\magnification 1200
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\centerline {\bf New Methods and Structures in the Theory}
\vskip 0.3cm
\centerline {\bf of the Multi-Mode Dicke Laser Model}
\vskip 1cm
\centerline {\bf by Giovanni Alli$^{a)}$ and Geoffrey L.
Sewell$^{b)}$}
\vskip 0.5cm
\centerline {\bf Department of Physics, Queen Mary and Westfield
College}
\vskip 0.3cm
\centerline {\bf Mile End Road, London E1 4NS, England}
\vskip 1.5cm
\centerline {\bf Abstract}
\vskip 0.5cm\noindent
We base a treatment of the dissipative, multi-mode version of
Dicke laser model on the theory of completely positive dynamical
semigroups and quantum Markov processes. This
leads to new results at both the mathematical and physical
levels. On the physical side, it provides a generalisation of the
Hepp-Lieb model that admits both chaotic and polychromatic laser
radiation. On the mathematical side, it extends the theory of
dynamical semigroups to a regime where the generators are
perturbed by unbounded derivations.
\vskip 0.5cm\noindent
PACS numbers: 02.20.Mp, 03.65.Db, 05.45.+b, 42.55.Ah
\vfill\eject
\centerline {\bf I. Introduction}
\vskip 0.3cm\noindent
In a seminal work on constructive non-equilibrium statistical
mechanics of open systems, Hepp and Lieb$^{1}$ (HL) proved that
the Dicke laser model, equipped with optical pumps and
reservoirs, undergoes a phase transition from normal to
coherent radiation when the pumping strength reaches a critical
value. This result substantiated earlier, more heuristic, ideas
of Graham and Haken$^{2, 3}$, concerning phase transitions far
from equilibrium. In a more general setting, subsequent works on
the mathematical structure of open quantum systems represented
their dynamics by completely positive (CP) one-parameter
contractive Markovian semigroups$^{4-7}$, which, under
suitable conditions, possess minimal dilations to quantum Markov
processes$^{8-10}$. Evidently, models of the HL type, that
are governed by singular couplings, are {\it prima facie}
candidates for such processes.
\vskip 0.2cm\noindent
In the present article, we provide a new, quantum stochastic
treatment of the {\it multi-mode} dynamical Dicke laser, or
maser, model, with the aim of extending both the physical picture
of HL and the theory of perturbations of CP semigroups. We note
that, since this is a non-equilibrium model, driven by pumps and
sinks, it is radically different from both the multi-mode
equilibrium model of ref. 11 and the internally driven dynamical
ones of refs. 12 and 13. Our main results may be summarised as
follows. On the physical side, we show that the model undergoes
transitions, far from equilibrium, to phases of chaotic and
polychromatic, as well as monochromatic, laser radiation. Thus,
it provides the framework for a theory comprising a rich variety
of radiative structures. On the mathematical side, we extend the
perturbation theory of one-parameter CP semigroups$^{14, 15}$ to
a regime where the generators are perturbed by {\it
unbounded} *-derivations of the algebra of observables.
\vskip 0.2cm\noindent
Our model, ${\Sigma}^{(N,n)},$ consists of $N$ two-level atoms
coupled to $n$ radiation modes, each atom and mode
interacting also with reservoirs that serve to provide optical
pumping and damping. The physics represented by this model is
thus essentially the same as that of HL, though our
mathematical formulation is quite different from theirs.
For, whereas the HL model was a {\it conservative} system,
comprising the atoms, radiation and reservoirs, with their
various interactions, the present model is an {\it open
dissipative} one, in which the action of the reservoirs is
incorporated into the structure of the dynamical semigroup,
governing the evolution of the matter and radiation only.
This formulation enables us to employ the technical machinery of
the theories of CP semigroups and quantum stochastic processes.
\vskip 0.2cm\noindent
We shall present the main structure of the theory in Sections
II-IV, leaving the technical constructions and proofs of
Propositions to Sections V and VI. Thus, in Section II, we shall
formulate the model ${\Sigma}^{(N,n)}$ as a $W^{\star}-$dynamical
system, whose evolution is governed by a CP semigroup of
contractions of its algebra of observables. This semigroup is
constructed from those of the component parts of the system, and
the microdynamics it engenders is specified by Props. 2.3 and
2.4. It will be seen that it provides a concrete example of a
non-quasi-free semigroup with unbounded generator.
\vskip 0.2cm\noindent
In Section III, we specify the macroscopic description of the
model.
Our main results here are given by Props. 3.4 and 3.5, which
serve to reduce the macro-dynamics of the model to a classical
deterministic form, in the limit where $N$ tends to infinity,
with $n$ fixed and finite. In fact, our resultant
phenomenological equations constitute a generalisation
of those of HL.
\vskip 0.2cm\noindent
Section IV encapsulates the physical content of this article at
the
macroscopic level. Here, we show that the phenomenological
equations yield dynamical phase transitions, not only from
incoherent to coherent monochromatic radiation, but also to more
complex phases of optical chaos and multi-mode activation,
which have been discussed in the optics literature$^{3, 16-18}$.
\vskip 0.2cm\noindent
Section V is devoted to the explicit construction of the
dynamical
semigroup of the model ${\Sigma}^{(N,n)},$ via the dilation of
this model to a quantum Markov process, and thence to the proofs
of the main Proposition of Section II. We remark here that, by
Stinespring's theorem$^{19}$, the advent of dilations is
essential
to CP maps.
\vskip 0.2cm\noindent
Section VI is devoted to the proofs of the main Propositions of
Section III. Here, the extraction of a classical, deterministic,
macroscopic law from the underlying quantum dynamics of the
system is achieved by a method, originally devised in the context
of certain hydrodynamical limits$^{20, 21}$.
\vskip 0.2cm\noindent
We conclude, in Section VII with a brief discussion of the
results obtained here and some outstanding problems.
\vskip 0.5cm
\centerline {\bf II. The Model}
\vskip 0.3cm\noindent
Our model is one of matter interacting with radiation, and may
be described as follows. The matter consists of $N$
identical two-level atoms and and radiation of $n$ modes. The
atoms are coupled to the radiation by dipolar interactions.
Furthermore, each element of the model, whether atom or mode, is
coupled to its own reservoir, an atomic reservoir consisting of
a pump and a sink, and a mode reservoir of a sink only (cf. ref.
1).
For physical purposes, we note that the radiation modes could be
photons or phonons, and that, in the latter case, they could be
of the accoustic or optical kind.
\vskip 0.2cm\noindent
We formulate the model as a quantum dynamical system
${\Sigma}=({\cal A},T,{\phi}),$ where ${\cal A}$ is a $W^{\star}-
$algebra of observables, ${\lbrace}T(t){\vert}t{\in}{\bf
R}_{+}{\rbrace}$ is a one-parameter semigroup of normal CP
contractions of ${\cal A},$ and ${\phi}$ is a normal state on
this algebra. We recall here that the generator of a {\it
strongly continuous} CP semigroup of contractions of a
$C^{\star}-$algebra takes the Lindblad form$^{7}$
$$L=i[H,.]_{-}+{\sum}(V_{j}^{\star}(.)V_{j}-
{1\over 2}[V_{j}^{\star}V_{j},.]_{+})\eqno(2.1)$$
where $H,$ which is self-adjoint, the $V_{j}'s$ and
${\sum}V_{j}^{\star}V_{j}$ are elements of the algebra. In the
present context, however, we shall generally be dealing with {\it
weakly continuous}, CP semigroups of contractions of
$W^{\star}-$algebras, with unbounded generators.
\vskip 0.2cm\noindent
We shall now build the model ${\Sigma}$ from its elements.
\vskip 0.3cm\noindent
{\bf A. The Single Atom.} We take the single two-level atom to
be a quantum dynamical system ${\Sigma}_{at}=({\cal
A}_{at},T_{at},{\phi}_{at}),$ with the following specifications.
\vskip 0.2cm\noindent
${\cal A}_{at},$ the $W^{\star}-$algebra of observables of the
atom, consists of the 2-by-2 matrices with complex entries, and
is therefore the linear span of the Pauli matrices
$({\sigma}_{x},{\sigma}_{y},{\sigma}_{z})$ and the identity $I.$
Its structure is thus given by the relations
$${\sigma}_{x}^{2}={\sigma}_{y}^{2}={\sigma}_{z}^{2}=I; \
{\sigma}_{x}{\sigma}_{y}=-{\sigma}_{y}{\sigma}_{x}=i{\sigma}_{z},
\ etc.\eqno(2.2)$$
We define the spin raising and lowering operators
$${\sigma}_{\pm}={1\over 2}({\sigma}_{x}{\pm}i{\sigma}_{y})
\eqno(2.3)$$
${\lbrace}T_{at}(t){\vert}t{\in}{\bf R}_{+}{\rbrace}$ is a
strongly continuous one-parameter semigroup of normal CP
contractions of ${\cal A}_{at},$ representing the dynamics of the
atom. We assume that its generator, $L_{at},$ is of the form
(2.1), where $H={\epsilon}{\sigma}_{z}/2,$ the $V's$ are of the
form $b_{\pm}^{1/2}{\sigma}_{\pm}$ and $b_{z}^{1/2}{\sigma}_{z},$
where ${\epsilon}$ and $b_{\pm}$ are positive constants and
$b_{z}$ is a non-negative one. Thus,
$L_{at}$ is given by the formula
$$L_{at}{\sigma}_{\pm}=-({\gamma}_{\perp}{\mp}i{\epsilon})
{\sigma}_{\pm};
\ L_{at}{\sigma}_{z}=-{\gamma}_{\parallel}({\sigma}_{z}-{\eta}I);
\ L_{at}I=0\eqno(2.4)$$
where
$${\gamma}_{\perp}={1\over 2}(b_{+}+b_{-})+2b_{z}; \
{\gamma}_{\parallel}=(b_{+}+b_{-}); \
{\eta}={b_{+}-b_{-}\over b_{+}+b_{-}}$$
One sees from these equations that $b_{+}$ represents the
strength of the interaction of the atom with a pump, while $b_{-
}, \ b_{z}$ represent its interactions with sinks. Further, the
formulae for ${\gamma}_{\perp}, \ {\gamma}_{\parallel}$ and
${\eta}$ imply that the values of these constants are constrained
by the conditions
$${\gamma}_{\parallel}{\leq}2{\gamma}_{\perp}; \
-1<{\eta}<1\eqno(2.5)$$
Finally, we take ${\phi}_{at}$ to be the unique
$T_{at}-$invariant state on ${\cal A}_{at},$ and this is given
by
$${\phi}_{at}({\sigma}_{z})={\eta}; \
{\phi}_{at}({\sigma}_{\pm})=0\eqno(2.6)$$
We note that, by (2.5), this state is faithful. Further, by
(2.6), the condition that it carries an inverted population is
that ${\eta}>0.$
\vskip 0.2cm\noindent
{\bf Note.} This atomic model is not the same as that of HL,
which was built from a pair of Fermi oscillators in such a way
that the damping constants ${\gamma}_{\parallel}$ and
${\gamma}_{\perp}$ were identical. Here, however, they are
generally different, as is standard in quantum optics.
\vskip 0.3cm\noindent
{\bf B. The Matter.} We now assume that the matter consists of
$N$ non-interacting copies of ${\Sigma}_{at},$ located on the
sites $r=1,. \ .,N$ of a one-dimensional lattice. Thus, to
each site $r,$ we assign a copy, ${\Sigma}_{r}=({\cal
A}_{r},T_{r},{\phi}_{r}),$ of ${\Sigma}_{at},$ and then represent
the matter as the $W^{\star}-$dynamical system
${\Sigma}_{mat}=({\cal A}_{mat},T_{mat},{\phi}_{mat}),$
where the elements of this triple are the tensor products of the
${\cal A}_{r}'s, \ T_{r}'s$ and ${\phi}_{r}'s,$ respectively.
${\cal A}_{mat}$ is therefore faithfully represented as the
linear transformations of the Hilbert space ${\cal H}_{mat}={\bf
C}^{2N}.$
\vskip 0.2cm\noindent
We identify the spin component ${\sigma}_{u,r} \
(u=x,y,z,{\pm}),$ of the atom at $r$ with the element of ${\cal
A}_{mat},$ given by the tensor product of $N$ elements of ${\cal
A}_{at},$ of which the r'th is ${\sigma}_{u}$ and the others are
$I.$ It follows from these specifications and (2.4) that the
generator, $L_{mat},$ of the semigroup $T_{mat}$ is given by
$$L_{mat}{\sigma}_{\pm,r}=-({\gamma}_{\perp}{\mp}i{\epsilon})
{\sigma}_{\pm,r}; \ L_{mat}{\sigma}_{z,r}=
-{\gamma}_{\parallel}({\sigma}_{z,r}-{\eta}I);\
L_{mat}I=0\eqno(2.7)$$
\vskip 0.3cm\noindent
{\bf C. The Radiation.} We assume that the radiation model,
${\Sigma}_{rad},$ corresponds to $n$ modes, with frequencies
${\omega}_{0},. \ .,{\omega}_{n-1},$ each mode being coupled to
its own sink. We formulate ${\Sigma}_{rad}$ as a $W^{\star}-$
dynamical system $({\cal A}_{rad},T_{rad},{\phi}_{rad}),$ as
follows.
\vskip 0.2cm\noindent
First, we construct ${\cal A}_{rad}$ as an algebra generated by
creation and destruction operators for the radiation modes
in the following standard way. We define the Hilbert space ${\cal
H}_{rad}$ and the closed, densely defined operators
${\lbrace}a_{l}^{\star},a_{l}{\vert}l=0,.. \ ,n-1{\rbrace}$ in
this space by the following conditions.
\vskip 0.2cm\noindent
(1) there is a unit vector, ${\Phi}_{rad}$ in ${\cal H}_{rad},$
such that $a_{l}{\Phi}_{rad}=0$ for $l=0,.. \ .,n-1;$
\vskip 0.2cm\noindent
(2) ${\cal H}_{rad}$ is generated by the application to
${\Phi}_{rad}$ of the polynomials in the $a^{\star}$'s; and
\vskip 0.2cm\noindent
(3) the $a'$s and $a^{\star}$'s satisfy the canonical commutation
relations
$$[a_{l},a_{m}^{\star}]_{-}={\delta}_{lm}I; \ [a_{l},a_{m}]_{-}=0
\eqno(2.8)$$
We then define ${\cal A}_{rad},$ the algebra of observables of
${\Sigma}_{rad},$ to be ${\cal L}({\cal H}_{rad}),$ the set of
bounded operators in ${\cal H}_{rad},$ and we take ${\phi}_{rad}$
to be the vacuum state $({\Phi}_{rad},.{\Phi}_{rad}).$
\vskip 0.2cm\noindent
We define the Weyl map $z=(z_{0},.. \
.,z_{n-1}){\rightarrow}W(z)$ of ${\bf C}^{n}$ into ${\cal
A}_{rad}$ by the standard prescription
$$W(z)={\exp}i(z.a+(z.a)^{\star}), \ with \
z.a={\sum}_{l=0}^{n-1}z_{l}a_{l}\eqno(2.9)$$
Thus, by (2.8), $W$ satisfies the Weyl algebraic relation
$$W(z)W(z^{\prime})=W(z+z^{\prime}){\exp}
(iIm(z,z^{\prime})_{n})\eqno(2.10)$$
where $(.,.)_{n}$ is the ${\bf C}^{n}$ inner product.
