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\begin{document}
\begin{center} {\Large\bf Berry's Phase and Fine Structure}\\
\vspace{.4in} {B{\sc ernd}~B{\sc inder}\footnote{\it email:
\href{mailto:binder@quanics.com} {binder@quanics.com}, {\it
Weildorferstr.22, 88682 Salem-Neufrach, Germany \copyright 2002}}}
\\{\tiny Date: 25.7.2002}\\
\end{center}
%====================================================================
\begin{abstract}\noindent
Irrational numbers can be assigned to physical entities based on
iterative processes of geometric objects. It is likely that
iterative round trips of vector signals include a geometric phase
component. If so, this component will couple back to the round
trip frequency or path length generating an non-linear feedback
loop (i.e. induced by precession). In this paper such a quantum
feedback mechanism is defined including generalized fine structure
constants in accordance with the fundamental gravitomagnetic
relation of spin-orbit coupling. Supported by measurements, the
general relativistic and topological background allows to propose,
that the deviation of the fine structure constant from 1/137 could
be assigned to Berry's phase. The interpretation is
straightforward: spacetime curvature effects can be greatly
amplified by non-linear phase-locked feedback-loops adjusted to
single-valued phase relationships in the quantum regime.
\end{abstract}
%====================================================================
\tiny PACS 03.65.Bz, 03.65.Vf, 06.20.Jr, 12.20.-m, 31.30.Jv \\
\small All pictures can be found in \cite{online1} \normalsize
\subsection*{Introduction}
A quantum mechanics of spin can not be complete without
considering the phase evolution of a wave function including
interference phenomena and geometric spin precession. Berry's
phase \cite{berry} can appear in purely classical situations such
as round trip excursions on curved surfaces. Since spatial phases
appear in any kind of wave propagation, different manifestations
of this extremely general phenomenon have been found in several
branches of physics from the high energy regime to the low. In
addition to a Hamiltonian-induced dynamic phase, a quantum state
evolving in parameter space on a trajectory that returns to the
initial state acquires an extra phase termed geometric phase. This
additional phase or angle depends only on the geometry of the
Hamiltonian's trajectory through parameter space and not on its
time evolution. Various manifestations of geometric phases exist
and are connected with names like Aharonov, Anandan, Berry, Bohm,
Pancharatnam, Simon, Thomas, Wilczek and many others, see e.g.
\cite{Batterman}. Although there are no widely recognized
practical applications of the nonabelian gauge theory, its
experimental observations have been reported in many fields of
science. But quantum electrodynamics (QED) was established long
before Berry's phase was discovered. The successful concept of QED
is perturbative and based on powers of the coupling constant
$\alpha$. QED handles (hyper)fine structure, Lamb shift, an spin
anomalies at the most accurate level while ignoring Berry's phase.
A quick search for Berry in this context over the last 10 years in
a physics archive returns almost now hits. The situation is
probably unbalanced regarding the connections of Dirac's theory
and Berry's phase. Dirac's equation and Dirac's theory of
monopoles \cite{Dirac-31} are very important for the foundations
of QED especially in the atomic range. A magnetic monopole as a
logical consequence of the Dirac theory is necessary to quantize
charge. Berry's phase is intimately related to Dirac magnetic
monopoles and arises naturally in the field of a monopole
\cite{berry}. But magnetic monopoles have a similar status like
Berry's phase: an abstract geometrical and topological feature,
where the topological structure of this abstract manifold is under
special circumstances observable and physical. But no monopole has
been found, at least been identified. This fact and the rather
exotic status seems to imply for many scientists that Berry's
phase could have no big impact on the foundations of quantum
mechanics manifested, i.e. in atomic spectra.\\
But arguments that support the role of geometric phases in atomic
physics can be found regarding the fine and hyperfine structure,
since transitions in this range (GHz, atomic clocks) depend on
exact frequency and phase relationships. These clocks usually
depend on phase-locked loops, where phase factors or phases
representing the `holonomy' provide for important boundary
conditions while reducing the degree of redundancy in variables.
Usually, controlled by external fields, the geometric phase has a
passive role, but in a phase-locking process the geometric phase
could also couple back to the dynamic phase. Consequently, it is
very interesting to consider round trips of vector signals
additionally constrained by an emerging geometric phases.
\subsection*{Generalized Berry phase}
The non-adiabatic generalization of \cite{ahanan} defines a
geometric phase factor for any cyclic evolution of a quantum
system (experimentally verified e.g. in \cite{suter88}). 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 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 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(\tau)] d\varphi(\tau),
\end{eqnarray}
the total area enclosed by the closed orbit (loop) is equal to
\begin{eqnarray}
\Omega = \oint_{\cal C} d\Omega :=
\int_0^T d\tau [1-\cos \theta(\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 precession
on the sphere
\begin{equation}
\varphi_{g}(T) = 2\pi J (1-\cos\theta), \quad \varphi(T)= 2\pi J
\labeqn{sv08},
\end{equation}
where $\theta$ is the vertex cone semiangle, $\varphi(T)= 2\pi J$,
$\varphi_{d}(T)= 2\pi J \cos\theta$. With $n$ parameters
$\lambda_i(t)$, $i=1,2,...,n$ that span a closed curve $\cal C$ in
the $T$-periodic parameter space $\lambda_i(0)=\lambda_i(T)$, the
Berry phase may be represented in terms of the `gauge potential'
\begin{equation} A_i = i\;\bra{\psi} {\diff \over \diff\lambda_i}
\ket{\psi} \labeqn{berry-connection},
\end{equation}
\begin{equation}
\varphi_{g}(T) = \oint_{\cal C} A = \int\limits_{\cal S_{\cal
C}} F , \quad F = dA
\labeqn{berry-geometrical},
\end{equation} where $A$ can be regarded as a winding number density, $\cal
S_{\cal C}$ is 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}.
