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\begin{document}

\title{{Generalized Wavefunctions for  \\
Correlated Quantum Oscillators III:\\
Chaos, Irreversibility.}}

\author{S. Maxson}
\email{smaxson@carbon.cudenver.edu}
\affiliation{Department of Physics\\University of Colorado at 
Denver\\Denver, Colorado 80217\\}

\date{\today}


\begin{abstract}

In this third of a series of articles, we continue the study of the
dynamical transformations of systems of correlated
quantized oscillators. By including generalized wave
function solutions to Schr{\"o}dinger's equation belonging 
to a rigged Hilbert space, and by considering the 
algebra of observables as a whole, the presence of hyperbolic 
quasi-invariant measures, torus actions, ergodicity  and entropy
generation associated to the
non-invertible decay of Gamow vectors and associated to Breit-Wigner
resonances is shown.  A weak (local) form of the second law of
thermodynamics is demonstrated through the decay of resonances.  Both
coherence formation and decoherence (decay) are associated with
irreversibility and may be associated with
entropy growth.  Hilbert space is the manifold of
non-ergodic stationary states.  There is a fractal structure associated with 
time evolution of resonances in the space of generalized states, and
the statistical nature of the exponential decay may be identified 
with quasi-trapping.  Equilibrium states may be regarded as strange
attractors with respect to dynamical time evolution.  



\end{abstract}


\pacs{05.30.-d,05.70.Ln,05.90.+m,47.53.+n}        %use showpacs class option

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\maketitle




\section{Introduction}\label{sec:intro}


\subsection{Motivation}\label{sec:motivation}

In this third installment, it is necessary to bear in mind that we
have in contemplation a formalism simultaneously applicable to two
types of system: the probabilistic description of a system which has
internal correlations which is {\em closed-in-fact}, and also we have
in mind the probabilistic description of a system which is merely {\em
formally-closed} due to reductionistic simplification.  In the
closed-in-fact category we might might include quantum
cosmology studies, such is implied in our fourth
installment~\cite{IV}, wherein it is shown how a hyperbolic quantum
theory over several fields (phrased in terms of canonical coordinates)  
can be related to hyperbolic dynamical behavior in space-time.  In the
formally-closed 
category, we would put most descriptions of resonant systems--the
micro-physical world is too complicated a place for us to describe in
complete and exhaustive detail, and so we necessarily simplify our
description of the system.  Hence, the use of a probabilistic
description, along the lines indicated in the first installment~\cite{I}.

Recall from~\cite{II} that we have an intrinsically hyperbolic
geometry for flows on our space of states by construction.
Because we have hyperbolic sensitivity to
initial conditions in our description of resonances
(see Appendix~\ref{sec:continuation}), we expect that our description
of resonances may be subject to the extreme sensitivity to
perturbation well known for chaotic classical dynamical systems.  As
is shown below, in the case of an exponentially decaying system that
is closed-in-fact, the
implication is that dynamical time evolution is an infinitely long
fractal meander towards equilibrium, when viewed as an evolution
through a linear space of probability distributions.  However, the
exponential decay commonly observed for everyday systems (which
are merely formally-closed) can be attributed to sensitivity to the
minute insults received at random time intervals
from sources external to our idealized
system~\cite{note0x5}.

Our choice of geodesic dynamical evolution in our
constructions of~\cite{II} conforms to Maxwell's ``continuous path''
alternative to Boltzmann's ergodic hypothesis.  Note also that our
invocation of geodesics and parallel transport implies the existence
of memory effects, including persistence
of those quantum numbers which characterize the coherences of quantum
fields we call particles, so that our description of resonances is of
dynamics as a non-Markovian process.  The explicit incorporation 
of correlation in our description (see~\cite{I,II}) further
implies sub-Poisson statistics
incompatible with notions of Markov process.  It is well
known that deviations from Poisson statistics may not dominate for
a long time.

The extension of many of the notions of classical non-linear dynamics
to quantum systems has been problematic~\cite{gutz,haake}.  What will
emerge from the present series of articles is a picture of quantum
dynamics which possesses many of the mathematical features which are
signatures of chaos and other interesting behaviors of classical
non-linear dynamics, such as 
sensitivity to initial conditions, hyperbolicity,
fractals and strange attractors, and so on.  It will also be apparent
that these  interesting non-linear dynamics
structures are categorically incompatible with the conventional
Hilbert space formalism.

We continue the lines of development begun in the first two installments in
this series~\cite{I,II}.
One major feature emerging (by construction)
from following this mathematical approach is the 
appearance of mathematical structures associated with the dynamical
evolution of
quantum resonant state vectors  which correspond to mathematical structures
(hyperbolic sets and measures) one associates with hard 
chaos in classical flows.  In place of the ``random'' wanderings of
points in phase space characteristic of classical chaos, one finds 
``random'' wanderings in the state space of probability amplitudes on
that phase space.  See Section~\ref{sec:span} and
Section~\ref{sec:ergodicity}.

Reflecting the maximal (complex) tori in the connected
(simple) complex Lie group derived 
of the {\em real} semi-simple Lie algebra of the group of symplectic
(=canonical) transformations one finds there exists 
a form of torus action on the analytically continued
representation space, with the evolution of
resonances being associated to dense windings generated on the
complex 1-tori of the complex covering group, 
and hence ergodic.  These complex tori are ``Hamiltonian 
quasi-invariant surfaces'' in the spaces of quantum states
in the present formalism, while the (real) KAM tori of classical non-linear
dynamics usually make their appearance as ``Lagrangian invariant
surfaces''.   The stability structure of these
hamiltonian complex tori is, however,
radically different from the stability structure of KAM tori.
Tori appear to play a much less significant role in our hamiltonian
formalism.  

Our construction preserves the ergodicity of the classical oscillator
and that of the quantum harmonic oscillator in the von Neumann
construction.  For us, ergodicity arises because dynamical evolution
is associated with continuous mappings of dense sets to dense sets,
i.e., via the exponential map of $\tau_\F$-closed, $\tau_\F$-continuous
operators.  Additionally, in the RHS
$\rhs$, there is an invariant (Lebesque) measure on $\HH$, but
$\F^\times$ supplies quasi-invariant measures (including hyperbolic
measures) for $\F$, and the two measures
are equivalent so that the quasi-invariant measure on $\F\subset \F^\times$ 
can be said to induce a Lebesque measure on $\F\cap\HH$.  
Both physically 
and mathematically this is a very subtle area and we will express no
definitive resolution of the issues today~\cite{note1x5}. 

Entropy
growth due to the dynamical evolution of quantum resonances which have
been represented by the Breit-Wigner resonances associated to 
Bohm-Gadella version of the Gamow vectors is
explicitly demonstrated, although prudent physical interpretation of
the mathematical structure suggests that only a weak (local) form of
the Second Law should be inferred from this
construction at the present level of development.  

In Section~\ref{sec:fractals} we see that our generalized space
of states has a fractal structure on it associated with the presence
of resonances and their dynamical evolution.  But the space of 
states is generated by
(representation of) an algebra of transformations which 
include transformations which are ergodic, and 
so possibly comprise an algebraic structure related to the von Neumann 
type III algebras, which are normally defined using conventional 
Hilbert spaces only, rather than on the ``auxiliary'' Hilbert spaces
of a RHS.  It is known that Von Neumann type III algebras may 
have non-integer dimension, but we shall not be diverted 
from our present discussions to explore these interesting issues.  
There are far reaching implications of
this which are too widely
beyond the scope of present discussions.  

An issue to be frequently re-encountered throughout (not
always with explicit announcement) is the lesson that an intrinsic 
irreversibility in micro-physical dynamics is associated with ``complexity'' 
in the sense that only when one considers a non-trivial algebra of 
observables is the mathematical representation of non-invertible dynamics 
possible.  Such a non-trivial algebra structure is provided by the 
``$SU(1,1)$ dissipative oscillator'' system of Feshbach and
Tikochinsky~\cite{ft} recapitulated in the appendix to~\cite{II}.  This 
is probably the simplest explicit example of non-invertible
dynamics just as (or precisely because) $Sp(2,\R )_\pm$ (which has a
algebra isomorphic to $\mathfrak{su}(1,1)_\pm$ )
is the simplest non-compact semi-simple Lie semi-group.  There is 
a lesson here to the effect that
reductionism can {\em always} be pursued to the point where
irreversibility is made to
disappears from the mathematical representation of dynamics.  This
present work thus gives explicit {\em partial} support of a long held
contention of Prigogine that irreversibility should be viewed as
intrinsic to dynamics.  This support is only partial because the
present work relies on {\em local} methods (algebra), and may in fact
be derivative of local approximations.  (See
Appendix~\ref{sec:status}.)  
Although a weak form of the Second Law can be deduced from the present
body of work (as indicated previously), there are 
possibilities for strengthening this weak Second Law beyond the
state of development we put forward here.
See Section~\ref{sec:entropy}.






