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\begin{document}

\title{{Generalized Wavefunctions for \\
Correlated Quantum Oscillators IV: \\
Bosonic and Fermionic Gauge Fields.}}

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

\date{\today}



\begin{abstract}

The hamiltonian quantum dynamical structures in the Gel'fand triplets
of spaces constructed in preceding installments in this series of
articles are shown to possess strictly
alternative representations by either fermionic or bosonic states.
A covering construction is given explicitly 
in terms of the properties of Clifford algebras appropriate to
different numbers of fields.  The unitary Clifford algebra
is constructed from the intersection of the orthogonal and common
symplectic Clifford (Weyl) algebras of the canonical phase space and
its complexification, establishing a well defined spin geometry for a
subset of the symplectic (dynamical) Clifford algebra. The unitary
Clifford algebra is used as the vehicle to define dynamical gauge
bundles and dynamical gauge semigroups for two, three and four fields.
The canonical dynamical gauge group for four fields is 
$SU(4)\times SU(3)$.  Charmed
quarks are thereby predicted in the fermionic representations of our
four field gauge theory.  An isomorphism  
exists from a subset of the gauge structure for four fields to a
subset of the orthogonal Clifford algebra of a non-trivial spacetime
with $(+,-,-,-)$ local signature, demonstrating probable compatibility
with general relativity.   

\end{abstract}



\pacs{11.15.-q,11.15.Tk,03.65.Fd,02.10.De}        %use showpacs class option

%\keywords{ }       %use showkeys classoption if keyword display desired

\maketitle


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

In this concluding installment concerning
correlations of quantum oscillators, we
will show how a field theory over canonical variables can be
associated to extended objects in a curved relativistic spacetime.  We
will demonstrate the mathematical and physical importance of
understanding that our constructions are in fact spin 
constructions, and show that it is
extremely useful to regard our constructions as associated with
representation of a new type of Clifford algebra, the unitary 
Clifford algebras, existing only for spaces possessing both
symplectic and orthogonal structure, such as phase space.  A 
Clifford algebra       
provides the covering structure to make this work well behaved
mathematically, and our generalized wave functions are a part of a
representation of this Clifford algebra of phase space and its
complexification.  

Spinors are associated with the Clifford algebras and their 
representations, and there are many subtleties we shall gloss and take an
optimistic view of in the present forum.  Spinor structures and spin
geometry can be problematic, but we have
grounds for feeling secure with respect to the key elements (the
unitary Clifford algebras) we depend on.  We would also suggest that
the unitary Clifford algebras provide a well defined ``nucleus'' which
may be extended to a full symplectic Clifford algebraic structures
without arbitrary conditions imposed to insure convergence of the
exponential map (as in the symplectic formal series type of Clifford
algebra).  There appear to be symplectic Clifford algebras containing
well defined semigroups of symplectic (=dynamical) transformations,
even though full groups do not seem available without additional
assumptions.  

We have a sufficiently well
defined spin structure as to this nucleus to have a well defined gauge
structure associated to it.  We will restrict ourselves to those
dynamical aspects of the spin geometry
issues of concern to field theorists, and leave broader dynamical
concerns for another day.  It is noteworthy, however, that formulating
our theory in a form capable of reflecting a dynamical arrow of time
(the semigroups of symplectic transformations provide a vehicle for
expressing the boundary and
initial conditions of an irreversible dynamical process) is
mathematically sufficient
to make our theory well defined.  (Whether such a semigroup formulation is also
necessary seems to touch on areas of great subtlety and complexity,
and will addressed in detail.)

As indicated in installment one~\cite{I}, our approach admits a 
field theoretic interpretation.  If
``dynamics is the geometry of behavior'', we would add that the most
suitable description of geometry seems to be in terms of geometric
(Clifford) algebra.  The present paper emphasizes Clifford algebra
issues and field theoretic interpretation of Clifford algebra
representations for various Clifford algebras associated with phase
spaces.  In conventional notation, the $p$ and $x$ of installment
two~\cite{II} become canonical variables $p$ and $q$, etc., in this 
part. 
After dealing with some left over matters from the preceding
installments, we will establish the Lie algebra valued connection
constructively, demonstrate the existence of generalized Yang-Mills
gauge structures, and show how the basic gauge group of the 
Electroweak Theory and Standard
Model, and also the canonical gauge structure for four fields, emerge
in a breathtaking and natural way, merely by looking at the canonical
transformations of appropriate numbers of oscillators (identified in 
the usual manner with fields).  The gauge groups are exact, however,
and there is no spontaneous symmetry breaking.  The special unitary
gauge group emerging from this unitary Clifford algebra approach
possesses both fermionic (even dimensional) and bosonic (odd
dimensional) representations, corresponding to whether we view $SU(N)$
as a subgroup of an orthogonal group, lying in an orthogonal Clifford
algebra (fermionic) or as part of a symplectic (semi-?)group, lying in
a symplectic (bosonic) Clifford algebra of some sort.  It is therefore
a gross mathematical error to mix bosons and fermions in our
hamiltonian formalism.  We suggest that there is some
correspondence between the even and odd dimensional representations,
since they do represent the same group,
so that, e.g., some identification exists between
bosonic color and fermionic (quark) flavor in a four color
generalization of QCD.  (This identification is probably related to
topological notions, but we will not attempt any justification at
present.)  This is seemingly where inescapable mathematical necessity
has led us in our pursuit of a hamiltonian quantum field theory, and
is either physically relevant or it isn't.  In any event, the
hamiltonian and Lagrangian approaches to field theory may lead to
significantly different end points. 

There are a couple of key ingredients in our geometric structure
that play essential roles in our construction.  Spinors are usually
frame dependent (this is the reason there is no spinor calculus 
analogous to the tensor calculus), and the unitary transformations will
leave our real Witt frames invariant (much like the orthogonal
transformations leave conventional frames invariant).  The real Witt
bases enable us to obtain a well defined differential geometry for our
symplectic spinors from the well known spin geometry of orthogonal
spinors (a special topic in Riemannian geometry~\cite{lawson}.)  There
is a simple mathematical trick using standard theorems of topology for
linear spaces which we invoke to obtain this result.  Secondly, we
have a weak symplectic form (see~\cite{II,III}), meaning that Darboux's
theorem does not apply, and our geometry can be other than locally
Euclidean, permitting non-trivial local invariants such as curvature.

A brief exercise will demonstrate that analytic continuation of the
traditional Hilbert space does not result in vectors possessing
Bose-Fermi symmetries which 
are well defined.  This follows because the energy spectrum for
vectors belonging to that analytically continued
space is not necessarily bounded from below.  Let us
consider energy eigenvectors $\vert a\rangle$ and $\vert
b\rangle$ belonging to some space for which there is a well defined
``vacuum'' or minimum energy eigenvector, $\vert 0\rangle$.  Then
there is some transformation $A$ such that $\vert a\rangle = A \vert
0\rangle$ and some transformation $B$ such that $\vert b\rangle =B
\vert 0 \rangle$.  Without loss of generality we may regard $A$
and $B$ as esa, and it follows that
\begin{equation}
\langle a \vert b\rangle = \langle 0\vert \;\; \frac{ AB+BA}{2} \; +\;
\frac {AB-BA}{2} \;\; \vert 0\rangle
\label{eq:decomp}
\end{equation}
The uniqueness of this decompositon into symmetric and antisymmetric
parts depends on the existence of a unique fiducial vector, such as
$\vert 0\rangle$.  When the energy spectrum is unbounded below, there
is no such fiducial vector, and many similar decompositions can exist,
with nothing to distinguish any particular one.  
This brief demonstration illustrates that the traditional form of the
boson-fermion superselection
rule does not apply to analytically continued systems, in which
the energy spectrum is not bounded from below.  However, for our
multicomponent state vectors there is a somewhat more complicated
situation than this naive calculation is relevant to, which we
elaborate in detail.

For the
multicomponent spinor formulation developed in preceding installments,
especially installment two~\cite{II}, and further specified below, 
any bilinear form on
phase space must be either strictly symmetric or strictly antisymmetric.
(This is a characteristic of Clifford algebras in general.)
This compels us to choose one or the other bilinear form for the
construction of our Clifford algebra, although there is a special
basis for phase space compatible with both the ordinary orthogonal
(symmetric) form and the (antisymmetric) symplectic form.  In this special
basis, the real Witt basis, we can simultaneously generate 
representations of either, enabling us to form the 
non-trivial intersection of the
orthogonal and symplectic Clifford algebras of phase space.  The
unitary Clifford algebra which results thus has a canonical basis in
which one may alternatively consider physical aspects associated with the
orthogonal perspective, such as fermionic representations of bulk
matter by Dirac spinors in a spacetime with local signature
$(+,-,-,-,)$, or those aspects associated with the symplectic perspective,
such as dynamics, forces, interactions, etc., associated with bosons,
represented by symmetric spinors.  We will refer to such choices of
representation as a
choice of perspective for our state vectors.  We can thus think of a
system as a bunch of fermions (particles) or as a bunch of bosons
(intermediaries of the forces--the dynamical entities), but must
consider a particle as either ``lumps of geometry'' or as ``lumps of
dynamical fields''.  These perspectives are alternative ways of
looking at one physical structure in alternative {\em single}
representations ,
according to our constructions of those representations of physical
structures using a unitary Clifford algebra.  In this view, 
the boson-fermion dichotomy is an artifact of the representation 
chosen, not of the fundamental structure being represented.  Our 
use of the terms boson and fermion may not, in consequence, exactly
correspond to the usual conventions of quantum theory.  

The decomposition of equation (\ref{eq:decomp}) can be said to be
unique in a unitary Clifford algebra such as we construct in
Section~\ref{sec:spinors} in the sense that each of the two terms is
non-trivial in one perspective only, each perspective being associated
with cofactors over ideals based
on one or the other of the bilinear forms which exist separately on
the space.  The two terms cannot mix to 
define a mixed bilinear form on our spin-vectors, which form
a representation of phase space (as part of the representation of the
Clifford algebra of phase space).  In our constructions, bosons and
fermions are associated with separate and distinct, 
unique bilinear forms, and each
bilinear form defines a perspective, but the perspectives (e.g.,
representations) may not
mix.  Thus, you may speak of the fermionic properties of bulk matter or you may
speak of bosonic forces and dynamical evolution, but you must change
perspective between these two alternatives, and really cannot properly
talk of both {\em simultaneously} without exceeding the bounds of
mathematical propriety.  To consider the electrodynamic
interaction of two electrons, for instance, one must consider each
electron as a ``particulate coherence of dynamical field'' in order to speak of the
exchange of photons (other ``particulate coherences of dynamical field'')
between them.  {\em Dynamics} is the exclusive jurisdiction of the
perspective associated with the symplectic form and factorization of
the tensor algebra over that form yields a representation of
bosons (symmetric spinors).  There is no superselection rule in our RHS spin
formulation in the same sense as such a rule is applied to
the conventional Hilbert space quantum theory.  Rather, there is a
selection between perspectives (representations).  

In terms of
equation (\ref{eq:decomp}), we would say that there are fermionic and
bosonic representations of the operators $A$ and $B$, appropriate to
the two alternative perspectives, and one or the other of the two
terms will vanish in a given perspective.  All is predicated on our choice
of the Witt basis, yet another instance of the basis dependence of
spinors.  The scalar product in
equation (\ref{eq:decomp}) will survive as
either symmetric or skew depending on whether the orthogonal
(symmetric) or symplectic (skew) form is chosen for factorization of
our tensor algebra.  The strictly alternative representations are also 
distinguishable from the thing being represented, since they are 
isomorphic to alternative topological completions of a set,
and, in any event, the representation isomorphisms are 
not natural isomorphisms so there is some 
inequivalence aside from any topological issues.





\section{Necessity of Spinor Structures}\label{sec:necessity}

In the following two subsections, we pursue the reasons for use of
spinors in our representation of the correlated combinations of
oscillators problem.  The puzzling structure motivating this is the
conjugacy of the (complex) symplectic transformations between $iY$ and
$Z$ seen in installment two~\cite{II}.  Of course, spinors figure in group
representations, providing the ``fundamental representations'', and
there are some technical mathematical reasons that make them
appealing (even mandatory), but 
there are strong physical reasons as well.  They make
our dynamical structure well defined.  We will first illustrate that a
partial version of our model may be constructed without use of the
structures we believe provide a full covering structure.


\subsection{Conjugacy of $iY$ and $-Z$: Version
1}\label{sec:conjugacy1}

{\em Semigroups which are tori are in fact groups}~\cite{hilgert}.  Therefore, 
in the context of the simple complex (semi-)group, there is no 
problem (locally) with conjugation 
by a single exponential transformation and its inverse, even if that 
transformation be a semigroup transformation.  Simply put, there is no
such thing as a $U(1)$ semigroup, because $U(1)\cong S^1$, the trivial
torus.  
For instance, in Section 4 of installment two~\cite{II}, one has the
simplification into a  
single  transformation
\begin{equation}
e^{i(\pi /2 ) Q_1} \, e^{i(\pi /2 )K_2} = e^{i (\pi /2 ) \{ Q_1 + K_2\}}
\label{eq:BCH2}
\end{equation}
because $[Q_1 ,K_2 ] =0$. 