Note if the Berry phase contains a string-like singularity (Dirac
string) somewhere on the surface of $\cal S$, the position of the
singularity may be arbitrary shifted by U(1) gauge
transformations. In the first step we will introduce a simple
feedback relation of the geometric phase to the dynamical phase.
\subsection*{Spin-orbit feedback loop}
The mostly important question is, what balances both parts of the
total phase, what is the phase boundary condition? It is likely
that iterative round trips of vector signals include a geometric
phase component. If so, this component will couple back to the
round trip frequency or path length generating an non-linear
feedback loop (i.e. induced by precession). A `rolling cone'
representing a vector state or signal is probably the simplest
model of spin-orbit coupling. Rotated once, the cone will change
its orbital orientation by a special angle, rotated $M$-times in
the quantum case, the cone will return to the initial position
with integral $M$ (providing for single-valuedness). But
precession will change the path of the rolling cone such, that the
number of conic sub-loops and the effective orbital frequency and
radius are changed. This is a non-linear feedback situation, since
a change in the number of loops couples back to the geometric and
dynamical phase. \\
Starting with this model, $M$ can divide the total phase range
into $M$ sub-loop intervals
$\triangle\varphi(T)=\triangle\varphi_{d}(T)+\triangle\varphi_{d}(T)$
where
\begin{equation}
\varphi_{d}(T) = M \triangle\varphi_{d}(T) ,\quad \varphi_{g}(T) =
M\triangle\varphi_{g}(T)
\labeqn{sv06}.
\end{equation}
Precession can be interpreted as a phase modulation of the orbital
path length and could couple to the number of sub-loops by
modulating the `rolling cone path'. It is very likely, that one or
more extra loops with range $\triangle \varphi_{d}(T)$ fit by an
integral quantum number $M_{g}$ within one vertex cone range
$2\theta$ providing for the feedback relation
\begin{equation} 2\theta={M_{g} \triangle
\varphi_{d}(T)}
\labeqn{sv011}.
\end{equation}
This is an orbital resonance condition regarding the dynamical
phase and the precession phase, in the next sections we will
analyze also a radial resonance of the geometric phase with the
precession phase, see fig.\fig{alpha9}. Now it is possible to find
with \eqn{sv08}, \eqn{sv06}, and \eqn{sv011} the optimum $\theta$
for a given $M$ and $J$, where
\begin{eqnarray}
M \theta = J M_{g} {\pi \cos\theta}
\labeqn{spinorbit10}.
\end{eqnarray}
As a test for $J=M_{g}=1$ and $M>0$, \eqn{spinorbit10} can be
solved by iteration
\begin{eqnarray}
\theta_{i+1} = \frac{\pi \cos\theta_{i}}{M}
\labeqn{spinorbit0}.
\end{eqnarray} After a few steps the algorithm converges (no problem for $J
M_{g} \ll M$).
Since $M_{g}$ contributes to the total number of sub-loops units
one could write $M = M_{d} + M_{g} = \pm 1, \pm 2, ...$. Now $M$
contains two parts: $M_{d}$ counts the sub-loops that contribute
to the dynamical phase for one circular loop over the phase range
$2\pi$, while $M_{g} =\pm 1, \pm 2, ...$ counts the extra number
of sub-loops induced by precession at vertex semiangle $\theta>0$.
Consequently, $M_{g}$ stands for the geometric contribution or
loop/sub-loop interaction. Requiring single-valuedness all three
numbers will be integer.\\
The criterion for the convergence of the nonlinear affine
iterative system or the asymptotic stability of the fixed point
will be visualized in the next sections showing bifurcations and
unstable regimes.
% ------------------------------------------
\begin{figure} [h]
%\center\includegraphics[scale=0.45] {pictures/resonances2.eps}
\caption{\small{On the left the orbital resonance of dynamical
phase and precession angle (for $M=5$ and $M_{g}=J=1$), in the
middle the effective frequency change, and on the right the radial
resonance of geometric phase and precession angle (quantum number
$N$).}} \labfig{alpha9}
\end{figure}
% ------------------------------------------
\subsection*{Generalized fine structure constants}
The coupling must be proportional to the dynamical part of the
phase interval $\triangle \varphi_{d}(T)/\varphi(T)$ and to spin
$J$. Therefore, generalized fine structure constants can be
defined by
\begin{equation}
\alpha(M) = { J \triangle\varphi_{d}(T) \over \varphi(T)} = { J
\varphi_{d}(T) \over M \varphi(T)} \labeqn{sv07}.
