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\pagestyle{myheadings}  %%{Authors}{Title}
\markboth{\small \em \quad Michael B.~Mensky \hfill}{\hfill \small \em 
Decoherence as a fundamental phenomenon  \quad}

\thispagestyle{plain}
\setcounter{page}{151}

\begin{document}
\sloppy 

\title{DECOHERENCE AS A FUNDAMENTAL PHENOMENON
IN QUANTUM DYNAMICS}
\author{MICHAEL B.~MENSKY}
\address{ P.N.~Lebedev Physical Institute, 117924 Moscow, Russia}


\maketitle

\abstracts{
The phenomenon of decoherence of a quantum system caused by the
entanglement of the system with its environment is discussed from
different points of view, particularly in the framework of quantum
theory of measurements. The selective presentation of decoherence
(taking into account the state of the environment) by restricted path
integrals or by effective Schr\"odinger equation is shown to follow
from the first principles or from models. Fundamental character of
this phenomenon is demonstrated, particularly the role played in it by
information is underlined. It is argued that quantum mechanics becomes
logically closed and contains no paradoxes if it is formulated as a
theory of open systems with decoherence taken into account. If one
insist on considering a completely closed system (the whole Universe),
the observer's consciousness has to be included in the theory
explicitly. Such a theory is not motivated by physics, but may be
interesting as a metaphysical theory clarifying the concept of
consciousness.
}

\vspace*{0.6cm}

\tableofcontents
\vfill
\section{Introduction}%\label{Sect-intro}

%\vspace*{-1mm}

It is honor for me to contribute to the volume in memory of my friend
Michael Marinov, particularly because this gives a good opportunity to
recall one of his early innovatory works which was not properly
understood at the moment of its publication. I mean the work started
in Refs.\,\protect\citebk{Marinov72deco}\,and\,\protect\citebk{Marinov74deco}
and continued later \mbox{in Ref.\,\protect\citebk{Marinov85deco}.}
In these papers Misha Marinov considered a sort
of nonunitary evolution of a quantum system converting a pure state
into a mixed state. Such a behavior appears in open quantum systems as
a consequence of the phenomenon which became known later as
decoherence. Decoherence was rediscovered several times in different
approaches (see for example the reviews\,\cite{Zeh-bk96}$^-$\cite{Men00bk}
%,Men98revEng,Men00bk} 
and references therein). One of
these approaches, based on restricted path integrals, will be
discussed in detail in the present paper.

Importance of the phenomenon of decoherence is now generally 
recognized, but the first two papers
by M.~Marinov\,\cite{Marinov72deco,Marinov74deco} evoked only perplexity.
Unitarity seemed to be necessary feature of any quantum evolution. Eventually
it became clear that the evolution of an open system may be non-unitary 
and must be thought of as a specific type of evolution. This 
idea was formulated  by various authors in various forms, often 
independently from each other. In many cases master equations, i.e. 
equations for density matrices, were used just as in the papers of 
M.~Marinov. The most general form of such an equation was suggested by 
G.~Lindblad.\cite{Lindblad76} However other approaches were also 
proposed for presentation of decoherence, among them: 
consistent histories,\cite{Griffiths84}$^-$\cite{Omnes94bk}
%{,GellMannHartle90,Omnes90,GellMannHartle93,Omnes94bk} 
quantum trajectories,\cite{Carmichael-bk93} stochastic 
equations,\cite{Gisin89stochast}$^-$\cite{Belavkin89stochast} 
%,Diosi89stochast,Belavkin89stochast} 
and restricted path integrals (called also quantum 
corridors).\cite{Men79a}$\!^-$\cite{Men98info,Men98revEng,Men00bk}
%\cite{Men79a,Men79b,MenBk93,Men98info,Men98revEng,Men00bk} 
The last two approaches provide the so-called selective description of 
the process in which a pure initial state remains pure despite of the 
decoherence.

I did not keep up with the literature on this subject until I
discovered that the formalism of restricted path integrals developed 
by me in order to describe continuous quantum measurements, represent 
in fact the process of gradual decoherence. It became clear later on 
that this approach to decoherence reveals very interesting conceptual 
features of the phenomenon. Below I shall discuss this in some detail.

The plan of the paper is following. In \Sect{Sect-entanglement}
it is demonstrated, with the help of the simplest model, how
entanglement of a system with its environment leads to the systems's
decoherence. In \Sect{SectRPI} the idea is presented of the
restricted-path-integral approach to continuous quantum measurements
or, what is the same, to gradual decoherence. In
\Sects{SectMonitor}{SectMonitorRealiz} monitoring an observable
is considered as a special but very important example of continuous
measurements. Fundamental aspects of decoherence are discussed in
\Sect{SectFundDecoher}. In \Sect{SectConscious} expansion of quantum
mechanics beyond the theory of decoherence is shortly commented. It is 
argued that such an expansion is not in fact physically motivated, but 
is interesting, particularly because it requires the observer's 
consciousness to be included in the theory explicitly. A short resume 
of the paper is given in \Sect{SectCoclusion}.

\section{Decoherence by entanglement with an
environment}\label{Sect-entanglement}

%\vspace*{-1mm}

A pure state of a quantum system may be described by the
wave function or state vector $\st{\psi}$. Instead, it may be
presented also by the density matrix $\rho=\oper{\psi}{\psi}$. If the
state $\st{\psi}$ is a superposition of some other states $\st{a}$ and
$\st{b}$,
\be
\st{\psi}=\alpha\st{a}+\beta\st{b}\,,
\label{pure-psi}
\ee
the density matrix takes the form
%\vspace*{-2mm}
\be
\rho=|\alpha|^2 \oper{a}{a} + \alpha\beta^*\oper{a}{b}
    + \beta\alpha^*\oper{b}{a}+ |\beta|^2 \oper{b}{b}
    = \left[ \begin{array}{cc}
    |\alpha|^2    &   \alpha\beta^* \\
    \beta\alpha^* &   |\beta|^2
    \end{array} \right].
\label{pure-rho}\ee
A mixed state may be presented only by a density matrix. The mixture
of the states $\st{a}$ and $\st{b}$ (with the weights
$|\alpha|^2$ and $|\beta|^2$) is presented by the density matrix
%\vspace*{-1mm}
\be
\rho=|\alpha|^2 \oper{a}{a} + |\beta|^2 \oper{b}{b}
    = \left[ \begin{array}{cc}
    |\alpha|^2    &   0         \\
    0             &   |\beta|^2
    \end{array} \right],
\label{mixture}\ee
differing from \eq{pure-rho} by the off-diagonal matrix elements
being zero.

