetic field is not
given externally, it is created by the orbital loop of the charge
in cylindrical symmetry of orbital spin and magnetic moment where
the geometry of the magnetic flux can be thought to be infinitely
long and straight thin with constant magnetic flux and mass
density. The magnet field is usually interpreted as the curvature
associated with the notion of parallel transport defined by the
vector potential. The Aharonov-Bohm phase
\begin{equation}
\theta_{AB}= {q \over \hbar} \oint_{\cal C_{AB}} {\bf A}_{AB}({\bf
r})\cdot d{\bf r}= 2\pi n {q\over e}{\Phi \over \Phi_0}
\end{equation}
collected by a charge $q$, moving in a closed path ${\cal C_{AB}}
$ about a line of magnetic flux $\Phi$, is purely topological. $n$
is the winding number of ${\cal C_{AB}} $ around the magnetic
flux, $e$ the elementary charge, and $\Phi_0$ the corresponding
flux unit $\Phi_0={2\pi\hbar /e}$. In contrast to the Berry phase,
the AB phase is independent of the shape of the path ${\cal
C_{AB}} $ and of the history of motion along it. In the case of
strong currents a flux tube model will have a non-vanishing mass
density and will introduce considerable topological coupling
effects in the motion of the scattered particles
\cite{Alvarezcone}.
% ------------------------------------------
\begin{figure}
\center\includegraphics[scale=0.7] {resonance5.ps}
\caption{\small{Aharonov-Bohm scattering of a wave on a cone with
radial resonance for $M=5$ and $Z=J=M_{g}=1$. The Berry phase is
shown by the short black arrow with length $\triangle
\varphi_{g}(T)$, the dynamical phase by the two green and longer
arrows with length $\triangle \varphi_{d}(T)$, the deficit angle
on the conic metric is $Z\pi\alpha$. In this picture the conic
AB-deficit angle and Berry's phase have been adjusted according to
\eqn{BerryAB} via iteration \eqn{spinorbit0}.}} \labfig{AB}
\end{figure}
% ------------------------------------------
The AB-scattering process is essentially two-dimensional. The
field of the string (here rather cosmic, later solitonic) can be
described in $2+1$ dimensions \cite{djt,dj}. As the third spatial
dimension can be taken to be parallel to the flux and will
decouple from the problem, the time-component of the metric does
not play any role. A convenient characterization of this conical
space is based on embedded coordinates \cite{dj}
\begin{eqnarray}
(dl)^{2}=(\alpha_{AB})^{2}(dr)^{2}+r^2(d\theta)^2, \nonumber \\
-\pi\leq\theta\leq\pi, \quad \alpha_{AB} = 1 - 4mG/(R_0 c^2)
\labeqn{conicmetric},
\end{eqnarray} where $G$ is a ``Newton-type" coupling constant,
$m$ the mass, and $R_0$ a characteristic orbital radius or horizon
(in the original work $1/\alpha_{AB} \rightarrow \alpha$). All of
the curvature of the cone is located at the tip of the cones but
is experienced indirectly by interference effects revealing the
deficit angle as a classical AB-scattering angle $\theta_{AB}$
\cite{Alvarezcone,djt,dj}, see fig.\fig{AB}:
\begin{eqnarray}
\theta_{AB\pm}=\pm \left({\alpha_{AB}\pi}-\pi\right)
\labeqn{ABscattering}.
\end{eqnarray}

%#############################################################
\section*{Coupling AB with Berry}
%#############################################################