\end{equation} With \eqn{sv06} and \eqn{sv07}, in \eqn{sv03}
\begin{equation}
{\varphi_{g}(T) \over 2\pi J }
= 1-{M \alpha \over J} \labeqn{sv091},
\end{equation}
the dynamical part of \eqn{sv091} with \eqn{sv07} in \eqn{sv08}
provides for
\begin{equation}
{M \alpha } = J\cos \({\theta}\) \labeqn{spinorbit11}.
\end{equation}
Comparing \eqn{spinorbit10} with \eqn{spinorbit11} the precession
cone vertex angle $2{\theta}$ of \eqn{sv08} equals the dynamical
phase of the spin-orbit interaction part in \eqn{sv06} with
\begin{equation}
{\theta} = \pi M_{g} \alpha
\labeqn{sv07b}.
\end{equation}
The two possible signs can be combined to $M/M_{g}>0$
\begin{equation}
{M \alpha } = J\cos \({\pi M_{g}\alpha }\) \labeqn{phase07},
\end{equation}
and for $M/M_{g}<0$
\begin{equation}
{M \alpha } = J\cos \({\pi - \pi M_{g}\alpha }\) \labeqn{phase08}.
\end{equation}
Results for $\alpha$ with variable $M$ for $M_{g}=J=1$ are shown
in table 1 and visualized in fig.\fig{alpha9} and \cite{online1}.
For a geometric phase contribution $\triangle\varphi_{g}(T)>0$ one
can expect $\alpha2$. The third row shows $N = |{\triangle\varphi_{d}(T) /
\triangle\varphi_{g}(T)}|$ or $|N_\pm|$ (bottom), known as winding
number on helical paths.}
\end{caption}\\
\begin{tabular}{p{40pt}p{120pt}p{120pt}}\\
\hline \hline \\
\textit{$M$}& \textit{$J/\alpha$}& \textit{$N$}\\
\hline & \\
3 & 4.13669 & 2.63924 \\
4 & 4.96178 & 4.15896 \\
5 & 5.82662 & 6.04873 \\
6 & 6.72097 & 8.32214 \\
7 & 7.6371 & 10.98727 \\
% 8 & 8.56944 & 14.0489 \\
% 9 & 9.51399 & 17.50997 \\
% 10 & 10.46789 & 21.37233 \\
% 11 & 11.42906 & 25.63716 \\
% 12 & 12.39597 & 30.30525 \\
% 13 & 13.36747 & 35.37714 \\
137 & \textbf{\small{137.03600941164}} & 3804.560912 \\ \hline
137 & \textbf{\small{137.03600998817}} & 3804.5\footnotemark[1] \\
137 & \textbf{\small{137.03600052556}} & 3805.5\footnotemark[1] \\
137 & \textbf{\small{137.03599106791}} & 3806.5\footnotemark[1] \\
137 & 137.03598161523 & 3807.5\footnotemark[1] \\
\hline \hline\\ \labtab{fsc1}
\end{tabular}\\
\small{\footnotemark[1] {The next hypocycloidal or epicycloidal
resonances for $M=137$ (see fig.\fig{alpha3}, \eqn{spinorbitN})
instead of free running ratio $N$ between dynamical and geometric
phase.}}
\end{table}
% ----------------------------------------------
\subsection*{Gravitomagnetic coupling}
The bridge to general relativity has already been built in the
previous sections by calculating the geometric phase via `parallel
transport' on a curved surface. Let $\omega_{M}$ be the orbital
frequency of a quantum particle on a circular path. The
spin-rotation coupling will act on the particle with mass-energy
$E$ and Compton wavenumber $k=E/\hbar c=\omega / c$ by generating
precession at frequency $\omega_{p}=E_{p}/\hbar$ and energy
$E_{p}$. The fine structure constants \eqn{sv07} and relative
dynamical coupling strengths can be defined by
\begin{equation}
{\alpha } = {J\omega_{M} \over \omega}
\labeqn{phase02d},
\end{equation}
a typical ratio orbital frequency divided by a Compton frequency
with particle spin $J$. If precession as a frequency ratio is
related to a geometric phase
\begin{equation}
{\omega_{p} \over \gamma \omega} = {\varphi_{g}(T) \over 2\pi J }
\labeqn{phase02g},
\end{equation}
and the coupling part to a dynamical phase according to
\eqn{sv091} and \eqn{phase02d}, the general expression for
spin-rotation coupling observed in the laboratory frame
(relativistic correction $\gamma$) can be assumed to
be\begin{equation}
{\omega_{p} \over \gamma \omega} = 1 - M {\omega_{M} \over \omega}
\labeqn{phase02},
\end{equation}
or
\begin{equation}
E_{p}=\gamma (E-M \hbar \omega_{M}) \labeqn{phase01},
\end{equation}
a form that corresponds exactly to the gravitomagnetic
spin-rotation coupling of Mashhoon \cite{mashhoon2}, a Lense -
Thirring effect \cite{Wheelergm} (also in
\cite{mashhoon0,mashhoon1}), where an integer $M$ covers scalar
and vector fields. Usually, the gravitomagnetic effect can be
hardly observed because of its tiny magnitude (tests with orbiting
gyroscopes are on the way, see gravity probe B news \cite{GPB}).
