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\begin{document}

\title{{Generalized Wavefunctions for \\
Correlated Quantum Oscillators II:\\
Geometry of the Space of States.}}

\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 second of a series of articles, a pair of quantized free 
oscillators is transformed into a resonant system of coupled
oscillators by analytic continuation which is 
performed algebraically by the group of complex symplectic
transformations, creating dynamical representations of numerous 
semi-groups from the Hamiltonian free system.  The free oscillators 
are the quantum analogue of action angle variable solutions for the 
coupled oscillators and quantum resonances, including Breit-Wigner 
resonances.  Among the exponentially decaying Breit-Wigner resonances 
are hamiltonian systems in which energy transfers from one oscillator 
to the other.  There are significant mathematical constraints in order 
that complex spectra be accommodate in a well defined formalism, which
may be met by using the commutative real algebra $\C (1,i)$ as the ring of
scalars in place of the field of complex numbers. By 
including distributional solutions to the Schr{\"o}dinger equation,
placing us in a rigged Hilbert space, and by using the 
Hamiltonian as a generator of canonical transformation of the space 
of states, the Schr{\"o}dinger equation is the equation for 
parallel transport of generalized energy eigenvectors, explicitly 
establishing the Hamiltonian as the generator of dynamical
time translations in this formalism.  The space of states is
explicitly constructed using a realization of the  Lie algebra of the
group of dynamical transformations as connections. 



\end{abstract}


\pacs{12.90.+b,11.30.Na,02.20.Sv,03.65.Db}        %use showpacs class option

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\maketitle


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


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


There are many lessons on many levels which emerge
when we adopt the attitude that, if classical dynamics is the study of
symplectic structures and symplectic transformations
on phase space, then quantum dynamics should be
based on the study of symplectic structures and symplectic
transformations associated with the
quantum mechanical space(s) of states.  We adopt such an attitude
herein, and we will discover it takes very specific mathematical
formalisms to insure our efforts are well defined: be forewarned that  
unitary transformations are a subgroup of the symplectic
transformations, and hermitean operators are a special (i.e.,
symmetric) case of the essentially self-adjoint operators, and we will
adopt the more general types of transformations to our uses, 
thereby violating an orthodoxy entirely appropriate in another
context.  We will not be in von Neumann's Hilbert space anymore!

This approach also presents many threads to many notions currently
under investigation in other contexts.  Thus, if decoherence is the 
conversion of quantum probability
into a classical probability, we showed in the first installment that
our quantum construction can be thought of as
the analytic continuation (complex symplectic transformations) of a
type of classical probability theory.   
In this view, there should naturally be complex symplectic 
transformations which will have the effect of realification of our
(spinorial) wave functions.  We thus present the view that quantum
theory is the result of the addition of coherence (e.g., correlation)
to a
particular form of classical probability theory, and the complimentary
procedure of ``decoherence'' is also within the mathematical scope of
the construction.  In the third installment of this series, we will study
many quantum 
analogues to classical dynamical systems, and their relation to
classical statistical mechanics and thermodynamics. 

From the combination of the seemingly simple problem of canonical
transformations of two harmonic
oscillators, and empowered by the mathematical tools of generalized
functions (distributions) 
in a RHS, an astonishingly rich structure emerges
exhibiting many of the features one associates with classical hard chaos 
and irreversible classical thermodynamics (in the $N=2$ limit).  The
methods used here have transparent generalizations to larger numbers
of oscillators.  

The constructions of this paper make use of four major elements which
distinguish it from the conventional Hilbert space treatment of
quantum systems:
\begin{enumerate}
\item{It includes the weak, or distributional, solutions to the
  Schr{\"o}dinger equation, mentioned above.  These were not available
  to von Neumann.}
\item{For our ring of scalars, rather than choosing the unit imaginary,
  $i$, as an element in the field of complex numbers, we will use $i$
  as a $90$ degree rotation in the complex plane, i.e., necessarily as
  an element of a real algebra.  The result is a complex hyperbolic
  (Lobachevsky) structure in the tangent space at every point in our
  complex spaces of generalized states (e.g., tangent bundle).We are
  building spaces (modules) whose ring of scalars are real algebras
  rather than spaces (modules) over fields.  This is the result of  a
  uniqueness consideration, based on the old 
  saw that complex conjugation is not a uniquely defined involution of
  the field of complex numbers regarded as an algebra--there is more
  than one possible real line structure in the field $\C$.  This also
  distinguishes the present work from other recent work using the RHS
  formalism.}
\item{We will use the {\em weak} symplectic structure available on
  $\F\times\F^\times$ (e.g., associated with the scalar product on
  $\F$) to reflect (represent) the canonical weak symplectic structure
  on $\g \times\g^\times$ for Lie algebra $\g$ (which is represented
  on $\F$).  For the complex Hilbert space of von Neumann $\HH$, there
  is a {\em strong} symplectic structure on $\HH\times\HH^\times$,
  e.g., associated the scalar product on $\HH$~\cite{note1}. You
  cannot have a non-trivial dynamics in an Euclidean geometry.}
\item{In order to proceed uniquely, we are required to work with the
  real form (symplectic form) of various complex Lie algebras (of
  complex simple Lie groups), and representations of those complex Lie
  algebras.   This will operate as a major determinant of structure
  for our spaces of states (which are, due to this mathematical
  necessity, spin spaces).}
\end{enumerate}

The basic program underlying the present work is conceptually quite austere,
although there are a myriad of details to the actual implementation.
We regard $i$ as, e.g.,  an element of the commutative real algebra $\C
(1,i) \equiv \R \oplus i\circ\R$~\cite{note2}.
We will regard the Hamiltonian of our oscillator system as a
realization of one of the generators of infinitesimal canonical 
transformations of the associated phase space. This makes 
it a generator in the Lie algebra of the appropriate symplectic group, 
e.g., for two pairs of canonical coordinates, $H$ belongs to 
$\mathfrak{sp}(4,\R )^\C$.  Since $Sp(4,\C )$ is simple, it's
representation provides a connected covering structure in which 
to exponentiate 
our continuous infinitesimal symplectic transvections.  (See
Section~\ref{sec:algresolution}.)  Our Hamiltonian can also be defined
to have an infinitesimal action on the spaces of states in such a way
that its action on the space(s) of states is symplectic as well.  This
will ultimately result in it (the Hamiltonian) being associated with
generating flows of hamiltonian (integrable) vector fields. In fact,  the
Schr{\"o}dinger equation would emerge as the analogue of Hamilton's
equation if we should chose a variational (i.e., Hamilton-Jacobi) 
treatment mathematically alternative to the approach adopted here.  

We are creating a forum in which
we may speak of a quantum dynamics evolving through dynamical
(=symplectic) transformations in a manner which follows much of the
spirit of the classical treatment of dynamical systems.  
(The ``quantum chaos'' associated with the flow of topologically
transitive hyperbolic affine transformations on the spaces of quantum
states, which also have a symplectic action on the spaces of states,
will be discussed in the third installment of this series~\cite{III}.)
These mathematical structures are not compatible with the conventional  
formulation of quantum mechanics using von Neumann's Hilbert space,
and so we are pursuing a description of a ``quantum dynamical system''
which is unique to our variant of the RHS formulation of quantum
mechanics.  In the course of these investigation of formulation of a
quantum dynamics, many insights emerge illuminating mathematical
structures that naturally arise when one formulates an {\em
hamiltonian} quantum field theory based on
harmonic oscillators dynamically evolving in our version of
the RHS formalism.  There is a lot of mathematical physics we will
show some consistent
connection to which is above and beyond the machinery we
actually invoke for our immediate purposes. 

Every complex semi-simple Lie algebra has some
complex Lie group, and on a semisimple Lie group with a complex Lie algebra, 
$exp$ is holomorphic~\cite{knapp}.  In consequence, the complex
simple Lie groups relevant to our problem are connected and locally
path connected.  Since $exp$ is holomorphic for
$\mathfrak{sp}(4,\R )^\C$, the exponential map of $H$ can locally be
identified with the transformation of 
parallel transport along geodesics in $Sp(4,\R )^\C_\pm$.
Formally, we work with the real (or symplectic) 
form of the Lie algebra, e.g., $\mathfrak{sp}(4,\R )^\C$ rather 
than $\mathfrak{sp}(4,\C )$ itself, in order
that adjoints be well defined for both the algebra and the group
simultaneously.  (See Section~\ref{sec:scattering}.)  We will
construct a representation of this Lie group/Lie algebra structure,
and, assuming that the potentials in the Hamiltonian are analytic,
we may make use of the creation and destruction operator formalism.
(See Section~\ref{sec:rhsresolution}.)

We construct our representation
spaces as modules over $\C (1,i)$, and corresponding to the
geodesics in $Sp(4,\R )^\C_\pm$ generated by $exp( \mathfrak{sp} (4,\R
)^\C )$ there will be geodesics of evolution
generated in our representation space by the representation of the Lie
group and associated Lie algebra: among the equations of parallel
transport along these geodesics, it is possible to find an equation equivalent 
to the Schr{\"o}dinger equation. 
We will (necessarily) allow weak solutions to these equations of
geodesic evolution on our representation space, so that our chosen
representation spaces include spaces of generalized functions, and thereby
our work naturally falls into the RHS formalism.  As part of our
construction, we endeavor to respect all canonical constructions,
including canonical inclusions (Section~\ref{sec:scattering}) and
canonical symplectic forms (Section~\ref{sec:poisson}
and Section~\ref{sec:scattering}).  The investigation of the
mathematical and physical structures resulting from this basic program
is the subject of the remainder of this series of papers.  We shall begin,
however,  under the guise of attacking the concrete problem of one
harmonic oscillator dissipating (transferring) 
energy to another oscillator, and our
objective is to describe and constructively illustrate the structures
and methods which figure in providing mathematically respectable
solutions to that problem.

On the space $\F$ of the Gel'fand triplet $\rhs$, essentially self
adjoint operators need not be also be symmetric, i.e., may be 
non-hermitean.  When working in these RHS's the
usual notions of hermiticity or anti-hermiticity are not controlling,
and we will
need a radically different notion of what is the proper form of
adjoint involution in order to provide dynamical (including unitary)
representations~\cite{note3} of the
connected Lie groups which our complex Lie algebras--and associated
representations--integrate into using the exponential map.  

Item 1 above, then gives us
a more general set of solutions to work with.  Item 2, we will see,
gives us a geometric context in which it is mathematically proper to
speak of the Hamiltonian as the generator of time translations (Item
4).  This interpretation of the Hamiltonian
does not automatically follow just because the
Hamiltonian has this meaning when restricted to another space (and
another topology).  This is also significant to establishing the
existence of Lie algebra valued connections, leading to a gauge
theoretic interpretation of some associated structures which we will
explore in the fourth installments of this series~\cite{IV}.
The third item is significant as indicating a possible source of
obstructions which must be avoided by proceeding carefully, and
our treatment demonstrates the inability of the standard Hilbert 
space formalism to address issues in the same
mathematical generality we have available in our RHS construction.  










\subsection{Symplectic transformations of oscillators}\label{sec:sptrans}

In the RHS formalism, one works in a subspace of the Schwartz space (of
functions of rapid decrease) for the function space realization of the
abstract space $\F$ of the RHS $\rhs$.  The Lie algebra and Lie group
of symplectic transformations used in this present work have been 
associated with problems in quantum optics.  One therefore infers 
the methodology of this paper is likely to find concrete
application with electromagnetic fields.  The group of 
symplectic transformations is in fact
the group of squeezing transformations
of quantum optics.  Hence, the present work can be considered the study of
the squeezing transformations of {\em generalized} Gaussian wave 
packets~\cite{note7}!  Physically, we are dealing with families 
of coherences of minimum
uncertainty states.  (If a field theoretic interpretation is adopted
for the oscillators, a natural candidate for the representation of a
``particle'' is a stable member of these families of minimum uncertainty
states.)   Two free oscillators are ``squeezed'' into
coherence by the symplectic transformations, and further ``squeezing''
results in resonant decay of the coupled oscillator
system.  Our formalism has strong relations to the formal treatment of
squeezed states in quantum optics.


  





\subsection{Order of proceeding}\label{sec:proceeding}
	
The appendix contain some important calculations.
Appendix~\ref{sec:DHO} recapitulates the Feshbach-Tikochinsky $SU(1,1)$
dissipative oscillator system calculation~\cite{ft} in slightly changed notation
(and with changed physical content, in order to put things in
a form which is not otherwise remarkable).  

We begin with a bit of naive algebra which requires substantial
justification, which we also begin in this section.  This is
a variant of the Feshbach-Tikochinsky calculation in which we take the
transformations to be part of a realization of the canonical
(symplectic) transformations for the two oscillator phase space and
{\em also define the action of the generators of the appropriate Lie
algebra to have a symplectic action on our spaces of
states} in this realization.  The simple
computation in Section~\ref{sec:scattering} has implications at
many levels.  It is shown that the dissipative oscillator system
can be obtained from a realization of the group of symplectic transformations
applied to the system of two free quantum oscillators.  Further
(complex) symplectic transformations yield vectors which decay
exponentially (without regeneration).  It is also
demonstrated that similar constructions can generate a representation
of $SU(2)$ in terms of creation and destruction operators, and further
complex symplectic extensions will yield a complex spectrum for it 
as well, indicating a
quite general process is involved.  

The present results ultimately must be compared to the analytic 
continuation used by Bohm to obtain his Gamow
vectors~\cite{bohm1,bohm2}.  The two methods  
both describe complex symplectic transformations, so the necessary and
sufficient ``very well  
behaved'' starting point for obtaining the Breit-Wigner resonance
poles to associate to Gamow vectors~\cite{gadella} for the dissipative
coherence of two oscillators problem is not the F-T 
system of coupled oscillators, but should probably be thought of as 
the system of two free oscillators.  

Section~\ref{sec:poisson} provides a summary
descriptions of aspects of well known structures which will play
important roles for us.  These establish a basic general mathematical
context in which we operate.
In Section~\ref{sec:poisson}, the subject is the multitude of
symplectic and Poisson structures associated with representation of
both real and complex
semi-simple Lie algebras in a RHS format.  The structure of the 
space $\F^\times$ of the RHS $\rhs$ is not completely known, but 
in the case
of a Lie algebra representation, the structure of
$\F^\times_{\g\pm}$ follows the structure of $\g^\times$ in many
important regards, and there is some elaboration on this.  

