\magnification 1200
\centerline {{\bf On Hyperbolic Flows and the Problem of Chaos
in Quantum Systems}\footnote*{Based on a Lecture at the
International Workshop on "Quantum Communication and Measurement"
held at Nottingham, 11-16 July, 1994}}
\vskip 0.5cm
\centerline {{\bf by Geoffrey L. Sewell}\footnote{**}{Partially
supported by European Capital and Mobility Contract No. CHRX-Ct.
92-0007}}
\vskip 0.5cm
\centerline {\bf Department of Physics, Queen Mary and Westfield
College}
\vskip 0.3cm
\centerline {\bf Mile End Road, London E1 4NS, England}
\vskip 1cm
\centerline {\bf Abstract}
\vskip 0.3cm
We briefly review the non-commutative generalisation, presented
in [1], of the theory of hyperbolic dynamical systems; and
then prove that hyperbolicity cannot be a paradigm for quantum
chaos, except possibly in a certain asymptotic sense.
\vskip 1cm
\centerline {\bf 1. Introduction}
\vskip 0.3cm
In a recent work [1], the theory of hyperbolic, or Anosov,
flows [2,3], which provides a paradigm for classical chaos, was
generalised to non-commutative dynamical systems. The
question naturally arises of whether such flows can prevail in
the standard Von Neumann model [4] of (finite) quantum systems,
and thus provide a paradigm for quantum chaos too. The object of
this note is to demonstrate that they cannot do so, except
possibly in an asymptotic sense, corresponding to the proximity
of a classical limit. This is quite in line with results
obtained on rather different bases about quantum chaos [5-7].
\vskip 0.2cm
In Section 2, I shall outline the generalised theory [1] of
hyperbolic flows, and in Section 3 I shall establish the
above-described results concerning quantum chaos. It will be seen
that the main result, i.e. Theorem 3, generalises one obtained
in [1], under special conditions, for the particular
case of the free dynamics of a particle on a manifold of constant
negative curvature.
\vskip 0.5cm
\centerline {\bf 2. Hyperbolic Flows}
\vskip 0.3cm\noindent
{\bf 2.1. The Classical Model.} In a standard way (cf. [3]), we
take our classical dynamical model, ${\Sigma}_{c},$ to be given
by a triple $(X,{\phi},{\mu}),$ where $X,$ the phase space, is
a compact, differentiable, Riemannian manifold,
${\lbrace}{\phi}_{t}{\vert}t{\in}{\bf Z} \ or \ {\bf R}{\rbrace}$
is a one-parameter group of diffeomorphisms of $X,$ representing
the dynamics of the system, and ${\mu}$ is a smooth,
${\phi}-$invariant probability measure on $X,$ corresponding to
a stationary state. We denote the tangent space at $x({\in}X)$
by $T(x)$ and, for fixed $t,$ we define
${\phi}_{t}^{\star}:=d{\phi}_{t}:T(x){\rightarrow}T({\phi}_{t}x).$
We shall employ the following definition of hyperbolic flows,
which is a restricted version of that of Anosov [2,3].
\vskip 0.3cm\noindent
{\bf Def. 2.1.} We term the dynamics of ${\Sigma}_{c}$ hyperbolic
if there are unit vector fields $V_{1},.. \ .,V_{m};$
$V_{m+1},.. \ .,V_{n}$ over $X,$ which are
linearly independent at each point of $X$ and satisfy the
following conditions.
\vskip 0.2cm\noindent
(a) $0m.$
\vskip 0.2cm\noindent
(c) The vector fields $V_{j}$ have globally integral
curves, generated by the action of one-parameter groups
${\lbrace}{\theta}_{j}(s){\vert}s{\in}{\bf R}{\rbrace}$
of diffeomorphisms of $X.$ Thus,
$$x_{j}(s)={\theta}_{j}(s)x\eqno(2.2)$$
is the unique global solution of the equation
$$x_{j}^{\prime}(s)=V_{j}(x_{j}(s)) \ {\forall}s{\in}{\bf R};
\ x(0)=x\eqno(2.3)$$
\vskip 0.3cm\noindent
{\bf Comments.} (1) It follows immediately from the condition (b)
that, generically, orbits emanating from neighbouring points
separate exponentially fast from one another. Hence the dynamics
is extremely unstable, i.e. chaotic.
