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\begin{document}

\title{{Generalized Wavefunctions for  \\
Correlated Quantum Oscillators I: \\
Basic Formalism and Classical Antecedants.}}

\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 first of a series of four articles, it is shown how a
hamiltonian quantum dynamics can be formulated based on a
generalization of classical probability theory using the
notion of quasi-invariant measures on the
classical phase space, based on
distributions rather than invariant Gibbs measures.  The 
first quantization is by functorial analytic
continuation of real probability amplitudes, effecting the
introduction of correlation between otherwise
independent subsystems, and whose physical consequence is the
incorporation of Breit-Wigner resonances associated to Gamow vectors into our
description of dynamics.  The resulting quantum
distribution density dynamics has a natural field theory
interpretation, the subject for future installments of this series.   

\end{abstract}



\pacs{11.10Ef, 3.65Ca, 3.65Db      }        %use showpacs class option

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\maketitle




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


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


This is the first installment of a four installment series of papers
describing various aspects of a hamiltonian description of quantum
fields~\cite{II,III,IV}.  We will create a field theory over
canonical position and momentum (phase space) variables, eventually
adopting the familiar creation and destruction operator formalism when
we deal with the dynamics of correlated (i.e., interacting)
fields.  We will incorporate Schr{\"o}dingers equation into the
theory, but
extend the allowable solutions to the Schr{\"o}dinger equation to
include weak (distributional) solutions.  This extension of the
allowable solutions necessarily involves mathematical developments
subsequent to von Neumann's Hilbert space (HS) formulation of quantum
mechanics~\cite{vonN}.  Herein, we will avail ourselves of a variant of the rigged
Hilbert space (RHS) formalism of Bohm and
Gadella~\cite{dirackets},
which is a natural generalization of the Hilbert
space formalism of von Neumann that accords with this enlarged
solution set, and which retains the HS formalism of
von Neumann for the description of stationary states.  (See the
review~\cite{bgm}.)  

The present series of papers is focused on four aspects of enlarging
the RHS formalism of Bohm and Gadella:
\begin{itemize}
\item{Adding solutions to concrete problems (which cannot even be
respectably addressed in the conventional HS formalism) to the RHS's
formal achievements in the area of intrinsic irreversibility
(including intrinsic irreversibility on the microphysical
level~\cite{qat1,qat2,qat3}).}
\item{Demonstrating how it is possible using our variant of
the RHS quantum formalisms
to incorporate genuine mathematical hyperbolicity into quantum
dynamics.  This leads to notions of quantum dynamics which have
attributes analogous to the non-linear dynamics, chaos, fractals,
etc., of the contemporary dynamical systems literature.  This is
categorically incompatible with von Neumann's HS formalism.}
\item{The variant of the RHS formalism which we adopt admits a field
theory interpretation, and from the dynamics alone of 2, 3 and 4
correlated fields respectively.}
\item{We will require a complex plane structure in our ring of
  scalars, making the algebraic field $\C$ unsuitable for technical
  mathematical reasons, including uniqueness.  This will be covered in
  length in installment two~\cite{II}.} 
\end{itemize}
The concrete problem addressed
is that of describing the dynamical evolution of
coupled (correlated) 
quantum oscillators, which physically is intrinsically
interesting.  This is a first description of this problem, and 
there are some quite deep conceptual concerns we will
touch on--we will try to illuminate many of these, but there is no
pretense of complete and definitive resolution on many key points.
Certainly there is much of at least formal interest, and much
structure will be revealed which is physically illuminating even if
one does not take the RHS picture to heart.  Whether
or not our developments are physically correct will have to await
later judgment, but our conclusions are largely generic, with a very
strong element of first principle deduction independent of the
physical 
makeup of the fields or the nature of their interaction.  For
instance, we will ultimately conclude that the gauge group of the
Standard Model is very similar to the most general gauge structure one 
might expect for three fields, independent of the precise nature of
those fields, but that the true gauge group must be exact and
without any spontaneous symmetry breaking.  If nature does not observe     
this exact gauge  structure, then this suggests
some special or non-generic physics is present, the possible
existence of more than three fields, etc.

This first installment is concerned with establishing that a well
defined mathematical formalism exists, and that a field theoretic
interpretation (for correlated fields) is legitimate.  The second
installment will demonstrate the mathematical steps which must be
taken in order to have mathematically well grounded complex spectra:
there are quite well defined points of departure from the usual
Hilbert space quantum formalism and also from the rigged Hilbert space
(distributional) generalization, although the rigged Hilbert space
(RHS) formalism will be parallelized in many regards in order to
accommodate the complex energies of Breit-Wigner resonances by
mathematically well defined means.  The third installment illustrates
how hyperbolic dynamics may be present in the field theory so
formulated--the exponential decay of Breit-Wigner resonances is an
example of exponential separation of trajectories, and may be
associated with an increase in entropy.  The fourth installment ties
up a lot of mathematical loose ends by defining a unique covering
structure in which our constructions are mathematically well defined,
and demonstrating how a uniquely defined gauge theory may be extracted
from dynamics alone.  This results in a largely generic construction,
the gauge group being dependent only on the number of fields , and
recovers an exact gauge group in all cases.  It will be shown that the
gauge structure for four fields over canonical variables may be
associated with structures in a non-trivial space-time with $(+,-,-,-)$
local signature.
There is an association of hamiltonian dynamics of canonical
variables, structures on non-trivial space-time, and possibly even to
irreps of the inhomogeneous Lorentz group (including possible 
dynamical links to PCT). 







\subsection{The present paper}\label{sec:present}

The canonical transformations of physics are the symplectic
transformations of mathematics.  In the theory of dynamical systems,
these symplectic transformations are the dynamical transformations.
See~\cite{souriau}.  Classically, dynamical evolution is associated with the
translation and reshaping of a region in phase space to which a
dynamical system is localized, and this dynamical evolution is by 
symplectic (e.g., ``area preserving'') maps.  

We will use a 
sophisticated mathematical structure herein, whose
implications we begin to relate in Section~\ref{sec:structure}.  For
instance, we can replace the classical description of a dynamical
system as a point in phase space with a more loosely localized
region in phase space to which
our dynamical system is localized by a ``bump function'' (function of
rapid decrease, e.g., a generalized gaussian) which is
properly normalized so that it may be regarded as a probability
amplitude.  We can then provide a similar ``bump function'' to represent
sensitivities and capture probabilities, etc., of a measurement
apparatus as another probability amplitude. (We might also use a
more exotic generalized function, or distribution, rather than 
a simple function here.)  There will be
symplectic (=dynamical) transformations on the function space made up
of all the mathematically acceptable ``bump functions,'' provided we
give this function space a symplectic structure during its construction.   
Those symplectic transformations should
represent dynamical evolution of these classical (i.e., {\em real})
probability amplitudes within the context of a classical probability
theory in which the outcome of a measurement is predicted by the
overlap of probability amplitudes, and this overlap should be able to
evolve with time: we are able to represent the time dependent
interaction of a non-stationary dynamical system with a non-stationary
measurement apparatus using this formalism.  We will defer
explicit  consideration of the
symplectic transformations until the second installment, and
concentrate on how the nature of the spaces involved lends itself to a
probability theory in this initial installment.  For us, probabilities
will be an intrinsic part of the construction, and not a subsequent
interpretation.

