Semiclassical theory of weak values
Atushi Tanaka
Department of Physics
Hachioji
Tokyo~192-0397
Japan
% for PLA, less than 50 words % Aharonov-Albert-Vaidman's weak values are investigated by a semiclassical method. % Examples of the semiclassical calculation that reproduces ``anomalous'' weak values are shown. % Furthermore, a complex extension of Ehrenfest's quantum-classical correspondence between quantum expectation values of the states with small quantum fluctuation, and classical dynamics, is shown. % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 1st submission to PLA %A semiclassical explanation of ``anomalies'' of %Aharonov-Albert-Vaidman's weak values with no invocation to quantum %interference is shown. % %By the semiclassical theory, the quantum-classical correspondence %between quantum expectation values of the states with small quantum %fluctuation, and classical dynamics, due to Ehrenfest's theorem is %extended to complex-valued classical trajectories. % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %A semiclassical theory of Aharonov-Albert-Vaidman's weak value is %developed. %% %Within the semiclassical accuracy $\Order (\hbar)$, it is shown that %the classical trajectories that appear in the semiclassical evaluations %of Feynman kernels agree with the corresponding weak values when the %interference among the classical trajectories are negligible. %% %This explains ``anomalous'' weak values, e.g. surprisingly enormous or %complex-valued, without relying on quantum interference phenomena. %% %Furthermore, this introduces a complex extension of the %quantum-classical correspondence that is provided by Ehrenfest's %theorem, i. e., the correspondence between the quantum expectation %values of the states with small quantum fluctuation, and the classical %trajectories.
quant-ph0203149.tex
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\begin{document}

%-------------------------------------------------------------------------
% Title
%-------------------------------------------------------------------------
%\begin{frontmatter}
\title{Semiclassical theory of weak values}
\author{Atushi Tanaka\\
 Department of Physics, Tokyo Metropolitan University,\\
 Hachioji, Tokyo~192-0397, Japan }
% \author{Atushi Tanaka}
% %\ead{tanaka@phys.metro-u.ac.jp}
% \address{Department of Physics, Tokyo Metropolitan University,   
%  Hachioji, Tokyo~192-0397, Japan }
\date{23 March 2002}
%\date{23 November 2001}
%\date{\ThisVersion}
\maketitle
\begin{abstract}
 % for PLA, less than 50 words
 %
 Aharonov-Albert-Vaidman's weak values are investigated by a
 semiclassical method. 
 %
 Examples of the semiclassical calculation that reproduces 
 ``anomalous'' weak values are shown.
 %
 Furthermore, a complex extension of Ehrenfest's quantum-classical
 correspondence between quantum expectation values of the states with
 small quantum fluctuation, and classical dynamics, is shown.
 %
 %
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 % 1st submission to PLA
 %A semiclassical explanation of ``anomalies'' of
 %Aharonov-Albert-Vaidman's weak values with no invocation to quantum
 %interference is shown.
 %
 %By the semiclassical theory, the quantum-classical correspondence
 %between quantum expectation values of the states with small quantum
 %fluctuation, and classical dynamics, due to Ehrenfest's theorem is
 %extended to complex-valued classical trajectories.
 %
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %A semiclassical theory of Aharonov-Albert-Vaidman's weak value is
 %developed.  
 %%
 %Within the semiclassical accuracy $\Order (\hbar)$, it is shown that
 %the classical trajectories that appear in the semiclassical evaluations
 %of Feynman kernels agree with the corresponding weak values when the
 %interference among the classical trajectories are negligible.  
 %%
 %This explains ``anomalous'' weak values, e.g. surprisingly enormous or
 %complex-valued, without relying on quantum interference phenomena.
 %%
 %Furthermore, this introduces a complex extension of the
 %quantum-classical correspondence that is provided by Ehrenfest's
 %theorem, i. e., the correspondence between the quantum expectation 
 %values of the states with small quantum fluctuation, and the classical
 %trajectories.
\end{abstract}
%
% \begin{keyword} % keywords here, in the form: keyword \sep keyword
% Weak measurements and weak values \sep Semiclassical method \sep
% Coherent state path integral \sep Complex-valued classical trajectories
% \sep Ehrenfest's theorem

% % PACS codes here, in the form: \PACS code \sep code
% % Foundations, theory of measurement, miscellaneous theories
% % (including Aharonov-Bohm effect, Bell inequalities, Berry's phase)  
% \PACS 03.65.Bz \sep
% 03.65.Sq \sep % Semiclassical theories and applications
% 03.65.-w \sep % Quantum mechanics
% %
% \end{keyword}
%\end{frontmatter}
%-------------------------------------------------------------------------




\section{Introduction}


%
%%%
% OLD head
%%%
% Recent conceptual investigations of quantum theory suggest us
% that we have a rich variety of ways to observe quantum phenomena: One
% direction involves multiple-time ``events''
% (e.~g. Aharonov-Bergmann-Lebowitz's rule~\cite{Aharonov:PR-134-B1410},
% consistent history approach~\cite{CSH}, and weak
% measurements~\cite{AAV:PRL-60-1351}) and another involves
% statistical nature of noisy readouts of measurement apparatus
% (e.~g. simultaneous measurement of noncommuting
% observables~\cite{SimMeasure}, and weak
% measurements~\cite{AAV:PRL-60-1351}).  
% %
% Besides the conceptual aspect of these works, they surely provide useful
% notions to intuitively grasp the hearts of complicated ``quantum
% measurement'' processes.  
%
%
%Among these works, this paper focuses on weak measurements and the
%corresponding ``expectation values'', {\em weak values}, which are
%introduced by  Aharonov, Albert and Vaidman~\cite{AAV:PRL-60-1351}. 
%


A standard approach of quantum theoretical description of the
interaction between a measurement apparatus and the system that is
subject to the measurement was introduced by von~Neumann.%
~\cite{Neumann:MGQ-1932}. 
%
Recently, it is revealed that the notion of 
{\em weak (expectation) values}%
~\cite{AAV:PRL-60-1351,AharonovVaidman:PRA-1990-41} emerges from
the von Neumann theory, in the limit that the influence of the
apparatus on the system is weak so as to avoid the collapse of
the state of the system.
%
The weak values and the conventional expectation values of quantum
theory coincide for the quantum ensembles that are specified by 
only the preselections (preparations) of initial states%
~\cite{AharonovVaidman:PRA-1990-41}.
%
However, once we consider the quantum ensembles that are specified not
only by the preselections of initial states, but also by the
postselections of finial states, the weak values depart from the
conventional expectation values%  
~\cite{Aharonov:PLA-124-199,AAV:PRL-60-1351,AharonovVaidman:PRA-1990-41}.
%
In particular, the weak values can take ``anomalous'' values that lie
outside of the range of the eigenvalues of the corresponding
operators. 
%
More precisely, there are two kinds of anomalous weak values,
surprisingly large weak values of bounded operators, and
complex-valued weak values of Hermite operators.
%
On one hand, Aharonov, Albert and Vaidman showed an experimental setup,
in which weak measurements of a component of the spin of the
spin-$\frac{1}{2}$ particle can turn out to be 100~%
\cite{AAV:PRL-60-1351} (see also, \cite{Aharonov:PLA-124-199}).
Experimental studies confirms the surprisingly large weak values%
~\cite{RSH:PRL-66-1107,Parks:PRSLA-454-2297};
%
On the other hand, Aharonov, Albert and Vaidman~\cite{AAV:PRL-60-1351}
mentioned that the weak values can be a complex number.
%
This is also shown to be experimentally accessible%
~\cite{Parks:PRSLA-454-2297}, and moreover, 
to provide useful theoretical notions%
~\cite{Parks:JPA-33-2555,Wang:ComplexNelson,Steinberg:PRL-74-1995}.