The fact that the momentum map for a
semi-simple Lie algebra $\g$ is an element of $\g^\times$ provides us
with the first indication of a mandatory topology change to a
weak-$^\times$ topology: viewing a formal ``adjoint'' complex symplectic 
transformation on $\g= \mathfrak{sp}(4,\R )$, taking ``$Ad_G \g$'', as
transitive and also having a symplectic action on a space requires us
to view it as giving rise to a {\em co-adjoint} representation of
$\g^\C$ in $(\g^\C)^\times$.  This co-adjoint orbit structure within
the individual symplectic sheaves making up the Poisson manifold
$(\g^\C)^\times$ is 
faithfully reflected by the associated orbits of transformations on the
representation spaces $\F_{\g\pm}$ and $\F^\times_{\g^\C\pm}$.  By
appropriate identifications, the symplectic (and therefore Poisson)
mapping of 
inclusion (of the representation of $\g$) into these co-adjoint orbit 
structures (lying inside the representation of $(\g^\C )^\times$) 
can be made to provide essentially self adjoint extensions of
generators {\em and} a representation of the associated
transformation (semi-)groups whereby the exponential mapping for $\g^\C$ 
can be identified properly with the exponential mapping of 
$(\g^\C )^\times$. 

The representations of the associated simple
complex semi-groups are defined below in such a way that the
transformations have a symplectic action on the representation spaces
themselves.  (See Section~\ref{sec:scattering}).


The algebraic structure are briefly summarized next in
Section~\ref{sec:algresolution}.

Following this, in Section~\ref{sec:rhsresolution}, function space issues 
are dealt with.  The key insight of this section is that a realization
of the Lie algebra consists of tangents vectors to the identity (if
there is a scalar product around), and, as such, is associated with a
derivation, ultimately leading to Lie algebra valued
connections uniquely defining geodesics.  
There is a construction which consists of
applying this creation-destruction operator realization of the Lie
algebra acting as a connection associated to differential operators 
in a manner exactly paralleling the standard construction of the 
Schwarz space itself.  Basically, in the case of parallel transport, we can
think of the connection as equivalent to a derivation.  The role of
the Lie algebra generating geodesics on its complex simple
Lie group can be extended
to the representation of a Lie algebra defining connections on the
representation space~\cite{note14}.
This shows that not only does some representation in a function 
space exist, but in fact there exists a function space 
representation which is appropriate for the
application of a mathematically rigorous analytic continuation, since it 
satisfies the necessary and sufficient criteria of the RHS
paradigm by construction.  Standard theorems provide that 
exponentiation of a continuous linear operator may lead 
to a semi-group on a function space~\cite{yosida}.  

In the Langlands type of inductive
construction, there seems to be a requirement for an elliptic problem
somewhere.  In the present construction, something similar is provided by the
Hamiltonian of the free harmonic oscillators, so we can be thought of
as having something in common.  We, however, are working on
algebras and spaces which have a complex hyperbolic structure, and we 
will ultimately wind up with structures
which include the orbits of hyperbolic operators on hyperbolic
spaces.  

In a subspace of Schwartz space, we will see how the 
representation of the realization of the complex 
covering group is instantly available from a representation of 
a creation destruction operator realization of a real semi-simple Lie
algebra.  (Section~\ref{sec:altconst}.)The construction also
demonstrates that the functionals (belonging 
to $\F^\times_{\g^\C \pm}$) on the representation space $\F_{\g^\C
\pm}$ are a $\sigma$-algebra of Borel sets, which will figure 
significantly in the measure 
discussions in Section~\ref{sec:observables} and in the fourth
installment of this series~\cite{IV}.

The overall procedures closely follow the prescription in~\cite{reed}.  
There is a topology change mandated by {\em{continuous}} 
transformations of analytic continuation, 
and this point is very subtle and easy to overlook, but after 
the continuous analytic continuation one's solutions have been
extended to the distributions and only 
a weak-dual topology is appropriate~\cite{gadella}.  (Once again,
we're not in Hilbert space anymore!)  









\section{``Scattering'' of Simple Oscillators}\label{sec:scattering}

In this section, we study the algebraic structure of the ``dissipative
oscillator'' system, but this section may also be interpreted as
a brief exercise undertaken to indicate the possibility of 
an algebraic theory of scattering, which is rigorous.  Such systems for the 
description of scattering have been considered before, e.g.,~\cite{gursey} uses
$SU(1,1)$ as the continuation of $SU(2)$, and uses $Sp(4,\R )$, to construct a 
partial theory of this sort.
Herein, a pair of free oscillators is subjected 
to canonical transformations to become an 
interacting (correlated or coupled) system in which one oscillator 
is ready to transfer 
energy to the other.  Further canonical transformations lead to
a dissipative oscillator system in which the flow of energy from one
oscillator (with higher energy) in the coupled system to the other
oscillator (with lower energy) is described by Gamow vectors which are 
energy eigenvectors with complex energy (and which therefore exhibit 
exponential growth or decay in their time evolution).  This is the result of 
the identification of the interaction Hamiltonian of the coupled system with 
a non-compact generator of the algebra $\mathfrak{su}(1,1)$, which is extended 
(analytically continued) from that real algebra to a complex algebra with the 
same generators and commutation relations defined (i.e., extended to a complex 
covering algebra).  This use of symplectic transformations is a
generalization of the F-T results recapitulated in
Appendix~\ref{sec:DHO}.  Any required additional justifications of the 
naive algebraic manipulations of 
the present section will be given in later sections.

The commutation relations useful to us 
for a unitary representation of $Sp(4,\RR )$ 
on Hilbert space, $\HH$, are~\cite{dirac} ($i,\, j,\, k=1,\, 2,\, 3$, sums implied):
\begin{eqnarray}
\left[J_i ,J_j\right] &=& \epsilon_{ijk} J_k   \label{eq:su2}   \\
\left[J_i,J_0\right] &=& 0                      \label{eq:zero} \\
\left[K_i,K_j\right] &=& -\epsilon_{ijk}J_k \\
\left[K_i,J_j\right] &=& \epsilon_{ijk}K_k \\
\left[Q_i,Q_j\right] &=& -\epsilon_{ijk}J_k \\
\left[Q_i,J_j\right] &=& \epsilon_{ijk}Q_k \\
\left[K_i,Q_j\right] &=& \delta_{ij}\, J_0 \\
\left[K_i,J_0\right] &=& Q_i \\
\left[Q_i,J_0\right] &=& -K_i	       \label{eq:lastcom}
\end{eqnarray}

An appropriate realization of this algebra in terms of two mode creation and 
annihilation operators is~\cite{dirac,kimandnoz}:
\begin{eqnarray}
iJ_1&=&{\frac{1}{2}}\;{\Big (}A^\dagger B + B^\dagger A{\Big )}   
							\label{eq:gen1} \\
iJ_2&=&-{\frac{i} {2}}\;{\Big (}A^\dagger B - B^\dagger A{\Big )}   
							\label{eq:gen2} \\
iJ_3&=&{\frac{1}{2}}\;{\Big (}A^\dagger A - B^\dagger B{\Big )}	  
							\label{eq:gen3} \\
iJ_0&=&{\frac{1}{2}}\;{\Big (}A^\dagger A + B B^\dagger {\Big )}   
							\label{eq:jzero} \\
iK_1&=&-{\frac{1} {4}}\;{\Big (}A^\dagger A^\dagger +  AA - B^\dagger       
	B^\dagger - BB   {\Big )}	\\
iK_2&=&{\frac{i}{4}}\;{\Big (}A^\dagger A^\dagger- AA +B^\dagger       
	B^\dagger - BB {\Big )}		\\
iK_3&=&{\frac{1}{2}}\;{\Big (}A^\dagger B^\dagger  + AB{\Big )}	\\
iQ_1&=&{\frac{i}{4}}\;{\Big (}A^\dagger A^\dagger -AA - B^\dagger       
	B^\dagger + BB   {\Big )} 	\\
iQ_2&=&-{\frac{1}{4}}\;{\Big (}A^\dagger A^\dagger +AA + B^\dagger       
	B^\dagger + BB   {\Big )}	\\
iQ_3&=&{\frac{i}{2}}\;{\Big (}A^\dagger B^\dagger  -AB{\Big )}  
							\label{eq:gen10}  
\end{eqnarray}

Identifying $X=iJ_1$, $Y=iJ_2$, and $Z=J_3$, we obtain the realization
of the $SU(1,1)$ Lie algebra generators used in Appendix~\ref{sec:DHO}.

The Baker-Campbell-Hausdorf relation  
\begin{equation}
e^B\, Ae^{-B} \, = \, \sum^\infty_{n=0} \;\frac{1}{n!} 
\left[ B,\,   \left[ B,\ldots \left[ B, A\right] \ldots \right] \right]
\end{equation}
containing $n$ factors of $B$ in each term, can be applied to semi-simple Lie 
groups and algebras, e.g., to $B = \mu X\in \mathfrak{su}(1,1) \subset
\mathfrak{sp} (4,\R )$, 
$A=Y \in \mathfrak{su}(1,1)\subset \mathfrak{sp}(4,\R )$, $\mu\in\R$, 
to yield:
\begin{equation}
e^{i\mu X}\left( iY \right) e^{-i\mu X}=\left( iY \right) \cos {\mu} - [X,Y] 
\sin {\mu}.
\end{equation}

For $\mathfrak{su}(1,1)$ and for pure imaginary $\mu$, the cosine becomes 
hyperbolic cosine (cosh) and sine becomes hyperbolic sine (sinh).  For the 
semi-simple group $SU(1,1)$ and its semi-simple algebra 
$\mathfrak{su}(1,1)$, it
 follows that:
\begin{equation}
e^{i\mu X}(iY) e^{-i\mu X}=(iY)\cos \mu -Z \sin \mu  \label{eq:adjoint}
\end{equation}
so that
\begin{equation}
e^{i({\pi}/{2}) X}(iY)e^{-i({\pi}/{2}) X}=-Z 
\end{equation}
or
\begin{equation}
Y=i\;e^{-i({\pi}/{2}) X}\, Z\, e^{i({\pi}/{2}) X}
\qquad Z=-i\;e^{i({\pi}/{2}) X}\,Y\,e^{-i({\pi}/{2}) X}   
\label{eq:F-T}
\end{equation}
(Note that we are not in Hilbert space, so that ``hermiticity'' is not
a relevant concept for us--it is not the case that 
conjugation of an hermitean operator by a seemingly
``unitary'' transformation has resulted in a non-hermitean operator.)  

Note that for $\mathfrak{su}(2)$, where 
$\left[ J_k,J_l \right] = \varepsilon_{klm} J_m$, for real $\mu$
\begin{equation}
e^{i\mu J_k} \left( i J_l \right) e^{-i\mu J_k} = \left( i J_l \right) 
\cosh \mu -  \left[ J_k,J_l\right] \sinh \mu \quad .
\label{eq:su2adjoint}
\end{equation}
For pure imaginary $\mu, $ the corresponding expression  is 
$\left( i J_l \right) \cos \mu +i \left[J_k,J_l\right] \sin\mu$.  This is 
consistent with the $\mathfrak{su} (1,1)$ results, because there is a (
``dangerous to use'') mapping $\mathfrak{su}(2) \longrightarrow 
\mathfrak{su}(1,1)$ 
given by $J_1 \longmapsto X= iJ_1$, $J_2 \longmapsto Y=iJ_2$, $J_3\longmapsto 
Z$.  The Baker-Campbell-Hausdorf relation applies to both compact and 
non-compact groups, and to pure real and pure imaginary coefficients.  

This transformation procedure (which is based on a form of the 
Baker-Campbell-Hausdorf relations) results in ``complex spectra for $SU(2)$'' 
just as a similar transformation resulted in ``complex spectra for 
$SU(1,1)$''. 
This illustrates that a general process is going on applicable to all locally 
compact semisimple Lie subgroups of a complex semisimple Lie covering group, 
and, in particular, applicable to both non-compact and compact 
subgroups alike.

Consider a system of two independent simple  
harmonic oscillators.  The Hamiltonian for this system is:
\begin{equation}
H = \frac{1}{2}\left( p^2_x + x^2 +p^2_y +y^2 \right).
\label{eq:Haf}
\end{equation}
This is the same operator as $iJ_0$, equation (\ref{eq:jzero}): $H=iJ_0$. 
If we subject this system to as ``preparation procedure'' the transformation:
\bea
&\, \alpha e^{i(\pi /2)J_1}&e^{i(\pi /2)K_2}\;(iJ_0)\; e^{-i(\pi /2)K_2}       
            \,\alpha e^{-i(\pi /2)J_1} \qquad 
	\nonumber	\\        
&\;&\qquad\qquad \;+\; \beta e^{i(\pi /2)Q_1}    \,e^{i(\pi /2)K_2}            
 \; (iJ_0)\; e^{-i(\pi /2)K_2}\, \beta e^{-i(\pi /2)Q1} = \quad                
         \nonumber   	\\   
\qquad &=&- \alpha e^{i(\pi /2)J_1} \, [K_2,J_0] \,                
		\alpha e^{-i(\pi /2)J_1} -\,         \beta e^{i(\pi /2)Q_1}  
		\, [K_2,J_0] \, \beta e^{-i(\pi /2)Q_1}               
	\nonumber   \\   
\qquad &=& +i\alpha e^{i(\pi /2)J_1} \; \left( iQ_2 \right) 		
		\;\alpha e^{-i(\pi /2)J_1}             
	\; +\; i \beta e^{i(\pi /2)Q1}\; \left( iQ_2 \right)
	\; \beta e^{-i(\pi /2)Q1}                 
		\nonumber  	\\    
\qquad&=& i\alpha^2 \; [J_1, Q_2]\;\; + \;\;              
	i\beta^2 \; [Q_1,Q_2] \nonumber  \\    
\qquad&=& i \alpha^2 \; (-\epsilon_{123}Q_3 ) \; +\; 
	i \beta^2 \; (- \epsilon_{123} J_3)                  
		\nonumber 	\\   
\qquad &=& -\alpha^2 \; \left( iQ_3 \right)  \; - \; \beta^2 \; 					\left( iJ_3 \right)         
\label{eq:adext} 
\eea
Choosing
\begin{equation}
\alpha^2 = -\frac{\G}{2} \qquad\qquad \beta^2 = -\Omega
\end{equation}
we have recovered the Hamiltonian of the $SU(1,1)$ dissipative oscillator 
system, equations (\ref{eq:ham1}), (\ref{eq:ham2}), (\ref{eq:ham3})  (See 
Appendix \ref{sec:DHO}).  Our Hamiltonian eigenstates have changed from 
$\{|n_A\rangle\}\oplus \{|n_B\rangle\}\in \F\cap\HH$ for the two free 
oscillators (energy/number representation) into two-component
state vectors representing coherences which also
provide a representation of the semi-groups $SU(1,1)_\pm\subset
Sp(4,\R)_\pm $.
The function space realization of this extension runs the from the 
space spanned by the direct sum of the 
``very well behaved'' free oscillator energy eigenstates representing 
the creation and destruction operator algebra, to a 
representation of $SU(1,1)\subset
Sp(4,\R)_\pm $,
and finally to a representation of $SL(2,\C )_\pm$, or some other
complex covering, as discussed below.