The analytic continuation--a complex symplectic transformation--of just
such a probabilistic description starting with {\em real} probability
amplitudes is mathematically equivalent to representation of quantum
resonances, including Breit-Wigner resonance poles (of the analytically
continued S-matrix) belonging to one of the spaces of a Gel'fand
triplet in our RHS formalism. Additionally, this accords--albeit
loosely--with the physical 
interpretation of the RHS formalism in~\cite{bgm}.  See
Section~\ref{sec:classical}. 

Our quantum theory will be obtained then by a functorial analytic
continuation of classical densities on phase space, and our
measurement apparatus throughout will be associated with distributions 
mathematically dual to those densities.  








\section{Structure of the Spaces Involved}\label{sec:structure}

There has been a recent revival of interest in 
working in a quantum paradigm involving use of generalized
wave functions in a rigged Hilbert space (RHS, which will be indicated
generically by the Gel'fand triplet of spaces $\rhs$).  This RHS
formalism involves applying
a mathematically rigorous methodology of analytic continuation, 
and leads to the unification of the vector space representation of 
quantum systems and the representation of resonances by poles of the 
analytically continued S-matrix.    In the RHS paradigm, 
resonances are represented by Gamow vectors associated to Breit-Wigner 
poles of the analytically continued S-matrix, with exponential decay, 
and the time evolution is by semi-groups.  These semi-groups provide the 
formalism with the ability to express the 
boundary conditions of an irreversible physical process without
regeneration \cite{qat1,qat2,qat3}.  

We follow the lead of Gadella~\cite{gadella}, who
refined the analysis issues in the function spaces relevant to our
work, 
and showed how to obtain a rigorous analytic continuation 
based on necessary and sufficient conditions.  Our concern, then, is
erecting particular geometric structures in the types
of spaces Gadella has shown us we must use.
We begin by very tersely describing the
interrelationships of the spaces involved in the RHS formalism, and
also providing references for further inquiry.  The RHS formalism
involves working with a hierarchy of triplets of spaces, with abstract
spaces and function space realizations of the abstract
spaces.

Distinguishing abstract spaces from their function space
realizations, and the analytic continuation of those spaces, 
is an essential, if tedious part of the RHS methodology.
This is physically
motivated in part.  There is considerable structure in the large
number of vector spaces involved, and there is a significantly
different physical content to each of the spaces: there is a
difference between the abstract Hilbert space ($\HH$) and its $L^2$
function space realization in the energy representation on the
half-line, $L^2[0,\infty
)$, and the space corresponding to the abstract $\HH$ realized
in a subspace of $L^2 (\R )$, also in the energy 
representation.  Whether or not the energy spectrum is bounded from
below has considerable practical physical implication 
as well as formal mathematical importance.

Formally, the RHS paradigm involves alternative topological
completions of a 
pre-Hilbert space, $\Psi$, to form the Gel'fand triplet of spaces
$\rhs$.  In the $\tau_\Phi$ topology set by a countable family of
semi-norms we have one topological completion to form $\Phi$.  Using
the scalar product to define a norm, we define a Hilbert space
topology, $\tau_\HH$, and the completion of $\Psi$ in this topology is
a Hilbert space $\HH$~\cite{note0x5}.  Dual to $\Phi$ is the conjugate
space $\Phi^\times$, which has a weak-dual topology, $\tau^\times$.
The resonances in the RHS formalism are obtained by analytic
continuation, which proceeds from
$\HH^\times\longrightarrow\tilde\Phi^\times$ \cite{gadella}, where
$\HH^\times\cong\HH$ (Riesz isomorphism), and $\tilde\Phi^\times$ is
the extension of $\Phi^\times$.  There are complex energy eigenvalues
on $\tilde\Phi^\times$, indicating exponential decay for quantum
systems represented by states there.  In terms of (mathematically
respectable) physical content, in the RHS formalism we have the
analytically continued S-matrix (and its poles), the Lippman-Schwinger
equation, M{\/o}ller operators, etc.  Although we will not deal with
these explicitly, in installment two~\cite{II} we will effectively see
M{\/o}ller wave operators as dynamical (=symplectic) transformations,
an algebraic Lippman-Schwinger equation, etc.  As a point of
distinction from prior work, previously the analytic continuation
(complex extension) was of the absolutely continuous scattering
spectrum, while the present work operates by complex extension (complex
symplectic transformation) of an operator algebra in which the only
spectrum initially apparent is discrete.  see installment
two~\cite{II}.  Although we will not be
dragging contours around in the complex plane, making explicit use of
the residue theorem, etc., nothing we will do will change the
necessary and sufficient conditions for the complex extension~\cite{gadella}
, or any of the other major structural features of the RHS
formalism.  (See the review~\cite{bgm} for more
details than it is possible to recapitulate here.)


The basic structure of the relationships of the abstract spaces and the
associated function spaces which occur in the RHS formalism are most
easily seen in the Gadella diagrams~\cite{bgm,bmlg}.  Conventionally one
envisions a measurement process having the
preparation procedure ending at time $t=0$, with preparation of the
in-state $\phi^+$ occurring during $t\le 0$ and observation of the effect
$\psi^-$ during $t\ge 0$.  The Gadella diagram for representing
the preparation of an in-state $\phi^+$ during $t\le 0$ is
\begin{equation}
\begin{CD}
{\phi^+\in}\;\; @.\F_-\subset@.\HH\subset@.(\F_-)^\times@.
{\ni{\ts^G}}\\
@.        @V\CU^- VV  @VV\CU^- V   @VV(\CU^-)^\times V@.      \\
{\<^+E|\phi^+\rangle\in}\qquad@.\CS\cap\HH^2_- \Bigm|_{\R^+}\subset@.
L^2[0,\infty)\subset@.
\Bigl(\CS\cap\HH^2_- \Bigm|_{\R^+}\Bigr)^\times\qquad@.      \\
@.        @V(\Theta_-)^{-1} VV  @.    @VV(\Theta_-^\times)^{-1} V@.   \\
        @.\CS\cap\HH^2_-\subset@.\HH^2_-\subset@.(\CS\cap\HH^2_-)^\times@.
{\ni -i\sqrt{\frac{\G}{2\pi }} {\frac{1} {E-z^\ast_R}} }
\label{eq:gadella-in}
\end{CD}
\nonumber
\end{equation}
and the corresponding diagram for the effect $\psi^-$ 
observed during $t\ge 0$ is
\begin{equation}
\begin{CD}
{\psi^-\in} \;\; @.\F_+\subset@.\HH\subset@.(\F_+)^\times@.
{\quad}{\ni \psi^G}\\
@.          @V\CU^+ VV   @VV\CU^+ V     @VV(\CU^+)^\times V@.\\
{\<^-E|\psi^-\rangle\in}\qquad@.\CS\cap\HH^2_+ \Bigm|_{\R^+}\subset@.
L^2[0,\infty)\subset@.\Bigl(\CS\cap\HH^2_+\Bigm|_{\R^+}\Bigr)^\times@.\\
@.      @V(\Theta_+)^{-1} VV @.       @VV(\Theta_+^\times)^{-1} V@.\\
       @.\CS\cap\HH^2_+\subset@.\HH^2_+\subset@.(\CS\cap\HH^2_+\,)^\times@.
{\ni i\sqrt{\frac{\G}{2\pi }} {\frac{1}{E-z_R}}}
\label{eq:gadella-out}
\end{CD}
\nonumber
\end{equation}
The basic ideas behind these stem from~\cite{gadella}; it or~\cite{dirackets} should be
consulted for details of construction for the various spaces and 
mappings, as well as the careful definition and meaning of the various vectors
identified in the left and right hand margins.  Each
level of each diagram contains a RHS or Gel'fand triplet of spaces,
and there are certain properties which each space inherits by virtue
of its relative position in a Gel'fand triplet.