% The anomalous weak values suggests us that the notion of weak
% values is completely foreign to classical concepts. It is often  
% suggested that the anomalous feature of the weak values are due to 
% quantum interference phenomena~\cite{Aharonov:PLA-124-199,Duck:PRD-40-2112}.
% %
% % Duck, Stevenson and Sudarshan~\cite{Duck:PRD-40-2112} demonstrate this
% % by examining the distribution function of apparatus' pointer by changing
% % the ``weakness'' of the measurement: In the ``strong measurement''
% % limit, the peaks of the distribution function corresponding to the
% % eigenvalues of the observables; As the measurement becomes weak, these
% % peaks merge into a single peak, which corresponds to a weak
% % value. Accordingly the pointer's state that indicates the weak value is 
% % the superposition of the states that point the eigenvalues of the
% % observable~\cite{Duck:PRD-40-2112}.  
% % %
% % From the consideration above, we can conclude that the anomalous weak
% % values are due to quantum interference phenomena.
% %
% However, it is important to remember that the notion of quantum
% interference essentially depends on the choice of the decomposition of
% a quantum wave into reference waves (e. g. expanding state vectors by
% basis). A convenient way to construct such reference waves is provided
% by semiclassical methods, which are particularly efficient when the
% quantum fluctuation around classical trajectories is small.
% %
% % In the famous double-slit experiment (Ref.~\cite{Landau:Quantum}, \S~1),
% % which is employed for the explanation of the essential feature of
% % quantum interference phenomena, it is feasible to say that the
% % oscillatory pattern (say, of $|\psi(x)|^2$) is due to the interference
% % between the two waves that pass the two slits, respectively. Our
% % intuition supports that these two waves, which are associated with two
% % spatially localized regions (i.e. slits), are much easier to understand,
% % than their superposition, which suggests nonlocality in space. 
% % %
% % Although this choice is natural, it should be emphasized that we have
% % arbitrariness of the choice. 
% % %
% % Another example is an electronic plane wave, which has a definite value
% % of momentum. This is a superposition of the eigenstates of the position
% % operator of the electron. Thus the plane wave is highly nonlocal in
% % space, and difficult to understand only by spatially localized
% % waves. Rather, we should optimize the representation in accordance with
% % the plane wave, i.e. it's easier to understand the plane wave in the
% % momentum representation rather than the position representation%
% % \footnote{Namely, the wavefunction of the momentum representation
% % concentrate in a point in the momentum space, whereas the wavefunction
% % in the position representation extends in the whole position space.}.
% % %
% % Similar to this, the weak values may have a much simple explanation
% % than the interpretation that rely on quantum interference phenomena.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Important Modification: Stop to say about quantum interference here.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
This letter reports a study on weak values by a semiclassical method.
%
The usefulness of the semiclassical approach for the weak values was
anticipated by Berry's asymptotic
(i.e. semiclassical~\cite{Berry:RPP-35-315}) analysis of
superosillations~\cite{Berry:FTF-1994}. 
%
Although the anomalous weak values suggests us that the notion of weak 
values is completely foreign to classical concepts, the semiclassical
theory is shown to reproduce examples of anomalous weak values.
This is achieved by extending the classical trajectories into
complex-valued phase space.
%
% %
% 1st PLA
% This letter reports an alternative explanation of the anomalous weak
% values in quantum ensembles with both pre- and post-selections, without
% relying on quantum interference phenomena, by extending the classical
% trajectories into complex-valued phase space with the help of
% a semiclassical method.   
% %
It is well-known that the semiclassical method employs 
complex-valued classical trajectories to describe classically forbidden
phenomena,  e.~g.  tunneling~\cite{Tunneling} and nonadiabatic
transitions~\cite{ComplexNonadiabatic,Miller:JCP-56-1972}.
%
Furthermore, the recent investigations of the classically forbidden
processes in ``quantum chaos'' (quantum phenomena in the systems
whose classical counterpart exhibit chaos)~\cite{Gutzwiller:CCQM-1990}
revealed that the complex extension of classical trajectory is
indispensable~\cite{Adachi:AP-195-45,ChaoticTunnel}.
%



In the following, I remind the basic point of the weak measurements.
%
Consider the quantum ensemble that is specified not only by a
preselection (preparation) of an initial state but also a 
postselection of a final state: let the state vectors of the initial
and the final states be $\ket{\psi'}$ (at time $t'$) and $\bra{\psi''}$
(at time $t''(>t')$), respectively. The corresponding ``expectation
value'' of an observable $\hat{A}$ at an intermediate time $t$ ($t' \le
t \le t''$) is called a {\em weak value}  $\Weak (\hat{A}, t)$, whose
definition is  
\begin{equation}
 \label{eq:DefWeakValue}
 \Weak (\hat{A}, t) \equiv 
  \frac{\bbraOket{\psi''}{\hat{U}(t'',t)\hat{A}\hat{U}(t,t')}{\psi'}}%
  {\bbraOket{\psi''}{\hat{U}(t'',t')}{\psi'}}
\end{equation}
where $\hat{U}(t_1,t_0)$ is the time evolution operator during the time
interval~$[t_0, t_1]$~\cite{AAV:PRL-60-1351}.


The weak values are experimentally accessible by
a kind of von Neumann type apparatus~\cite{Neumann:MGQ-1932} whose
``pointer'' position has large quantum fluctuation. After the apparatus
weakly contact with the system and the system is succeedingly
postselected, we can gain the information of the corresponding weak
value by examining the statistics of the pointer. 
%
At each readout, the pointer value, which is obtained by the
standard quantum measurement that invokes wavepacket reduction, is very
noisy due to the initial fluctuation of the pointer and the weakness of
the interaction with the system. Accordingly we need to
accumulate the readouts of the pointer to make sense. 
%
The peak of the pointer position distribution determines the real part 
of the weak value. In addition, the imaginary part of the weak value is
determined by the peak of the momentum (i.e. the conjugate quantity of
the pointer position) distribution of the pointer. Thus both the real
and the imaginary parts of weak values are experimentally accessible%
~\cite{AAV:PRL-60-1351}.


