\magnification 1200
\centerline {{\bf Recent Developments in Macroscopic Quantum
Electrodynamics}\footnote*{Based on lectures given at the
Symposia on Mathematical Physics at Gdansk and Torun, 4-8
December, 1995}}
\vskip 0.4cm
\centerline {{\bf by Geoffrey L. Sewell}\footnote{**}{Partially
supported by European Capital and Mobility Contract No. CHRX-CT-
0007}}
\vskip 0.4cm
\centerline {\bf Department of Physics, Queen Mary and Westfield
College, London E1 4NS}
\vskip 1cm
\centerline {\bf Abstract}
\vskip 0.3cm\noindent
We review two recent developments in constructive macroscopic
quantum electrodynamics. These concern the derivation of the
large-scale dynamical properties of plasma and laser models from
their underlying quantum structures. In the case of the plasma
model, consisting of a non-relativistic system of electrons
coupled to a quantised electromagnetic field, we show that the
macroscopic dynamics is governed by classical Vlasov-Maxwell
equations and supports transitions from deterministic to
stochastic flows. In the case of the laser model, which is a new
version of that of Hepp and Lieb [HL], recast within the
framework of quantum dynamical semigroups, we obtain a
generalisation of the HL theory and show that it supports
optically chaotic phases, as well as the usual quiescent and
coherent ones.
\vskip 1cm
\centerline {\bf 1. Introduction}
\vskip 0.3cm\noindent
This article will be devoted to a review of two recent
developments in constructive macroscopic quantum
electrodynamics. These are concerned with the large-scale
dynamical properties of models of a plasma and of a laser.
\vskip 0.2cm\noindent
The plasma model is a system of non-relativistic electrons,
coupled via regularised interactions to a quantised
electromagnetic field and a passive neutralising positively
charged background. Our treatment of this model, which we present
in Section 2, is based on an extension of the methods devised
previously [Se1,2] for the case where the interactions were
purely electrostatic. As in that case, we establish that, in a
suitable large-scale limit, the dynamics of the plasma is
governed by classical Vlasov-cum-Maxwell equations, and exhibits
a hydrodynamical phase transition from a deterministic Eulerian
flow to a stochastic one, when the initial conditions become
sufficiently non-uniform.
\vskip 0.2cm\noindent
The laser model is a new version [AS] of that of Dicke [Di] and
Hepp and Lieb [HL], recast within the framework of quantum
dynamical semigroups and stochastic processes. Thus,
whereas the HL model is a conservative system, consisting of
matter, radiation and reservoirs (pumps and sinks), ours is an
open dissipative system of matter and radiation, whose dynamics
is governed by a one-parameter semigroup, which incorporates the
action of the reservoirs. The recasting of the model in this way
enables us to take advantage of the theory of quantum Markov
processes [AFL, HP, Ku], and thereby to gain some new
perspectives on the theory. Our treatment of the model, which we
present in Section 3, leads to a generalisation of the HL
results. In particular, it yields a macroscopic dynamics that
supports optically chaotic phases, as well as the coherent and
quiescent ones.
\vskip 0.2cm\noindent
Both of the models presented here reduce to mean field theories.
In Section 4, we shall briefly discuss the common features of our
treatments of them, as well as the possible extension of our
methods to more realistic models.
\vskip 0.5cm\noindent
\centerline {\bf 2. The Plasma Model}
\vskip 0.3cm\noindent
{\bf 2.1 The Model.}
\vskip 0.2cm\noindent
This is a system, ${\Sigma}^{(N,L)},$ consisting of $N$ non-
relativistic electrons and their radiation field in a
three-dimensional periodic cube, ${\Omega}^{(L)},$ of side $L,$
which carries a fixed, uniform, neutralising charge background.
Thus, ${\Omega}^{(L)}=({\bf R}/L{\bf Z})^{3},$ and
the particle number density, $n_{0},$ and classical plasma
frequency, ${\omega}_{p},$ of the model are given by the formulae
$$n_{0}=N/L^{3},\eqno(2.1)$$
and
$${\omega}_{p}=(n_{0}{\epsilon}^{2}/m)^{1/2}\eqno(2.2)$$
where $m$ and ${\epsilon}$ are the electronic mass and charge,
respectively. We denote points in ${\Omega}^{(L)}$ by $X,$
sometimes with indices $j$ or $k,$ and the gradient operator in
this space by ${\nabla}^{(L)}.$ Components of ${\bf R}^{3}$
vectors will generally be indicated by suffixes ${\mu}$ or
${\nu}.$
\vskip 0.2cm\noindent
We represent the positions and momenta of the electrons by the
standard multiplicative and differential operators
${\lbrace}X_{j},P_{j}=-i{\hbar}{\nabla}^{(L)}{\vert}j=1,. \
.,N{\rbrace},$ acting on the Hilbert space, ${\cal
H}_{el}^{(N,L)},$ of antisymmetric, square integrable functions
on $({\Omega}^{(L)})^{N}.$
\vskip 0.2cm\noindent
The interactions are assumed to be the standard electromagnetic
ones. These comprise the electrostatic two-body potential,
${\epsilon}^{2}U^{(L)},$ and the quantum field of a transversely
gauged vector potential, $A,$ corresponding to the magnetic field
$B:$
$$B={\nabla}^{(L)}{\times}A\eqno(2.3)$$
In view of the uniform charge background, the Coulomb potential
$U^{(L)}$ may be expressed in the form [BP]
$$U^{(L)}(X)=L^{-3}{\sum}_{Q{\in}
(2{\pi}L^{-1}{\bf Z})^{3}{\backslash}{\lbrace}0{\rbrace}}
{{\exp}(iQ.X)\over Q^{2}},\eqno(2.4)$$
while $A$ and its canonical conjugate, $F,$ the transverse
electric field, are defined as Hermitian distribution-valued
operators in a Fock-Hilbert space, ${\cal H}_{rad}^{(L)},$ by the
following conditions.