Along the top line of each diagram are abstract spaces, the middle
line gives the function space realizations of the spaces in the top
line conforming to the necessary and sufficient mathematical conditions
for performing analytic continuation in a
unique fashion (energy picture, physical energy bounded from below
with boundary conventionally taken as $E=0$), and the bottom line
shows the function spaces resulting from the analytic continuation~\cite{gadella}.
The lack of connecting link between the $L^2 (\R_+)$ realization of $\HH$
on the middle level and the subspace of
$L^2 (\R )$ on the lower level reflects the
lack of unique extension between these spaces.

Considering only the top diagram briefly, we have an abstract
(exponential formation culminating at $t=0$) Gamow
vector $\ts^G$ belonging to the rightmost abstract space $(\F
)^\times$ on the top line being associated to a Breit-Wigner 
resonance pole in the the rightmost
function spaces of the lowest line, which has a complex energy
eigenvalue $z^\ast_R =
E_R + i\Gamma /2$.  As a practical matter, the physically 
prepared abstract state denoted $\phi^+$
is input into this hierarchy as a ``very well behaved'' element of the
function space lying in the intersection of the Schwarz and Hardy
class functions (from below) over the half line 
$\CS\cap\HH^2_- \Bigm|_{\R^+}$, e.g., as a physically determined
``very well behaved'' energy
distribution of a beam actually prepared in an accelerator $\langle^+ E| 
\phi^+ \rangle = \phi^+ (E)$.  The $\R_+$ indicated with the spaces
corresponds to the physical energy
spectrum, which is bounded from below.  The abstract
prepared state $\phi^+$ ($+$ superscript physical notation) is an
represented by an element of the
space of Hardy class functions over the half line from below, $\HH^2_-
\Bigm|_{\R_+}$ ($-$ subscript mathematical notation).  

The second diagram for $t\ge o$ is similar, with an abstract Gamow
vector of pure exponential decay associated with a Breit-Wigner
resonance pole whose complex energy is  $z_R =
E_R - i\Gamma /2$, etc.

In the sequel, whether the given use of 
a rigged Hilbert space $\rhs$ is intended generically or as a
particular Gel'fand triplet of spaces often
will depend on the context of use.  The spaces usually will be indicated as 
$\F$ or as $\F_{\g\pm}$, or as $\subset \CS\cap\HH^2_\pm$, etc., 
depending on whether one is concerned with a generic rigged Hilbert space
structure (abstract or function space realized), 
its Lie algebra representation structure, or the space of 
``very well behaved'' vectors in the 
intersection of the Schwartz and Hardy class functions (from above or
below).   









\section{A classical system extended}\label{sec:classical}

For a real elliptic Hamiltonian (which the analytic continuation 
procedure must start with)--such as the Hamiltonian for two free
oscillators--there exist {\em real} eigenfunctions.  Indeed, 
this is the way
the function space realizations of the vectors of quantum physics are
usually encountered the first time, and it is only later 
that issues of time evolution and phase freedom for these special 
functions are discussed in detail.  From a
mathematical perspective, it would be reasonable to consider the
abstract spaces $\F_\pm$as real, and to also consider only the purely
real subspaces of the
associated (complex) function spaces identified in the preceding Gadella
diagrams.  See also~\cite{note10}.  

In order to understand some aspects of the present treatment of this
analytic continuation of a Lie algebra representation, it is useful to
adopt an idiosynchratic perspective, and to consider extending the
real representation of the real algebra to a representation of its
complexification, both in terms of the abstract representation space
of a group whose Lie algebra is $\g$,$\F_{\g\pm}$, and the proper
function  space realization lying in the
intersection of the Schwarz and Hardy class functions.  At first, this
must be 
regarded as idiosynchratic since the exponential map need not be a
unitary transformation on a real Hilbert space if one follows
the von Neumann model of construction, so that 
probabilities appear not to be Noetherian conserved quantities of the 
evolutionary flow there, etc.  This associated 
{\em real} Hilbert space may not seem very interesting
physically at first, but will offer more promise once it is clear we use
a different Hilbert space than the one von Neumann constructs .  (The
elementary constructions of 
the appendix to the second installment of these articles also
make use of a structure much like that
in this ``idsiosynchratic perspective''.)  Further,  
nonconservation of the probability of observing a
resonance--essentially, its survival probability--suggests decay of the 
non-surviving system, which is not repugnant to us in any way.  We
require the conservation of the {\em total} probability, also
including the 
probability of the system which comes into being as the product of the 
decay of the resonance system which does not survive.  (We may not be 
able to show total probability conservation constructively, but we will be able to
show the existence of the equilibrium towards which the system decays
in the third installment of this series, from which conservation of
total probability may be inferred.)

This idiosynchratic perspective arises simply from noting that there
is nothing in Gadella's construction which requires the top two levels
of the Gadella diagrams to involve complex spaces!   There is a field
of classical mechanics which can be called distribution density
dynamics.  (See, e.g., the appendix 14 to~\cite{arnold} and references
therein.)  There is thus a physical context in which the
mathematically permissable choice of using real spaces on the top two
lines of the Gadella diagrams makes sense.  
We can define a classical dynamical system on real spaces $\F_\pm$ and
$\F^\times_\pm$, and the associated physical context is, in effect, 
an averaging over an infinite number of classical trajectories 
(e.g., in phase space) using a distribution density.  Such a 
construction would have abstract spaces and
function space realizations, corresponding to the top two levels of
our idiosynchratic Gadella diagram.  The transition from the middle to
the bottom levels of the idiosynchratic diagram
via analytic continuation represents the (functorial)
first quantization of the classical distribution density dynamics
method of description of a dynamical system, and results in 
the description of a dynamical system by 
a recogniable quantum theory which includes Breit-Wigner
resonances~\cite{note11}.  

Most recent physics using the RHS formalism has been preoccupied with the
energy representation, since the primary recent interest has been in
describing irreversible time evolution using the formalism's ability
to express the boundary conditions of an irreversible process, without
regeneration~\cite{qat1,qat2,qat3}.  There is also a momentum space
picture of the Gamow 
states~\cite{handm,mandh}.  For our present idiosynchratic purposes, it
is interesting to consider starting from a classical phase space, since
when we do so many similarities appear between the RHS methodology
advanced herein and a version of classical statistical mechanics.  The
(real) spaces for the phase space variables 
can be constructed using methods similar to the methods
given explicitly in the construction of the spaces in the second installment.