The compact transformations are invertible, 
because, e.g., the transformation of $(iJ_0 )$ by
the transformation $exp \{ i (\pi /2) J_1\}$ 
is the equivalent to a one-quarter 
revolution of the circular path generated on the $J_0$
complex 1-torus.  The flows generated by $J_0$ are symplectic
(Poisson), and these transformations are symplectic in their actions
on the representation space
because of the way we defined adjoints in~\cite{II}, so we must
understand that the $J_0$ complex 1-tori is associated
with the group of complex symplectic transformations, lying in 
a coadjoint orbit in dual of the Lie algebra of complex symplectic
transformations.  ($J_0$ represents a
symplectic transformation on phase space.)  In this case, three 
additional iterations results in the 
inversion.  The extension of the oscillator Hamiltonian by conjugation by single 
non-compact transformations and combinations of compact
transformations is well defined on the complex covering (semi-)group,
but in the first instance this only has a {\em local} meaning with
respect to the flow structures generated by the semigroup algebra.  

The conjugacy of the essentially self adjoint operator
$iY$ and the operator $-Z$ as operators locally 
generating paths on tori has 
local stability implications.  If two flows winding on a classical real 
torus are conjugate 
under a homeomorphism and one is stable (e.g., compact in the
applicable topology), then the other is 
``sufficiently close'' to be viewed as stable
also.  In the present case,
this identification is only local (not completely continuous), 
because, e.g., at some $|j,m\rangle\in \CS_{\mathfrak{sp}(4,\R)^\C\pm}$ the 
(coadjoint) 
orbits of the representations of the local $Y$-torus and the local 
$Z$-torus are not separated in the coarse topology applicable to
$\CS^\times_{\mathfrak{sp}(4,\R )^\C\pm}$.   
The resulting appearance of stability (based on analogy to classical
stability considerations) is only a local aspect of the 
evolutionary 
cycle--the semiflow is quasi-stable, but no truly global stability is
present. One interpretation is that one is merely incapable of
distinguishing between an unstable quantum system and a stable quantum
system in this situation, based on purely local information: you
cannot predict when the decay is going to occur based on local
information only.  

Of course the two tori possess a common identity, so
are not disjoint locally in the group, and their images 
overlap locally in the representation space.  But, finite
displacements along the geodesic directions determined by the
non-compact generator $iY$ and the compact generator
$-Z$ generate paths in orthogonal subspaces of the abstract
representation space $\F_{\mathfrak{sp}(4,\R )^\C\pm}$ (according to 
Hilbert space topology and also by virtue of 
the complex structure on $\F_{\mathfrak{sp}(4,\R )^\C\pm}$ itself), so 
the identification among the elements of 
$\F^\times_{\mathfrak{sp}(4,\R )^\C\pm}$ which (locally) identifies
the orbit of a compact generator with the rotated
orbit of a non-compact generator is of major consequence.  Quasi-stability
is the physical interpretation of this mathematical
structure.

It is the lack of topological closedness on the algebraically dual space
$\F^\times_\g$ which prevents
completion of the dual algebra generated semigroup structure there into 
a full ``dual group'' structure,  even though at first glance 
it seems analytic.  Thus, sub-semigroup ``$U(1)^\C_\pm$'' structures may
be invertible on $\F$ in some sense (as Abelian tori, and therefore
groups, when considered individually).
However, a larger structure than $\F^\times$ must be appealed to in order for
invertibility to take place in the context of $\F$ and $\F^\times$.  If
$g\in \g$ is the generator of a $U(1)$ subgroup, we have the sequence of
spaces  (whose meaning should be self-explanatory):
\begin{equation}
\F_g \subset \F_{\g\pm} \subset \HH \subset \F^\times_{\g\pm} \subset
	\F^\times_g \;\; .
\label{eq:heirarchy}
\end{equation}
Invertibility in the restricted sense used here
is really a feature provided by an auxilliary space,
$\F^\times_g$, where there is a proper extension from 
a subspace of $\F_{\g\pm}$ and an
adoption of a weak dual topology as a necessary consequence of the
continuous extension process,
followed by a subsequent restriction from $\F^\times_g$ back 
to $\F^\times_{\g\pm}$ (also in
the weak-dual topology).  This conjugacy relationship is thus well defined
even though there is no invertibility on the spaces $\F$ and $\F^\times$
themselves.  Note that (in the language of the Gadella diagrams in
the first installment of this series~\cite{I}) 
$\CU^\pm\F_{\g\pm}=\CS_{\CG\pm}$ taken as a set contains all of the
elements needed for the adjoint transformation to be well defined on it 
(e.g., $\F\cap \HH$ is compatible with inverses), but it is the 
compulsory change to the weak-$^\times$ topology in 
$\F^\times_{\g\pm}$ which obstructs invertibility and compels us to
think of the ``adjoint'' transformation as actually being associated
with a coadjoint structure on $\F^\times_{\g\pm}$. 

Hence, the conjugacy of $iY$ and $-Z$ is defined only locally
and is not a global and complete relationship (i.e., is not 
continuous).  In consequence, 
the esa extension of $iY$ is only associated with quasistability on 
$\F^\times_{\mathfrak{sp}(4,\R )^\C\pm}$.  The abelian torus (group) 
structure is only complete as to a larger space than 
$\F^\times_{\mathfrak{sp}(4,\R )^\C\pm}$,
and only in this larger structure is the conjugacy of $iY$ and $-Z$
completely defined as a continuous transformation.  There
are obstructions to the unification of the semigroup structures into group
structures on $\F_{\mathfrak{sp}(4,\R )^\C\pm}$ and
$\F^\times_{\mathfrak{sp}(4,\R )^\C\pm}$ since one involves the duals
and their coarse topology
in the determination of continuity.  The inexactitude in the conjugacy
means that the Gamow vectors (and associated Breit-Wigner resonance
poles) are not stable by analogy to the theorem
mentioned above, but the {\em local} indistinguishability of their precise
relationship from a stable conjugacy relationship means that the
physical decay event being represented need not be
instantaneous either: the mathematical result corresponds to what is 
physically interpreted as quasi-stability
or quasi-stationary behavior in the evolution of a resonance.  The
resonance is quasi-invariant under evolution and yet may evolve towards an
equilibrium (e.g., perhaps as the result of ergodicity). 
We have a system with
sensitivity to initial conditions and also to external 
perturbations, so that even though evolution may be Hamiltonian we are
forced to use probabilistic methods to describe it.

The complex tori occuring in the RHS treatment of
symmetries are {\em not} invariant tori such as arise in classical
non-linear dynamics: they are quasi-invariant tori!  They are also
associated with an Hamiltonian formalism, whereas the KAM tori arise
most naturally in a Lagrangian formalism.  We can associate
non-compact operators with dense  orbits on either kind
of torus.  Where things get
complicated is after analytic continuation has taken place, when one must
use a weak topology.  Locally this weak topology 
may fail to clearly distinguish the stability behavior of the orbits of
compact and non-compact generators.   

The algebraic analytic continuation of
the second installment of this series of articles produces a
pair of Gamow vectors associated to Breit-Wigner resonance poles--note
there are only two possible unstable pure states (up to equivalence)
which result from each individual family of esa extensions to the
duals.  The treatment of this subsection and followed throughout this
work adopts the position that orbits of compact generators (e.g.,
generators with discrete spectra) are associated with stability
and quantizability (perhaps generalizing the notion of compact orbit
quantization).  The complex tori
obtained herein can be thought of as being obtained from the
deformation of trivial tori ($U(1)$ subgroups), and it seems likely
that there is some association with the instability which occurs in
the deformation of KAM tori.  The fact that a generator is
noncompact and the coarseness of the weak topology associated with
evaluating the decay parameters of orbits on representations of
these complex tori seem to be the key ingredients in resolving these
stability issues, but there is an obvious and deeply felt need for
greater understanding in this area. 








\subsection{Conjugacy of $iY$ and $-Z$: version 2}\label{sec:conjugacy2}

Although we may have uncertainties about the best way to interpret
stability implications of this conjugacy, at least
there is a covering structure in which the conjugation is well
defined, without regard to the appearance of an apparently undefined
inverse semigroup transformation in it.  This is because we are
working with spinors: our spaces possess an orientation by virtue of
their symplectic structure and $exp$ is holomorphic for us, so the first 
two Stieffel-Whitney classes vanish, making our 
generalized spaces of quantum states spin spaces by
construction.  Following these preliminary inquiries, we will
address those mathematically necessary spin issues, and we will find
the related Clifford algebras revealing. 

Our representation spaces $\F_{\mathfrak{sp}(4,\R )^\C \pm}$ possess a
complex symplectic structure (since their automorphism group is $Sp(4,\R
)^\C$.)  Spinors are ideals of Clifford algebras, so we
conclude that our representation space is part of the representation of a
symplectic Clifford algebra~\cite{crumeyrolle}.

The spinors we most often encounter arise in the representation of
orthogonal Clifford algebras.  Typical spinors of physics
are associated with
a Clifford algebra for some space $V$ which possesses an orthogonal
structure (symmetric bilinear form, elliptic scalar product, etc.),
denoted $Cl(V)$.  This universal Clifford algebra for $V$ contains a
group, the (orthogonal) Clifford group $\mathscr{G}_O (V)$.  The
(orthogonal) Clifford group contains a spin group, $Spin(V)$, which in
turn provides a double cover for the group of orthogonal
transformations on $V$, $O(V)$.  If we represent an orthogonal
transformation on $V$ belonging to $Cl(V)$ by an exponential, $\left(
e^{iX\theta /2} \right)_{O(V)} \in O(V)$, then the orthogonal rotation
of a vector $A\in V\subset Cl(V)$ about the direction given by the
vector $X$ is represented as the conjugation
\begin{equation}
\left( e^{iX\theta /2} \right)_{O(V)} A \left( e^{-iX\theta /2}
\right)_{O(V)} \quad .	
\label{eq:orthrot}
\end{equation}
This exact same rotation may be represented in the $Spin(V)$ covering
structure as
\begin{equation}
\left( e^{iX\theta } \right)_{Spin(V)} A \quad ,
\label{eq:spinrot}
\end{equation}
i.e., an operation from the left to right, without conjugation.  
In other words, a semigroup orthogonal
rotation which is performed by conjugation (and therefore has only a
conditional local meaning, as indicated in the preceding section)
determines a spin transformation which acts from the left only,
thereby defining a unique geometric structure having a global 
meaning extracted from our local identification.  

There is a similar series of groups in the case of the symplectic
Clifford algebras~\cite{crumeyrolle}, namely a symplectic Clifford group,
$\mathscr{G}_S (V)$ covering the spinoplectic (or toroplectic) 
group $Sp_2 (V)$ which is a non-trivial
double cover of the symplectic group $Sp(2n,V )$,
where $V$ is here assumed to be a real space and
$dim(V)=2n$.  There is also a metaplectic group, and higher covering
spinoplectic groups, $Sp_q (V)$, $q>2$.  There are subtleties with the
symplectic Clifford algebras which we overlook for the moment, since
the common symplectic Clifford algebra is a polynomial algebra so that
additional structure must be added in order for the exponential map to
be defined.  There are a number of alternatives for this extra structure.  
See~\cite{crumeyrolle}.  

We will indicate the construction of a symplectic Clifford algebra in
Section~\ref{sec:spinors} which is distinguished from the symplectic
Clifford algebras in ~\cite{crumeyrolle} (which may contain some of
those algebras) which is better adapted to
our purposes and in which the exponential map is well
defined, although described not in full and careful detail.  Thus, our
conjugation by an element of the semigroup of symplectic
transformations has a covering structure of orbits of spinoplectic
transformations which act from the left only, and whose semigroup
meaning is not qualified or restricted in any way once identified
(uniquely) with the appropriate spinoplectic
covering structure.  Our locally defined conjugation operation
fixes a spinoplectic (semi?)-group covering structure which is global
in nature, analogous to the orthogonal case.  (We still associate
dynamics with the symplectic
transformations, which we will take to be a semigroup herein, and not 
with their covering structure, which may or may not be a full group.)
We have a very strong motivation, therefore, for working in
Clifford modules: we know instantly that
we have a covering structure in which our constructions are well
defined and have a global meaning.  

Clifford covering structures and spinors will be addressed further in
the Section~\ref{sec:spinors}.  There we will show
that our choice of ``creation and destruction operator basis'' for the
complexification of phase space (refer
back to Section 2 of~\cite{dirackets}) leads to the ability to
simultaneously represent bosons and fermions using the same abstract
structures, without explicitly invoking supersymmetry, as such.  Since
the existence of Clifford algebras depends on the existence of an
orthogonal or symplectic form from which a frame may be defined, our
notions of fermion and boson are frame dependent, both as to the type
of frame (orthogonal or symplectic) and as to the type of basis used
in that frame.  For the special unitary groups, whether our 
representations are fermionic or bosonic will also depend on the
dimension of the representation, so that the representation of an
orthogonal Clifford algebra must be even dimensional (fermionic
spinors) and the representation of a symplectic Clifford algebra must
be odd dimensional (bosonic spinors). 