The relative phase of the complex coefficients $\alpha$ and $\beta$ in
the superposition \eq{pure-psi} or \eq{pure-rho} is essential. In
order to underline this fact, the state \eq{pure-psi} or \eq{pure-rho}
is sometimes called {\em coherent superposition}. On the contrary, no 
phase at all appears in the definition of mixture \eq{mixture}. The 
process converting the pure state \eq{pure-rho} into the mixed state 
\eq{mixture} may be described as nullifying the off-diagonal elements 
of the density matrix. As a result, the density matrix becomes 
diagonal in the given basis. This process is called {\em decoherence}. 
We consider here only the simplest example of decoherence, but it is 
enough to demonstrate the idea.

Decoherence may occur in the course of interaction of the system
$\S$ with its environment $\E$. This means that the interaction
results in diagonalizing the systems's density matrix in a certain
basis. Of course, the concrete basis in which the density matrix
becomes diagonal depends on the features of the interaction. Consider
this in the same very simple example.

Let the initial state of the system $\S$ be $\st{\psi}$ from \Eq{pure-psi}
and the initial state of its environment $\E$ be $\st{\Phi}$. If the system
interacts with the environment during some time, the initial state
$\st{\psi}\st{\Phi}$ of the total system (composed of $\S$ and  $\E$
) goes over to the state $U\st{\psi}\st{\Phi}$ with some unitary
evolution operator $U$ (depending on the interaction). Let the states
$\st{a}$, $\st{b}$ be conserved by the interaction so that
\be
U\st{a}\st{\Phi} = \st{a}\st{\Phi_a}, \quad
U\st{b}\st{\Phi} = \st{b}\st{\Phi_b}\,.
\label{meas-interact}\ee
Such an interaction realizes a {\em measurement} of the system (by its
environment) distinguishing between the states $\st{a}$ or $\st{b}$. 
Indeed, observing the state $\st{\Phi_a}$ or $\st{\Phi_b}$ of the
environment $\E$ provides the information on what of the states
$\st{a}$ or $\st{b}$ the system $\S$ was in before the interaction 
(and stays after it). The interaction of this type takes place if the 
interaction Hamiltonian commutes with the observables $\oper{a}{a}$, 
$\oper{b}{b}$.

If the interaction between the subsystems $\S$ and $\E$ satisfies
the condition \eq{meas-interact}, then, owing to the linearity of
the operator $U$, the initial state $\st{\psi}\st{\Phi}$ of
\eq{pure-psi} goes over (after the interaction) to the state
\be
\st{\Psi}=U\st{\psi}\st{\Phi}
=\alpha\st{a}\st{\Phi_a}+\beta\st{b}\st{\Phi_b}\,.
\label{entangled}\ee

The state \eq{entangled} is said to be an {\em entangled state} of the 
two subsystems $\S$ and $\E$. This term underlines that the state 
cannot be presented in the form of the product of state vectors of 
each of these subsystems. Entanglement of two subsystems leads to 
decoherence of each of them. This may be shown as follows.

If the state of the compound system $\S + \E$ is presented by
the vector $\st{\Psi}$ (or, what is the same, by the density matrix
$\oper{\Psi}{\Psi}$), then the state of its subsystem $\S$ is
described by the reduced density matrix equal to the trace of
$\oper{\Psi}{\Psi}$ over the degrees of freedom of the subsystem $\E$:
%\ba
%&&\!\!\!\!\!\!\rho = \tr_{\Phi} \oper{\Psi}{\Psi} 
%\!=\!
%|\alpha|^2 \oper{a}{a}\! +\! \alpha\beta^*\braket{\Phi_b}{\Phi_a}\oper{a}{b}
%\! + \!\beta\alpha^*\braket{\Phi_a}{\Phi_b}\oper{b}{a}\!+\! |\beta|^2 \oper{b}{b}
%                                                   \nonumber \\[2mm]
%&&=
%\left[\!\! \begin{array}{cc}
%|\alpha|^2                   &   \alpha\beta^*\braket{\Phi_b}{\Phi_a} \\[1.5mm]
%\beta\alpha^*\braket{\Phi_a}{\Phi_b}             &   |\beta|^2
%    \end{array} \!\!\right] .
%\label{reduced-rho}\ea
\ba
\rho
\!\!\!&=&\!\!\!\tr_{\Phi} \oper{\Psi}{\Psi}  \nonumber \\[2mm]
\!\!\!&=&\!\!\!
|\alpha|^2 \oper{a}{a} + \alpha\beta^*\braket{\Phi_b}{\Phi_a}\oper{a}{b}
 + \beta\alpha^*\braket{\Phi_a}{\Phi_b}\oper{b}{a}+ |\beta|^2 \oper{b}{b}
                                                   \nonumber \\[2.5mm]
\!\!\!&=&\!\!\!\!
\left[\!\! \begin{array}{cc}
|\alpha|^2                   &   \alpha\beta^*\braket{\Phi_b}{\Phi_a} \\[1.5mm]
\beta\alpha^*\braket{\Phi_a}{\Phi_b}             &   |\beta|^2
    \end{array} \!\!\right] .
\label{reduced-rho}\ea
Here the states $\st{\Phi_a}$, $\st{\Phi_b}$ are taken to be
normalized. Besides, the modulus of the scalar product of these states
is smaller than unity. Therefore, the off-diagonal matrix elements of
the matrix \eq{reduced-rho} are smaller than those of the matrix
\eq{pure-rho}. This means that a partial decoherence of this system
occurred as a result of the interaction with the environment. The
complete decoherence occurs if the states $\st{\Phi_a}$, $\st{\Phi_b}$
of the environment are orthogonal. This always takes place if the
environment is macroscopic and these states are macroscopically
distinct. The complete decoherence takes place also in the mesoscopic 
situation provided these states are orthogonal.

We compared only the initial (before decoherence) and final (after
decoherence) states of the system. Typical is the situation when
decoherence develops in a very short time.\cite{Zurek91deco} Then
there is no reason (at least from practical point of view) to consider
time evolution of the system during this short period. If however
decoherence is slow, it is important to describe it as a process
developing in time. Evidently, this may be described as time
dependence of the density matrix of the system. In most cases the
process is Markovian, i.e. the state (density matrix) at some moment
completely determines the state in all future moments. In this case 
time dependence of the density matrix may be characterized by a 
differential equation including the time derivative. This sort of 
equation was used by
M.\,Marinov\,\cite{Marinov72deco}$\!^-$\cite{Marinov85deco} who added the 
double-commutator term to the usual von Neumann equation:
\be
\frac{d\rho}{dt}=-\,\ioverh\,[\hat H,\rho]
   -\tau\frac{1}{2\hbar^2}\,[\,\hat H,[\hat H,\rho]\,]+{\rm O}(\tau^2)\,.
\label{marinov-eq}\ee
A more general equation for the density matrix was investigated by
G.\,Lindblad.\cite{Lindblad76} Such equations are usually called {\em
master equations}.