By working with probability amplitudes in the RHS
format, rather than with probability amplitudes on the constant energy
surfaces in phase space, we are defining probability measures on phase
space itself.  When we cause those probability amplitudes to evolve
dynamically, we avoid the many of the
pitfalls of time evolving probabilities in
the Boltzmann and Gibbs approaches based on discrete
partitioning of phase space.  (See, e.g.,~\cite{italian}.)  

In place of the energy representation of the prepared state $\langle E
|\phi^{in}\rangle = \langle^+ E |\phi^+\rangle \in\CS \cap\HH^2_-
\Bigm|_{\R^+}$, we will have a similarly well behaved $\langle \pi |
\phi^{in}\rangle = \langle^+\pi | \phi^+\rangle \in \overline{
\CS\cap\HH^2_-}$, where by $\pi$ we indicate the canonical phase space
variables $(p_x, p_y, q_x, q_y)$, and by $\overline{\CS\cap\HH^2_-}$
we indicate real valued functions on $(\R^2 )^\times \oplus i \circ
\R^2$ (see~\cite{II}),
which are understood to be properly normalized as probability
amplitudes.  For instance, for an equilibrium ideal gas in a
container, $\langle^+\pi | \phi^+\rangle$ might be Maxwellian
(gaussian) velocity distributions and functions of compact support
over the position as components of a multi-component spinor
(see~\cite{II}).   

Similarly, in place of $\langle E|\psi^{out}\rangle = \langle^-
E|\psi^-\rangle \in \CS\cap \HH^2_+ \Bigm|_{\R^+}$ for the measurement
result of our measuring apparatus (i.e., a measure on phase
space), we will have $\langle \pi |\psi^{out}\rangle = \langle^-
\pi  |
\psi^-\rangle \in\overline{\CS\cap\HH^2_+}$, meaning a real valued and
very well behaved function of our $\pi$-variables, also
normalized.
Both of these sets of generalized functions will be defined for
restricted time 
domains just as the functions on the energy surfaces within these
function spaces have restricted time domain of definition, and
dynamical evolution (including dynamical time evolution) will be by
semigroups, meaning we continue to have the ability to express the
boundary conditions of an irreversible process referred to earlier.
(Because we include distributions, we have the conceptual ability to
include ``distributional probability measures'', such as measures
which will yield Boolean value 0 or 1 representing the outcome of a
yes-no experiment.)  Because of the hyperbolic structure inherent on
properly constructed
complex spaces, we we later see it is possible to construct hyperbolic
probability measures, e.g., the probability of observing a
Breit-Wigner resonance decreases hyperbolically with time.

This formalism is also compatible with the notion of extended objects,
because, e.g., $\delta x \ne 0$ in general, so no point
localization is assumed anywhere, although we do have the mathematical
machinery (Dirac measures) to accomodate point localization if
desired.  The idiosynchratic perspective contemplates concurrent 
position and velocity measurements, and this is perfectly okay--we're
not in Hilbert space anymore!  If we 
had a gaussian classical momentum distribution (e.g., a 
a Maxwellian velocity distribution)
and gaussian classical position distributions, then we would
recover a classical equivalent of the Heisenberg uncertainty relation,
based on the widths of the gaussians,
notwithstanding that Planck's constant and the full implications 
of the quantum mechanical
complementarity principle are not incorporated into the system.

From this $\pi$-representation in terms of $(p,q)$, analytic
continuation takes us $(p,q) \longmapsto (p,ip,q,iq)$. Following a
simple change of coordinates
\begin{equation}
A = ( q+ip)/\sqrt{2} \qquad A^\dag = (q-ip)/\sqrt{2}
\label{eq:cdops}
\end{equation}
so that now $(p,ip,q,iq) \longmapsto (A,iA,A^\dag ,iA^\dag )$ provides
a basis for the complexified phase space.  We are thus an eyeblink
away from the creation and destruction operator formalism used in the
rest of this series of papers.  In installment four, we will see how
obtaining a  real Witt basis from this basis
enables the representation of either bosons or
fermions without a change of basis (spinors are notoriously basis
dependent).

Joint position and momentum probability distributions do not exist for
any quantum state represented by an element of $L^2 (\R^n )$. (This is the
motivation for the Wigner transform, for instance.)  Both position and
momentum cannot simultaneously be represented by continuous operators
on $L^2 (\R^n )$.  Similarly, on $L^2 (\R^n )$ it cannot be the case
that both the creation and destruction operators are both continuous
operators, and so, for instance, 
one has to use care in defining the coherent states.
However, in the RHS formalism position and momentum are both
represented by operators which are $\tau_\Phi$-continuous and
$\tau_\Phi$-closed.  Likewise, both creation and
destruction operator are $\tau_\Phi$-continuous and $\tau_\Phi$-closed
operators (see chapter two of~\cite{dirackets}), and so 
these restrictions do not apply.  Thus, after the analytic
continuation we change basis to the real Witt basis and think in
terms of eigenvector probability densities of the creation and
destruction operators, since they are $\tau_\Phi$-continuous
$\tau_\Phi$-closed operators.  Continuity and closedness in $\Phi$ and
$\HH$ need not agree, and this has far reaching consequences, which we
are exploring.  








\section{Quasi-Invariant Measures}\label{sec:quasiinvariant}

The space $\FT$ provides quasi-invariant measures for the space $\F$ in
the Gel'fand triplet $\rhs$~\cite{genfun4}.  Because we use operators
which are not symmetric, the left and right quasi-invariant measures
will differ, e.g., in the resolution of the identity provided by the
spectral theorem, the dyads $\vert \phi\rangle \langle {\tilde{\phi}}
\vert$ are not symmetric, and in particular $\vert \phi\rangle$ and 
$\langle {\tilde{\phi}}\vert$ will be defined for different time
domains as a consequence of our use of semigroups of dynamical time
evolution.  This will have physical consequences we will demonstrate
in installment three~\cite{III}.  This also will add a minor wrinkle to
Liouville theory, should you wish to apply Liouville theory (rather
than the Schr{\"o}dinger equation)  in a distributional
context, because in the conventional development time evolution
is measure preserving, whereas when distributions are involved the
dynamical time evolution is only quasi-invariant.  In installment
three~\cite{III}, we will see that dynamical evolution can be
hyperbolic, and these quasi-invariant measures are hyperbolicly
evolving as well, but that overall probability is conserved.









\section{Physical Interpretation}\label{sec:interpretation}

In the present series of articles, we are exploring a dynamical system
of oscillators in which there is correlation between the oscillators.
We are using probability amplitudes rather than point particle
localization, and in particular we allow the use of densities and
distributions for those probability amplitudes.  This involves us with
not the $L^2$ functions but with a subset of 
the Hardy class functions which
partition $L^2$, $L^2 = \hh^2_+ \oplus \hh^2_-$, according to the
Paley-Wiener theorem.  