The plan of this letter is as follows. In Sec.~\ref{sec:semilcassical}, I
develop a theory to evaluate weak values by a semiclassical method. 
Examples are shown in Sec.~\ref{sec:example}; We encounter 
two kinds of ``anomalous'' weak values, complex-valued weak values
(Sec.~\ref{subsec:complexEx}), and surprisingly large weak values
(Sec.~\ref{subsec:largeEx}).  
%
It is shown that they provide an ``anomalously-extended''
classical-quantum correspondence. 
%
I discuss the limitation of the present semiclassical argument
in Sec.~\ref{sec:limit}. Finally, Sec.~\ref{sec:SummaryAndOutlook}
summarizes this letter.
% and give an outlook, where I emphasize the
%relationship between the notion of weak values and the complex-valued
% trajectories that appear in the semiclassical method.




\section{Semiclassical evaluation of weak values}
\label{sec:semilcassical}

For brevity, I employ a one degree-of-freedom system, whose
position and momentum operators are $\hat{q}$ and $\hat{p}$, respectively.
%
Let us consider the quantum ensemble specified by an initial state
$\ket{\psi'}\equiv\ket{q'}$ at $t=t'$ and a final state
$\bra{\psi''}\equiv\bra{q''}$ at $t = t''$ ($>t'$), where $\ket{q}$ is 
the $\hat{q}$'s eigenvector whose eigenvalue is $q$.
In the evaluation of weak value $\Weak(\hat{A}, t)$, the following
generating functional is useful: 
\begin{equation}
 \label{eq:weakGF}
 Z(\zeta(\cdot), \hat{A}) \equiv 
  \langle\psi''|%
  \Texp\bigl\{-\frac{i}{\hbar}\int_{t'}^{t''} 
  \left(\hat{H} - \hat{A}\zeta(t)\right)dt
  \bigr\}%
  |\psi'\rangle
\end{equation}
where $\displaystyle\Texp(\cdot)$ is the time-ordered exponential and
$\hat{H}$ is the Hamiltonian of the system.
It is straightforward to show that
%
\begin{equation}
 \label{eq:weakValueByGF}
  \Weak (\hat{A}, t) =
  -\left. i\hbar\, 
    \frac{\delta\ln Z(\zeta(\cdot),\hat{A})}{\delta\zeta(t)}
  \right|_{\zeta(\cdot) = 0}
\end{equation}
holds. This is evaluated by a semiclassical method in the following.

Let $A(q,p)$ and $H(q,p)$ be the classical counterparts of the operator
$\hat{A}$ and the Hamiltonian $\hat{H}$ of the system, respectively. I
ignore the operator ordering problem, since it changes the result only
$\Order (\hbar)$, i.e., within the accuracy of the following
semiclassical argument.     



% Correct a typo.
In the evaluation of $Z(\zeta(\cdot),\hat{A})$~(\ref{eq:weakGF}),
I employ the semiclassical approximation that evaluates the Feynman path
%I employ the semiclassical method that evaluates the Feynman path
integral representation of $Z$ by the stationary phase
method~\cite{Schulman:TAPI-1981}.
%
In order to carry out the semiclassical evaluations, I introduce an
important assumption: for infinitesimally small values of
$\zeta(\cdot)$, quantum interference between multiple classical
trajectories do not present in the semiclassical evaluation. Namely, the
semiclassical generating functional have a contribution only from one
classical trajectory $(q(t), p(t))$, which satisfies the Hamilton
equation with the classical Hamiltonian $H(q, p) - A(q, p)\zeta(t)$ and
the boundary condition $q(t') = q'$ and $q(t'') =q''$,
which are specified by the initial and the
final states $\ket{q'}$ and $\bra{q''}$, respectively%
\footnote{The Hamilton equation with the boundary condition $q(t') = q'$
and $q(t'') = q''$ can have multiple solutions. If $t''-t'$ is
small enough, it is proved that the boundary-value problem has
a unique solution. See, e.g., Ref.~\cite{Schulman:TAPI-1981}, Chap.~12.%
}. 
%
In other words, $Z$ is assumed to be in a single-term form%
\footnote{I omit Maslov's index~\cite{Gutzwiller:JMP-8-1979}, since this
is irrelevant to the present argument.} 
\begin{equation}
 \label{eq:ZqSingle}
 Z \simeq E \exp (i S /\hbar)
\end{equation}
where $E$ and $S$ are the amplitude factor and classical action,
respectively, 
\begin{eqnarray}
 \label{eq:Eq}
 E &\equiv& \frac{1}{\sqrt{2\pi \hbar\;\partial q''/\partial p'}}\\
 \label{eq:Sq}
 S &\equiv& \int_{t'}^{t''} 
  \{p(t) \dot{q}(t) - H(q(t), p(t)) \nonumber\\
 &{}& {}\quad\quad+ A(q(t), p(t)) \zeta(t)\}dt 
\end{eqnarray}
and $p' \equiv p(t')$~\cite{VanVleck:PNASUSA-14-178}.
%
The single-term condition~(\ref{eq:ZqSingle}) holds when $\hbar$ is
small or the time scale in question is short. The details are discussed
in Sec.~\ref{sec:limit}.
%
The single-term condition~(\ref{eq:ZqSingle}) implies
\begin{equation}
 \Weak (\hat{A}, t) 
  = \left. \frac{\delta S}{\delta\zeta(t)}\right|_{\zeta(\cdot) \equiv 0} 
 + \Order (\hbar).
\end{equation}
Applying~(\ref{eq:Sq}) to this, the main result is obtained:
\begin{equation}
 \label{eq:WKBWeakq}
  \Weak (\hat{A}, t)  = A(q(t), p(t)) + \Order (\hbar)
\end{equation}
where $\zeta(\cdot)\equiv 0$ is imposed on $(q(t), p(t))$.
%
The first term of eq.~(\ref{eq:WKBWeakq}) is a ``classical'' quantity:
It persists in the classical limit $\hbar\to0$ and is almost independent
of $\hbar$, in general.
%
The weak values are accordingly determined, with an error of
$\Order (\hbar)$, by the classical trajectory $(q(t), p(t))$ that
composes the semiclassical evaluation of the Feynman kernel 
$\braOket{\psi''}{\hat{U}(t'',t')}{\psi'} = Z(\zeta(\cdot)\equiv 0)$.

I emphasize the significance of the single-term
assumption~(\ref{eq:ZqSingle}), which optimize the shape of
semiclassical kernel, by using a simple wave~(\ref{eq:ZqSingle}) that is
composed by a single classical trajectory. The resultant
estimation~(\ref{eq:WKBWeakq}) accordingly exclude the effect of 
the quantum interference phenomena among multiple semiclassical
amplitudes. Namely, eq.~(\ref{eq:WKBWeakq}) establishes a correspondence
between a weak value and a single classical trajectory.

