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% Classical Mechanics as Quantum Mechanics with Infinitesimal $\hbar$
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\begin{document}
\begin{titlepage}
\begin{center}
{\huge\bf
Classical Mechanics as \\
Quantum Mechanics with\\
Infinitesimal \raise 7pt\hbox{--}\kern-12pt h\\}
\vskip20pt by \vskip20pt
{\bf
R.F. Werner $^{(*)}$ and M.P.H. Wolff $^{(**)}$}
\end{center}
\vskip 1 truecm \vfill
\noindent{\bf Abstract.\ }
We develop an approach to the classical limit of quantum theory
using the mathematical framework of nonstandard analysis. In this
framework infinitesimal quantities have a rigorous meaning, and the
quantum mechanical parameter $\hbar$ can be chosen to be such an
infinitesimal. We consider those bounded observables which are
transformed continuously on the standard (non-infinitesimal) scale
by the phase space translations. We show that, up to corrections of
infinitesimally small norm, such continuous elements form a
commutative algebra which is isomorphic to the algebra of classical
observables represented by functions on phase space. Commutators of
differentiable quantum observables, divided by $\hbar$, are
infinitesimally close to the Poisson bracket of the corresponding
functions. Moreover, the quantum time evolution is infinitesimally
close to the classical time evolution. Analogous results are shown
for the classical limit of a spin system, in which the half-integer
spin parameter, i.e.\ the angular momentum divided by $\hbar$, is
taken as an infinite number.
\vskip12pt\vfill
%
% \noindent {\bf Physics and Astronomy classification scheme PACS
% (1994):}
% \class 03.65.Sq Semiclassical theories and applications *
% \class 02.10.By Logic and foundations*
% \class 03.65.Db Functional analytical methods *
%
%
\noindent {\bf Mathematics Subject Classification (1991):}
\class 81Q20 Semiclassical techniques including WKB and Maslov
methods*
\class 46S20 Nonstandard functional analysis*
\class 81S30 Phase space methods including Wigner
distributions*
% \class 47S20 Nonstandard operator theory*
% \class 26E35 Nonstandard analysis*
% \class 46M07 Ultraproducts*
% \class 81R30 Coherent states*
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\begin{flushleft}
(*)\
FB Physik, Univ.\ Osnabr\"uck, D-49069 Osnabr\"uck, Germany\\
E-mail: {\tt reinwer@dosuni1.rz.uni-osnabrueck.de}\\
(**)\ Math.\ Institut, Univ.\ T\"ubingen,
Auf der Morgenstelle 10, \\D-72076 T\"ubingen, Germany\\
E-mail: {\tt manfred.wolff@uni-tuebingen.de}
\end{flushleft}
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\section{Introduction}
The classical limit of quantum mechanics is often identified with
the WKB \cite{Maslov,BurdHJ} method. While this approach gives a
good picture of the asymptotic behaviour of solutions of the
Schr\"odinger equation as $\hbar\to0$, it does not give a
satisfactory explanation why in this limit the non-commutativity of
quantum observables suddenly turns into the commutativity of
classical observables. The same is true of approaches based on
Feynman integrals \cite{Albeverio}, and on the limits of coherent
states \cite{Hepp,Hagedorn}. An approach to the classical limit
emphasizing the limit of observables and their algebraic structure
has recently been developed in \cite{CLQ} (compare also
\cite{Rieffel,Emch}). This approach makes rigorous the intuitive
criterion for deciding which observables in quantum theory may
effectively be treated classically: {\it classical observables
should not change too much under small position or momentum
translations}, where, due to the relation $p=\hbar k$, a small
momentum translation might still correspond to a large translation
in terms of wave numbers.
The aim of this paper is to show that a full theory of the classical
limit can be based on this single physical idea. We make use of
nonstandard analysis \cite{Robinson,AlbevBook} because it allows us
to describe an infinite separation of scales inside a single
mathematical structure. The main idea is that our usual ``standard''
quantities can be embedded into a larger structure containing also
infinitely small and infinitely large quantities (this embedding is
to a certain extent comparable to the embedding of the reals into
the complex numbers). Like almost all results of nonstandard
analysis our results can be translated into statements about
ordinary limits (in an abstract sense), e.g.\ an infinitesimal
(i.e.\ an infinitely small) number can be seen as nothing but a
maximally detailed description of how a sequence of real numbers can
go to zero.