It is  natural to avail ourselves of the spinoplectic
covering structure or perhaps
even to use the representation of the full Clifford algebra
itself.  This has the further virtue of making our representation
structure into Clifford modules, which are well known and well
studied~\cite{note7}.








\section{Spinors and Clifford algebras}\label{sec:spinors}


This brings us to an interesting juncture, which we illustrate with
the simple case of the phase space over a single pair of canonical
variables, which we will call $q$ and $p$.  Our phase space, which we
denote $T^\times \R$, has a basis $e_q ,e_p$.  If we perform an
analytic continuation of functions on $T^\times \R$, we will get a
space of functions over an ``analytically continued'' phase space
upon which we construct both an orthogonal and a complex
symplectic structure (which we take in its real or symplectic
form), and which we may label $\widetilde{T^\times \R} \equiv
T^\times \R \oplus i \circ T^\times \R$.  We can define a
basis for $\widetilde{T^\times \R}$ which is simultaneously orthogonal and
symplectic (compatible with both orthogonal and symplectic forms in
their standard form), just as $e_q$ and $e_p$ provide such a basis for
${T^\times \R}$.  That basis is nothing more nor less than the
creation and destruction operators or $\pm$ the unit imaginary times a
creation or destruction operator (borrowing directly from
\cite{crumeyrolle}, page 247):
\begin{equation}
\textrm{Define}\;\; a= \frac{(e_q+ie_p)}{\sqrt{2}} \;\; 
\textrm{and} \;\; a^\dag= \frac{(e_q-ie_p)}{\sqrt{2}}
\label{eq:crumops}
\end{equation}
so that the basis we choose for $\widetilde{T^\times \R}$ is
\bea
\epsilon_1=& ia\sqrt{2}   \qquad \epsilon_{1\ast } = a^\dag \sqrt{2}\nonumber \\
\epsilon_2=& a\sqrt{2} \qquad \epsilon_{2\ast}= ia^\dag \sqrt{2}\quad .
\label{eq:sobasis}
\eea
This means we have broken $\widetilde{T^\times \R}$ down into transverse
hyperbolic spaces with bases $\{ \epsilon_\alpha\}$ and $\{
\epsilon_{\alpha \ast}\}$, $\alpha =1,2$, respectively.  These satisfy
the relations
\bea
(\epsilon_\alpha ,\epsilon_\beta)=& (\epsilon_{\alpha\ast}
	,\epsilon_{\beta\ast} )=0 \qquad (\epsilon_\alpha,
		\epsilon_{\beta\ast} ) = \delta_{\alpha\beta}
			\nonumber \\
\omega (\epsilon_\alpha ,\epsilon_\beta)=& \omega (\epsilon_{\alpha\ast}
	,\epsilon_{\beta\ast} )=0 \qquad \omega (\epsilon_\alpha,
		\epsilon_{\beta\ast} ) = i\delta_{\alpha\beta}
\label{eq:spotests}
\eea
where $(\cdot , \cdot )$ is the symmetric form (e.g., associated with
the anticommutator and symmetric scalar product), and
where $\omega$ is the skew symmetric form (e.g., associated with the
commutator and skew symmetric scalar product).
Geometrically, $\epsilon_1 \perp \epsilon_2$ amd $\epsilon_{\ast 1}
\perp \epsilon_{\ast 2}$, because of the orthogonality of real and pure
imaginary components.  The other relations are straightforward.  The
$\epsilon_\alpha$ and $\epsilon_{\alpha\ast}$ thus form a real Witt basis
for the metric of $\widetilde{T^\times \R}$, the complexification of
the phase space $T^\times \R$.

Note that to implement the constructions of~\cite{II}, it is necessary to
use the {\em commutative real algebra} $\C (1,i)$ for the ring of
scalars of our Clifford algebras, rather than the {\em field} $\C$,
for the reasons given in~\cite{II}.  We will also adopt the notion of
involution given there for our adjoint transformations.  This is
what~\cite{porteous} calls a $L^\alpha$ Clifford algebra, $L$ being a
commutative algebra and $\alpha$ being an involution.

Of related importance to us is the notion of ``correlation''.   The
scalar product $\langle \psi^{\textrm{out}} \vert \phi^{\textrm{in}}
\rangle$ establishes the correlation between the prepared state
$\phi^{\textrm{in}}$ and the observed effect $\psi^{\textrm{out}}$.
The matrix elements of quantum theory are representations of
correlations of this sort.  With our
multicomponent spin vectors there are more exotic correlations which
it is possible to calculate.  If on our space of states there is a symmetric
(orthogonal) form 
\begin{equation}
Q=\begin{pmatrix} 0 & \II \\ \II & 0 \end{pmatrix} \quad ,
\nonumber
\end{equation} 
where $\II$ is the appropriate unit operator, and if there is a symplectic form 
\begin{equation}
F=\begin{pmatrix} 0 & \II \\ -\II & 0 \end{pmatrix}
\nonumber
\end{equation}
in addition to the scalar product
$\langle \psi \vert \phi \rangle$ we can form the symmetric (bosonic)
correlation $\langle Q\psi \vert \phi\rangle$ and the skew (fermionic)
correlation $\langle F\psi \vert \phi\rangle$~\cite{I}.  The unit
imaginary 
``$i$'' is associated with a skew symmetric form as well, providing a
further source of skew correlation.  The significance of
the unit imaginary ``$i$'' is that it is associated in our
constructions with a complex hyperbolic (e.g., Lobachevsky) geometry,
rather than some real hyperbolic structure--both are associated with a
symplectic form~\cite{II}.  The unit imaginary is a ``correlation
map''~\cite{porteous}. 

The point of this preliminary bit of algebra is that in our construction in
part two of this series~\cite{II}, we used the creation and destruction
operators as generators of (geodesic) transformations.  Their role as
vectors there is fully equivalent to their use within this paper 
above: {\em this} 
$a$, $ia$, $a^\dag$ and $ia^\dag$ are exactly the same as {\em that} $A$ and 
$A^\dag$, etc.  Here they provide a basis for the base
space of a symplectic Clifford algebra and at the same time
they can provide a basis for the base space of
an orthogonal Clifford algebra for the
same space (the complexification of phase space).  Both 
orthogonal and symplectic Clifford algebras of a single space
exist simultaneously in this basis!  In~\cite{II}, the creation and
destruction operators were identified with the generators of
translations (vectors), and in the present work we see those
translations are in coordinate directions.

The orthogonal Clifford algebras can be formally defined using the tensor algebra of a
space.  Thus, given a (real) space $E$ space with a quadratic form $Q$
defined on it, the orthogonal Clifford algebra $Cl_O (E)$ is defined as
the quotient of the tensor algebra $\times (E)$ by the two sided
ideal ${\cal {N}} (Q)$ generated by elements of the
form~\cite{crumeyrolle}, p. 37:
\begin{equation}
x \times x - Q(x) \; , \quad x\in E \subset \times (E) \; .
\label{eq:cl0}
\end{equation}
Due to invocation of the tensor algebra, existence is a fairly trivial
issue.  The orthogonal Clifford algebras contain the orthogonal
Clifford group, ${\cal G}_O$, which contains the familiar pin, 
spin and orthogonal
groups as subgroups.  The orthogonal Clifford algebras are associated with the
representations of fermions.  (The familiar Dirac algebra is the even
subalgebra of the orthogonal Clifford algebra for Minkowski space with
signature $(+,-,-,-),$~\cite{hestenes0},~\cite{hestenes}, pp 67, 75.)

Similarly, for $E$ an $n$-dimensional real vector space, and $F$ an
antisymmetric bilinear form, we define the common symplectic Clifford algebra
$Cl_S (E)$ by the quotient of the tensor algebra $\times (E)$ by the
two-sided ideal ${\cal{N}} (F)$ generated by the elements~\cite{crumeyrolle}, p. 233,
\begin{equation}
x\times y - y\times x - F(x,y) \; , \quad x,y \in E \; .
\label{eq:cls}
\end{equation}
The common symplectic Clifford algebras (Weyl algebras) are
essentially polynomial algebras, for which the exponential map is not
closed, and
comprise subsets of a number of other
symplectic Clifford algebras.  Of particular interest to us are the
formal symplectic Clifford
algebras over $K((h))$ containing a symplectic Clifford
group, ${\cal G}_S$, which contains the toroplectic (metaplectic), 
spinoplectic and familiar symplectic groups all as subgroups.  We
identify the ring $K$ as the commutative real algebra $\C (1,i)$, and
$h$ is Planck's constant.  (See~\cite{crumeyrolle} for details.)
The symplectic Clifford algebras are associated with the
representation of bosons.)

If $Cl_O (E) = \times (E) / {\cal{N}} (Q)$ and $Cl_S (E) = \times (E)
/ {\cal{N}} (F)$, are both defined for a space $E$,
there are obvious generalizations, such as, in particular,
\begin{equation}
Cl_U (E) = \times (E) / \left \{ {\cal{N}} (Q) \cup {\cal{N}} (F)  
	\right\} \quad .
\label{eq:clu}
\end{equation}
It is straightforward that
\begin{equation}
Cl_U (E) = Cl_O (E) \cap Cl_S (E) \ne \emptyset
\label{eq:clualt}
\end{equation}
$Cl_U (E)$ contains the space $E$ and $\II_E$, so is non-trivial.  We will
call it the unitary Clifford algebra, and note that its existence
depends on the space $E$ having a real Witt basis (such as phase space
or its complexification) and both symmetric and antisymmetric forms
(such as phase space or its complexification).  At this point, we must
regard it as a set in the tensor algebra having algebraic properties,
although not necessarily a fully endowed ``algebra'', and in
particular the exponential map is not closed on it
(since this is the case for the Weyl algebras).

The orthogonal Clifford algebras have an exponential map which is
complete as a by product of their ring of scalars being the reals,
effectively using the same norm topology as $\R^n$.
See~\cite{crumeyrolle}.  Choosing this same separating norm topology
for the completion of sequences formed of elements from $Cl_U (E)$, 
we can form what we will temporarily call the
orthogonal completion (o-completion) of the unitary Clifford algebra,
which we denote $Cl_{U-O}(E)$.

The common symplectic Clifford algebras (Weyl algebras) do not contain
Lie groups since they are basically polynomial algebras and
the exponential map is not complete in them.  Some
form of completion may be imposed on them in order to obtain an
augmented symplectic Clifford algebra in which the exponential map
converges.  A linear topological space is an algebra plus a scalar
product, so it is no great additional assumption 
if we treat $Cl_{U-O} (E)$ as a topological
vector space complete in the o-topology as indicated above.
We may freely regard our base spaces--phase space and its
complexification using the commutative ring
$\C (1,i)$--as scalar product spaces.  Since the
space $Cl_{U-O} (E) = Cl_{U-O} (\widetilde{T^\times \R^n})$ is now a  
linear topological space, it
has a neighborhood of $0$ of sets complete complete in $Cl_{U-O}
(\widetilde{T^\times \R^n})$ in any finer topology than the
o-topology previously
chosen for the completion of $Cl_S (\widetilde{T^\times \R^n})
\cap  Cl_U  (\widetilde{T^\times \R^n})$, and
we obtain a complete linear topological space in the finer topology.

Thus, the Kobayashi
semidistance on the complex hyperbolic space $\widetilde{T^\times
\R^n}$, $n\ge 2$, provides a (effectively seminorm) topology 
for $Cl_U  (\widetilde{T^\times
\R^n})$~\cite{lang}.  The notion of geodesic is well defined in this
topology~\cite{lang}, which we will call the s-topology.  
Convergence of symplectic (=dynamical) transformations in
semigroups is thus a sufficient condition for us to talk about
bosons, e.g., a dynamical  arrow of time is a sufficient condition for us to
talk about bosonic fields in this construction.  

By these standard theorems, $ Cl_{U-O}
(\widetilde{T^\times \R^n})$  will also be complete in the finer seminorm
s-topology~\cite{schaefer}.  Local convexity, and so on, easily follow from
this.  In the sequel, we will work with the algebraic set 
$Cl_U  (\widetilde{T^\times \R^n})$ as a completed linear
topological space $Cl_{U-O} (\widetilde{T^\times
\R^n})$, with alternative normed o-topology and seminorm 
s-topology completions\cite{note2}. We will distinguish the individual
completions, as necessary, by indicating the form of completion thus: 
$ Cl_{U-O} (\widetilde{T^\times \R^n})$ and $ Cl_{U-S}
(\widetilde{T^\times \R^n})$.  

Putting matters slightly differently, we can use the real Witt basis defined
above as the basis for a unitary Clifford algebra, the intersection of 
the orthogonal and symplectic Clifford algebras of the
complexification of phase space:
\begin{equation}
Cl_U (\widetilde{T^\times \R^n}) = Cl_O (
	\widetilde{T^\times \R^n}) \cap
		Cl_S (\widetilde{T^\times \R^n}) \quad n\ge 2 \quad.
\label{eq:cluphasespace}
\end{equation}
From the perspective of o-topology associated with the orthogonal
form, we may identify $Cl_{U-O} (\widetilde{T^\times \R^n})$ with
representations of fermions.  From the perspective of s-topology
associated with the symplectic form, we may identify this same
$Cl_{U-S} (\widetilde{T^\times \R^n})$ with the representation of
bosonic 
fields.  The complexification of phase space, $\widetilde{T^\times
\R^n}$, $n\ge 2$, is used for the representation of correlated
dynamics over $T^\times \R^n$.  We are dealing with the algebraic
treatment of correlated dynamics from alternative perspectives by
using the vehicle of Clifford algebras.