\vskip 0.2cm\noindent
(2) As prototype examples of hyperbolic systems, we cite (cf. [3])
\vskip 0.2cm\noindent
(a) the 'Arnold Cat' model, whose dynamics corresponds to
iterations of an automorphism of the two-dimensional torus; and
\vskip 0.2cm\noindent
(b) geodesic flow over a compact manifold of negative curvature.
\vskip 0.3cm
The following key lemma, concerning hyperbolic systems, will be
proved in the Appendix.
\vskip 0.3cm\noindent
{\bf Lemma 2.2.} {\it For any hyberbolic system,} ${\Sigma}_{c},$
$${\phi}_{t}{\theta}_{j}(s){\phi}_{t}^{-1}=
{\theta}_{j}(s.{\exp}({\lambda}_{j}t))\eqno(2.4)$$
\vskip 0.3cm\noindent
{\bf 2.2. Algebraic Formulation of ${\Sigma}_{c}.$} As a first
step towards a quantum generalisation of the above model, we
reformulate it as an {\it abelian} $W^{\star}-$dynamical system
$({\cal A}_{c},{\alpha}_{c},{\omega}_{c}),$ where ${\cal A}_{c},$
the algebra of observables, is $L^{\infty}(X,d{\mu}), \ {\lbrace}
{\alpha}_{c}(t){\vert}t{\in}{\bf Z} \ or \ {\bf R}{\rbrace}$ is
the one-parameter group of automorphisms of ${\cal A}_{c}$
induced by ${\phi},$ i.e.
$$({\alpha}_{c}(t)f)(x){\equiv}f({\phi}_{t}^{-1}x)\eqno(2.5)$$
and ${\omega}_{c}$ is the normal state on ${\cal A}$
corresponding to ${\mu},$ i.e.
$${\omega}_{c}(f){\equiv}{\int}fd{\mu}\eqno(2.6)$$
Further, we denote by ${\sigma}_{c,j}(s)$ the
automorphism of ${\cal A}$ induced by ${\theta}_{j},$
i.e.
$$({\theta}_{c,j}(s)f)(x){\equiv}
f({\theta}_{j}(-s)x)\eqno(2.7)$$
It follows immediately from (2.5) and (2.7) that the
hyperbolicity condition (2.4) is equivalent to
$${\alpha}_{c}(t){\sigma}_{c,j}(s){\alpha}({-t})=
{\sigma}_{c,j}(s.{\exp}({\lambda}_{j}t))
\eqno(2.8)$$
\vskip 0.3cm\noindent
{\bf 2.3. Generalisation to Quantum Systems} [1]. Now let
${\Sigma}=({\cal A},{\alpha},{\omega})$
be an arbitrary $W^{\star}$-dynamical system, with ${\cal A}$ an
algebra of observables,
${\lbrace}{\alpha}(t){\vert}t{\in}{\bf Z} \ or
\ {\bf R}{\rbrace}$ a one-parameter group of automorphisms of
${\cal A},$ and ${\omega}$ a normal, ${\alpha}-$invariant
state on ${\cal A}.$ Thus, ${\Sigma}$ provides a generalisation
of the model ${\Sigma}_{c}$ to quantum systems. For these, ${\cal
A}$ is non-abelian and conforms to the canonical
commutation relations.
\vskip 0.3cm\noindent
{\bf Def. 2.3.} We term the system ${\Sigma}$ {\it hyperbolic}
if ${\cal A}$ is equipped with weakly continuous one-parameter
groups of automorphisms ${\sigma}_{1}({\bf R}),.. \
.,{\sigma}_{m}({\bf R}); \ {\sigma}_{m+1}({\bf R}),..
\ .,{\sigma}_{n}({\bf R}),$ with $0m.$
\vskip 0.3cm\noindent
{\bf Comments.} (1) The present definition is less restrictive
than that of [1] in that it does not require the
${\sigma}_{j}-$invariance of ${\omega}.$
\vskip 0.2cm\noindent
(2) In the classical case, the derivations ${\delta}_{j}$ are
those corresponding to the vector fields $V_{j}.$
\vskip 0.2cm\noindent
(3) In general, by contrast with the classical case, Def. 2.3
does not provide a specification of $n$ in terms of the
structure of ${\cal A}.$ We would hope that this deficiency could
be remedied in the future. For present purposes, however, it
suffices that $0