\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}}
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\centerline {\bf Department of Physics, Queen Mary and Westfield
College} 
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\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) $0<m<n=Dim(X) \ or \ Dim(X)-1,$ according to whether the time
scale is discrete or continuous.
\vskip 0.2cm\noindent
(b) $${\phi}_{t}^{\star}V_{j}(x)=V_{j}({\phi}_{t}x){\exp}
({\lambda}_{j}t)\eqno(2.1)$$
where the Liapounov exponents ${\lambda}$ are constants, with
${\lambda}_{j}$ positive for $j{\le}m$ and negative for $j>m.$
\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])
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(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.
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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 $0<m<n,$ such that
\vskip 0.2cm\noindent
(a) the generators, ${\delta}_{j}$ of these groups are
linearly independent, and
\vskip 0.2cm\noindent
(b) the condition corresponding to (2.8) is fulfilled, i.e.
$${\alpha}(t){\sigma}_{j}(s){\alpha}(-t)=
{\sigma}_{j}(s.{\exp}({\lambda}_{j}t))\eqno
(2.9)$$
where the ${\lambda}'$s are constants, with ${\lambda}_{j}$
positive for $j{\le}m$ and negative for $j>m.$
\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 $0<m<n.$ 
\vskip 0.3cm\noindent
{\bf Proposition 2.4.} [1] {\it If ${\Sigma}$ is hyperbolic,
then the domain $D({\delta}_{j})$ of the derivation
${\delta}_{j}$ is stable under ${\alpha}({\bf R}),$}
$${\alpha}(t){\delta}_{j}{\alpha}(-t)={\delta}_{j}
{\exp}({\lambda}_{j}t)\eqno(2.10)$$
and
$${\Vert}{\delta}_{j}{\alpha}(t)A{\Vert}=
{\Vert}{\delta}_{j}A{\Vert}
{\exp}(-{\lambda}_{j}t) \
{\forall}A{\in}D({\delta}_{j}), \ t{\in}
{\bf R}\eqno(2.11)$$
\vskip 0.3cm\noindent
{\bf Comments.} (1) Equation (2.10) is the differential form of
(2.9), and reveals the exponential time dependence of the evolute
${\alpha}(t){\delta}_{j}{\alpha}(-t)$ of ${\delta}_{j},$ with
Liapounov exponent ${\lambda}_{j}.$  
\vskip 0.2cm\noindent
(2) Since ${\delta}_{j}A$ represent the rate at which $A$ moves
along the orbit generated by the action of ${\sigma}_{j},$ this
Proposition establishes a precise sense in which the dynamics is
hyperbolic.
\vskip 0.3cm\noindent
{\bf Examples of Hyperbolic Quantum Systems.} (cf. [1].)
\vskip 0.2cm\noindent
(1) A quantum version of the 'Arnold Cat'.
\vskip 0.2cm\noindent
(2) A Wightman relativistic quantum field in Minkowski space, as
viewed by a uniformly accelerated observer. 
\vskip 0.5cm
\centerline {\bf 3. Suppression of Chaos by Quantisation}
\vskip 0.3cm
In view of the extreme sensitivity of hyperbolic systems to
perturbations of initial conditions (cf. Prop. 2.4), we shall
take these as our paradigm of chaotic dynamics. The examples
cited at the end of ${\S}2$ thus provide us with specific models
of non-commutative chaotic systems.
\vskip 0.2cm
However, neither of these corresponds to a finite quantum system,
in the standard sense [4], for the following reasons.