According to this prescription, all the spaces in our Gel'fand
triplets of spaces in the 
Gadella diagrams are built by using alternative topological
completions of Clifford algebra representations.  However, the
Kobayashi semidistance adapted to provide a seminorm above does not
produce a countable family of seminorms, such as involved in the
construction of our representation spaces in~\cite{I}.  








\subsection{Why focus on the unitary Clifford  algebra?}\label{sec:focus}

Our ultimate goal is to define a set of structures in which every
space in the associated Gadella diagrams is a spin space, since we
have already seen in installment two~\cite{II} the presence of
multicomponent vectors (which in fact satisfy technical requirements
for being spinors).  The unitary Clifford algebras are significant 
because they contain the
relevant unitary group (or semigroup) as transformations groups, but
they play a special role for us because they contain the relevant 
special unitary
group (or semigroup).  The unitary groups preserve the Witt bases
which are the foundation of our construction.  Because we have complex
{\em simple} Lie groups, and not merely semisimple Lie groups, we are
assured of unique spin structures~\cite{lawson}.  

More particularly,
$SU(N)$ is simply
connected, and spin manifolds are manifolds with simply connected
structure groups, relevant to having well defined spinor bundles
associated to the $SU(N)$ generated dynamical flow structure which we
will use to set up a gauge theory in the following
section.  Also,
$SU(N)$ has a bi-invariant Riemannian (symmetric) metric, and this
metric is identifiable with harmonic forms, meaning that the group
(and associated flows enerated by it) will have a well defined
harmonic structure, with kernels and propagators, etc., well defined
(initially in the o-topology).  Likewise, one parameter subgroups are
geodesic.  $Cl_{U-O} (\widetilde{T^\times \R^n})$ is thereby a very
well behaved linear space, with well defined flows and harmonic
structure on $\widetilde{T^\times \R^n}$.

Spin geometry for $Cl_O(E)$ is a specialized branch of Riemannian
geometry~\cite{lawson}.  For $Cl_U(\widetilde{T^\times \R^n})$, when
the unitary Clifford 
algebra is completed in the category of topological linear spaces
using a seminorm topology, the real Witt basis gives us a vehicle 
to obtain a well defined spin
geometry for $Cl_{U-S} (\widetilde{T^\times \R^n})$ as follows.  The
well defined spin structure on $Cl_{U-O} (\widetilde{T^\times \R^n})$
derives from the spin structure on  $Cl_{O} (\widetilde{T^\times
  \R^n})$.  The first and second Stieffel-Whitney classes are trivial
on $Cl_{U-O} (\widetilde{T^\times \R^n})$, and are homotopy invariants
(characteristic classes).  (For proper homotopy theory, we must use
groups and not semigroups.  See Section~\ref{sec:spinbundles} below as
to $SU(N)_\pm$ having no obstructions on any set of positive measure
to extapolation to a full group structure.)
It follows that this spin structure survives
the change to a finer topology so that $Cl_{U-S}
(\widetilde{T^\times \R^n})$ also has a well defined spin structure
(and harmonic structure, etc.).

Because $exp$ maps 
dense sets to dense sets (topological notions!), even for 
non-compact (e.g., hyperbolic)
generators, the $Cl_{U-S}(\widetilde{T^\times \R^n})$ nucleus of the
common symplectic Clifford algebra  $Cl_S(\widetilde{T^\times \R^n})$ 
can be extended to define a nuclear space, with nuclear topology,
etc., for a covering space of the Weyl algebra 
$Cl_S (\widetilde{T^\times \R^n})$.  By this
device, we obtain a complete linear topological space
$\overline{Cl}_S (\widetilde{T^\times \R^n})$ for which the exponential
map is complete.  This suggests that there is a well 
defined spin geometry on all or at least parts of
our nuclear symplectic Clifford algebra,
$\overline{Cl}_S (\widetilde{T^\times \R^n})$, and that the exponential map of
the Lie algebra of the symplectic semigroups $Sp(2n,\R )^\C_\pm$ 
is holomorphic with respect to our seminorm completion,
and as extended in this seminorm topology $\overline{Cl}_S(\widetilde{T^\times
  \R^n})$ is also a Clifford algebra containing the common symplectic
Clifford algebra (Weyl algebra) as a sub-algabra.

As a noteworthy aside, since our Clifford algebras 
above are completed in the category of linear topological spaces, they
also possess their own Clifford algebras.  Thus, we have the
possibility of constructing towers of algebras, and these have
properties of interest also.~\cite{coxeter} 

As to the unitary Clifford algebras, it is reasonably straightforward
to obtain fiber bundles with unitary structure groups, much in the
manner typical for frame bundles obtained from the base space and
orthogonal transformations of an orthogonal Clifford
algebra.  In similarly straighforward and well known manner, one may
obtain bosonic and fermionic principal fiber bundles with special unitary
groups as the structure group.  (We will
discuss these and their physical relevance below.)  There is also a
suggestion of a type of ``dynamical principal fiber bundle'' with 
semigroups of symplectic transforms as structure (semi-)group.  We 
will provide a description of the spinor bundle structures of
immediate relevance in Section~\ref{sec:spinstructure}. 

Note that  in general the Hamiltonian does not commute with the full
symplectic Lie algebra, so that energy is not a constant of all 
possible dynamical evolutions (i.e., it is possible to represent
open systems), and the energy
eigenstates do {\em not} provide an irrep of the group--typical for
spinor representations of groups.  In order to include complex
spectra in a mathematically well defined formalism, we have been 
led by {\em mathematical necessity} to representations
which are neither unitary (they are ``dynamical'', or more general)
nor irreducible (they are spin)!

There are also other possible implications of defining the unitary and
extended symplectic Clifford algebras as we have.  The orthogonal
Clifford algebras are associated with commutative
geometry~\cite{connes,garcia-bondia}.
Because the symplectic Clifford algebras associated with the skew
symmetric symplectic form rather than the symmetric orthogonal form,
it is possible (and we will so conjecture) that our s-topology
completion of the unitary Clifford algebra and the extended Clifford
algebra obtained from it are associated with noncommutative geometry.
The Clifford algebras (all types) are $\mathbb{Z}_2$ graded and thus are
superalgebras; being topologically completed as spaces they are
superspaces as well.  Although beyond the scope of these present 
inquiries, we will conjecture that most of the machinery of
noncommutative geometry, superalgebras and superspaces, Hopf
algebras, etc., is fairly close to hand even though not presently
revealed.  If these conjectures are true, the unitary Clifford
algebras possess both commutative and noncommutative geometric
structures, depending on the choice of perspective (choice of bilinear
form and topological completion).  The unitary Clifford algebras seem
to be a regime in which a lot of mathematical machinery is
exceptionally well behaved, connecting a lot of disparate
methodologies by having them defined over the same sets.

The spinor discussions in Section~\ref{sec:conjugacy2} refer to
invertibility of what are nominally semigroups in the context of a
covering structure which is spin.  In the context of the orthogonal
transformations, the well known spin groups provide the simply
connected covering structure for obtaining equation (\ref{eq:spinrot})
from equation (\ref{eq:orthrot}), even in the case of semigroups of
orthogonal transformations--simple connectedness is the key.  The
spinoplectic {\em groups} provide the analogous simply connected
covering structure for the semigroups of symplectic
transformations~\cite{crumeyrolle}.  Thus, even though our use of a
seminorm topology for the complex symplectic Clifford algebras
formally results in semigroups of transformations, there are no
obstructions on sets of positive measure to our extrapolating simply
connected sub-semigroups of the unitary semigroup
into full group structures--we can convert
special unitary sub-semigroups such as
$SU(4)_\pm$ semigroups into full groups.  These special unitary
(effective) groups may be thought of as subgroups of symplectic and
spinoplectic groups (in yet another topology!)  We therefore conclude
that the unitary sub-semigroups of our extended symplectic Clifford
algebras effectively provide a spin representation of the special
unitary group (and possibly the unitary group) in both s-topology and
o-topology completions of the algebraic set $Cl_U (\widetilde{T^\times
  R^n})$.  







\subsection{Physical consequences}\label{sec:consequences}

We will ultimately adopt a gauge field interpretation for these
constructions, and this approach has some interesting physical
consequences.  Thus, dynamics should be mediated by bosons, as
represented by the spinors which in turn belong to 
a representation of some form of a symplectic Clifford algebra.  On
the other hand, bulk matter
(fermions) should be represented by spinors representing an orthogonal
Clifford algebra.  Because there are more generators for $Sp(2n,\R )$
than for $U(n)$, there is the formal possibility there could be bosons 
(intermediaries for dynamical forces) with no direct coupling to bulk 
matter properties--i.e., there are no fermionic photons!  
Likewise, we infer there are
aspects of bulk matter not immediately associated with dynamics--i.e.,
the gravitational force does not depend on the kind of bulk matter,
but on the quantity of mass.  The true quantum
geometrodynamics is contained only in the intersection of geometry and
dynamics, the unitary Clifford algebra.  We are working in a formal
system in which there is only a limited overlap in which we can
concurrently talk about all of the issues which are important to
us.  We are constrained to two separate perspectives, dynamics or
geometry, which do not completely overlap.  We must
choose one or the other perspective exclusively when we choose to speak
carefully, since there is no mathematically respectable way
of speaking from both perspectives at once.  There is,
however, a domain of strong correspondences in which a single
structure (an abstract spinor) may have alternative fermionic (skew
symmetric matrix) and bosonic (symmetric matrix) representations.  A
reminder that the representative is not the thing itself.

Our abstract unitary Clifford algebra possesses both symmetric and
skew symmetric representations, corresponding to the bosonic and
fermionic perspectives.  In our carefully constructed mathematical
structures, the notions of boson and fermion correspond to field and
particle perspectives, respectively, but they no longer retain all of
their traditional meaning (which we would argue arises from working in
a mathematical formalism incompatible with resonances).










\section{Spin Bundle Structures}\label{sec:spinstructure}




\subsection{Non-trivial dynamics}\label{sec:nontrivial}

At this juncture, let us recapitulate the road to the mathematically
well-defined covering structure for our hamiltonian
quantum field theory, incorporating resonances and other
coherence structures, and which is treated as a form of 
dynamical system.  We based this on a probability rather than point
particle localization description of a dynamical system in phase space
(e.g., over canonical coordinates).  We tacitly assume that there is a
lot of freedom in the dynamical system and also that
we have some uncertainty in
our specification of the initial state.  A probabilistic field theory
is the result.  We added the
notion of correlation maps (injective embedings in the dual--which are
related mathematically to conjugation and associated notions of
coadjoint orbits, and related dynamically to momentum maps).  For
real symplectic (=dynamical) correlations of a simple two component
system, we 
found that the system was either stationary or exponentially decays to
some equilibrium configuration.  There are more elaborate
correlations--complex symplectic (=dynamical)
correlations--which make more complex
behavior possible, with complex spectra and possible oscillatory time
evolution or damped oscillations occuring in the dynamical time
evolution of the probability amplitudes.  

If we conjecture well behaved algebraic and topological properties,
with proper algebras and topological linear spaces for the above, we
are led to multicomponent representations, our function spaces
representing the probability amplitudes are $L^2$ spaces, and this
coupled with the
complex spectra forces us into a variant of the rigged Hilbert space
formalism outlined in ~\cite{I} and ~\cite{II}.  The lesson of this
fourth installment is that these multicomponent vectors are indeed
spinors, and we identified the special roles played by the
unitary Clifford algebras in providing a very well defined
mathematical structure which forms a nucleus which may be enlarged to
provide covering structures so that all of the relevant dynamics and
geometry may be (reasonably) well defined mathematically.

The classical function space realizations
of our abstract RHS, $\rhs$, was shown by Gadella to belong to the
intersection of the Schwartz space ($\CS$) and the spaces of Hardy 
class functions from above and below ($\HH^2_\pm$)~\cite{gadella}.
There are Clifford analogues of $\CS$ and
$\HH^2_\pm$~\cite{brackx,delange,gilbert}, and so the
function space realizations will be well defined if the abstract
spaces are also well defined.  (Recall the Gadella diagrams of
installment one~\cite{I}, and references therein.)

Gadella's use of van Winter's theorem~\cite{gadella} still provides the
necessary and sufficient conditions for ``analytic continuation''.  
Just as in the work of Bohm~\cite{bohm1,bohm2}, there will be contours at
infinity in integrals.  We have performed our construction in such a
way as to preserve the complex hyperbolic (Lobachevbsky) geometry of
the tangent space to our space(s) of states.
The functions spaces used are classic Schwartz and Hardy
spaces as to their components, and so the necessity proof of 
Gadella-van Winter suffices even in the spinorial RHS paradigm.