\subsection{Background}\label{sec:background}


It has long been known that analytic continuation is associated with
``sensitivity to initial conditions''~\cite{mandf1}, and the present procedures also
seem associated to the issues of stability of solutions, e.g., a faithful 
mathematical representation of decaying states must be unstable in some sense. 
It is also known that hyperbolic problems (and hyperbolic structures) 
emerge from the analytic continuation of elliptic
problems~\cite{garabedian,lewy}.  The 
presence of hyperbolic structures and sensitivity to initial conditions
suggests that hard chaos may be present {\em{somewhere}} in an analytically
continued system: in the analytic continuation (extension) of a space of 
quantum states, such hard chaos could possibly represent actual
hard chaos in a quantum 
system.  The exponentially decaying abstract Gamow states and
associated Breit-Wigner resonance poles of~\cite{bohm1,bohm2} attract immediate 
attention as offering a possible quantum analogue to the 
exponential separation of 
trajectories in classical phase space which is a signature in classical hard 
chaos.  

We have a generic sensitivity to initial conditions in our methods.
Symplectic transformations of a Hausdorf space are
transitive~\cite{souriau}.  We have dense sets of periodic orbits associated with the
$U(1)$ cycles of equilibria, among other possibilities for dense sets
of periodic points.  We thus have DeVaney's elements of chaos present
in our space(s) of quantum states, and we can further label this as
deterministic chaos since evolution is Hamiltonian, notwithstanding an
element of apparent randomness appearing in the decay of resonances.









\subsection{Order of proceeding}\label{sec:proceeding}
	
In Appendix~\ref{sec:continuation}, many analytic continuation issues
not otherwise addressed are dealt with at the level of undergraduate
math.  This appendix
includes exploration of the possibility that 
hyperbolicity and chaos may be generic to analytic continuation
(whenever it is performed as a {\em continuous} process).  It is 
in this section that the importance 
of continuous transformations and continuous structures becomes apparent in 
setting the foundations for a quantum analogue of classical non-linear 
dynamics.  A construction based on traditional complex
analysis at the undergraduate level of difficulty explicitly
demonstrates how many of the features of hard chaos are generic 
to analytic continuation, including sensitivity to initial conditions,
the presence of tori, and the presence of ``stable'' and ``unstable''
manifolds (hyperbolicity).  There is also very strong suggestion of
the emergence of semi-groups, although those are not visible at the
undergraduate complex analysis level.  (At a more advanced level, not
dealt with but readily apparent after reading the rest of the paper,
is the observation that the analytically continued Green's functions
are distributions called propagators, which as distributions must be
associated to semi-groups of evolution over restricted domains of time
definition.  For example, we infer in the theory of radiation it would be a
serious mathematical error to define an advanced propagator for $t\ge
0$, and so by pursuing mathematical rigor we are led to avoid the 
well known physical problems of causality attending such a poor
mathematical choice.)

Appendix~\ref{sec:status} discusses the status of a physical theory cast in a 
rigged Hilbert space along the lines used herein.  It is
pointed out that the entire paradigm may not be based merely on local
methods (e.g., the algebra of infinitesimal generators), but may even
be based on a {\em local approximation} in the case of semi-simple
transformation symmetries involved in those predictions.  This
provides 
a substantial reason for restraint in interpreting the breadth of
our version of the Second Law.  (We are evading the issue of
whether construction which is locally hyperbolic at every point means
globally hyperbolic evolution is {\em necessary} in any compelling
sense partly because our dual spaces need not be
manifolds.)  

As a part of Section \ref{sec:span}, it is shown that the pure 
exponential decay of the quantum states represented by 
Gamow vectors (and their associated Breit-Wigner resonance poles) 
is the quantum analogue of the 
classical exponential separation of trajectories in phase space.  
Because of the impossibility of pure exponential decay in Hilbert 
space within the domain of definition of an hermitean
operator~\cite{khalfin,fonda,note1}, 
these structures could only be approximated in the Hilbert space 
topology.  The nature of the structures produced is explored,
largely geometrically.  Of special interest are the equismooth 
semi-flows generated by the semi-groups of transformations on the 
representation spaces, and the local action of each generator 
taken individually (e.g., in its role 
as a generator of a sub-semi-group of the complex covering semi-group).  
The ergodicity of the semi-flows of a generator, and the associated 
Gamow vectors, are discussed, and the role of the Gamow vectors in 
generating entropy as they evolve is shown.  This is the crux 
of the entire construction, fairly demonstrating the existence of
hyperbolic dynamical evolution, issues of exact meaning aside. 




 






\section{Hyperbolicity}\label{sec:hyperbolicity}


The spaces of states constructed in~\cite{II} are composed of spinors which
are associated with representation of a Clifford Algebra of the phase
space for the classical oscillator problem.  (In the fourth
installment~\cite{IV}, we will provide greater particularity as just exactly
which Clifford algebra this is.)  Because the phase space possesses
both orthogonal and symplectic structures (and induced symmetric and
skew-symmetric scalar products), our spaces of states correspondingly  
possess representations of orthogonal and symplectic structures (and
induced symmetric and skew-symmetric scalar products--see~\cite{I}).
In installment two~\cite{II}, the action of the representation of the
semi-groups $Sp(4,\R )_\pm$ on 
$\F$ was defined in such a way that the symplectic form
on $\F$ was preserved .

By the Paley-Wiener Theorem, $L^2 = \HH^2_+ \oplus \HH^2_-$,
indicating that the $L^2$ function space realization of our Hilbert
space $\HH$ is bifurcated (in Lebesque measure) along the lines of
semi-groups of time evolution.  The Schr{\"o}dinger equation gives us 
$U_1 (t)_\pm$ semi-groups of time
evolution on $\F_{\mathfrak{sp}(4,\R )\pm}$ (and on the function space
$\CS_{\mathfrak{sp}(4,\R )} \cap \HH^2_\pm |_{\R^+}$), and  exponential
growth/decay on Gamow vectors (energy eigenvectors belonging to)  
$\F_{\mathfrak{sp}(4,\R)}^\times $, e.g., on Breit-Wigner resonances
of the dual function space
$(\CS_{\mathfrak{sp}(4,\R )} \cap \HH^2_\pm |_{\R^+})^\times $.
(Refer back to the Gadella diagrams in installment one~\cite{I}.)

As a consequence of the representation of the Clifford algebra of
phase space (see installment four~\cite{IV}), 
there is a symplectic structure (with all of its attributes,
such as a skew-symmetric scalar product) on $\F_{\mathfrak{sp}(4,\R
)} \subset \HH_{\mathfrak{sp}(4,\R )\pm}$ (and on
$\CS_{\mathfrak{sp}(4,\R )} \cap \HH^2_\pm |_{\R^+}\subset 
L^2_{\mathfrak{sp}(4,\R )\pm}$).  There is only a set of Lebesgue
measure zero obstructing the completion of the semi-groups of dynamical
evolution (including dynamical time evolution)
on the Hilbert spaces $\HH_{\mathfrak{sp}(4,\R )\pm}$ into full
groups, e.g., extrapolating the $U(1)$ semigroup of dynamical time
evolution defined for $t\ge 0$ backwards over $t\le 0$ to obtain a
group of dynamical time evolution over $t \in \R$~\cite{ludwig1,ludwig2}.
Thus, we can fairly write that our $\F_{\mathfrak{sp}(4,\R )\pm}
\subset \HH_{Sp(4.\R )}$,
where $\HH_{Sp(4,\R )}$  is a representation Hilbert space for the
full group $Sp(4,\R )$.  Equivalently, given our strictly
infinitesimally generated semi-groups whose sum is the full group (up
to sets of Lebesque measure zero), we may signify the representation
of the full group on symplectic transformations on $\HH$ thus:
\begin{equation}
\HH_{Sp(4,\R )} \cong \left(\CS_{\mathfrak{sp}(4,\R )} \cap \HH^2_+
\right) \,\, \oplus \,\,
\left( \CS_{\mathfrak{sp}(4,\R )} \cap \HH^2_- \right) \quad .
\label{eq:hypdecomp}
\end{equation}
The transformations generated by the essentially self adjoint
Hamiltonian with  
complex eigenvalues, e.g., the semi-groups of time evolution $U_t
(1)_\pm = e^{\pm iHt}$, $t\lessgtr 0$, 
have bifurcated solution spaces.  For times $t\le 0$, we have the
preparation  
of the state $\phi^+$ belonging to the abstract space $\F_-$.  For
$t\ge 0$,  
one observes the effect $\psi^-$, belonging to the abstract space
$\F_+$ 
~\cite{gadella, bgm,bmlg,gadella,dirackets,qat1,qat2,qat3}.  
The two spaces $\F_-$ and $\F_+$, and the corresponding function space  
representations, $\left. \CS\cap\HH^2_- \right|_{\R_+}$ and $\left. 
\CS\cap\HH^2_+ \right|_{\R_+}$ respectively, may be viewed as
alternative {\em continuous} linear 
space structures obtained on the same set in the present
constructions.   