Given the possibility of free oscillators, we expect some sort of
direct sum structure (sort of like a Foch space), except our dynamical
transformations may also mix the components.  We will see in
installment two~\cite{II} that these multi-component probability
amplitudes are of mathematical necessity of a certain form and a
sufficient form for them is put forward.  On our phase space, and on
the abstract and function space representations of it (using these
densities and distributions), there is a symmetric form $Q$ and a
skew-symmetric form $J$:
\begin{equation}
Q= \begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix} \qquad \textrm{and} \qquad 
J= \begin{pmatrix} 0 & 1\\  -1 & 0  \end{pmatrix}  \quad .
\label{eq:qjmatrices}
\end{equation}
Each of these forms induces a scalar product, so that there are, in
effect, three possible scalar products.  Assuming 
\begin{equation}
\vert \phi ) = \begin{pmatrix} \phi_1 \\ \phi_2 \end{pmatrix} \quad
\textrm{and} \quad 
\vert \psi ) = \begin{pmatrix} \psi_1 \\ \psi_2 \end{pmatrix} 
\label{eq:vectors}
\end{equation}
we may represent these three scalar products as:
\bea
(\psi | \phi ) &=& \psi_1^\times \circ\phi_1 + \psi_2^\times \circ\phi_2 \nonumber \\
\langle \psi | \phi \rangle &=& ( Q \psi | \phi ) = \psi_2 ^\times \circ
     \phi_1 + \psi_1^\times \circ\phi_2  \nonumber   \\
\{ \psi | \phi \} &=& ( J \psi | \phi ) = \psi_2^\times \circ\phi_1 -
     \psi_1^\times \circ\phi_2  
\label{eq:3scalar}
\eea
where $\psi_2^\times  \circ \phi_1$ indicates a ``single component'' scalar
product.  We will return to this in installment four~\cite{IV}, but
the important ovservation for now is the types of correlation shown in
the above alternative scalar products.  Thus, the complex unit scalar
$i$ is identified as a correlation mapping (a type of injective
embedding in the dual)~\cite{porteous}.








\subsection{Collateral Implications}\label{sec:collateral}

The foregoing ``idiosynchratic perspective'' suggests at once fairly
straightforward and conceptually obvious modifications are possible to
the Grad moment expansion~\cite{grad,muller}, but such pursuits are very
wide of our present concerns--possibly it will be taken up nother day,
establishing a connection between the present formalism and a
generalized ``thermodynamics''.  In place of a moment expansion about
the Maxwellian (e.g., gaussian) velocity distribution, we generalize
to functions of rapid decrease, and similarly interpret the moments as
thermodynamic quantities.  This also has implications for the Meyer
cluster expansion and its graphical representation by Feynman
diagrams.

There are also implictions for the theory of solution of differential
equations.  Thus, the second installment is concerned with the
Lie-Poisson bracket of vector fields on phase space, and the fourth
installment is concerned with the tensor algebra of phase space.

There is a breathtakingly direct analogy between a
probabilistically oriented classical description and the mathematics
of the present formalism.  We find a description of the classical
treatment in~\cite{arnold} Appendix 14, page 457:
\begin{quote}
``Jacobi realized that the (classical) Poisson brackets of the first
integrals of any hamiltonian system could be considered as a Poisson
structure [reference in original].

The construction of a Poisson structure on the dual space of a Lie
algebra leads to a new Lie algebra.  This construction may then be
repeated, leading to a whole series  of new (infinite dimensional)
Poisson structures.  More generally, suppose that one is given any
Poisson structure on a manifold.  Then the space of functions on that
manifold carries the structure of a Lie algebra.  This implies that
the dual space of this function space carries its own Poisson
structure.  Elements of this dual space may be interpreted as
distribution densities on the original manifold.  Thus, the space of
distributions on a Poisson manifold (for example, on a symplectic
phase space) has a natural Poisson structure.  This structure makes it
possible to apply the hamiltonian formalism to equations of Vlasov
type, which describe the evolution of distributions of particles in
phase space under the action of a field which is consistent with the
particles themselves.''
\end{quote}

Physically, this means we may be working in a quantum paradigm whose
classical analogue includes 
a field theoretic treatment of a large number of
particles whose paths are locally hamiltonian, and whose interactions
are dealt with in a self consistent way~\cite{note3}. There are other
applications the formalism is also well equipped for of less immediate
interest. such as working with a few systems each having
a very large number of possibilities for their self-consistent
evolution, under the guiding 
influences of a mutual field of interaction. 









\appendix

\section{Complex Spectral Theorem}\label{sec:compspectthm}

From the starting point of a real semisimple Lie algebra, we
undertake to show the role of the complex covering algebra in the 
construction of a representation space $\F$.  From the universal embedding 
algebra, one has the 
existence of a complete set of commuting operators.  Let us assume this 
c.s.c.o. of essentially self adjoint operators is
$\{ A_1, \, A_2,\, \dots , A_N\}$.  Let $\Lambda_i$ be the Hilbert space
spectrum of the operator $A_i$, $i =1,\, 2, \dots ,N$, and let
$\Lambda = \Lambda_1 \times \Lambda_2\times \cdots \times \Lambda_N$
be the Cartesian product of the $\Lambda_i$.  Then the general 
Gel'fand-Maurin Theorem (also called general Nuclear Spectral Theorem) 
asserts there exists a rigged Hilbert space $\rhs$ such that there exists
a uniquely defined positive measure on $\Lambda$ such that~\cite{bgm}:
\begin{enumerate}
\item{$\F$ has a topology determined by a countable family of semi-norms.}
\item{$A_1,\, A_2, \dots , A_N$ are esa and are 
$\tauf$-continuous on $\F$.}
\item{For any $(\lambda_1 ,\, \lambda_2 , \dots ,\lambda_N )$ in 
$\Lambda$ there exists a generalized eigenvector in $\F^\times$, 
$|F_\lambda\rangle =|\lambda_1 , \lambda_2 , \dots ,\lambda_N \rangle$ 
such that

\begin{enumerate}
\item{$A^\times_i |F_\lambda \rangle = 
\lambda_i |F_\lambda \rangle$ for 
almost (with respect to $\mu$) all $i = 1,2, 
\dots , N$.  $A^\times_i$ denotes the extension of $A_i$ to $\F^\times$, the 
dual space to$\F$.  If the $A_i$ have no singular spectrum, ``almost all''
can be replaced by ``all''.}
\item{For any pair of vectors $\phi , \psi \in\F$ and any well defined 
function $f$ of $N$ variables, one has}

\end{enumerate}}
\end{enumerate}

\begin{equation}
 \left(  \phi ,  \psi \right) = 
 	 \int_\Lambda 	\langle \phi  | F_\lambda  \rangle 
		\langle F_\lambda |\psi\rangle 	\; 
			d\mu (\lambda_1 ,\, \lambda_2 , \dots ,\lambda_N )
\label{eq:partunity}
\end{equation}
\begin{equation}
 \left(  \phi , f(A_1,A_2,\dots ,A_N ) \psi \right) = 
 	\int_\Lambda f(\lambda_1 ,\, \lambda_2 , \dots ,\lambda_N )  
		\langle \phi |F_\lambda \rangle       
			\langle F_\lambda | \psi \rangle 	\; 
			d\mu (\lambda_1 ,\, \lambda_2 , \dots ,\lambda_N )
\label{eq:rspectres}
\end{equation}
The positive measure is unique up to equivalence of the null set.  In
general, the space $\F$ of the RHS may not be unique~\cite{note12}.