We identified in ~\cite{III} many possible linkages with the classical
treatments of dynamical systems, and in particular possible
relationships with various notions of complex systems, statistical
mechanics (and related thermodynamic ideas), fractals, etc., 
which seem compatible with the formalism,
but which emerged as a by-product of largely mathematical, 
considerations in our incorporation of correlation and associated
resonances into a classical probability theory.  The question arises
then whether this is another instance in
the long string of unreasonable effectiveness of mathematics in
physics described by Wigner many years ago, or if we have wandered off
the path somehow.  In the following subsection,
we will adapt this structure to exhibit principal bundle structures
associated to our constructions, and interpret this structure
as a gauge theory, setting the stage for calculations which make 
predictions which will ultimately 
tell us if this is a toy theory or has some relevance to the real
world.  







\subsection{Special unitary spinor bundles}\label{sec:specialunitary}

The unitary groups are compact and locally path connected,
while the special unitary groups are simply connected, with geodesic
subgroups.  It might be supposed that because our sought after 
spin structure is obtained from a seminorm topology, there is no
invertibility, notwithstanding that $SU(N)$ is compact and simply
connected. The inverses used in conjugation are in one sense
basically pullbacks along a single fiber to the nucleus (our base
space), and so are well defined individually and locally.
We will examine the existence of spin structures further below, 
but all these structures (and, for present inquiries, especially
the spin structure) depend on both the existence of and
choice of a special basis, the real Witt basis, and we must suppose
that there would not be invertibility of any sort
in a general basis.  Given the extreme dependence on the choice of
basis, there is an interesting interaction between seminorm
topology (and associated semigroups), spin conjugation and momentum
maps worthy of much further inquiry.  Our spin conjugation is not an
``inner automorphism'', but is a momentum map, involving the dual,
reinforcing our choice of completion of the unitary Clifford
algebraic set as a linear topological space, with scalar product and
dual.  The existence of any principal bundle structure depends on the
triviality of the structures, or (equivalently) the existence of
sections. 

With respect to the special unitary semigroup orbits in
 $\F_{\mathfrak{su}(N)\pm}\subset \F_{\mathfrak{sp}(2N, \R )\pm}$,
notwithstanding the seminorm topology, there is no obstruction to
invertibility on any set of positive measure: we may regard
$\F_{\mathfrak{su}(N)\pm}$ as simply connected
fiber liftings of a simply connected base space.
Conventionally, if given 
a space $E$ with base space $B$ and and whose fiber $F$ is isomorphic 
to group $G$, a principal fiber bundle structure associated to $E$ has
a global section (making both $P(E)$ and $E$ trivial).  This means
there is a continuous mapping
\begin{equation}
s:B \; \longrightarrow \; E
\label{eq:sectionmap2}
\end{equation}
which is invertible, i.e., there also exists a projection $\pi$ such
that
\begin{equation}
\pi s(x) = x \quad , \forall x\in B \;\; .
\label{eq:projectionmap2}
\end{equation}
This means, in effect, that $s=\pi^{-1}$.  See, e.g.,~\cite{nash}.  The
lifting $s=\pi^{-1}$ is in fact all we really have {\em globally for the
full symplectic group} in the
present case.  The projection $(\pi )$ is not defined as a continuous
transform on $\F_{\mathfrak{sp}(2N,\R )^\C_\pm}$, due to topological 
obstruction associated with our seminorm topology.  

Maximal compact subgroups are homotopy equivalent to the
Lie groups that contain them, e.g., $U(N)$ is homotopy equivalent to
$Sp(2N,\R )$.  However, homotopy is based on groups, and semigroups
won't do!  Thus, there are finite dimensional UIR's of $U(N)$, but
none of the noncompact $Sp(2N,\R )$, recalling 
Wigner's definition of
noncompact groups.  This suggests, once again, that we should think
naturally of semigroups of symplectic transformations, and not of
groups--else, from this homotopy equivalence, one would expect   
there to be finite dimensional representations of
$Sp(2N,\R )$ which are merely lifts of of finite dimensional $U(N)$
UIRs. (Similar statements could be made for other noncompact groups
containing compact subgroups.)

However, there is no obstruction to invertibility as to the special
unitary subgroup of the symplectic group itself.  We have simply connected
fibers and a simply connected base space: the lifting of the base space
are $1:1$ and onto the ``sections'', and so are isomorphisms and
invertible~\cite{note25}.
Thus, identifying (both abstractly and as to the related very well
behaved function space realizations)
\bea
B &\Longleftrightarrow& \left\{ |n_A\rangle\right\} \oplus
	\left\{ |n_B\rangle\right\} \nonumber \\
G&\Longleftrightarrow& {exp} \; \mathfrak{su}(N)^\C=SU(N)^\C 
	\nonumber \\
F&\Longleftrightarrow& \textrm{span} \left\{\left( \theta\circ SU(N)^\C \right)
	\circ b \right\}  , \quad b\in B \nonumber \\
E&\Longleftrightarrow& \F_{\mathfrak{su}(N)^\C}
\label{eq:indentify}
\eea
where $\theta$ is the representation mapping $\theta : SU(N)^\C
\longrightarrow 
\textrm{Aut} (E)$, and the representation {\em must} be framed in
terms of creation and 
destruction operators.  We have chosen to represent $B$ by using the
simple harmonic oscillator number states as a basis, e.g., the energy 
representation. 
$E$ locally has the structure $B\times F$ by construction, and we can
readily invert the represention homeomorphism to identify an element
of $SU(N)^\C$, so that we have $[\II\times\theta^{-1}] : B\times F
\longrightarrow B\times SU(N)^\C$, $[\II\times \theta^{-1}]\circ \left( x,
g(x)\right) \longmapsto \left( x,g \right)$, $x\in B$, $g\in SU(N)^\C$.  
If in our candidate
for $P(E)$ we consider $s(x) \in SU(N)^\C$ and $g\in SU(N)^\C$, then $gs(x)$
belongs to the fiber over $x\in B$.  $P(E)$ thus
has the global structure of a product between the base space $B$ and a
fiber $SU(N)^\C$.  

Because we have no obstructions (on sets of positive measure)
to extrapolating the semigroup
structure due to the seminorm topology, as to the $SU(N)^\C_\pm$
subsemigroup extrapolated into a group structure we have a candidate
for a dynamical homotopy group for the base space.  Reiterating, there
is no such thing as a
homotopy based on semigroups (possible distributional measures, such
as Dirac measures confound the notions of continuity, analyticity,
etc.), and
the property of being spin is determined by characteristic classes,
which are homotopy invariants.  Our base is spin, our fibers are spin,
and so we may properly talk of spinor bundles, and principal spinor
bundles in particular, only as to the special unitary orbits within
the overall dynamical structure.  As indicated earlier, these may be
represented by either bosonic or fermionic spinors when we do take a
representation, depending on
choice of bilinear form and associated topology (``perspective''). 











\subsection{Dynamical spinor bundles}\label{sec:spinbundles}

The $\bigcup_{\beta\in\R}\CS^\times_{(\beta )}$ used in the
construction of the very well behaved representation spaces of~\cite{II}
form a union over a one-parameter family of 
transformations for the case $G=U(1)$, e.g., for a $U(1)_\pm$
sub-semigroup.  Thus, the 
$\CS^\times_{(\beta )}$ defined there could be associated with the
formation of a 
simple line bundle for $\CG =U(1)$, provided 
$\beta$ were permitted to run freely over all $\R$, and not merely
over $\R_+$.  

If there were full groups 
generally available for the full symplectic semigroups in this 
construction, there would be no problem 
thinking of the symplectic transformations of, for instance, the function
space of energy eigenfunctions of two free quantum harmonic oscillators,
which we will call $\CS_{\hh \pm}$, which produces a family of function spaces 
we will call
$\CS_{\mathfrak{sp}(4,\R)^\C \pm}$.  This space may be  
associated in some very loose (and unspecified) sense
to a principal G-bundle, e.g., an $Sp(4,\R )^\C$-bundle with
$\CS_{\hh\pm}$ as base space.  However, the only initial suggestion of
invertibility is with the $U(1)$ sub-semigroups 
which have local actions on $\F_{\mathfrak{sp}(4,\R)^\C \pm}$ (and the
related very well behaved spin-function spaces).
Invertibility generally does not extend to sub-semigroups larger than
$SU(N)^\C$.  From extending the $SU(N)$ nucleus by a representation of
the $Sp(2N,\R)^\C$ algebra, it seems as if there should exist
something like a spinor bundle (vector bundle of spin type), but the
topological obstruction which keeps our semigroups from being
extrapolated into full groups also prevents formation of a full spinor
bundle structure, notwithstanding a proper spinor bundle is contained
somewhere within
this extended structure.  We do not have homotopy equivalence, and the
characteristic classes which define the property of being ``spin'' are
not preserved in arbitrary mathematical operations.  Properly, we have
special unitary spin bundles, with improper homotopies extending this
to a larger multicomponent ``symplectic bundle'' which is not spin.  

The liftings taken as a whole 
are not isomorphisms, as in the conventional treatment of principal 
fiber bundles, since generally there is no invertibility.  It 
is precisely this lack of isomorphic 
liftings of the ``paths along flows in phase space''--lifting of vector 
fields composed of state vectors--which enables us to 
convert two free oscillators
into a pair of coupled oscillators.  If we identify this construction 
with particle-fields, pair production is not $1:1$, so is not what we 
would think of as an isomorphism either.  Dynamical pair production or
destruction is not $1:1$ in ${before}:{after}$, so in order to incorporate
such processes into our overall dynamical structure we have lost the
use of invertibility and dynamical evolution as an isomorphism.  The
compact generators of $SU(N)$ take us from ``island of stability'' to
``island of stability'', while the noncompact generators of the full
symplectic semigroup take these ``islands of stability'' and make
resonances out of them, which will evolve dynamically towards another
``island of stability''.  Bifurcations are possible, in much the same
sense as that term is used in classical dynamical systems, except that
probability may flow along both paths of the bifurcation, e.g., there
may be pair production.

This lack of isomorphism in the ``fiber'' liftings raises questions as
to the extent to which we may reasonably think of the constructions
over the full symplectic semigroups 
as principal fiber bundles, merely involving semigroups rather than groups
for the fiber of the principal bundle.  
The lack, in general, of invertibility in our structural
semigroups, e.g., $Sp(4,\R )^\C_\pm$, is reflected in the flow
structure of the representation space and provides a novel meaning to
the term connection.  As in the standard principal G-bundle
construction, our Lie algebra provides the ``connection''.  The path of
$exp \,\g$ connects ``sections'' in both cases, e.g., $e^{-iHt}$ for
$t\ge 0$ is the semigroup which transports you from section (time
slice) to section (time slice) in generalized state space.  The
Schr{\"o}dinger equation is the equation for geodesic transport
(parallel transport in this case), giving us a constant of the motion:
energy is conserved.  There seems to be a clear sense of meaning here,
and clear mathematical analogies.

These are physically appealing notions which need substantial work to
make them mathematically respectable. 







\subsection{Gauge bundles}\label{sec:gaugebundles}

We will consider four fields with the fullest correlation structure
envisioned in our conservative constructions, based on the special
unitary groups.  (We will suggest
possible larger constructions with possible enhanced physical interest
in Section~\ref{sec:canonical}.)  The full dynamical structure thus 
has structure semigroup isomorphic to $Sp(8,\R )^\C$, and we may think 
of our base as the phase space of four pairs of oscillators
complexified, and the function space representative of the base space 
being of the form $B \oplus i\circ B$, where $B$ may be, e.g., 
$B = \{ | n_a\rangle \}
\oplus\{ | n_b\rangle \} \oplus\{ | n_c\rangle \} \oplus\{ |
n_d\rangle \} $, a very well behaved spin space representing the
number states of the four oscillators.  (We may think of $B$ as a Foch 
space, but the components will be mixed by dynamical correlations as
the symplectic semigroup runs.  Also, we need not choose energy 
eigenstates for a basis.)

There is direct mathematical analogy to this structure in the Whitney
sum construction, in which if space $E$ has gauge group $G$, then 
$E\oplus E$ has gauge group isomorphic to $G\times G$.  This analogy 
follows because the orbits of 
$e^{\alpha A}$ and $e^{i\beta B }$ are isomorphic, 
$\alpha , \beta \in\R_+$, $A,B \in \mathfrak{g}$, the Lie algebra of
$G$, and
$\mathfrak{sp}(2n,\R )$ and $i\circ\mathfrak{sp}(2n,\R )$ are isomorphic. 

Identifying  only ``stable'' gauge transformations, 
which will, e.g., take stable states to stable states, one associates 
to each ``block diagonal'' subspace not $Sp(8,\R )_\pm$, but the 
largest compact sub-semigroup of transformations in $Sp(8,\R )_\pm$,
or $U(4)$.  We arrive at $S[U(4)^\C ]$ from $U(4)^\C$ by any of a number of routes:
by insisting only on unimodular (unit Jacobean) transformations so that
one avoids (for now)
imputing any physical content to scale changes or inversions of
coordinate orientations, or in order to preserve the normalized probability
measure, or to obtain a simple connected sub-semigroup (which is
thereby really a group), one identifies a representation of the maximal compact
subgroups $S[U(4)\times U(4)]$ as the ``second quantized'' gauge 
transformations for the four correlated oscillator system whose
structure (semi-)group was $Sp(8,\R )^\C_\pm$.  