The spaces  
$\left. \CS\cap\HH^2_- \right|_{\R_+}$ and $\left. \CS\cap\HH^2_+ 
\right|_{\R_+}$ can be uniquely extended to $\CS\cap\HH^2_-$ and 
$\CS\cap\HH^2_+$ due to their Hardy class properties.
See~\cite{gadella} and~\cite{dirackets} and
references therein.  

Again, because of the Paley-Wiener theorem, $L^2 =\HH^2_+\oplus \HH^2_-$, 
and so the abstract spaces $\F_{\g +}$ and 
$\F_{\g -}$ are orthogonal (transverse), as are their function space
realizations.  The function space 
$\left(\CS_{\mathfrak{sp}(4,\C )^\R }\cap\HH^2_+\right)^\times$ 
corresponds to what would 
be called an unstable set in classical nonlinear dynamics and the space 
$\left(\CS_{\mathfrak{sp}(4,\C )^\R } \cap\HH^2_-\right)^\times$ would be 
called the stable set in nonlinear dynamics 
terminology, and one might even regard $\psi_G (0)$ as a repellor,
and so on.  From the eigenvalues of the Hamiltonian, we identify the
function spaces
$\CS_{\mathfrak{sp}(4,\C )^\R}\cap \HH^2_ \subset
\left(\CS_{\mathfrak{sp}(4,\C )^\R}\cap \HH^2_- \right)^\times$ as an expanding 
subspace and $\CS_{\mathfrak{sp}(4,\C )^\R } \cap \HH^2_+  \subset
\left(\CS_{\mathfrak{sp}(4,\C )^\R }\cap \HH^2_+ \right)^\times$ as a
contracting subspace of an hyperbolic set $\HH\cong L^2 = \HH_+ \oplus \HH_-$, 
and the corresponding duals  
$\left( \CS_{\mathfrak{sp}(4,\C )^\R }\cap \HH^2_\pm \right)^\times$, 
(by reflexitivity) can be
thought of as defining hyperbolic measures on $\CS_{\mathfrak{sp}(4,\C
)^\R} \cap \HH^2_\pm$~\cite{hirsch,zimmer}.







\section{The Structure Spanned by the Gamow Vectors}\label{sec:span}

\subsection{Complex 1-tori}\label{sec:complextori} 

To obtain KAM tori in classical mechanics, it is necessary to address
many subtle and delicate mathematical issues.  See, e.g.,~\cite{moser}.   
The presence of symmetries in the present class of hamiltonian
problems leads quite directly to 
the presence of tori in the problem, and toral actions on the 
representation spaces.  The maximal tori in the group will provide us
with a ``torus action'', although this is
not at all the same as the torus action on classical KAM tori.  
The correspondences
to the irrational windings on KAM (real) tori occurring on 
these maximal tori (e.g., dense, non-compact torus actions) seem to be 
associated with ergodicity (see Section~\ref{sec:ergodicity}) and
entropy production (see Section~\ref{sec:entropy}), but have
stability implications which are distinguishable from the stability
considerations associated with classical tori and classical torus
actions.  E.g., contrast the widths of the resonances developed in
the second installment of this series~\cite{II}, which
relate to the half life of the decay with classical developments
in~\cite{siegel}, Section III, which relates to more general type of
stability against perturbation.  In the present developments, {\em
complex tori} are involved: rational ``winding numbers'' of 
paths on the complex tori will commonly possess a 
non-zero complex energy component for
the dissipative oscillator system, meaning there is exponential growth
or decay associated even with rational
``winding numbers'' on these complex
tori.  We will later see that some of these orbits are ergodic.

Ergodicity
of a toral winding is a characteristic determined by whether the
winding is dense on that torus, and whether one has a real torus or a
complex torus has no immediate bearing on ergodicity: what matters is
during evolution an averaging takes place over a dense set of
points of the torus~\cite{petersen,zimmer}.  Whether or not a set is
dense is of course a topological issue.

Any possible comparison between real and complex tori based on``winding
numbers'' should therefore not be taken as governing.  This is because the
complex plane is a complex
hyperbolic space, and the meaning of a winding
(geodesic) has a different meaning in the appropriate topologies for
the two spaces: in standard metric topologies, 
a geodesic is a minima in the
real plane but may be a maximum distance in the complex plane.
Nevertheless, there are loose similarities, and further detailed
investigations may prove worthwhile.
 
There is a very important theorem related to the preceding:
\begin{Thm}(~\cite{kandn2}, page 159)
The complex tori are the only compact parallelizable manifolds which admit 
K{\"a}hler metrics.
\end{Thm}
This means that complex tori, such as the complex 1-tori of the 
complex simple Lie groups (and their strictly infinitesimally generated 
sub-semi-groups) are the exclusive (compact complex)
domain upon which one may speak of 
parallelism in the dynamical evolution of 
resonances, e.g., represent resonant evolution with 
the conservation of some quantum numbers, and other related conservation 
effects which are the physical consequence of the 
geometric parallelism permitted of 
the mathematical theory.  (Complex tori are similarly
the exclusive domain for the study of geometric phase of resonances 
arising through dynamics.)   

From the perspective of algebra, these 
complex 1-tori are commutative and possess an identity, and so 
they and their faithful representations are Noetherian semi-rings, which will 
lead to induced Noetherian rings~\cite{hungerford}, page 387, when working in the Hilbert 
space topology.  Compare Section \ref{sec:ergodicity}.  The ability to
establish a geodesic structure (using $exp$) on the  
complex 1-tori (and their faithful representations) is the source of 
a form of Noetherian conservation, and the geodesic 
semi-flows on these complex 1-tori should be related to the 
currents of Noetherian theory (which would exist in the Hilbert space 
completion of these structures) in a straightforward manner.  One also
infers
from this that the present formalism should be translatable into some 
form of ``current algebra'' formalism.













\section{Ergodicity Associated with the Gamow 
Vectors}\label{sec:ergodicity}

Given rigged Hilbert space is taken 
to contain a dynamical representation of a group 
on $\F_{\g\pm}\subset\HH$ (meaning
$\F_{\g\pm}\cap\HH$), the ergodic nature of the transformations 
in the complex covering group represented on the
analytically continued space $\F_{\g^\C\pm}\subset\F^\times$ has 
two consequences:
\begin{enumerate}
\item{These transformations define a quasi-invariant measure on 
$\F_{\g^\C\pm}\subset\F^\times_{\g^\C\pm}$ equivalent to the 
invariant (Lebesque) measure on 
$\F_{\g^\C\pm}\cap\HH$.}
\item{As a necessary and sufficient condition of their ergodicity, 
these transformations induce unitary transformations on the appropriate 
$L^2$ space, e.g., including
on the complex Hilbert space or on the Hardy class
function space realization of the abstract space $\F$.}
\end{enumerate}
We find a quasi-invariant measure associated with the resonances and 
complex spectra of esa operators on $\F_{\g^\C\pm}\subset\F^\times$, 
which induces an invariant measure on the manifold 
$\F_{\g^\C\pm}\cap\HH$ which is associated with the hyperbolic sets.  


Any possible clarity of the meaning of a compact or non-compact complex
torus action on
the abstract space $\F_{\g^\C\pm}$ or its function space realization
$\CS_{\g^\C\pm}$ is missing when working in the corresponding dual 
spaces.  The esa
extension of generators to the dual spaces may result in an unbounded
operator there, and additionally there is a coarser topology at work
which may fail to separate the actions of the extended (esa)
generators.  Any ergodicity in the present setting must {\em not} have
any special reliance on torus actions.  Of course, the fact that we
have non-trivial {\em continuous} transformations representing time
evolution is a substantial clue that we will find that time evolution
to be ergodic, unless we destroy ergodicity somewhere along the way.

The representation $\theta :G \longrightarrow Aut (\F )$ maps dense 
sets to dense sets.  One can thus associate a dense orbit of $exp$ in $G$ 
with some dense orbit of some exponential map on $\F$.   