It is important for our aim that the slow gradual decoherence may be
interpreted as a {\em continuous measurement}. The idea is following.
Interaction of a quantum system with its environment changes the state
of the environment. As a result, a certain information about the state 
of the system is contained in the state of the environment. The 
interaction may therevore be considered as a sort of prolonged 
(continuous) measurement. We shall see below that this enables one to 
describe the process in terms of {\em restricted path integrals}.

The question naturally arises about what are conditions for the 
decoherence to be slow. We mentioned already that a macroscopic 
environment leads usually to a very fast decoherence. However 
decoherence caused by a mesoscopic environment is often slow. This is 
the case for example in quantum-optical models of 
measurements.\cite{RaimondBruneHaroche97} Another example of a slow 
decoherence may become practically important. It is decoherence in 
quantum computers. In an ideal case no decoherence at all should occur 
in a device working as a quantum computer. In practice, decoherence in 
a quantum computer is inevitable but has to be very slow.

\section{Restricted path integrals}\label{SectRPI}

%\vspace*{-1mm}

Decoherence of a quantum system is caused by an interaction with its
environment. Therefore the description of decoherence may be derived
from the consideration of a concrete model of the environment and the
interaction with it. It is interesting however that such a
description may also be obtained without any model, from general
principles. This may be achieved with the help of the path-integral
formulation of quantum mechanics suggested by
R.\,Feynman\,\cite{FeynBk65eng} and the formalism of {\em restricted path
integrals} developed by the present
author\,\cite{Men79a}$\!^-$\cite{MenBk93,Men00bk} after the idea of
Feynman.\cite{Feynman48}

In the Feynman's formulation of quantum mechanics the probability 
amplitude for a system to propagate during a time interval 
$[t',t'']$ along a path
$$
[p,q]=\{\,[p(t),q(t)]\ |\ t'\!<\!t\!<\!t''\,\}
$$
in the phase space of this system is equal to
\be
U[p,q]=\exp\left[
\ioverh \int_{t'}^{t''} dt\, (p\dot q - H(p,q,t))
\right].
\label{phase-path-ampl}\ee

If, under  conditions of the given experiment, there is no way to find
out what path is taken by the system, the complete propagator between
the points $q'$ and $q''$ must be
calculated as the sum (integral) over all paths $[p,q]$. This leads to
the propagator in the form of the {\em Feynman path integral} in the
phase-space representation:
%\vspace*{-3mm}
\be
U(q''\!, t''\,|\,q', t') = \int d\,[p,q] \, \exp\left[
\ioverh \int_{t'}^{t''} dt\, (p\dot q - H(p,q,t))
\right],
\label{Feyn-prop}\ee
where the integral is meant over $[p,q]\!=\![p]\,[q]$ with all paths $[p]$
in the momentum space and those paths $[q]$ in the configuration space
which satisfy the boundary conditions $q(t')=q'\!$, $q(t'')=q''\!$.

The measure of the path integration (in symbolic form, see details
in Refs.\,\protect\citebk{FeynBk65eng}\,and\,\protect\citebk{Men00bk})
is\,\footnote{\,Here a one-dimensional system
is considered for simplicity. In case of many degrees of freedom an
analogous measure must be taken for each of them.}
\be
d[p,q]=\prod_{t=t'}^{t''}\!\frac{dq(t)\,dp(t)}{2\pi\hbar}\ .
\label{phase-path-measure}\ee
The propagator \eq{Feyn-prop} satisfies Schr\"odinger equation,
\be
\left(\!\frac{d}{dt''}+\ioverh \hat H''\!\right) U(q''\!, t''\,|\,q', t') = 0\,.
\label{Schr-eq}\ee
The two primes of the operator $\hat H''$ mean here that this operator
acts on the variable $q''\!$.

If a {\em continuous measurement} is performed during the time interval
$[t',t'']$, then the result (readout) $\al$ of this measurement
contains certain information about what path $[p,q]$ the system
propagates along. In this case the path integral has to be restricted
in accord with this information. In the simplest case the measurement
readout being $\al$ means that the path taken by the system lies in
a certain subset $J_\al$ of paths. Then the integration in the path
integral has to be restricted by this subset of paths. In a
more general (and more realistic) case the information is expressed
by some weight functional $w_\al[p,q]$ in the space of paths. Then
the path integral has to be taken over all paths but with this
functional in the integrand:
\be
U_\al(q''\!, t'' \,|\, q', t') = \int\!d[p,q] \, w_\al[p,q] \exp\left[
\ioverh \int_{t'}^{t''} \hspace{-3mm}dt\,\big[p\dot q - H(p,q,t)\big]
\right].
\label{weighted-path-int}\ee

This gives {\em partial propagators} $U_\al(q''\!, t'' \,|\, q', t')$ in the form of
a {\em restricted path integral}\,.\cite{Men79b,MenBk93,Men00bk} The system's
evolution is presented then by the set of {\em partial evolution operators}
$U_\al$ having the partial propagators as their kernels. Namely, the initial
state $\st{\psi}$ is converted into the state 
\be
\st{\psi_\al(t'')}=U_\al\st{\psi(t')}
\label{partial-evolution-psi}\ee
in the evolution during the time interval $[t',t'']$ under the 
condition that the continuous measurement during this interval had 
been performed and gave the readout $\al$. The same may be expressed 
in terms of the density matrix:
\be
\rho_\al(t'')=U_\al\rho(t')U_\al^\dagger\,.
\label{partial-evolution-rho}\ee
The latter form of the evolution law is applicable also to a mixed 
initial state.

The state vector \eq{partial-evolution-psi} and the density matrix
\eq{partial-evolution-rho} are not normalized. Instead, the
square norm of the state vector or the trace of the density matrix is
the {\em probability density of the measurement readout} $\al$
(assuming the initial state is normalized). Thus,
\be
\int_{\!\!\cal A}\!d\al \, \tr \rho_\al(t'')
\label{prob-A}\ee
is the probability that the measurement readout $\al$ belongs to
the set ${\cal A}$.

The formulas (\ref{partial-evolution-psi}) and (\ref{partial-evolution-rho})
provide a {\em selective description} of the evolution which takes
into account the readout given by the measurement. If the measurement
is performed but its readout is not known, then summation
(integration) must be performed over all possible readouts:
\be
\rho(t'')
=\int d\al \, \rho_\al(t'')=\int d\al \, U_{\!\al}\,\rho(t')\,U_\al^\dagger\,.
\label{total-evolution}\ee
The measure $d\al$ has to be chosen in such a way that the following
{\em generalized unitarity condition} be valid:
%\vspace*{-3mm}
\be
\int d\al \, U_\al^\dagger U_\al = {\bf 1}.
\label{gen-unitarity}\ee
Then the total density matrix $\rho(t'')$ is normalized provided the
initial density matrix $\rho(t')$ is. This guarantees conservation of
probability in the evolution under the continuous measurement.