The generalization of the result~(\ref{eq:WKBWeakq}) to various
initial and final states (e.g. the eigenstates of the
momentum operator, and coherent states) can be obtained
straightforwardly with the help of the semiclassical
algebra~\cite{Miller:ACP-25-69,Weissman:JCP-76-4067}, as long as the 
single-term approximation~(cf. (\ref{eq:ZqSingle})) holds for
the corresponding semiclassical generating function $Z$.    

The ``variance'' of the weak value of $\hat{A}$ is 
\mbox{$\Weak (\{\hat{A} - \Weak (\hat{A})\}^2)$}~%
\cite{AharonovVaidman:PRA-1990-41}. According to the semiclassical
evaluation of weak values~(\ref{eq:WKBWeakq}), the ``weak variance'' is
$\Order (\hbar)$, when the single-term approximation~(\ref{eq:ZqSingle})
for the Feynman kernel holds.  



\section{Examples: ``anomalous'' weak values}
%\section{The semiclassical theory explains ``anomalous'' weak values}
\label{sec:example}

\subsection{Coherent state path integral:
An ``anomalous'' extension of quantum-classical correspondence into
complex-valued phase-space}
%
%\subsection{Coherent state path integral:
%  Complex-valued weak values and complex-valued classical trajectories}
\label{subsec:complexEx}

Firstly, I show an example that the semiclassical theory above
reproduces complex-valued weak values. 
% 1st PLA
%Firstly, I show that the semiclassical argument above explains 
%complex-valued weak values. 
%
The semiclassical evaluation
(\ref{eq:WKBWeakq}) suggests that we encounter complex-valued weak
values for classically forbidden phenomena (e.g. tunneling phenomena
and nonadiabatic transitions). One of the simplest ways to investigate 
the classically forbidden phenomena is to study semiclassical coherent
state path integrals, which are generically composed of complex-valued
classical trajectory~\cite{SCPI}.
The present argument accordingly provides an extension of
quantum-classical correspondence into complex-valued phase-space.




I use the coherent states~\cite{KlauderSkagestram:CS-1985} that are
characterized with the help of a complex symplectic
transformation~\cite{KMS:GTIA-3-249,Weissman:JCP-76-4067}
\begin{equation}
 \label{eq:KMSTrans}
 \left[ \begin{array}{c} Q\\ P \end{array} \right]
 = 
  \left[
   \begin{array}{cc} 
    1/\sqrt{2}& -i/\sqrt{2}\\
    -i/\sqrt{2}& 1/\sqrt{2}\\
   \end{array} 
 \right]
 \left[ \begin{array}{c} q\\ p \end{array} \right]
\end{equation}
where $q$ and $p$ are the position and its canonical conjugate momentum
of the system, respectively. The quantized operators $\hat{Q}$ and 
$\hat{P}$ are creation and annihilation operators, respectively, of a
harmonic oscillator. 
%
The coherent state $\ket{q'p'}$ that is employed here is 
$\hat{P}$'s eigenstate whose eigenvalue is \mbox{$(p'-iq')/\sqrt2$}.




The semiclassical evaluation of $K(q''p''t'';q'p't') \equiv
\braOket{q''p''}{\hat{U}(t'',t')}{q'p'}$, which is the Feynman kernel in
the coherent state representation, is obtained by the stationary phase
evaluation of the coherent state path integral representation of
$K$~\cite{DaubechiesKlauder:JMP-26-2239}.
%
The boundary condition (at $t = t', t''$) of the classical trajectories
for semiclassical coherent state path integral is obtained by
Klauder~\cite{SCPI} 
\begin{eqnarray}
 \label{eq:CSKBC}
  \matrix{
 P(t') &=& P' \left(\equiv (p'-iq')/\sqrt{2}\right) \cr
 Q(t'') &=& Q'' \left(\equiv (q''-i p'')/\sqrt{2}\right)
    }
\end{eqnarray}
%
One way to explain the Klauder's boundary condition~(\ref{eq:CSKBC}) is
to remember the fact that $\ket{q'p'}$ is a right-eigenvector of $\hat{P}$
and $\bra{q''p''}$ is a left-eigenvector of $\hat{Q}$: The corresponding
eigenvalues determine the values of classical variables $P(t')$ and
$Q(t'')$. 
%
The classical trajectory $(q(t),p(t))$ during the time interval
$t'<t<t''$ is complex-valued in general, except the case that the
real-valued classical time evolution carries the point in the phase
space $(q',p')$ at time $t'$ to $(q'',p'')$ at time $t''$.


The single-term condition (cf. (\ref{eq:ZqSingle}))
holds for semiclassical Feynman kernel in the coherent state
representation, when the time interval $t'' - t'$ is small enough.
%
Accordingly the semiclassical estimation~(\ref{eq:WKBWeakq}) implies
\begin{equation}
 \label{eq:Wqp_sc}
  \left(\Weak(\hat{q}, t), \Weak(\hat{p}, t)\right) 
  = \left(q(t), p(t)\right) + \Order (\hbar).
\end{equation}
Namely, the weak values $\Weak(\hat{q}, t)$ and $\Weak(\hat{p}, t)$
approximately obey the classical equation of motion.
Furthermore, since $\left(q(t), p(t)\right)$ are generally
complex-valued as is explained above, so are the weak values
$\Weak(\hat{q}, t)$ and $\Weak(\hat{p}, t)$. 
%
% 1st PLA: Removed 
%
% Note that the present argument that explains complex-valued weak value
% do not involve neither quantum interference nor quantum entanglement,
% which are the sources of typical quantum effect. This is thanks to the
% semiclassical method, which organizes the Feynman kernel into a
% simple form~(\ref{eq:ZqSingle}).



The estimation~(\ref{eq:Wqp_sc}) provides an extension of a
correspondence between real-valued classical trajectories and the
expectation values of quantum systems.
%
The Ehrenfest theorem implies that the expectation values of quantum
system obeys the corresponding classical theory, as long as the quantum
fluctuation in a phase space representation is
small~\cite{EhrenfestTheorem}.  Although this 
concerns only for the real-valued, ``ordinary'' correspondence, the
semiclassical theory of weak values developed in this letter extends the
argument to the ``anomalous''-valued (i. e. complex-valued) trajectories.  
%
Note that the complex extension~(\ref{eq:Wqp_sc}) is carried out
without any discrimination between real- and complex-valued
trajectories.
%
This suggests that the distinction between real (``normal'') and complex
(``anomalous'') trajectories is only superficial in the framework of
the weak measurements.


Let us consider an example, the vanishing Hamiltonian $H=0$, which is
the simplest, yet nontrivial example in the studies of semiclassical coherent
state path integral~\cite{SCPI}. I remind that the semiclassical
evaluation for this system is exact. Let the initial  ($t =t'$) and
final ($t = t''$) states be $\ket{q'p'}$ and $\bra{q''p''}$,
respectively. During the time interval  $t' < t < t''$, 
the position and the momentum of the classical trajectory in
the semiclassical Feynman kernel are complex-valued in
general~\cite{SCPI}:  
\begin{eqnarray}
 q &=& \frac{1}{2}(q'' + q') - \frac{i}{2}(p''-p')\\
 p &=& \frac{1}{2}(p'' + p') + \frac{i}{2}(q''-q')
\end{eqnarray}
These are nothing but complex-valued weak values $\Weak(\hat{q}, t)$ and
$\Weak(\hat{p}, t)$ during the time interval  $t' < t < t''$.