Physically, the free oscillator system has been transformed by the
introduction of correlation into a
coherence composed of two oscillators.  That coherence, in turn, is able
to undergo an internal transformation which redistributes the energy
between the two oscillators.  See Appendix~\ref{sec:DHO}.  Note that
the coherence acts as a (formally) closed system in which both source
and sink for the ``dissipation'' are dealt with.

In~\cite{lindblad}, use is made of rigged Hilbert spaces for
representation of $SU(1,1)$, and considerable care is taken to choose
a topology which insures any anomalous complex eigenvalues for this
real group are properly avoided.   
Herein, the anomalous complex eigenvalues are not excluded, meaning 
that at a minimum we must 
be working in some complex covering group, such as
$SL(2,\C )$ or $Sp(4,\R )^\C$, for which complex
scalars are defined--and we note that analytic continuation is a
complex symplectic transformation, which narrows our choices.  Now 
making an  
additional extension of the {\em full} Hamiltonian of the dissipative  
oscillator system to $\F^\times_{\mathfrak{sp}(4,\R )_\pm}$ using the 
$e^{i\mu K_3}$ map used in 
Appendix \ref{sec:DHO} (as an exemplar of the extension
process---there are nine other possible generators for the liftings
and other possible scalar coefficients), for the value $\mu =\pi /2$
we find: 
\bea
e^{i(\pi /2)K_3} \, H \, e^{-i(\pi /2)K_3} &=& \alpha^2           
	e^{i(\pi /2)K_3}\; \left( iQ_3 \right) \; e^{-i(\pi /2)K_3}
	     \nonumber   	\\       
& &\qquad +\, \beta^2 e^{i(\pi /2)K_3}\; \left( i J_3 \right) \;
	     e^{-i(\pi /2)K_3}                      
		\nonumber 	\\     
&=&- \alpha^2 \, [K_3, Q_3 ] \, - \, \beta^2 [K_3, J_3 ] 
		\nonumber 	\\     
&=& - \alpha^2 J_0 \; - \, \beta^2 (0) 
		\nonumber 	\\     
&=& i \alpha^2 \; \left( iJ_0 \right)   
\label{eq:Htransform}
\eea
$( iJ_0 )$ is the same as the operator $iZ$ previously identified in the 
$SU(1,1)$ algebra, and this recovers the Feshbach-Tikochinsky result, 
equation (\ref{eq:F-T2}) or equation
(\ref{eq:F-T}).  There are other liftings, however, for which the commutator 
which results from the lifting of $H_0$ does not vanish.  
There are other decay 
constants (and decay processes) besides the single one identified in the F-T 
construction, although not all of these relate simply to the 
$\mathfrak{su}(1,1)$ algebra which the F-T procedure seems associated with.
(Reiterating, it is $\mathfrak{sp}(4,\C ) \supset\mathfrak{sl}(2,\C )\supset
\mathfrak{su}(1,1)$ which sets the overall structure.  This is only a
substructure of that larger $\mathfrak{sp}(4,\C ))$ structure.)~\cite{note16}

The essentially self adjoint extensions of the {\em generators} of the
realization of the Lie algebra can be understood as, for instance:
\bea
(iY)^\times = iY &=& e^{-i(\pi /2)K_3}\cdot\left[            
	e^{+i(\pi /2)K_3} \;(iY)\; e^{-i(\pi /2)K_3}  \right]          
	\cdot e^{+i(\pi /2)K_3}                      
		\nonumber  	\\     
&=& -\, e^{-i(\pi /2)K_3}Ze^{+i(\pi /2)K_3}
\eea
This amounts to identifying the simple inclusion of an operator
belonging to $\g_\pm$ into $(\g^\C_\pm )^\times$ with a certain point
on the co-adjoint orbit of another operator also within $(\g^\C_\pm
)^\times$, which can be explained as the failure of the
weak-$^\times$ topology to separate them.

Beyond those justifications for such an interpretation already given, 
we might add
that this is of the form $gAg^{-1}$, $g\in Sp(4,\C )$, $A\in
\mathfrak{sp} (4,\C )$, and that on the semi-group representation space
for which $g$ is continuous, the operator $g^{-1}$ cannot 
be a continuous operator.  (The conjugacy of $iY$ and $Z$ will be
addressed at length in the fourth installment~\cite{IV} of these
articles, when we discuss the nature of the covering structure which
makes all this well defined, and of which the structure spanned by the
Gamow vectors and their associated Breit-Wigner resonances is a part.) 
From the perspective of $\F\subset\F^\times$, esa operators need not
be  
symmetric, contrary to the situation in $\HH$.  As to $\F$ and
$\F^\times$, there is no violation of any sense of hermiticity in the 
above.  (Again, we're not in Hilbert space anymore, so insisting on
applying notions of hermiticity or anti-hermiticity is unduly
restrictive.)

Many readers may be asking why we do not just go ahead and speak of
anti-self adjoint (anti-self dual, or asd) extensions.  The generators 
of the 
Lie algebra $\g$ are also canonically generators of the dual Lie
algebra $\g^\times$ due to the canonical inclusion
$\g\subset\g^\times$.  The generators of the algebra are taken
to be identified with generators of infinitesimal translations
(directional derivatives), and these have
invariant geometrical meaning as a certain tangent vector.  If we wish   
to identify representations of these generators 
in the Lie algebra with connections
associated to covariant derivatives, i.e., identify $U(1)$
subgroups as geodesic subgroups, then we must understand that 
the semi-simple Lie
algebras we are working with here may be associated with a group or 
with either of a pair of semi-groups: we are speaking of the same
generator in all cases.  (Because generators are first order 
derivatives, on analytic spaces the generator is locally 
independent of the direction of travel along the path of $exp$.)
Hence, because $\g \subset \g^\times$, the generators of $\g$ are
canonically (essentially) self adjoint (or self dual).  To
define a dynamical inner product transformation structure, on the other
hand, requires an anti-self adjoint (e.g., anti-self dual or asd)
extension of the Lie algebra~\cite{note17}.

It is necessary of symplectic transformations that their group obey
the dynamical law.  I.e., to say that a transformation $t$ belongs to
the group of symplectic transformations of a complex symplectic
space necessarily implies that $t\cdot t^\times$ must be the identity 
transformation~\cite{porteous}.
Therefore, there is really no other alternative for the form of
adjoint involution other than the alternative chosen here, assuming we
persist in the requirement that our Hamiltonian generate infinitesimal
dynamical (=symplectic) transformations.

For unitary transformations on Hilbert
space, one conventionally thinks of the dual transformation in terms
of hermitean (i.e., complex) conjugation, e.g.,
\begin{equation}
\left( e^{-iHt}\right)^\dagger = e^{(-i)^\ast H^\dagger t} =
	e^{+iH^\dagger t } = e^{+iHt}
	= \textrm{``}\left( e^{-iHt}\right)^{-1}\textrm{''}
\label{eq:trad}
\end{equation}
On $\F$ and $\F^\times$, there one must think in terms of, e.g.,
\begin{equation}
\left( e^{-iHt}\right)^\times = \left( e^{- (iH) t}\right)^\times =
	e^{-(iH)^\times (-t)} = e^{+(iH)t} = e^{+iHt}
	= \textrm{``}\left( e^{-iHt}\right)^{-1}\textrm{''}
\label{eq:alttrad}
\end{equation}
because it must then also be the case that
\begin{equation}
\left( e^{Ht} \right)^\times = e^{H^\times (-t)} = e^{-Ht}
	= \textrm{``}\left( e^{Ht}\right)^{-1}\textrm{''}
\label{eq:realmod}
\end{equation}
in order that the group scalar product be properly
defined~\cite{note18,note19}.  
(The quotations on the right hand sides
above are cautions that care must be exercised interpreting these
relations, because, e.g., defining the rotation group by $RR^\dagger =
\II$ or by $RR^{-1}=\II$ leads to different spin groups covering the
rotation group.)  This should look more natural when we look at the
spinorial nature of these constructions in the fourth
installment of this series of papers~\cite{IV}.

Recapitulating, in order to accomplish this properly dynamical
construction, we see from the above that must:
\begin{itemize}
\item{Deal with the complex Lie algebra as a {\em{real}} algebra.
Note that there is a unique decomposition of complex 
operators into operators whose algebra is isomorphic to
the complex number algebra times the components of
the complex algebra against the basis provided by this first algebra,
and finally times a real number~\cite{teichmuller,handb,adler}.  
Thus, we distinguish $H$ and $(iH)$ as
distinct generators of a real operator algebra containing $H$ itself
as a generator, because there is a complex plane structure present.}
\item{Accompanying the dual extension of this real operator algebra
must be an involution, transforming the real algebra used as 
semi-group ring scalars in such a way
that the resulting semi-group transformations satisfy the tests for
symplectic action.}
\end{itemize}

This form of extension which is self-dual (esa) as to the generators
of the realization of the semi-algebra but anti-self-dual as to the 
realization of elements of the Lie algebra
and associated semi-group
sets up the connections due to the ``$U(1)^\C_\pm$'' sub-semi-groups 
which figure in the identification
of a generalized Yang-Mills gauge structure associated to the
group representations in generalized spaces used in 
this present body of work (see installment four~\cite{IV}
of this series),
and also figures in setting up the quasi-invariant measures which are
part of establishing the ergodicity of the Gamow vectors.  (See
installment three of this series~\cite{III}.) 

This esa extension makes possible the geometric identification of the
roles of the Lie algebra
generators in their role as generators of infinitesimal
translations in the semi-group.  Their being esa also
enables us to apply the nuclear spectral theorem to the generators of
the Lie algebra, meaning when we proceed in the manner chosen for
the adjoint involution, the generators
can be associated with physical observables.  Thereby, we are
simultaneously 
creating an identification between dynamical quantities and physical
observables by means of careful mathematical construction.

The existence of a complex
structure on $\F_{\mathfrak{sp}(4,\R )^\C_\pm}$ (induced by the 
representation of complex Lie algebras 
$\mathfrak{sp}(4,\R )^\C\pm$) means,
for instance, that $Y$ and $iY$ define {\em{different}} connections
associated to different directional derivatives along transverse geodesics
on $\F_{\mathfrak{sp}(4,\R )^\C \pm}$, and each must be 
given an esa extension
individually.  (This is shown constructively for the function space
realizations in Section~\ref{sec:altconst}, bearing in mind that
$\CS_{\g\pm} \subset\CS^\times_{\g\pm}$ so that inclusion offers a form of 
extension.)  It is obviously not possible to accommodate this structure
in the traditional Hilbert space formalism.  

There are {\em multiple} one parameter families
of symplectic esa extensions of the generators
from $\F$ to $\F^\times$, since there are 
other generators of symplectic transformations and the lifting 
parameters for all are continuous.  This is a 
radically different structure from  
the structure one is more familiar with in Hilbert space.  These group
extensions satisfy the often encountered and standard criteria for
``unitary representations'', e.g., if $\theta : G \longrightarrow
GL(V)$, then as a standard criterion for a unitary representation of 
$G$ on $V$ we require
$\langle \, \theta\circ g \, v \, , \, \theta\circ g \, w \, \rangle =
\langle \, v,w\, \rangle$, $\forall g \in G,\; \forall\, v,w\in
V$.  See, e.g.,~\cite{grillet}.  However, in the present context we see them as
dynamical (=symplectic) in their action rather than as unitary, since
the theorem of Porteous~\cite{porteous} applies to all the symplectic
transformations, of which the unitary transformations are a subpart.

For symmetric esa extensions (e.g., of hermitean operators 
on Hilbert space),
the eigenvalues in the unitary representation of the group are
necessarily unimodular, and the categorization of the present extensions as
quasi-invariant rather than invariant is intended to clearly signal that  
the eigenvalues of the group
representation need not be unimodular.  The scalar 
product is only ``quasi-invariant'' (see~\cite{fandh}, page 311), even though
a test similar to the familiar test for unitarity is 
satisfied.  Following  the complexification--dual-extension, the
physical expectation values may be 
different from those previously obtained, since in general the 
extended operators
are no longer symmetric.  Eigenvalues for the quasi-invariant
semi-group
representation are not necessarily unimodular any longer, so that the
semi-group operator $e^{-iHt}$ responsible for time evolution during
$t\ge 0$ need no
longer contribute a simple unimodular phase oscillation
for energy eigenvectors.

Many details will not be covered exhaustively herein, such as the 
existence of different left and right quasi-invariant measures, which
will be touched on very lightly
in the third and fourth installment of this series~\cite{III,IV}.








\subsection{Physical interpretation.}\label{sec:interp}

What is physically important are the matrix elements emerging from
this structure.  For a connected complex semi-simple Lie group $G$,
such as $Sp(4,\C )$, integration of forms on the 
group is equivalent to integration on forms on the algebra.  From bi-duality, 
one associates the scalar product $\langle g,h \rangle$ on $G_\pm $ and 
$G^\times_\pm$ with a scalar product on the representation space $V$, 
$\langle \bullet , \bullet \rangle_V$, $\theta : G_\pm 
\longrightarrow Aut(V)$, $\theta : \g_\pm \longrightarrow\mathfrak{aut} (V)$, 
by~\cite{marsden}:
\begin{equation}
\left\langle \theta (\tau ) v ,\theta (\tau ) w \right\rangle_V= 
	\left\langle v,w \right\rangle_V \; ,\, \forall w\in V^\times \; , 
		\forall v\in V \; , \forall \tau \in G_\pm \; .
\label{eq:bidualityrep}
\end{equation}

For $G$ complex, there is a canonical scalar product on $V$ and 
$V^\times$ which is ``unitary'' (quasi-isometric):
\begin{equation}
\left\langle v,w \right\rangle_V = \int_{G\pm} \,
\left\langle \theta (\tau ) v ,\theta (\tau ) w \right\rangle_V \; 
d \tau
\label{eq:unitarygprod}
\end{equation}
This is a trivial extension of~\cite{marsden}, p 151-153, to more general function 
spaces and to semi-groups.  Note that for the typical $e^{\pm iHt}$ time 
evolution semi-groups, this scalar product would be phrased
\begin{align}
\int^\infty_0 \left( e^{-iHt} v\right)^\times \;\left( e^{-iHt} w \right) 
	d \mu_t    
& =\int^\infty_0 \left[ \left( e^{-iHt} \right)^\times v^\times \right]
	\;\left( e^{-iHt} w \right) 
	d \mu_t  				\nonumber 	\\
&= \left\langle v,w \right\rangle_V \;\; \int^\infty_0 \; d\mu_t 
	=\left\langle v,w \right\rangle_V
\label{eq:u1norm}
\end{align}
where $d\mu_t$ is the left invariant 1-form on the one parameter
time evolution sub-semi-group, properly oriented and normalized.  For a 
{\em{real}} semi-group generated by the essentially self adjoint operator $A$,  
one must make the association $\left( e^{\alpha A} v\right)^\times = 
 v^\times e^{-\alpha A}$, $\alpha \in\R_+$, in order for this to also result 
in a dynamical transformation of the inner product, i.e., physically one 
should think of the above integral as
\begin{equation}
\int_{G+} \, \left( e^{-iHt} v\right)^\times\, \left( e^{-iHt} w\right) \,
		 d\om_t 
		=\int_{G+}  v^\times e^{-iH^\times (-t)}\circ e^{-iHt} w\; 
			d \mu_t
\label{eq:ftnorm}
\end{equation}
or as the ``time reversed scalar product'' of F-T~\cite{ft} in some sense.

