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\pagestyle{myheadings} \markright{\rm O. V. Prezhdo and V. V. Kisil:
Mixing Quantum and Classical Mechanics \hfill}
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\title{Mixing Quantum and Classical Mechanics}
\author{Oleg V. Prezhdo\thanks{E-mail: \texttt{oleg@czar.cm.utexas.edu}}\\
\emph{Department of Chemistry}\\
\emph{and Biochemistry}\\
\emph{University of Texas at Austin}\\
\emph{Austin, Texas 78712, USA}
\and
Vladimir V. Kisil\thanks{Current address: Vakgroep Wiskundige Analyse,
Universiteit Gent,
Galglaan 2, B-9000,
Gent, BELGIE.
E-mail: \texttt{vk@cage.rug.ac.be}} \\
\emph{Institute of Mathematics}\\
\emph{Economics and Mechanics}\\
\emph{Odessa State University}\\
\emph{ul. Petra Velikogo, 2,}\\
\emph{Odessa-57, 270057, UKRAINE}
}
%\date{PREPRINT Revision: \today}
\date{October 11, 1996}
\begin{document}
\maketitle
\setlength{\baselineskip}{7mm}
\begin{abstract}
\setlength{\baselineskip}{7mm}
Using a group theoretical approach we derive an equation of motion for
a mixed quantum-classical system.
The quantum-classical bracket entering the equation preserves the
Lie algebra structure of quantum and classical mechanics:
The bracket is antisymmetric and satisfies the Jacobi identity, and,
therefore, leads to a natural description of interaction between quantum
and classical degrees of freedom. We apply the formalism to coupled quantum
and classical oscillators and show how various approximations,
such as the mean-field and the multiconfiguration mean-field approaches,
can be obtained from the quantum-classical equation of motion.
\vspace{3mm}
\noindent PACS numbers: 03.65.Sq, 03.65.Db, 03.65.Fd
\end{abstract}
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\section{Introduction}\label{sec1}
Many phenomena in nature are described by quantum mechanics at a
fundamental level
and with high precision.
Yet, there exist numerous situations where mixed quantum-classical
models are needed. In some cases the phenomena are too complex to
allow for a fully quantum approach, in others a consistent quantum
theory is lacking.
Classical mechanics often provides a more suggestive description and
a clearer picture of physical events. Applications of various
quantum-classical approaches range from biochemical and condensed
matter chemical reactions, where the large dimensionality of the
systems of interest requires approximations, to the evolution of the
universe and cosmology, where no theory of quantum
gravity has been established.
%(see, however, reference~\cite{ChamConnes96b}).
The issue of treating quantum and classical degrees of freedom within
the same formalism was recently discussed in a number of
publications~\cite{Maddox95,Anderson95,Boucher88}. The interest
was spurred by the cosmological problem of defining the backreaction
of quantum matter fields on the classical space-time background,
where classical variables should be independently correlated with
each individual quantum state. The traditional approach fails
to satisfy the last requirement.
(For a fully quantum approach to cosmology see
reference~\cite{ChamConnes96b}.)
Somewhat earlier a similar situation
was encountered in chemical physics, where quantum-classical
trajectory methods were employed to model gas phase scattering
phenomena and chemical dynamics in
liquids~\cite{Diestler83,Aleksandrov81,%
Miller80,Tully76,Pechukas69a,Pechukas69b}.
It was noticed in these studies that asymptotically distinct quantum
evolutions should correlate with different classical trajectories.
The first relationship between quantum and classical variables
is due to Ehrenfest~\cite{Ehrenfest27}
who showed that the equation of motion for the average values of
quantum observables coincides with the corresponding classical
expression. (Surprisingly, the first mathematically rigorous treatment
on the subject was not carried out until 1974, see
reference~\cite{Hepp74}.)
Ehrenfest's result leads to the mean-field approach, where classical
dynamics is coupled to the evolution of the expectation values of
quantum variables~\cite{Mott31,Mittelman61,Delos72a,Delos72b}. The
mean-field equations of motion possess all of the properties of the
purely classical equations and are exact as far as the mean
values of quantum operators are concerned.
However, an expectation value does
not provide information of the outcome of an individual process.
The mean-field approach gives a satisfactory
description of the classical subsystem as long as changes within
the quantum part are fast compared to the characteristic
classical time-scale.
If classical trajectories depend strongly on a particular
realization of the quantum evolution, the mean field approximation is
inadequate. The problem can be corrected, for instance by
introduction
of stochastic quantum hops between preferred basis states with
probabilities determined by the usual quantum-mechanical
rules~\cite{Tully71,Tully90,OPrezhdo96b}.
Similarity between the algebraic structures underlying quantum and
classical mechanics provides a consistent way of improving upon the
mean-field approximation, as explored in
references~\cite{Anderson95,Boucher88,Aleksandrov81}.
\comment{references~\cite{Anderson95,Anderson95a,
Salcedo95,Boucher88,Aleksandrov81}.}
In those studies the aim was to derive a quantum-classical bracket
that reduces to the quantum commutator and the Poisson bracket in
the purely quantum and classical cases. In addition to the reduction
property the bracket should satisfy other criteria so as to
give physically meaningful pictures of quantum-classical
evolutions. In particular, an antisymmetric bracket conserves
the total energy and a bracket satisfying the Jacobi identity ensures
that the Heisenberg uncertainty principle is not violated.
Recently, one of us (VVK) proposed~\cite{Kisil96a} a natural
mathematical
construction, which we name \p-mechanics,
enveloping classical and quantum mechanics.