The algebra of polynomials in
${\lbrace}W(z){\vert}z{\in}{\bf C}^{n}{\rbrace}$ is therefore
just their linear span, and is ultraweakly dense in ${\cal
A}_{rad}.$
\vskip 0.2cm\noindent
We formulate the dynamical semigroup $T_{rad}$ by
Vanheuverszwijn's prescription$^{22, 23}$ for normal quasi-free
CP semigroups of contractions on the CCR algebra. In this scheme,
the action of $T_{rad}(t)$ on $W(z)$ is given by
$$T_{rad}(t)[W(z)]=W({\xi}(t)z){\exp}(-{\theta}(t))\eqno(2.11)$$
where ${\xi}(t):{\bf C}^{n}{\rightarrow}{\bf C}^{n}$ and
${\theta}:{\bf R}_{+}{\rightarrow}{\bf R}_{+}$ are defined in
terms of the frequencies ${\omega}_{l}$ and damping constants
${\kappa}_{l}$ of the modes by the formulae
$$({\xi}(t)z)_{l}=z_{l}{\exp}(-(i{\omega}_{l}+{\kappa}_{l})t)
\ for \ l=0,.. \ .,n-1\eqno(2.12)$$
and
$${\theta}(t)={1\over 2}({\Vert}z{\Vert}_{n}^{2}-
{\Vert}{\xi}(t)z{\Vert}_{n}^{2})
\eqno(2.13)$$
where ${\Vert}.{\Vert}_{n}$ is the ${\bf C}^{n}$ norm. The
generator of the semigroup $T_{rad}$ is
$$L_{rad}={\sum}_{l=0}^{n-1}
(i[{\omega}_{l}a_{l}^{\star}a_{l},.]_{-}
+2{\kappa}_{l}a_{l}^{\star}(.)a_{l}-
{\kappa}_{l}[a_{l}^{\star}a_{l},.]_{+})\eqno(2.14)$$
We note that this is an unbounded version of a Lindblad generator
(cf. (2.1)).
\vskip 0.3cm\noindent
{\bf D. The Interacting System.} This is the system,
${\Sigma},$ formed by coupling ${\Sigma}_{mat}$ to
${\Sigma}_{rad},$ by interactions specified below. We formulate
${\Sigma}$ as a $W^{\star}-$dynamical system $({\cal
A},T,{\phi}),$ as follows.
\vskip 0.2cm\noindent
The algebra of observables ${\cal A}$ is the tensor product,
${\cal A}_{mat}{\otimes}{\cal A}_{rad},$ of those of the matter
and radiation. Thus, ${\cal A}$ is an algebra of operators in the
Hilbert space ${\cal H}={\cal H}_{mat}{\otimes}{\cal H}_{rad}.$
We shall identify ${\cal A}_{mat}{\otimes}I_{rad}$ and
$I_{mat}{\otimes}{\cal A}_{rad}$ with ${\cal A}_{mat}$ and ${\cal
A}_{rad},$ respectively, thus rendering them intercommuting
subalgebras of ${\cal A}.$ Correspondingly, if
$A_{mat}{\in}{\cal A}_{mat}$ and $B_{rad}{\in}{\cal A}_{rad},$
we denote the tensor product $A_{mat}{\otimes}B_{rad}$ by
$A_{mat}B_{rad}.$
\vskip 0.2cm\noindent
We take the state ${\phi}$ to be the tensor product
${\phi}_{mat}{\otimes}{\phi}_{rad}.$ Thus,
$${\phi}(A_{mat}A_{rad})={\phi}_{mat}(A_{mat})
{\phi}_{rad}(A_{rad}) \ {\forall}A_{mat}{\in}
{\cal A}_{mat},A_{rad}{\in}{\cal A}_{rad}\eqno(2.15)$$
We denote by ${\cal N}({\cal A})$ the set of normal states
on ${\cal A}.$ Since ${\cal A}={\cal L}({\cal H}),$ each normal
state ${\psi}$ corresponds to a unique density matrix
${\rho}_{\psi}$ in ${\cal H},$ with
${\psi}{\equiv}Tr({\rho}_{\psi}.).$
Thus, denoting by ${\cal H}_{HS}$ the Hilbert-Schmidt space
${\lbrace}A{\in}{\cal L}({\cal
H}){\vert}Tr(A^{\star}A)<{\infty}{\rbrace},$
with inner product $(A,B)_{HS}=Tr(A^{\star}B)$ and corresponding
norm
${\Vert}.{\Vert}_{HS},$ we may represent ${\psi}$ by
the vector ${\rho}_{\psi}^{1/2}$ in this space, according to the
formula
$${\psi}(A)=({\rho}_{\psi}^{1/2},A{\rho}_{\psi}^{1/2})_{HS} \
{\forall}A{\in}{\cal A}\eqno(2.16)$$
Hence, ${\psi}$ has a canonical extension$^{24}$ to the unbounded
affiliates, $Q,$ of ${\cal A},$ for which
$Q{\rho}_{\psi}^{1/2}{\in}
{\cal H}_{HS},$ i.e.,
$${\psi}(Q)=({\rho}_{\psi}^{1/2},Q{\rho}_{\psi}^{1/2})_{HS}
\eqno(2.16)^{\prime}$$
\vskip 0.2cm\noindent
Turning now to the dynamics, we assume that the coupling between
the matter and radiation is dipolar and corresponds to an
interaction Hamiltonian of the form
$$H_{int}=iN^{-1/2}{\sum}_{r=1}^{N}{\sum}_{l=0}^{n-1}
{\lambda}_{l}({\sigma}_{-,r}a_{l}^{\star}
{\exp}(-2{\pi}ik_{l}r)-h.c.),\eqno(2.17)$$
where $k_{l}$ is the wave-number of the $k'$th mode and the
${\lambda}'s$ are real-valued, $N-$independent coupling
constants. We shall provide further specifications of $k_{l}$ in
${\S}3.$
\vskip 0.2cm\noindent
We note here that the unboundedness of $H_{int}$ presents serious
problems when one seeks to construct the dynamical semigroup,
$T,$ for ${\Sigma}.$ One might envisage, of course, that this
interaction leads simply to a contribution $i[H_{int},.]$ to the
generator of $T,$ so that this takes the form
$$L=L_{mat}+L_{rad}+L_{int}\eqno(2.18)$$
where
$$L_{int}=i[H_{int},.]\eqno(2.19)$$
and $L_{mat},L_{rad}$ are identified with
$L_{mat}{\otimes}I,I{\otimes}L_{rad},$ respectively. However, in
view of the fact that both $L_{rad}$ and $L_{int}$ are unbounded,
it is not clear from existing results on one-parameter semigroups
(e.g. refs. 14, 15, 25, 26) whether the $L$ of this formula would
generate one. In fact, we shall show in ${\S}5$ that it is indeed
the generator of a CP semigroup, $T,$ by a construction based on
the theory of quantum Markov processes$^{9}$.
\vskip 0.2cm\noindent
We shall denote by $T^{\star}({\bf R}_{+})$ the one-parameter
semi-group of transformations of ${\cal N}({\cal A})$ dual to
$T,$ and by ${\psi}_{t}$ the image of a normal state ${\psi}$
under $T^{\star}(t).$ Thus,
$${\psi}_{t}(A){\equiv}(T^{\star}(t){\psi})(A)={\psi}(T(t)A) \
{\forall}A{\in}{\cal A}, \ t{\in}{\bf R}_{+}\eqno(2.20)$$
\vskip 0.2cm\noindent
In order to formulate the evolution of ${\Sigma}$ in states
sufficiently regular to yield well-defined photon statistics,
we introduce the following definitions.
\vskip 0.2cm\noindent
{\bf Definition 2.1.} (1) We define ${\cal M}$ to be the set
of all monomials $a_{l_{1}}^{\sharp}.. \ .a_{l_{m}}^{\sharp},$
where $m$ is an arbitrary positive integer and $a_{l}^{\sharp}$
is either $a_{l}$ or $a_{l}^{\star}.$
\vskip 0.2cm\noindent
(2) We define ${\cal D}^{(0)}$ to be the set of vectors of
${\cal H},$ that lie in the domain of the operators ${\cal M},$
and, for ${\Psi}{\in}{\cal D}^{(0)}$ and fixed $m,$ we define
$M_{m}({\Psi})$ to be the maximum value of
${\lbrace}{\Vert}a_{l_{1}}^{\sharp}.. \
.a_{l_{m}}^{\sharp}{\Psi}{\Vert}{\vert}l_{1}. \
.l_{m}{\in}[0,n-1]{\rbrace}.$
\vskip 0.2cm\noindent
(3) We define ${\cal D}$ to be subset of ${\cal D}^{(0)},$ whose
elements, ${\Psi},$ satisfy the condition that
$${\sum}_{m=1}^{\infty}M_{m}({\Psi})v^{m}/m!<{\infty} \
{\forall}v{\in}{\bf R}_{+}\eqno(2.21)$$
Thus, ${\cal D}$ is dense in ${\cal H}.$
\vskip 0.2cm\noindent
{\bf Definition 2.2.} We define ${\cal F}({\cal A})$ to be the
$^{\star}-$algebra of polynomials in the elements of ${\cal
A}_{mat},$ the Weyl operators $W_{rad}(f)$ and the creation and
destruction operators $a,a^{\star}.$
\vskip 0.2cm\noindent
The following Proposition will be proved in ${\S}5,$ following
the construction there of the semigroup $T.$
\vskip 0.2cm\noindent
{\bf Proposition 2.3.} {\it Let ${\Psi}{\in}{\cal D}$ and let
${\psi}=({\Psi},.{\Psi})$ be the corresponding pure state on
${\cal A}.$ Then the evolute, ${\psi}_{t},$ of ${\psi}$ has a
canonical extension to ${\cal F}({\cal A})$ and satisfies the
equation of motion}
$${d\over dt}{\psi}_{t}(Q)={\psi}_{t}(LQ) \
{\forall}Q{\in}{\cal F}({\cal A})\eqno(2.22)$$
\vskip 0.3cm\noindent
{\bf Extension to Initial Mixed States.} This last result is
easily generalised to initial mixed states by the following
procedure. We define ${\cal D}_{HS}$ to be the subset of ${\cal
H}_{HS},$ whose elements, ${\Theta},$ satisfy the analogue of
(2.21), obtained by replacing ${\Psi},{\Vert}.{\Vert}$ by
${\Theta},{\Vert}.{\Vert}_{HS},$ respectively, in Def. 2.1. In
view of this definition of ${\cal D}_{HS}$ and the representation
(2.16) of mixed states, it is a straightforward matter to
establish the following counterpart of Prop. 2.3 for such states.
\vskip 0.3cm\noindent
{\bf Proposition 2.4.} {\it Let ${\psi}$ be a normal state on
${\cal A}$ corresponding to a density matrix ${\rho},$ which
satisfies the regularity condition that
${\rho}_{\psi}^{1/2}{\in}{\cal
D}_{HS}.$ Then the evolute, ${\psi}_{t},$ of ${\psi}$ has a
canonical extension to ${\cal F}({\cal A}),$ which satisfies the
equation of motion (2.22).}
\vskip 0.5cm\noindent
\centerline {\bf III. The Macroscopic Dynamics}
\vskip 0.2cm\noindent
We formulate the macroscopic description of the model in terms
of the global intensive observables
$$s_{l}^{(N)}=N^{-1}{\sum}_{r=1}^{N}
{\sigma}_{-,r}{\exp}(-2{\pi}ik_{l}r); \ l=0,.. \
.,n-1\eqno(3.1)$$
and
$$p_{l}^{(N)}=N^{-1}{\sum}_{r=1}^{N}
{\sigma}_{z,r}{\exp}(-2{\pi}ik_{l}r); \ l=0,.. \
.,n-1\eqno(3.2)$$
together with the operators
$${\alpha}_{l}^{(N)}=N^{-1/2}a_{l}; \ l=0,.. \ .,n-1\eqno(3.3)$$
corresponding to a scaling of the number operators
$a_{l}^{\star}a_{l}$ in units of $N$ (cf. HL). We denote the set
${\lbrace}s_{l}^{(N)},s_{l}^{(N){\star}},p_{l}^{(N)},p_{l}^{(N
){\star}},{\alpha}_{l}^{(N)},{\alpha}_{l}^{(N){\star}}{\vert}l
=0,1,. \ .,n-1{\rbrace}$ by ${\bf M}^{(N)}.$ To simplify the
model, we choose
$$k_{l}={l\over n}\eqno(3.4)$$
so that ${\bf M}^{(N)}$ is a Lie algebra w.r.t. commutation.