\vskip 0.2cm\noindent
(1) Ex.1 is a quantised version [8] of the classical
hyperbolic system given by iterations of the automorphisms of a
torus, $T^{2}.$ Specifically, the quantisation amounts to the
replacement of $T^{2}$ by a {\it non-commutative} torus, in the
sense of Connes [9], as defined by a Weyl algebra,
${\cal A},$ over the dual group, ${\bf Z}^{2},$ of $T^{2}.$ 
The dynamics is quasi-free, and induced by the same
transformation of ${\bf Z}^{2}$ as in the classical case, and the 
equilibrium state is the tracial one on ${\cal A}.$ Now this model 
does {\it not} correspond a standard finite quantum system, 
since ${\cal A}$ is not a type $I$ factor. In fact, it is either 
of type $II_{1}$ or a tensor product of $L^{\infty}(T^{2})$ and 
the algebra of $n-by-n$ matrices, with n finite, according to 
whether the parameter representing the non-commutativity of the 
torus is irrational or rational. Thus, in the first case, ${\cal A}$ 
is isomorphic with the algebra of an infinitely extended Heisenberg 
ferromagnet in its infinite temperature state; and, in the second case, 
it is essentially classical.
\vskip0.2cm\noindent
(2) Ex.2 is manifestly a model of a system with an infinite
number of degrees of freedom, and its algebra of observables is
of type $III.$
\vskip 0.3cm
In view of these observations, it is natural to ask whether
chaos, as characterised by hyperbolicity, can occur in standard
finite quantum systems. For example, does the hyperbolic property
survive canonical quantisation in the case of the free motion of
a particle on a compact manifold of constant negative curvature?
The answer we shall provide to these questions is negative. In
other words, we shall establish that quantum mechanics suppresses
chaos. This does {\it not} mean that it destroys it, but simply
that a weaker condition than strict hyperbolicity is needed to
represent any viable definition of quantum chaos. We shall
discuss this matter further in the Comment following Theorem 3.
\vskip 0.3cm
We base our treatment of finite quantum systems on the standard
model ${\Sigma}=({\cal A},{\alpha},{\omega}),$ where ${\cal A}$
is the set, ${\cal L}({\cal H}),$ of bounded operators in a
separable Hilbert space, ${\cal H}, \ {\alpha}({\bf R})$ is
implemented by a unitary representation of ${\bf R}$ in ${\cal
H},$ generated by a Hamiltonian operator, $H,$ with discrete
spectrum, i.e.,
$${\alpha}(t)A={\exp}(iHt/{\hbar})A{\exp}(-iHt/{\hbar}) \ 
{\forall}A{\in}{\cal A},t{\in}{\bf R}\eqno(3.1)$$
and ${\omega}$ is a normal, ${\alpha}-$invariant state on ${\cal
A}.$  
\vskip 0.2cm\noindent
{\bf Note.} The above conditions, including the discreteness of
the spectrum of $H,$ are fulfilled not only by finite systems of
particles, with realistic interactions, in bounded regions of a
Euclidean space or on a lattice [10, P.65], but also by the
canonically quantised version of the classical hyperbolic system,
given by the free dynamics of a particle on a compact manifold
of constant negative curvature [11]. It was proved by [1] that
the latter model is not hyperbolic, subject to the assumption
that at least one of the automorphism groups ${\sigma}$ is
unitarily implementable in ${\cal H}.$ The following theorem
generalises this result to all finite systems, without any such
assumption. 
\vskip 0.3cm\noindent
{\bf Theorem 3.} {\it The above-specified model of a
finite system is never hyperbolic.}
\vskip 0.3cm\noindent
{\bf Comment.} This does not rule out the possibility of a
quantum chaos that is represented by a weakened form of the
hyperbolicity condition of Def. 2.3. For example, it remains an
open question whether the free quantum dynamics of a particle on
a compact manifold of constant negative curvature fulfills the
condition (2.9) in an appropriate classical limit. Since the
dimensionless constant representing quantum effects in this model
is ${\gamma}:={\hbar}/(mE)^{1\over 2}R,$ where $m$ and $E$ are
the mass and energy, respectively, of the particle, and $R$ is
the radius of curvature of the manifold, such a classical limit
would imply that the dynamics was close to hyperbolicity when
${\gamma}<<1.$ In this case, it could well be that the motion of
the particle could still appear extremely unstable, and thus be
deemed to be chaotic, over physically relevant time scales.