\section{Monitoring of an observable}\label{SectMonitor}

%\vspace*{-1mm}

In the preceding section we considered a generic continuous
measurement and denoted its readout symbolically by $\al$.
Let us consider now a concrete and very important 
continuous measurement, the {\em monitoring of an observable}\,\footnote{\,For
simplicity we shall consider a single observable, but actually $A$ may be
multicomponent with commuting components. The generalization on the
monitoring of many non-commuting observables is preliminary discussed
in Ref.\,\protect\citebkcap{Men00bk}. The definition of path integrals may need
further elaboration in this case.}
$A=A(p,q,t)$.
A readout obtained in the course of the monitoring is presented by the
curve $[a]\!=\!\{ a(t)\,|\,t'\!<\!t\!<\!t''\}$ so that $a(t)$ is an estimate
for the value of $A$ at the time moment $t$.

A finite precision $\Delta a$ of the measurement has to be taken
into account. This means that the actual value of $A$ at time $t$
may differ from $a(t)$ but not more than by $\Delta a$. Therefore the
readout denoted as $[a]$ may be in a more adequate manner presented by
the {\em corridor} $J_{[a]}$ of curves $[a']$ close to the curve $[a]$.
Namely, $J_{[a]}$ may be the corridor of the width $2\Delta a$ with
the curve $[a]$ in its center. Then the partial propagator or partial
evolution operator $U_{[a]}$ will be equal to the path integral over
the paths $[p,q]$ such that $A\,[p(t),q(t),t\,]$ is a curve lying in the
corridor $J_{[a]}$ for each of these paths.

This definition of the partial propagator is palpable but not quite
realistic. More realistic would be a {\em `corridor with fuzzy 
boundaries'} that may be presented by a weight functional 
$w_{[a]}[p,q]$. The functional must be close to unity for the curve 
$A(p(t),q(t),t)$ near $a(t)$, and close to zero for these two 
curves far from each other. The simplest (and also quite realistic, 
see \Sect{SectMonitorRealiz}) choice of the weight functional is {\em 
Gaussian functional}:
\be
w_{[a]}[p,q]=\exp\left[ -\kappa \int_{t'}^{t''} dt \,
\Big( A(p(t),q(t),t)-a(t)\Big)^2
\right].
\label{Gauss-weight}\ee

Here $\kappa$ is a parameter inversely proportional to the `width' of
the Gaussian corridor. Therefore it defines the precision of the
measurement. A more obvious characteristic, the `width' $\Delta a$ of
the fuzzy corridor may be defined by the formula
\be
\kappa=\frac{1}{\Delta a^2 (t''-t')}\ .
\label{cor-width}\ee
It is important that $\Delta a$ depends on the duration $T=t''-t'$ of
the continuous measurement so that a longer measurement is more
precise (for constant $\kappa$).

A corridor of paths as well as the corresponding restricted path
integral may be called {\em quantum corridor}. The palpable image of a
corridor the path integral is taken over, justifies applying the term
`quantum corridor' to the restricted path integral even in the most
general case.

The specific Gaussian form \eq{Gauss-weight} of the weight functional
$w_{[a]}$ in case of monitoring enables one to introduce an effective
Hamiltonian and the corresponding effective Schr\"odinger equation for
the partial propagator $U_{[a]}$ (and therefore for the `partial wave
function' $\st{\psi_{[a]}}$). Indeed, usage of
Eqs.\,\eq{Gauss-weight} and\,\eq{weighted-path-int}
gives for a partial propagator
$U_{[a]}$ the form of the Feynman path integral
%\vspace*{-3mm}
\be
U_{[a]}(q''\!, t''\,|\,q', t') = \int \! d\,[p,q] \,
\exp\!\left[
\ioverh \int_{t'}^{t''} \hspace{-2mm} dt\, \big[p\dot q - H_{[a]}(p,q,t)\big]
\right],
\label{Eff-Ham-path-int}\ee
but with the {\em effective Hamiltonian} $H_{[a]}$ which has to
be defined as follows:
\be
H_{[a]}(p,q,t)
= H(p,q,t)-i\kappa\hbar\,\Big(\!A(p,q,t)-a(t)\!\Big)^2.
\label{Eff-Ham-class}\ee
The corresponding quantum operator is
\be
\hat H_{[a]}
= \hat H-i\kappa\hbar\,\Big(\!\hat A-a(t)\!\Big)^2.
\label{Eff-Ham-qu}\ee

The partial propagator satisfies therefore the following {\em
effective Schr\"odinger equation}:
\ba
\hspace{-5mm}\left( \frac{d}{dt''}+\ioverh {\hat H''}_{[a]}\right)
\hspace{-3mm}&&\hspace{-3.5mm}
U_{[a]}(q''\!, t'' \,|\, q', t') \nonumber\\[2mm]
=&&\hspace{-6mm} \left( \frac{d}{dt''}
+\ioverh \hat H''
+\kappa\,\Big(\!\hat A-a(t)\!\Big)^2 \right)
U_{[a]}(q''\!, t'' \,|\, q', t') = 0\,.
\label{Eff-Schr-eq-prop}\ea
The same equation is satisfied by the wave function evolving
according to the partial propagator:
\be
\left( \frac{d}{dt}+\ioverh \hat H_{[a]}\!\right)
\psi_{[a]}(q, t)
= \left( \frac{d}{dt}
+\ioverh \hat H
+\kappa\,\Big(\!\hat A-a(t)\!\Big)^2 \right)
\psi_{[a]}(q, t) = 0\,.
\label{Eff-Schr-eq-state}\ee

This equation gives a {\em selective description} for the continuous
monitoring of an observable. Such a description takes into account the
measurement readout $[a]$. This enables one to efficiently explore
this type of continuous measurements.

Both the probability distribution of the measurement readouts and the
evolution of the system given the measurement readout may be found.
The procedure is following. One has to fix an initial state $\psi(t')$
and choose various curves $[a]$. For each choice of $[a]$ one has to
solve \Eq{Eff-Schr-eq-state} and find $\psi_{[a]}(t'')$. Then the
square norm $\|\psi_{[a]}(t'')\|^2$ is the probability density of the
measurement readout $[a]$ for the system initially in the state
$\psi(t')$. The time-dependent state $\psi_{[a]}(t)$, $t'\!<\!t\!<\!t''\!$,
presents the evolution of the system under the condition that the
measurement readout is $[a]$. The wave functions $\psi_{[a]}(t)$ may
be normalized if necessary.

This gives a selective description of the process of decoherence in
the course of monitoring the observable $\hat A$. Application of
this approach for monitoring various observables in a two-level
system may be found in Ref.\,\protect\citebk{Men00bk}.