Specifically, by (3.1)-(3.4), its non-zero Lie brackets
are the following ones, and their adjoints.
$$[s_{l}^{(N)},s_{m}^{(N){\star}}]=-N^{-1}p_{[l-m]}^{(N)}; \
[s_{l}^{(N)},p_{m}^{(N)}]=2N^{-1}s_{[l-m]}^{(N)};$$
$$[s_{l}^{(N){\star}},p_{m}^{(N)}]=
-2N^{-1}s_{[l+m]}^{(N){\star}}; \
[a_{l}^{(N)},a_{m}^{(N){\star}}]=N^{-1}I{\delta}_{lm}\eqno(3.5)$$
where $[l{\pm}m]=l{\pm}m \ (mod \ n).$ Thus, the observables
${\bf M}^{(N)}$ become classical in the limit
$N{\rightarrow}{\infty}.$ Further, by (2.2), (2.3), (3.1) and
(3.2),
$${\Vert}s_{l}^{(N)}{\Vert}=1; \ {\Vert}p_{l}^{(N)}{\Vert}=1
\ for \ l=0,. \ .,n-1\eqno(3.6)$$
and
$$p_{0}^{(N){\star}}=p_{0}^{(N)}; \ and \ p_{l}^{(N){\star}}
=p_{n-l}^{(N)} \ for \ l=1,. \ .,n-1\eqno(3.7)$$
\vskip 0.2cm\noindent
By (2.17) and (3.1)-(3.3), the interaction Hamiltonian $H_{int}$
is a function of the macro-observables only, i.e.,
$$H_{int}^{(N)}=iN{\sum}_{l=0}^{n-1}{\lambda}_{l}
({\alpha}_{l}^{(N){\star}}s_{l}^{(N)}-
{\alpha}_{l}^{(N)}s_{l}^{(N){\star}})\eqno(3.8)$$
\vskip 0.2cm\noindent
Our objective will be to extract the dynamics of ${\bf M}^{(N)}$
from the microscopic equation of motion (2.22), in a limit where
$N{\rightarrow}{\infty}$ and $n$ remains fixed and finite. Since
$N$ is not fixed here, we shall indicate the dependence of
${\Sigma},{\cal A},T,{\phi},{\psi},L,H_{int}$ and ${\cal D}$ on
this number by the superscript $(N).$
\vskip 0.2cm\noindent
For finite $N,$ the macroscopic dynamics is governed by the
action of $L^{(N)}$ on ${\bf M}^{(N)}.$ This is given by the
following equations, which ensue from (2.17)-(2.19) and
(3.1)-(3.4).
$$L^{(N)}{\alpha}_{l}^{(N)}=A_{l}^{(N)}; \ L^{(N)}s_{l}^{(N)}
=S_{l}^{(N)}; \ L^{(N)}p_{l}^{(N)}=P_{l}^{(N)}\eqno(3.9)$$
where
$$A_{l}^{(N)}=-(i{\omega}_{l}+{\kappa}_{l}){\alpha}_{l}^{(N)}
+{\lambda}_{l}s_{l}^{(N)}\eqno(3.10a)$$
$$S_{l}^{(N)}=-(i{\epsilon}+{\gamma}_{\perp})s_{l}^{(N)}
+{\sum}_{m=0}^{n-1}{\lambda}_{m}p_{[l-m]}^{(N)}
{\alpha}_{m}^{(N)}\eqno(3.10b)$$
and
$$P_{l}^{(N)}=-{\gamma}_{\parallel}(p_{l}^{(N)}-
{\eta}_{l}^{(N)}I)-2{\sum}_{m=0}^{n-1}{\lambda}_{m}
({\alpha}_{m}^{(N){\star}}s_{[l+m]}^{(N)}+
{\alpha}_{m}^{(N)}s_{[m-l]}^{(N){\star}})\eqno(3.10c)$$
where
$${\eta}_{l}^{(N)}={\eta}{\delta}_{l,0}+O(N^{-1})\eqno(3.10d)$$
\vskip 0.2cm\noindent
We shall assume that the initial state, ${\psi}^{(N)},$ of
${\Sigma}^{(N)}$ corresponds to a vector ${\Psi}^{(N)}$ in ${\cal
D}^{(N)},$ and that the number of photons it carries does not
increase faster than $N,$ i.e. that, for some finite constant
$B,$
$${\psi}^{(N)}({\alpha}_{l}^{(N){\star}}{\alpha}_{l}^{(N)})**From the last equation, we see that $p_{t,l}^{\prime}$ decays to
zero, and so the stability of $p_{t}$ is guaranteed. To test for
stability of ${\alpha}_{t},s_{t},$ it suffices to look at the
solutions of the first two equations of (4.1) for which
${\alpha}_{t,l}^{\prime}$ and $s_{t,l}^{\prime}$ are constant
multiples of ${\exp}(-{\zeta}_{l}t),$ with ${\zeta}_{l}$ a
complex constant. Thus, by (4.1), ${\zeta}_{l}$ is determined by
the roots of the quadratic equation
$$({\zeta}_{l}-{\kappa}_{l}-i{\omega}_{l})
({\zeta}_{l}-{\gamma}_{\perp}-i{\epsilon})-
{\lambda}_{l}^{2}{\eta}=0\eqno(4.2)$$
The real parts of the roots of this equation both positive, i.e.
${\alpha}_{t,l}^{\prime}$ and $s_{t,l}^{\prime}$ decay to zero,
provided that ${\eta}$ is less than the critical value
${\eta}_{l}^{(c)},$ given by
$${\eta}_{l}^{(c)}={{\kappa}_{l}{\gamma}_{\perp}\over
{\lambda}_{l}^{2}}
[1+{({\epsilon}-{\omega}_{l})^{2}\over
({\kappa}_{l}+{\gamma}_{\perp})^{2}}]\eqno(4.3)$$
Hence, the fixed point $x^{(0)}$ is stable provided that ${\eta}$
is less than $min_{l}{\lbrace}{\eta}_{l}^{(c)}{\rbrace}.$
\vskip 0.2cm\noindent
We now assume that this minimum, which we denote by
${\eta}^{(1)},$ is attained at just one value, $k,$ of $l.$ Then,
if ${\eta}$ is increased beyond ${\eta}^{(1)},$ the real part of
a root of (4.2) becomes positive, for $l=k,$ and so the
components ${\alpha}_{k},s_{k}$ of the phase point $x^{(0)}$
become unstable. On the other hand, for ${\eta}>{\eta}^{(1)},$
the equations of motion (3.20) have a periodic solution
$${\alpha}_{t,l}={\alpha}_{k}{\delta}_{l,k}
{\exp}(-i{\nu}t); \
s_{t,l}=s_{k}{\delta}_{l,k}{\exp}(-i{\nu}t); \ p_{t,0}=
{\eta}^{(1)}{\delta}_{l,0}\eqno(4.4)$$
where
$${\nu}={{\gamma}_{\perp}{\omega}_{k}+{\epsilon}{\kappa}_{k}
\over {\gamma}_{\perp}+{\kappa}_{k}}\eqno(4.5)$$
$${\vert}{\alpha}_{k}{\vert}={1\over 2}[{{\gamma}_{\perp}
({\eta}-{\eta}^{(1)})\over {\kappa}_{k}}]^{1\over 2}
\eqno(4.6)$$
and
$$s_{k}=-{{\kappa}_{k}({\gamma}_{\perp}+{\kappa}_{k})+
i({\omega}_{k}-{\epsilon})){\alpha}_{k}\over
{\lambda}_{k}({\gamma}_{\perp}+{\kappa}_{k})}\eqno(4.7)$$
Further, by the Hopf bifurcation theory$^{27, 28}$, this periodic
orbit is stable for ${\eta}-{\eta}^{(1)}$ sufficiently small.
Evidently, it corresponds to coherent laser light in the mode
$l=k.$ Thus, the following result, obtained by HL for the single
mode case, prevails here too.
\vskip 0.3cm\noindent
{\bf Proposition 4.1.} {\it Under the specified conditions,
including the initial pure phase assumption of Prop. 3.5, there
is a Hopf bifurcation in ${\cal K},$ corresponding to a phase
transition in ${\Sigma}$ from a non-radiant state to a coherently
radiating one, as ${\eta}$ increases past the critical value
${\eta}^{(1)}.$}
\vskip 0.3cm\noindent
{\bf B. Further Transitions: Polychromatic and Chaotic Laser
Light.} In the single mode case, it was proved by HL that, in the
particular case that
${\gamma}_{\perp}={\gamma}_{\parallel}={\kappa},$ the periodic
orbit of ${\cal K}$, and hence the coherent monochromatic
radiation of ${\Sigma},$ is stable for all ${\eta}>{\eta}^{(1)}.$
\vskip 0.2cm\noindent
In general, however, it is clear from the theory of classical
dynamical systems$^{29, 30}$ that the equations of motion (3.19)
and (3.20) for ${\cal K}$ provide the framework for further
bifurcations, of the kinds arising in hydrodynamics, as ${\eta}$
is increased. Specifically, we have the following possibilities.
\vskip 0.3cm\noindent
(I) There could be a bifurcation of the simply periodic orbit of
${\cal K}$ into a strange attractor$^{29}$ at some value
${\eta}^{{(2)}}(>{\eta}^{(1)})$ of ${\eta}.$ In this
case, there would be a transition from monochromatic to chaotic
radiation when ${\eta}$ increased beyond the value
${\eta}^{(2)}$ (cf. refs. 16-18).
\vskip 0.3cm\noindent
(II) There could be a succession of bifurcations, corresponding
to the activation of different modes, according to the Landau
mechanism for the onset of turbulence$^{31}$. This would
then correspond to polychromatic radiation, and would simulate
optical chaos when the number of activated modes became very
large.
\vskip 0.3cm\noindent
In fact, we shall now adapt an argument of Haken$^{16}$ to the
present model to show that the scenario (I) is achieved even
when there is only one mode and when the resonance condition
${\epsilon}={\omega}$ is fulfilled. Thus, assuming these
conditions, we note first note that, by (4.3),
$${\eta}^{(1)}={{\kappa}{\gamma}_{\perp}\over
{\lambda}^{2}}\eqno(4.8)$$
Thus, introducing a transformation of variables,
$({\alpha}_{t},s_{t},p_{t}){\rightarrow}(e_{t},c_{t},f_{t}),$ by
the prescription
$$e_{t}=2({{\kappa}\over {\gamma}_{\parallel}({\eta}-
{\eta}^{(1)})})^{1/2}{\alpha}_{t}{\exp}(i{\epsilon}t); \
c_{t}=2{\lambda}({\kappa}{\gamma}_{\parallel}({\eta}-
{\eta}^{(1)}))^{-1/2}s_{t}{\exp}(i{\epsilon}t); \
f_{t}={p_{t}\over {\eta}^{(1)}}\eqno(4.9)$$
where the subscript $l(=0)$ has been omitted, we see from (4.8)
and (4.9) that the equations of motion(3.19) and (3.20) take the
following form.
$${de_{t}\over dt}=-{\kappa}(e_{t}-c_{t}); \ {dc_{t}\over dt}
=-{\gamma}_{\perp}(c_{t}-e_{t}f_{t}); \
{df_{t}\over dt}=-{\gamma}_{\parallel}(f_{t}-g+(g-1)
Re(e_{t}{\overline c}_{t}))\eqno(4.10)$$
where
$$g={{\eta}\over {\eta}^{(1)}}\eqno(4.11)$$
By the first two equations of (4.10),
$$({d\over dt}+{\kappa}+{\gamma}_{\perp})
Im(e_{t}{\overline c}_{t})=0$$
which signifies that $Im(e_{t}{\overline c}_{t})$ is a transient
quantity, which decays exponentially to zero. Hence, for the
long-time dynamics, we may take $e_{t}{\overline c}_{t}$ to be
real, i.e. we may assume that $e_{t}, \ c_{t}$ have the same
phase:-
$$e_{t}=e_{t}^{(0)}{\exp}(i{\beta}_{t}); \
c_{t}=c_{t}^{(0)}{\exp}(i{\beta}_{t})\eqno(4.12)$$
where $e_{t}^{(0)}, \ c_{t}^{(0)}, \ {\beta}_{t}$ are real.
Further, it is a simple consequence of the uniqueness Proposition
3.4(1)
that ${\beta}_{t}$ must be constant. Hence, equns. (4.10)
reduce to the following form for the real-valued variables
$e_{t}^{(0)}, \ c_{t}^{(0)}, \ f_{t}.$
$${de_{t}^{(0)}\over dt}=-{\kappa}(e_{t}^{(0)}-c_{t}^{(0)})
\eqno(4.13a)$$
$${dc_{t}^{(0)}\over dt}=-{\gamma}_{\perp}(c_{t}^{(0)}-
e_{t}^{(0)}f_{t})\eqno(4.13b)$$
$${df_{t}\over dt}=-
{\gamma}_{\parallel}(f_{t}-g+(g-1)(e_{t}^{(0)}c_{t}^{(0)}))
\eqno(4.13c)$$
These are the single-mode Maxwell-Bloch equations. As shown by
Haken$^{16}$, these transform, under a simple linear
transformation to the equations of motion for a Lorenz attractor,
which is known to be undergo a transition to chaos for
appropriate values of the control variables. Specifically, these
Maxwell-Bloch equations support such a transition if (cf. ref.
16)
$${\kappa}>{\gamma}_{\perp}+{\gamma}_{\parallel}; \ and \
{{\eta}\over {\eta}^{(1)}}-1>
{({\gamma}_{\perp}+{\gamma}_{\parallel}+{\kappa})
({\gamma}_{\perp}+{\kappa})\over {\gamma}_{\perp}
({\kappa}-{\gamma}_{\perp}-{\gamma}_{\parallel})}\eqno(4.14)$$
Since these conditions are compatible with the demands
of equns. (2.5), we have the following result.
\vskip 0.3cm\noindent
{\bf Proposition 4.2.} {\it The single-mode model, satisfying the
resonance condition ${\epsilon}={\omega},$ exhibits chaotic
behaviour when the control parameters ${\gamma}_{\perp}, \
{\gamma}_{\parallel}, \ {\kappa}, \ {\lambda}$ lie in a certain
domain.}
\vskip 0.3cm\noindent
{\bf Comment.} The point to emphasise here is that, {\it even in
the single-mode case,} there is a chaotic regime. In the
multi-mode case, the prevalence of additional degrees of freedom
would be expected to be favour chaos still further$^{29, 30}$.
\vskip 0.2cm\noindent
{\bf Semantic Note.} There is a terminological paradox here,
since all the states arising in these scenarios, including the
chaotic ones, are coherent in the sense that they satisfy
Glauber's$^{32}$ factorisation condition in the limit
$N{\rightarrow}{\infty}.$
\vskip 0.5cm
\centerline {\bf V. Construction, via Quantum Stochastics, of
the Semigroup $T$}
\vskip 0.3cm\noindent
We shall now present our construction of the dynamical semigroup
$T.$ Here, in accordance with Stinespring's representation$^{19}$
of CP transformations of $C^{\star}-$algebras, we construct $T$
as the reduction to ${\cal A}$ of a semigroup of isomorphic
mappings of this algebra into a larger one. Thus, we start by
dilating ${\Sigma}_{mat}$ and ${\Sigma}_{rad}$ to quantum
stochastic processes, in the sense defined by Accardi, Frigerio
and Lewis$^{9}$. These have dynamics corresponding to
isomorphisms
${\cal J}_{mat}(t), \ {\cal J}_{rad}(t)$ of ${\cal A}_{mat}, \
{\cal A}_{rad}$ into larger algebras, ${\hat {\cal A}}_{mat}, \
{\hat {\cal A}}_{rad},$ whose reductions to the latter algebras
are $T_{mat}(t), \ T_{rad}(t),$ respectively. We couple
these processes via the interaction Hamiltonian $H_{int},$ and
thereby obtain a process whose dynamics corresponds to isometries
${\cal J}(t)$ of ${\cal A}$ into ${\hat {\cal A}}={\hat {\cal
A}}_{mat}{\otimes}{\hat {\cal A}}_{rad}.$ We then define $T(t)$
to be the reduction of ${\cal J}(t)$ to ${\cal A}$ and prove, in
Prop. 5.4, that $T({\bf R}_{+})$ is indeed a CP semigroup.