\subsection{Spin coherent state path integral:
an example of anomalously large weak values}
% \subsection{Spin coherent state path integral:
% Why the result of a weak measurement of a component of the
% spin of the spin-$\frac{1}{2}$ particle turn out to be 100}
\label{subsec:largeEx}

The weak values investigated in ref.~\cite{AAV:PRL-60-1351} 
for a spin-$\frac{1}{2}$ system can be reproduced
by~(\ref{eq:WKBWeakq}) the semiclassical theory of spin coherent state
path integral~\cite{SCPI}. 
%
Let $(\theta,\phi)$ be the orientation of the spin in the polar
coordinate, which parameterizes spin coherent states $\ket{\theta,\phi}$%
~\cite{KlauderSkagestram:CS-1985}. The classical
trajectory $(\theta(t),\phi(t))$ that composes the semiclassical
evaluation of a Feynman kernel
$\braOket{\theta'',\phi''}{\hat{U}(t'',t')}{\theta',\phi'}$ satisfies
Klauder's boundary 
condition~\cite{SCPI}  
\begin{equation}
 \label{eq:SpinKlauder}
  \matrix{
   e^{i \phi'} \tan (\theta'/2) &=&
    e^{i \phi (t')} \tan (\theta (t')/2) \cr
   e^{-i \phi''} \tan (\theta''/2) &=&
    e^{-i \phi (t'')} \tan (\theta (t'')/2)
    }
\end{equation}
as well as the classical equation of motion. Note that the classical
variables $\theta(t)$ and $\phi(t)$ are complex-valued in general.
%
The semiclassical weak values of the ensemble that is specified by an
initial state $\ket{\theta',\phi'}$ and a final state
$\bra{\theta'',\phi''}$ are obtained with the help of the semiclassical
formula~(\ref{eq:WKBWeakq}), if we do not encounter multiple classical
trajectories. 

For example, when the Hamiltonian of the system vanishes 
(i.e. $H\equiv 0$), the weak values of
the all components of the spin of the spin-$\frac{1}{2}$ particle
%$(\hat{\sigma}_x,\hat{\sigma}_y,\hat{\sigma}_z)$ 
exactly agree with the values obtained by the semiclassical
evaluation~(\ref{eq:WKBWeakq}).
%
I demonstrate an example%
\footnote{For brevity, I simplified the example
studied in ref.~\cite{AAV:PRL-60-1351}.}%
$(\theta', \phi') = (2\alpha, 0)$, and
$(\theta'', \phi'') = (\frac{\pi}{2}, \pi)$, where $0<\alpha<\pi/2$.
The weak values, during the time interval $t'<t<t''$, are
\begin{eqnarray}
 \Weak(\hat{\sigma}_x) &=& -1\\
 \Weak(\hat{\sigma}_y) &=& 
  -i\; \frac{\cos\alpha + \sin\alpha}{\cos\alpha - \sin\alpha}\\
 \Weak(\hat{\sigma}_z) &=& 
  \frac{\cos\alpha + \sin\alpha}{\cos\alpha - \sin\alpha}
\end{eqnarray}
%
The value of $\Weak(\hat{\sigma}_x)$ is determined by the postselection;
$\Weak(\hat{\sigma}_y)$ is an example of complex-valued weak value;
$\Weak(\hat{\sigma}_z)$ can take surprisingly large values.


%
% COMMENT OUT for 2nd submission to PLA
%
% Hence, a possible explanation why the result of a weak measurement of a 
% component of the spin of the spin-$\frac{1}{2}$ particle turn out to
% be 100~\cite{AAV:PRL-60-1351} is that the spin just obeys the classical
% equation of motion and the boundary condition of the 
% classical trajectory~(\ref{eq:SpinKlauder}). 
% %
% Note that the corresponding classical trajectory $(\theta(t), \phi(t))$ 
% is complex-valued in the present example.
%
% At the same time, I emphasize again that the present explanation of
% ``anomalous'' weak value does not involve quantum interference effect,
% whose definition heavily depends on the way to see wave phenomena
% (e.g. the choice of representations). The semiclassical construction
% succeeds to optimize the Feynman kernel to eliminate the interference
% effect, as long as the single-term assumption holds. 




\section{Limitation of the semiclassical argument}
\label{sec:limit}

In order to obtain the semiclassical formula~(\ref{eq:WKBWeakq}), 
the single-term assumption~(cf.~(\ref{eq:ZqSingle})) is essential.
In this section, I discuss about the breakdown of this assumption.
%
% Note that this limitation is common to the conventional trajectory
% approaches for quantum theory, since the source of the difficulty 
% comes from the quantum interference between the waves associated to
% (classical) trajectories. Accordingly, the trajectory approaches invented
% by Bohm~\cite{Bohm:PR-85-166} and Nelson~\cite{Nelson:PR-150-1079} 
% will not help this point, in place of the semiclassical method.
%
I focus on the coherent state representation discussed in
Sec~\ref{subsec:complexEx}. 
%~\cite{Adachi:AP-195-45,AT:Preprint-1999}.  
Concerning to other pairs of initial and final states, the similar
phenomena occur (more precisely, see ref.~\cite{Littlejohn:JSP-68-7}).
%
When $t''-t'$ is small, the semiclassical Feynman kernel have a
significant contribution only from the single classical trajectory that
are placed around the real-valued classical trajectory. Since (phase
space) caustics \cite{Adachi:AP-195-45,Caustics},
which are bifurcation points of classical trajectories, exist far from
the real-valued trajectory, the influence from the caustics is small.
The single-term assumption accordingly holds. 
%
As $t''-t'$ become larger, several caustics approaches to the
real-valued trajectory~\cite{AT:Preprint-1999}. Consequently the
influence from the caustics to the Feynman kernel become significant to
produce the quantum interference phenomena between multiple classical
trajectories. Hence the single-term assumption breaks
down~\cite{Adachi:AP-195-45}.
%
Furthermore, the caustics induce the divergence of the semiclassical
amplitude factor (cf. eq.~(\ref{eq:Eq})). Such divergence induces the
divergence of $\Order (\hbar)$ contribution to
$\Weak(\hat{A})$~(\ref{eq:WKBWeakq}) as well as the divergence of the
semiclassical evaluation of weak variance 
$\Weak(\{\hat{A}-\Weak(\hat{A})\}^2)$.  
%
Although the semiclassical method itself does not breakdown even in the
presence of the quantum interference, the semiclassical expressions of
the weak values become complicated in general. 

In summary, at the breakdown of the single-term assumption, the
semiclassical evaluation encounters large fluctuations due to the
caustics. After the large fluctuations, the interference between the
multiple classical trajectories emerge. 