Formulated within the framework of operator algebras,
the \p-mechanical equation of motion reduces to the appropriate
quantum or
classical equations under suitable representations of the algebra
of observables. In this paper we extend the ideas of \p-mechanics
to incorporate mixed
quantum-classical descriptions. In particular, we derive the
quantum-classical bracket and explicitly show that it satisfies the
properties common to quantum and classical mechanics.
\comment{To our knowledge, this is the most consistent
derivation to date. It provides a clear understanding of the
interaction between quantum and classical variables and allows for
further generalizations.}
Using the technique described it is possible to construct families of
mixed
quantum-classical approaches, each having a specific set of
properties.
\changeone{
The format of this paper is as follows: In Section~\ref{sec2}
we summarize \p-mechanics and introduce the essential mathematical
definitions. In Section~\ref{sec3} we
construct the simplest \p-mechanical model that adopts two distinct
sets of variables associated with quantum and classical degrees of
freedom.
By taking the appropriate
representation we derive the quantum-classical bracket and show that
it is antisymmetric and obeys the Jacobi identity, that is, it
possesses the two major properties shared by the quantum and
classical brackets. In Section~\ref{se:2oscilators}
we work out the case of coupled
classical and quantum harmonic oscillators.
\comment{as an example of application of the suggested formalism}
Finally, in Section~\ref{sec4} we discuss how various
approximations to the general quantum-classical
description can be obtained, including the mean-field and
the multiconfiguration mean-field approaches.}
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\section{\pp-mechanics}\label{sec2}
\comment{In this section we summarize the unified approach to
classical and
quantum mechanics. Starting
with an algebraic construction it allows to derive classical
and quantum mechanical laws of evolution by selecting a
representation
for the generalized equation of motion.}
\subsection{The Elements of \pp-mechanics}
We recall the constructions from
references~\cite{Kisil96a,Kisil94e} together with appropriate
modifications.
\begin{defn}\label{de:p-mech}\textrm
An operator algebra \algebra{P} gives a \emph{\p-mechanical
description}~\cite{Kisil96a} of a system if the following
conditions hold.
\begin{enumerate}
\item\label{it:representation} The set $\widehat{\algebra{P}}$ of all
irreducible representations $\pi_h$ of \algebra{P} is a disjoint
union of subsets
$\widehat{\algebra{P}}=\sqcup_{p\in P} \widehat{\algebra{P}}_p$
parameterized by the elements of a set $P$. The elements of the set
$P$
are associated with different values for the Planck constant.
We refer to this set as the
\emph{set of Planck constants}. If for
$p_0$ the set $\widehat{\algebra{P}}_{p_0}$ consists of only
commutative (and, therefore, one-dimensional) representations, then
$\widehat{\algebra{P}}_{p_0}$ gives a \emph{classical} description. If
$\widehat{\algebra{P}}_{p_0}=\{\pi_{p_0}\}$ consists of a single
non-commutative representation $\pi_{p_0}$, then
$\widehat{\algebra{P}}_{p_0}$ gives a purely \emph{quantum} model.
Sets
$\widehat{\algebra{P}}_p$ of other types provide \emph{mixed}
(quantum-classical) descriptions.
\item\label{it:topology} Let $\widehat{\algebra{P}}$ be equipped with
a
natural operator topology (for example, it may be the Jacobson
topology~\cite{Dixmier69} or the *-bundle
topology~\cite{DaunHof68,Hofmann72}). Then $P$ has a natural
\emph{factor
topology} induced by the partition $\widehat{\algebra{P}}=\sqcup_{p\in
P}$.
\item\label{it:dynamics}(Dynamics) The algebra \algebra{P} is equipped
with the one-parameter semigroup of transformations
$G(t):\algebra{P}\rightarrow\algebra{P}$, $t\in\Space{R}{+}$. All sets
$\widehat{\algebra{P}}_p$, $p\in P$ are \emph{preserved} by $G(t)$.
Namely, for any $\pi\in \widehat{\algebra{P}}_p$ all new
representations
$\pi_t=\pi\circ G(t)$ again belong to $\widehat{\algebra{P}}_p$.
\item \label{it:correspond}(The Correspondence Principle) Let
$S:p\mapsto S(p)\in \algebra{P}_p$ be an operator-valued section
continuous in the *-bundle topology~\cite{DaunHof68,Hofmann72} over
$P$.
Then for any $t$, i.e., at any moment of time the image
$S_t(p)=G(t)S(p)$ is also
a section due to statement~\ref{it:dynamics}. In the *-bundle
topology
the sections $S_t(p)$ are \emph{continuous for all} $t$.
\end{enumerate}
\end{defn}
Having listed these quite natural conditions, we do not yet know how
to construct
\p-mechanics. Next, we describe an important particular case
of \emph{group quantization}~\cite{Kisil94e}. All components of
\p-mechanics (operator
algebra, partition of representations, topology) readily arise there.
\begin{const}\rm \emph{Group quantization} consists of the following
steps.
\begin{enumerate}
\item Let $\Omega=\{x_j\}, 1\leq j\leq N$ be a set of physical
variables defining the state of a classical system. Classical
observables are
real-valued functions on the states.
The best known and the most important case is the set
$\{x_j=q_j,x_{j+n}=p_j\},\
1\leq j\leq n, N=2n$ of coordinates and momenta of classical particles
forming an $n$ degree of freedom system. The observables are real
valued functions on
\Space{R}{2n}. We will use this example throughout this Section.
\item We complete the set $\Omega$ with
additional variables ${x_j}, N