But the tiny magnitude of the gravitomagnetic field in a classical
measurement does not necessarily mean, that the magnitude of the
emerging geometric phase and related quantum mass--energy currents
in feedback loops must be tiny. 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. If gravitomagnetic spin-rotation coupling
\eqn{phase01} controls the feedback loop in combination with
\eqn{phase07}, the mass-energy current could increase to a level
that is only limited by damping and (multipole) radiation
effects, probably a level characterized by electromagnetism.\\
% ------------------------------------------
\begin{figure} [h]
%\center\includegraphics[scale=0.6] {pictures/alpha3pm1.eps}
\caption{\small{The geometric and dynamic phase evolution at
resonance: on the left hypocycloids $N_{+}=4.5$ ($N=4$), in the
middle epicycloids with $N_{-}=-3.5$ ($N=4$). On the right both
are adjusted with $N_+=4.5$ ($N=4$) and $N_-=-4.5$ ($N=5$), but in
this case epicycloids and hypocycloids have different radii. }}
\labfig{alpha3}
\end{figure}
% ------------------------------------------
In this context there are some noteworthy comments from Mashhoon:\\
The coupling of intrinsic spin with rotation reveals the
rotational inertia of intrinsic spin. The phase perturbation
arising from spin-rotation coupling can be developed as a natural
extension of the celebrated Sagnac effect \cite{mashhoon0}.
Spin-rotation coupling, however, violates the underlying
assumption of locality in special relativity: that the results of
any measurement performed by an accelerating observer (in this
case the measurement of frequency) are locally equivalent to those
of a momentarily comoving inertial observer, but agrees with an
extended form of the locality hypothesis. This is a nontrivial
axiom since there exist definite acceleration scales of time and
length that are associated with an accelerated observer
\cite{mashhoon0}.
Note, that orbital precession of the geometrical phase provides
for a change in the frequency ratio
\begin{equation}
{\omega \over M \omega_{M}}={1 \over \cos\theta}=
{\varphi_{d}(T)+\varphi_{g}(T) \over \varphi_{d}(T)}
\labeqn{sv010}.
\end{equation}
Because of the additional geometric phase $\varphi_{g}(T)$ leading
to precession, the Compton frequency $\omega$ is a little more
than $M=137$ times the orbital Bohr frequency $\omega_{M}$ of the
electron. In the rolling cone picture, the `rolling' Compton wave
number or cone base radius rolling at distance $R$ is given by
$\lambda=R'J/M=R J\omega_{M} / \omega$.
\subsection*{Hypocycloidal and epicycloidal dynamics}
Additionally to the coupling of dynamical phase and precession a
radial resonance of geometric phase and precession angle could be
induced, see fig.\fig{alpha9}. These resonances can be projected
to a planetary gear model in plane showing hypocycloidal and
epicycloidal dynamics with equivalent phase evolution, see
fig.\fig{alpha3}. A counter-rotating hypocycloidal geometric phase
($+$) generates more dynamical sub-loops on the total loop than in
the epicycloidal case ($-$). In the most simple case the total
phase rotates by $\pm 4\pi N$ while the geometric phase rotates by
$2\pi$. Subtracting Berry's phase from the total phase of one
closed loop gives $|N_\pm| = N \pm {1 \over 2}$, the total
dynamical phase in the hypocycloidal and epicycloidal paths shows
$N_{+}>0$ and $N_{-}< 0$ sub-loops, respectively. The
correspondent quantum number $N_{\pm}$ includes two cases labelled
by the $\pm$ sign of the geometric phase evolution with respect to
the dynamical phase evolution with
\begin{equation}
- N_{\pm} = M {\gamma \omega_{M} \over
\omega_{p} } = {\varphi_{d}(T) \over \varphi_{g\pm}(T)}
= {\varphi(T) \over \varphi_{g\pm}(T)} + 1
\labeqn{epihypo01}.