This mathematical structure (which is forced upon us in order to
provide both uniquely defined adjoints and
a proper dynamical representation of the connected complex
covering group)
has a sensible physical interpretation, as on our generalized space of
states it effectively measures 
the overlap of a state (=probability amplitude representing a dynamical
system) prepared during $t\le 0$ with the results 
of a measurement (=probability amplitude representation of a second
dynamical system used as a measurement apparatus) 
during $t\ge 0$.  Working in a rigged Hilbert space 
suggests it may be reasonable to adopt an untraditional 
but physically meaningful physical interpretation of evolution 
operators appearing in the integrals of evolving state vectors.  
This is an unexpected ancillary result of these 
quasi-invariant measures.

See also~\cite{butzer,bgm,bmlg} for extensive 
discussions of these semi-groups of time evolution and their physical 
interpretation.  This present work finally establishes they truly deserve to
be called semi-groups of dynamical
time evolution, and that they are not merely
generalizations gotten from the $U(1)$ groups of time evolution
obtained with the Schr{\"o}dinger equation in the HS formulation,
and for which the same physical meaning is conjectured.  Note that the
{\em form} for the semi-groups of time evolution and their duals
obtained by formal analysis in~\cite{butzer,bgm,bmlg} is here obtained 
by an alternative means 
based on careful algebra and geometry.  

In an {\em algebraic} theory of scattering based on the RHS and this
procedure, the M{\o}ller operators $\Om^\pm$ would be represented by sums 
and products of canonical transformations representing the actual dynamical 
transformation 
processes, such as in (\ref{eq:adext}).  The procedures dealt with here 
certainly do not constitute a complete theory of scattering, 
but do suggest that the formulation of quantum mechanics may be 
enlarged from a Hilbert space to 
a RHS to provide the vehicle from which a rigorous algebraic theory of 
scattering may be developed. 

The present algebraic approach to scattering is distinguished
by:
\bee
\item{Representation of the preparation and registration processes as 
continuous quantum dynamical transformations (which preserve some 
internal symmetry of the system).  There 
is no classical apparatus anywhere, and so this is a generalization of the 
``Copenhagen Interpretation''.}
\item{Dynamics can be separated into internal and external components, 
and both are based on symmetries.  The external evolutionary 
impetus takes the form of symplectic transformations which preserve 
some internal symmetry.  Internally, the identity of the oscillators
remains fixed by the internal symmetry (creation and destruction
operator algebra) which remains unbroken.  
The interpretation of this seems consistent with ``noisy'' external 
perturbations of the internal system and deliberate action upon the internal 
system both being represented by canonical transformations. }
\item{There are no infinite reservoirs in the present construction,
which involves a closed system of two oscillators dynamically
evolving,  and there is no privileged role for 
any apparatus or observer.  There is no ``collapse of the wave function''.  }
\item{The preparation or registration process is treated as a true process, 
evolving continuously according to the $U(1)$ semi-group of dynamical 
time evolution.  Thus, for the 
preparation procedure represented by $e^{i\alpha A}$, the preparation advances 
with the transformation parameter $\alpha$.}
\item{Only essentially self adjoint operators are used as
infinitesimal generators, and although the self 
adjoint Hamiltonian is transformed by the preparation procedure (and so may 
depend parametrically on the preparation process), it has no explicit time 
dependence.}
\een
There are substantial open questions concerning the formalism, particularly 
relating to interpretation of the physical meaning of the mathematics, and, 
conversely, how to phrase a given physical situation mathematically.   
For instance, systems seem to evolve 
independently of any preparation or registration apparatuses, so one must 
distinguish the decay of a resonance from its observation, yet both seem to 
be representable by the same dynamical evolution parameter and 
transformation.  Apparently, one must encounter and properly interpret 
both active and passive canonical transformations, e.g., the decay
process itself might be considered passive and the registration of the
decay event by an experimental apparatus might be considered active.
A fuller account of stability against small perturbations also needs
to be provided for these quantum states which decay, and this will 
begin in the third installment of this series~\cite{III}.  








\section{Poisson and Symplectic Structures}\label{sec:poisson}

The choice of spaces upon which to realize an 
operator representation 
of the algebras and semi-groups is very significant.  By a theorem of 
Marsden, Darboux's theorem does not hold for weak symplectic
structures~\cite{marsden}.
Thus, during the course of representing a Lie algebra structure 
(which participates in defining a weak symplectic form on
$\g\times\g^\times$) on a subspace of 
the Schwartz space (and its 
dual), Darboux's theorem is of no use to construct a local euclidean grid 
which is complete, e.g., there exists no way to completely diagonalize
the matrix elements of that representation
space and its dual (which is based on Darboux's
theorem).  This has the physical implication that there is a generic
possibility of mixing occurring during transformations on a space with
a weak symplectic structure.  (Mixing is a condition precedent for
entropy increase, discussed in installment three~\cite{III}.)

Semi-simple operators locally may be diagonalizable
individually (since they are ``linear''), 
but in the present situation 
there is no possibility of a complete set of commuting 
operators being obtained using Darboux's theorem.  (A complete 
set of commuting operators may still be obtained at the even higher
level of the universal enveloping algebra in the case of 
semi-simple groups.)   This weak symplectic structure
also permits the faithful representation of some 
sub-semi-groups which may contain elements not describable by the exponential 
mappings of single generators~\cite{hilgert}, meaning we may add
certain sets (e.g., countable sets which are therefore of
measure zero) to our spaces without adverse effect on our mathematics.  

Marsden's Theorem means that there are non-trivial
local invariants (such as curvature or torsion) for 
the structure on $\F\times\FT$.  The existence 
of non-trivial local invariants
means the geodesics which represent the evolution of a state in the
space of states can be non-trivial geometrically, and the 
physical evolution of the systems they idealize may be non-trivial as well.








\subsection{Canonical Poisson and symplectic
  structures}\label{sec:canonical} 

This section is a brief recapitulation and reinterpretation
of standard results for Lie
algebras and their duals from the perspective of Lie-Poisson
structures.  See, e.g.,~\cite{mandr}.  It will serve as
an indication of the structure of $\F^\times_{\g\pm}$, which 
reflects the structure of $\g^\times$ in important
regards.  

There is a canonical symplectic structure on the direct product
of a Lie algebra with its dual, i.e., on $\g\times\g^\times$.  There
is a similar canonical symplectic structure on $\F_{\g\pm}\times
\F^\times_{\g\pm}$ and other pairings of locally convex 
vector spaces and duals, e.g., associated with the scalar product 
viewed as a Cartesian pair.  These are categorized as {\em weak}
symplectic structures.  Note that the {\em real} Hilbert space $\HH$
is associated with
a weak symplectic structure, but the complex Hilbert space
$\HH^\C$ is associated with a {\em strong} symplectic form defined by the
Hermitean product on $\HH^C \times \HH^\C$, which coincides
with the symplectic form on $\HH^\C \times (\HH^\C )^\times$,
so a complex Hilbert space is associated with a strong symplectic form while a
real Hilbert space possesses a weak one.  (See, e.g.,~\cite{mandr}, chapter 2.)

The dual of a Lie algebra is a Poisson manifold, e.g., $\g^\times$ is
Poisson, and co-adjoint orbits in $\g^\times$ individually 
possess a symplectic
manifold structure: $\g^\times$ can be thought of as a union of these
co-adjoint-orbit-symplectic-manifolds (but need not be a symplectic
manifold itself, since those unions may be disjoint, meaning there is
no global symplectic form).  On $\g^\times$, we have a Poisson
structure associated with the Lie-Poisson brackets on $\g^\times$, and
a symplectic structure associated with the Lie-Poisson structure on
the co-adjoint orbits.  We thus have symplectic sheaves in the Poisson
manifold $\F^\times_{\g^\C\pm}$, e.g., ``$Ad_{Sp(4,\C )}$'' applied to the
realization of $\mathfrak{sp}(4,\R )^\C_\pm$ on
$\F_{\mathfrak{sp}(4,\R )^\C \pm}$
forms a series of symplectic sheaves (co-adjoint orbits) 
in $\F^\times_{\mathfrak{sp}(4,\C
)\pm}$.  Inclusion is a Poisson mapping.  We can also think of
$\g^\times$ as $T^\times G/G$, where $G$ is obtained by integration of
$\g$, and $\g^\times $ must be even dimensional~\cite{mandr}, page 293 and
Chapter 13.  For $\g$ the value of the {\em momentum mapping}
is always an element in $\g^\times$, and under the momentum mapping a
Poisson (e.g., including symplectic)
action of a connected Lie group $G$ is taken as the co-adjoint
action of $G$ on the dual algebra $\g^\times$.  See~\cite{arnold}, Appendix 5,
page 374.  

We thus have an association of transitive action of a
connected Lie group,  the momentum map, Poisson action of a group on a
space, and a 
topology change to a weak-$^\times$ topology in the dual of the Lie 
algebra of that Lie group.  The transitive symplectic action (momentum
map) of a
connected Lie group is taken as the co-adjoint action of that Lie group
on the appropriate dual.  We will see this structure mirrored every
time when we address the issue of why what appear to be
adjoint orbits on some representation space 
$\F_{\mathfrak{sp}(4,\R ) \pm}$ are in fact
co-adjoint orbits in the representation space
$\F^\times_{\mathfrak{sp}(4,\R )^\C \pm}$.  It is appropriate to begin
addressing that issue now.  The scalar product 
$\langle\,\bullet\,\vert\,\bullet\,\rangle$
on $\F\times\F^\times$ is what matters here, both physically and
mathematically.  There is a canonical symplectic structure here
(e.g., when viewed as a Cartesian pair), 
and it exists canonically on $\g\times\g^\times$ as well (provided
those transformations have well defined adjoints).  This 
means that, e.g., even a seemingly 
``adjoint transformation on $\g$'' can, if transitive, 
be thought of as an element of the group of
symplectic transformations on $\g\times\g^\times$, and that group of
symplectic transformations may include complex and even quaternionic 
transformations.  For instance, even
the mere choice of a ladder operator basis for a real group may amount to a
complex symplectic transformation in precisely this sense.  See, e.g.,
the $SO(3)$ and $SO(2,1)$ comparison, yielding a complex spectrum for
$SO(2,1)$ in this regard, used as an example in~\cite{bgm}, Section
2.

Gadella has given the necessary and sufficient mathematical conditions
for analytic continuation in the function space realizations of the
RHS structure~\cite{gadella}, and in particular shows that the analytic
continuation proceeds from the realization of $\HH^\times$ to the
function space realization of $\F^\times$.  H{\"o}rmander~\cite{hormander}, gives
a prescription for the analytic continuation of an elliptic Green's
function which demonstrates how one must proceed with the abstract
spaces.  The fundamental solutions are analytically continued, but a
topology change is mandated since this analytic continuation is to the
distributions.  Hence, a complex symplectic transformation of a real
algebra which works an extension to the complexes must be
accompanied by a topology change to a weak-$^\times$ topology.
Analytic continuation is a momentum mapping (!) in the jargon of
dynamical systems.  Tied
up in this general topology change shown by H{\"o}rmander are issues
in harmonic analysis necessary for the existence of Green's functions,
resolvents (whose poles provide the spectrum), etc.  All
the mathematical machinery necessary for us to do physics depends 
on us making
this topology change, and this topology change is not optional, but is
necessary for well defined mathematics.




\subsection{Symplectic structures and forms of algebras}\label{sec:spstructures}

A complex structure is a type of symplectic structure.  It is
important in the present context to think of the complex numbers in
their symplectic form, i.e., as a real algebra $\R \oplus
i\circ\R$.
This is because if we think of $\C$ as a {\em field}, the operation of
involution in the form of complex conjugation cannot be 
uniquely be defined without the addition of more structure than 
the axioms of a field provide.  For example, if one thinks only 
in terms of complex conjugation of the field of complex numbers
on a conventional Hilbert space, the
representation of a complex Lie group obtained from exponentiating
a complex Lie algebra (using an esa realization of the generators of
the algebra, so as to make them identifiable with physical observables)
would make the scalar product of that space
not be uniquely defined, since exponentials of complex operators 
(e.g., with mixed real and imaginary parts) would not have unique
adjoints!  Those operators of the form $H=H_0 +iH_1$
could no longer be characterized as either hermitean or
anti-hermitean, and so any effort to accommodate complex operators and
complex spectra on von Neumann's Hilbert space is destined for very
serious mathematical difficulties. 

The usual insistence on hermitean (or anti-hermitean)
representations does not stem
from any structure of the Lie algebra, but from the need for unitarity 
in commonly met examples involving the von Neumann Hilbert space, with
the hermitean scalar product.  The hermitean conjugation there has a
symplectic (=dynamical) action on the von Neumann Hilbert space.
The extension of this form of hermitean scalar product
to our complex group/algebra situation on our different
spaces would not make the transformations represented complex
symplectic, however, which may be even more serious than being
non-hermitean and non-unitary, since this
relates to integrability in a more general way, and would also be
contrary to our current understanding of what is ``dynamical''.    

If, however, one thinks
of $\C$ as a real algebra (i.e., considers it in its symplectic form
$\R\oplus i\circ\R$), then one may define an involution for that real
algebra uniquely.  There may be many possible alternative ways of 
defining involution for a given real algebra, but you must choose 
one and stick to it~\cite{note19}.  In the interests of proceeding
uniquely in the present algebraically oriented treatment, it is
critical to think of ``analytic continuation'' not in terms of 
``field extension'' but in terms of (transitive)
transformations extending a real algebra into the symplectic form of an 
enlarged algebra.  We further insisted on defining the action of
representations 
of groups of transformations on our spaces in such a way that their
action on our representation spaces is symplectic~\cite{note20}.  
(See  again Section~\ref{sec:scattering}.)