\section{Summary}
\label{sec:SummaryAndOutlook}

The present argument establishes an intimate relationship between weak
values and classical trajectories that appear in the semiclassical
evaluations of Feynman kernels, when the quantum interference between
multiple classical trajectories are negligible. 
%
In particular, it is shown that the semiclassical theory has an ability
to reproduce complex-valued or surprisingly large, ``anomalous'', weak
values, with the help of the complex-valued classical trajectories.

%
%This explains complex-valued or surprisingly large, ``anomalous'', weak
%values, without relying on quantum interference phenomena. 
%
%From the semiclassical point of view, we do not have any way to
%discriminate ``anomalous'' weak values from ``normal'' ones.


% Finally, in order to give an outlook, I mention about an attempt to find
% the counterpart of complex-valued classical trajectories, in nature.
% %
% It is well-known that the complex-valued classical trajectories
% are indispensable in the semiclassical studies of various physical
% phenomena (see, for example, recent developments of the studies on
% ``quantum chaos''~\cite{Gutzwiller:CCQM-1990} of classically
% forbidden processes~\cite{Adachi:AP-195-45,ChaoticTunnel}).
% % %
% % When we encounter quantum phenomena that includes classically forbidden
% % processes, e. g. tunneling~\cite{LandauLifshits:QM-Tunnel} and
% % nonadiabatic transitions~\cite{ComplexNonadiabatic,Miller:JCP-56-1972}, 
% % the complex-valued classical trajectories play crucial roles. Recent
% % studies elucidate that this approach enable us to introduce the
% % knowledge of dynamical systems to investigate ``quantum chaos''
% % \cite{Gutzwiller:CCQM-1990} in classically
% % forbidden processes \cite{Adachi:AP-195-45,ChaoticTunnel}. Furthermore,
% % the complex-valued classical trajectories are also employed in exact
% % studies of quantum
% % theory~\cite{BalianBloch:AP-85-514,Wang:ComplexNelson}.    
% % The complex-valued classical trajectories thus provide a theoretical
% % framework for the explanations of many physical phenomena.
% %
% Nonetheless, it is presumed that such complex-valued
% trajectories do not have any counterpart in nature and are just
% artifacts due to approximate theories (e. g. the stationary phase
% evaluations of Feynman path integrals).
% %
% The result obtained in the present letter suggest us that the weak
% measurement enable us to approximately ``observe'' complex-valued
% classical trajectories, as statistical expectation values.
% %
% %Namely, we may find the counterparts of
% %complex-valued classical trajectories in nature.
% %
% %
% %One point is a statistical nature of weak values. 
% %
% %With the conventional interpretation of quantum theory, a weak value is
% %a property of a statistical ensemble, which is specified by initial and
% %final states. Hence it is not clear that the notion of weak value
% %concerns about physical reality.
% %
% % In this respect, Vaidman~\cite{Aharonov:qph-0105101}
% % suggests that the relevance of weak value and the many-worlds
% % interpretation of quantum theory~\cite{Everett:RMP-29-454} to enable us
% % to talk about weak values without relying on statistical ensembles.   
% %
% %However, this idea has a difficulty that the semiclassical
% %formula~(\ref{eq:WKBWeakq}) is just an approximate one. Namely,
% %complex-valued classical trajectories are observable only
% %approximately. However, there are several systems whose semiclassical
% %theories are exact. In particular, the examples presented in this letter
% %are also exact. I point out that the situation is 
% %
% Although one may argue that the relationship between the complex-valued
% classical trajectory and weak values has semiclassical error of
% $\mathcal{O}(\hbar)$, real-valued classical trajectories have the same
% drawback.  
% %
% Namely, in short time scales, we can find the counterparts of both real-
% and complex-valued classical trajectories, in nature, statistically as
% well as approximately, although for much longer time scales, we lose
% such a correspondence.
% %
% In other words, the ``degrees of the physical reality'' of real- and
% complex-valued classical trajectories are qualitatively the same. 
% %
% Thus we have no reason to discriminate against complex-valued classical
% trajectories.



%-----------------------------------------------------
\section*{Acknowledgments}
%-----------------------------------------------------
A part of this work is carried out at Institute of Physics,
University of Tsukuba, where this work is supported by University of
Tsukuba Research Projects.