\bibliographystyle{apsrev}
\bibliography{qho1}


\end{document}
\end


\vfill\eject


\bigskip
\centerline{\bf References}


1.  A. Bohm, M. Gadella and S. Maxson, Computers Math. Applic., {\bf
34}, 427 (1997).

2.  A. Bohm, S. Maxson, M. Loewe, M. Gadella, Physica A{\bf  236}, 485, 
(1997). 

3.	 Note the use of the symplectic form on
$\F\times\F^\times$ and $\HH\times\HH^\times$ is associated with a
scalar product.  In the usual physical notation for the scalar
product, $(\bullet \vert \bullet )$ or $\langle
\bullet\vert\bullet\rangle$, the ket $\vert\bullet\rangle$ would be
identified with the space and the bra $\langle \bullet \vert$
identified with the dual space.  For the time being, we will follow
the mathematical notation regarding symplectic forms also in regards
to these scalar products, in effect defining a scalar product which
is the complex conjugate of the usual scalar product of the physics
literature.  Note that when working with dual pairs of spaces, $(\chi
,\chi^\times )$, many things are freed from dependence on the
particular choice of topology on $\chi$, meaning that many of the
properties we will deal with by proceeding in this way are
topologically invariant.  See, e.g., L. Narici and E. Beckenstein,
{\em Topological Vector Spaces}, (Marcel Dekker, New York, 1985). Thus, the choice of seminorm topology for $\F$ or the
choice of the norm topology for $\HH$ may actually have less effect on
the final results than one might at first suspect.  The use of the RHS
formalism for the description of resonances has the advantage of a
transparent mathematical rigor, whereas alternative treatments--such as using 
Hilbert space operators outside of their apparent domain of
definition--often seem lacking in justification even if mathematical care is
used.  It is possible that careful examination of the RHS formalism
could lead justification of many hueristic devices used to describe 
resonances over the years.

4.	The essential purpose of making an alternative choice to the 
field $\C$ is the need for a geometric structure (in numerous places)
which reflects the structure of the complex plane, and the requirement
for a real algebra structure in order to have well defined adjoints,
as will be discussed below.  The commutative
real algebra $\C (1,i)\equiv \R\oplus i\circ \R$ 
has such a structure, and will be used for
now.  There is, however, another possible choice which we will later
see is probably conceptually preferable: the complex field $\C$ is
isomoprphic to the real Clifford algebra $Cl^+_2 \cong \R\oplus \bigwedge^2
\R^2$, a proper subalgebra of $Cl_2$, the universal Clifford algebra of
the real plane $\R^2$.  The substitution of 
$Cl^+_2$ for $\C (1,i)$ would make no
difference in our developments, and will be discussed in the
second section of the second installment of this series.  The algebra
$\C (1,i)$ is obtained by adding further structure to the field $\C$,
in order to make it into a field.
The (real) Clifford algebra $Cl^+_2$ has a unit imaginary which does
not commute with vectors (in the real plane), so there is some
structure which some readers will find problematic (while others
should have
no objections).  For now, it is merely important for the reader to
understand that either real algebra may be used in place of the field
$\C$.  In the second installment of this series, we will see that
there is an interesting concordance of structures when we choose the
Clifford algebra alternative, but such a choice is probably not
mandatory.  

5.  Since we use operators for which the
notion of hermiticity is not governing, our dynamical analogue of 
unitarity is defined using
the dual rather than the hermitean conjugate: $U^\times U = UU^\times
=\II$ replaces the familiar Hilbert space 
$U^\dagger U =UU^\dagger =\II$.  Because we work with spaces having a
complex symplectic structure, this form of adjoint transformation
insures the transformations have a symplectic (=dynamical) action on
our spaces of states, i.e., $U$ so defined is a dynamical
transformation on $\F$ of $\rhs$.  

6.  H. Feshbach, Y. Tikochinsky, Trans. N.Y. Acad. Sci., Ser. II, 
{\bf 38}, 44 (1977).

7.  By a semialgebra, denoted 
$\g_+$ or $\g_-$, the author means the Lie algebra $\g$
together with an associated continuity structure such that the algebra
exponentiates to the connected part of a semigroup, rather than a
group.  (This useage may or may not differ from other useages of the
term semialgebra.)  We will be considering only semisimple  
Lie algebras $\g$ such that
$\g^\C$ is the Lie algebra of a connected semisimple group.  For us
then, complex semialgebras integrate into a strictly infinitesimally 
generated semigroup.  For the representations, we likewise mean
strictly infinitesimally generated semigroups, although with the
representations we also mean ``equicontinuous semigroups of class
$C_0$''.  The strictly infinitesimally generated semigroups possess
identity, and so may be categorized as monoids as well. 

8.  By Schr{\"o}dinger equation, in this context we
think not in terms of 
$$
i\hslash \frac{\partial\;}{\partial t} \psi
=H\psi  \;\; ,
\nonumber
$$ 
but in terms of 
$$
\left[ \frac{\partial\;}{\partial t} + (\frac{i}{\hslash} \, H)
\right] \; \psi =0
\nonumber
$$
which occurs as the equation of parallel transport
(parallel time translation) for energy eigenvectors.

9. J. -M.Souriau, {\em Structure of
Dynamical Systems}, Birkh{\"a}user (Boston, 1997). 


10.  When one does quantum field theory using 
careful mathematics, distributions are usually involved.  In this 
case, a rigged Hilbert space and semigroups seems quite appropriate in the
first instance, and gauge groups seem likely to be a consequence
obtained of the semigroups.  In our third installment, we explore the
gauge semigroup versus gauge group relationship.  The gauge structures
derived there are generic, and are shown to depend only on the
interaction being analytic, and so they do not require the specific oscillator
form.  Thus, knowing whether or not neutrinos have mass may tell you
something about specific interactions, and should not be presumed to 
follow from merely knowing the gauge group.


11.  If nothing else, the present line of developments may supply
some of the mathematical rigor occasionally lacking in the Lagrangian
path integral formalism.  Thus, one sometimes meets with the {\em
assumption} of Markovian paths in the course of a careful Lagrangian
path integral exposition: the ergodic geodesics obtained in the present
Hamiltonian based work are Markovian extrema.  

12. Technically, the present procedure deals with
a generalization of the gaussian pure states and the transformations between
these types of states by (inhomogeneous) symplectic 
transformations.  For a description of the gaussian pure state 
formalism this present work generalizes, see R. 
Simon, E.C.G. Sudarshan and N. Mukunda, Phys. Rev. {\bf A 37}, 3028 (1988),
and references therein.