A complex manifold 
with a hermitean metric whose imaginary part is closed (e.g., symplectic) is 
called a K{\"a}hler manifold.  On the complexification
of the real phase space $T^\times \R^n\cong\R^{2n}_{p,q}$, which will be 
denoted (abusively)
$\widetilde{T^\times \R^n} \cong \widehat{\R^{2n}_{p,q}}\cong \C^n$, 
one has an hermitean metric:
\begin{equation}
h({\boldsymbol \xi},{\boldsymbol \eta}) = 
	\left({\boldsymbol\xi},{\boldsymbol \eta}\right) - i 
		{\boldsymbol \om}({\boldsymbol\xi},{\boldsymbol \eta}) =   
\left({\boldsymbol \xi},{\boldsymbol\eta} \right) 
	- i \left[ {\boldsymbol \xi},{\boldsymbol \eta} \right] \; .
\end{equation}  
That $\boldsymbol \om$ is closed on $\R^{2n}_{p,q}$, $d{\boldsymbol \om} =0$, 
is a consequence of the existence of a symplectic (Darboux) coordinate system, 
i.e., Euclidean structure on $\R^{2n}$.  In such a coordinate system the
torsion vanishes, meaning (pseudo-)Riemannian connections can be defined for 
the manifold which possesses a complex structure~\cite{nash}, page
167f.  We may take the vanishing of the torsion for 
as the integrability condition for integrating Lie algebras to the 
(connected part of the) Lie group, i.e., so that torsion free
connections can be said to exist on the complex simple Lie group.

Given that the work herein involves the use of hermitean metrics on
complex manifolds, i.e., involves (weak) K{\"a}hler
metrics, the vanishing of the torsion is not a trivial 
matter, and depends on the fact that $\R^{2n}$ is torsion free, and that the 
complex semi-simple Lie algebras are torsion free.  In a generalized
Riemannian geometry setting, the skew symmetric part of
the metric is associated with possible torsion (and some corrections 
to the curvature tensor arising from the torsion), and the hermitean 
and K{\"a}hler metrics have just such a skew symmetric part, as shown
above.  The torsion itself is not an especially  
serious obstruction to integrability (e.g., we are untroubled by
holonomy), but the conditions necessary for 
the existence of torsion also makes 
possible {\em shear}, and the formal possibility of essential
discontinuities on a set of positive measure can present serious
problems!  Local integrability is possible because
the possibility of shear on sets of positive measure is avoided through
satisfaction of the integrability condition on the group.  We may view
shear as being restricted to, at most, a set of zero measure, which we
understand to be reflected in the branch cut (if any) 
and countable set of resonance poles associated with the analytic
continuation.  The branch cut represents a bifurcation of solution sets  
by discontinuities in the second and higher derivatives, and so does not
represent shear in the ordinary sense.  Hence, shear occurring during
the dynamical time evolution in our spaces of generalized states, if
present at all, exists only on a set of measure zero, the countable
set of Gamow vectors, or their associated countable set of
Breit-Wigner resonance poles~\cite{note21}. 






\subsection{Hamiltonian symplectic actions (integrability)}
	\label{sec:hamiltonian}

Because we have defined a new type of Lie algebra and Lie group
representation, it is appropriate to take an aside to demonstrate that
our constructions are non-pathological.  We have constrained ourselves
by insisting that the 
representation have a symplectic action on the representation spaces,
and also that there be weak symplectic forms, etc., so some further
explanation is necessary.

The representation of evolutionary flows generated
by semisimple symmetry transformations are defined over a variety of spaces
representing quantum mechanical states, e.g., all the spaces in the
Gadella diagrams of installment one~\cite{I}.  There are a variety of
integrability conditions which must mesh everywhere throughout this
structure.  The conditions for the Lie algebra structure of the
infinintesimal generators of these transformations to be integrable
(e.g., into the connected part of a group structure) are reflected in
their representation counterparts as conditions for the flows to have
``single valued hamiltonian functions''.

For a semisimple 
symmetry group $G$, by limiting consideration
consideration to semi-groups and sub-semi-groups of $G$
which are strictly infinitesimally generated, we require in the first
instance that the group 
is the union of the two strictly infinitesimally generated 
semi-groups, which are unique: $G= G_+ \cup G_-$~\cite{hilgert},
p. 378. The 
representation of these strictly infinitesimally generated 
sub-semi-groups will be represented by
the Lie algebra as a subscript, as in $\CS_{\g\pm}$.  The 
complete spaces $\CS_{\CG\pm}$ and $\CS^\times_{\CG\pm}$ in 
Section~\ref{sec:altconst} are not necessarily
restricted by this limitation of consideration, so that we must think 
$\CS_{\g\pm}\subseteq \CS_{\CG\pm}$.  We are interested in the
intersection of the Schwartz space and space of Hardy class functions,
$\CS_{\g\pm} \cap \HH^2_\pm$ and $\CS_{\CG\pm}\cap\HH^2_\pm$,  
and with the $\HH^p$ spaces one cannot be assured that 
sufficiently many functionals  
exist to separate a point from a subspace~\cite{duren}.  The spaces of
physical interest, $\CS_{\CG\pm}$ and $\CS_{\g\pm}$, are likely 
to differ at most on a set of measure zero.  See 
Section~\ref{sec:altconst}.   

A Lie algebra $\g$ when viewed as a linear 
vector space possesses a dual $\g^\times$, and one normally defines     
an inner product $\langle g , \xi \rangle$, $\xi \in \g^\times$, 
$g\in \g$.  (A linear space is an algebra plus a scalar product.) On
representation spaces,  
one conventionally reflects the bi-duality of this product 
(reflexitivity of $\g$ and $\g^\times$ as spaces) by appropriate 
definition of product modules over the representation spaces, 
and there should also be reflexitivity of the representation spaces
and duals~\cite{note21x5}.  For semi-groups and their algebra of generators, a 
similar scalar product can be defined, although one works with
semi-group rings rather than rings.  See Section~\ref{sec:scattering} and 
installment three of this series.  See also, e.g.,~\cite{yabogr}, page
398.

The continuous extensions of infinitesimal generators  $A \longrightarrow 
A^\times$, which are taken to be essentially self adjoint, is adopted
because the operator 
$A^\times$ dual to a generator $A$ is equal to the weak-$^\times$ 
generator of the dual semi-group.

When the $G$-action of a semi-simple group $G$ 
preserves the symplectic form $\boldsymbol \om$, all of the
fundamental vector fields of the action of $\g$ on $G$ are locally 
hamiltonian (locally integrable)~\cite{moser}, p. 47, ~\cite{siegel}.  The action 
of the strictly infinitesimally generated semi-simple (sub-)semi-groups 
thus preserve the symplectic form and all of the fundamental 
vector fields are thereby locally hamiltonian.  This follows because
$\g\subset\g^\times$, fixing the symplectic form on all of the
symplectic sheaves of $\g^\times$ (even though there may be no global
symplectic form on $\g^\times$ itself.)   

The Lie algebra of generators of 
the strictly infinitesimally generated Lie semigroups (as ray
semi-groups~\cite{hilgert}), 
therefore exponentiate to generate locally hamiltonian 
semi-flows on the (connected part of the semi-simple) semi-groups $G_\pm$.  
This locally hamiltonian structure should be carried forward by any 
non-pathological representation mapping.  There may even be global
integrability, but at least
local integrability is assured for all semi-simple semi-groups and
their non-pathological representations.

At this juncture, we see torsion free structures on the complex
semi-simple Lie algebras whose groups are therefore connected (although
not necessarily simply connected).  Elsewhere in this paper (e.g.,
Section~\ref{sec:altconst}), we develop various locally convex
abstract representations 
and associated function space realizations which mirror this
structure.  We have symplectic structures on the direct product of
(complex) spaces and their duals.  We are thus ready to invoke the
result that the symplectic actions of semi-simple Lie algebras on
symplectic manifolds are Hamiltonian~\cite{kandn2}, page 354.  In 
fact, as to
the symplectic transformations on $\g\times\g^\times$ (e.g., on
$\mathfrak{sp}(4,\C )\times\mathfrak{sp}(4,\C )^\times$) we have a
textbook Hamiltonian action~\cite{arnold}, page 346, example (e):
\begin{quote}	
The co-adjoint action of $G$ on a co-adjoint orbit in $\g^\times$
with the $\pm$ orbit symplectic structure has a momentum map which is
the $\pm$ inclusion map\ldots This momentum map is clearly
equivariant, thus providing an example of a globally Hamiltonian
action which is not an extended point transformation.  
\end{quote}
Equivariance of the mapping is a condition for integrability.
See also Section~\ref{sec:scattering} and installment three~\cite{III} 
of this series.  The Singular Froebenius Theorem permits 
integration in systems involving distributions.

This globally 
Hamiltonian action in the abstract Lie algebra dual is reflected
in the representations.  In particular, the Hamiltonian vector fields
on the (weak) symplectic representation manifold have symplectic
flows upon which ``energy'' is conserved~\cite{arnold}, page 566.  This will
enable us to, e.g., make statements of a thermodynamic character
appropriate to an energetically isolated system of quantum resonances
represented by Gamow states, as we will do in the third installment
of this series.  This also
makes conservation principles available generally.

This globally Hamiltonian action may also exhibit the 
sensitive dependence on initial conditions (inherent in analytic
continuation~\cite{mandf1}).  Thus, in the present instance
the global integrability which the
momentum map promises may not be usable as a practical matter, and
in the case of analytic continuation, only local
integrability may be obtained as a useful result.






\section{Algebraic Resolution}\label{sec:algresolution}

As a standard result, the complex simple Lie algebra 
$\mathfrak{sl}(n,\C )= {\mathfrak{a}}_{n-1}$ is identified with
the complexification 
of the algebras $\mathfrak{su}(n)$, as well as $\mathfrak{sl}(n,\R )$, 
$\mathfrak{su}(p,q),\;p+q=n$, $\mathfrak{sl}(n,\C )^\R$ and 
$\mathfrak{su}^\ast (2n)$.

Similarly, the complex simple Lie algebra $\mathfrak{sp}(2n,\C)=
{\mathfrak{c}}_n$ is
associated with the complexification of the Lie algebras $\mathfrak{sp}(2n)$, 
as well as $\mathfrak{sp}(p,q),\; p+q=n, \; p\ge q$,\;
$\mathfrak{sp}(2n,\R )$, and $\mathfrak{sp}(2n,\C )^\R$. 









\section{Analytic Resolution in a Rigged Hilbert Space}
\label{sec:rhsresolution}

\subsection{Standard construction of ${\boldsymbol \CS}$ and 
${\boldsymbol \CS^\times}$}\label{sec:Sconst}

This subsection and the following subsection are concerned with
construction of subspaces of the Schwartz space $\CS$ for use in the
representation of symmetries.   Ultimately, we wish to use a
representation of the Lie algebra itself to generate the
representation.  There is a standard construction of $\CS$ over
$\R^n$, e.g., a  construction of $\CS$ over $\R^4$, which will be
called $\CS_{\R^4}$, perhaps over the phase space for the dissipative
oscillator system.  

There is a transparent extension of that method to
{\em locally compact} manifolds besides $\R^n$,  
and of particular interest is the extension of the standard
construction to the   
connected manifold of a locally compact group, $\CG$.  The standard
construction can be generalized to  
generate abstract representations of locally compact groups which have
been realized
in terms of creation and destruction operators, or their equivalents
in terms of the equivalent phase space variables ($x$ and $\partial
/\partial x$).  The present 
initial emphasis is on showing the dynamical realizations 
of the algebra of $\CG$ over $\CS_{\CG}$, and the actions of both {\em
real} and {\em pure imaginary} generators will be shown in the context
of realizations of the real form of a complex algebra 
included in $\CS_{\CG}$.  

The standard construction uses a countable 
family of semi-norms on $\CS$ and the construction 
defines $\CS$ and $\CS^\times$ based on 
unions and intersections~\cite{genfun4}. 
In this section, this construction will first be 
recapitulated in terms of phase 
space variables $\{ x,p_x,y,p_y\}\cong \RR^4$.  We first set the (multi-index) 
notation:
\bea
D & = \left( \frac{\partial\;}{\partial z_1},      
	\frac{\partial\;}{\partial z_2},      \cdots , 
		\frac{\partial\;}{\partial z_n} \right)		\\
D^\alpha & = {\frac{\partial^{|\alpha |}}{\partial z^{\alpha_1}_1           
	{\partial z^{\alpha_2}_2}}       
		\cdots {\partial z^{\alpha_n}_n}}		\\
|\alpha | & = \alpha_1 + \alpha_2 + \cdots +\alpha_n		\\
\alpha ! & = \alpha_1 ! \alpha_2 ! \cdots \alpha_n !		\\
z^\alpha & = z^{\alpha_1}_1z^{\alpha_2}_2 \cdots z^{\alpha_n}_n.
\eea

The standard countable family of semi-norms on $\CS$, define topologies, e.g., 
convergence of (Cauchy) sequences:
\begin{equation}
\nor \phi \nor_m =\sup_{x, |\alpha |\le m} \left( 1+|x|^2 \right)^m     
	\left| D^\alpha \phi (x)\right|,\quad 
	\phi\in\CS,\;\;m=0,1,\cdots
\label{eq:topa}
\end{equation}
$\CS^{(m)}_{\R^n}$ is defined as the subset of $C^\infty (\R^n )$ for which 
the $m$-th semi-norm (\ref{eq:topa}) is bounded.  $\CS^{(m)}$ is the 
completion of $\CS$ with respect to the $m$th semi-norm, and belongs to a 
family of separable Banach spaces with $D$ dense on them with respect to the 
$m$-th (semi-)norm.  (The $\CS^{(m)}$ are also called countably Hilbert spaces 
in~\cite{genfun2,genfun4}.)  Then, as standard results,
\begin{equation}    
m \nor \phi \nor_m \le \nor \phi \nor_{m+1} 
\label{eq:mnorm}
\end{equation}
and
\begin{equation}     
\CS^{(0)}\supset \CS^{(1)}\supset\cdots
\label{eq:inclusions}
\end{equation}
Hence, we can define 
\begin{equation}     \CS       =\bigcap_{m\ge 0}  \CS^{(m)}
\label{eq:scap}
\end{equation}
If $\CS^{(m)\times}$ is the dual to $\CS^{(m)}$, then:
\begin{equation}     
\CS^\times =\bigcup_{m\ge 0} \CS^{(m)\times}
\label{eq:unions}
\end{equation}

If $N$ is the order of $F\in \CS^\times$, we can define the decreasing 
sequence of norms on $F\in \CS^{(N)\times}$ :
\begin{equation}
\nor F\nor_{-m}=\sup_{\nor\phi\nor =1,\phi\in\CS^{(m)}}          
	\langle\phi|F\rangle\; ,       \quad m=N,N+1,\cdots
\end{equation} 

This is a standard construction for the Schwartz space $\CS$ from 
$C^\infty (\R^n )$ and also of its dual $\CS^\times$.  