\vskip 0.2cm\noindent
{\bf Comment.} Since ${\cal J}$ represents the
conservative evolution of a composite of ${\Sigma}$ and another
system, ${\tilde {\Sigma}},$ one might think of regarding the
latter as a heat bath. From a physical point of view, however,
this is rather unnatural, since some limit proceedure, e.g. that
of Van Hove (cf. ref. 4), is generally required in order that a
system may evolve according to a Markovian semigroup as a result
of its coupling to a thermal reservoir; and, even in the
appropriate limit, this reservoir is generally quite different
from the one arising in the above-described dilation scheme. We
therefore regard that scheme as a mathematical one, designed for
the construction of the semigroup $T.$
\vskip 0.2cm\noindent
Since, in this Section, we shall deal only with the
structure of ${\Sigma}^{(N)}$ for fixed $N,$ we shall lighten the
notation here by dropping all the superscripts $(N)$ referring
to this system.
\vskip 0.3cm\noindent
{\bf A. The Material Process.} By K\"ummerer's
construction$^{10}$,
the system ${\Sigma}_{mat},$ formulated in ${\S}2B,$ has
a minimal dilation to a conservative $W^{\star}-$dynamical
system, ${\hat {\Sigma}}_{mat}=({\hat {\cal A}_{mat}},{\hat
T}_{mat},{\hat {\phi}}_{mat}),$ where ${\hat {\cal A}}_{mat}$
is a $W^{\star}-$algebra, ${\hat {\phi}}_{mat}$ is a faithful
normal state on ${\hat {\cal A}}_{mat}$ and ${\hat T}_{mat}$
is a weakly continuous representation of ${\bf R}$ in ${\hat
{\cal A}},$ such that the following conditions are fulfilled.
\vskip 0.2cm\noindent
(1) ${\hat {\cal A}}_{mat}$ is the $W^{\star}-$tensor product,
${\cal A}_{mat}{\otimes}{\tilde {\cal A}}_{mat},$ of ${\cal
A}_{mat}$ and another $W^{\star}-$algebra, ${\tilde {\cal
A}}_{mat}$. We define the injection ${\cal I}_{mat}$ and, for
$t{\in}{\bf R},$ the isomorphism ${\cal J}(t)$ of ${\cal
A}_{mat}$ into ${\hat {\cal A}}_{mat}$ by the formulae
$${\cal I}_{mat}A=A{\otimes}{\tilde I} \ {\forall}
A{\in}{\cal A}_{mat}\eqno(5.1)$$
and
$${\cal J}_{mat}(t)={\hat T}_{mat}(t){\circ}{\cal I}_{mat}
\eqno(5.2)$$
We shall be concerned exclusively with the restrictions of ${\cal
J}$ and ${\hat T}_{mat}$ to ${\bf R}_{+}.$
\vskip 0.2cm\noindent
(2) ${\hat {\phi}}_{mat}$ is a ${\hat T}_{mat}-$invariant state
on ${\hat {\cal A}},$ which takes the form
${\phi}_{mat}{\otimes}{\tilde {\phi}}_{mat},$ where ${\tilde
{\phi}}_{mat}$ is a normal state on ${\tilde {\cal A}}.$ We
define the projection
${\cal P}_{mat}:{\hat {\cal A}}_{mat}{\rightarrow}{\cal A}_{mat}$
by the formula
$${\cal P}_{mat}(A{\otimes}{\tilde A})=
{\tilde {\phi}}_{mat}({\tilde A})A \ {\forall}A{\in}{\cal
A}_{mat},{\tilde A}{\in}{\tilde {\cal A}}_{mat}\eqno(5.3)$$
\vskip 0.2cm\noindent
(3) $T_{mat}$ is the reduction of ${\hat T}_{mat}$ to ${\cal A},$
given by
$$T_{mat}(t)={\cal P}_{mat}{\circ}{\hat T}_{mat}(t)
{\circ}{\cal I}_{mat}{\equiv}{\cal P}_{mat}{\circ}
{\cal J}_{mat}(t)\eqno(5.4)$$
\vskip 0.2cm\noindent
(4) The triple ${\Pi}_{mat}=({\hat {\cal A}}_{mat},{\cal
J}_{mat},{\hat {\phi}}_{mat})$ is a stationary Markov process,
with conditional expectations (CE's), over ${\cal A}.$ Thus,
defining ${\cal A}_{t,mat}$ to be ${\cal J}_{mat}(t){\cal
A}_{mat},$ and ${\cal A}_{t{\pm},mat}$ to be the
$W^{\star}-$algebras generated by ${\lbrace}{\cal
A}_{u,mat}{\vert}u{\in}[t,{\infty}) \ (resp. \ [0,t]){\rbrace},
\ {\Pi}_{mat}$ is equipped with CE's
${\lbrace}E_{t,mat}:{\hat {\cal A}}_{mat}{\rightarrow}{\cal
A}_{t-,mat}{\vert}t{\in}{\bf R}_{+}{\rbrace},$ that are
compatible with ${\hat {\phi}}_{mat}$ (i.e. ${\hat
{\phi}}_{mat}={\hat {\phi}}_{mat}{\circ}E_{t,mat}$)
and satisfy the Markov condition
$$E_{t,mat}({\cal A}_{t+,mat}){\subset}{\cal A}_{t,mat}$$
\vskip 0.2cm\noindent
Since ${\tilde {\phi}}_{mat}$ is faithful and normal, we may
assume, without loss of generality, that ${\tilde {\cal
A}}_{mat}$ acts in a Hilbert space, ${\tilde {\cal H}}_{mat},$
and that the state ${\tilde {\phi}}_{mat}$ is that of a cyclic
and separating vector, ${\tilde {\Phi}}_{mat},$ in this space.
Thus, ${\tilde {\cal A}}_{mat}{\subset}{\cal L}({\tilde {\cal
H}}_{mat})$ and ${\tilde {\phi}}_{mat}=({\tilde
{\Phi}}_{mat},.{\tilde {\Phi}}_{mat}).$ Correspondingly, ${\hat
{\cal A}}_{mat}{\subset}{\cal L}({\hat {\cal H}}_{mat}),$ where
${\hat {\cal H}}_{mat}={\cal H}_{mat}{\otimes}{\tilde {\cal
H}}_{mat}.$
\vskip 0.3cm\noindent
{\bf B. The Quantum Wiener Process.} We assume that the
radiative modes are subjected to the damping and fluctuating
forces imposed by a Bose field, ${\tilde w},$ corresponding to
a quantum Wiener process. To formulate ${\tilde w}$ (cf. ref.
33),
we start by defining $H$ to be the complex ('single-particle')
Hilbert space,
${\lbrace}f:{\bf R}_{+}{\rightarrow}{\bf C}^{n}{\vert}
{\int}_{0}^{\infty}{\Vert}f(t){\Vert}_{n}^{2}dt<{\infty}
{\rbrace},$ with inner product
$$(f,g)_{H}={\int}_{0}^{\infty}
dt(f(t),g(t))_{n}\eqno(5.5)$$
where, as in ${\S}2, \ (.,.)_{n}$ is the ${\bf C}^{n}$ inner
product. We define the field ${\tilde w}$ and the Fock space
${\tilde {\cal H}}_{rad}$ by the following standard conditions
(cf.(1)-(3) of ${\S}2C). \ {\tilde {\cal H}}_{rad}$ is a complex
Hilbert space and ${\tilde w}$ is a map of $H$ into the
closed, densely-defined operators in ${\tilde {\cal H}}_{rad},$
such that
\vskip 0.2cm\noindent
(1) ${\tilde {\cal H}}_{rad}$ contains a normalised vector
${\tilde {\Phi}}_{rad},$ which is annihilated by the operators
${\tilde w}(f);$
\vskip 0.2cm\noindent
(2) ${\tilde {\Phi}}_{rad}$ is cyclic w.r.t. the polynomial
algebra in ${\lbrace}{\tilde
w}^{\star}(f){\vert}f{\in}H{\rbrace};$ and
\vskip 0.2cm\noindent
(3) ${\tilde w}$ satisfies the CCR,
$$[{\tilde w}(f),{\tilde w}(g)^{\star}]_{-}=(g,f)_{H}I; \
[{\tilde w}(f),{\tilde w}(g)]=0\eqno(5.6)$$
We define the algebra ${\tilde {\cal A}}_{rad}$ to be ${\cal L}
({\tilde {\cal H}}_{rad}),$ and ${\tilde {\phi}}$ to be the state
on this algebra represented by the vector ${\tilde
{\Phi}}_{rad},$ i.e., ${\tilde {\phi}}_{rad}=({\tilde
{\Phi}}_{rad},.{\tilde {\Phi}}_{rad}).$
\vskip 0.2cm\noindent
By our definition of $H,$ this space may be
canonically identified with $L^{2}({\bf R}_{+})^{n},$ each
element $f$ of $H$ being given by a sequence
$(f_{0}, \ .,f_{n-1})$ of $L^{2}({\bf R}_{+})$ vectors. Under
this identification,
$$(f,g)_{H}={\sum}_{l=0}^{n-1}(f_{l},g_{l})
\eqno(5.7)$$
where $(.,.)$ is the $L^{2}({\bf R}_{+})$ inner product. Further,
by linearity, the application of ${\tilde w}$ to $f=(f_{0},. \
.,f_{n-1})$ serves to define linear maps ${\tilde w}_{0},. \
.,{\tilde w}_{n-1}$ from $L^{2}({\bf R}_{+})$ into the operators
in ${\tilde {\cal H}}_{rad}$ by the formula
$${\tilde w}(f_{0},. \ .,f_{n-1})={\sum}_{l=0}^{n-1}{\tilde
w}_{l}(f_{l})
\ {\forall}f_{0},. \ .,f_{n-1}{\in}
L^{2}({\bf R}_{+})\eqno(5.8)$$
It follows immediately from this equation and (5.6) that these
maps,
${\tilde w}_{l},$ satisfy the CCR
$$[{\tilde w}_{k}(p),{\tilde
w}_{l}(q)^{\star}]_{-}=(q,p){\delta}_{kl}; \
[{\tilde w}_{k}(p),{\tilde w}_{l}(q)]_{-}=0 \ {\forall}p,q{\in}
L^{2}({\bf R}_{+})\eqno(5.9)$$
\vskip 0.2cm\noindent
We formulate structure of ${\tilde {\cal A}}_{rad}$ in terms of
the Weyl map ${\tilde W}:H{\rightarrow}{\cal L}({\tilde {\cal
H}}_{rad}),$ defined by the formula
$${\tilde W}(f)={\exp}i({\tilde w}(f)+{\tilde w}(f)^{\star}) \
{\forall} f{\in}H\eqno(5.10)$$
Thus, by (5.6), we may express the CCR as the Weyl relations
$${\tilde W}(f){\tilde W}(g)={\tilde W}(f+g)
{\exp}i(Im(f,g)_{H}) \ {\forall}f,g{\in}H\eqno(5.11)$$
Hence, the algebra of polynomials in the Weyl operators ${\tilde
W}(f)$ is simply their linear span. Further, by standard
arguments, its weak closure is ${\tilde {\cal A}},$ and the state
${\tilde {\phi}}_{rad}$ is completely specified by the formula
$${\tilde {\phi}}_{rad}({\tilde W}(f))=
{\exp}(-{1\over 2}{\Vert}f{\Vert}_{H}^{2}) \ {\forall}
f{\in}H\eqno(5.12)$$
{\bf Note.} It follows easily from equns. (5.7), (5.11) and
(5.12) that, if $f,g$ are elements of $H,$ the intersection of
whose supports is of Lebesgue measure zero, then ${\tilde W}(f)$
and ${\tilde W}(g)$ intercommute and are uncorrelated
in the state ${\tilde {\phi}}_{rad},$ i.e. ${\tilde {\phi}}_{rad}
({\tilde W}(f){\tilde W}(g))={\tilde {\phi}}_{rad}({\tilde W}(f))
{\tilde {\phi}}_{rad}({\tilde W}(g)).$
\vskip 0.2cm\noindent
In order to specify the temporally local structure of ${\tilde
{\cal A}}_{rad},$ we introduce the set, ${\Gamma},$ of closed
intervals in ${\bf R}_{+},$ and, for $J{\in}{\Gamma},$ we define
$J_{c}$ to be the closure of ${\bf R}_{+}{\backslash}J.$ We then
define $H_{J}$ and $H_{J_{c}}$ to be the subspaces of $H$ given
by ${\lbrace}f{\in}H{\vert}supp(f){\in}J \
(resp. \ J_{c}){\rbrace},$ and correspondingly, for $K=J$ or
$J_{c},$ we define $({\tilde {\cal H}}_{K,rad},
\ {\tilde {\Phi}}_{K,rad}, \ {\tilde {\phi}}_{K,rad},
\ {\tilde {\cal A}}_{K,rad})$ to be the objects obtained by
replacing $H$ by $H_{K}$ in the above definitions of $({\tilde
{\cal H}}_{rad}, \ {\tilde {\Phi}}_{rad}, \ {\tilde
{\phi}}_{rad}, \ {\tilde {\cal A}}_{rad}).$ Thus, ${\tilde {\cal
H}}_{rad}, \ {\tilde {\Phi}}_{rad}, \ {\tilde {\cal A}}_{rad},
\ {\tilde {\phi}}_{rad}$ may be canonically identified with the
tensor products ${\tilde {\cal H}}_{J,rad}{\otimes}{\tilde {\cal
H}}_{J_{c},rad}, \ {\tilde {\Phi}}_{J,rad}{\otimes}{\tilde
{\Phi}}_{J_{c},rad}, \ {\tilde {\cal A}}_{J,rad}{\otimes}{\tilde
{\cal A}}_{J_{c},rad}, \ {\tilde {\phi}}_{J,rad}{\otimes}{\tilde
{\phi}}_{J_{c},rad},$ respectively.