The action of the representation of $\g$ on 
$\F_{\g\pm}$ induces a 
positive, countably additive measure $\mu$ in 
$\F^\times_{\g\pm} $~\cite{genfun4}, page 311.  
This is a quasi-invariant measure.  
The hermitean scalar product on 
$\F\subset\F^\times$ defines a Gaussian 
measure which is quasi-invariant, in the sense that $\psi\in\F$ induces 
$F_\psi \in\F^\times$ via the scalar product $(\phi ,\psi )$, $\phi \in\F$, 
such that $(F_\psi ,\phi )= (\phi ,\psi )$ is the canonical anti-linear 
embedding $\psi\longrightarrow F_\psi$ of $\F $ into
$\F^\times$~\cite{genfun4}, page 355.
This quasi-invariant measure is equivalent to the Lebesque measure, but there 
is no quasi-invariant measure on $\HH$.  These quasi-invariant
measures equivalent to Lebesque measures exist within each Gel'fand
triplet (each level) in the Gadella diagrams in the first installment
of this series~\cite{I}.

This equivalence to the Lebesque 
measure can be interpreted to mean 
that the quasi-invariant measure ``induces'' a Lebesque measure 
on $\HH$. This induced measure is $G_\pm$ invariant (e.g., Lebesque) on 
$\F\cap \HH$, since $G$-translations carry sets of measure zero to sets of 
measure zero on $\HH$~\cite{note22x5}.  The necessary 
and sufficient condition for any transformations to be ergodic
is that they induce an unitary transformation on the
appropriate space of Lebesque square integrable functions, e.g., 
on the complex Hilbert space $\HH^\C$ and its associated 
function space realizations~\cite{petersen}, page 43.  The Hardy class spaces of
interest to us are $L^2$ spaces, and so the Gadella diagrams of
installment two~\cite{II} establish the necessary and sufficient
conditions for ergodicity have been met.  Considerations of (complex)
tori have no special meaning here.  Of course it is well known
that both classical and quantum oscillators are
ergodic.  We have not impaired that ergodicity, and
have shown that the time evolution of the related gausian pure
states is ergodic as well.  This ergodicity guarantees the existence
of a unique equilibrium state toward which dynamical time evolution
is directed, according to standard theorems of ergodic theory.
We will confirm that dynamical time evolution is in fact
toward this unique equilibrium--see Section~\ref{sec:entropy}.

The formulation of the abstract 
$G_\pm$-transformations in a form which is ergodic 
on the abstract space $\F$ means that a $G_\pm$-transformation 
on $\F$ induces an unitary transformation on a subspace of the 
abstract space $\HH$, and this set of unitary semi-group
transformations on $\HH$ may be made into a unitary group
representation on $\HH$ by extrapolation
backwards from the semi-group and by ignoring, at most, 
a set of measure zero (as done in~\cite{ludwig1}, see~\cite{ludwig2}.)          

Separation of considerations of the Haar measure
on the group (see~\cite{barut,duren}) by 
artful definition of (semi-)group actions on the various spaces 
involved enables this 
structure of two equivalent measures.  




6









\section{Noninvertibility of the Transformations}
\label{sec:noninvertibility}

Recall that the constituent complex 1-tori which 
result from the complexification of a $U(1)$ sub-semi-group of a semi-group 
are not semi-groups when taken in isolation from the full semi-group 
structure~\cite{hilgert}.  This is because the complexification of 
a {\em single} $U(1)$ subgroup is an
abelian torus semi-group, i.e., there are no obstructions (on sets of
positive measure) to restoring the 
full group structure to the ``$U(1)^\C_\pm$'' sub-semi-groups individually.
Intrinsic micro-physical irreversibility~\cite{qat1,qat2,qat3,bgm,bmlg}
results 
only when the fuller algebra is 
invoked, and the physical interpretation of this mathematical structure 
is that temporal irreversibility emerges from the complexity 
of concurrent evolutionary processes.  A single complex spectrum is not 
enough, and it is the use of the full non-trivial algebra of
observables which precludes the extrapolation from semi-group to full
group~\cite{note22}.

A key observation to be made here is that the usual process
of reductionism until one is considering only a single evolutionary process
in isolation should seemingly always lead to a reversible picture of dynamics,
since in that limit the semi-group structure becomes ephemeral and
can almost certainly be made into a group structure by perfectly 
reasonable means.  See, e.g.,~\cite{ludwig1}.  There is a measure of complexity
required before you can begin to speak of irreversibility in a
meaningful way.  The
semi-groups and the complex spectra are associated not with a single
generator but with a single generator as 
but one element of an algebra of (non-commuting) observables. 

The previously shown ergodicity is a necessary but not sufficient condition 
for non-invertibility.  Non-invertibility (e.g., semi-groups of time
evolution equivalent to the quantum arrow of
time, QAT) is an attribute coming from
use of the full algebra, which includes the Hamiltonian $H$ and other
non-commuting observables.  

Non-invertibility, in turn, is necessary but not
sufficient for the entropy of a system to increase from its initial
value.  See, e.g.,~\cite{mackey}.  In the next subsection, it will be
demonstrated that the quantum arrow of time which is dynamical in
origin~\cite{qat1,qat2,qat3,bgm,bmlg}, 
is also associated with the non-invertible entropy increase due to the
decay of the Gamow vectors (and their associated Breit-Wigner
resonance poles).  It is significant that there were only a countable
number of Gamow states constructible using the prescription of
installment two~\cite{II}, since this means they may possibly be
excepted from the Poincar{\"e} recurrence theorem.

Note also that the 
non-invertible coupled oscillators can be solved by the device of 
action angle variables, which provides invertible (time reversible) 
solutions in the free oscillators, as the two dimensional problem of 
coupled oscillators is converted into
a pair of independent one dimensional problems. 
The use of action angle variables eliminates
correlation (coherence) between the oscillators, and so 
the {\em system defined by the action angle variables represents a
dynamically (and hence physically)
inequivalent system}.  Correlation--and possibly even coherence--are 
(in this instance at least) to be
associated with non-invertibility.  Removal of the correlation
(decoherence) of the coupled oscillators produces an invertible 
equilibrium system!
The canonical transformations are much more than
a formal device for
first obtaining solutions to one dimensional problems and then 
constructing solutions to more complicated problems.  







\section{Maximum Entropy Due to the Gamow Vectors}\label{sec:entropy}

Ergodicity establishes the existence of a unique equilibrium state for the
system, toward which the system can evolve.  Noninvertibility 
establishes that entropy growth due to the decay
of the Gamow vectors is possible. The obvious question at this point is
whether the dynamical evolution is toward that equilibrium.  The
answers to that question depend on entropy increasing or not, but in
order to address this question it is in order that we address some
issues related to the density matrix, in order to generalize the von
Neumann entropy.

Throughout we have used operators which are esa but not symmetric
(i.e., not hermitean).  This was necessitated by mathematical
requirements, since continuity existed only in semi-groups and not in
groups, compelling us to clearly distinguish time domains of definition.
The prepared state $\phi^{in}$ has well defined dynamical time
evolution only for $t\le 0$ and the observed effect $\psi^{out}$ has well defined
dynamical time evolution only for $ t\ge 0$.  Our scalar product
$\langle \psi^{out} \vert \phi^{in}\rangle$ is composed of two parts,
each with a different domain of time definition, so that any
projection operator $\vert \lambda\rangle\langle\tilde\lambda\vert$
must likewise consist of elements with differing domains of time
definition.  Hence, the genuine need for the tilde or some other
notational device to indicate time domain of definition.
Thus, if we insert the projector 
$\vert \lambda\rangle\langle\tilde\lambda\vert$ 
into the matrix element 
$\langle \psi^{out} \vert \phi^{in}\rangle$ 
\begin{equation}
\langle \psi^{out} \vert \phi^{in}\rangle \longmapsto 
\langle \psi^{out} \vert
\lambda\rangle\langle\tilde\lambda\vert\phi^{in}\rangle  
\nonumber
\end{equation}
it is required that $\langle \psi^{out}\vert$ and
$\langle\tilde\lambda\vert$ both be defined form $t\ge 0$ only, and
that $\vert\lambda\rangle$ and $\vert\phi^{out}\rangle$ be defined for
$t\le 0$ only.  If $\vert \lambda\rangle\langle\tilde\lambda\vert$ is
to be 
thought of as a measure, then we would say that the left and right
(quasi-) invariant measures are different (it is not a symmetric
operator), with $\vert \lambda\rangle$
and $\langle \tilde\lambda \vert$ having different time domains for
which they are defined.