13.  Strictly speaking, one could and should
extend this formalism to include inhomogeneous symplectic
transformations.  For instance, the present procedure creates 
a representation of a realization of the $\mathfrak{sl} (2,\C )_\pm$ 
algebra in terms of creation and 
destruction operators by using symplectic transformations of the
Heisenberg algebras.  $SL(2,\C )$ is of course the complex covering group 
of the proper homogeneous Lorentz group as well as of $SU(1,1)$ and $SU(2)$.  
An inhomogeneous procedure would create a representation of the complex
covering (semi-)group  $ISL(2,\C )$ of the inhomogeneous
Lorentz group, or Poincar{\'e} (semi-)group, and could represent 
Poincar{\'e} resonances.  

14.  G. Ludwig, {\em An Axiomatic Basis of Quantum Mechanics}, volume
I, (Springer-Verlag, Berlin, 1983).  

15.  G. Ludwig, {\em An Axiomatic Basis of Quantum Mechanics}, volume
II , (Springer-Verlag, Berlin, 1987).

16.  A. Bohm, J. Math. Phys. {\bf 22}, 2813 (1981); 
A. Bohm, Lett. Math. Phys. 
{\bf 3}, 455 (1978).

17.  E. Celeghini, M. Rasetti, M. Tarlini and G. Vitiello, 
Mod. Phys. Lett. {\bf 3}, 1213 (1989).  

18.  E. Celeghini, M. Rasetti, and G. Vitiello, Ann. Phys. 
(New York) {\bf 215}, 156 (1992).

19. There are many perspectives from which
we can view our mathematics, and all descriptions ultimately 
must be compatible.
Thus, we can speak of ring (semigroup algebra) extension, mapping into
another space, or work with endomorphisms of a covering
space. 

20.  M. Gadella, J. Math. Phys. {\bf 24}, 1462 (1983).

21.  A. B\"ohm, M. Gadella, {\it Dirac Kets, Gamow Vectors and 
Gel'fand Triplets}, Lecture Notes in Physics {\bf 348}, (Springer-Verlag, 
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24. A. Perelomov, {\it Generalized Coherent States and Their 
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34.  G. Lindblad and B. Nagel, Ann. Inst. Henri Poincar\'e (N.S.) 
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35.  The
symplectic transformations form a larger and more general class than
the unitary transformations, and are associated with quasi-invariance
rather than invariance in the present construction.  The symplectic
transformations are also traditionally associated with dynamical
transformations.  See, e.g., [3.5].  The
conventional complex Hilbert space of von Neumann, and the
definition of unitary transformations on that space are such that
it does not possess a complex symplectic structure associated with
the flows of unitary transformations on itself.  This is partly
because, as a space, it is a space over the {\em field} of complex 
numbers, rather than over the real algebra $\C (1,i)$ (which
we adopt herein, and so produce a Hilbert space which is not
dynamically equivalent to von Neumann's).
Hence, the unitary transformations on von Neumann's Hilbert space 
do not {\em necessarily} have a symplectic 
(dynamical) action.

36.  One way of
viewing this construction is to note that our space of states is
obtained by constructing all possible evolutionary sequences which are
geodesic.  See
Section~\ref{sec:altconst}.  Thus, all possible (including
hypothetical) consistent histories in the sense of Griffiths are
included in the construction of the space of states, although our dual
space also includes resonances whose ergodic evolution--described
in the second installment of this series--makes many of the standard
notions of the consistent histories approach
problematic.

37.  J.K. Yosida, {\em Functional Analysis}, sixth edition, 
(Springer-Verlag, Berlin, 1980).

38.  If group $G$ contains semigroup 
$S$ with Lie semialgebra $\mathfrak{s}$, and $S$ contains 
a maximal strictly infinitesimally 
generated sub-semigroup $S_0$ (e.g., also over $\mathfrak{s}$), the group
arrived at from $S$ and from $S_0$ is the same [32].

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41.  The extension of the $e^{-iHt}$ time evolution group on $\HH$ to
transformations of the same $U(1)$ form on $\F$ and $\F^\times$ is not
a trivial matter, and more is involved than the generalization of the
form of formal solutions to the Schr{\"o}dinger equation.  It is a
mathematically and physically reasonable step to do so (discussed in
[1] and [2]), but one of the lessons of the present work is that even
a time evolution operator of apparently ``non-unitary'' form on
$\F\subset\F^\times$ could be ergodic and induce a unitary time
evolution operator on $\HH$ (in effect inducing the Schr{\"o}dinger
description of time evolution for pure states in $\HH$).  This will be
addressed in the second installment of this series.

42.  V.I. Arnold, {\em Mathematical Methods of Classical Mechanics}, second 
edition (Springer-Verlag, New York, 1989).

43.  	We will not here address the issue of whether
	the bottom line of this idiosynchratic diagram can 
	simultaneously serve as the bottom level of a conventional 
	Gadella diagram, in effect determining the complex spaces 
	above this bottom line.  See [15] for
	the discussion of the mappings.

43.1.	E. Hernandez and A. Mondragon, Phys. Rev. {\bf C29}, 722
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43.2.	A. Mondragon and E. Hernandez, Ann. Der Physik (Leipzig) {\bf
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43.7.	H. Grad in {\em Handbuch der Physik} Band XII, Thermodynaiik
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44.  This statement of the complex spectral theorem stems from [9]
       and [1].  The present treatment is distinguishable in a variety
       of ways, and one of the consequences of those distinctions is
       that our work involves different left and right
       (quasi-)invariant measures.  Our partition of unity is of the
       form $\II = \Sigma \, \vert F_\lambda \rangle \langle
       {\tilde{F_\lambda}} \vert$, where $\vert F_\lambda \rangle$ and
       $\langle {\tilde{F_\lambda}} \vert$ have different time domains
       of definition.  This will be addressed in the second
       installment of this series, and is the result of attention to
       boundary conditions.  

45.   Marsden, J. Am. Math. Soc. {\textbf 32}, 590 (1972).

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47.   L. H{\"o}rmander, {\em The Analysis of Linear Partial Differential 
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48. For a discussion of the subtleties
of complex conjugation versus involution in Pertti Lounesto, {\em Clifford
Algebras}, London Mathematical Society Lecture Notes Series 239,
(Cambridge University Press, Cambridge, 1997).

49.  It will be seen that the
uniqueness requirement for the adjoint involution of our complex Lie
algebra/group representation means we must work with multicomponent
vectors which are in fact spinors.  This issue
will be addressed in the planned fourth installment of this series.

50.  C. Nash and S. Sen, {\em Topology and Geometry for 
Physicists}, (Academic Press, NY, 1983).

51.  The fact that these Gamow vectors make up a set of zero measure has
physical significance for the ergodicity discussions in the second
installment of this series.  If they had formed a set of positive
measure, they and the resonances they represent could not be
associated with entropy growth, for instance.  See, e.g., N.S. Krylov,
{\em Works on the Foundations of Statistical Physics}, Princeton
University Press (princeton, N.J., 1979).

52.  We regard the physical interpretation of these state 
vectors which
continuously evolve in their statistical sense, i.e., as determining
continuously changing probabilities of the outcome of a measurement.
This interpretation corresponds well to what is conventionally meant
in speaking of exponential decay, for instance.  