\vskip 0.2cm\noindent
Now, it follows from the Note after equn. (5.12) that, if the
interiors of $J,K$ are mutually disjoint and $A,B$ belong to
${\tilde {\cal A}}_{J,rad}$ and ${\tilde {\cal A}}_{K,rad},$
respectively, then ${\tilde {\phi}}_{rad}(AB)$ factorises into
the product ${\tilde {\phi}}_{rad}(A){\tilde {\phi}}_{rad}(B).$
Hence, by our specifications of its local structure, the process
${\tilde w}$ is equipped with conditional expectations (CE's)
${\lbrace}{\tilde E}_{J,rad}:{\tilde {\cal
A}}_{rad}{\rightarrow}{\tilde {\cal
A}}_{J,rad}{\vert}J{\in}{\Gamma}{\rbrace},$ defined by the
formula
$$({\tilde {\Psi}}_{1},{\tilde E}_{J,rad}(A){\tilde {\Psi}}_{2})=
({\tilde {\Psi}}_{1}{\otimes}{\tilde {\Phi}}_{J_{c}},A
({\tilde {\Psi}}_{2}{\otimes}{\tilde {\Phi}}_{J_{c}})), \
{\forall}A{\in}{\tilde {\cal A}}_{rad},
\ {\tilde {\Psi}}_{1},{\tilde {\Psi}}_{2}{\in}{\tilde {\cal
H}}_{J,rad}\eqno(5.13)$$
or, equivalently,
$${\tilde E}_{J,rad}({\tilde W}(f))={\tilde W}({\chi}_{J}f)
{\exp}(-{1\over 2}{\Vert}(1-{\chi}_{J})f{\Vert}_{H}^{2}) \
{\forall}f{\in}H, \ J{\in}{\Gamma}\eqno(5.13)^{\prime}$$
where ${\chi}_{J}$ is the index function for $J,$ acting
multiplicatively on $H.$ These CE's evidently satisfy the
projective condition
$${\tilde E}_{J,rad}{\tilde E}_{K,rad}=
{\tilde E}_{J{\cap}K,rad}\eqno(5.14)$$
and are compatible with ${\tilde {\phi}}_{rad},$ i.e., ${\tilde
{\phi}}_{rad}{\equiv}{\tilde {\phi}}_{rad}{\circ}{\tilde
E}_{J,rad}.$
\vskip 0.3cm\noindent
{\bf C. The Radiation Process.} To couple the field ${\tilde w}$
to the radiation modes of ${\S}2C,$ we define ${\hat {\cal
H}}_{rad}={\cal H}_{rad}{\otimes}{\tilde {\cal H}}_{rad}, \ {\hat
{\cal A}}_{rad}
={\cal A}_{rad}{\otimes}{\tilde {\cal A}}_{rad}{\equiv}
{\cal L}({\hat{\cal H}}_{rad}),$ and identify the operators
$a,W(z),$ in ${\cal H}_{rad},$ and ${\tilde w}(f),{\tilde W}(f),$
in ${\tilde {\cal H}}_{rad},$ with $a{\otimes}{\tilde I},
\ W(z){\otimes}{\tilde I}, \ I{\otimes}{\tilde w}(f),
I{\otimes}{\tilde W}(f),$ respectively, in ${\hat {\cal
H}}_{rad}.$ We then define
the injection
${\cal I}_{rad}:{\cal A}_{rad}{\rightarrow}{\hat {\cal A}}_{rad}$
and the projection ${\cal P}_{rad}:{\hat {\cal
A}}_{rad}{\rightarrow}{\cal A}_{rad}$ by the equations
$${\cal I}_{rad}A=A{\otimes}{\tilde I}
\ {\forall}A{\in}{\cal A}_{rad}\eqno(5.15)$$
and
$${\cal P}_{rad}(A{\otimes}{\tilde A})={\tilde {\phi}}_{rad}
({\tilde A})A \ {\forall}A{\in}{\cal A}_{rad},{\tilde A}{\in}
{\tilde {\cal A}}_{rad}\eqno(5.16)$$
We assume the following Langevin equation of motion for the
oscillators, under the action of the Wiener field ${\tilde w}.$
$$a_{l}(t)-a_{l}+(i{\omega}_{l}+{\kappa}_{l})
{\int}_{0}^{t}dsa_{l}(s)=(2{\kappa}_{l})^{1/2}
{\tilde w}_{l}({\chi}_{[0,t]})\eqno(5.16)$$
The solution of this equation is
$$a_{l}(t)=a_{l}{\exp}(-({\kappa}_{l}+i{\omega}_{l})t)+
(2{\kappa})^{1/2}{\tilde w}_{l}(h_{t,l})\eqno(5.17)$$
where
$$h_{t,l}(s)={\chi}_{[0,t]}(s){\exp}(-({\kappa}_{l}
+i{\omega}_{l})(t-s))\eqno(5.18)$$
Hence, defining $h_{t}=(h_{t,0},. \ .,h_{t,n-1}),$ it follows
from (2.9), (2.12), (5.10) and (5.16) that the transformation
$a{\rightarrow}a(t)$ sends $W(z)$ to $W_{t}(z),$ as defined by
the formula
$$W_{t}(z)=W({\xi}(t)z){\otimes}
{\tilde W}([z.h_{t}]) \ with \
[z.h_{t}]_{l}=z_{l}h_{t,l}\eqno(5.19)$$
We define ${\cal A}_{t,rad}$ and ${\hat {\cal A}}_{rad}$
to be the $W^{\star}-$algebras generated by ${\lbrace}
W_{t}(z){\vert}z{\in}{\bf C}^{n}{\rbrace}$ and ${\lbrace}
W_{t}(z){\vert}z{\in}{\bf C}^{n}, \ t{\in}{\bf R}_{+}{\rbrace}
\ ({\equiv}{\lbrace}{\cal A}_{t,rad}{\vert}t{\in}{\bf
R}_{+}{\rbrace}),$ respectively. In fact, ${\hat {\cal A}}_{rad}$
is the tensor product ${\cal A}_{rad}{\otimes}{\tilde {\cal
A}}_{rad},$ since the linear span of
${\lbrace}[z.h_{t}]{\vert}z{\in}{\bf C}^{n},t{\in}{\bf
R}_{+}{\rbrace}$ is dense in $H$, and
consequently ${\lbrace}{\tilde W}([z.h_{t}]){\vert}z{\in}{\bf
C}^{n},t{\in}{\bf R}_{+}{\rbrace}^{{\prime}{\prime}}=
{\tilde {\cal A}}_{rad}.$
\vskip 0.2cm\noindent
We define ${\cal J}_{rad}(t)$ to be the mapping of ${\cal
A}_{rad}$ onto ${\cal A}_{t,rad}$ given by
$${\cal J}_{rad}(t)W(z)=W_{t}(z)\eqno(5.20)$$
Thus, by (5.17) and (5.18), ${\cal J}_{rad}(t)$ is an
isomorphism. By equns. (2.11), (5.12), (5.14), (5.15) and (5.20),
its relation to $T_{rad}(t)$ is given, analogously with (5.4),
by the formula
$$T_{rad}(t)={\cal P}_{rad}{\circ}{\cal J}_{rad}(t)\eqno(5.21)$$
We define the semigroup, ${\hat T}_{rad}({\bf R}_{+}),$ of
isomorphisms of ${\hat {\cal A}}_{rad}$ by the formula
$${\hat T}_{rad}(t)[({\cal J}_{rad}(t_{1})A_{1}).. \ .
({\cal J}_{rad}(t_{m})A_{m})]=$$
$$({\cal J}_{rad}((t_{1}+t)A_{1}).. \ .({\cal J}_{rad}
(t_{m}+t)A_{m}) \ {\forall}A_{1},. \ .,A_{m}{\in}
{\cal A}_{rad}, \ t_{1},. \ .,t_{m}{\in}{\bf R}_{+}\eqno(5.22)$$
Thus, the dynamical system ${\hat {\Sigma}}_{rad}=({\hat {\cal
A}}_{rad},{\hat T}_{rad},{\hat {\phi}}_{rad})$ is a minimal
dilation of ${\Sigma}_{rad}.$ One checks easily from our
specifications that ${\hat T}_{rad}(t)$ is weakly continuous in
$t$ and that ${\hat {\phi}}_{rad}$ is ${\hat T}_{rad}-$invariant.
\vskip 0.2cm\noindent
To specify the local properties of ${\hat {\Sigma}}_{rad},$ we
define ${\cal A}_{J,rad}$ to be ${\cal A}_{rad}{\otimes}
{\tilde {\cal A}}_{J,rad},$ for $J{\in}{\Gamma},$ and
${\lbrace}E_{J,rad}:{\hat {\cal A}}_{rad}{\rightarrow}
{\cal A}_{J,rad}{\vert}J{\in}{\Gamma}{\rbrace}$ to be the
conditional expectations given by the formula
$$E_{J,rad}=I{\otimes}{\tilde E}_{J,rad}\eqno(5.23)$$
To lighten the notation a little, we shall denote ${\cal
A}_{[0,t],rad}$ and ${\cal A}_{[t,{\infty}),rad}$ by ${\cal
A}_{t{\mp},rad},$ respectively, and put
$$E_{t,rad}{\equiv}E_{[0,t],rad}\eqno(5.24)$$
It follows now from equations (5.13), (5.14), (5.19), (5.23) and
(5.24) that these CE's are compatible with ${\hat {\phi}}_{rad}.$
\vskip 0.3cm\noindent
{\bf Proposition 5.1.} {\it The CE's $E_{t,rad}$ possess the
Markov property
$$E_{t,rad}({\cal A}_{t+,rad}){\subset}{\cal A}_{t,rad}
\eqno(5.25)$$
Thus, by the ${\hat T}_{rad}-$invariance of ${\hat
{\phi}}_{rad},$ the process ${\Pi}_{rad}=({\hat {\cal
A}}_{rad},{\cal J}_{rad},{\hat {\phi}}_{rad})$ is stationary and
Markovian.}
\vskip 0.3cm\noindent
{\bf Proof.} For $t_{1},. \ .,t_{m}{\ge}t$ and $z^{(1)},. \
.,z^{(m)}{\in}{\bf C}^{n},$ it follows from equns. (5.11),
(5.13)$^{\prime},$ (5.19) and (5.23) that
$E_{t,rad}[W_{t_{1}}(z^{(1)}).. \ .W_{t_{m}}(z^{(m)})]$
is a scalar multiple of
$W_{t}({\sum}_{j=1}^{m}{\xi}(t_{j}-t)z_{j})$ and therefore lies
in ${\cal A}_{t,rad}.$ Hence, as the algebra ${\cal
A}_{t_{j},rad}$ is generated by $W_{t_{j}}({\bf C}^{n}),$ the
normality of $E_{t,rad}$ ensures that the Markov condition (5.25)
is fulfilled.
\vskip 0.2cm\noindent
{\bf Note.} Since the vectors ${\Phi}_{rad}$ and ${\tilde
{\Phi}}_{rad}$ lie in the domains of the polynomials in
$a,a^{\star}$ and ${\tilde w}(f),{\tilde w}(f)^{\star},$
respectively, it follows from (5.13), (5.15), (5.16), (5.23) and
(5.24) that ${\cal I}_{rad}, \ {\cal P}_{rad}$ and $E_{t,rad}$
have canonical extensions to polynomials in $a,a^{\star}$ and
their images under ${\lbrace}{\cal J}(t){\rbrace}.$
\vskip 0.3cm\noindent
{\bf D. The Uncoupled Matter-cum-Field Process.} Let
${\Sigma}^{(0)}=({\cal A},T^{(0)},{\phi})$ be the composite of
${\Sigma}_{mat}$ and ${\Sigma}_{rad}$ when these latter systems
are uncoupled. Thus,
$$({\cal A}={\cal A}_{mat}{\otimes}{\cal A}_{rad},
\ T^{(0)}=T_{mat}{\otimes}T_{rad}, \ {\phi}={\phi}_{mat}{\otimes}
{\phi}_{rad}),$$
and hence, by Sections VA,B, ${\Sigma}^{(0)}$
has a minimum dilation to
$${\hat {\Sigma}}^{(0)}=({\hat {\cal A}}=
{\hat {\cal A}}_{mat}{\otimes}{\hat {\cal A}}_{rad}, \
{\hat T}^{(0)}={\hat T}_{mat}{\otimes}{\hat T}_{rad}, \
{\hat {\phi}}={\hat {\phi}}_{mat}{\otimes}{\hat {\phi}}_{rad}).$$
Evidently, ${\hat T}^{(0)}$ is a weakly continuous one-parameter
semigroup of isomorphisms of ${\hat {\cal A}}.$ It follows from
the definitions of ${\S}2$ and the present Section that ${\hat
{\cal A}}={\cal A}{\otimes}{\tilde {\cal A}}$ and ${\hat
{\phi}}={\phi}{\otimes}{\tilde {\phi}},$ where
${\tilde {\cal A}}={\tilde {\cal A}}_{mat}{\otimes}{\tilde {\cal
A}}_{rad}$ and ${\tilde {\phi}}={\tilde
{\phi}}_{mat}{\otimes}{\tilde {\phi}}_{rad}.$ Thus, ${\tilde
{\cal A}}$ and ${\hat {\cal A}}$ are $W^{\star}-$algebras of
operators in the Hilbert spaces ${\tilde {\cal H}}={\tilde {\cal
H}}_{mat}{\otimes}{\tilde {\cal H}}_{rad}$ and ${\hat {\cal
H}}={\hat {\cal H}}_{mat}{\otimes}{\hat {\cal H}}_{rad},$
respectively.
\vskip 0.2cm\noindent
We define the injection ${\cal I}:{\cal
A}{\rightarrow}{\hat {\cal A}},$ the projection ${\cal
P}:{\hat {\cal A}}{\rightarrow}{\cal A}$ and the isomorphisms
${\cal J}^{(0)}(t)$ of ${\cal A}$ into ${\hat {\cal A}}$ to be
${\cal I}_{mat}{\otimes}{\cal I}_{rad}, \ {\cal
P}_{mat}{\otimes}{\cal P}_{rad}$ and ${\cal
J}_{mat}(t){\otimes}{\cal J}_{rad}(t),$ respectively. Hence, by
these definitions and those of ${\S}'s$ 5A,C,
$${\cal I}A=A{\otimes}{\tilde I} \ {\forall}A{\in}{\cal A}
\eqno(5.26)$$
$${\cal P}(A{\otimes}{\tilde A})=
{\tilde {\phi}}({\tilde A})A \ {\forall}A{\in}
{\cal A},{\tilde A}{\in}{\tilde {\cal A}}\eqno(5.27)$$
and
$$T^{(0)}(t)={\cal P}{\circ}{\cal J}^{(0)}(t)\eqno(5.28)$$
For $t{\in}{\bf R}_{+},$ we define ${\cal A}_{t}^{(0)}$
to be ${\cal J}^{(0)}(t){\cal A}$ and ${\cal A}_{t{\pm}}^{(0)}$
to be the $W^{\star}-$algebras generated by ${\lbrace}{\cal
A}_{u}^{(0)}{\vert}u{\geq}(resp. \ {\leq}t){\rbrace}.$ For
$s,t(>s){\in}{\bf R}_{+},$ we define ${\cal A}_{[s,t]}^{(0)}$ to
be the $W^{\star}-$algebra generated by ${\cal
A}_{u}{\vert}s{\leq}u{\leq}t{\rbrace}.$
\vskip 0.2cm\noindent
The dynamics of ${\hat {\Sigma}}^{(0)}$ is given by the
stochastic process ${\Pi}^{(0)}=({\hat {\cal A}},{\cal
J}^{(0)},{\hat {\phi}})$ over ${\cal A},$ which inherits the
stationarity property from its components ${\Pi}_{mat}$ and
${\Pi}_{rad}.$ Evidently, ${\Pi}^{(0)}$ is equipped with
conditional expectations
${\lbrace}E_{t}^{(0)}(=E_{t,mat}{\otimes}E_{t,rad}):{\hat {\cal
A}}{\rightarrow}{\cal A}_{t-}^{(0)}{\vert}t{\in}{\bf
R}_{+}{\rbrace},$ possessing the following standard properties.