Aharonov et al.\cite{AharonovBohmBerry} have investigated the
phase accumulated by a charged particle in an extended quantum
state and revealed by topological analysis a possible interplay
between the AB phase and Berry phase. In contrast to the
non--abelian Berry phase on a local path ${\cal C} $ the abelian
AB phase is independent of the shape of the path ${\cal C_{AB}}$,
it has a nonlocal character. This allows in the AB case, to adjust
a path ${\cal C_{AB}}$ that supports a special Berry phase
constellation forced by topology. The interplay between the Berry
phase and the AB effect can essentially be described as a reaction
of the curvature to the additional geometric phase and vice versa.
One could argue, that the AB effect is due to the non triviality
of the holonomy (Dirac phase) even though the curvature of the
connection $F_{\mu \nu}$ vanishes. The problem with this reasoning
is that the wave function is not single-valued
\cite{CorichiPierri}. To restore single-valuedness and to generate
a closed string state, curvature could generate a deficit angle
that exactly compensates the Berry phase in a non--linear
iterative process. In other words: in contrast to the flat metric
the conic metric provides for single-valuedness, where the conic
opening angle $\theta_{AB}$ is directly related to the Berry phase
leading to the precession angle $\theta_{B}$. The coupling is
driven by the spatial extension of the subloop pattern. Regarding
the scattering geometry shown in figs.\fig{AB}, those arguments
can directly be
transformed into a geometrical resonance condition as follows:\\
{\it
%------------------------------------------------------------
The cone vertex angle of Berry precession multiplied by spin
equals the absolute value of the cone deficit angle of
AB-scattering on the conic 2+1 dimensional metric in polygonal
axial symmetry, where}
%------------------------------------------------------------
\begin{equation}
\theta_{AB\pm} = \mp J \theta_{B} , \quad \alpha_{AB} = 1 -
 Z\alpha
\labeqn{BerryAB}.
\end{equation}
The difference between the two couplings for the same regular
$M$-gonal symmetry is obviously given by the angle $\pi$ and a
geometric coupling factor $M_{g} = Z/J$. Now, \eqn{BerryAB}
provides with \eqn{conicmetric} for the Newton-type coupling
\begin{equation}
Z\alpha = {4 m G \over R_0 c^2} =\mp {J \theta_{B} \over \pi },
\quad Z = JM_{g} \labeqn{coupling}.
\end{equation}
\eqn{coupling} reveals the role of $Z$ as a charge unit that is
integral with integral $M_{g}$ and $J$. The factor
$\cos\theta_{B}$ provides for the radial projection from the
z-offset on $S^2$ onto the polar plane, see
fig.\fig{ABconic1}. \\

Proceeding with \eqn{BerryAB} and \eqn{ABscattering} in
\eqn{spinorbit11} provides for the conic iteration
\begin{eqnarray}
\mp\theta_{B,i+1} = \frac{M_{g}}{M}\pi\cos\theta_{B,i}
 \labeqn{spinorbit0},
\end{eqnarray} where the sign of the conic deficit angle $\theta_{AB\pm}$
of \eqn{ABscattering} related to \eqn{sv08} can realize a
bi-stable configuration that can be characterized by two states:
\begin{eqnarray}
 \theta_{AB+}: \frac{M_{g}}{M}>0, \quad \theta_{AB-}: \frac{M_{g}}{M}<0.
\end{eqnarray}
This non--linear iteration converges for integral $|M/M_{g}|>2.4$
outside the chaotic regime after a few steps to a special fixed
point $\theta_{B\pm}$ \cite{alpha137MN,alpha137top,
alphacomp,online1}. In any case, it provides for a nice and fast
converging non--linear iterative system where the scattered wave
packet couples back to the scattering conic singularity in
resonance with the orbital Berry phase evolution. The iterative
solution provides for a AB deficit angle that compensates the
Berry phase, where the conic metric provides for single-valuedness
despite of the phase shift.

%#############################################################
\section*{Solitons and monopoles}
%#############################################################