53.  P.L. Duren, {\em  $H^p$ Spaces}, (Academic Press, New York, 1970).

54.  F.W. Warner, {\em Foundations of Differentiable Manifolds and Lie
Groups}, (Springer-Verlag, New York, 1983).

55.  S. Kobayashi and K. Nomizu, {\em Foundations of Differential
Geometry}, Volume II, (Interscience (Wiley), New York, 1969);  
volume I of the same work, (Interscience (Wiley), New York, 1964). 

56.	Pierre Grillet, {\em Algebra}, (Interscience (Wiley), New
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57.  W. Fulton and J. Harris, {\em Representation Theory},
(Springer-Verlag, New York, 1991).

58.  P.L. Butzer and H. Berens, {\em Semi-Groups of  Operators and 
Approximation}, (Springer-Verlag, New York, 1967).

59.  N.M.J. Woodhouse, {\it Geometric Quantization}, second edition, 
(Clarendon Press, Oxford, 1992). 

60.  M. Audin, {\em The topology of Torus Actions on Symplectic
Manifolds}, (Birkh{\"a}user Verlag, Basel, 1991).

61.  P.M. Morse and H. Feshbach, {\em Methods of Theoretical Physics},
volume I, (McGraw-Hill, New York, 1953).

62.  Y. Alhassid, F. G{\"u}rsey, F. Iachello, Ann. Phys. (NY) {\bf 148}, 346 
(1983).

63.  P.A.M. Dirac, J. Math. Phys. {\bf 4}, 901 (1963); generators
relabelled.

64.  Y.S. Kim and M.E. Noz, {\it Phase Space Picture of Quantum M
echanics}, (World Scientific, Singapore, 1991); and references
therein.

65.  This procedure can be given the physical interpretation of 
representing the preparation procedure of an experiment 
as a combination of continuous canonical transformations, and the continuation 
along similar lines to represent the decay and measurement processes 
as canonical 
transformations is straightforward.  There could ultimately be 
free oscillators at the outgoing end of the process.  The 
equation (\ref{eq:Htransform}) can be interpreted as the (active) canonical 
transform, relating to the measurement process, or as merely setting up a 
formal evaluation of the constant $\G$ which characterizes the random decay  
process.  Not all of the interpretation issues can be resolved at this
time.  

66.  Because $\g\subset \g^\times$, there is a canonical
$\pm$-inclusion of $\g$ into $\g^\times$.  To respect the canonical 
symplectic form on $\g\times\g^\times$ demands we chose the 
minus-inclusion, because on the representation space, in order 
for our transformations to be symplectic in their action on the
representation space (e.g., be dynamical transformations on the
representation space), it is necessary that they
must satisfy the test analogous to the test
for ``unitarity'' of group transformations.
See Ian R. Porteous, {\em Clifford Algebras and the Classical
Groups}, Cambridge University Press (1995).  (Recall also that the
unitary transforms comprise a subgroup of the symplectic transforms.)
This, in turn, leads quite canonically to the
dynamical group representation indicated below being simultaneously
being defined on the Lie algebra representation space!  The
representation space is complete, so that Cauchy sequences defined by
$exp$ of the realization of the algebra acting on elements of 
the representation space also yields elements of the space.  Our asd
Lie algebra representation with esa generators yields proper
dynamical semigroup representations as well.  The action of our group of
symplectic transformations is symplectic on the spaces as well, which
are Hausdorf spaces, establishing that they are topologically
transitive, so that one of the prerequisites for their association
with chaos is satisfied.  Chaos issues will be addressed in the second
installment of this series.  Conversely, since the flows of von
Neumann's definition of unitary
transformations on the conventional Hilbert space do not define a
complex symplectic structure on that space (i.e.,
if $e^{-iHt}$ is a unitary
transformation, then $e^{-Ht}$ is not), their action is not
necessarily symplectic as to that space (and so the Schr{\"o}dinger
equation does not necessarily reflect a proper {\em transitive} dynamical
transformation of the conventional Hilbert space).

67.  This means there is really no exponential catastrophe for the
Gamow vectors when the scalar product as a ``length function'' 
is properly defined.  The ``alternative normalization that always
seems to work'' of V. I. Kukulin, V.M. Krasnopol'sky, 
J. Hor{\'a}{\v c}ek, {\it Theory 
of Resonances} (Eng. transl.), (Kluwer, 1989), is in fact
mathematically principled and proper.  Note this also involves
identification of the scalar product conjugate of a
Gamow vector with the Schwartz reflection principle conjugate.

68.  The chosen forms mean that
the $H$ in equation (\ref{eq:trad}) may differ from that in equations
(\ref{eq:alttrad}) and (\ref{eq:realmod}).  These equations must be
understood as indicating a {\em way to think} in different contexts,
and emphasize that different rules may apply in diferent spaces.  One
major point is that on $\F$, both $H$ and $iH$ may both be esa
generators simultaneously.  We will follow the form 
of (\ref{eq:realmod}) and
the definition of the adjoint involution operation, with the
understanding that $H=H_0+iH_1$, reflecting the fact that the
Hamiltonian of our analytically continued system 
now is an element of a realization of the real form of a
complex Lie algebra.  The Hilbert space obtained of the RHS 
containing the $\F$ 
and $\F^\times$ constructed using the present definitions of the
adjoint involution is certainly a different space from the usual
Hilbert space of the von Neumann formalism!  Also note that in the
exponential mapping, the time parameter $t$ does not change as $H
\longrightarrow H_0 +iH_1$, and the complex energy eigenvalues occur.

69.  O. Teichm{\"u}ller, J. Reine Angew. Math., {\bf 174}, 73 (1935);
L. Horowitz aand L. Biedenharn, Annals of Physics {\bf 157}, 432
(1984); S.L. Adler, {\em Quaternionic Quantum Mechanics and Quantum
Fields}, Cambridge University Press, (Cambridge, 1995).

70.  B. Simon, {\em
Representations of Finite and Compact Groups}, (American Math Society,
Providence, Rhode Island, 1996).

71.  I.M.Gel'fand and G.E. Shilov, {\it Generalized Functions}, 
vol 2, (Academic Press, 1968), chapt 4.

72.  A. Bohm, {\em Proceedings of the Symposium on the Foundations of
Modern Physics, Cologne, June 1, 1993}, P. Busch and P. Mittelstaedt,
Editors, (World Scientific, 1993), page 77; A. Bohm, I. Antoniou,
P. Kielanowski, Phys. Lett. {\bf A189}, 442 (1994); A. Bohm, I. Antoniou,
P. Kielanowski, J. Math. Phys. {\bf 36}, 2593 (1995).

73.  J. Moser,
{\em Stable and Random Motions in Dynamical Systems}, (Princeton
University Press, Princeton, N.J., 1973).

74.  C.L. Siegel and J.K. Moser, {\em Lectures on Celestial
Mechanics}, (Springer-Verlag, Heidelberg, 1995).

75.  A.W. Knapp, {\em Lie Groups Beyond an Introduction}, 
(Birkh{\"a}user, Boston, 1996).




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