\vskip 0.2cm\noindent
(CE1) $$E_{t}^{(0)}(AB)=E_{t}^{(0)}(A)B \ {\forall}A{\in}
{\hat {\cal A}},B{\in}{\cal A}_{t-}^{(0)}$$
\vskip 0.2cm\noindent
(CE2) $$E_{t}^{(0)}E_{s}^{(0)}=E_{s{\wedge}t}^{(0)},\ with \
s{\wedge}t=min{\lbrace}s,t{\rbrace} \ (projectivity)$$
(CE3) $${\hat {\phi}}{\circ}E_{t}^{(0)}={\hat {\phi}} \
(compatibility)$$
Furthermore, the process ${\Pi}^{(0)}$ inherits the Markov
property from its constituents ${\Pi}_{mat}$ and ${\Pi}_{rad},$
i.e.
\vskip 0.2cm\noindent
(CE4) $$E_{t}^{(0)}({\cal A}_{t+}^{(0)}){\subset}
{\cal A}_{t}^{(0)}$$
\vskip 0.2cm\noindent
{\bf Note.} It follows from Def. 2.1 and the observation at the
end of ${\S}5C$ that ${\cal I}, \ {\cal P},$ ${\cal J}^{(0)}(t),
\ T(t)$ and $E_{t}^{(0)}$ have canonical extensions to the
algebra ${\cal F}({\cal A}).$ In particular, by (2.18), the
time-translate of the element $H_{int}$ of this algebra is
$$H_{int}(t)={\cal J}^{(0)}(t)H_{int}=$$
$$iN^{-1/2}{\sum}_{r=1}^{N}{\sum}_{l=0}^{n-1}
{\lambda}_{l}({\sigma}_{-,r}(t)a_{l}^{\star}(t)-h.c.)
\eqno(5.29)$$
with
$${\sigma}_{+,r}(t)={\cal J}_{mat}(t){\sigma}_{+,r} \
and \ a_{l}(t)={\cal J}_{rad}(t)a_{l}\eqno(5.29)^{\prime}$$
this last quantity being given by (5.17).
\vskip 0.3cm\noindent
{\bf E. The Interactive Dynamics.} We shall now formulate the
dynamics of the system ${\hat {\Sigma}},$ formed by coupling
${\hat {\Sigma}}_{mat}$ and ${\hat {\Sigma}}_{rad}$ by the
interactions $H_{int}.$ For this, we employ an interaction
representation, in the form of a two-parameter family
${\lbrace}V(t,s){\vert}t,s{\in}{\bf R}_{+}{\rbrace}$ of unitary
transformation of ${\hat {\cal H}},$ which implement a Markovian
cocycle of automorphisms of ${\hat {\cal A}},$ corresponding to
this coupling.
\vskip 0.3cm\noindent
{\bf Definition 5.2.} (1) For $L{\in}{\bf N},$ we define ${\tilde
{\cal D}}_{L,rad}$ to be the subspace of ${\tilde {\cal
H}}_{rad}$ generated by application to ${\tilde {\Phi}}_{rad}$
of all polynomials of degree $L$ in ${\lbrace}{\tilde
w}(f)^{\star}{\vert}f{\in}H{\rbrace}.$ This is simply the
subspace in which the number of quanta of the field ${\tilde w}$
does not exceed $L.$
\vskip 0.2cm\noindent
(2) We define ${\tilde {\cal D}}_{L}$ to be the subspace ${\tilde
{\cal H}}_{mat}{\otimes}{\tilde {\cal D}}_{L,rad}$ of ${\tilde
{\cal H}},$ and ${\hat {\cal D}}_{L} \ ({\subset}{\hat {\cal
H}})$ to be ${\cal D}{\otimes}{\tilde {\cal D}}_{L},$
respectively,
where ${\cal D}$ is the domain of ${\cal H}$ specified in Def.
2.1.
\vskip 0.2cm\noindent
(3) We define ${\hat {\cal D}}$ to be the dense domain
${\cup}_{L{\in}{\bf N}}{\hat {\cal D}}_{L}$ of ${\hat {\cal H}}.$
\vskip 0.3cm\noindent
{\bf Proposition 5.3.} {\it Let
$$V_{m}(t,s)={\int}_{s}^{t}dt_{1}.. \ .{\int}_{s}^{t}dt_{m}
{\cal T}[H_{int}(t_{1}). \ .H_{int}(t_{m})], \
{\forall}m{\in}{\bf N}\eqno(5.30)$$
with ${\cal T}$ the time-ordering operator, and let
$$V(t,s)={\sum}_{m=0}^{\infty}{(-i)^{m}V_{m}(t,s)\over m!}
\eqno(5.31)$$
Then this sum is strongly convergent on ${\hat {\cal D}}$, and
defines $V(t,s)$ as an isometry of ${\hat {\cal D}}$ into ${\hat
{\cal H}}.$ Moreover, this operator extends by continuity to a
unitary element of ${\cal A}_{[s,t]}^{(0)},$ that is strongly
continuous in both its arguments and satisfies the conditions
$$V(u,t)V(t,s)=V(u,s) \ for \ s{\leq}t{\leq}u\eqno(5.32)$$
and}
$${\hat T}^{(0)}(u)V(t,s)=V(t+u,s+u) \ for \ t,s({\leq}t),u{\in}
{\bf R}_{+}\eqno(5.33)$$
\vskip 0.3cm\noindent
{\bf Definition 5.4.} (1) We define ${\lbrace}{\cal
V}(t,s){\vert}t,s({\leq}t){\in}{\bf R}_{+}{\rbrace}$ to be the
two-parameter family of automorphisms of ${\hat {\cal A}},$
implemented by $V$ according to the formula
$${\cal V}(t,s)=V(t,s)^{\star}(.)V(t,s)\eqno(5.34)$$
\vskip 0.2cm\noindent
(2) We define ${\lbrace}{\hat T}(t){\vert}t{\in}{\bf
R}_{+}{\rbrace}$
to be the family of isomorphisms of ${\hat {\cal A}}$ given by
the formula
$${\hat T}(t)={\cal V}(t,0){\hat T}^{(0)}(t) \
{\forall}t{\in}{\bf R}_{+}\eqno(5.35)$$
\vskip 0.2cm\noindent
(3) We define ${\lbrace}T(t){\vert}t{\in}{\bf R}_{+}{\rbrace}$
to be the family of transformations of ${\cal A}$ by the formula
$$T(t)={\cal P}{\circ}{\hat T}(t){\circ}{\cal I}{\equiv}
{\cal I}^{-1}{\circ}E_{0}^{(0)}{\circ}{\hat T}(t){\circ}{\cal I}
\ {\forall}t{\in}{\bf R}_{+}\eqno(5.36)$$
\vskip 0.3cm\noindent
{\bf Proposition 5.5.} {\it (1) The automorphisms ${\cal V}$
are weakly continuous in both arguments and satisfy the relation
$${\cal V}(t,s){\cal V}(u,t)={\cal V}(u,s) \ for \
s{\leq}t{\leq}u\eqno(5.37)$$
and the cocycle condition
$${\cal V}(t+s,t){\hat T}^{(0)}(t)=
{\hat T}^{(0)}(t){\cal V}(s,0) \
{\forall}s,t{\in}{\bf R}_{+}\eqno(5.38)$$
\vskip 0.2cm\noindent
(2) The isomorphisms ${\hat T}$ form a weakly continuous
one-parameter semigroup, i.e.,}
$${\hat T}(s){\hat T}(t)={\hat T}(s+t) \ {\forall}s,t{\in}
{\bf R}_{+}\eqno(5.39)$$
\vskip 0.2cm\noindent
(3) $T({\bf R}_{+})$ is a weakly continuous, one-parameter
semigroup of contractions of ${\cal A}.$
\vskip 0.3cm\noindent
{\bf Proof of Prop. 5.5, assuming Prop. 5.3.} (1) The
relation (5.37) is an immediate consequence of equns. (5.32) and
(5.34), while (5.38) follows from (5.33)-(5.35). Since
$V(t,s)$ is continous, it follows from (5.34) that ${\cal V}$ is
weakly continuous, in both its arguments.
\vskip 0.2cm\noindent
(2) Since ${\hat T}^{(0)}$ is a one-parameter semigroup, it
follows from equns. (5.35) and (5.36) that so too is ${\hat T}.$
Further, by (5.34) and (5.35), the weak continuity of ${\hat
T}(t)$ follows from that of ${\hat T}^{(0)}(t),$ and the strong
continuity of the unitary $V(t,0).$
\vskip 0.2cm\noindent
(3) Since ${\hat T}(t)$ is an isomorphism of ${\hat {\cal A}},$
it follows immediately from (5.27) and (5.36) that $T(t)$ is a
CP contraction of ${\cal A}.$ To prove the semigroup property of
$T,$ we note that, by (5.35) and (5.36),
$$T(t)T(s)A={\cal I}^{-1}E_{0}^{(0)}{\cal V}(t,0){\hat
T}^{(0)}(t)
E_{0}^{(0)}{\cal V}(s,0){\hat T}^{(0)}(s)(A{\otimes}{\tilde I})
\
{\forall}A{\in}{\cal A}\eqno(5.40)$$
Further, by the stationarity of the process ${\Pi}^{(0)},$
together with the compatibility condition (CE3), ${\hat
T}^{(0)}(t)E_{0}^{(0)}{\equiv}
E_{t}^{(0)}{\hat T}^{(0)}(t),$ and therefore, by equns. (5.33)
and (5.34)
and the semigroup property of ${\hat T}^{(0)},$
$${\hat T}^{(0)}(t)E_{0}^{(0)}{\cal V}(s,0){\hat T}^{(0)}(s)
(A{\otimes}{\tilde I})
{\equiv}E_{t}^{(0)}{\cal V}(s+t,t){\hat T}^{(0)}(s+t)
(A{\otimes}{\tilde I})$$
and this belongs to ${\cal A}_{t}^{(0)},$ by the Markov property
of ${\Pi}^{(0)}.$ Hence, as $V(t,0){\in}{\cal A}_{[0,t]},$
it follows from the fact that $E_{t}^{(0)}$ is a CE from ${\hat
{\cal A}}$ onto ${\cal A}_{[0,t]}$ that
$${\cal V}(t,0)E_{t}^{(0)}{\cal V}(s+t,t){\hat T}^{(0)}(s+t)
(A{\otimes}{\tilde I}){\equiv}E_{t}^{(0)}{\cal V}(t,0)
{\cal V}(s+t,t){\hat T}^{(0)}(s+t)(A{\otimes}{\tilde I})$$
$${\equiv}E_{t}^{(0)}{\cal V}(s+t,0){\hat T}^{(0)}(s+t)
(A{\otimes}{\tilde I}), \ by \ (5.37)$$
Consequently, the r.h.s. of (5.40) is
$${\cal I}^{-1}E_{0}^{(0)}E_{t}^{(0)}{\cal V}(s+t,0){\hat
T}^{(0)}(s+t)(A{\otimes}{\tilde I})$$
and since $E_{0}^{(0)}E_{t}^{(0)}=E_{0}^{(0)},$ by the
projectivity of the CE's, it follows from (5.26) that this is
equal to $T(t+s)A,$ as required.
\vskip 0.3cm\noindent
{\bf Proof of Prop. 5.3.} By equns. (5.17) and (5.29), Defs. 2.1
and 5.4, together with the boundedness of the spins
${\sigma}_{r}$ and the elementary properties of creation and
annihilation operators $a^{\star},{\tilde w}(f)^{\star},a$ and
${\tilde w}(f),$ the action of the operator $H_{int}(t_{1}). \
.H_{int}(t_{m})$ on ${\hat {\cal D}}$ is continuous in all the
$t's,$ and hence, by (5.30), $V_{m}(t,s)$ is continuous there in
both $s$ and $t.$ In particular, the application of $V_{m}(t,s)$
to vectors ${\Psi}{\otimes}{\tilde {\Psi}},$ with
${\Psi}{\in}{\cal D}$ and ${\tilde {\Psi}}{\in}{\tilde {\cal
D}}_{L},$ yields the estimate
$${\Vert}V_{m}(t,s)({\Psi}{\otimes}{\tilde {\Psi}}){\Vert}<
B^{m}{\sum}_{r=0}^{m}{m!\over (m-r)!r!}
M_{r}({\Psi})[L(L+1). \ .(L+m-r)]^{1/2}(t-s)^{m}$$
where $M_{r}({\Psi})$ is defined in Def. 2.1 and $B$ is a finite
constant, whose value depends on $N$ and
the ${\lambda}'s.$ Hence,
$${\sum}_{m=0}^{\infty}{{\Vert}V_{m}(t,s)
({\Psi}{\otimes}{\tilde {\Psi}}){\Vert}\over m!}<
{\sum}_{r=0}^{\infty}M_{r}({\Psi})(B(t-s))^{r}/r! \
{\sum}_{m=0}^{\infty}[(L+1). \ .(L+m)]^{1/2}/m!$$
and, by Def.2.1 (3), this converges, uniformly w.r.t. $s$ and
$t,$ on the compacts in ${\bf R}_{+}^{2}.$ In view of (5.31),
this implies that the continuity of $V_{m}(t,s),$ in both its
arguments, on the vectors ${\Psi}{\otimes}{\tilde {\Psi}}$,
implies that of $V(t,s).$ Hence, by linearity, this operator is
continuous w.r.t. $s$ and $t$ on ${\hat {\cal D}}.$ Further, it
is a straightforward matter to establish the same result for the
operators $H_{int}(t)V(t,s)$ and $V(t,s)H_{int}(s).$
\vskip 0.2cm\noindent
Having established the convergence and continuity properties of
$V$ on ${\hat {\cal D}},$ we infer from (5.30) and (5.31) that
$$V(t,s)=I-i{\int}_{s}^{t}duH_{int}(u)V(u,s)=
I+i{\int}_{s}^{t}duV(t,u)H_{int}(u)$$
on this domain. Hence, by the continuity of the integrands
here,
$$s:{d\over dt}V(t,s)=-iH_{int}(t)V(t,s); \ and \
s:{d\over ds}V(t,s)=iV(t,s)H_{int}(s)\eqno(5.41)$$
on ${\hat {\cal D}}.$ Consequently,
$${d\over dt}(V(t,s){\hat {\Psi}}_{1},V(t,s){\hat {\Psi}}_{2})=0
\ {\forall}{\hat {\Psi}}_{1},{\hat {\Psi}}_{2}{\in}
{\hat {\cal D}}$$
which implies that
$$(V(t,s){\hat {\Psi}}_{1},V(t,s){\hat {\Psi}}_{2})
=({\hat {\Psi}}_{1},{\hat {\Psi}}_{2}) \
{\forall}{\hat {\Psi}}_{1},{\hat {\Psi}}_{2}{\in}
{\hat {\cal D}}$$
since $V(s,s)=I.$ Thus, $V(t,s)$ maps ${\hat {\cal D}}$
isometrically into ${\hat {\cal H}},$ and therefore extends by
continuity to an isometry of ${\hat {\cal H}}.$
\vskip 0.2cm\noindent
To show that this extension, which we also denote by $V(t,s),$
is unitary, it suffices to prove that its adjoint,
$V^{\star}(t,s),$
is also isometric. Now it follows from (5.30) and (5.31) that the
restriction of $V^{\star}(t,s)$ to ${\hat {\cal D}}$ is
$${\sum}_{m=0}^{\infty}
{i^{m}\over m!}{\int}_{s}^{t}dt_{1}.. \ .