The relations for the phase shift and scattering angle are derived
under fairly general conditions based on fundamental topological
principles. Topological solitons are spatially confined
(localized), non-dispersive and non-singular solution of a
non--linear field theory. In 2+1-dimensional gauge vortex
scattering it follows from purely geometric considerations that
the head-on scattering of $M$ topological solitons (like
monopoles, vortices, skyrmions, ...) distributed symmetrically
around the point of scattering (relative angular separations
$2\pi/M$) is by an angle $\pi/M$, independent of various details
of the scattering \cite{MacKenzie}. In this case the initial
configuration has the symmetry group of a regular $M$-gon, the
``moduli space'' of $M$ vortices, ${\cal M}_M$\cite{moduli}.\\
%With
%metric \eqn{conicmetric} the motion of vortices at low energy in
%the Bogomol'nyi limit is geodesic with metric determined by the
%original field theory action, applied to configurations in ${\cal
%M}_M$.%
The Berry and AB phases are intimately related to Dirac magnetic
monopoles and arise naturally in the field of a monopole
\cite{berry}, \cite{Dirac-31,tH81} with $M$-gonal scattering
symmetry where $M = 137$ ('t Hooft-Polyakov and Dirac monopoles
will be discussed elsewhere). In this case the Newton-type
coupling constant of \eqn{coupling} is given by $\alpha \approx
1/137.03600941164$ and fits within error range to a neutral and
theory independent determination of the Sommerfeld fine structure
constant, see the Java simulation \cite{online1}. This provides
for evidence that $\alpha$ is indeed a topological spin-orbit
coupling constant. The origin of special $M$-values is up to this
stage unclear but could be found in non--linear topological
quantum field theory. A non--linear non--pertubative quantum field
theory seems to be generally relevant for information processing,
storage, and transport \cite{alphacomp}.

%#############################################################
\section*{A scalar field for the deficit angle}
%#############################################################

The conic metric \eqn{conicmetric} corresponds to a
2+1-dimensional closed-loop a priori open in the $z$-direction.
Recently, \cite{camacho1} discussed coupling gravitomagnetism-spin
and Berry's phase and pointed out, that the geometric phase
changes should depend exclusively upon the solid angle of a field,
and not on the strength of the field. In this context it should be
possible to decouple angular properties from the coupling field
strength and localize it on the $z$-axis with a confinement
potential. For this purpose, there are candidates for soliton-type
solutions based on a non--pertubative non--linear topological
quantum field theory of a scalar field. In this case the field
must describe the coupling or charge density $\phi(z)$, i.e. via
$\theta_{B} (z) = \mp \alpha \phi(z)$ as a travelling geometric
precession (or local conic deficit angle) moving with radial
symmetry in $z$-direction according to \eqn{coupling} with $M_{g}
= \phi(0)$. In 1+1--dimensional space--time there are two
non--trivial minimal quantum field theories which describe
non--perturbative phenomena: the sine--Gordon (SG) model
\cite{Remoissenet} and the massive Thirring model \cite{Thirring}
(a self-coupled Dirac field, see the Lagrangians
\cite{Remoissenet}), both are intimately related \cite{Coleman}.
These types emerge from the most simple relation between field
strength and potential, where the potential is given by the square
of the $z$ gradient of $\phi(z)$. Based on the relative precession
frequency and energy \eqn{freq1} at $z=z_0$ the potential $V(z)$
of a relativistic SG kink is given by
\begin{eqnarray}
\hbar\omega_{p} \rightarrow V(z) = {1\over 2}\({\partial \phi
\over
\partial z}\)^2 = {k_z^2 \over \alpha^2} \[1 - \cos (\alpha
\phi)\] \labeqn{Sine-Gordon01},
\end{eqnarray}\\
where $k_z$ is a characteristic wave number and $E = {8 k_z /
\alpha^2}$ the self-energy. The equation of motion is given by the
SG equation
\begin{eqnarray}
  \Box \phi + {k_z^2\over \alpha} \sin (\alpha \phi) = 0
\labeqn{Sine-Gordon02}.
\end{eqnarray}
Boosting the static solution gives the indestructible moving
solution
\begin{eqnarray}
  \phi_{\pm}(z) = \pm 4 \arctan\( \exp\[{\gamma(z-z_0-vt) \over \sqrt{\alpha}}\]\),\nonumber\\
  \quad \gamma={1\over\sqrt{1-v^2}}
\labeqn{Sine-Gordon03}.
\end{eqnarray}
For the SG system colliding solitons preserve their $(z)$-shape, a
desirable property regarding the stability of $\alpha$ and the
iteration. For $\alpha \rightarrow 0$ or $M \rightarrow \infty$
the relativistic 1-d wave equation is recovered. Solitons can
annihilate with anti--solitons, many--soliton solutions obey
Pauli's exclusion principle. As pointed out by Skyrme
\cite{Skyrme} this can be interpreted as a fermion--like behavior.
Regarding this work, there is a relation between the real Thirring
coupling parameter $g>0$ to the SG parameter $\alpha$
\cite{Coleman} given by
\begin{eqnarray}
\frac{4\pi}{\alpha^2} = 1 + \frac{g}{\pi} \labeqn{Thirringcoupl}.
\end{eqnarray}
Reduced to the basic mechanism it is oversimplified regarding
electrodynamics. But it is possible to generalize to non--abelian
group structures and their physical properties hidden in nonlinear
interactions to a group theoretic effective field theory which
generalizes the sine-Gordon theory to the non--abelian cases on a
coset $SU(N)/U(N-1)$ in terms of a deformed gauged
Wess-Zumino-Witten action \cite{ParkShin}. The $SU(2)/U(1)$ case
can be identified with the so-called complex SG theory that is
applied to the phenomenon of anomalously low energy loss in
coherent optical pulse propagation. In this case of nonlinear
optics there are coupled nonlinear partial differential equations
in 1+1-dimensional spacetime given by the Maxwell and Bloch
equations with two complex fields and one real field. The optical
Bloch equation can be used to describe dipole spin precession
\cite{bloch}. The chain of reasoning can be closed by observing,
that the Maxwell equation determines the strength of the torque
vector along the $z$-axis. This agrees not only with the
conventional mechanical interpretation of the SG equation as a
continuum limit of the infinite chain of coupled pendulum
equations, it allows also to interpret the SG field or pendulum
amplitude as an precession angle, in our case
$\theta_{B}=\mp\alpha\phi=\mp \theta_{AB}/J$.