The actual construction of quantum theory with infinitesimal $\hbar$
is completely trivial, thanks to a powerful principle of nonstandard
analysis, called the Transfer Principle. It states that whatever can
be formulated correctly in standard terms is immediately true or
defined, respectively, in the nonstandard world. In contrast to
standard analysis the key here is not to go to the limit, but to
extract from the limiting theory the ``standard part''. Here the
above mentioned physical idea gives an immediate criterion: we only
need to restrict the theory to observables which are {\em continuous
on the standard scale}, and then to neglect infinitesimal terms.
This formulation corresponds completely to the physical intuition.
Moreover it is much more compact than the formulation in
conventional mathematical terms \cite{CLQ} on which it is based. At
the same time it retains full mathematical rigour. Due to its
extreme simplicity the nonstandard formulation is also more
suggestive of further generalizations. Another bonus is that some
proofs are simplified, but this is not our main point, and, in fact,
we draw heavily on the standard techniques of proof.
We briefly review some ideas and notation relating to nonstandard
analysis. There are quite good introductions to the subject
\cite{Lindstrom,AlbevBook,HurdLoeb}, including undergraduate
calculus courses based on it \cite{Keisler}, and we refer to these
for more detailed information. Whenever it is convenient we denote
an entitiy of our standard mathematical world by a prefix
``$\,\star\,$'' if it is considered in the nonstandard universe. For
example the real line is denoted by $\star\Rl$. It contains elements
of the form $\star r$, where $r\in\Rl$ is an ordinary real number,
but by far not all elements of $\star \Rl$ are of this form. In
particular, there are infinitesimals $\varepsilon\in\star\Rl$ with
the property that $\varepsilon>0$, but $\varepsilon<\star r$ for
every $r>0$. When $x,y\in\star\Rl$, we write ``$x\approx y$'' for
``$x-y$ is infinitesimal''. The infinitesimals have a decent
arithmetic and, for example, the inverse of an infinitesimal is an
infinite number, which is larger than any standard real $\star r$ in
accordance with our intuition.
\section{Classical limit for phase space observables}
The first structure to which we will apply the Transfer Principle is
an irreducible representation of the canonical commutation relations
in Weyl form with $d<\infty$ degrees of freedom, i.e.\ we consider
on the Hilbert space $\H={\cal L}^2(\Rl^d)$ the unitary operators
$\Weyl(x,p)$ of phase space translations given by
\begin{equation}
\Bigl(\Weyl(x,p)\psi\Bigr)(y)
=\exp\left\lbrace{-i{x\cdot p\over 2\hbar}
+i{p\cdot y\over\hbar}}\right\rbrace \psi(y-x)
\quad.\end{equation}
Alternatively, these operators can be written as
\begin{equation}
\Weyl(x,p)=e^{ix\cdot{\bf P}+iq\cdot{\bf Q}}
\quad,\end{equation}
where ${\bf P}$ and ${\bf Q}$ denote the usual momentum and position
operators, and the dot stands for the scalar product in $\Rl^d$. It
will often be convenient to denote phase space points $(x,p)$ by a
single letter $\xi$. The Euclidean length of $\xi$ is written as
$\vert\xi\vert=(x\cdot x+p\cdot p)^{1/2}$. Observables are described by
bounded operators $A$ on the Hilbert space, written as $A\in\B(\H)$.
On the observables the Weyl operators implement the phase space
translations $\alpha_\xi$ via
\begin{equation}
\alpha_{\xi}(A)=\Weyl(\xi)\,A\,\Weyl(\xi)^*
\quad.\end{equation}
By the transfer principle all these formulas make sense also in the
nonstandard world, i. e. for observables $A\in\star\B(\H)$,
for phase space points $\xi=(x,p)\in\star\Rl^d$, and for any value of
the constant $\hbar\in\star\Rl$, including infinitesimal or
infinite numbers. (We refrain from taking the number $d$ of degrees
of freedom to be an infinite number in $\star\Nl$). The norm is a
well-defined $\star\Rl$-valued function on $\star\B(\H)$, and, of
course, it can be infinitesimal, as well as infinite.
We will call $A \in\star\B(\H)$ {\it infinitesimal}, writing
$A\approx0$, if $\Vert{A}\Vert\approx0$, and we will say that
$A\in\star\B(\H)$ is {\em finite}, if there is an ordinary real
number $r\in\star\Rl$ such that $\Vert{A}\Vert\leq\star r$.
We can now state the basic idea of this paper: inside this
monstrously large object $\star\B(\H)$ we will single out those
operators that are well-behaved in the sense of the classical limit.