{\int}_{s}^{t}dt_{m}{\tilde {\cal T}}
[H_{int}(t_{1}). \ .H_{int}(t_{m})]$$
where ${\tilde {\cal T}}$ is the anti-chronological ordering
operator. Thus, proceeding as in the passage from (5.31) to
(5.40), we obtain the equations
$$s:{d\over dt}V^{\star}(t,s)=
iV^{\star}(t,s)H_{int}(t); \ and \
s:{d\over ds}V^{\star}(t,s)=
-iH_{int}(s)V^{\star}(t,s) \ on \ {\hat {\cal D}}\eqno(5.42)$$
The isometric property of $V^{\star}(t,s)$ now follows from these
equations, by the same argument that we used to establish it for
$V(t,s).$
\vskip 0.2cm\noindent
To show that $V$ satisfies the condition (5.32), we note that,
by (5.41) and (5.42)
$${d\over dt}(V^{\star}(u,t){\hat {\Psi}}_{1},
V(t,s){\hat {\Psi}}_{2})=0 \ {\forall}
{\hat {\Psi}}_{1},{\hat {\Psi}}_{2}
{\in}{\hat {\cal D}}$$
Hence, as $V^{\star}(u,u)=I,$
$$(V^{\star}(u,t){\hat {\Psi}}_{1},V(t,s){\hat {\Psi}}_{2})=
({\hat {\Psi}}_{1},V(u,s){\hat {\Psi}}_{2})$$
for ${\hat {\Psi}}_{1},{\hat {\Psi}}_{2}$ in ${\hat {\cal D}}$,
and hence, by continuity, in ${\hat {\cal H}}.$
This is equivalent to (5.32).
\vskip 0.2cm\noindent
To obtain (5.33), we first infer from equns. (5.30) and (5.31),
together with the definition of $H_{int}(t)$ as ${\cal
J}^{(0)}(t)H_{int}{\equiv}{\hat
T}^{(0)}(t)[H_{int}{\otimes}{\tilde I}],$ that
$${\hat T}^{(0)}(u)V(t,s)=V(t+u,s+u)$$
on ${\hat {\cal D}}$ and hence, by continuity, of ${\hat {\cal
H}}.$
\vskip 0.2cm\noindent
Further, since, as noted above, the restriction of $V(s,t)$ to
${\hat {\cal D}}$ is continuous in both $s$ and $t,$ the same is
true for this operator throughout the space ${\hat {\cal H}}.$
\vskip 0.2cm\noindent
Finally, as $H_{int}(t)(={\hat T}^{(0)}(t)H_{int})$ is
affiliated to ${\hat {\cal A}}_{[s,t]}^{(0)},$ it follows from
(5.29) and (5.30) that, since $V(t,s)$ is bounded, it must belong
to this algebra.
\vskip 0.3cm\noindent
{\bf F. Proof of Proposition 2.3.} Let ${\tilde {\Phi}}={\tilde
{\Phi}}_{mat}{\otimes}{\tilde {\Phi}}_{rad},$ where the two
components of this tensor product are as defined in ${\S}'s$
5A,B. Then, since ${\psi}=({\Psi}.,.{\Psi}),$ it follows from
equns. (2.20), (5.27) and (5.36) that
$${\psi}_{t}(A)={\langle}V(t,0)({\Psi}{\otimes}{\tilde {\Phi}}),
[{\hat T}^{(0)}(t)(A{\otimes}{\tilde I})]V(t,0)
({\Psi}{\otimes}{\tilde {\Phi}}){\rangle}
\ {\forall}A{\in}{\cal A}\eqno(5.43)$$
where ${\langle}.,.{\rangle}$ denotes the ${\hat {\cal H}}$
inner product. By a simple adaptation of the argument, employed
in the proof of Prop. 5.3, to establish that the restriction of
$V(t,s)$ to ${\hat {\cal D}}$ is continuous in both $s$ and $t,$
one readily establishes the same thing for $({\hat
T}^{(0)}(t)(Q{\otimes}{\tilde I})V(t,0)),$ where $Q{\in}{\cal
F}({\cal A}).$ Thus, by (5.43), ${\psi}_{t}$ extends to a state
on ${\cal F}({\cal A}),$ and ${\psi}_{t}(Q)$ is continuous in
$t$ for all $Q{\in}{\cal F}({\cal A}).$
\vskip 0.2cm\noindent
To derive the equation of motion (2.22), we note that, for such
$Q,$ it follows from (5.43) that
\vfill\eject
$$h^{-1}[{\psi}_{t+h}(Q)-{\psi}_{t}(Q)]=$$
$${\langle}[{V(t+h,0)-V(t,0)\over h}]({\Psi}{\otimes}
{\tilde {\Phi}}),[{\hat T}^{(0)}(t+h)(Q{\otimes}{\tilde I})]
V(t+h,0)({\Psi}{\otimes}{\tilde {\Phi}}){\rangle}+\eqno(5.44a)$$
$${\langle}V(t,0)({\Psi}{\otimes}{\tilde {\Phi}}),
[{\hat T}^{(0)}(t+h)(Q{\otimes}{\tilde I})]({V(t+h,0)-V(t,0)\over
h})
({\Psi}{\otimes}{\tilde {\Phi}}){\rangle}+\eqno(5.44b)$$
$${\langle}V(t,0)({\Psi}{\otimes}{\tilde {\Phi}}),
[({{\hat T}^{(0)}(t+h)-{\hat T}^{(0)}(t)\over
h})(Q{\otimes}{\tilde I})]
V(t,0)({\Psi}{\otimes}{\tilde {\Phi}}){\rangle}\eqno(5.44c)$$
In our treatment of these three terms, as in the proof of Prop.
5.3, we shall repeatedly utilise the fact that, on ${\hat {\cal
D}},$ the polynomials in the operators $a^{\sharp}(t)$ and
$V(t,s)$ are continuous in all their arguments.
\vskip 0.2cm\noindent
Thus, these continuity properties, together with (5.41), imply
that, as $h{\rightarrow}0,$
$$Term \ (a){\rightarrow}-{\langle}iH_{int}(t)V(t,0)
({\Psi}{\otimes}{\tilde {\Phi}}),
[{\hat T}^{(0)}(t)(Q{\otimes}{\tilde I})]
V(t,0)({\Psi}{\otimes}{\tilde {\Phi}}){\rangle}$$
$${\equiv}i{\langle}V(t,0)({\Psi}{\otimes}{\tilde {\Phi}}),
[{\hat T}^{(0)}(t)(H_{int}Q{\otimes}{\tilde I})]
V(t,0)({\Psi}{\otimes}{\tilde {\Phi}}){\rangle}$$
since $H_{int}(t)=T^{(0)}(t)(H_{int}{\otimes}{\tilde I}),$
and, by (2.20) and (5.36), this expression is equal to $
i{\psi}_{t}(H_{int}Q).$ Similarly, the term (b) tends to
$-i{\psi}_{t}(QH_{int})$ as $h{\rightarrow}0,$ and consequently,
by (2.19),
$$(a)+(b){\rightarrow}{\psi}_{t}(L_{int}Q) \ as \
h{\rightarrow}0\eqno(5.45)$$
Further, by our definitions of the conditional expectations
$E_{t}^{(0)},$ and in view of the continuity properties, on
${\hat {\cal D}},$ of polynomials in the operators
$a^{\sharp}(t)$ and $V(t,s),$ term (c) may be re-expressed as the
${\cal H}$ inner product
$${\langle}{\Psi},{\cal I}^{-1}E_{0}^{(0)}[V(t,0)^{\star}
({{\cal J}^{(0)}(t+h)Q-{\cal J}^{(0)}(t)Q\over
h})V(t,0)]{\Psi}{\rangle}$$
Further, in view of the properties (CE1-3) of $E_{t}^{(0)},$
specified in ${\S}5D,$ this is identical to
$${\langle}{\Psi},{\cal I}^{-1}E_{0}^{(0)}E_{t}^{(0)}
[V(t,0)^{\star}({{\cal J}^{(0)}(t+h)Q-{\cal J}^{(0)}(t)Q\over h})
V(t,0)]{\Psi}{\rangle}$$
$${\equiv}{\langle}{\Psi},{\cal I}^{-1}E_{0}^{(0)}[V(t,0)^{\star}
E_{t}^{(0)}[{{\cal J}^{(0)}(t+h)Q-{\cal J}^{(0)}(t)Q\over h}]
V(t,0)]{\Psi}{\rangle}$$
$${\equiv}{\langle}{\Psi},{\cal I}^{-1}E_{0}^{(0)}[V(t,0)^{\star}
[{\hat T}^{(0)}(t)E_{0}^{(0)}({{\cal J}^{(0)}(h)Q-Q\over h})]
V(t,0)]{\Psi}{\rangle}$$
and, by (5.36), this is equal to
$${\langle}{\Psi},T(t){\cal P}
({{\cal J}^{(0)}(h)Q-Q\over h}){\Psi}{\rangle}$$
Hence, as ${\cal P}{\circ}{\cal
J}^{(0)}(h)=T_{mat}(h){\otimes}T_{rad}(h),$ it follows that
$$term \ (c){\rightarrow}{\langle}{\Psi},[T(t)(L_{mat}+L_{rad})Q]
{\Psi}{\rangle}{\equiv}{\psi}_{t}((L_{mat}+L_{rad})Q)
\ as \ h{\rightarrow}0\eqno(5.46)$$
The required result (2.22) is an immediate consequence of equns.
(5.44)-(5.46).
\vskip 0.5cm
\centerline {\bf VI. The Macroscopic Classical Limit.}
\vskip 0.3cm\noindent
We begin with two lemmas, that will be needed for the proofs of
Props. 3.2 and 3.4; and we shall then prove those Propositions
in reverse order, in accordance with their logical relationship.
\vskip 0.3cm\noindent
{\bf Lemma 6.1.} {\it Let ${\xi}$ be a continuous, positive-
valued function on ${\bf R}_{+},$ which satisfies the equation
$${\xi}(t)={\xi}(0){\exp}(-kt)+{\int}_{0}^{t}ds
{\theta}(s){\exp}(-k(t-s))\eqno(6.1)$$
where $k$ is a positive constant and ${\xi}(0),\ {\theta}$
satisfy the conditions
$${\xi}(0){\leq}B_{1}\eqno(6.2)$$
and
$${\vert}{\theta}(s){\vert}{\leq}B_{2}[{\xi}(s)]^{1/2}
\eqno(6.3)$$
and $B_{1}, \ B_{2}$ are finite constants. Then there are
finite positive constants $B, \ C,$ whose values depend on $k,
\ B_{1}, \ B_{2}$ only, such that
\vskip 0.2cm\noindent
(1) the function ${\xi}$ is majorised by $B;$ and
\vskip 0.2cm\noindent
(2)} $${\xi}(0){\leq}({\xi}(t)+C){\exp}(kt) \ {\forall}t{\in}
{\bf R}_{+}\eqno(6.4)$$
\vskip 0.3cm\noindent
{\bf Proof.} By (6.1)-(6.3),
$${\xi}(t){\leq}B_{1}{\exp}(-kt)+B_{2}{\int}_{0}^{t}ds
{\exp}(-k(t-s))[{\xi}(s)]^{1/2}\eqno(6.5)$$
and
$${\xi}(0){\exp}(-kt)-B_{2}{\int}_{0}^{t}ds
{\exp}(-k(t-s))[{\xi}(s)]^{1/2}{\leq}{\xi}(t)\eqno(6.6)$$
Hence, defining $M_{t}$ to be
$max{\lbrace}{\xi}(s){\vert}s{\in}[0,t]{\rbrace},$ it follows
from (6.5) that
$$M_{t}{\leq}B_{1}{\exp}(-kt)+k^{-1}B_{2}M_{t}^{1/2}
(1-{\exp}(-kt)){\leq}B_{1}+k^{-1}B_{2}M_{t}^{1/2}$$
and consequently that
$$M_{t}^{1/2}{\leq}{1\over 2}
(k^{-1}B_{2}+[k^{-2}B_{2}^{2}+B_{1}]^{1/2})$$
This implies the result (1), with $B$ equal to the square of the
r.h.s. of this last inequality.
\vskip 0.2cm\noindent
It follows now from this result and (6.6) that
$${\xi}(0){\exp}(-kt){\leq}{\xi}(t)+k^{-1}B_{2}B^{1/2}
(1-{\exp}(-kt)){\leq}{\xi}(t)+k^{-1}B_{2}B^{1/2}$$
This implies the required result (2).
\vskip 0.3cm\noindent
{\bf Lemma 6.2.} {\it Assuming the equation of motion (2.22) and
the initial condition (3.11), the time-dependent expectation
value of ${\alpha}^{(N){\star}}{\alpha}^{(N)}$ is uniformly
bounded w.r.t. $N$ and $t,$ i.e., for some finite constant $D,$}
$${\psi}_{t}^{(N)}({\alpha}_{l}^{(N){\star}}{\alpha}_{l}^{(N)})G;$ and, in this
case, by (3.19), it also satisfies (3.20). This establishes the
existence of a solution of the latter equations. To prove its
uniqueness, we proceed as in our treatment of (6.10), and show
that, for ${\cal K},$ too, the evolution $x{\rightarrow}x_{t}$
maps compacts, $K,$ into compacts, $K^{\prime},$ for all positive
$t.$ Therefore, since, by (3.19), the functions $A,S,P$ and their
derivatives of all orders are uniformly bounded on the compacts,
it follows from standard fixed point methods that the solution
$x_{t}$ of (3.20) is both unique and differentiable w.r.t. the
initial data $x_{0}.$ Evidently, this solution defines a one-
parameter semigroup ${\tau}({\bf R}_{+})$ of transformations of
$X$ according to the prescription $x_{t}={\tau}(t)x_{0}.$
\vskip 0.2cm\noindent
(2) On applying Lemma 6.1(2) to equn. (6.12), and then using
equns. (6.9) and (6.11), we see that, for $t{\in}{\bf R}_{+},$
the inverse image, under ${\tau}(t),$ of any compact region of
$X$ is itself compact. Hence, as we have seen that ${\tau}(t)x$
is differentiable w.r.t. $x,$ it follows that ${\cal
C}_{0}^{(1)}(X)$ is stable under the transformations
$f{\rightarrow}f{\circ}{\tau}(t).$ We define
$$f_{t}=f{\circ}{\tau}(t)eqno(6.14)$$
and
$$F(s,t)={\int}_{X}dm_{s}f_{t}\eqno(6.15)$$
In view of the semi-group property of ${\tau}({\bf R}_{+}),$ it
follows from equns. (3.17), (3.18), (6.14) and (6.15) that
$${{\partial}\over {\partial}t}F(s,t)=
{{\partial}\over {\partial}s}F(s,t)=
{\int}_{X}dm_{s}{\cal L}f_{t} \ {\forall}
f{\in}{\cal C}_{0}^{(1)}(X); \ s,t{\in}{\bf R}_{+}$$
from which it follows that $F(s,t){\equiv}F(t+s,0),$ and
therefore that $F(t,0){\equiv}F(0,t).$ Thus, by (6.14) and
(6.15),
$${\int}_{X}dm_{t}f={\int}_{X}dm_{0}f{\circ}{\tau}(t)$$
for $f{\in}{\cal C}_{0}^{(1)}(X)$ and hence, by continuity, for
$f{\in}{\cal C}_{0}(X).$
\vskip 0.3cm\noindent
Our proof of Prop. 3.2 will be based on the method devised in
refs. 20, 21 for other models. Here, we shall omit details of
some straightforward, but rather tedious, manipulations employed
in the proof.
