\magnification 1200 %\baselineskip=1.5\normalbaselineskip \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 \centerline {\bf References} \vskip 0.3cm\noindent $^{a)}$ E-mail: G.ALLI@QMW.AC.UK \vskip 0.2cm\noindent $^{b)}$ E-mail: G.L.SEWELL@QMW.AC.UK \vskip 0.2cm\noindent $^{1}$K. Hepp and E. H. Lieb, Helv. Phys. Acta {\bf 46}, 573, (1973); and Pp. 178-208 of "Dynamical Systems, Theory and Applications", Springer Lecture Notes in Physics {\bf 38}, Ed. J. Moser, (Springer, Heidelberg, Berlin, New York, 1975) \vskip 0.2cm\noindent $^{2}$R. Graham and H. Haken, Z. Phys. {\bf 237}, 31, (1970) \vskip 0.2cm\noindent $^{3}$H. Haken, Handbuch der Physik, Bd. XXV/2C, (Springer, Heidelberg, Berlin, New York, 1970) \vskip 0.2cm\noindent $^{4}$E. B. Davies, Commun. Math. Phys. {\bf 39}, 91, (1974) \vskip 0.2cm\noindent $^{5}$E. B. Davies, "Quantum Theory of Open Systems", (Academic Press, London, New York, 1976) \vskip 0.2cm\noindent $^{6}$V. Gorini and A. Kossakowski, J. Math. Phys. {\bf 17}, 1298, (1976) \vskip 0.2cm\noindent $^{7}$G. Lindblad, Commun. Math. Phys. {\bf 48}, 119, (1976) \vskip 0.2cm\noindent $^{8}$L. Accardi, Adv. in Math. {\bf 20}, 329, (1976) \vskip 0.2cm\noindent $^{9}$L. Accardi, A. Frigerio and J. T. Lewis, Publ. RIMS {\bf 18}, 97, (1982) \vskip 0.2cm\noindent $^{10}$B. K\"ummerer, J. Funct. Anal. {\bf 63}, 139, (1985) \vskip 0.2cm\noindent $^{11}$M. Fannes, P. N. M. Sissons, A. Verbeure and J. C. Wolfe, Annals of Physics {\bf 98}, 38, (1976) \vskip 0.2cm\noindent $^{12}$E. B. Davies, Commun. Math. Phys. {\bf 33}, 187, (1973) \vskip 0.2cm\noindent $^{13}$R. Honneger and A. Rieckers, Publ. RIMS {\bf 30}, 111, (1994) \vskip 0.2cm\noindent $^{14}$T. Kato "Perturbation Theory for Linear Operators", (Springer, New York, 1966) \vskip 0.2cm\noindent $^{15}$A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations", (Springer, New York, Berlin, Tokyo, 1983) \vskip 0.2cm\noindent $^{16}$H. Haken, Phys. Lett. {\bf 53}A, 77, (1975) \vskip 0.2cm\noindent $^{17}$R. G. Harrison and D. J. Biswas, Nature {\bf 321}, 394, (1986) \vskip 0.2cm\noindent $^{18}$T. Milonni, M. Shih and J. R. Ackerhalt, "Chaos in Laser- Matter Interactions", (World Scientific, Singapore, 1976) \vskip 0.2cm\noindent $^{19}$W. F. Stinespring, Proc. Amer. Math. Soc. {\bf 6}, 211, (1955) \vskip 0.2cm\noindent $^{20}$H. Narnhofer and G. L. Sewell, Commun. Math. Phys. {\bf 79}, 9, (1981) \vskip 0.2cm\noindent $^{21}$G. L. Sewell, Helv. Phys. Acta {\bf 67}, 4, (1994) \vskip 0.2cm\noindent $^{22}$P. Vanheuverzwijn, Ann. Inst. H. Poincare A {\bf 29}, 123, (1978); Erratum {\it ibid} {\bf 30}, 83, (1979) \vskip 0.2cm\noindent $^{23}$B. de Moen, P. Vanheuverzwijn and A. Verbeure, Rep. Math. Phys. {\bf 15}, 27, (1979) \vskip 0.2cm\noindent $^{24}$G. L. Sewell, J. Math. Phys. {\bf 11}, 1868, (1970) \vskip 0.2cm\noindent $^{25}$E. B. Davies, "One-Parameter Semigroups", (Academic Press, London, New York, San Francisco, 1980) \vskip 0.2cm\noindent $^{26}$A. S. Kholevo, Russian Acad. Sci. Dokl. Math. {\bf 47}, 161, (1993) \vskip 0.2cm\noindent $^{27}$E. Hopf, Berl. Math.-Phys. Kl. S\"achs Acad. Wiss. Leipzig {\bf 94}, 1, (1942) \vskip 0.2cm\noindent $^{28}$R. Jost and E. Zehnder, helv. Phys. Acta {\bf 45}, 258, (1972) \vskip 0.2cm\noindent $^{29}$D. Ruelle and F. Takens, Commun. Math. Phys. {\bf 20}, 167, (1971) \vskip 0.2cm\noindent $^{30}$J. P. Eckmann and D. Ruelle, Rev. Mod. Phys. {\bf 57}, 617, (1985) \vskip 0.2cm\noindent $^{31}$L. D. Landau and E. M. Lifshitz, "Fluid Mechanics", (Pergamon, Oxford, New York, Toronto, Sydney, Paris, 1984 ), sec. 27 \vskip 0.2cm\noindent $^{32}$R. J. Glauber, Phys. Rev. {\bf 130}, 2529, (1963) \vskip 0.2cm\noindent $^{33}$R. Hudson and K. R. Parthasaraty, Commun. Math. Phys. {\bf 93}, 301, (1984) \vskip 0.2cm\noindent $^{34}$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.