The Weyl operators themselves are an example of badly behaved
operators, since they oscillate wildly on an infinitesimal scale. In
contrast, operators that we can imagine as ``observables'' ought to
depend continuously on ${\bf P}$ and ${\bf Q}$. We define the
algebra $\good\subset\star\B(\H)$ of ``good'' observables as those
finite elements $A\in\star\B(\H)$, such that $\alpha_\xi(A)$ is
{\it continuous on the standard scale}.
Nonstandard analysis offers two equivalent ways of making this
phrase precise. The first is to say that for all standard
$\varepsilon\in\Rl$ we can find a standard $\delta\in\Rl$ such that
$\vert\xi\vert<\star\delta$ implies
$\Vert{\alpha_\xi(A)-A}\Vert\leq\star\varepsilon$. The second is to
say that
\begin{equation} \label{cont}
\xi\approx0
\qquad\hbox{implies}\qquad
\alpha_\xi(A)\approx A
\quad.\end{equation}
For example, it is seen immediately that the Weyl operators fail
this definition, whereas, if we scale down their oscillations by a
factor $\hbar$, they do become continuous. Indeed, due to the
commutation relations of the Weyl operators, we have
\begin{equation}
\alpha_\xi\bigl(\Weyl(\hbar\eta)\bigr)
=e^{i\sigma(\xi,\eta)}\ \Weyl(\hbar\eta)
\quad,\end{equation}
where $\sigma((x,p),(x',p'))=p\cdot x'-x\cdot p'$ is the symplectic
form on phase space. Since the exponential factor is continuous in
$\xi$ on the standard scale, we have $\Weyl(\hbar\eta)\in\good$,
if $\eta$ is finite.
Also, every infinitesimal operator is in $\good$. Thus $\good$ is
every bit as non-commutative as $\star\B(\H)$. The surprise is,
however, that we only need to neglect infinitesimal terms to see in
it the observable algebra of classical mechanics.
\proclaim Theorem.
Let $\good\subset\star\B(\H)$ be the algebra of continuous
observables defined above, and let $\good_\approx$ denote the same
algebra, but with all infinitesimal elements identified with $0$.
Then $\good_\approx$ is canonically isomorphic to the algebra of
bounded uniformly continuous functions on phase space.
We sketch the rather simple proof because it uses only ideas
well-known from the physics literature, and gives an explicit
description of the isomorphism claimed in the Theorem. Let $\Omega$
denote the ground state wave function of the oscillator Hamiltonian
$H=({\bf P}^2+{\bf Q}^2)/2=(-\hbar^2\Delta+{\bf Q}^2)/2$. Then for
any $A\in\B(\H)$, we define a function on phase space by
\begin{eqnarray}
(\upsym A)(\xi)&= \langle\Weyl(\xi)\Omega,\,
A\,\Weyl(\xi)\Omega\rangle
\nonumber\\
&= \langle\Omega,\,\alpha_{-\xi}(A)\Omega \rangle
\quad.\end{eqnarray}
This is variously called the lower symbol \cite{Simon}, a smeared
Wigner function \cite{Cartwright}, the Husimi function
\cite{Takahashi}, or the convolution with a coherent state
\cite{QHA} of the operator $A$. In the other direction, we have the
upper symbol \cite{Simon}, or P-representation \cite{Klauder}
(also going by many other names) which assigns an operator to each
bounded measurable function $f$ via
\begin{equation}
\dnsym f=\int\!\!\!{dx\,dp\over (2\pi\hbar)^d}\
f(x,p)\, \alpha_\xi\bigl(
\vert\Omega\rangle\langle\Omega\vert\bigr)
\quad.\end{equation}
Unlike the Wigner-Weyl isomorphisms between operators and phase
space functions, these operators map positive elements into positive
elements. \break
Moreover, $\upsym\dnsym$ and $\dnsym\upsym$
are both operators which just average over translations with a
Gaussian weight. Explicitly,
\begin{equation}
\dnsym\upsym(A)=\int\!\!\!{dx\,dp\over (2\pi\hbar)^d}\
e^{\textstyle -\xi^2/(2\hbar)}\
\alpha_\xi(A)
\quad.\end{equation}
By transfer, all these formulas remain valid in the nonstandard
case. However, since $\hbar$ is infinitesimal, the Gaussian factor
is a nonstandard representation of a Dirac $\delta$-Function. Hence
for continuous operators $A\in\good$,
$\dnsym\upsym(A)\approx A$. Identifying infinitesimally close
elements we find that $\upsym$ and $\dnsym$ become inverses of each
other. On the other hand, it is clear that, for $A\in\good$, the
function $\upsym(A)$ is also finite and continuous in the sense of
equation (\ref{cont}), $\alpha_\xi$ being interpreted as the phase
space translation of functions. This is the same as saying that
$\alpha_\xi$ is uniformly continuous up to infinitesimals. Hence
$\good_\approx$ is isomorphic to the space of bounded uniformly
continuous functions on phase space. Because $\upsym$ and $\dnsym$
both preserve positivity, these isomorphisms respect the ordering as
well. This implies that they are also algebraic isomorphisms.