\vskip 0.3cm\noindent
{\bf Proof of Prop. 3.2.} (1) Since, by equations (3.5) and
(3.6), the norms of all the commutators of the Lie algebra ${\bf
M}^{(N)}$ are $O(N^{-1}),$ it follows easily from equations
(2.22), (3.9), (3.10), (3.14) and (3.15) that, for
$(z,w,v)$ in a compact region of $X,$
$${{\partial}{\mu}_{t}^{(N)}\over {\partial}z}=
i{\psi}_{t}^{(N)}(U^{(N)}{\alpha}^{(N)})+O(N^{-1})\eqno(6.16a)$$
$${{\partial}{\mu}_{t}^{(N)}\over {\partial}w}=
i{\psi}_{t}^{(N)}(U^{(N)}s^{(N)})+O(N^{-1})\eqno(6.16b)$$
$${{\partial}{\mu}_{t}^{(N)}\over {\partial}v}=
i{\psi}_{t}^{(N)}(U^{(N)}p^{(N)})+O(N^{-1})\eqno(6.16c)$$
and
$${{\partial}{\mu}_{t}^{(N)}\over {\partial}t}=
i{\psi}_{t}^{(N)}(U^{(N)}
([z.A^{(N)}+w.S^{(N)}+h.c.]+v.P^{(N)}))+O(N^{-1})
\eqno(6.17)$$
where the arguments $z,w,v$ of $U^{(N)}$ and
${\mu}_{t}^{(N)}$ have been omitted. In view of Lemma 6.2 and the
unitarity of $U^{(N)},$ it follows from equns. (3.10), (6.16) and
(6.17) that both ${\mu}_{t}^{(N)}$ and its derivatives w.r.t.
$z,w,v$ and $t$ are uniformly bounded on the $X-$compacts. Hence,
by the Arzela-Ascoli theorem, ${\mu}_{t}^{(N)}$ converges
pointwise to a function ${\mu}_{t}$ on $X,$ as $N$ tends to
infinity over some sequence of integers, the convergence being
uniform on the compacts. We shall generalise this result to one
of convergence over ${\bf N}$ at the end of the proof of part (2)
of this Proposition.
\vskip 0.2cm\noindent
It also follows from (3.3), (3.6), (3.14) and (3.15) that
$${\mu}_{t}^{(N)}(z-z^{\prime},w-w^{\prime},v-v^{\prime})=
{\psi}_{t}^{(N)}(U^{(N){\star}}(z^{\prime},w^{\prime},v^{\prime})
U^{(N)}(z,w,v))+O(N^{-1})\eqno(6.18)$$
Further, the positivity of ${\psi}_{t}^{(N)}$ ensures that, for
any
complex numbers $c_{1},. \ .,c_{m},$ and elements
$(z_{1},w_{1},v_{1}),. \ .,(z_{m},w_{m},v_{m})$ of $X,$
$${\sum}_{j,k=1}^{m}{\overline c}_{j}c_{k}{\psi}_{t}^{(N)}
(U^{(N){\star}}(z_{j},w_{j},v_{j})U^{(N)}(z_{k},w_{k},v_{k}))
{\geq}0$$
By equation (6.18), this inequality reduces to the following one
in the limit $N{\rightarrow}{\infty}.$
$${\sum}_{j,k=1}^{m}{\overline c}_{j}c_{k}{\mu}_{t}(z_{k}-z_{j},
w_{k}-w_{j},v_{k}-v_{j}){\geq}0$$
Therefore, since ${\mu}_{t}$ is a continuous function on $X,$
which, by (3.14) and (3.15), reduces to unity when its argument
is zero, it follows from Bochner's theorem that ${\mu}_{t}$ is
the characteristic function of a probability measure, $m_{t},$
on $X,$ in accordance with (3.16).
\vskip 0.2cm\noindent
(2) We now re-employ the method we have just used to establish
the convergence of ${\mu}_{t}^{(N)},$ in order to show that the
derivatives of this function w.r.t. $t,z,w,v$ converge
subsequentially to the corresponding ones of ${mu}_{t}.$ The
argument runs as follows. By equns. (2.22), (3.5), (3.9), (3.10),
(3.14) and (3.15), the derivatives w.r.t. $z,w,v,t$ of the
r.h.s.'s of equns. (6.16) and (6.17), barring the $O(N^{-1})$
terms, all take the form
$${\psi}_{t}^{(N)}(U^{(N)}Q)+O(N^{-1})\eqno(6.19)$$
on the $X-$compacts, where $Q$ is a polynomial in the elements
of ${\bf M}^{(N)},$ whose coefficients are $N-$ independent
functions of $z,w,v,$ and which is of second degree in
${\alpha}^{(N)}, \ {\alpha}^{(N){\star}}.$ Hence, as the
commutators $[U,{\alpha}^{(N){\sharp}}]$ are $O(N^{-1}),$ by
(2.8), (3.3) and (3.14), the expression (6.19) reduces, up to
$O(N^{-1}),$ to a finite sum of terms of the forms
$${\psi}_{t}^{(N)}({\alpha}_{l}^{(N){\sharp}}U^{(N)}
R_{1}{\alpha}_{m}^{(N){\sharp}}), \
{\psi}_{t}^{(N)}(U^{(N)}R_{2}{\alpha}_{l}^{(N){\sharp}}) \
and \ {\psi}_{t}^{(N)}(U^{(N)}R_{3})$$
where $R_{1}, \ R_{2}, \ R_{3}$ are polynomials in $s^{(N)}, \
s^{(N){\star}}$ and $p^{(N)}$ only. Hence, by Lemma 6.2 and
the Arzela-Ascoli theorem, it converges pointwise to a limit,
uniformly
so on the compacts, as $N$ tends to infinity over some sequence
of
integers.
\vskip 0.2cm\noindent
On combining this result with that of (1), we see that
${\mu}_{t}^{(N)}$ and its derivatives w.r.t. $z,w,v,t$ converge
to ${\mu}_{t}$ and its corresponding derivatives, uniformly on
the compacts, as $N$ tends to infinity over some sequence of
integers; and further, ${\mu}_{t}(y)$ and its first derivatives
are continuous in $y$ and $t.$ Thus, by (6.16),
$${{\partial}{\mu}_{t}\over {\partial}z}=
{\lim}_{N\to\infty}i{\psi}_{t}^{(N)}(U^{(N)}{\alpha}^{(N)})
\eqno(6.20a)$$
$${{\partial}{\mu}_{t}\over {\partial}w}=
{\lim}_{N\to\infty}i{\psi}_{t}^{(N)}(U^{(N)}s^{(N)})
\eqno(6.20b)$$
and
$${{\partial}{\mu}_{t}\over {\partial}v}=
{\lim}_{N\to\infty}i{\psi}_{t}^{(N)}(U^{(N)}p^{(N)})
\eqno(6.20c)$$
Similarly, we obtain the following formulae for the second
derivatives of ${\mu}_{t}$ that we shall require.
$${{\partial}^{2}{\mu}_{t}\over {\partial}z_{l}{\partial}v_{m}}
=-{\lim}_{N\to\infty}
{\psi}_{t}^{(N)}(U^{(N)}{\alpha}_{l}^{(N)}p_{m}^{(N)})
\eqno(6.21a)$$
$${{\partial}^{2}{\mu}_{t}\over
{\partial}{\overline z}_{l}{\partial}w_{m}}
=-{\lim}_{N\to\infty}
{\psi}_{t}^{(N)}(U^{(N)}{\alpha}_{l}^{(N){\star}}s_{m}^{(N)})
\eqno(6.21b)$$
$${{\partial}^{2}{\mu}_{t}\over
{\partial}z_{l}{\partial}{\overline w}_{m}}
=-{\lim}_{N\to\infty}
{\psi}_{t}^{(N)}(U^{(N)}{\alpha}_{l}^{(N)}s_{m}^{(N){\star}})
\eqno(6.21c)$$
\vskip 0.2cm\noindent
We now use these formulae to derive the equation of motion (3.17)
by integrating (6.17) over the time interval [0,t] and against
suitable test-functions on $X,$ and then passing to a
(subsequential) limit $N{\rightarrow}{\infty}.$ Thus, denoting
by ${\cal Z}(X)$ the space of Fourier transforms of the Schwartz
space ${\cal D}(X),$ we infer from equns. (3.13), (3.16) and
(6.14) that
$${\int}_{X}dm_{t}(x)f(x)-{\int}_{X}dm_{0}(x)f(x)=$$
$${\lim}_{N\to\infty}i{\int}_{0}^{t}du{\int}_{X}dy{\hat f}(y)
{\psi}_{u}^{(N)}(U^{(N)}
([z.A^{(N)}+w.S^{(N)}+h.c.]+p.P^{(N)})) \ {\forall}
f{\in}{\cal Z}(X)\eqno(6.22)$$
Hence, as the Fourier transformation ${\hat f}{\rightarrow}f$ of
${\cal Z}(X)$ onto ${\cal D}(X)$ converts the multipliers
$iz,i{\overline z},iw,i{\overline w},iv$ into the differential
operators ${\partial}/{\partial}{\alpha}, \
{\partial}/{\partial}{\overline {\alpha}},
{\partial}/{\partial}s, \ {\partial}/{\partial}{\overline s},
{\partial}/{\partial}p,$ it follows from (3.16), (3.18), (3.19)
and (6.20)-(6.22) that
$${\int}_{X}dm_{t}(x)f(x)-{\int}_{X}dm_{0}(x)f(x)=
{\int}_{0}^{t}du{\int}_{X}dy{\mu}_{u}
{\widehat {{\cal L}f}}{\equiv}
{\int}_{0}^{t}du{\int}_{X}dm_{u}{\cal L}f\eqno(6.23)$$
for $f{\in}{\cal Z}(X)$. This result can be extended by
continuity to functions $f$ in the Schwartz space ${\cal S}(X)$
and thence to those of class ${\cal C}_{0}^{(1)}(X).$ Further,
for $f{\in}{\cal C}_{0}^{(1)}, \ {\int}_{X}dm_{t}{\cal L}f$ is
continuous in $t,$ for the following reasons. On the one hand,
by Def. 3.1(3) and equns. (3.19), ${\cal L}f$ is the limit, in
the ${\cal C}_{0}(X)$ topology, of a sequence ${\lbrace}{\cal
L}f_{n}{\rbrace},$ with $f_{n}{\in}{\cal S}(X);$ while, on the
other hand, the uniform boundedness and continuity of ${\mu}_{t}$
in $t$ ensure that ${\int}_{X}dm_{t}f_{n}$ is continuous in this
variable. Therefore, when $f{\in}{\cal C}_{0}^{(1)}(X),$ we may
differentiate equn. (6.22) w.r.t. $t$ and thereby obtain the
required result (3.17).
\vskip 0.2cm\noindent
Finally, we note that since, by Prop. 3.4, the solutions of
equns. (3.18) and (3.20) are unique, the above compactness
arguments, which led to the convergence of ${\mu}_{t}^{(N)}$ and
its derivatives over subsequences of integers, can now be
extended to establish their convergence over ${\bf N}.$
\vskip 0.5cm
\centerline {\bf VII. Conclusion}
\vskip 0.3cm\noindent
We have cast the theory of the multi-mode Dicke laser model
within the framework of quantum dynamical semigroups and
stochastic processes, and have thereby obtained a number of new
results. On the physical side, these are encapsulated by the
generalisation to the HL theory provided by Props. 3.4, 3.5, 4.1
and 4.2. In particular, we see from this last proposition and the
discussion of ${\S}4B$ that the present model admits the
phenomena of both chaotic and polychromatic laser radiation.
\vskip 0.2cm\noindent
On the mathematical side, we have extended the theory of
one-parameter CP semigroups to a regime where the generators are
perturbed by unbounded *-derivations (Prop. 2.3).
\vskip 0.2cm\noindent
We shall conclude by noting two outstanding problems. The first
is that of generalising the theory to a continuum of radiation
modes, in the limit $N{\rightarrow}{\infty}$. This could be
formulated by putting $n=N,$ and scaling the interaction
Hamiltonian by $N^{-1},$ as in refs. 11-13, rather than
$N^{-1/2}.$
In this case, the theory of ${\S}'s$ 2 and 5 would still be
applicable, and the remaining problem would be to carry through
the limit procedures of ${\S}6.$
\vskip 0.2cm\noindent
The second outstanding problem, of course, is to obtain a general
characterisation of the conditions that favour the onset of
chaotic or polychromatic laser radiation. This is evidently a
very deep problem, similar to that of turbulence.
\vskip 1cm\noindent
\centerline {\bf Acknowledgments}
\vskip 0.3cm\noindent
The authors would like to thank John Lewis
and Hans Maassen for some clarifying remarks on quantum
stochastic process. G. A. wishes to thank the SERC for financial
support, and the work of both authors was partially supported by
European Capital and Mobility Contract No. CHRX-Ct. 92-0007.
\vfill\eject
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$^{34}$The idea behind the
introduction of $g_{.}$ is that it corresponds to
a classical limit of ${\sum}_{l=0}^{n}(4s_{l}^{(N){\star}}
s_{l}^{(N)}+p_{l}^{(N){\star}}p_{l}^{(N)}),$ which is a Casimir
operator when $N$ is an integral multiple of $n.$ Hence, one
anticipates that the rate of change of $g_{t}$ is governed by the
dissipative part of the r.h.s. of equns. (3.20); and, by (3.19)
and (6.9), this is indeed the case.
**