Exactly such a non-symmetric esa operator occurs in the complex
spectral 
theorem (partition of the identity) for the Gamow vectors
\begin{equation}
\sum_j \;\; \vert \psi^G_j\rangle\langle \ts^G_j\vert = \II
\nonumber
\end{equation}
The sum over bound states and the geometric phase (holonomy)
term (also known as the background term) have been
ignored (refer back to the Appendix to installment one~\cite{I}.)  The   
$\vert \psi^G_j\rangle$ is defined for $t\ge 0$ only, and $\langle
\ts^G_j\vert$ is defined for $t\le 0$ only, because, e.g.,
\begin{equation}
\langle \psi^{out} \vert \phi^{in}\rangle =
\sum_j \;\; \langle \psi^{out} 
\vert \psi^G_j\rangle\langle \ts^G_j\vert \phi^{in}\rangle \;\;\; .
\label{eq:partid}
\end{equation}

To take the trace of the esa $\vert
\lambda\rangle\langle\tilde\lambda\vert$ involves something like
$\sum_i \langle \tilde\psi_i\vert
\lambda\rangle\langle\tilde\lambda\vert \psi_i\rangle$, where the
$\psi_i$ are elements of the dual space providing normalized duals to
out basis.  Then the $\langle \tilde\psi_i\vert$ form a  
basis in the dual space (or at least for a subspace of the dual space).
The spaces which $\vert\psi_i\rangle$
and $\langle\tilde\psi_i\vert$ provide a basis (and dual) for
can be determined by a brief inquiry.  In the
standard von Neumann Hilbert space, if we define the density matrix 
$W = \vert\phi\rangle\langle\tilde\phi\vert$, then the probability or
expectation of observing the physical state represented by the density
matrix $W$ is defined as
\begin{eqnarray}
\textrm{Tr} \; W = \textrm{Tr} \vert\phi\rangle\langle\tilde\phi\vert &=&
\sum_i \langle
\tilde\psi_i\vert\phi\rangle\langle\tilde\phi\vert\psi_i\rangle \nonumber \\
&=& \sum_i \langle\tilde\phi\vert\psi_i\rangle \langle
\tilde\psi_i\vert\phi\rangle\nonumber \\
&=& \langle\tilde\phi\vert \, \II \, \vert \phi\rangle  \nonumber \\
&=& \langle\tilde\phi\vert\phi\rangle \;\;\; .
\label{eq:vNtrace}
\end{eqnarray}
Note that if a complex vector $\phi$ is obtained by the complex
symplectic transformation $T$ of a real vector $\phi_0$, $\phi =
T\phi_0$ 
according to the prescription offered in installments one and two
~\cite{I,II}, then $\langle \tilde\phi\vert\phi\rangle =
\langle\tilde\phi_0 \vert \phi_0\rangle \in \RR$, since $TT^\times =
\II$. 

Although the partition of the identity is symmetric in von Neumann's
formalism, it is still the case there that
$\langle\tilde\phi_i\vert \ne \vert\phi_i\rangle$, i.e., the dual
of a vector is not the same as the vector itself.  (Recall, in the
complex function space realization of von Neumann's Hilbert space, 
$\langle\psi_i\vert = \psi^\times_i = \psi^\ast_i$.)

In the present situation, we assume a density matrix of the form
$W=\vert\phi\rangle\langle\tilde\phi\vert$, where $\vert\phi\rangle$
is defined for $t\ge 0$ only , and $\langle\tilde\phi\vert$ is defined
for $t\le 0$ only.  However, $\langle\tilde\phi\vert$ is not
generally identifiable with $\vert\phi\rangle^\ast$ due to the nature
of our choice of adjoint involution in~\cite{II}.  ($W$ is also not
symmetric--hence the added relevance of the tilde over the $\phi$ as a 
label--and also this is a spin representation.)

Our time dependent density matrix is
\begin{equation}
W(t) = \overrightarrow{e^{-iHt}} \, \vert \phi (0)\rangle
\langle \tilde\phi (0) \vert \, \overleftarrow{e^{+iHt}}
\label{eq:densitymat}
\end{equation}
where $\overrightarrow{e^{-iHt}} \, \vert \phi (0)\rangle$ exists for
$t\ge 0$ only, and $\langle \tilde\phi (0) \vert \,
\overleftarrow{e^{+iHt}}$ exists for $t\le 0$ only.

Considering, for simplicity sake, only a single
Gamow vector, we can denote our simplified partition of the identity
as $W^G = \vert \psi^G\rangle\langle\ts^G\vert$.  Then, aligning the
$t\le 0$ portion point-wise with the $t \ge 0$ portion,
so as to simply use $t\ge 0$ as a common label for both time domains
of definition:
\begin{equation}
W^G (t) = e^{-\Gamma (2m+1) t} \; |\psi^G \rangle \langle \ts^G | = 
	e^{-\Gamma (2m+1) t}\;  W^G (0) \;\; , \textrm{ for} \,\,t\ge 0.
\label{eq:densitymatrix}
\end{equation}
The $m$ quantum number is identified in the second installment of this
series.  From this it follows that
\begin{eqnarray}
\textrm{Tr} \, W(t) &=& \textrm{Tr} \, \left\{W^G\, W(t) \, W^G
	\right\}	\nonumber \\
&=& e^{-2\Gamma (2m+1) t}\; \textrm{Tr} \; W^G
(0) \, \left\vert \langle\ts^G \vert\phi (0)\rangle\right\vert^2 \;\;
, \textrm{ for} \,\, t\ge 0.
\label{eq:tracedensity}
\end{eqnarray}

If $\textrm{Tr}\, W^G (0) =1$~\cite{note22x5}, then
\begin{eqnarray}
&\textrm{Tr} \; W(0)\, \textrm{ln}\, W(0) \; 
	&= \textrm{Tr}\, 
			\left\{ 
		\left\vert \langle\ts^G \vert\phi (0)\rangle 
		\right\vert^2   \; W^G (0)
			\right\} 
		\times \textrm{ln} 
			\left\{ 
		\left\vert \langle \ts^G  \vert \phi (0) \rangle 
		\right\vert^2  \; W^G (0) 
			\right\} 		\nonumber \\
	&=& 	\left\vert \langle\ts^G \vert\phi (0)\rangle
		\right\vert^2 \;
		\textrm{Tr}\, W^G(0) \;  
			\left\{
		\textrm{ln}  \left\vert \langle\ts^G \vert\phi (0)\rangle
			     \right\vert^2  \;+\; \textrm{ln} W^G (0) \;
			\right\}		 \nonumber \\
	&=&  \left\vert \langle\ts^G \vert\phi (0)\rangle\right\vert^2
	\; \textrm{ln} \left\vert \langle\ts^G \vert\phi
	(0)\rangle\right\vert^2  
\label{eq:tracegamow}
\end{eqnarray}

Using our density matrix, the time dependent von Neumann entropy for
our generalized wave-functions is 
\begin{eqnarray}
S \left[ W(t)\right] &=& - \textrm{Tr}\, \; W(t) \; ln\, W(t)
	\nonumber \\
	&=& e^{-2\Gamma (2m+1)t}
		\left\vert \langle\ts^G \vert\phi (0)\rangle\right\vert^2
	\; \textrm{ln} \left\vert \langle\ts^G \vert\phi
	(0)\rangle\right\vert^2 \;\;.
\label{eq:Neumannentropy}
\end{eqnarray}

If $\phi (0)$ is not an equilibrium state, $S (0) \ne 0$ and we deduce
that 
\begin{equation}
S(t) = e^{-2\Gamma (2m+1)t}\; S(0)
\label{eq:timedepent}
\end{equation}
The ${\frac{\Gamma}{2}} (2m+1)$ half width of the Gamow states means 
there can never be a true zero width to resonances represented by these Gamow 
vectors, and so the above expression is always defined in a manner
which makes it seem genuinely relevant to real physical processes, 
and there is
no non-trivial zero to entropy.  Taking $m\ge 0$ as 
characterizing a decay channel, the entropy due to the resonant decay of a 
state with initial quantum numbers $j,m$ via a Gamow vector resonance is thus 
increasing to zero as time $t\longrightarrow +\infty$.  Note that the
generalized von Neumann entropy converges to zero at approximately
twice the rate of the decay process itself, fixed by $\Gamma (2m+1)$.
There are two components to the system, with each subsystem converging
to equilibrium with its correlated sibling, so this definition is
extensive.
 
The instant calculation demonstrates that entropy evolves following
the culmination of the preparation of a  
resonance (conventionally taken as occurring at time $t=0$), and during 
the ensuing time ($t\ge 0$) an initially negative
value of entropy exponentially decreases in
magnitude (exponentially increases to-wards zero) as time passes,
indicating that the evolution of the decaying Gamow state is to-wards
an equilibrium, and, although we only have confidence as to the local
increase in entropy, we expect that this dynamical time evolution will
produce an entropy of
zero in the limit $t\longrightarrow +\infty$.  Equilibrium (a state of zero
entropy) is thus obtained in the limit, since evolution is toward
this limit at every instant along the way.  This $\textrm{Tr}\; W(t)$ is
the time dependent probability of the outcome of observing , i.e., a
form of probability of survival for the system represented by $W(t)$.

Exponential decay of this 
(survival) probability of a resonance presents no problem in
accounting for the
conservation (continuity of existence) of the resonant system.
Entropy increasing to zero means there is an equilibrium forming as
the resonance decays, so no probability vanishes magically in this
description of the decay process.  Considering only the decreasing
(survival) probability of the decaying resonances represented by Gamow
vectors (associated to Breit-Wigner resonances) does not mean that
some probabilistic anomaly occurs over time--there is also an
equilibrium state (not automatically constructed by the paradigm) whose
(excitation) probability experiences complimentary increases over time,
so that total probability is conserved.