Another way to selectively describe decoherence is to make use of {\em
stochastic wave
equations}\,\cite{Gisin89stochast}$\!^-$\cite{Belavkin89stochast}
(a short account of this approach is given in Ref.\,\protect\citebk{Men00bk}).
This is equivalent\,\cite{PreOnoTam96,AlbeverioKolSmol97,Men00bk} to the
effective Schr\"odinger equation discussed above. The characteristic
features of the stochastic equation method are the following: (i)~the wave
function presenting the system's evolution is normalized; (ii)~the
stochastic equation is non-linear; (iii)~the function presenting
influence of the environment has in this approach standard statistics
of white noise but no direct physical interpretation. Contrary to
this, in our approach the effective Schr\"odinger equation is linear
and the function $a(t)$ (presenting influence of the environment)
has the direct interpretation as an {\em information recorded in the 
environment about the state of the system}. The latter is important 
from the conceptual point of view, and we shall discuss this below.

If being not interested in the description being selective, one may go 
over to the {\em non-selective description} by integrating over 
measurement readouts $[a]$. According to the general formula 
\eq{total-evolution}, in case of monitoring this gives
\be
\rho\,(t'')
=\!\!\int\!\!d[a] \, \rho_{[a]}(t'')=\!\int\!\!d[a] \,
U_{[a]}\rho\,(t')\,U_{[a]}^\dagger\,.
\label{non-select-monitor}\ee
The resulting density matrix $\rho(t)$ may be shown\,\cite{Men00bk}
to satisfy the equation
\be
\frac{d\rho}{dt}=-\,\ioverh\,[\hat H,\rho]
   -\frac{\kappa}{2}\,[\,\hat A,[\hat A,\rho\,]\,]\,,
\label{master-eq-A}\ee
a special case of Lindblad equation.\cite{Lindblad76} The equation
\eq{marinov-eq} exploited by M.\,Marinov in his
papers\,\cite{Marinov72deco}$\!^-$\cite{Marinov85deco} is evidently a
special case of \eq{master-eq-A} for $\hat A = \hat H$. More
complicated continuous measurements lead in the same way to a more
general class of master equations.\cite{Men00bk}

In the preceding consideration $a(t)$ was an estimate of the value of
the observable $A$ at certain precisely known instant $t$. Very often
this may be accepted as a good approximation. We have seen that this
leads to a differential equation \eq{master-eq-A} for the density
matrix. Therefore this approximation is Markovian.

A more precise description of the monitoring may be given in the 
following way. For $[a]$ being a measurement readout, $a(t)$ is 
interpreted as an estimate of the value of the observable $A$ averaged 
over certain time interval containing the moment $t$. For example it 
may be the average of $A$ over the time interval of duration $\tau$ 
around moment $t$. The duration $\tau$ is then the {\em 
`time resolution' of the measurement}. This type of monitoring may 
also be formulated and explored in terms of restricted path 
integrals.\cite{Men00bk} This supplies a {\em non-Markovian 
approximation} in presentation of monitoring, still in the form of 
restricted path integrals. The measurement of this type is not local 
in time and therefore cannot be expressed with the help of a 
differential equation with time derivatives. 

\section{Quantum monitoring and quantum Central Limiting
Theorem}\label{SectMonitorRealiz}

%\vspace*{-1mm}

In the preceding sections a description of a continuous measurement,
or continuous decoherence, is derived from the Feynman path-integral
formulation of quantum mechanics. Such a model-independent derivation,
based only on first principles, has advantages which will be discussed
later. However there are people for whom such a derivation seems to be
not convincing. This is why a number of concrete models of systems and
their environments were explored by usual quantum-mechanical
methods\,\cite{KonMenNamiot93}$\!^-$\cite{Men97timeResol,Men00bk} and
the results were compared with those obtained in the framework of the
restricted-path-integral approach. Coincidence of the results of both
approaches was confirmed in each case.

There is one more problem solved in this series of papers. We saw in
the preceding consideration that a quantum continuous measurement
may be presented by restricted path integrals if a set $\{ \al \}$ of
possible measurement readouts and associated weight functionals
$\{ w_\al\}$ are chosen. In case of observable's monitoring the
Gaussian form of weight functionals \eq{Gauss-weight} was suggested.
The question naturally arises whether the Gaussian weight functional
describes any feasible measurement. Another formulation of the
question is whether any environment leads to the decoherence described
by the Gaussian weight functional.

The answer to this question is that `Gaussian decoherence' is not only
feasible but also typical. Namely, it arises each time when
decoherence is caused by the action of a large number of very weak
influences independent of each other. The situation resembles
Central Limiting Theorem in probability theory so that a sort of
a {\em quantum analogue of Central Limiting Theorem} seems to be
valid. This generalization has not been proven or even precisely 
formulated, but the consideration of a class of continuous 
measurements confirmed this suggestion in the following way.

The demonstration of this feature of continuous measurements was given
in Refs.\,\protect\citebk{AuMenNam97scat}\,and\,\protect\citebk{Men00bk}.
A two-level system was considered
which periodically briefly and weakly interacts with a subsidiary
device called {\em `meter'}. Thus, the two-level system undergoes a
long series of weak (fuzzy) {\em observations}. The whole series of
observations may be interpreted as a single continuous measurement.
Each interaction brings the meter into one of two orthogonal states.
The probabilities of them slightly depend on the state of the
two-level system. After each interaction the state of the meter is
measured and the meter is returned into the initial state to be ready
for the next interaction.

Each interaction of the meter with the system gives a vague
information about the state of the system and slightly changes the
system's state. In the long series of interactions the information about
the system as well as the back influence of the interaction onto the
system become significant. It is shown that the whole process,
including both probability distribution of measurement readouts and
the evolution of the system's state, are correctly described by the
effective Hamiltonian and the effective Schr\"odinger
equation \eq{Eff-Schr-eq-state}.

The latter shows that the character of continuous measurements or
of the precesses of gradual decoherence is universal. A concrete 
scheme of a continuous measurement (the monitoring of an observable in 
our case) may be complicated, particularly it may depend on many 
parameters. However, the final description of the process depends only 
on the observable $A$ being monitored and the precision $\kappa$ of 
the monitoring. It was shown\,\cite{Men00bk}
%in Ref.\,\protect\citebk{Men00bk}
how $A$ and $\kappa$ are 
determined by the parameters of the interaction of the two-level 
system with the meter.

A continuous measurement (realized by a series of weak interactions or
in any other way) may be used for obtaining information about quantum
processes `in real time', i.e. without interrupting the processes. Of
course, back influence on the process is unavoidable. This can be
demonstrated\,\cite{Men00bk} in a simple case where the energy of a
two-level system is monitored.