\end{equation}
The sign convention is adjusted to the sign of the charge and to
$|N_{+}|=|N_{-}|+1$ for a given ratio of loop/sup-loop radius, by
definition, the sign of $N_{\pm}$ is opposite to the sign of $M$
in \eqn{sv06}. Half integral values of the quantum number
$|N_\pm|$ in \eqn{epihypo01} indicate an interference of the
precession frequency with both, the orbital rotation and particle
Compton frequency. $-{\varphi(T) / \varphi_{g}(T)} = N_{\pm}-1$ is
known as the winding number on helical photon paths, note that
according to \eqn{berry-geometrical} Berry's potential $A$ is a
winding number density. The number characterizes both, a standing
wave in radial and orbital dimension. The ratio dynamical to total
phase evolution is given by
\begin{eqnarray}
\cos(\theta) = {2N_\pm \mp 1 \over 2N_\pm \pm 1}
\labeqn{spinorbitN},
\end{eqnarray}
where $N_0=|N_\pm| \mp {1 \over 2} > 0$ is in the case of
hypocycloidal and epicycloidal phase-locking the next higher
available integral number with respect to the free running
irrational number $N$ based on $M$ and $M_{g}$ where
$-|N_\pm|>N_0$, see table 1. The dynamical coupling strength for
any $N$ provides for the ratio
\begin{eqnarray}
\left|{\varphi_{d_+}(T) \over \varphi_{d_-}(T)}\right| = {2N + 1
\over 2N - 1}= {r_a + r_b \over r_a - r_b} \labeqn{spinorbitC},
\end{eqnarray}
where the circle of radius $r_b$ rolls on or in the circle of
radius $r_a$. Now there are two principle integral parameters $M$
and $N$ providing for single-valuedness on the closed loop, where
$N$ is a consequence of $M$ in the case of
hypocycloid/epicycloidal resonances.\\
% ------------------------------------------
\begin{figure} [h]
%\center\includegraphics[scale=0.7] {pictures/lambtransition3.eps}
\caption{\small{The $N_\pm$ hypocycloidal or epicycloidal levels
for the relevant $N$-values in the case $M=137$ where
$N_0=3806$.}} \labfig{alpha6}
\end{figure}
% ------------------------------------------
\subsection*{Charge}
The modern interpretation of the fine-structure constant defines
$\alpha$ as the coupling constant for the electromagnetic force.
Solving \eqn{phase07} or \eqn{phase08} by iteration provides for
the balance of dynamical and geometric phase given by $\alpha$
subject to a given number of sub-loops $M$, coupling loops
$M_{g}$, sub-loop spin $J$, and eventually $N$ characterizing a
hypocycloidal or epicycloidal resonance. The coupling is polar
since a positive and negative $M/M_{g}$ corresponds to the
repulsive and attractive case, respectively, in the negative case
the coupling phase interval and precession of \eqn{sv07b} is
negative with respect to the total phase in \eqn{sv06}.
Consequently, the hypocycloidal and epicycloidal character with
negative and positive curvature can be assigned to a negative and
positive coupling sign, respectively, see fig.\fig{alpha3}. In
fig.\fig{alpha3} it is shown how both types must be combined in
circular symmetry: the positive charge must `roll' inside as a
hypocycloidal with $N$, and the negative charge outside as a
epicycloidal with $N+1$ sub-loops. This could be helpful for an
interpretation of natures symmetry breaking regarding positive and
negative charges, especially `charged' fine structure measurements
with a shift based on $|N_-| + 1 = |N_+|$. Since $M_{g}$ is a
spin-independent coupling number and the effective coupling should
be proportional to spin, it is straightforward to introduce
$Z_{e}=J M_{g}$ as a charge number. This enables to determine the
same $\alpha/J$ in \eqn{phase07} for different $Z_{e}$ since
$M/Z_{e}$ is $\alpha$-scale invariant and counts the sub-loops
necessary to generate one basic coupling loop. As shown by Berry,
a geometric phase is produced in the field of a magnetic monopole
\cite{berry}. The geometrical phase in quantum mechanics is
ultimately related to the construction of an Abelian monopole
since this is the only topologically non trivial object which
arises when the structure group is U(1)\cite{Zakharov}. For
electrodynamics, the gauge group is $U(1)$ which has the topology
of a circle, on which the homotopy classes of closed curves are
labelled by their winding or loop numbers, and where the magnetic
charge is quantized taking integral values \cite{Dirac-31}. The
topological nature of the monopole charge is by definition
discrete and invariant under continuous deformations. Dirac
\cite{Dirac-31} showed that the existence of magnetic monopoles
can explain the quantization of electric charge and that a
monopole must carry a magnetic charge which is an integral
multiple of $68.5$. According to these monopole properties and the
U(1) relation to the Berry phase for $J={1 \over 2}$, $M=137$ is
the topological candidate to generate the quantum monopole charges
of magnetism and electrostatics. All-in-all, it suggests to take
$\alpha_0=\alpha(M=137)=1/137.036009412...$ as a candidate for the
free or neutral Sommerfeld-Dirac fine structure constant, see
tables 1 and 2. Variations are likely since external fields could
force hypocycloidal or epicycloidal resonances depending on the
polarity of the coupling. Spin-orbit coupling based on linear
phase relations suggests to visualize precession by cones rolling
in or on cones.\\
Visualizing Thomas precession and aberration (angle $\theta$
obtained by infinitesimal Lorentz boosts) \cite{Malykin1} already
pointed out, that the geometric phase can be found in classical
mechanics with a gyroscope or point-like compass as a solid-body
turn during conical movement \cite{Ishlinski}.