There are substantial questions not resolved regarding many of the
traditional tools often used in the study of irreversibility, such as the
applicability and meaning of a generalization of the concept of
{\em exact} transformations~\cite{mackey,mane,sinai}, also called strong markov
transformations~\cite{misra}, and strong (global) versus weak (local) 
forms of the Second
Law.  And, even if these transformations were to satisfy a generalized 
mathematical exactness criteria (strong form of convergence to an 
equilibrium state in $\F_\pm\subset\HH$) 
there must still be nagging concerns in our physical interpretation of this
because the whole present formalism can seemingly be based on local
approximations only (see
Section~\ref{sec:status})~\cite{note2}.
These present results seem to clearly establish a weak form of the Second
Law for resonances associated with representations of locally compact 
semi-simple symmetries in a RHS.  The circumstances under which a strong
form of the Second Law as a possible outcome of the application of the
RHS paradigm to symmetry problems should be regarded as 
an open question~\cite{note23}.

Indeed, the entropy growth is associated with our exponentially
decaying elements of our dual spaces, and these dual spaces do not
necessarily possess a global symplectic form (dynamical structure), as
was discussed in installment two~\cite{II}.  Hence, we should show restraint
in assigning a global entropy implication to consequences of
mathematically local dynamical behavior.  We really need to think
through the meaning of relating entropy growth associated to the
various symplectic sheaves of $\F^\times$, even though a single sheaf
is all that is involved in time evolution associated with a single
dynamical configuration so long as it persists (i.e., up until the
decay event).  A deeper understanding of stability in this formalism
is needed as well.

There are also conceptual difficulties with a strong Second Law arrived at after
computing matrix elements of the effect 
with the forming Gamow vectors $\ts^G$, since the system is 
being prepared during $t\le 0$ and it is hard to think of it as 
isolated in an absolute sense 
then: although the action of the preparation apparatus may be represented as 
a continuous transformation within the rigged Hilbert space formalism, there 
is still the unresolved potential for an ``external'' system 
being able to diminish the entropy of the system being acted upon 
without accounting for the effects on the total entropy
~\cite{note3}.  There is need for further careful study to provide
assurances that the results are consistent with expectations for this case.









\subsection{The weak Second Law}\label{sec:secondlaw}

At the
macro-physical scale, where $\hslash\approx 0$, it is not transparently
clear that these micro-physical accumulations of entropy should have
observational significance.  Of course, $\hslash$ is not truly zero,
and there may be very large numbers of events whose entropy
contribution is scaled by $\hslash \omega$, for large $\omega$, 
and not by $\hslash$
only.  Secondly, there is an analogy from the numeric simulation of
classical chaotic systems which seems apt for quantum chaotic systems
also. 

It is well known that many classical systems can be shown analytically
to be chaotic, but defy numeric simulation due to round off
errors occurring in the computer. The finite number of significant
figures stored in the computer memory effectively ``damps'' chaos
attributable to accumulations below a certain scale.  Sometimes you
obtain a faithful simulation of the underlying chaos, sometimes chaotic
behavior is seen which is not faithful to that known analytically 
to exist, and
sometimes no chaos is visible in the simulation.  Presumably, the
macroscopic observation of quantum chaos associated with 
intrinsic micro-physical irreversibility of quantum resonances will
sometimes reflect the micro-physical chaos and entropy growth, but
there may be times when micro-physically irreversible 
events are not observed faithfully at the macroscopic scale.  Effects
of order $\hslash$ may or may not be macroscopically observable, and
it seems probable that ``not'' is the general rule.  
Perhaps it may be possible to classify
which interactions will be ``rounded off'' and which will have
macroscopic consequences.

Note that most of the more familiar examples of entropy growth (e.g.,
in chemistry) seem to arise in the context of the evolution of
correlations (e.g., formation or breaking of chemical bonds).  The
micro-physical correlations in the present paper seem to offer a
promise of generalization to meso-
and macroscopic scales.  The process of decoherence (e.g., really
meaning decorrelation, such as occurs during
diffusion) also seems to have some conceptual links to present work.  

There also seem to be substantial links
between the present developments and the existing body of work on
quantum chaos.  There is
already a substantial body of work on extending a micro-physical quantum
chaos paradigm to a quantum theory on the mesozoic scale.  See, e.g.,
~\cite{hurt}.  






\section{Fractals Everywhere?}\label{sec:fractals}

It seem to be accepted that to generate fractals requires an affine
hyperbolic transformation, and this usually means a contractive
affine transformation.  The most commonly met hyperbolic affine
transformations would seem to involve complex transformations of a
space with a complex structure, i.e., a type of 
symplectic transformation on a
type of symplectic space.  This is equivalent to saying that some of
our (hyperbolic) symplectic affine
transformations will lead to generation of fractals, since the
generators of the symplectic transformations include generators of
hyperbolic transformations, and the symplectic groups are geodesic.
We used hyperbolic transformations in the 
construction of our Gamow vectors/Breit-Wigner resonances in the second
installment of this series.  We anticipate then (and this should
probably be confirmed by specific calculation or other inquiry) that
exponential decay is the result of a time evolution history (path in
state space) that wanders through state space along a fractal path.

Formally, in our space of states there may be no ``genuine
stability''.  When one works in the standard Hilbert space formalism
with energies unbounded from below, stability disappears
completely--matter is unstable.  There is really nothing to suggest
the situation is different in the RHS formalism, which differs only in
the infinitesimal completion of Cauchy sequences from the von Neumann
Hilbert space formalism.  Indeed, for us the annihilation operator is
$\tau_\Phi$-continuous and $\tau_\Phi$-closed, 
giving an energy spectrum unbounded below.  However, the
presence of fractals in our formalism suggests another (unproven)
possibility: quasi-traps, or fractal trapping.  Pure traps are
excluded classically in area preserving dynamics, and so should be 
missing here as well.  Ergodicity in a classical system
precludes the presence of traps and
island structures in classical phase space, and the exact same
mathematical conditions should hold for us as well, at least near
resonances or the regions of state space dynamically accessible to
them.  We 
deduce there are not many eternal or absolutely stable structures in
our generalized spaces of states, excepting, perhaps in the
Hilbert space as a consequence of Hegerfeldt's theorem~\cite{note4},
implying that our quantum dynamical system will exhibit the dynamical
time evolution either of an open system, of a resonant system or of a
static system.  

We can even anticipate that there might be self-similarity
associated with the exponential decay process, since it involves a
contractive hyperbolic affine mapping--the standard way of generating
self-similar fractals.  Physically, this seems to have the consequence
that the exponentially decaying quantum system evolves in a manner in
which is seems to appear much the same as before (in a self-similar
way), with its state at one instant much like its state at the
next~\cite{note24}.

On the other hand, the equilibrium states towards which our Gamow
vectors and Breit-Wigner resonances evolve ergodically are reached by
a fractal path, and must be viewed as the equivalent of the classical
strange attractors.  (Recall that these ``equilibrium states'' may be
thought of as a ``bump function'' probability amplitude on phase space
rather than the usual classical point localization.)  The associated
fractal traps 
can be thought of as segments of the history of the system in which
the dynamical state wanders around in a fractal maze for 
arbitrarily long and unpredictable periods of time.  (We infer on
similar reasonings that the related non-compact Lie groups generators can
generate fractals in their associated non-compact Lie groups.  Perhaps
this is why the study of non-compact groups has always been so
difficult!)  In a genuinely closed system, we would expect that the
decay of a resonance is an infinitely long process.  The fact that we
do observe decay events occurring is evidence that the system is not
closed and receiving subtle perturbations from
elsewhere~\cite{note0x5}.  The representation of resonances by Gamow
vectors and 
associated Breit-Wigner resonance poles using a RHS (for mathematical
rigor) seems to identify them formally
as closed systems infinitely far from equilibrium,
so to understand them as in fact open systems only weakly interacting
with their environment is a major clarification in our understanding.









\section{Additional Discussion}\label{sec:discussion}

The present paper is a exercise in the use of a topological completion to the 
space of states alternative to the Hilbert space topological completion.  
Since the topology cannot be determined experimentally~\cite{bohm1,bohm2,bmlg}, presumably 
there 
is some license to consider alternative topological completions.  The
fact 
that only infinitesimals separate the two approaches to completion suggests 
that whatever exact predictions are made by one approach
probably can be approximated to high precision in 
the other.  Thus, the rigged Hilbert space can include elements with pure 
exponential decay, while only approximately exponential decay is possible in 
Hilbert space.  Because of this possible unobservability of distinguishing  
measurements, some utilitarian advantage in computation or superior adherence 
to some underlying principles, or the like, should be shown in order to make 
the appeal to a rigged Hilbert space anything other than a formal exercise.  
The rigged Hilbert space previously has been shown to be superior to a Hilbert 
space as a vehicle for expressing the boundary conditions for an irreversible 
process~\cite{qat1,qat2,qat3,bgm}.  The present paper adds further hints of other substantial
physical reasons for using a rigged Hilbert space (at least sometimes).