Let a two-level atom undergo the so-called {\em Rabi oscillation}
(i.e. periodically transits from one level to the other and backward)
under influence of the resonance laser radiation. If simultaneously
the {\em energy of the atom is monitored}, the picture of Rabi
oscillations as developing in time may be derived from the measurement
readout. The oscillation is distorted because of the back influence of
the measurement. It is interesting how this distortion depends on the
strength (resolution) of the monitoring.

If the monitoring is very weak, or very fuzzy ($\kappa$ is small), the
back influence is also very weak so that the Rabi oscillation is not
significantly distorted. However the measurement gives in this case a
vague information about the state of the atom. In the opposite case
when $\kappa$ is large (strong or highly precise monitoring), the
information about the state of the atom is quite definite. However the
back influence of the measurement is in this case so strong that the Rabi
oscillation is strongly damped. In the limiting case the Rabi oscillation
is completely prevented and the system is frozen in the initial state.
This phenomenon is known as {\em quantum Zeno effect} or quantum Zeno
paradox.\cite{MisraSudar77-Zeno,ChiuSudarMisra77-Zeno}

The regimes of monitoring with large and small $\kappa$ may be called
correspondingly {\em Zeno- and Rabi-regimes}. They are most evidently
characterized by the time parameter
\be
T_{\rm lr}=\frac{1}{\kappa\Delta E^2}\ ,
\label{decoher-time}\ee
where $\Delta E$ is the difference between two energy levels of the
atom. The para\-meter $T_{\rm lr}$ may be called {\em `level resolution
time'}. If monitoring of energy $A=H_0$ ($H_0$ being a free Hamiltonian
of a two-level atom) with the strength $\kappa$ lasts as long as
$T_{\rm lr}$, the system will transit in one of the energy eigenstate
(on one of the two energy levels) even if it is in an arbitrary
superposition initially.  The regime of the measurement depends
on the relation between $T_{\rm lr}$ and the period $T_{\rm R}$ of the
Rabi oscillation. In case of $T_{\rm lr}\!\ll\!T_{\rm R}$ the monitoring
is performed in the Zeno regime, while for $T_{\rm lr}\!\gg\!T_{\rm R}$
the Rabi regime is realized.

The most interesting is the intermediate regime when $T_{\rm lr}$ is
of the order of $T_{\rm R}$. In this case the Rabi oscillation is not
completely prevented (although somewhat damped) and the information
about the state of the atom (supplied by the monitoring) is more or
less definite. This is the only regime in which {\em transitions
between levels may be observed not as instantaneous quantum jumps
but as prolonged processes} gradually developing in time. Of course,
observing such a process influences it. The result of this back
influence is a decrease of the rate.\cite{Men00bk}

\section{Fundamental character of
decoherence}\label{SectFundDecoher}

%\vspace*{-1mm}

The restricted-path-integral presentation of both continuous
measurements and the accompanying process of decoherence quite clearly
demonstrates fundamental character of these phenomena. The
following features point out to this: (i)~theory of continuous
measurements may be derived from the first principles; (ii)~it reveals
the dynamical role of information and (iii)~it makes quantum mechanics
conceptually close.

(i)  The first feature ({\em derivability from the first principles})
was demonstrated in \Sect{SectRPI}. The starting point for the
theory was quantum mechanics formulated in terms of Feynman path
integrals. No additional assumptions or axioms were necessary to
construct theory of measurements. The main instrument used for 
constructing the theory was the ordinary quantum-mechanical
rules of dealing with probability amplitudes.

This construction gives in fact a {\em new type of evolution}
described by restricted path integrals instead of unlimited Feynman
integrals. From the physical point of view this evolution differs in
that it takes into account continuous decoherence. Such an
evolution includes {\em both quantum and classical elements}. Namely,
the alternatives $\al$ are classical (and this is why they are
characterized by probabilities) while each partial propagator or
evolution operator $U_\al$ presents purely quantum evolution inside
the given classical channel $\al$ (single paths inside each
alternative $\al$ may be associated only with probability amplitudes,
not probabilities).

Evolution of a quantum system under decoherence (continuous
measurement) may be derived from the usual quantum-mechanical 
consideration of a wider system including not only the system of 
interest but also its environment (see \Sect{SectMonitorRealiz}). 
However, it is wonderful and exciting that it may also be derived 
without explicit consideration of the environment, from the first 
principles of quantum mechanics applied to the system itself. This is 
an evidence of the fundamental character of the phenomenon.

(ii) One more exciting feature of decoherence is that the {\em back
action of the environment onto a decohering system depends only on
the information} about this system recorded in the environment.
Different environments interacting with the system in different ways
will give the same evolution of this system if the information
recorded in them is the same. For example, if the system's coordinates
at the successive time moments are recorded in any form in the
environment (so that knowledge of the environment's state enables one
to calculate these coordinates), then the interaction with the
environment may be said to realize the monitoring of this coordinate.
The system's evolution is presented then by the path integrals
restricted on the corridors in the space of this coordinate (see
\Sect{SectMonitor}). The width of the corridors depends on the
precision with which the coordinate at each moment may be
reconstructed from the state of the environment.

It is this feature that provides a universal character of the
evolution of the decohering system (for example Gaussian decoherence
in case of monitoring). Of course this is closely related to the
previously mentioned derivability of the evolution from the first
principles. From practical point of view, this makes it possible to
predict the behavior of the decohering system even if details of its
interaction with the environment are not known.

(iii) In the long-lasting discussion about conceptual problems of quantum
mechanics it was often claimed that quantum mechanics is not
conceptually closed. According to this point of view, quantum
mechanics describes evolution of a quantum system between any two
(instantaneous) measurements but it cannot describe the measurements 
of the system. What happens with the system in the course of a 
measurement cannot, according to this opinion, be presented by 
quantum-mechanical methods and requires a radically different 
instrument, for example von Neumann's projection postulate. In other 
words, quantum theory of measurements must be added to quantum 
mechanics as an independent counterpart of the complete quantum 
theory.

We saw however that {\em continuous measurements are naturally
described in the framework of quantum mechanics} if the latter is taken
in the Feynman path-integral version including theory of probability 
amplitudes (which was developed in the most complete and consistent 
form by Feynman). The theory constructed in such a way includes 
measurements not separately from the quantum evolution (as in the 
usual theory), but in the non-separable unity with it. Classical 
elements which are necessary to describe measurements (as has been 
pointed out already by N.\,Bohr) need not be added to quantum theory 
but arise in a natural way inside it (see the above-mentioned remarks 
about classical character of the alternatives $\al$).