Since the product of functions is commutative, we have that
$AB-BA\approx0$ for $A,B\in\good$. For sufficiently smooth
observables we can even determine the precise order of this
infinitesimal. We say that $A\in\good$ is twice differentiable, if
it has ``partial derivatives'' $A_i,A_{ij}\in\good$ such that
$\xi\approx0$ implies
\begin{equation}
\alpha_{\xi}(A) = A + \sum_1^{2d} \xi_i A_i
+ \sum_{i,j =1}^{2d} \xi_i \xi_j A_{ij}
+ \xi^2 D(\xi)
\quad,\end{equation}
where $D(\xi) \approx 0$. Here $\xi_i$ denotes the components of
$\xi$, i.e.\ the $d$ position and the $d$ momentum coordinates.
Applying $\upsym$ and $\alpha_{\xi+\eta}=\alpha_\xi\alpha_\eta$ we
see that $\upsym(A)$ is twice differentiable with uniformly
continuous bounded derivatives. Then, if $A,B\in\good$ are twice
differentiable, we get
\begin{equation} \label{Poisson}
{i\over\hbar}\lbrack A,B\rbrack
\approx \dnsym\lbrace \upsym(A),\upsym(B)\rbrace
\quad,\end{equation}
where the braces on the right hand side denote the Poisson bracket
of the phase space functions. The operators $\upsym$ and $\dnsym$
just effect the isomorphism between functions and operators given by
the Theorem. The proof of formula (\ref{Poisson}) is rather involved,
but completely parallel to the standard proof of the analogous
result in \cite{CLQ}, so we will omit it.
An instructive example is the case of Weyl operators with
slowed-down oscillation, i.e.\ $\Weyl(\hbar\eta)$ for finite $\eta$.
Their classical limits are the exponential functions
\begin{equation} \label{upweyl}
\bigl(\upsym\Weyl(\hbar \eta)\bigr)(\xi)
=e^{i\lbrace\xi,\eta\rbrace -\hbar\eta^2/4}
\approx e^{i\lbrace\xi,\eta\rbrace }
\quad,\end{equation}
with the ``symplectic form''
$\lbrace(x_1,p_1),\, (x_2,p_2)\rbrace= p_1\cdot x_2-p_2\cdot x_1$.
Commutators become
\begin{equation}
{i\over\hbar} \bigl\lbrack \Weyl(\hbar\xi),\, \Weyl(\hbar\eta)
\bigr\rbrack
={2\over\hbar} \sin\bigl({\hbar\over2}
\lbrace\xi,\eta\rbrace\bigr)
\Weyl(\hbar(\xi+\eta))
\quad.\end{equation}
Because $\hbar\approx0$, and $\xi,\eta$ are finite, the factor is
$\approx\lbrace\xi,\eta\rbrace$, from which equation (\ref{Poisson})
is verified by applying $\upsym$ to each Weyl operator.
By equation (\ref{Poisson}) the quantum mechanical equations of
motion are infinitesimally close to the classical ones. The same is
also true for the {\it solutions} of these equations: If $H\in\good$
is a hermitian operator, we define the time evolution of an
observable $A\in\good$ as usual by
\begin{equation} \label{tevolq}
A(t)= e^{\textstyle itH/\hbar}A e^{\textstyle itH/\hbar}
\quad.\end{equation}
Then one can prove as in the standard case \cite{CLQ} that
\begin{equation} \label{tevol}
(\upsym A(t))(\xi)\approx(\upsym A)({\cal F}_t\xi)
\quad.
\end{equation}
Here ${\cal F}_t\xi$ denotes the solution of Hamilton's equation of
motion with Hamiltonian function $\upsym H$ for the time interval
$t$ with initial condition $\xi$.