The analytic continuation
is done by continuous transformations and analogues 
to classical chaos appear in the resulting structures.  Further, it is 
possible to represent preparation and measurement 
as continuous processes within the confines of the theory itself when
one works 
in a rigged Hilbert space, without any appeal to any classical apparatus or 
the like.  Ergodicity, hyperbolicity and the non-invertibility
of ``$U(1)^\C_\pm$'' sub-semi-groups are possible in the rigged Hilbert  
space where a quasi-invariant measure is available, while only the invariant 
(Lebesque) measure is available in a traditional von Neumann
Hilbert space, and the invariant measure prevents exact analogues 
to classical chaotic structures and true non-invertibility there.  There
is a weak form of the Second Law available whenever semi-simple 
symmetries are involved.

Analytic continuation is not a conventional unitary transformation, 
notwithstanding that the operator performing the continuation is 
of a form appropriate to an unitary transformation on $\HH$, such as
we saw in installment two~\cite{II}.  We have
seen the remarkable result that the extension must
be accompanied by a mandatory topology
change from the perspective of the function spaces (Gadella~\cite{gadella}), 
from the perspective of analysis (H{\"o}rmander's Greens functions 
~\cite{hormander}) and from the perspective of the momentum mapping (~\cite{mandr}).  
Following analytic continuation, there emerges an hyperbolic problem with
semi-groups of evolution (Appendix~\ref{sec:continuation}).  The
transitive actions (momentum maps) on a semi-simple Lie algebra $\g$ belong
to  $\g^\times$, and when one views the torus actions on $\F_{\g\pm}$
as transitive, e.g., as active evolutionary processes, then one must
view them as part of $\F^\times_{\g\pm}$ (at a mimimum by using the
canonical inclusion, which is a Poisson mapping).  This profoundly 
limits our choice of mathematical formalism to be used 
to describe non-trivial evolution of quantum systems.
Hegerfeld's Theorem~\cite{note4} places constraints on our ability to
represent evolving systems in the conventional von Neumann
Hilbert space itself.  One of the
lessons presented, then, is that there seem to be limitations in the
mathematical vocabulary available to us for the representation of
dynamical evolution: in Hilbert space only stationary states whose
dynamical evolution is trivial makes physical sense out of what is
mathematically available, while in the rigged Hilbert space formalism 
evolution is by dynamical semi-groups and non-invertible in general.  








\appendix

\section{Analytic Continuation}\label{sec:continuation}


\subsection{Analytic continuation of elliptic operator systems}
\label{sec:ellipticcontinuation} 

The fundamental solutions (Green's functions) of elliptic operators are the 
only fundamental solutions which may be analytically continued (to the 
distributions)~\cite{hormander}.  Hence, as one extends the algebra with its elliptic 
operators from $\R$ to $\C$, one may extend the eigenvalues and 
eigenvector space simultaneously.  The analytic continuation of the two free 
oscillator problem in the second installment of this series~\cite{II}, by the
use of continuous algebraic 
means is thus sound mathematically from this perspective, but in the present 
context it would be easy to place too much emphasis on the existence of 
{\em{some}} elliptic problem to view as the starting point for the
analytic continuation.  The constructions given herein 
show that for subgroups of the complex semi-simple Lie groups 
of symplectic transformations one can convert any realization of a
semi-simple operator into some elliptic operator by a continuous symplectic 
transformation.  The controlling issue is not the existence of some 
``elliptic ancestor'', which seems always to exist, but whether or not the 
elliptic ancestors which surely exist are physically interesting, and perhaps 
whether or not there are ``very well behaved'' ancestral eigenfunctions
which have physical relevance.





\subsection{Is chaos generic to analytic continuation?}\label{sec:chaos}

This subsection illustrates many features generic to analytic continuation, 
done as a {\em continuous} transformation.  The momentum 
representation is chosen 
to simplify the presentation.  Illustrated are the doubling (resonance
bifurcation) of solution sets (vector fields), dimension doubling (due
to analytic continuation), strong hints 
at the emergence of semi-groups 
due to pairs of hyperbolic propagators, and the introduction 
of a limited form of shear 
(discontinuities in the second derivatives along the branch cut and
countable set of resonance poles).  Sensitivity 
to initial conditions is a normal consequence in analytic continuation~\cite{mandf1}.   
The two sheeted Riemann surface which results from analytic
continuation represents a complex torus in the problem.  We now 
demonstrate the generic presence of hyperbolic
measures and what would be called stable and unstable sets in
nonlinear dynamics jargon.  

In this section, we will consider from a different perspective the analytic 
continuation of the Hamiltonian for two free simple oscillators:
\begin{equation}
H = \frac{1}{2} \left( p^2_x + p^2_y \right) \; + \;	
		\frac{1}{2} \left( x^2 + y^2 \right) 
\nonumber
\end{equation}
$H$ is elliptic as a positive definite quadratic form, and the 
eigenvalue problem $H\psi = E\psi$ has {\em{real}} analytic solutions, 
e.g., the familiar oscillator energy eigenstates in direct sum
$\left\{ |n_x\rangle \right\} \oplus \left\{ |n_y\rangle \right\}$.  It will 
be shown that $H$ is transformed when the problem is analytically
continued, and so the elliptic (compact) problem transforms into finding 
the eigenvalues and eigenvectors of a non-compact (hyperbolic) operator.  

Consider the analytic continuation of the momentum representation 
$\psi (p) \longrightarrow \psi (k+i\kappa )$, where $p=k+i\kappa $ is the 
complex momenta.  For $x = i \frac{\partial\,}{\partial p_x}$ and $y=i
\frac{\partial\,}{\partial p_y}$, the eigenvalue problem can be written in the 
form
\begin{equation}
\psi_{xx} + \psi_{yy} \, = \, \left( p^2_x + p^2_y - E \right) \psi
\label{eq:elliptic}
\end{equation}
where $\psi_{xx} \equiv \frac{\partial^2}{\partial p^2_x} \psi$.  Note the 
function in parenthesis on the right hand side is an entire function.
Going to complex momenta 
\begin{alignat}{2}
p_x\in\R \longrightarrow & p_x = k_x +i \kappa_x & \qquad	
		& \overline{p_x} = k_x -i \kappa_x 	\nonumber \\
p_y\in\R \longrightarrow & p_y = k_y +i\kappa_y  &\qquad	
	& \overline{p_y} = k_y -i\kappa_y
\label{eq:complexmomenta}
\end{alignat}
we have the set of differential operators $\frac{\partial\,}{\partial p_x},\; 
\frac{\partial\,}{\partial p_y},\;\frac{\partial\,}{\partial \overline{p_x}},
\;\frac{\partial\,}{\partial \overline{p_y}}$.

Analyticity of $\psi$ in the complex momentum domain means that the 
Cauchy-Riemann equations
\begin{equation}
{\frac{\partial \psi}{\partial \overline{p_x}}} =0 \qquad\qquad
	{\frac{\partial \psi}{\partial \overline{p_y}}} =0
\label{eq:cauchyriemann}
\end{equation}
are fulfilled continuously on the domain on which $\psi(p)$ is holomorphic.  
Therefore,
\begin{equation}
{\frac{\partial^2 \psi}{\partial k^2_x}} \; +\;	
	{\frac{\partial^2 \psi}{\partial \kappa^2_x}}\; = 
		\; 0\qquad\qquad\left( 	{\frac{\partial \;}{\partial p_x}}
	{\frac{\partial \psi}{\partial \overline{p_x}}} \; = \; 0\right) 
\label{eq:holomorphic1} 
\end{equation}
\begin{equation}
{\frac{\partial^2 \psi}{\partial k^2_y}} \; +\;	
	{\frac{\partial^2 \psi}{\partial \kappa^2_y}}\; = 
		\; 0\qquad\qquad\left( 	{\frac{\partial \;}{\partial p_y}}
	{\frac{\partial \psi}{\partial \overline{p_y}}} \; = \; 0\right) 
\label{eq:holomorphic2}
\end{equation}
i.e., $\psi (p)$ is a solution to these Laplace's equations:
\begin{equation}
\psi_{k_x k_x} +\psi_{k_y k_y} \, =\, \left( p^2_x + p^2_y -E \right) \psi
\label{eq:laplace}
\end{equation}
as the real part of equation (\ref{eq:elliptic}).  Note that the previously 
real $\psi$ (e.g., modulo phase) is now continued to the complex plane as well.
Equation (\ref{eq:laplace}) minus equation (\ref{eq:holomorphic2}) says
\begin{equation} 
\psi_{k_x k_x} -\psi_{\kappa_y \kappa_y}  =
	\left( p^2_x + p^2_y - E \right) \psi
\label{eq:hyperbolic1}
\end{equation}
and quation (\ref{eq:laplace}) minus equation (\ref{eq:holomorphic1}) says
\begin{equation}
-\psi_{\kappa_x \kappa_x} + \, \psi_{k_y k_y} =
	\left( p^2_x + p^2_y - E \right) \psi \; .
\label{eq:hyperbolic2}
\end{equation}
Both equations (\ref{eq:hyperbolic1}) and (\ref{eq:hyperbolic2}) 
are hyperbolic differential equations~\cite{garabedian,lewy}.  