%#############################################################
\section*{Conclusion}
%#############################################################

Orbital ($x,y$)-stability is guaranteed by the single-valuedness
requirement with $\alpha$-iteration that could be assigned to a
regular $M$-gonal coupled soliton system. The strength of the
field is given by deficit angle induced by the closed-loop around
the $z$-axis, a rotating target surely "sees" a deficit modified
by its relative angular velocity with two possible signs. The
deficit strength is clearly quantized with $\phi$ mod $2\pi$, so
it seems that electromagnetic properties could be traced down to
topological phases. The AB effect can now be understood as a
path-dependent deficit angle that can be found by passing on two
different sides of a ``deficit flux'', the Berry phase is the
deficit induced by parallel transport. Both deficits can be
combined, the interplay is nonlinear and can restore
single-valuedness. Topological solitons of the scalar coupling
field emerge as localized, non-dispersive and non-singular
solutions of the sine-Gordon equation with a relation to the
Thirring coupling constant. SG solitons could serve as carriers of
gauge dependent conic ``deficit energy'' originated by topological
phases.

%............................................................#
\begin{thebibliography}{40}

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(1984).

\bibitem{AharonovBohmBerry}Y. Aharonov, S. Coleman, Goldhaber, S. Nussinov, S. Popescu, B. Reznik, D. Rohrlich, and L. Vaidman, Phys. Rev. Lett. {\bf 73}, 918 (1994).

\bibitem{ahanan} Y. Aharonov, J. Anandan, Phys. Rev. Lett. {\bf 58}, 1593 (1987).

\bibitem{Zakharov} F.V. Gubarev, V.I. Zakharov, Int. J. Mod.
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\bibitem{Alvarezcone} M. Alvarez, "Aharonov-Bohm scattering on a cone";
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\bibitem{djt} S.~Deser, R.~Jackiw, G.~'t Hooft, Ann. Phys. {\bf 152}, 220
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\bibitem{CorichiPierri} A. Corichi, M. Pierri, Phys.Rev. {\bf D51}, 5870 (1995);
\\\href{http://xxx.lanl.gov/abs/gr-qc/9412006}{gr-qc/9412006}.%

\bibitem{alpha137MN} B. Binder, `Berry's Phase and Fine Structure' (2002);
\\\href{http://philsci-archive.pitt.edu/documents/disk0/00/00/06/82/index.html}{PITT-PHIL-SCI00000682};
\href{http://www.quanics.com/alfa137MN6.pdf}{plus minor
corrections}.

\bibitem{alpha137top} B. Binder, `Geometric Phase Locked in Fine Structure'
(2002);
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\end{document}
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