\vskip 0.3cm\noindent
{\bf Proof of Theorem 3.} We shall employ a {\it reductio ad
absurdum} argument to prove that the hyperbolic property (2.9)
is incompatible with the discreteness of the spectrum of $H.$
Thus, we suppose that (2.9) is valid for one of the ${\sigma}'$s,
i.e. that
$${\alpha}(t){\sigma}(s){\alpha}(-t)=
{\sigma}(s.{\exp}({\lambda}t))\eqno(3.2)$$
where ${\lambda}$ is a real, non-zero constant, and the suffix
$j$ has been dropped. Hence, it follows from the
${\alpha}-$invariance of ${\omega}$ that 
$${\int}dsf(s){\langle}{\omega};{\sigma}(s){\alpha}(-t)
A{\rangle}={\int}dsf(s){\langle}{\omega};{\sigma}
(s.{\exp}({\lambda}t))A{\rangle} \ {\forall}A{\in}
{\cal A}, \ f{\in}{\bf C}_{0}({\bf R}), \ t{\in}{\bf R}\eqno(3.3)$$
where ${\bf C}_{0}({\bf R})$ denotes the continuous functions on
${\bf R}$ with compact support. By the dominated convergence
theorem, we see that 
$$R.H.S. \ of \ (3.3){\rightarrow}c{\omega}(A), \ with
 \ c={\int}dsf(s), \ as \
{\lambda}t{\rightarrow}-{\infty}\eqno(3.4)$$ 
On the other hand, 
$$L.H.S. \ of \
(3.3){\equiv}{\psi}({\alpha}(-t)A)\eqno(3.5)$$ 
where
$${\psi}={\int}ds{\sigma}^{\star}(s){\omega},\eqno(3.6)$$
and ${\sigma}^{\star}(s)$ is the dual of ${\sigma}(s).$ Thus
${\psi},$ like ${\omega},$ is normal.
\vskip 0.2cm
Now let ${\lbrace}{\phi}_{r}{\vert}r{\in}{\bf N}{\rbrace}$ be an
orthogonal basis in ${\cal H},$ consisting of eigenvectors of
$H,$ and let ${\lbrace}F_{mn}{\vert}m,n{\in}{\bf N}{\rbrace}$
be the operators in this space defined by
$$F_{mn}{\phi}_{r}:={\delta}_{nr}{\phi}_{m}\eqno(3.7)$$
These operators thus form a basis set for ${\cal A}.$ Moreover,
on applying (3.1) to the case where $A=F_{mn},$ it follows from
(3.5) that 
$$L.H.S. \ of \
(3.3)={\psi}(F_{lm}){\exp}(-i{\nu}_{mn}t)\eqno(3.8)$$
where ${\nu}_{mn}$ is the difference in the energy levels of
${\phi}_{m}$ and ${\phi}_{n},$ in units of Planck's constant.
Further, it follows from (3.1), (3.7) and
the ${\alpha}-$invariance of ${\omega}$ that, if
${\nu}_{mn}{\neq}0,$ then ${\omega}(F_{mn})=0.$ Consequently, the
compatibility of (3.4) with (3.8) implies that
$${\psi}(F_{mn})=0 \ when \ {\nu}_{mn}{\neq}0$$
By (3.1) and the normality of ${\psi},$ this signifies that
${\psi}$ is ${\alpha}-$invariant. Hence, by(3.6), together with
the ${\alpha}-$invariance of ${\omega}$ and the hyperbolicity
condition (3.2), 
$${\int}dsf(s){\langle}{\omega};{\sigma}(s)A{\rangle}
={\int}dsf(s){\langle}{\omega};{\sigma}(s){\alpha}(t)A
{\rangle}$$
$${\equiv}{\int}dsf(s){\langle}{\omega};{\alpha}(t)
{\sigma}(se^{{\lambda}t})A{\rangle}{\equiv}
{\int}dsf(s){\langle}{\omega};{\sigma}(se^{{\lambda}t})A{\rangle}
 \ {\forall}A{\in}{\cal A}, \ f{\in}{\bf C}_{0}({\bf R}) \ 
t{\in}{\bf R}\eqno(3.9)$$
Now let ${\lbrace}f_{n}{\rbrace}$ be a positive sequence in 
${\bf C}_{0}({\bf R}),$ whose integrals are all unity, and whose
supports are all contained in a single compact and converge to 
some point $u.$ Then, on replacing $f$ by
$f_{n}$ in (3.9), and passing to the limit
$n{\rightarrow}{\infty},$ we obtain the equation
$${\omega}({\sigma}(u)A)={\omega}({\sigma}(ue^{{\lambda}t})A)
 \ {\forall}A{\in}{\cal A}, \ t,u{\in}{\bf R}$$
Hence, replacing $A$ by ${\sigma}(-u)A$ here, and defining
$$v:=u(e^{{\lambda}t}-1),$$
$${\omega}(A)={\omega}({\sigma}(v)A) \ {\forall}A{\in}{\cal A}
 \ v{\in}{\bf R},\eqno(3.10)$$
which signifies that ${\omega}$ is ${\sigma}-$invariant.