Convergence of the exponential map of the generators of (semi-)groups 
of transformations can be shown using only the properties of the 
generators of the transformations as closed linear
operators~\cite{yosida},
although additionally
we should think of them as continuous operators when we
address topological vector space issues.  This does not yet
demonstrate how fundamentally these orbits of 
generators of transformations participate in the generation of the spaces by 
a ``bootstrap construction'' based on the procedure given below.  We can make 
the identification of them in their Lie group operator sense with 
differential operators on the representation manifold $\F$.  These generators 
are not merely linear operators, and their status as linear 
{\it differential} operators makes a difference in the role they play in the 
topology of the space.  This will be explored in the next subsection. 





\subsection{Construction of ${\mathbf \CS}$ and ${\mathbf\CS^\times}$ over 
semi-groups}\label{sec:altconst} 

In this section, the construction of $\CS_{\R^n}$ and its dual is extended to 
$\CS (\CM )$ for a particular class of manifolds $\CM$ possessing a 
differentiable structure and local charts.  The 
generalization of interest here is extension of the standard construction 
to use realizations of the generators of symmetry transformations in terms
of polynomials in creation and destruction operators (or,
equivalently, polynomials of coordinates and derivatives).  This
subsection illustrates
both abstract constructions and the associated function space analogues.
The creation and annihilation operators $a^\dagger$ and $a$ are closed on 
$\CS_{\R^1}$.  Any polynomial in 
position and momenta (or the equivalent creation and annihilation operators) 
is bounded on $\CS_{\R^n}$.  

Identifying a realization of
the generators of a semi-simple complex Lie algebra of the complex
semi-simple Lie group with directional derivatives results in a procedure
similar to the use of ordinary derivatives in the preceding
subsection.  A semi-simple locally compact group  
will be assumed, and its realization will be split into two
semi-groups, as by, e.g., positive or negative real parameters in the
exponential mapping.   For the 
present no concern will be given to any parts of any (semi-)group manifold 
not connected to the identity.  

The motivation for this is that the 
unitary dynamical time evolution group of Hilbert space quantum
mechanics, $e^{-iHt}$, splits into two dynamical semi-groups (for $t\ge
0$, and $t\le 0$) during the course of the analytic continuation which
takes place in the extension from the Hilbert space $\HH\cong
\HH^\times $ to $\F^\times$ in the rigged Hilbert 
space formulation of quantum
mechanics~\cite{dirackets,bohm1,bohm2,gadella}.  The present
construction uses the derivational properties of $H$ as the
infinitesimal generator of a symmetry transformation in order to
construct a representation of that symmetry in a RHS.  Geometrically,
if $H$ is an element of the Lie algebra of the group of dynamical
transformations, then it is identified with a vector tangent to a
geodesic at the identity of the group; vectors tangent to geodesics
are further identifiable with covariant derivations, and 
the connection may be identified with the covariant derivation in the
case of parallel transport.  We will apply this
geometric notion of derivation in the place of the analytic notion of
derivation used in the preceding subsection.

The order of proceeding is to first justify the semi-group
representation constructions of $\CS_{\g \pm}  
\subseteq \CS_{{\CG \pm} ({\R^n} )}\subset\CS_{\R^n}\subset 
C^\infty(\R^n )$.  In this subsection, the notation $\CS_{\CG \pm}$ 
will be used and the specific realization of $\CG$ on $\R^n$ not 
explored: a faithful realization based on generators which are bounded
and continuous 
operators on $\CS_{\R^n}$ is assumed, such as is seen in 
Section~\ref{sec:scattering}.  It will be seen that this construction
ultimately involves the complexification of $\CG$ to a connected
complex group, meaning for the present problem we will ultimately be
dealing with a simple complex Lie group (a geodesically connected
group) acting as complex covering group.  

If we split $\CG$ into strictly 
infinitesimally generated semi-groups based on parameters of the
exponential  mapping of {\em real} algebras $\g_\pm$, the
decomposition  $\CG \longmapsto {\CG_+} \cup {\CG_-}$ reflects a
direct  sum decomposition~\cite{hilgert}.  (This becomes the source of
transversality in our hyperbolic structures in installment three of
this series~\cite{III}.) Similarly, if $\CG^\C_\pm$ denotes the
(connected) complex semi-simple semi-groups which are strictly
infinitesimally generated by $\g^\C_\pm$, there is a similar direct
sum decomposition, $\CG^\C = \CG^\C_+ \oplus \CG^\C_-$.  Once the
group has been strictly  
split into semi-groups, the procedure for generating $\CS_{\CG \pm}$ 
parallels standard theorems, see, e.g.,~\cite{hilgert,yosida}.

We define an alternative countable 
family of semi-norms, equivalent to the previous countable family of semi-norms 
by identification of the realization of the
generators as directional differentiations, and then 
using them to define $n$th order differential operators in analogy to the 
preceding:
\begin{equation}
{\cal D}^n_\beta = exp (\beta B)^{(n)}
\end{equation}
where $\beta B =\beta_1 B_1 + \beta_2 B_2 +\cdots$, $\beta_i\in\R_+$ and 
finite, and $B_i$ a continuous and closed operator realization of the
generator of an algebra, such as 
$\mathfrak{su}(1,1)$, or $\mathfrak{sp}(4,\R )^\C$.  
Note that by choosing $\beta_i\in\R_+$ at this point, we restrict 
ourselves temporarily to real algebras only. The construction over 
$\R_-$ is identical. 

The use of 
$exp(\beta B)$ in the majority of this section is only an exemplar, 
and an identical construction then follows for $exp(i\beta B )$.  I.E., 
pure imaginary $\beta_i$ would create continuous sets of dynamical 
transformations transverse to the transformations of $exp(\beta B)$
for real $\beta_i$.  For $\beta\in\R_+$, $B$ a continuous
closed operator on $\CS_{\R^n}$ serving as the 
realization of a generator of a real algebra $\g^\C$), 
both $exp (\beta B)$ and $exp (i\beta B)$ may be taken as continuous
closed operators on $\CS_{\R^n}$.  We may immediately generalize the
following to construct a representation
space of the complex simple Lie algebra and associated semi-groups,
$\CS_{\CG\pm}$ as a subspace of $\CS_{(\R^n)^\C}$, carefully noting
our chosen ring of scalars is $\C (1,i)$.  The symplectic form (real
form) of the complexified real 
algebra $\g^\C$, $(\g^\C_\pm )^\R \equiv \g_\pm \oplus i \circ \g_\pm$,
has a representation $\CS_{\g^\C \pm}$ which faithfully separates both of these
transverse subparts: both $exp(\g_\pm )$ and $exp(i\circ\g_\pm )$ 
are continuous and closed in $\CS_{\CG^\C \pm}$, but 
on separate subspaces, by straightforward generalization of
the construction of this section.  

By $exp (\beta B)^{(n)}$ is meant the first $n$ terms of 
the series expansion for the exponential.  The finite powers of the 
``directional derivative'' $B^n$ are closed, because 
the $B_i$ are closed (continuous)
on $\F\subseteq \CS$, so that $(\beta B)^n$ is closed and continuous 
for finite $\beta$.  Then, 
$\frac{1}{|n |!} {(\beta B)^n }$ is bounded, and we can say that
\begin{equation}
\nor \phi \nor_{m, \beta } = \sup_{x, |n|\le m}       
	\left( 1+|x|^2 \right)^m |{\cal D}^n_\beta \phi (x) |, \quad           
		\forall \phi \in \F \subset \CS, \;\; m = 0,1,\cdots
\end{equation}
is a countable family of semi-norms equivalent to the previous (standard) 
countable family of semi-norms for fixed and finite $\beta\in\R_+$.  

For fixed $\beta$, the preceding defines
a series of $\CS^{(m)}_{(\beta )}$ as in (\ref{eq:inclusions}) and 
$\CS^{(m) \times }_{(\beta )}$ as in (\ref{eq:scap}) for each value of 
$\beta$:
\bea
\CS_{(\beta )} & = \bigcap_{m\ge 0} \CS^{(m)}_{(\beta )} \nonumber\\
\CS^\times_{(\beta )} & = \bigcup_{m\ge 0} \CS^{(m)\times}_{(\beta )} 
	\;\; .
\label{eq:bunions}
\eea
The $\CS_{(\beta )}$ represent a family of complete spaces parametrized
by $\beta$, and the construction reveals that the running of the
multi-index parameters $\beta$ represents a continuous translation
along the continuity structures on the space as traced out by the
exponential map of differential generators--i.e., generates continuous 
flow structures on the space.

Similarly, for fixed $m$ one can define the spaces complete with respect
to the $m$-th semi-norm and their duals:
\bea
\CS^{(m )}_\pm & = \bigcap_{\beta \in\R_\pm} \CS^{(m)}_{(\beta )} \nonumber\\
\CS^{(m)\times}_\pm & = \bigcup_{\beta \in\R_\pm} \CS^{(m)\times}_{(\beta )} 
	\;\; .
\label{eq:munions}
\eea

Extending the standard procedure in an obvious way, we take the complete 
intersections and unions to form the orbits of the generators of the semi-groups
\begin{equation}
\CS_{\CG_\pm} = \bigcap_{m\ge 0} \bigcap_{\beta \in\R_\pm} 
		 \CS^{(m)}_{(\beta )}
	=\bigcap_{m\ge 0} \CS^{(m)} = \bigcap_{\beta\in\R_\pm}\CS_{(\beta )}
\label{eq:Sb}
\end{equation}
and their duals
\begin{equation}
\CS^\times_{\CG_\pm} = \bigcup_{m\ge 0} \bigcup_{\beta\in\R_\pm }   
	\CS^{(m)\times}_{(\beta )}= \bigcup_{m\ge 0} \CS^{(m)\times}
		= \bigcup_{\beta\in\R_\pm } \CS^\times_{(\beta)} 
\label{eq:Sbdual}
\end{equation}
thereby specializing the standard construction of $\CS$ and 
$\CS^\times$ to make use of the directional rather than ordinary 
derivatives.  The options regarding the choice of $\beta\in\R_\pm$ or
$\beta\in\R_\mp$ in the last expression should be regarded as still
fluid and not fixed.  Further definition will be given in
Section~\ref{sec:hardy}.  

Note that the full semi-group realizations exist 
as transformations on the complete space
$\CS_{\CG \pm}$, and that $\CS^\times_{\CG \pm}$ is made up of full 
orbits of realizations of the dual-semi-group transformations. 
The spaces $\CS_{\CG\pm}$ have been {\em completed}, and may 
possibly have more elements than a strictly infinitesimally generated  
semi-group might span in its orbit on the space.  For the
directional derivative construction of this section, as for the
standard construction preceding, it is the $m$-th semi-norm completions
which tell us   
that at most a countable number of points have been added to the span of
the orbits of the strictly infinitesimally generated semi-groups.  

The structure of the semi-groups follows into the representation here,
and so $\CS_{\CG +}$ and $\CS_{\CG -}$ may possibly
be two different linear structures 
(spaces) constructed over the same set, with differing notions of convergence 
and continuity.  For a 
representation of $g\in\mathfrak{g}$, even though a representation of its 
transformations is defined on all of $\CS$, the dense orbits of the 
exponential map of its representation which belong to
$\CS_{\mathfrak{g}^\C +}$ 
and to $\CS_{\mathfrak{g}^\C -}$ respectively trace out different continuity 
structures on 
the very same sub-set in $\CS_\g^\C$.  The ``torus 
action'' of $g\in\g$ on $\CS_\g^\C$ generates a compact set which 
can be thought of as having a toral 
structure; one may wind around the torus in only one direction 
in each of the disjoint continuity structures upon it.  
The second installment of this series will address further comments to
the maximal tori of complex groups. 

Use is now made of the standard theorems of Hille  
and Yosida on the properties of equicontinuous semi-groups of class $C_0$
on locally convex linear topological sequentially complete spaces to 
construct the real representation subspaces $\CS_{\CG \pm}  \subset \CS_{\R^n}
\subset C^\infty(\R^n)$.  The object is now to show
that
\begin{Thm}\label{thm:MAIN}
Any algebraic realization of the real algebra $\g$ 
of a real semi-simple Lie group $\CG$
in terms of polynomials and derivatives (or, equivalently, in terms of 
creation and destruction operators) can be used to generate a semi-group
representation on some subspace $\CS_{\CG+} \subseteq \CS_{\R^n}$.  
\end{Thm}
The key steps in the construction beyond those given are:

The family of mappings  on $\CS_{\R^n}$ are equi-continuous
semi-groups under the identification $\CG \ni g=e^{tB}
\Longleftrightarrow T_t$, since for
$\phi\in\CS_{\R^n}$:

\begin{enumerate}
\item{The polynomial realization of a generator $B$ is continuous 
and equi-continuous on $\CS_{\R^n}$.}
\item{$B^k$ is therefore continuous and equicontinuous on $\phi$: $B\phi$
is bounded on any $\CS^{(m)}$, so $\nor B\phi \nor_m $ is finite.  This 
means that $\nor B^n \phi \nor_m$ is also finite, and there must then
be some $(q-1)$ such that
\begin{equation}
\nor B^n\phi \nor_m\le (q-1)\nor \phi \nor_m\le \nor\phi\nor_q
\label{eq:bounded}
\end{equation}
by equation (\ref{eq:mnorm}) and the fact that $\nor \phi \nor_m$ is
bounded.}
\item{Therefore, $e^{tB}$ converges as a Cauchy sequence on $\CS_{\R^n}$, and 
also as a Cauchy sequence on the complete space $\CS_{\CG\pm}$.}
\item{Therefore, $\nor e^{tB}\phi \nor_m$ is bounded all $\phi\in\CS_{\R^n}$ 
and all $m$.  An argument identical to that leading to equation 
(\ref{eq:bounded}) shows equicontinuity in the identification 
$T_t\longleftrightarrow e^{tB}$.}
\end{enumerate} 

For additional generators, one merely adds additional  intersections 
to the construction of $\CS_{\CG +}$ and additional unions to the 
construction of $\CS^\times_{\CG +}$.  

The representation of 
equi-continous semi-groups of class $C_0$ defined this way provide 
an identification of the representation mapping $\theta :
\CG_\pm\longrightarrow Aut (\CS_{\R^n})$ which satisfies the definition of a 
representation on $\CS_{\R^n}$.  The same definition for a semi-group 
representation is used as is used for a group representation, except  
for mapping a semi-group rather than a group onto the automorphisms of a space. 
Since the two strictly infinitesimally generated semi-groups
$\CG_+$ and $\CG_-$ have the same generators, there is no need 
at present to 
distinguish the two semi-group representation mappings notationally.  

The fact that $\CS_{\CG\pm}\subset\CS_{\R^n}\subset C^\infty (\R^n )$ played 
a key role in 
showing convergence of the exponential mapping here.  To converge in 
semi-groups is a stronger statement that simple Cauchy convergence.
Ordinarily first derivatives
don't depend on direction for $C^\infty$ manifolds (e.g., 
$\delta x / \delta y = (-\delta x) /(-\delta y)$) where there is an
identification of the tangent space locally with the manifold itself.