The general case of the correct compact
gauge group is deduced from the relationship $U(N)\equiv Sp(2N,\R )
\cap SO(2N)$.  The largest subgroup of unimodular (unit Jacobean),
simply connected group of gauge
transformations is $S[ U(2)\times U(2)]$ for the two oscillator
system.  A similar construction involving three oscillators 
will result in an algebra representation with gauge semigroup 
$S[U(3)_\pm \times U(3)_\pm]$, and above we showed that
four oscillators will yield $S[U(4)_\pm \times U(4)_\pm]$.  The
pattern is obvious. 

The full range of these 
gauge groups is not available for any given transformation, but they 
provide the overall framework for such transformations.  This is because the 
transformations are hamiltonian and operate by geodesic transport (see again the 
constructions in~\cite{II}, and also the generators of the special
unitary groups act geodesically), and so along any particular 
evolution trajectory in the generalized state space
not all quantum numbers can change.  For instance,
there must be some non-zero component along some eigenvector in the
spectral resolution (of the generator of the generalized gauge 
transformation) which is non-vanishing under the gauge transformation, and so
there must be some quantum number which is conserved.  (This is
analogous to saying that $e^{-iHt}\psi$ is not identically zero unless
$\psi$ is orthogonal to {\em all} energy eigenvectors, which requires
that $\psi \equiv 0$ since the energy eigenvectors form a complete
set.)

The maximum allowable group of gauge transformations which are
transitive and dynamical in the case of four fields is thus 
from $S[U(4)\times U(3)]$, for
three fields, $S[U(3)\times U(2)]$, and for two fields $S[U(2)\times
  U(1)]$, all further restricted by the constraint that dynamical
evolution be geodesic.  See Section~\ref{sec:canonical} for more details.  

The implications of this section should be obvious to anyone familiar with the 
Electroweak and Standard Models.  Whether or nor there is any deep lesson 
here for field theory remains to be seen~\cite{note1}.  It is possible 
to construct some 
representations of the Poincar{\'e} group in a rigged Hilbert
space~\cite{nagel}.  There is also a construction for relativistic
Gamow vectors~\cite{antoniou}.  Given the highly generic nature of
our methods, we seem to have confirmation that the Standard Model
has the most general gauge structure one would expect from three
fields in the absence of some new and special non-generic
physics, although these gauge groups represent exact symmetries,
with no suggestion of ``spontaneous symmetry breaking''.  They thus
differ, at least in some details, from the Electroweak Theory and
Standard Model.  We will address the gauge structure further in
Section~\ref{sec:canonical}~\cite{note8}. 








\section{Canonical Variables to Spacetime}\label{sec:spacetimevariables}

Starting with a phase space for four pairs of conjugate variables, we
can construct {\em real} probability amplitudes (densities) 
over it , as sketched in the preceding Section~\ref{sec:spinbundles}.
We have thus constructed a very well behaved spin representation of 
the maximal compact subgroup 
subgroup ($U(4)^\C\equiv U(4)\oplus i\cdot U(4)$)
of the the group of dynamical (=symplectic) transformations on the
complexification of phase space for four pairs of canonical variables
($Sp(8,\R )^\C$), whose action on the
representation space is symplectic (=dynamical) as well.  
We have two structures to relate to spacetime.  We have the four
canonical position coordinates to relate to spacetime coordinates, and
we have a non-trivial dynamical structure containing resonances over
the complex extensions (analytic continuation) of our phase space to
relate to a similar evolution structure over spacetime.  The
most interesting subset of this dynamical structure over phase space
is a special unitary group orbit.

The Clifford algebra of the 
space spanned by the four canonical position
coordinates can be associated
with spacetime structures via the isomorphism $Cl_{O(4,0)} \cong
Cl_{O(1,3)}$~\cite{note3}.
For the dynamical structure, we require a longer chain of
associations.  The largest compact subgroup of $Sp(8,\R )^\C$ is
isomorphic to $U(4)\oplus U(4)$, and we can identify this with a gauge
structure~\cite{note4}.  The Whitney sum rule~\cite{whitney} 
and the restriction to unimodular
transformations gives us a gauge group isomorphic to $S[U(4)\times
U(4)]$, which may be identified with $SU(4,4)\cap U(4)$ up to
isomorphism~\cite{knapp}.  We can consider the
restriction of $SU(4,4)$ to $SO(4,4)$, and note we can represent
$SO(4,4)$ in $Mat(2,\BH )$, treating the quaternions as $4\times 4$
real matrices.  But, $Mat(2,{\BH} ) \cong Cl_{O(1,3)}$, and so we see
that $Cl_{U-O} (\widetilde{T^\times \R^4} ) \cap Cl_{O(1,3)} \supset 
Cl_{U-S} (\widetilde{T^\times \R^4} ) \cap Cl_{O(1,3)} 
\supset S[U(4)\times U(4)]\cap Cl_{O(1,3)} \ne \emptyset $.  We have a
subset of correlated
hamiltonian dynamics over four pairs of canonical variables identified via
isomorphism with a subset of the universal Clifford algebra (geometric
algebra) of a spacetime with local $(+,-,-,-)$
signature.

The $R^{1,3}$ which is the foundation for the $Cl_{O(1,3)}$ above can
be a general riemannian spacetime, and not merely a Minlowski
spacetime, suggesting our formalism is
compatible with general relativity~\cite{snygg}.  By considering the
relatively straightforward chain of textbook isomorphisms above, one
can identify at least a subset of our quantum canonical field theory's
dynamics with ``geometrodynamics'' of a riemannian
spacetime.  What is lacking in the above is any account of hyperbolic
dynamics on either level, or even the demonstration of the existence
of non-trivial connections on this spacetime associated to our
hyperbolic dynamics.  The linkage here needs much more careful exploration than
we will attempt in the present forum. 

The nature of the unanswered questions in this formulation is
illustrated by considering the problems posed in describing ``falling
quantum rocks''.  Quantum dynamical time evolution appears to be 
geodesic in the
space of states as a consequence of Schr{\"o}dinger's equation.  (But,
recall our caution in attributing group properties to the noncompact
generator orbits, because our Lie algebra may in fact generate
something less than a riemannian, or pseudoriemannian, connection.)
However, the dynamical evolution of a falling rock in general
relativity is along an orthogonal to a geodesic.  Thus, our
isomorphism between the full scope of
quantum dynamics and spacetime geometrodynamics
must certainly map dense sets to dense sets, but apparently need not
necessarily preserve {\em all} geodesics.  

There are also conceptual problems
with reductionism in which a subpart of a large system is approximated
as an isolated system, and we should prefer that there be some sort of
analogy, at least, between the process of simplification and
reductionism in our quantum canonical variable dynamics and the
related description in spacetime geometrodynamics.  We have also 
shown a possible
linkage between energy-centric Hamiltonian dynamics and mass-centric
general relativity.












\section{``$SU(4)$ canonical gauge gravity''}\label{sec:canonical}

The intersection of $Sp(8,\R )$ and $O(8)$ is $U(4)= U(1)\times
SU(4)/\mathbb{Z}_4$, so that $U(4)$ has four sheets, just as the
inhomogeneous Lorentz group does.  As indicated previously, there may
be means of extending our well behaved $SU(4)$ structure to larger
structures, and it is interesting to speculate that such an extension
may provide a dynamical analogue to PCT, especially given the
associations with spacetime shown in the previous section.  We shall,
however, choose the path of simplicity for the present and concentrate
on $SU(4)$ only, and identify our basic gauge structure
modulo PCT as $SU(4) \times SU(3)$,   
which we further simplify to consideration of $SU(4)$ or $ SU(3)$
only. 

$SU(4)$ is one of the first quantum symmetries investigated, being
used in nuclear physics~\cite{wigner1, wigner2, wigner3, hund, franzini,
  gursey}.  It also figured as a ``spontaneously broken'' symmetry in an
early competitor to the Standard Model (and also a
GUT)~\cite{pati1, pati2, pati3, pati4, pati5, pati6}, has been 
explored as a spectrum generating algebra~\cite{bohm6} (and references
therein), and has relevance to 
current string orbifold theories~\cite{hanany}.  Our treatment is
dynamically based, the above symmetries are exact, and we abjure any
spontaneous miracles.  Our exact $SU(4)$ gauge theory should have a
lot in 
common with the exact $SU(3)$ symmetry of QCD, and we will conjecture 
that $SU(4)$ is associated with a color chromodynamics of its own.
In place of the ``three color separation'' of the RGB of QCD, we have
a ``four color separation'' we may label CMYK (cyan, magenta, yellow, 
carmine), based on analogy the the color separations of the printing 
industry:  The labels are, of course, arbitrary.  Being a special
unitary group, the even dimensional irreducible representations are
fermionic and the odd dimensional irreducible representations are
bosonic.  We will discuss salient features of the bosonic
representations first: 
\begin{itemize}
\item{Associated with the $SU(4)$ gauge symmetry, we conjecture there
  are 15 color carrying gauge 
  bosons--gluons--with zero rest mass and spin 1.  } 
\item{Being spin representations of $SU(4)$, our bosonic spinors are a
  direct sum of inequivalent bosonic (i.e., odd dimensional) irreps of
  $SU(4)$.  (These irreps need not be 
  UIR's, since they are dynamical, a superset of the unitary irreps.)}
\item{There are four possible $SU(3)$'s contained in $SU(4)$, so
  it is possible to identify the $SU(3)$ gluons of the strong
  interaction with $SU(4)$ gluons more or less directly, e.g.,
  $R\leftrightarrow M$, $G \leftrightarrow Y$ and $B\leftrightarrow
  C$, is typical of the four 
  alternative $SU(4)$ to $SU(3)$ decompositions.  We envisage two,
  three and four color combination particles may be possible.}
\item{Gluonium, glueballs, etc, exist for four colors (and four
  anti-colors) just as they exist for the three colors of standard
  QCD.  We expect like colors to repel and color-anticolor to attract,
  in analogy to the electromagnetic charge case. }
\item{The massive $W^\pm$ and $Z^0$ bosons cannot be gauge bosons,
  both because of their mass and their association with a broken
  symmetry.  We would suggest the $SU(2)$ spontaneous symmetry
  breaking is dynamical and not ``spontaneous'', and related to the
  $SU(4)$ covering symmetry in some way. They must be some
  sort of composite particles--glueballs--in the bosonic perspective.}   
\item{The massless spin 2 graviton, if it exists, may be
  some sort of glueball, or perhaps stem from some special
  representation of $SU(4)$. Note there are
  possible repulsive color interactions, so there 
  may be repulsive field boson interactions attributable to
  the fourth (gravitational) field as well as the attractive
  interaction 
  intermediated by the graviton.   To make the graviton massless
  limits the possible candidates considerably.  }
\item{In QCD, a significant percentage of mass of nucleons is
  associated with quark-gluon plasma.  Although the mixed notions of
  fermionic quark and bosonic gluon violate our notion of separation
  of perspectives, an identification of a bosonic analogue of the
  quark is indicated below.  We would conjecture that
  all mass is the result of color interactions, principally
  intermediated by 
  gluons.  (The other known source of mass, spontaneous symmetry
  breaking, does not seem available in this exact symmetry model.)}  
\item{In our dynamical treatment, we will find only bosons.  There is
  no proper place for quarks in this perspective, although there may
  be bosonic structures related to them (see below).  The bosonic and  
  fermionic $SU(4)$ representations need to be studied for their
  structure, and there should be similarities and correspondences
  between many, if not all, of the structures in the respective
  bosonic and fermionic perspectives.} 
\end{itemize}
As indicated in the last item above, the need for rigorous separation
of perspectives means that our chromodynamics will differ
significantly from QCD, and not just in the addition of one more color
with the attendant possibility of four color combinations in addition
to two and three color combinations already familiar in QCD.  For
instance, the notion of a ``quark-gluon plasma'' seems an oxymoron, 
given the enforced separation of bosonic and fermionic perspectives
(based, in part, on dimensionality of the representations). 

In the fermionic perspective, we adopt quark flavor as the smallest
fermionic analogue to the bosonic color charge, e.g., the
fermionic counterpart of the two color gluon, the two quark meson,
is constructed from the four udsc quark flavors in direct
analogy to the role of the four bosonic  cmyk colors in the gluon
construction, suggesting an identification between structures in the
alternative perspectives exists.  On general principles, it seems like
there should be some sort of identification between bosonic color and 
fermionic flavor, but this seemingly irrestible identification
really does not inescapably mean (at this level of development) that
gluon=meson, although we do favor this simplistic 
identification at this early stage of elaboration of the $SU(4)$ gauge 
theory.  This is an
identification between energy centric gauge bosons associated with
hamiltonian dynamics, and mass centric gauge fermions associated with
orthogonal (Riemannian) geometry.  Both types of
spinors are associated with representations of $SU(4)$.  At a minimum,
this identification does seem to set the basic observables in each
perspective, and establish a sort of invariance as to those basic
observables given the mass-energy equivalence well known from
relativity.  

There is extensive
discussion of the $SU(4)$ symmetry in the charmed baryons article in
the Particle Data Book~\cite{charm} and in the quark model section
also~\cite{quark}.  It is interesting to speculate if the photon and
graviton correspond to neutral mesons at the centers of the central
planes of the two 16-plets in~\cite{quark}.  In any event, the basic
16-plet and 20-plet structures we have pointed to in the Particle Data
Book references should have counterparts in both bosonic and fermionic
representations of $SU(4)$.  