So-called instantaneous measurements (which appear for example in von
Neumann's projection postulate) are in fact continuous but very short
ones. The concept of a strictly instantaneous measurement may if
necessary be derived from theory of continuous measurements in the
limit of null duration.\cite{Calarco95} Free evolution is of course
also a limiting case of the evolution under a continuous measurement. 
It appears if the strength of the continuous measurement tends to 
zero. Therefore the situation usually considered in quantum mechanics 
(periods of free evolution between instantaneous measurements) may be 
derived from theory of continuous measurements. This completes 
proof of {\em closeness of quantum mechanics}.\\[4mm]\indent
%\quad
Thus, quantum theory of continuous measurements is naturally contained
in (Feynman form of) quantum mechanics, and this is especially evident
in the restricted-path-integral approach to quantum measurements. The
price paid for this conceptual advantage is that the {\em theory deals
with open rather than closed systems}. Indeed, besides a Hamiltonian,
the system in such a theory is characterized by the set $\{\al\}$ of
classical alternatives. Each of these alternatives is interpreted as
an information about the system recorded in its environment after the
interaction with the system is over. Therefore, besides the system
itself some environment of this system is assumed to exist and to 
interact with the system. The system is therefore open. A closed
system may be considered as a limiting case when there is only one
alternative $\al$ with the weight functional $w_\al$ identically equal
to unity.

In the conventional quantum mechanics one is used to deal with closed
systems. Nevertheless theory of open systems is not only more general
(as it has been argued above) but also more realistic than theory of
closed systems. Indeed, on one hand there is in fact no strictly
closed system except total Universe and on the other hand one may
consider an arbitrary wide open system. One more advantage of the
restricted-path-integral theory of open systems (including
decoherence) is that it resolves known quantum-mechanical paradoxes or
rather these paradoxes do not arise in such a theory.\cite{Men00ufn}

\section{The role of consciousness in quantum 
mechanics}\label{SectConscious}

%\vspace*{-1mm}

As it has been argued above (\Sect{SectFundDecoher}), the theory of 
open quantum systems including decoherence and formulated in terms of 
restricted path integrals is a consistent and logically closed quantum 
theory. However in some aspect (not at all necessary from the point of 
view of a physicist) this theory may seem not quite satisfactory. We 
mean the two rather abstract and in fact metaphysical requirements 
that are not met by the theory of open systems. This theory is not 
sufficient 1)~if one desires to include the whole Universe as an 
object of the theory and 2)~if one desires to explain how selection of 
a single measurement readout occurs.

The {\em whole Universe} may be considered only as a {\em closed 
system}, and this is impossible in a theory dealing only with open 
systems. If the theorist requires that Universe could be considered as 
one of the objects of his theory, then the theory of decohering open 
systems is inappropriate. Of course there is no reason in physics that 
leads to this requirement. All questions which may be correctly 
formulated in physics, can be answered in the theory of open systems. 
Therefore a theory of closed systems is not necessary as a physical 
theory. It may however be considered on philosophical grounds.

The second argument for extending quantum mechanics beyond the theory 
of decohering open systems is also metaphysical. In the theory of 
decoherence all possible readouts $\{ \al\}$ of a measurement 
performed on an (open) system may be considered, the probability 
distribution over the set of these alternative readouts calculated and 
evolution of the system conditioned by any of these readouts found. 
This is what actually is necessary for a physicist. This is enough to 
make (probabilistic) predictions for any measurement or to find out 
the behavior of any (open) system.

What may one need beyond this? For a physicist the answer is: nothing.
However from metaphysical (philosophical) point of view it may seem
desirable to answer the next question. Even if being able to describe
in a probabilistic way all alternative ways of evolution of the open
system, one may ask how a concrete single alternative is selected or
{\em what is the reason (mechanism) of selection}.

These metaphysical questions or requirements are not necessary (or not
correct) from any practical (physical) point of view. Nevertheless,
from the theoretical point of view, it is interesting to consider
them, because they lead to a quite novel and exciting line of 
reasoning. The point is that {\em both theory of closed systems and 
theory of selection must include consciousness of an observer} as an 
object of consideration. This is one of the reasons why unphysical 
questions of this type have always been discussed and are still under
discussion now.\cite{Squires94}$\!^-$\cite{Whitaker00}\\[4mm]\indent
%\quad
The concept of measurement presupposes existence of an observer
who reads out the measurement results or can in principle read out
them. However, the boundary (so-called {\em Heisenberg's cut}) between
the {\em measured system} on one hand and the {\em measuring device
plus observer} on the other hand are to some extend arbitrary. A part
of the measuring device or even a part of the observer's body may be
included in the subsystem called measured system. This shifts the
Heisenberg's cut in the direction of the observer.

In the framework of the theory of open decohering systems the
measuring device and observer are not explicitly considered (although
are assumed to exist). Shifting the Heisenberg's cut to the direction 
of the observer corresponds in this theory to widening the open system 
which is under consideration. One may widen this system (measured 
system) more and more. If however one wish to remain in the framework 
of the theory of open decohering systems, he must leave something 
outside the measured system. The part of Universe left outside has to 
be capable of distinguishing between alternative measurement results. 
This may be the observer's brain or even some structures in his brain 
which are responsible for perceiving difference between the 
alternatives. Important is however that something must be outside the 
measured system.

On the other hand, if one tries to deal with an actually closed system,
one has to include the whole Universe into it. Then the observer
together with his brain and everything that may be responsible for
perceiving alternatives are parts of the considered system. The theory
then has to deal with the observer's consciousness. Any theory of 
closed systems must include consciousness.

The same may be said about any theory of selection. In the theory of
decoherence of open systems the alternatives are considered together
with the probability distribution over these alternatives. The
mechanism of selection of one of these alternatives is not considered,
and the question about this mechanism has to be estimated as
incorrect. Physics well does without this mechanism. If however one
insists (on some metaphysical grounds) on necessity to explicitly
consider the mechanism of selection, then he is obliged to include
into consideration everything that may in principle distinguish
between the alternatives. Therefore the observer's consciousness (or
consciousness' material carrier) which evidently distinguishes between
them must be included in the theory with necessity.\\[4mm]\indent
%\quad
Thus, not only a theory of closed systems but also any theory of
individual alternatives (theory of selection) must include the
observer's consciousness. The most interesting theory of this type is
so-called {\em many-world interpretation of quantum mechanics} put
forward by H.\,Everett.\cite{Everett57}$\!^-$\cite{Everett73}

In this theory a closed system (coinciding therefore with the whole
Universe) is considered. There is no decoherence in the theory, so
that pure state remains pure. If an initial (before the measurement)
pure state is a superposition of the states corresponding to the
alternative measurement readouts, then the final (after the
measurement is over) state is a superposition of macroscopically
distinct states (as it is shown in \Sect{Sect-entanglement} in the
case of a simple model). The specific assumption of Everett (or at 
least one of its formulations) was that all components of such a 
superposition are realized but `in different worlds'. This means that 
each of these components describes a possible state of Universe 
including the corresponding state of the observer (observers). All 
these states are equally real. Any two states from this set describe 
the two states of Universe corresponding to different measurement 
results. The observer's state in each of these `components of the 
superposition' describes him as perceiving the corresponding 
measurement readout.