Because $\CS^\times_{\CG_\pm}$ is 
formed by a union, it is presumably free to be a disjoint union, so that e.g., 
there is no mandate of fully group behavior there.  The real result of this 
theorem, in light of the later construction, is the definition of 
$\CS_{\CG_\pm}$ and $\CS^\times_{\CG_\pm}$ over half-spaces of real
coefficients of the exponential mapping, $\Om_\pm = 
\big\{ (a_1,a_2, \cdots , a_n) \mid a_i \gtrless 0 \big\}\subset \R^n$, 
which we call $\CS_{\CG\pm}$ and $\CS^\times_{\CG\pm}$. 
The original definition of $\CS_{\CG\pm}$ by equation 
(\ref{eq:Sb}), is defined as the non-empty ($\ni \{ 0\}$) intersection  of 
dense spaces.  Superficially, it would appear to always be possible to 
mathematically re-unite the two semi-groups of the above theorem into a group 
on $\CS_{\CG_\pm}$.  However, there are analyticity impediments in the 
dual space, which constrains continuity in $\CS_{\R^n}$, but note there also 
may be faithful reflection of physical impediments to doing so, such as the 
example of the ``quantum arrow of time'' due to causality (``a state must 
be prepared
before it can be observed'') which necessarily restricts the functions on 
$\F$ so that they are defined only over a restricted domain of time, resulting 
in semi-groups of time evolution which reflect the temporal
boundary conditions (i.e., causality) of a real physical
process~\cite{qat1,qat2,qat3,bgm,bmlg}.   

The result is the extension of (realizations of) 
operators $\{ A_i\}\in \CG$ defined 
on all of the set $\CS_{\CG}$ to (realizations of) $\{A^\times_i\}$ 
defined on $\CS^\times_{\CG}$, but which are not necessarily bounded on
$\CS^\times_{\CG}$.  This is because, e.g., the 
dual space of a subspace is larger than the dual of the space containing 
that subspace.  In 
the case of time evolution, this lapse in analyticity (boundedness) 
on $\F^\times_\pm$ was shown to result in semi-groups of time 
evolution on $\F^\times_\pm$. 
It can be shown that there is a general bifurcation of $\F_\pm$ and
$\FT_\pm$ according to semi-groups of evolution
along the lines of the initial and boundary conditions for both 
temporal--e.g., in-states during $t\le0$ or out-effects during $t\ge 0$--and 
dynamical--e.g., forming or decaying--evolution
of the solution set~\cite{qat1,qat2,qat3}.  

This leads to the very important corollaries about 
complexification of semi-groups and their
representations:
\begin{Cor}
Given an algebraic realization of a real semi-simple Lie algebra $\g$
in terms of polynomials and derivatives (or, equivalently, in terms of
creation and destruction operators), and the representation  
$\CS_{\CG\pm}\subseteq 
\CS_{\R^n}$ generated by this realization using transformations of
the form $exp (\g )$ according to Theorem 
\ref{thm:MAIN}, there also exists a representation of $\CG$ generated
by transformations of the form $exp (i\g)$ defined on dense subspaces
of $\CS_{\R^n}$.
\end{Cor}
and
\begin{Cor}
Given an algebraic realization of a real semi-simple Lie algebra $\g$ in terms 
of polynomials and derivatives (or, equivalently, in terms of creation and 
destruction operators), and the representation $\CS_{\CG\pm}\subseteq 
\CS_{\R^n}$ generated by this realization by the preceding Theorem 
\ref{thm:MAIN}, there also exists a representation of the orbit under the 
exponential map of the semi-simple complex Lie algebras $\g^\C_\pm$ 
(call it $\CS_{\CG^\C \pm} $) defined on dense subspaces of $\CS_{\R^n}$,  
\end{Cor}
Proof: Because $\CS_{\R^n}$ is closed under multiplication by complex scalars, 
if $B$ is bounded on a subspace of $\CS_{\R^n}$ so is $iB$, although not 
necessarily on the same subspace.  By an equivalent procedure, we can also
construct an equi-continuous semi-group of class $C_0$ on $\CS_{\R^n}$
using $e^{itB}=e^{t(iB)}$, $t\in\R^+$.  

$\CS_{\R^n}$ can contain a representation of the complexification of the
real semi algebra, or, equivalently, the complexification of the real 
semi-group, $\CG^\C_\pm$, so that 
$\CS_{\R^n}\supset \CS_{\CG^\C \pm}  \supseteq \CS_{\CG\pm}$.  The 
exponential maps can be thought of as convergent Cauchy sequences
in $\CS_{\R^n}$, and, because in the present context 
$exp$ is holomorphic and maps dense sets to dense sets,
$\CS_{\CG^\C\pm}$ is connected and locally path connected
(it may have non-zero curvature).  
Hence, $\CS_{\CG^\C \pm}$ is a representation space for the complex 
covering semi-group of the real algebra $\g_\pm$.  The
strictly infinitesimally generated $\CS_{\g^\C\pm}$ of the rest 
of this paper correspond to the complete spaces
$\CS_{\CG^\C \pm}$ of this subsection which may contain an additional
set of zero (Lebesque) measure.  See also
Section~\ref{sec:hardy} regarding the
extension of the energy spectrum from the half line to the whole
line. 

This construction for $\CS_{\CG_\pm}$ uses 
continuous operators for representing the generators of the locally compact 
group.  
The construction is in sub-domains of $C^\infty (\R^n )$.  The results 
are not inconsistent with the constructions for elliptic operators on $L^p$ 
spaces using the theorems of Langlands~\cite{vogan,robinson}; the use
of a more restrictive  
space gives convergence for more general operators, including hyperbolic and
parabolic realizations of the generators of the algebras.  Hyperbolic 
operators are an integral part of the analytic continuation of an elliptic 
operator problem [11][12], and the hyperbolicity of the dissipative oscillator 
Hamiltonian is perfectly in keeping with this general result.
Alternatively (as is relevant for our bundle discussions in installment
four~\cite{IV}), one can
continuously change the operators and their  
eigenfunctions simultaneously when one starts with an elliptic problem, such 
as the two free oscillators here.  See~\cite{reed}, Theorem 4.4.3 and
commentary following, and also see Section~\ref{sec:canonical}.
The familiar
eigenfunctions of the free oscillators are ``very well behaved'', so that 
the ``very well behaved'' requirement is satisfied in the present case by 
viewing the free oscillators as the elliptic starting point of the whole 
procedure: there is an elliptic ancestor for the operator and corresponding 
``very well behaved'' eigenfunctions for the spectral resolution, enabling 
all to be analytically continued.  See also the Appendix to
installment three~\cite{III}.

The definitions 
of the spaces $\CS^{(m)}_{(\beta )}$ used herein are loosely comparable to the 
definitions used in~\cite{vogan} based on the domains of powers of operators.  For 
notational simplicity, the ``$D$ of $D$'' notation required for 
specifying domains of 
directional derivatives was not used.  The $exp$ map is used, 
and the transparent device of restriction of the range of admissible 
parameters 
of the $exp$ map to obtain semi-groups provided convergence using the 
Hille-Yosida Theorem; Langland's theorems give the closure of infinite powers 
of elliptic operators on $L^p$ spaces, and the Hardy class spaces are 
subspaces of $L^p$ spaces, so that $\HH^2_\pm \subset L^2$.

We shall identify the function space realization of $\F_{\g\pm}$
($\CU^\pm\F_{\pm}$ of the Gadella diagrams of
Section ref{sec:structure}) with
$\CS_{\g\pm}$ in all that follows.  The necessary Hardy class
properties will be addressed in Section~\ref{sec:hardy}.






\subsection{Hardy class properties}\label{sec:hardy}

The algebra realizations, equations (\ref{eq:gen1}) to
(\ref{eq:gen10}), include antisymmetric (hyperbolic)
combinations of what nominally appear to be Bose creation and 
destruction operators.  Presumably, it is safe to anticipate that 
our construction may contain the span of direct sums of families 
of Bose oscillator number states, $\{ |n_A\rangle\}
\oplus \{ |n_B\rangle \}$.  This means that even though our
intersection $\CS\cap\HH^2_\pm |_{\R^+}$ is dense, it does not
necessarily follow that the orbit of our realizations of the
symplectic transformations must be interesting.  However, there are
several points that show that these hyperbolic operators have non
trivial orbits, and that our construction is in fact
physically interesting.

There are many ways of approaching this, all in harmony on this
point.  In chapter 2 of~\cite{dirackets}, the construction of a RHS
for the creation and destruction operators, for which both are
continuous bounded operators, is given.  It is also known that
hyperbolic problems emerge from the analytic continuation of elliptic
problems~\cite{garabedian,lewy}.  (See 
also the appendix to the second installment of these articles.)  Thus,
the transition from the middle to the bottom layer of the Gadella
diagrams of the first installment~\cite{I} is {\em always} 
accomplished by hyperbolic transformations--complex hyperbolic
transformations.  Ado's
theorem also assures us of the ability to create representations of
our relevant Lie algebras and associated (semi-)groups in terms of
creation and destruction operators~\cite{barut}.   
A partial key to understanding the non-triviality of the
present construction, then, seems to be to understand that we are
dealing with multicomponent component vectors, and that there
is in fact mixing between the two components by these hyperbolic
operators.  From~\cite{dirackets} 
we have dense component spaces which are 
part of our construction, and in
particular, we have states for which {\em both} the creation and
destruction operators are $\tf$-continuous and $\tf$-bounded operators.
Ado's theorem says we can mix those component spaces up using 
hyperbolic operators. 

The space constructed in the second chapter of~\cite{dirackets} for
which both creation and destruction operators are continuous and
bounded, and for which it follows the energy 
spectrum is unbounded from below, is both dense and non-trivial.  This
shows that the representations of the complexified algebras
obtained using spaces constructed by the procedures of
Section~\ref{sec:altconst} are dense and non-trivial, and the strictly
infinitesimally generated semi-group flows going from this initial base
space are dense and non-trivial also.

Other than identifying the starting point for the analytic continuation of 
as being predicated on our function space realizations  
lying in the intersection of the Schwartz and Hardy 
class functions, no additional mention has been made of the preservation of 
those Hardy class properties.  The procedure begins with a 
``very well behaved'' energy representation over the non-negative
physical energy spectrum, in $\left. \CS\cap \HH^2_\pm \right|_{\R_+}$ and
the dual $\left( \CS\cap \HH^2_\pm \mid_{\R_+} \right)^\times$.  The procedure
extends the energy spectrum to the whole line, and the space becomes $ \CS\cap 
\HH^2_\pm $ and the dual $\left( \CS\cap \HH^2_\pm  \right)^\times$. 
See~\cite{gadella,bohm1,bohm2}.

Hardy class functions from above and 
below, $\HH^2_\pm$, are subspaces of $\HH \cong L^2$. The construction
of Section~\ref{sec:altconst} shows how very well behaved quantum
oscillator eigenstates can be transformed to provide a representation
of $Sp(4,\C )^\R$.  The uniqueness of that extension is provided by
the well behaved nature of $exp$, which is holomorphic in the case at
hand.  Thus, the oscillator eigenstates clearly lie within the subspace
of the Schwartz space constructed in Section~\ref{sec:altconst} using
the creation-destruction operator realization of $\mathfrak{sp}(4,\R
)$ identified in Section~\ref{sec:scattering}.  
The $Sp(4,\C )^\R$ representation space obtained by the bootstrap
construction of Section~\ref{sec:altconst} will also lie in the Hardy
class functions if the resulting representations are defined in such a
way that the represented group actions are dynamical on the
space.  The spaces are quasi-invariant under the transformations in
this case (since they are dynamical as to the spaces themselves), 
and the Hardy class properties are preserved.  (One can
think of this as a fiber bundle, with each section along the fiber
being a Hardy space, and the Hardy class properties also existing
along a vertical fiber.  See also installment four~\cite{IV}.)

The esa ``extension'' of the generators of the realization of the Lie
algebra $\mathfrak{sp}(4,\R )^\C$ is just the result of inclusion, and
so exists.  The construction of the dual space in
equation(\ref{eq:Sbdual}) is the union of spaces over the parameter
$\beta\in\R_\pm$.  The only detail we need to address to provide the
dynamical family of product modules required to preserve the
Hardy class properties with the extension is to refine the $\pm$
choices available to $\beta$.  The choice
\begin{equation}
\CS^\times_{{\cal G}\pm} = \bigcup_{\beta\in\R_\mp} \CS^\times_{(\beta
)}
\label{eq:bdual-2}
\end{equation}
means that $\left( \CS_{\mathfrak{sp}(4,\C )^\R\pm}\cap\HH^2_\pm \right) \times
\left( \CS_{\mathfrak{sp}(4,\C )^\R\pm}\cap\HH^2_\pm \right)^\times$
will have sections reflecting the module structure of the scalar
product of the initial representation space for the creation
destruction operators.
Once this family of modules structure is defined, it is apparent that
the Hardy class properties are satisfied on each section and along
each fiber--in fact globally--and so the resulting 
representaion structure does in fact
lie within the intersection of the Schwartz and Hardy class functions,
as required by the Gadella-van Winter necessary and sufficient
conditions.  We have a series of holomorphic, symplectic and
integrable (hamiltonian) transformations deforming the initial simple
harmonic oscillator representation spaces, and the resulting global 
structure is ``very well
behaved'', just as the initial structure was.

More subtle and not covered 
explicitly here is the extension of the energy spectrum until it is 
unbounded below: see, e.g., Section~\ref{sec:altconst} and
Chapter 2 of~\cite{dirackets} for proof that in a rigged 
Hilbert space both the creation and destruction operators can be continuous 
operators, and in such a way that the energy spectrum can be unbounded
below.  Note that in
our Section~\ref{sec:scattering} both $a$ and $a^\dagger$ have effectively
been given esa extensions (!), so the energy spectrum is no longer
bounded from below.  Extension of the representation space
$\CS_{\g\pm}\cap\HH^2_\mp |_{\R^+}$ to
$\CS_{\g\pm}\cap\HH^2_\mp$ results from analytic continuation in energy, so
that $\g_\pm \longrightarrow \g^\C_\pm$, corresponding to the continuous
deformation of the contour of integration in~\cite{bohm1,bohm2}.  