%==============================================================================
\begin{thebibliography}{99}
%-----------------------------------------------------------------------------
 \bibitem{Neumann:MGQ-1932}{J. von~Neumann,
	 Mathematische grundlagen der quantenmechanik
	 (Springer Verlag, Ber\-lin, 1932), \S~VI.3.}
 \bibitem{AAV:PRL-60-1351}{Y. Aharonov, D. Z. Albert and L. Vaidman,
	 Phys. Rev. Lett. 60 (1988) 1351.}
 \bibitem{AharonovVaidman:PRA-1990-41}{Y. Aharonov and L. Vaidman,
	 Phys. Rev. A 41 (1990) 11.}
 \bibitem{Aharonov:PLA-124-199}{%
	 Y.~Aharonov, D. Z. Albert, A. Casher and
	 L. Vaidman, Phys. Lett. A 124 (1987) 199.}
 \bibitem{RSH:PRL-66-1107}{%
	 N. W. M. Ritchie, J. G. Story and R. G. Hulet, 
	 Phys. Rev. Lett. 66 (1991) 1107.}
 \bibitem{Parks:PRSLA-454-2297}{%
	 A. D. Parks, D. W. Cullin, and D. C. Stoudt,  
	 Proc.~R.~Soc.~Lond. A 454 (1998) 2997.}
 \bibitem{Parks:JPA-33-2555}{A. D. Parks, J. Phys. A. 33 (2000) 2555.}
 \bibitem{Wang:ComplexNelson}{M. S. Wang, 
	 Phys. Rev. Lett. 79 (1997) 3319; Phys. Rev. A 57 (1998) 1565.}
 \bibitem{Steinberg:PRL-74-1995}{A. M. Steinberg, 
	 Phys. Rev. Lett. 74 (1995) 2405.}
%
 \bibitem{Berry:RPP-35-315}{M V Berry and K E Mount, 
	 Rep. Prog. Phys. 35 (1972) 315.}
 \bibitem{Berry:FTF-1994}{M Berry, 
	 in Quantum Coherence and Reality, 
	 eds. J S Anandan and J L Safko 
	 (World Scientific, Singapore, 1994) p.~55.}
% \bibitem{Duck:PRD-40-2112}%
%	 {I. M. Duck, P. M. Stevenson and E. C. G. Sudarshan
%	 Phys. Rev. D. 40 (1989) 2112.}
	 %
 \bibitem{Tunneling}{See, e.g., ref.~\cite{Bohm:QuantumTheory}, Chap.~12.}
 \bibitem{Bohm:QuantumTheory}{D.~Bohm, 
	 {\em Quantum theory} (Dover, New York, 1989).}
	 %
 \bibitem{ComplexNonadiabatic}{%
	 L. Landau, Physik. Z. Sowjet. 2 (1932) 46;
	 E. C. G. Stueckelberg, Helv. Phys. Acta 5 (1932) 369;
	 J-T. Hwang and P. Pechukas, J. Chem. Phys. 67 (1977) 4640.}
 \bibitem{Miller:JCP-56-1972}{W. H. Miller and T. F. George,
	 J. Chem. Phys. 56 (1972) 5668.}
	 %
 \bibitem{Gutzwiller:CCQM-1990}{M. C. Gutzwiller, 
	 Chaos in Classical and Quantum Mechanics
	 (Springer-Verlag, New York, 1990).}
 \bibitem{Adachi:AP-195-45}{S. Adachi, Ann. Phys. 195 (1989) 45.}
 \bibitem{ChaoticTunnel}{%
	 A. Shudo and K. S. Ikeda, Phys. Rev. Lett. 74 (1995) 682;
	 K.~Takahashi and K.~S.~Ikeda, Ann.~Phys. (N.~Y.) 283 (2000) 94;
	 T. Onishi et al., Phys. Rev.~E 64 (2001) 025201.}
	 %
 \bibitem{Schulman:TAPI-1981}{L. S. Schulman, 
	 Techniques and applications of path integration,
	 (Wiley, New York, 1981).}
 \bibitem{Gutzwiller:JMP-8-1979}{M. C. Gutzwiller, 
	  J. Math. Phys. 8 (1967) 1979.}
 \bibitem{VanVleck:PNASUSA-14-178}{J. H. van Vleck, 
	  Proc. Natl. Acad. Sci. USA 14 (1928) 178.}
 \bibitem{Miller:ACP-25-69}{W. H. Miller, 
	 Adv. Chem. Phys. 25 (1974) 69.}
 \bibitem{Weissman:JCP-76-4067}{Y. Weissman, 
	 J. Chem. Phys. 76 (1982) 4067.}
 \bibitem{SCPI}{J. R. Klauder, 
	 in Path Integrals, edited by G. J. Papadopoulos and J. T. Devreese, 
	 NATO Advanced Summer Institute (Plenum, New York, 1978), p.~5; 
	 in Random Media, edited by G. Papanicolauou, 
	 (Springer-Verlag, New York, 1987), p.~163.} 
 \bibitem{KlauderSkagestram:CS-1985}{J. R. Klauder and B.-S. Skagerstam,
	 Coherent states (World Scientific, Singapore, 1985).}
 \bibitem{KMS:GTIA-3-249}{P. Kramer, M. Moshinsky and T. H. Seligman,
	 in Group Theory and Its Applications, 
	 edited by E.~M.~Loebl (Academic Press, New York, 1975),
	 Vol.~III, p.~249.}
 \bibitem{DaubechiesKlauder:JMP-26-2239}{%
	 I. Daubechies and J. R. Klauder, 
	 J. Math. Phys. 26 (1985) 2239.}
	 %
 \bibitem{EhrenfestTheorem}%
	 {See, e.g.,  ref.~\cite{Bohm:QuantumTheory}, \S~9.26.}
	 %
 \bibitem{Littlejohn:JSP-68-7}{R. G. Littlejohn, 
	 J. Stat. Phys. 68 (1992) 7.}
 \bibitem{Caustics}{Ref.~\cite{Schulman:TAPI-1981}, Chap.~15.}
 \bibitem{AT:Preprint-1999}{Atushi Tanaka, (unpublished).}
% \bibitem{BalianBloch:AP-85-514}{R. Balian and C. Bloch,
%		  Ann. Phys. (N.~Y.) {\bf 85}, 514 (1974).}
%
% \bibitem{Aharonov:PR-134-B1410}{Y.~Aharonov, P. G. Bergmann, and
%		  J.~L. Lebowitz, 
%		  Phys. Rev. {\bf 134}, B1410 (1964)}
% \bibitem{Rubin:AP-241-212}{A. Rubin and J. R. Klauder, {Ann. Phys.}
%		  {\bf 241}, 212 (1995).}
%  \bibitem{CSH}{R.~B.~Griffiths, J. Stat. Phys.~{\bf 36}, 219 (1984);
% 		  Phy. Rev. A {\bf 57}, 1604 (1998),
% 		  R.~Omn{\'e}s, Rev. Mod. Phys.~{\bf 64}, 339 (1992),
% 		  M.~Gell-Mann and J.~B. Hartle, Phys. Rev.~D {\bf 47},
% 		  3345 (1993)}
%  \bibitem{SimMeasure}{%
% 		  E.~Arthurs and J. L. Kelly, 
% 		  Bell Syst. Tech. J. {\bf 44}, 725 (1965);
% 		  S. L. Braunstein, C. M. Caves, and G. J. Milburn,
% 		  Phys. Rev. A {\bf 43}, 1153 (1991);
% 		  S. Stenholm, Ann. Phys. (NY) {\bf 218}, 233 (1992)}
%  \bibitem{Aharonov:qph-0105101}{Y. Aharonov and L. Vaidman,
% 		  arXiv:quant-ph/0105101 (2001).}
%  \bibitem{Everett:RMP-29-454}{H. Everett, III, Rev. Mod. Phys. {\bf 29},
% 		  454 (1957)}
%  \bibitem{Bohm:PR-85-166}{D. Bohm, {Phys. Rev.} {\bf 85}, 166 (1952);
% 		  {Phys. Rev.} {\bf 85}, 180 (1952).}
%  \bibitem{Nelson:PR-150-1079}{E. Nelson, {Phys. Rev.} {\bf 150}, 1079
% 		  (1966)} 
%
\end{thebibliography}
%==============================================================================