The existence of the charmed quark and the associated extension of
$SU(3)$ flavor to $SU(4)$ supports our proposed four field gauge
structure.  This suggests there is in fact evidence for a quantized
gravitational field at much lower energies than had been thought
possible.  If there are also top and bottom quarks, then either 
there are six fields (a straightforward inference according to this
construction),or the quarks are not the elemental fermionic building
block, or perhaps there is something going on with our fermionic
$SU(4)$ representations we haven't noticed.  Perhaps gravity is field
number five or six on the energy scale, and so on.  At least things
seem compatible with standard QCD and the Electroweak Theory, but the
translation from standard QCD to our four field QCD is not trivial,
and the devil is in the details.

There are a lot of intriguing hints of how things may work out when an
exact four color QCD is fully developed with strict separation of
perspectives.  That is, however, another major undertaking which we
will postpone completion of for another date.  We have gone on long
enough, so will conclude with a
couple of sections delimiting our work in various ways.











\section{Possible Quaternion Structure?}\label{sec:possible}

We began with a pair of spaces in direct sum, each element of which is
a sum of states associated with four correlated oscillators.  If we
were to allow full expression of the
correlations possible between four oscillators, 
each of the states for four
correlated oscillators would have a quaternionic structure.  We have
also made use of an isomorphism involving $Mat(2,{\BH} ) \cong
Cl_{O(1,3)}$, further suggesting a possible 
quaternionic structure.  We will
not make further investigation of quaternionic structures, and
dynamics with quaternionic correlation structures, in this forum, but
will note in passing that quaternionic quantum mechanics is a fairly
mature field, with many points of interest. See, e.g.,~\cite{adler}.
Our reason for avoiding quaternions
is that quaternionic notions would raise tensions, if not
outright conflict, within our fundamental structures.

The equivalence principle of Einstein is related to Mach's principle,
and all that is required for the equivalence principle (``your can't have
mass $A$ without also having a mass $B$'', to paraphrase Wheeler's
well known version) is that $| A\rangle \oplus |B\rangle$ lie in a
space wherein there is a correlation between the $|A\rangle$ and
$|B\rangle$ components.  A Foch space structure alone will not
suffice for the desired correlation.  In the present context, we 
envision a complex symplectic
structure between $|A\rangle$ and$|B\rangle$ components, reflecting 
the real form of our correlated dynamical semigroups, 
$Sp(2n,\R )^\C_\pm$.  (A real symplectic structure might possibly 
suffice, however, 
or, indeed, even an orthogonal correlation).  In light of the
quaternion issues with respect to $|A\rangle$ and $|B\rangle$
suggested above, if $|A\rangle$ and $|B\rangle$ were quaternionic then
the existence of a complex symplectic structure
(complex correlation) between them would raise the specter of
non-associative octonionic structures, with consequent loss of
functoriality of mappings, categoriality within the meaning of the
mathematical theory of categories, and ultimately even the boolean
structure underlying our probability
interpretation~\cite{cox}. vanishes  









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

The concordance between spacetime geometry and the non-trivial
dynamics of quantum fields
shown above concerns that sub-aspect of all possible
dynamics which is identifiable with hyperbolic dynamical
evolution, implying that hyperbolic dynamical evolution is possible in
spacetime isomorphic to the hyperbolic dynamics possible of our canonical
variables.  There are undoubtedly some possible aspects of even
non-trivial dynamics not associated
with influencing the geometry of spacetime (e.g., wholly internal
irreversible transitions of a resonance, at least in toy models and
gedanken experiments), and possible aspects of spacetime geometry which do not have
any non-trivial dynamical significance (e.g., stable or strictly periodic
phenomena). This requires that our cosmological thinking ought to
include consideration of hyperbolic dynamics such as arise in non-conservative 
and/or open cosmologies, should we consider the wavefunction of the
universe.  Possible conservative, closed or cyclic
cosmologies seem excluded by the experimental observations of accelerating
expansion~\cite{perlmutter,garnavich}, and are also in tension 
with the observation of (near) flatness
on cosmological scales of distance. 

The formalism we outline in this series of articles
makes no assumptions about the nature of
particles, their numbers or their conservation, only that there were
some structures on phase space which could be localized in some sense
(so that our measures are finite)
and which possessed non-trivial dynamics~\cite{note6}.
Since we are working with
distributions, evolution is by semigroups, and there is an intrinsic
arrow of time to the formalism (subsequent to the analytic
continuation, which physically introduces correlation into our considerations and
mathematically necessarily places us in the distributions).
Conversely, this arrow of time requires that our quantum desriptions
make use of distributions and the rigged Hilbert space
structure~\cite{qat1,qat2,qat3}, such as we use here.  Since we see
no evidence for stationary or strictly
cyclic cosmological dynamical behavior (in fact, quite
the contrary), and since we do see expansion, we must expect that the
rate of expansion increases exponentially (hyperbolically)--as
apparently it actually does~\cite{perlmutter,garnavich}.  
We have also illustrated
how these hyperbolic dynamical system
results may be obtained from a quantum description of the
universe, with bosons and fermions.  






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%\begin{references}

\bibitem{arnold}
Almost any reasonably advanced book on nonlinear dynamics covers
asymptotic stability, also refrerred to as Lyapunov stability.  See,
e.g., V.I. Arnold, {\em Mathematical Methods of Classical Mechanics}, second 
edition (Springer-Verlag, New York, 1989).  It would be miraculously
unlikely for a conservative dynamical system universe to exhibit both 
uniform cosmological flatness and expansion, the signatures of
relative freedom from randomness and fluctuation (which a conservative
expansion would {\em not} damp out).  

\bibitem{note1}
The author's came to this realization during a
rereading of G. Nicolis and I. Prigogine, {\em Exploring Complexity},
W. H. Freeman, New York (1989).  See page 197f.  The within
adaptation of their presentation would have been unthinkable in 1989.

\bibitem{ref1}
Perlmutter, {\em et al}, {\em Nature}, 1 January, 1998.

\bibitem{ref2}
Garnavich, {\em et al}, {\em Ap. J. Letters}, February 1, 1998.

\bibitem{crumeyrolle}
Albert Crumeyrolle, {\em Orthogonal and Symplectic Clifford Algebras},
Kluwer, Dordrecht (1990).

\bibitem{brackx}
There are Clifford analogues of the
Schwartz and Hardy class functions, which we chose to work with.  
 F. Brackx, R. Delanghe and F. Sommen, {\em Clifford Analysis},
    Research Notes in Mathematics 76, (Pitman Books Ltd., London,
1982).   R. Delange, F. Sommen and
V. Sou{$\check{\textrm{c}}$}ek, {\em Clifford Algebras and Spinor Valued
Functions}, Kluwer, Dordrecht,( 1992).

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

\bibitem{bgm}
See the review A. Bohm, M. Gadella and S. Maxson, {\em Computers Math. Applic.}, {\bf
34}, 427 (1997), and references therein.

\bibitem{note2}
Note that we are
{\em not} in Hilbert space, and $a^\dag$ is not necessarily
the dual or adjoint
operator of $a$.  They are independent continuous operators
(corresponding to the operator representation of a real Witt basis), still
obeying the familiar commutation relations.  Similarly, given that the
Hamiltonian is our generator of time translations according to the
Schr{\"o}dinger equation, $H$ and $iH$ generate transverse orbits,
reflecting to the direct sum structure of our dynamical group,
$Sp(8,\R )^\C$.  This will be elaborated
later. 

\bibitem{note3}
The choice of real spinors corresponds to
the election to use real algebras, which is necessary for uniquely
defined involutions, and taking adjoints may be defined so as to be an
involution, due to the canonical inclusion of Lie algebras into their
duals, $\mathfrak{g} \subset \mathfrak{g}^\times$.  This choice has
far-reaching implications in terms of physical interpretation, but is
essentially just a mathematical uniqueness requirement for the most
general form of operators available (esa but non-symmetric, i.e.,
non-hermitean, operators, since we are not in Hilbert space anymore).
Space precludes full discussion here, but analytic continuation is a
complex symplectic transformation, and not a unitary transformation,
and requires the use of {\em continuous} esa but non-symmetric
operators, {\em on their domain of definition}, in order to properly 
exit the Hilbert
space into the space of distributions.  From \cite{gadella}, we know
analytic continuation must necessarily exit the Hilbert space for the
distributions. 

\bibitem{porteous}
Ian R. Porteous, {\em Clifford Algebras and the Classical
Groups}, Cambridge University Press (1995).

\bibitem{barut}
 A. Barut and R. R{\c{a}}czka, {\em Theory of Group Representations
and Applications}, second edition, World Scientific, Singapore, (1986).

\bibitem{note3x5}
The classical and quantum oscillators are ergodic.  Therefore, the
result of their non-trivial time evolution will be a unique
equilibrium state towards which they are evolving, i.e., the result of
their dynamical time evolution will be an asymptotically stable
state.  In consequence, any non-conservation of probability for the
{\em real} Hilbert space associated with this approach is not a cause
for concern, as it represents the disappearance of the resonance.  The
only failure is to identify or construct the asymptotic equilibrium
whose probability increases complimentary to the decaying probability
of the resonance.  (Note that analytic continuation is associated with
sensitivity to initial conditions.   P.M. Morse and H. Feshbach, 
{\em Methods of Theoretical Physics},
volume I, McGraw-Hill, New York, (1953).  We may not be able to
predict the exact asymptotic state which will occur.)


\bibitem{note4}
This isomorphism is unique up to an
equivalence.  If there are isomorphisms from some aspect of our
dynamical structure into two separate $Cl_{O(1,3)}$ algebras, then
their substructures are isomorphic to each other not only because of
the isomorphism of algebras, but also
by way of their isomorphism to the geometric
algebra of the canonical coordinate space, and so any alternative
structures can differ by only the choice of frame used to describe
them. 

\bibitem{note5}
We can treat the space as if it were obtained
by lifting direct sums of independent states, e.g., number states of a
special sort,for which both
creation and destruction operators are continuous.  The existence of
such states is known.  See chapter 1 of A. B\"ohm, M. Gadella, 
{\em Dirac Kets, Gamow Vectors and 
Gel'fand Triplets}, Lecture Notes in Physics {\bf 348}, Springer-Verlag 
(1989).  These are really semigroups
of gauge transformations, since the distributions are involved, and
groups are not continuously defined.

\bibitem{whitney}
If space $E$ has gauge froup $G$, then $E\oplus E$ has gauge group
isomorphic to $G\times G$.

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

\bibitem{note6}
Since we have
a spacetime, presumably there is a representation of the inhomogeneous
Lorentz group available somewhere which meets Wigner's definition of a
particle. What we have represented is a dynamical structure for
particles, which includes the gauge structure of the
Electroweak and Standard Models as literally arising from
field dynamics.  It also has
similarities and significant differences from the first unified model,
the $SU(4)\times SU(4)$ of J. Pati and A. Salam, {\em
Phys. Rev. Lett.} {\bf 31}, 275 (1973).

\bibitem{bohm}
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, {\em Phys. Lett.} {\bf A189}, 442 (1994); A. Bohm, I. Antoniou,
P. Kielanowski, {\em J. Math. Phys.}
 {\bf 36}, 2593 (1995).

1.  

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

4.  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.  

5. 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.

6.  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.  

7.  G. Ludwig, {\em An Axiomatic Basis of Quantum Mechanics}, volume
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8.  G. Ludwig, {\em An Axiomatic Basis of Quantum Mechanics}, volume
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8.  A. Bohm, J. Math. Phys. {\bf 22}, 2813 (1981); 
A. Bohm, Lett. Math. Phys. 
{\bf 3}, 455 (1978).

10.  P.M. Morse and H. Feshbach, {\em Methods of Theoretical Physics},
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11.  P.R. Garabedian, {\em Partial Differential Equations}, (Wiley, 
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12.  H. Lewy, Math. Ann., {\bf 101}, 609 (1929).   

13.  E. Celeghini, M. Rasetti, M. Tarlini and G. Vitiello, 
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14.  E. Celeghini, M. Rasetti, and G. Vitiello, Ann. Phys. 
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15.  M. Gadella, J. Math. Phys. {\bf 24}, 1462 (1983).

16.  A. B\"ohm, M. Gadella, {\it Dirac Kets, Gamow Vectors and 
Gel'fand Triplets}, Lecture Notes in Physics {\bf 348}, (Springer-Verlag, 
1989).

17.  A.V. Safanov, A.N. Starkov and A.M. Stepin, Chapter 4 in 
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1995).

18.  R. Feynman, Phys Rev. {\bf 84}, 108 (1951).

19. J. Schwinger, Phys Rev. {\bf 91}, 728 (1953).

20. A. Perelomov, {\it Generalized Coherent States and Their 
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21. I.A. Pedrosa and B. Baseia, Phys. Rev. {\bf D30}, 765 (1984), and 
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22. R.G. Newton, Annals of Physics, {\bf 124}, 327 (1980). 