Consequently, since this state is also ${\alpha}-$invariant, it
follows from [1, Th.3.3] that the spectrum of ${\cal H}$
covers the real line. However, this result, which stems from the
supposition that ${\Sigma}$ is hyperbolic, conflicts with the
specification that the spectrum of $H$ is discrete. We conclude
therefore that ${\Sigma}$ cannot be hyperbolic.
\vskip 0.5cm
\centerline {\bf Appendix A}
\vskip 0.3cm\noindent
{\bf Proof of Lemma 2.2}. In order to simplify the notation, we
shall drop the suffixes $t,j$ from the symbols
${\phi},V,{\theta},{\lambda},$ and define
$${\kappa}={\exp}({\lambda}t)\eqno(A.1)$$
for fixed $t.$ Thus, by equations (2.1)-(2.3),
$${\phi}^{\star}V(x)={\kappa}V({\phi}x);\eqno(A.2)$$
and 
$$x(s)={\theta}(s)x\eqno(A.3)$$
is the unique global solution of
$$x^{\prime}(s)=V(x(s)), \ {\forall}s{\in}{\bf R}; \
x(0)=x\eqno(A.4)$$
Hence, defining
$$x_{\phi}(s):={\phi}{\theta}
({\kappa}^{-1}s){\phi}^{-1}x,\eqno(A.5)$$
it follows from (A.2-4) that
$$x_{\phi}^{\prime}(s)={\kappa}^{-1}{\phi}^{\star}
({\phi}^{-1}x)^{\prime}[{\kappa}^{-1}s]=$$
$${\kappa}^{-1}{\phi}^{\star}V(({\phi}^{-1}x)[{\kappa}^{-1}s])=
V({\phi}({\phi}^{-1}x)[{\kappa}^{-1}s])=
V({\phi}{\theta}[{\kappa}^{-1}s]{\phi}^{-1}x)$$
i.e.
$$x_{\phi}^{\prime}(s)=V(x_{\phi}(s)),$$
which signifies that $x_{\phi}(.)$ satisfies (A.4). Hence, by the
uniqueness of the solution of that equation, 
$$x_{\phi}(s)=x(s) \ {\forall}s{\in}{\bf R},$$
i.e., by (A.3) and (A.5), 
$${\phi}{\theta}(s){\phi}^{-1}{\equiv}{\theta}({\kappa}s),$$
which is the required result.
\vskip 0.5cm
\centerline {\bf References}
\vskip 0.3cm\noindent
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\vskip 0.2cm\noindent
2.D. V. Anosov: Proc. Inst. Steklov {\bf 90}, 1 (1967)
\vskip 0.2cm\noindent
3. V. I. Arnold and A. Avez: "Ergodic Problems of Classical
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\vskip 0.2cm\noindent
4. J. Von Neumann: "Mathematical Foundations of Quantum
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\vskip 0.2cm\noindent
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\vskip 0.2cm\noindent
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\vskip 0.2cm\noindent
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\vskip 0.2cm\noindent
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\vskip 0.2cm\noindent
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\vskip 0.2cm\noindent
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\vskip 0.2cm\noindent
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