\end{document}


quant-ph0203149.extracted_bib
\begin{thebibliography}{99}
%-----------------------------------------------------------------------------
 \bibitem{Neumann:MGQ-1932}{J. von~Neumann,
	 Mathematische grundlagen der quantenmechanik
	 (Springer Verlag, Ber\-lin, 1932), \S~VI.3.}
 \bibitem{AAV:PRL-60-1351}{Y. Aharonov, D. Z. Albert and L. Vaidman,
	 Phys. Rev. Lett. 60 (1988) 1351.}
 \bibitem{AharonovVaidman:PRA-1990-41}{Y. Aharonov and L. Vaidman,
	 Phys. Rev. A 41 (1990) 11.}
 \bibitem{Aharonov:PLA-124-199}{%
	 Y.~Aharonov, D. Z. Albert, A. Casher and
	 L. Vaidman, Phys. Lett. A 124 (1987) 199.}
 \bibitem{RSH:PRL-66-1107}{%
	 N. W. M. Ritchie, J. G. Story and R. G. Hulet, 
	 Phys. Rev. Lett. 66 (1991) 1107.}
 \bibitem{Parks:PRSLA-454-2297}{%
	 A. D. Parks, D. W. Cullin, and D. C. Stoudt,  
	 Proc.~R.~Soc.~Lond. A 454 (1998) 2997.}
 \bibitem{Parks:JPA-33-2555}{A. D. Parks, J. Phys. A. 33 (2000) 2555.}
 \bibitem{Wang:ComplexNelson}{M. S. Wang, 
	 Phys. Rev. Lett. 79 (1997) 3319; Phys. Rev. A 57 (1998) 1565.}
 \bibitem{Steinberg:PRL-74-1995}{A. M. Steinberg, 
	 Phys. Rev. Lett. 74 (1995) 2405.}
%
 \bibitem{Berry:RPP-35-315}{M V Berry and K E Mount, 
	 Rep. Prog. Phys. 35 (1972) 315.}
 \bibitem{Berry:FTF-1994}{M Berry, 
	 in Quantum Coherence and Reality, 
	 eds. J S Anandan and J L Safko 
	 (World Scientific, Singapore, 1994) p.~55.}
% \bibitem{Duck:PRD-40-2112}%
%	 {I. M. Duck, P. M. Stevenson and E. C. G. Sudarshan
%	 Phys. Rev. D. 40 (1989) 2112.}
	 %
 \bibitem{Tunneling}{See, e.g., ref.~\cite{Bohm:QuantumTheory}, Chap.~12.}
 \bibitem{Bohm:QuantumTheory}{D.~Bohm, 
	 {\em Quantum theory} (Dover, New York, 1989).}
	 %
 \bibitem{ComplexNonadiabatic}{%
	 L. Landau, Physik. Z. Sowjet. 2 (1932) 46;
	 E. C. G. Stueckelberg, Helv. Phys. Acta 5 (1932) 369;
	 J-T. Hwang and P. Pechukas, J. Chem. Phys. 67 (1977) 4640.}
 \bibitem{Miller:JCP-56-1972}{W. H. Miller and T. F. George,
	 J. Chem. Phys. 56 (1972) 5668.}
	 %
 \bibitem{Gutzwiller:CCQM-1990}{M. C. Gutzwiller, 
	 Chaos in Classical and Quantum Mechanics
	 (Springer-Verlag, New York, 1990).}
 \bibitem{Adachi:AP-195-45}{S. Adachi, Ann. Phys. 195 (1989) 45.}
 \bibitem{ChaoticTunnel}{%
	 A. Shudo and K. S. Ikeda, Phys. Rev. Lett. 74 (1995) 682;
	 K.~Takahashi and K.~S.~Ikeda, Ann.~Phys. (N.~Y.) 283 (2000) 94;
	 T. Onishi et al., Phys. Rev.~E 64 (2001) 025201.}
	 %
 \bibitem{Schulman:TAPI-1981}{L. S. Schulman, 
	 Techniques and applications of path integration,
	 (Wiley, New York, 1981).}
 \bibitem{Gutzwiller:JMP-8-1979}{M. C. Gutzwiller, 
	  J. Math. Phys. 8 (1967) 1979.}
 \bibitem{VanVleck:PNASUSA-14-178}{J. H. van Vleck, 
	  Proc. Natl. Acad. Sci. USA 14 (1928) 178.}
 \bibitem{Miller:ACP-25-69}{W. H. Miller, 
	 Adv. Chem. Phys. 25 (1974) 69.}
 \bibitem{Weissman:JCP-76-4067}{Y. Weissman, 
	 J. Chem. Phys. 76 (1982) 4067.}
 \bibitem{SCPI}{J. R. Klauder, 
	 in Path Integrals, edited by G. J. Papadopoulos and J. T. Devreese, 
	 NATO Advanced Summer Institute (Plenum, New York, 1978), p.~5; 
	 in Random Media, edited by G. Papanicolauou, 
	 (Springer-Verlag, New York, 1987), p.~163.} 
 \bibitem{KlauderSkagestram:CS-1985}{J. R. Klauder and B.-S. Skagerstam,
	 Coherent states (World Scientific, Singapore, 1985).}
 \bibitem{KMS:GTIA-3-249}{P. Kramer, M. Moshinsky and T. H. Seligman,
	 in Group Theory and Its Applications, 
	 edited by E.~M.~Loebl (Academic Press, New York, 1975),
	 Vol.~III, p.~249.}
 \bibitem{DaubechiesKlauder:JMP-26-2239}{%
	 I. Daubechies and J. R. Klauder, 
	 J. Math. Phys. 26 (1985) 2239.}
	 %
 \bibitem{EhrenfestTheorem}%
	 {See, e.g.,  ref.~\cite{Bohm:QuantumTheory}, \S~9.26.}
	 %
 \bibitem{Littlejohn:JSP-68-7}{R. G. Littlejohn, 
	 J. Stat. Phys. 68 (1992) 7.}
 \bibitem{Caustics}{Ref.~\cite{Schulman:TAPI-1981}, Chap.~15.}
 \bibitem{AT:Preprint-1999}{Atushi Tanaka, (unpublished).}
% \bibitem{BalianBloch:AP-85-514}{R. Balian and C. Bloch,
%		  Ann. Phys. (N.~Y.) {\bf 85}, 514 (1974).}
%
% \bibitem{Aharonov:PR-134-B1410}{Y.~Aharonov, P. G. Bergmann, and
%		  J.~L. Lebowitz, 
%		  Phys. Rev. {\bf 134}, B1410 (1964)}
% \bibitem{Rubin:AP-241-212}{A. Rubin and J. R. Klauder, {Ann. Phys.}
%		  {\bf 241}, 212 (1995).}
%  \bibitem{CSH}{R.~B.~Griffiths, J. Stat. Phys.~{\bf 36}, 219 (1984);
% 		  Phy. Rev. A {\bf 57}, 1604 (1998),
% 		  R.~Omn{\'e}s, Rev. Mod. Phys.~{\bf 64}, 339 (1992),
% 		  M.~Gell-Mann and J.~B. Hartle, Phys. Rev.~D {\bf 47},
% 		  3345 (1993)}
%  \bibitem{SimMeasure}{%
% 		  E.~Arthurs and J. L. Kelly, 
% 		  Bell Syst. Tech. J. {\bf 44}, 725 (1965);
% 		  S. L. Braunstein, C. M. Caves, and G. J. Milburn,
% 		  Phys. Rev. A {\bf 43}, 1153 (1991);
% 		  S. Stenholm, Ann. Phys. (NY) {\bf 218}, 233 (1992)}
%  \bibitem{Aharonov:qph-0105101}{Y. Aharonov and L. Vaidman,
% 		  arXiv:quant-ph/0105101 (2001).}
%  \bibitem{Everett:RMP-29-454}{H. Everett, III, Rev. Mod. Phys. {\bf 29},
% 		  454 (1957)}
%  \bibitem{Bohm:PR-85-166}{D. Bohm, {Phys. Rev.} {\bf 85}, 166 (1952);
% 		  {Phys. Rev.} {\bf 85}, 180 (1952).}
%  \bibitem{Nelson:PR-150-1079}{E. Nelson, {Phys. Rev.} {\bf 150}, 1079
% 		  (1966)} 
%
\end{thebibliography}