\subsection*{Overload}
% ------------------------------------------
\begin{figure} [h]
\center
%\includegraphics[scale=0.4]{pictures/bifurcationtiff1.eps}
\caption{\small{Chaotic occupation density of geometric phase
space for a variable coupling number $M_{g}=Z_{e}/J$. Stable
regions with one fixed $\varphi_{g}(T)$-value and bifurcations
occur periodically. Stable regions for $J={1 \over 2}$ and $M=137$
exist with $M_{g}$ values 1-114, 260-310, 541-568. Feigenbaum's
number $\approx 4.669$ characterizes the branching sequence of
bifurcations. The dashed blue rectangle shows $M_{g} \le 137$ for
$J={1 \over 2}$.}} \labfig{alphaZ2}
\end{figure}
% ------------------------------------------
Increasing $|M_{g}/M|$ will increase the precession semiangle
$|{\theta}|$ for a given dynamical phase of the sub-loop
$\triangle\varphi_{d}(T)$. With variable $M_{g}$, \eqn{phase07}
and \eqn{phase08} characterize a complex one-dimensional system
that can show chaotic dynamics and quasiperiodicity
\cite{Feigenbaum82}. It is a cosine map related to the circle and
sine map \cite{Devaney87} and as an iterative system it shows
asymptotic stable and converging regimes but also bifurcations and
unstable regimes for special feedback coupling strengths $M_{g}$.
The geometric part and precession will become more and more
dominant with increasing $M_{g}$, blocking or occupying phase
space as a charge-dependent screening effect, see
fig.\fig{alphaZ2}. If the nonlinear system characterizes Coulomb
coupling and fine structure, the chaotic dynamics should be found
in charged nuclei where $M_{g}$ reaches the critical value. Since
the production cross section was found to be rapidly decreasing
with the atomic number near 114 \cite{Heenen}, it was concluded
that it would be very difficult to reach still heavier elements.
In fig.\fig{alphaZ2} the first stable regions with one fixed
$\varphi_{g}(T)$-value end s near $M_{g}=114$, the next
bifurcations occur periodically. This indicates, that the Coulomb
coupling generating chaotic electrodynamics could to some extend
be responsible for the instability of the nucleus.
\subsection*{Measurement}
% ------------------------------------------
\begin{figure} [h]
%\center\includegraphics[scale=0.8] {pictures/ameasurement1.eps}
\caption{\small{Regarding the most accurate measurements of the
last years there is almost no overlap between the three different
setups given by neutron (free running $\alpha_0$), electronic
(epicycloidal $\alpha_-$), and protonic (hypocycloidal $\alpha_+$)
couplings. Dashed are combined values of different measurements.}}
\labfig{alpha4}
\end{figure}
% ------------------------------------------
Over the years there was a discussion about the value of the fine
structure constant. Different values measured with comparable
accuracy disagree in different directions by several standard
deviations. The additional quantum condition \eqn{spinorbitN}
selecting the hypocycloid or epicycloid character could force
extra shifts $\approx N^{-2}$, for $M=137$ about $3806^{-2}\approx
6.9 \cdot 10^{-8}$ (see table 1 and fig.\fig{alpha6}). Such shifts
are likely to be observed in the charged case if the Compton
wavelength couples to the precession frequency with asymmetry $ N
\pm {1 \over 2}$ in \eqn{spinorbitN}. So it remains to check if
measurements based on externally forced loops in modulated fields
or internal forced loops due to mass-to-charge coupling could
favor the correspondent resonances leading to hypocycloidal or
epicycloidal phase evolutions. And there is a strong evidence
regarding the probably most often cited and accurate measurements
of the fine structure constant over the last years. It seems, that
measurements can be grouped into three $\alpha$-categories:
$\alpha_0$, $\alpha_-$, and $\alpha_+$, see fig.\fig{alpha4}:
\begin{itemize}
\item The free running value $\alpha_0$ fits very well to
$h/m_{n}$ neutron experiments \cite{CODATA1998,KNW-99}. The value
is also supported by a CODATA $h/m_{n}$ evaluation $\alpha_0^{-1}
= 137.036 0084(33)$ \cite{CODATA1998} that is a combination of the
PTB, IMGC, and NRLM results. Over the years this range was
confirmed by repeated PTB experiments providing for $\alpha_0^{-1}
= 137.03601144(498)$ \cite{KNW-99, KNW-86, KNW-95, KNW-98}, see
table 3. The neutron values have been obtained by measuring the de
Broglie wave length of a beam of neutrons and the Bragg reflection
in a perfect silicon crystal.
\item The epicycloidal $\alpha_-$ corresponds to measurement based on
single electrons.
The prominent electron
$g-2$ value $\alpha_-^{-1} = 137.03599958(52)$ in
\cite{Kinoshita2} fit's within 5 ppb (relies on extensive QED
calculations and does strongly contribute to the 1998 CODATA
evaluation \cite{MohrTaylor,CODATA1998,NIST2}), see table 3. To
this type also fits the mean value $\alpha_-^{-1}=137.0360008(30)$
of two well known experiments obtained by the quantum Hall effect
with $\alpha_-^{-1}= 137.03599790(320)$ in \cite{Cage1} and
$\alpha_-^{-1}=137.0360037(27)$ in \cite{Jeffery1}. The von
Klitzing NIST-97 value $R_{K}$ $\alpha_-^{-1}= 137.036 0037(33)$
\cite{CODATA1998} fits also to the electronic value.