23.  A.J. Bodner, J. Math. Phys. {\bf 38}, 3247 (1997).

24.  A. Mostafazadeh, J. Math. Phys. {\bf 38}, 3489 (1997).

25.  M.A. Naimark, {\it Linear Differential Operators}, Part II, (Engl.
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26.  On extension of symmetric but not self-adjoint operators in physics:
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27.  J.S. Howland, in {\it Mathematical Methods and Applications in 
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28.  M. Ban, J. Math. Phys. {\bf 33}, 3213 (1992).

29.  B. Simon, Intern. J. Quant. Chemistry {\bf XIV}, 529 (1978).

30  G. Lindblad and B. Nagel, Ann. Inst. Henri Poincar\'e (N.S.) 
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31.  JK. Yosida, {\em Functional Analysis}, sixth edition, 
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32.  J. Hilgert, K.H. Hofmann and J.D. Lawson, {\em Lie Groups, 
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33.  M. Reed and B. Simon, {\it Methods of Mathematical Physics, 
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35.  

36.  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
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37.  A. Barut and R. R{\c{a}}czka, {\em Theory of Group Representations
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38.   

39.  J.E. Marsden and T.S. Ratiu, {\em Introduction to Mechanics
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40.   L. H{\"o}rmander, {\em The Analysis of Linear Partial Differential 
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41.  

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

43.  F.W. Warner, {\em Foundations of Differentiable Manifolds and Lie
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44.  

45.  W. Fulton and J. Harris, {\em Representation Theory},
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46.  P.L. Butzer and H. Berens, {\em Semi-Groups of  Operators and 
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47.  

48.  M. Audin, {\em The topology of Torus Actions on Symplectic
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49.  Y. Alhassid, F. G{\"u}rsey, F. Iachello, Ann. Phys. (NY) {\bf 148}, 346 
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50.  P.A.M. Dirac, J. Math. Phys. {\bf 4}, 901 (1963); generators
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51.  Y.S. Kim and M.E. Noz, {\it Phase Space Picture of Quantum M
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52.  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
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53.  B. Simon, {\em
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54.  I.M.Gel'fand and G.E. Shilov, {\it Generalized Functions}, 
vol 2, (Academic Press, 1968), chapt 4.

55.  J. Moser,
{\em Stable and Random Motions in Dynamical Systems}, (Princeton
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57.  A.W. Knapp, {\em Lie Groups Beyond an Introduction}, 
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58.  J. Frank Adams, {\em Lectures on Lie Groups}, (University of
Chicago Press, Chicago, 1969).

59.  T.W. Hungerford, {\em Algebra}, Springer-Verlag, New York, 1974.

60.  V.I. Arnold, {\em Geometrical Methods in the Theory of Ordinary
Differential Equations}, (Springer-Verlag, New York, 1988).

61.  There is a complement to this disucssion in regarding the
difficulty in stabilizing a quantum resonance based on the distance of
its winding numbers from the rationals, e.g., a resonance with a
``Golden Mean'' winding number should be especially difficult to
stabilize.  The complex and real tori are homeomorphic, at least, so
that the classical analysis of stability against perturbation 
of the windings on KAM tori seems likely to have generalizations to
considerations of destabilization of a stable quantum state and  also 
the converse process of
stabilization against resonant decay: controlling quantum chaos.  

62. I.M. Gel'fand and N.Ya. Vilenkin, {\em Generalized Functions}, 
volume 4 (AcademicPress, New York, 1964).

63.  A. Pazy, {\em Semigroups of Linear Operators and Applications to
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64.  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).

65.  See, e.g., D.W. Robinson, {\it Elliptic Operators and Lie 
Groups}, (Clarendon Press, Oxford, 1991).

66.  D.A. Vogan, Jr., {\em Representations of Real Reductive
Lie Groups}, (Birkh{\"a}user, Boston, 1981).

67.  M. Gadella has
privately communicated to the author and others that there could be 
convergence concerns if these sums were not finite. 

68.  W. Rudin, {\it Functional Analysis}, (McGraw-Hill, 1973).

69.  

70.  

71.  R. Abraham, J.E. Marsden and T. Ratiu, {\em Manifolds, Tensor
Analysis and Applications}, (Springer-Verlag, New York, 1988).  

72.   

73.  Ya. G. Sinai, {\em Topics in Ergodic Theory}, (Princeton 
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(Springer-Verlag, New York, 1982).  

75.  R. Ma{\~n}{\'e}, {\em Ergodic Theory of Differentiable 
Dynamics}, (Springer-Verlag, Berlin, 1987).  

76.  Hegerfeldt's theorem, Phys. Rev. Lett. {\bf 72}, 596 (1994), discussing 
excitation probabilities in Hilbert space shows that, for reasonable
observables, either
the probability ${\cal{P}} (t) \equiv 0$ or that ${\cal{P}} (t)$ is
almost always 
non-zero.  ${\cal{P}} (t) \equiv 0$ means that the Hilbert space is
stationary, and this seems the physically preferable alternative [2].   The 
present results support that preference explicitly. 

77.  M.W. Hirsch and S. Smale, {\em Differential Equations, 
Dynamical Systems and Linear Algebra}, (Academic Press, San Diego, 1974).

78.  R.J. Zimmer, {\em Ergodic Theory and Semisimple Groups},
(Birkh{\"a}user, Boston, 1984).

79.  Ya. G. Sinai, Dokl. Akad. Nauk. {\bf{124}}, 768 (1959).

80.  Petersen, reference [72], gives a list of transformations
isomorphic to Bernoilli shifts, with references.

81.  B. Misra, I. Prigogine and M. Courbage, Physica {\bf 98A},
1 (1979).

82.  The use of mathematically local approximations may permit physical
interpretation as physically local approximation as well.  The classical
possibility of invertible systems having seemingly non-invertible subsystems
also comes to mind.  See chapter 9 of [70].   Consider additionally the
symplectic transformations which in some circumstances may reflect a lack of
total isolation during the preparation process during $t\le 0$.  The
possibility of quantum generalizations of Maxwell's demons may be open
if one is careless.
Any physically reasonable interpretation of the mathematics as 
expressing a strong form of the Second Law requires a far more careful
formulation than will be attempted here, and only a weak (local) form of the
Second Law should be inferred from present results.

83.  

84.  N.E. Hurt, {\em Quantum Chaos and Mesoscopic Systems}, (Kluwer
Academnic Publishers, Dordrecht, The Netherlands, 1997).

85.  

88.  

89. 

90.  

91.  

92.

93. B. Yurke, S.L. McCall and J. Klauder, Phys. Rev. A {\bf{33}}, 
4033 (1986).

94.  R.T. Cox, {\em The Algebra of Probable Inference}, (John Hopkins
Press, Baltimore, 1961).  

95.  The Gamow-Jordan vectors were first described in an unpublished
work of I. Antoniou and M. Gadella.  A description will be found in
A. Bohm, M. Gadella, M. Loewe, S. Maxson, P. Patuleanu, C. P{\"u}ntmann, 
J. Math. Phys., {\bf 38}, 6072, (1997).

96.  M.L. Goldberger and K.M. Watson, {\em Collision 
Theory},( Wiley, New York, 196)4.

97.  C. Doran, D. Hestenes, F. Sommer and N. van Archer,
J. Math. Phys., {\bf 34}, 3642, (1993).

98.  H. B. Lawson, Jr., and M-L Michelson, {\em Spin Geometry},
(Princeton University Press, 1989).








\end{document}

\end



jmp4 refs

In the construction of Section~\ref{sec:construction}, the exponential
map was holomorphic, meaning we can think of the resulting complex Lie
semigroupps $Sp(8,\R )^\C_\pm$ as simply connected.  Thus, the first
Chern classes of the semigroups are equal to $0$ (mod 2), and the
semigroup is a {\em spin} manifold inducing a spin structure on the
representation manifold [18, page 87].  (Fixing a spin structure on
the components $E$ and $E^{'}$ fixes a spin structure on their sum
$E^{''}=E\oplus E^{'}$.)
\bigskip
\centerline{\bf References}

1.  S. Maxson, ``Generalized
    Wavefunctions for Dissipative Coherences of Quantum Oscillators I:
    Ergodicity and Chaos'', XXXXXXX.

2.  S. Maxson, ``Generalized 
    Wavefunctions for Dissipative Coherences of Quantum Oscillators II:
    Non-Abelian Gauge Structures'', XXXXXXXX.

3.  See, e.g., I.L. Kantor and A.S. Solodovnikov, {\em Hypercomplex
    Numbers, An Elementary Introduction to Algebras}, (Springer-Verlag,
    Berlin, 1989).

4.  S.L. Adler, {\em Quaternionic Quantum Mechanics and Quantum
    Fields}, (Cambridge University Press, Cambridge, 1995).

5.  R.T. Cox, {\em The Algebra of Probable Inference}, (John Hopkins
    Press, Baltimore, 1961). 

6.  Adler [4] notes that there has been no success in attempts to
    construct a normalized probability measure over the octonians.

7.  O. Teichm{\"u}ller, J. Reine Angew. Math., {\bf 174}, 73 (1935);
    L. Horowitz aand L. Biedenharn, Annals of Physics {\bf 157}, 432
    (1984). 

8.  J.P. Crawford, ``Hypergravity I'' in Bayliss, ed., 
     {\em Clifford (Geometric) Algebras}, (Birkhauser, Boston, 1996),
     pages 341--451.

9.  J.P. Crawford, ``Hypergravity II'' in Bayliss, ed., 
     {\em Clifford (Geometric) Algebras}, (Birkhauser, Boston, 1996),
     pages 353--354.

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

11.  F. Brackx, R. Delanghe and F. Sommen, {\em Clifford Analysis},
    Research Notes in Mathematics 76, (Pitman Books Ltd., London, 1982).

12.  J. Gilbert and M.A.M. Murray, {\em Clifford Algebras and Dirac
     Operators in Harmonic Analysis}, (Cambridge University Press,
     Cambridge, 1991).

13.  See the construction in [1] based on a realization of the
     generators of a non-Abelian Lie algebra in their role as
     derivations on the group.  

14.  Pertti Lounesto, {\em Clifford Algebras}, London Mathematical 
     Society Lecture Notes Series 239, (Cambridge University Press, 
     Cambridge, 1997).

15.  A. B\"ohm, M. Gadella, {\it Dirac Kets, Gamow Vectors and 
     Gel'fand Triplets}, Lecture Notes in Physics {\bf 348}, 
     (Springer-Verlag, 1989).

16.  A. Bohm, M. Gadella and S. Maxson, Computers Math. 
     Applic., {\bf 34}, 427 (1997) or A. Bohm, S. Maxson, M. Loewe, 
     M. Gadella, Physica A{\bf  236}, 485, (1997). 

17.  Compare page 31 of F. G{\"u}rsey and Chia-Hsing Tze, {\it On the
     Role of Division, Jordan and Related Algebras in Particle
     Physics}, (World Scientific, Singapore, 1996).  

18.  H. B. Lawson, Jr., and M-L Michelson, {\em Spin Geometry},
     (Princeton University Press, 1989).

19.  D. Hestenes and G. Sobczyk, {\em Clifford Algebras to Geometric
     Calculus}, (Reidel Pub. Co., Dordrecth, 1984).

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

21.  A. Barut and R. R{\c{a}}czka, {\em Theory of Group Representations
     and Applications}, second edition, (World Scientific, Singapore, 1986).

22.  C. Nash, {\it Differential Topology and Quantum 
     Field Theory}, (Academic Press, 1991).

23.  M. Nakahara, {\it Geometry, Topology and Physics}, (Adam Hilger, 
     Bristol, 1990). 

24.  J.E. Marsden and T.S. Ratiu, {\em Introduction to Mechanics
     and Symmetry}, (Springer-Verlag, NewYork, 1994).   

25.  The importance of initial conditions has been emphasized by
     R. Peierlls and his school.  R. Peierls, {\em Surprises in
     Theoretical Physics}, (Princeton University Press, 1979, chapter
     3.8, and {\em More Surprises in Theoretical Physics}, (Princeton
     University Press, 1992), chapter 5.3.

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

27.  See, e.g., the review A. Bohm, M. Gadella and S. Maxson,
     Computers Math. Applic., {\bf34}, 427 (1997), and references therein.

28.  Note the seeming possible implications for the present work in
     light of the work of Baird, {\it
     Harmonic Maps with Symmetry, Harmonic Morphisms and Deformations
     of Metrics}, Pitman Research Notes in Mathematics (Pitman Books
     Ltd., London, 1983).  There, by an harmonic analysis approach it
     is shown that only for 1, 2, 4, and 8 dimensions can you define
     both orthogonal multiplication and produce harmonic morphisms,
     with $\R$, $\C$, $\BH$ and $\BO$ as the associated scalars.  Our
     $\BH\oplus\BH$ looks very much like a real form of $\BO$, as
     indicated in the text, and our symplectic transformations would
     preserve a weak conformal structure on the $\BO$-linear space (which
     does not admit normalizeable probability measures globally).  Orthogonal
     multiplication is associated with first quantization, e.g.,
     $\langle j,m \vert j^{'} m^{'} \rangle =0$ unless some integer
     relationships hold.  Harmonic morphisms are associated with
     Green's functions, and Green's functions have a resolvent whose
     poles provide the spectrum, and also may be analytically
     continued to define a ``propagator''.




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