Another subtle point is that although complex structures are features of both 
the representation space and its dual, the {\em{continuous}}
extension process of analytic continuation (e.g., a process by which
the contour of integration is continuously deformed) proceeds necessarily
from representation space to dual.  This reflects the fact that 
the momentum map of a Lie algebra $\g$ belongs to its dual, 
$\g^\times$, and as is explicit in the work of 
Gadella~\cite{gadella}, and implicit in the work of H{\"o}rmander 
showing the continuation 
of fundamental solutions is to the distributions~\cite{hormander}.  Also, the pure 
exponential decay of the Gamow vectors says they probably
should not be regarded as 
elements of any Hilbert space~\cite{note22}.  Hence, one must take this as mandating a 
change of topology during the continuous 
extension, $\HH^\times \longrightarrow \FT_\pm$,
notwithstanding any possible preservation of the $L^2$ scalar product from a 
transformed vector and its transformed dual.  Proper definition
of the group scalar product on the representation structures insures
that the extensions are quasi-isometric.  This quasi-isometric extension
of the group product to the group product of the complex covering group 
is the reason there
is a quasi-invariant measure equivalent to the invariant (e.g., Lebesque)
measure rather than there being simply another invariant measure.  See 
Section~\ref{sec:scattering}.








\section{Predicted Observables}\label{sec:observables}

There are several general areas in which possible observable consequences of 
this model may emerge, and these will be pursued in detail separately from 
this paper.  The most obvious is the quantization of the width (or, 
equivalently, the decay rate) as characterizing the resonances described 
here.  Widths are usually difficult to measure, so the actual experimental 
resolution of this effect may not be easy.  Note, however, that the $Z$ 
eigenvalue appearing in the width is also a characteristic of the 
$|j,m\rangle$ prior to analytic 
continuation.  The realization of the $SU(1,1)$ algebra used herein (and also 
the $SU(2)$ algebra) has an association with interferometry in quantum 
optics~\cite{yurke}, and perhaps sufficient sensitivity can be attained by the
use of non-linear optical phenomena, such as four wave mixing, etc.  
The indicated decay rate is inversely
proportional to the total energy in the system (including vacuum zero
point energy) and independent of the energy gap between the two
oscillators (or fields). 

Because the
energy is analytically continued to include values from the negative real
axis, the excitation numbers of the modes can be negative.  (Recall that
$m=n_a +n_B$ is a characteristic quantum number
of the eigenstates of $SU(1,1)$ prior to the
analytic continuation which produced the Gamow states.)  For vacuum states,
$n_A +n_B =m =0$, there is still a finite decay probability for the
process transferring energy from oscillator $A$ to oscillator $B$.  This
provides an explicit representation of, e.g., the well known vacuum
fluctuations of quantum optics.   

The present work also implies the possibility of observing quantization
effects in the rate of decay of any two field 
resonant decay processes.  If one interprets the two field
interferometry constructions of~\cite{yurke} quite literally, then at the
appropriate energy scales quite general combinations of fields 
should show analogues to the familiar
interferometry process in quantum optics.  Thus, one infers that there
may be electroweak analogues to Fabry-Perrot interferometers, four
wave mixing, etc., at the electroweak unification scale of energy.    

It is also predicted that the characterization of an irreversible process 
can take a particular form: pure exponential decay.  In the derivation of 
the complex spectral resolution, Bohm obtained
two terms~\cite{bohm1,bohm2}, rather than the single sum over Gamow vectors obtained here.
Thus, for the prepared state $\phi^+$, one has formally
\begin{equation}
\phi^+ = \sum^n_{i=1} \psi^G_i \langle \ts^G_i | \phi^+\rangle	+ 
\int^{-\infty_{II}}_0  dE \;\; |E^-\rangle	\langle^+E|\phi^+ \rangle  
\;\; .
\label{eq:spectresolution}
\end{equation}
The first term on the right hand side corresponds to the result of the
present paper.  The second term on the right hand side (sometimes called
the ``background term'') has the geometric meaning of an holonomy 
contribution: there is no observable holonomy in the present two oscillator 
system since the evolution does {\em not} generate a 
trajectory which closes in the curved state space.  The 
plain geometric implication is that even if one should 
manage to return to what might be called the initial configuration, the various
symmetry operations representing the necessary preparation steps and 
the necessary evolution steps will be 
by geodesic transport over curved space and one expects that some geometric 
phase (holonomy) will be accumulated, making the ``background term'' no 
longer zero.  Indeed, there may possibly be some general identification
between simple irreversible dynamical time evolution and holonomy.

It is to be expected that careful examination of the 
``background term'' should disclose the role of the history of the system 
(non-Markovian dynamical time evolution--discussed in installment
three~\cite{III}) in the irreversibility of the system  as expressed by
the semigroups of evolution of that system.  One is thus  
led to expect deviations from exponential decay only in systems which
have a history, e.g., correlations among events between subsystems.  The
formalism predicts, e.g., that ``spontaneous'' decay should be
exponential, the result of some stochastic external perturbation
(see~\cite{III}), and further one might observe 
deviations from the exponential when there exists correlations
(coherence) between the
decay events.  This agrees with physical intuitions, and 
the principles seem to have weak (non-quantitative)
experimental support.  See~\cite{note22}.

The gauge interpretation of the symplectic transformations of this
paper which will be undertaken in the fourth installment of this
series should also have physical observables~\cite{IV}.

Finally, in resonant quantum microsystems, geometric phase (holonomy)
accumulating during system evolution seems inextricably associated
with intrinsic microphysical irreversibility.  For resonant quantum
quantum microsystems, there is perhaps some intrinsic association
between the accumulation of geometric phase (holonomy) and entropy
growth.  










\appendix
\section{The $SU(1,1)$ Dissipative Oscillator}\label{sec:DHO}

This section is a partial recapitulation of the $SU(1,1)$ dissipative
oscillator, first described in the paper of Feshbach and Tikochinsky~\cite{ft}.  In 
addition to changes in emphasis and mathematical detail, in the present work 
there are changes of form, e.g., only operators 
which are ``symmetric under time reversal'' will be used
(contrary to the original).  From these 
operators which are time reversal symmetric, irreversible time evolution will 
emerge in the form of exponentially decaying Gamow vectors.  

The equation of motion of the damped simple oscillator is:
\begin{equation}
m{\ddot{x}} + R \dot{x}+kx=0
\end{equation}
Canonical quantization requires a Lagrangian, and the appropriate Lagrangian 
requires an auxiliary variable $y$:
\begin{equation}
L=m\dot{x}\dot{y} +\frac{1}{2} R(x\dot{y}-\dot{x}y)-kxy
\end{equation} 
The equation of motion for $y$ becomes:
\begin{equation}
m\ddot{y}-R\dot{y}+ky=0
\end{equation}
The Hamiltonian $H$ is
\begin{equation}
H=\frac{1}{ m} p_xp_y + \frac{R}{2m} (yp_y-xp_x) + 
	\bigl( k - \frac{R^2}{2m} \bigr) xy
\label{eq:Hint}
\end{equation}
The $[a,\,a^\dagger ]=\II =[b,\,b^\dagger ],\;[a,\, b]=0$, etc. commutation 
relations are equivalent to the $[p,\,q]=-i\hbar\,\II$, etc., Heisenberg 
relations, and canonical quantization is straightforward. After canonical 
quantization and translation of variables into destruction and creation 
operators for two modes, $a=(\sqrt{m} \Om x+ip_x/\sqrt{m})/\sqrt{2\Om}$,and 
$b=(\sqrt{m} \Om y+ip_y/\sqrt{m})/\sqrt{2\Om}$, rearrangement suggests the 
further change of variables:
\begin{equation}
a=\frac{1}{\sqrt{2}} (A+B)\;\qquad b = \frac{1}{\sqrt{2}} (A-B)
\end{equation}  

We then have a splitting of the Hamiltonian
\begin{equation}
H=H_0+H_1     
\label{eq:ham1}
\end{equation}
where
\begin{equation}
H_0= \Om (A^\dagger A-B^\dagger B) \qquad\qquad \Omega =k-\frac{R^2}{2m}    
 \label{eq:ham2}
\end{equation}
and
\begin{equation}
H_1=i\frac{\G}{ 2} (A^\dagger B^\dagger -AB)\qquad \qquad \G =\frac{R}{m}     
\label{eq:ham3}
\end{equation}
There is a natural system--reservoir interpretation of the above, and in the 
limit $R\rightarrow 0$ the eigenstates of $H$ reduce to those of the simple 
undamped harmonic oscillator provided one considers only states annihilated 
by $B$~\cite{ft}. 

The operator $H_0$ is simply related to the ${\cal C}^2$ Casimir 
of $SU(1,1)$.  The operator $H_1$ together with two other operators form a 
realization of the algebra of $SU(1,1)$, useful in determining the 
eigenvalues of the Hamiltonian:
\begin{equation}
iX=\frac{1}{2} (A^\dagger B^\dagger +AB) 
\label{eq:su11x}
\end{equation}
\begin{equation}
iY=\frac{i}{2}\; (A^\dagger B^\dagger -AB) 
\label{eq:su11y}
\end{equation}
\begin{equation}
iZ=\frac{1}{2} (A^\dagger A +BB^\dagger  )  
\label{eq:su11z}
\end{equation}
{obeying the algebra}
\begin{equation}
[X,Y]=Z
\label{eq:xyz}
\end{equation}
\begin{equation}
[Z,Y]=X
\label{eq:zyx}
\end{equation}
\begin{equation}
[X,Z]=Y
\label{eq:xzy}
\end{equation}

$iZ$ is essentially the Hamiltonian for the two mode simple oscillator, with 
eigenvalues $2m+{1}$, $m=({1/2}) (n_A +n_B)$, while we may label the 
eigenvalues of $H_0$ by $2\Om j$, where $j={1/2}\, (n_A-n_B)$.  

The Baker-Campbell-Hausdorf relation can 
be applied to $B = i\mu X\in SU(1,1)$, $A=Y \in SU(1,1)$, $\mu\in\R$, to yield:
\begin{equation}
e^{i\mu X}iYe^{-i\mu X}=iY\cos {\mu} - [X,Y] \sin {\mu}.
\end{equation}
For the semi-simple (e.g., non-solvable) group $SU(1,1)$and its semi-simple 
algebra $\mathfrak{su}(1,1)$, it follows that:
\begin{equation}
e^{i\mu X}iYe^{-i\mu X}=iY\cos \mu -Z \sin \mu
\label{eq:adjoint2}
\end{equation}
so that
\begin{equation}
e^{i({\pi}/{2}) X}iYe^{-i({\pi}/{2}) X}=-Z 
\end{equation}
or
\begin{equation}
Y=i\;e^{-i({\pi}/{2}) X}\, Z\, e^{i({\pi}/{2}) X}\qquad 
	Z=-i\;e^{i({\pi}/{2}) X}\,Y\,e^{-i({\pi}/{2}) X}   
\label{eq:F-T2}
\end{equation} 

Because the non-unitary dynamical transformations (analytic continuation)
work a complexification of the algebra 
$\mathfrak{su}(1,1) \longrightarrow \mathfrak{su}(1,1)^{\C}= 
\mathfrak{sl}(2,\C)$, the adjoint transformations of $(iY)$ exit 
the realization 
of $\mathfrak{su}(1,1)$ to become a realization of $\mathfrak{sl}(2,\C )$; 
the transformed eigenvectors have left the representation space for 
$\mathfrak{su}(1,1)_\pm$ into a representation space of 
$\mathfrak{su}(1,1)^{\C}_\pm= \mathfrak{sl}(2,\C )_\pm$ as well.  $H_0$ 
of the Hamiltonian is in fact proportional to the angular momentum operator 
$J^2$ of $\mathfrak{su}(2)$, and there is also 
the ``dangerous'' so-called analytic continuation of
$\mathfrak{su}(2)\longrightarrow \mathfrak{su}(1,1)$ given by 
$J_1\mapsto iJ_1=X$, 
$J_2\mapsto iJ_2=Y$, and $J_3 \mapsto Z$, so that $Z$ can also be thought of 
as an $\mathfrak{su}(2)$ generator, and thereby the source of a 
discrete spectrum; because $Z\in \mathfrak{sl}(2,\C )$, so is $iZ$, and 
hence the appropriateness of 
complex eigenvalues for $iZ$ on the eigenvector of $J_3$.  The complex 
eigenvalue is appropriate in $\mathfrak{sl}(2,\C )$ because it is a complex 
algebra.  

If the eigenstates of $Z$ are $|j,m\rangle\in \CS$, the Schwartz space, as 
above, the eigenstates of $Y$ resulting from the extension of these vectors 
to $\F^\times$ using the $Ad_{exp(\mu X)}$ map corresponding to 
$\mu=-{\pi}/{2}$ are $e^{-i({\pi}/{2})X}\,|j,m\rangle$:
\begin{equation}
\left( iY\right) \,e^{-i({\pi}/{2})X}\,|j,m\rangle=i\,(m+{1}/{2} )\,
	e^{-i({\pi}/{2})X}  \,|j,m\rangle 
\label{eq:la}
\end{equation}
Because $m\ge 0$, this is a positive pure imaginary eigenvalue.  There is 
also a negative pure imaginary eigenvalue corresponding to an eigenstate 
$e^{i({\pi}/{2})X}\,|j,m\rangle$ associated with $\mu=+i{\pi}/{2}$:
\begin{equation}
\left( i Y\right) \,e^{i({\pi}/{2})X}\,|j,m\rangle=	
	-i\,(m+{1}/{2} )\,e^{i({\pi}/{2})X}\,  |j,m\rangle 
\label{eq:lastar}
\end{equation}

Since $H_1=\G Y$, we therewith have complex eigenvalues for the Hamiltonian $H=
H_0+H_1$, and the eigenvectors of $H$ are in fact Gamow vectors (belonging to 
$\F^\times$~\cite{dirackets,bohm1,bohm2,gadella}) which exhibit pure exponential growth or
decay, depending on the sign of $\pm m$:
\begin{equation}
\psi^{G\;\pm}_{j,m}(t)=e^{\mp i({\pi}/{2})X}\,|j,m\rangle \,
	e^{-2i\Om jt\pm (\G /2)(2m+1)t}
\label{eq:gamow}
\end{equation}
Because $iY\sim H_1$ is in the familiar form of
a symmetric (hermitean) operator on Hilbert space, 
$e^{\pm i(\pi /2)X}\,|j,m \rangle$ cannot belong to the Hilbert space 
$\HH$ domain of the hermitean operator $H_1$, as proven by these complex 
eigenvalues of the hermitean $H_1 = iY$.  (One of the lessons of the
main body of this paper is that these Gamow vectors are associated
with semigroups of time evolution, with restricted time domains of
definition.  Thus, $\psi^{G+}_{j,m}\equiv \ts^G$ is defined only for
$t\le 0$ and $\psi^{G-}_{j,m}\equiv \psi^G$ is defined only for $t\ge
0$.) 

 These complex eigenvalues are 
totally unacceptable in the real algebra of a real group such as $SU(1,1)$, or 
in a representation of the same.  This is because $i \, \la \II \notin 
\mathfrak{su}(1,1)$, since the real algebra (and its associated group) 
is defined over the field of real scalars only.  






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