\item The protonic
hypocycloidal $\alpha_+$ was found with measurement error smaller
40 ppb \cite{CODATA1998} by the NIST-89 shielded proton
gyromagnetic ratio $\Gamma'_{p-90}(lo)$ $\alpha_+^{-1}=
137.0359880(51)$. A recent Penning trap value obtained with Cs$^+$
$\alpha_+^{-1} = 137.0359922(40)$ \cite{Bradley1} fits also very
well, see table 3, but does not contribute to the CODATA
recommended value \cite{CODATA1998}.
\end{itemize}
% ----------------------------------------------
\begin{table} [h]
\begin{caption}
\newline \small{A simple comparison matrix in standard deviation units
$\sigma_{i}$
comparing three different types of the most relevant measurements
\cite{Kinoshita2} based on neutron \footnotemark[1], electron
$g-2$ \footnotemark[2], proton Cs$^+$ \footnotemark[3]
measurements, see fig.\fig{alpha4}.} The diagonal values indicate
a significant correlation to the calculated values.
\end{caption}\\
\begin{tabular}{p{90pt}p{105pt}p{105pt}p{105pt}}\\
\hline \hline \\
& $137.03601144(498)$\footnotemark[1] & $137.03599976(50)$\footnotemark[2]
& $137.03599220(400)$\footnotemark[3]\\
\hline & & \\
$137.03600941164$\footnotemark[4] & \textbf{\small{-0.4$\sigma_1$/-1.5
$\cdot 10^{-8}$}} & +19 $\sigma_2$/+7.0 $\cdot 10^{-8}$ & +4.3
$\sigma_3$/+13 $\cdot 10^{-8}$ \\
$137.03600052556$\footnotemark[5] & {-2.2 $\sigma_1$/-8.0 $\cdot 10^{-8}$}
& \textbf{\small{+1.5 $\sigma_2$/+5.6 $\cdot 10^{-9}$}} & +2.1
$\sigma_3$/+6.0 $\cdot 10^{-8}$ \\
$137.03599106791$\footnotemark[6] & {-4.1 $\sigma_1$/-15 $\cdot 10^{-8}$}
& -17 $\sigma_2$/-6.3 $\cdot 10^{-8}$ & \textbf{\small{-0.3 $\sigma_3$/-8.3
$\cdot 10^{-9}$}} \\
\hline \hline\\
\end{tabular}\\
\small{\footnotemark[1] {1998 Neutron value \cite{KNW-99}}},
\small{\footnotemark[2] {1999 Codata approx. electron $g-2$
\cite{CODATA1998,NIST2}}}, \small{\footnotemark[3] Penning trap
\cite{Bradley1}},
\\\small{\footnotemark[4] {free running
solution of table 1}}, \small{\footnotemark[5] {3805.5 resonance
of table 1}}, \small{\footnotemark[6] {3806.5 resonance of table
1}} \labtab{fsc2}
\end{table}
% ----------------------------------------------
\subsection*{Conclusion}
The probably most prominent fundamental constant can be within
measurement uncertainty reproduced by iterative phase
relationships that obey the single-valuedness requirement. If this
paper provides for a correct approach to the nature of fine
structure and charge, the strength and sign of coupling is
controlled by Berry's phase. Especially the effect generated by
the coupling sign is highly interesting, Berry's phase can evolve
against or with the dynamical phase depending on the sign of
curvature. Measuring the correspondent shifts could play a crucial
role since they would indirectly confirm (a) the existence of
magnetic monopoles and (b) the coupling of dynamical and geometric
phase. The present experimental situation supports the proposals
of this work, especially the relevance of hypocycloidal or
epicycloidal resonances. There is a strong evidence that
$\alpha_0=\alpha(M=137)$ is a candidate for the Sommerfeld-Dirac
fine structure constant. Without knowing the driving mechanism
that leads to coupling and curvature beyond general relativity,
single-valuedness of $M$ on the closed path (and $N$ subject to
external fields) can guide to exact results governed by a number
of physics relations that are:
\begin{itemize}
\item $\alpha_0$ is based on a plausible connection between the geometric
and dynamical
phase,
\item the Berry phase screening $1-M\alpha_0/J$ of electrodynamic
coupling provides the necessary magnetic monopole component to
quantize charge,
\item $\alpha_0$ is perfectly compatible with gravitomagnetic coupling,
\item the correspondence principle is satisfied by `classical'
counterparts,
\item chaotic dynamics and instability could fit to known instabilities of
superheavy
charged nuclei, see fig.\fig{alphaZ2}, outside the chaotic regime the
coupling mechanism is flexible, self-consistent, regenerative, and
self-balancing
subject to external distortions,
\item positive and negative curvatures and related sign in Berry's phase
requires to introduce $\alpha_+$ and $\alpha_-$
based on the hypocycloidal and epicycloidal character, respectively,
\item based on $N_0=3806$ the shifts $\approx 3806^{-2}\approx 6.9 \cdot
10^{-8}$ are supported by highly rated measurements, see fig.\fig{alpha4},
where
the epicycloidal $\alpha_- \approx 1/137.03600052556$ with $N_-=3805.5$
fits within a view ppb to the indirect QED electron $g-2$
determination, the `free' neutron measurements with $\alpha_0 \approx
1/137.03600941164$ to $N \approx 3804.56$,
and the proton dominated measurements to $N_+=3806.5$ with $\alpha_+
\approx
1/137.03599106791$.