From practical point of view this theory has no advantage. It only 
provides a different interpretation, but the physical contents of the
theory are the same as in the theory of decoherence. However the 
Everett's theory is interesting because it can shed light on the 
concept of consciousness.

Note that all alternatives are present in the Everett's theory on 
equal foot just as in theory of decoherence. However each alternative 
is now presented by the state of the whole Universe (of the closed 
system). Besides, the observers's state and specifically the 
observer's consciousness is a characteristic of the corresponding 
alternative. In a sense, this is a description of the mechanism of 
selection. 

As the next (also metaphysical) step in the analysis of this theory, 
one may assume\,\cite{Men00ufn} that {\em the concept of selection is 
identical to the concept of consciousness} (or, more precisely, the 
concept of comprehension). To make this identification more clear, one 
may say that consciousness is ability to see only a single 
alternative, only a single component in the superposition of 
macroscopically distinct states. One more formulation: {\em 
consciousness is ability to live in a single classical reality} 
even if many classical realities are included in the superposition. 
The two very complicated concepts (selection in quantum physics and 
consciousness in psychology) are then identified and therefore 
explained in terms of each other: a psychological explanation of the 
physical phenomenon of selection and a physical explanation of the 
psychological phenomenon of consciousness.

This may open new prospects both in theory of consciousness and in
quantum theory. For example one may put the question whether
the observer's consciousness can influence the result of
selection.\cite{Wigner83mind,Men00ufn} It was shown\,\cite{Men00ufn}
%in Ref.\,\protect\citebk{Men00ufn}
that the positive answer to this question (existence of {\em `active
consciousness'} capable to influence selection) may be considered to
be consistent if one is ready to essentially change the methodological 
principles usually accepted in science. It may be formulated even as 
{\em changing the criteria of reality}.

Without going into detail, let us mention that the status of {\em
personal (individual) experience} must become in this case higher than 
it is usually accepted. An individual experience may in principle be 
different from what is called `common experience'. It is only the 
common experience (but not the individual one) that can be confirmed 
or refuted by statistics. Usually only the common experience 
(confirmed by statistics) is accepted as objective reality. In the new 
methodology (which is necessary if the hypothesis of active 
consciousness is accepted) the individual experience must also be 
considered as objective.\cite{Men00ufn} In fact, this is natural in 
the framework of the Everett's theory because there are many equally 
real worlds in this theory (instead of only one real world usually 
presupposed). 

Of course, this change of methodology is radical and hardly acceptable 
for modern science based upon repeated experiments as the only way to 
prove something. Therefore, the hypothesis of active consciousness 
cannot be included into what is now called science. It has to be 
considered either as a non-scientific concept or as one leading to a 
qualitatively different (more wide) understanding of what is science. 
The new science (or the new level of science) may then include some 
spiritual phenomena which are considered, up to now, as being 
inconsistent with science. If accepting this line of thinking, one 
obtains a more wide or more general view of the world than the view 
given by the modern science. This may give a chance to include in the 
same framework those spiritual areas which are considered now as being 
quite distant from each other or even opposite, for example {\em 
science and religion}.
%\vfill

\section{Conclusion}\label{SectCoclusion}

%\vspace*{-1mm}

In the preceding sections a specific view of an important phenomenon of
decoherence was surveyed. Let us underline the main points of it.

Decoherence of a quantum system is a process caused by interaction of
the system with its environment. In the course of decoherence the
state of the system acquires classical features. This manifests itself
in the conversion of a pure state into a mixed state in the course of
decoherence. This is why decoherence is usually described by a density
matrix instead of a wave function (state vector). Typically evolution
of the density matrix in the course of decoherence is presented by the
differential equation called master equation.

A decohering system must be described by a density matrix (instead of
a state vector) if the state of the environment is undetermined. This
type of description is called non-selective. If the state of the
environment is explicitly taken into account in some way or another,
the decohering system may be described by a state vector. Such a
description is called selective.

Restricted path integrals enable one to take into account the state of
the environment in terms which are characteristic of the decohering
system itself. Namely, the influence of the environment onto the
decohering system is expressed in this approach in terms of the
information about this system which is recorded in the state of the
environment. This means that the process of decoherence may be
considered as continuous (prolonged in time) measurement of the
system.

More concretely, the information about the system recorded in the
environment (the measurement readout) may be presented as a quantum
corridor, i.e. a set of paths $[p,q]$ in the phase space of the system 
or, in a more general case, as a functional in the space of paths 
$[p,q]$. Evolution of the system under the influence of the 
environment (i.e. under the back influence of the measurement) is 
presented by the propagator equal to the path integral over the 
quantum corridor (in general case, the path integral with the 
corresponding functional in the integrand). The propagator depends on 
the information recorded in the environment (on the measurement 
readout). Complete description of the dynamics of the decohering 
(measured) system is given by a set of these partial propagators 
corresponding to all possible measurement readouts. In a special case 
of monitoring an observable, the dynamics of the measured system is 
described by an effective Schr\"odinger equation.

The formalism of restricted path integrals or (in a special case)
effective Hamiltonians allows one to construct a self-consistent
theory of open quantum systems which is fundamental i.e. derivable
from the first principles. Quantum mechanics of open systems
formulated in this language is logically closed, contains quantum
theory of measurements and leads to no paradoxes.

A decohering open system may evolve in various classical channels
corresponding to various measurement readouts. Selection of one of
these alternative channels is random and may be characterized by the
probability distribution. This is why the evolution of an open system
is a stochastic process. In the restricted-path-integral approach this 
process is described by a set of `underunitary' partial evolution 
operators instead of a single unitary evolution operator. This 
approach underlines the dynamical role played by information.

Any theory of closed quantum systems as well as a theory explaining
the mechanism of selection should explicitly include consciousness of
observers. Such a theory is not necessary for solving those problems
which are typical for physics, but it may be motivated by metaphysical
arguments. The most interesting attempt to construct a theory of this
type is the many-world interpretation of quantum mechanics suggested
by Everett. Metaphysical as it is, such a theory nevertheless evokes
high interest because its further elaboration may shed light upon the
nature of consciousness directly connecting the physical concept of 
selection and psychological concept of consciousness.


\section*{Acknowledgements}

%\vspace*{-1mm}

The author is obliged to late Yu.~A.~Golfand for many very interesting 
discussions on quantum theory of measurements and conceptual problems 
of quantum mechanics.

\section*{References}

%\vspace*{-1mm}
%\bibliography{decohere,rpi,cont-mes,qu-meas,path-int,qm-cncpt}
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\end{thebibliography}


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