\section{Classical limit for spin systems}
The same basic idea for obtaining a classical limit works in a
number of other contexts. We present here the case of spin systems,
restricting attention to a single spin, for simplicity of
presentation. The basic relation here is that angular momentum,
which is the quantity remaining meaningful in the classical limit
equals $\hbar s$, where $s$ is the usual integer or half-integer
spin parameter labelling the irreducible representations of
$\hbox{SU(2)}$. Hence in this context $\hbar\to0$ simply means
$s\to\infty$. The nonstandard approach to this limit is to take an
an irreducible representation $U$ of $\star\hbox{SU(2)}$ with
infinite spin $s\in{1\over2}\star\Nl$, where $\star\Nl$ denotes the
nonstandard version of the natural numbers. The nonstandard notions
of ``representation'' and ``irreducibility'' are again defined by
the Transfer Principle. Since standard rotations $R\in\hbox{SU(2)}$
are contained in $\star\hbox{SU(2)}$ in the form $\star
R\in\star\hbox{SU(2)}$, and since the nonstandard representation
space is also a complex vector space with scalar multiplication
restricted to standard numbers, $\hbox{SU(2)}$ is also represented
by the infinite spin representation, but, of course, this
representation is highly reducible as a standard representation.
The analogue of the Theorem in the previous section is proved
precisely along the same lines, with the Weyl operators replaced by
the representing unitaries $U_R$, and the coherent state $\Omega$
replaced by the eigenvector of the rotations around the $3$-axis
with maximal eigenvalue (namely $s$). The analogue of the phase
space is the orbit of the one-dimensional projection
$\vert\Omega\rangle\langle\Omega\vert$ under rotations, which is
isomorphic in an obvious way to the (nonstandard) sphere. The
measure ``$d\xi$'' is thus replaced by the integration over the
sphere. The operators $\upsym,\dnsym$ are also defined analogously,
and that $\dnsym$ maps the constant function into the identity
operator is obvious from the irreducibility of $U$.
Only one thing really has to be verified, namely that the kernel
describing the operator $\dnsym\upsym$ is infinitesimal outside the
so-called monad $\lbrace{\phi:\, \phi \approx 0 \rbrace}$ of the
north pole, i.e.\ if
\begin{equation}
R=\pmatrix{\cos(\phi/2)&\sin(\phi/2)\cr
-\sin(\phi/2)&\cos(\phi/2)}
\end{equation}
is the $\hbox{SU(2)}$-rotation taking the north pole to latitude
$\phi\leq2\pi$,
\begin{equation} \label{phinitesi}
\left\vert{\langle\Omega,U_R\Omega\rangle}\right\vert^2
\approx0
\quad,\quad\hbox{unless}\quad\phi\approx0
\quad.\end{equation}
This can be seen without computation by realizing the spin-$s$
representation as the subrepresentation of a tensor product of $2s$
copies of the defining (spin-$1/2$) representation, namely the
representation on the completely symmetric (Bose-) subspace. In this
subspace $\Omega$ is identified with the product of $2s$ ``spin
up'' vectors. The matrix element we have to compute is thus equal to
$(\cos(\phi/2))^{2s}$, and the estimate (\ref{phinitesi}) follows.
Scaled commutators of the form $is\lbrack A,B\rbrack$ become Poisson
brackets as before. The ``phase space'' on which this bracket lives
is the sphere, which is slightly unusual in that it does not contain
unbounded momenta. The Poisson bracket is uniquely determined by the
condition that the three coordinate functions on the sphere (which
are the limits of the angular momentum components divided by $s$)
satisfy angular momentum ``commutation'' relations. In differential
geometric terms this is saying that the symplectic form is the
surface $2$-form of the sphere. Moreover, the dynamics converges in
the same sense as before.
For the proof of these statements it is helpful once again to look
at the spin-$s$ representation as a representation on the
permutation symmetric subspace of $2s$ spins-$1/2$. One can then
appeal to the theory of mean-field lattice systems in which
permutation symmetry is the key ingredient, and in which these
results already exist. We refer the reader to \cite{MFRW,MFD} for
the standard version of this theory, and to \cite{nsMF,nsMFF} for
its nonstandard form.
Generalizations to compact Lie groups other than $\hbox{SU(2)}$, or
to several interacting spins are straightforward using the
appropriate coherent states and symbol maps \cite{Simon}. For a
discussion of phase measurements in the classical limit, also using
nonstandard methods, see \cite{Ozawa}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\let\REF\doref
\Acknow
The work reported in this paper was done during a visit of R.F.W.\
in T\"ubingen. He would like to thank the members of the
mathematical physics groups in both departments for their warm
hospitality. R.F.W.\ acknowledges financial support from the DFG
(Bonn).
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\end{itemize}
\end{document}