\end{itemize}
Spin-orbit coupling with coupled phase conditions (e.g. between a
global and a local phase) can be found in many cyclic quantum
systems, especially in highly symmetric or degenerated mesoscopic
systems showing helical or regular $M$-polygonal structure. Spin
precession of electrons in cyclic motion can lead to various
interference phenomena such as oscillating persistent current and
conductance \cite{YiQianSu}. For solids, atoms, or in nuclear
physics it should be possible to define more complex and
generalized fine structure constants that characterize the back
reaction of the geometric phase to the dynamical phase for more
complex paths, probably involving lattice interaction, especially
in situations where interaction between vibrational and electronic
states happens in the degenerate state (i.e. Jahn-Teller effect).
In this case the phenomena of superconductivity and bose-einstein
condensates could probably also benefit from special feedback
relationships between the geometric and dynamical phase, supported
by regeneration and revival processes characterized by generalized
fine structure constants. I suspect that the type of fine
structure constants tabulated in table 1 could appear only in the
simplest cases of one-dimensional systems, i.e. in fullerene rings
or tubes or in Heisenberg spin chains, especially if neighboring
units generate cooperative spin resonance phenomena. Although, the
mathematics that leads to the number $M=137$ is open, topology and
geometry in the $\alpha$-theory part of this work could guide
experimentalists and theorists to new connections between
electrodynamics and (loop-)gravity. Regarding the identical
coupling relation predicted by gravitomagnetism, the tiny
magnitude of the gravitomagnetic field in a classical measurement
does not necessarily mean, that the magnitude of the emerging
geometric phase and related quantum mass--energy currents in
feedback loops must be tiny. A loop-gravitational gravitomagnetic
field magnitude reaching the electromagnetic level would shift the
Planck scale to the nuclear range. Regarding the $\alpha$-powers
produced with $N^2$
\begin{eqnarray}
\alpha^2 \approx { 2 \over \pi^2 N}
\labeqn{conclusion01},
\end{eqnarray}
the Berry contribution with coupling change ${\triangle \alpha /
{\alpha}} $ proportional to $1/N^2$ has lowest order
$\alpha^4$-terms. It should be interesting to note, that
hyperfine, fine structure, and Lamb shift are usually assigned to
the same $\alpha$-power dependency. But geometric phase
contributions are totally missing in almost all QED evaluations of
the atomic spectra.
\subsection*{Summary}
This article starts by a quick evaluation of the role and status
of Berry's phase. Although, Berry's phase especially the
Aharonov-Bohm phase can be related to a gauge potential that is
very similar to a magnetic monopole potential, (hyper)fine
structure and Lamb shift theories handling the most accurate
measurements seem to ignore Berry's phase. The rather exotic
status seems to imply for many scientists that Berry's phase could
have no big impact on the quantum mechanics of atomic energy
spectra. This is quite normal, since Berry's phase is quite young.
To provide for examples that support the proposal of this article,
Sommerfeld fine structure constant $\alpha$ is successfully
defined based on Berry's geometric phase coupling back to the
hamiltonian-induced dynamical phase in a phase-locked spin-orbit
system. $\alpha$ identified as the ratio of orbital to Compton
wave number fits exactly to gravitomagnetic interaction (general
relativity). Moreover, the resulting nonlinear affine iteration
$\alpha =\cos(\pi \alpha )/M$ provides for a free running solution
outside the chaotic regime, where the Berry phase screening
component $1-M\alpha$ of electrodynamic coupling can be assigned
to magnetic monopoles on SU(2)/U(1) = S$^2$ with $M=137$ as
required by Dirac. In the chaotic regime the onset of bifurcations
could fit to known instabilities of superheavy charged nuclei. A
sub-quantization of $\alpha$ based on a integral winding number
shows three basic cartegories of solutions: the free neutral value
$1/\alpha$=137.03600941164, and the next protonic (hypocycloidal)
and electronic (epicycloidal) resonances at 137.03599106791 and
137.03600052556, respectively. This could resolve the disagreement
of $\alpha$-values obtained by different measurements using
neutrons, electronic, and protonic interactions. Almost all of the
current available, most accurate, and most often cited
measurements and evaluations of $\alpha$ fit within standard
deviation (a view ppb's) to the correspondent values. The results
of this article shows, that it possible to built a bridge via
Berry's phase between classical physics, quantum electrodynamics,
and general relativity and that the exotic status of Berry's phase
has to be replaced by the probably most fundamental status of
mathematical physics.
\newpage
\footnotesize
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