Suppose $\psi (p_x,p_y)$ is the 
solution to equation (\ref{eq:elliptic}) on some domain $\Omega 
\subseteq \R^2$.
Call the solution to the hyperbolic equation (\ref{eq:hyperbolic1}) 
$\psi_y (k_x,k_y,\kappa_y ) $and the solution to (\ref{eq:hyperbolic2}) 
$\psi_x (k_x, \kappa_x ,k_y)$.  The initial conditions to the solutions 
$\psi_y (k_x,k_y,\kappa_y )$ on $\Omega \subseteq \R^2$ are:
\begin{align}
\psi_y (k_x,k_y,0)& = \psi (k_x,k_y) 		\nonumber 	\\
{\frac{\partial \;}{\partial \kappa_y}}\psi_y (k_x,k_y,0)&
	= i {\frac{\partial\;}{\partial k_y}}\psi (k_x,k_y )
\label{eq:ic1}
\end{align}

For fixed $k_y$, (\ref{eq:hyperbolic1}) and (\ref{eq:ic1}) specifies a two 
dimensional Cauchy problem, determining $\psi (k_x,k_y,\kappa_y )$ a 
sufficiently small (``characteristic'') triangle parallel to the 
$(\kappa_x ,\kappa_y )$ plane, the base of which lies in $\Omega$ and whose 
two remaining sides are characteristics.  At a minimum, there exists a method 
of successive approximations to arbitrary precision 
(``Picard's procedure'' [11]) which gives an 
analytic function $\psi(k_x,k_y,\kappa_y )$ throughout a neighborhood U in 
$\R^3 = \left\{ ( k_x,k_y,\kappa_y ) \right\}$ ).  

There exists a similar 
Cauchy problem for solving the hyperbolic problem covering 
$\psi (k_x,\kappa_x ,k_y )$ over some neighborhood V in $\R^3 = 
\left\{ (k_x, \kappa_x ,k_y )\right\}$:
\begin{align}
\psi_y (k_x,0,k_y) &= \psi (k_x,k_y) 		\nonumber 	\\
{\frac{\partial \;}{\partial \kappa_x}}\psi_x (k_x,0,k_y)&= 
	i {\frac{\partial\;}{\partial k_x}}\psi (k_x,k_y )
\label{eq:ic2}
\end{align}
The key fact here is that when the real solutions $\psi$ of the
elliptic problem are analytically continued, a pair of hyperbolic problems 
emerge, with complex solutions.  There are two distinct sets of solutions 
defined over separate domains which are incompatible near the domain boundary 
(branch cut on the real axes) in their second and higher derivatives.  The 
Schwartz reflection principle gives the relationship between the solution 
sets which have been bifurcated: $\psi (p) = [\psi (p^\ast )]^\ast$.  In 
effect, there is a separation of the solutions along the lines of the real 
axes: $\psi_x (p)$ and $\psi_y (p)$ each regard the complex part of the other 
as constant, because of the relations
\begin{equation}
{\frac{\partial^2 \psi_x }{\partial \kappa^2_y}} = 0\qquad\qquad
	{\frac{\partial^2 \psi_x }{\partial \kappa^2_x}} \ne 0 \; ,	
\label{eq:laplace1} 
\end{equation}
and
\begin{equation}
{\frac{\partial^2 \psi_y }{\partial \kappa^2_x}} = 0\qquad\qquad
	{\frac{\partial^2 \psi_y }{\partial \kappa^2_y}} \ne 0 \;\; .	
\label{eq:laplace2}
\end{equation}
There are effectively two one-dimensional problems to join smoothly.  The 
restriction to smooth solutions must be regarded as physically motivated, and 
it should be pointed out that holomorphic distributions are 
ordinary holomorphic functions~\cite{rudin}.  

There are well defined Green's functions which relate the 
real solutions to the elliptic problem over the real axes to the values of 
the complex eigenfunctions over each holomorphic domain of the
hyperbolic problem: for $\psi_x$ there is an analytically continued 
or generalized Green's function which extends $\psi_x (k_x,0,k_y )$ to 
$\psi_x (k_x,\kappa_x ,k_y )$ for $\kappa_x > 0$, and there is a
similar Green's function extending $\psi_y (k_x, 0,k_y )$ to 
$\psi_x (k_x,\kappa_x ,k_y)$ for 
$\kappa_x <0$.  Another pair of generalized Green's functions 
extends 
$\psi_y (k_x,k_y,0)$ to $\psi_y (k_x,k_y,\kappa_y )$ for the domains 
$\kappa_y >0$ and $\kappa_y <0$ respectively.  

Working with the classical fields on a classical
phase space having no initial complex structure 
(e.g., with $p_x \thicksim \frac{\partial\;}{\partial x}$ or 
$x\thicksim\frac{\partial\;}{\partial p_x}$), the analytic continuation of the 
classical flows generated by the elliptic free oscillator Hamiltonian would 
similarly result in an hyperbolic problem~\cite{garabedian,lewy}, and ultimately
exhibit many features common to the preceding treatment over the complexified 
quantum phase space.  Formally, it seems one may end up with alternative 
chaotic structures on the same formal space.  







\section{Status as a Physical Theory}\label{sec:status}

The algebra of physical observables may be represented by operators on 
$\F_\pm $ and $\F^\times_\pm$ which are mathematical horrors.  However, for 
the semi-simple complex covering group of the present construction, 
one has a series of compact finite dimensional representation sub-spaces
for the local torus action of each operator acting on a given state
vector, meaning that any arbitrary {\em{linear}} operator acting on
any given state vector may be locally approximated on each 
sub-space by a semi-simple operator.  Individual {\em linear} 
operators will locally
have a ``one dimensional action'' which may be approximated to arbitrary
precision by a semi-simple operator over some neighborhood.

An example of a semi-simple approximation is provided by the higher order
Gamow vectors.  (See~\cite{bmlg,note5} and references therein for a description.)  For 
these vectors associated to higher order poles of the S-matrix, the 
Hamiltonian takes irreducible Jordan block form.
This Hamiltonian is not a semi-simple operator, since it is not possible
to locally diagonalize it.  The physical and mathematical difficulty in 
distinguishing second order poles from a redundant 
pair of first order poles is well 
known~\cite{goldberger}, reflecting the ability of semi-simple operators (Hamiltonians 
associated to the first order poles and their eigenvectors) to locally 
approximate a non-semi-simple operator (a Jordan block Hamiltonian and its 
eigenvectors).   This approximation to arbitrary precision works both
directions: the mixing behavior so apparent in a Jordan block
system of higher order poles should be locally associated 
to arbitrary precision
with the ergodicity and entropy growth of an appropriate first order system 
based on the present construction.  Conversely redundant poles arising
in the present methodology should be identifiable to arbitrary precision
with some irreducible Jordan block system provided one chooses a sufficiently
small neighborhood.

The lesson of this section is that the present paper may be founded
not merely on local methods (e.g., the algebra of infinitesimal
generators), but on local approximation.



\vskip1cm \hrule
The evolution of quantum resonances can (at
least sometimes) be associated with ``windings'' on complex tori (as
is discussed in the second installment of this series), in a
context suggesting that there is a fairly limited set of mathematical
tools available for representing the {\em regulation} of the evolution
of quantum micro-systems.  For instance, it would seem that the
natural vehicle for the ``control'' of quantum chaos (meaning resonant
decay) would be to
substitute a realization of one symmetry for another, e.g.,
``breaking'' one symmetry and ``installing'' an alternative
realization of a symmetry.  The algebra of this process is that of
symplectic transformations, and has associations with interferometry,
suggesting there are likely to be concrete physical implementations
associated with these mathematical idealizations.  Shrinking the
present femtosecond laser pulse technology to even shorter time
domains, and associating the resulting pulsed laser with the output of
an accelerator (for instance) may provide an experimental vehicle
for uniquely directing the electroweak decay process.





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