\vskip 0.2cm\noindent
(1) $A$ and $F$ satisfy the canonical commutation relations
$$[A_{\mu}(X),F_{\nu}(X^{\prime})]=
i{\hbar}D_{{\mu}{\nu}}^{(L)}(X-X^{\prime})
\eqno(2.5)$$
where $D^{(L)}$ is the divergence-free part of the product of the
unit tensor in ${\bf R}^{3}$ and the Dirac distribution in
${\Omega}^{(L)},$ i.e., its Fourier coefficients are
$${\hat D}_{{\mu}{\nu}}^{(L)}(Q)={\int}_{{\Omega}^{(L)}}dX
D_{{\mu}{\nu}}^{(L)}(X){\exp}(2{\pi}iQ.X)={\delta}_{{\mu}{\nu}}
-{Q_{\mu}Q_{\nu}\over Q^{2}}(1-{\delta}_{Q,0})
\ {\forall}Q{\in}({2{\pi}{\bf Z}\over L})^{3}$$
\vskip 0.2cm\noindent
(2) ${\cal H}_{rad}^{(L)}$ is the vacuum sector of the free
transverse electromagnetic field with Hamiltonian
$${1\over 2}{\int}_{{\Omega}^{(L)}}:(F(X))^{2}+
c^{2}({\nabla}^{(L)}{\times}A(X))^{2}):dX,$$
the colons denoting Wick ordering.
\vskip 0.2cm\noindent
Thus, we represent the electronic and radiative field observables
by the self-adjoint operators in ${\cal H}_{el}^{(N,L)}$ and
${\cal H}_{rad}^{(L)},$ respectively. We now define ${\cal
H}^{(N,L)}:={\cal H}_{el}^{(N,L)}{\otimes}
{\cal H}_{rad}^{(L)},$ and canonically identify operators $R,$
in ${\cal H}_{el}^{(N,L)}$, and $S,$ in ${\cal H}_{rad}^{(L)},$
with $R{\otimes}I$ and $I{\otimes}S,$ respectively. We take the
observables and states of whole system ${\Sigma}^{(N,L)}$ to be
the self-adjoint operators and density matrices, repectively, in
${\cal H}^{(N,L)}.$
\vskip 0.2cm\noindent
We assume that the Hamiltonian, $H^{(N,L)},$ for the model is of
the standard form for non-relativistic particles with the above
electromagnetic interactions. Thus [BFS],
$$H^{(N,L)}={\sum}_{j=1}^{N}
{1\over 2m}(P_{j}-{\epsilon}(A_{\kappa}(X_{j})^{2})+
{\epsilon}^{2}{\sum}_{k,l(>k)=1}^{N}U^{(L)}(X_{k}-X_{l})+$$
$${1\over 2}{\int}_{{\Omega}^{(L)}}dX:(F(X))^{2}+
c^{2}({\nabla}^{(L)}{\times}A(X))^{2}:\eqno(2.6)$$
where the replacement of $A$ by $A_{\kappa}$ in the first term
represents a cut-off obtained by removing from $A$ its Fourier
components whose wave-vectors have magnitude greater than
${\kappa}:={\hbar}/mc.$
\vskip 0.2cm\noindent
Our aim now is to investigate the dynamics of the model on the
length scale $L,$ in a limit where $L$ and $N$ tend to infinity
and the particle density $n_{0}$ remains fixed and finite.
\vskip 0.3cm\noindent
{\bf 2.2. The Rescaled Description.}
\vskip 0.2cm\noindent
We take our macroscopic description of the model to be the
'large' scale one, where the unit of length is $L.$ Since we know
from phenomenological considerations that the corresponding time
scale is ${\omega}_{p}^{-1},$ we effect this description by
rescaling the variables of ${\Sigma}^{(N,L)}$ so that its units
of mass, length and time are $m, \ L$ and ${\omega}_{p}^{-1},$
respectively. In this scaling, Planck's constant is
$${\hbar}_{N}={{\hbar}\over mL^{2}{\omega}_{p}}
{\equiv}{{\hbar}\over m{\omega}_{p}}
({n_{0}\over N})^{2/3}\eqno(2.7)$$
and the speed of light is
$$c_{0}=c/L{\omega}_{p}\eqno(2.8)$$
\vskip 0.2cm\noindent
We formulate our macroscopic description of ${\Sigma}^{(N,L)}$
by mapping it onto a system ${\Sigma}^{(N)}$ of $N$ particles and
its radiation field in the unit periodic cube ${\Omega}:=({\bf
R}/{\bf Z})^{3}.$ Thus, we define ${\cal H}^{(N)}$ to be the
Hilbert space of square integrable, antisymmetric functions on
${\Omega}^{N},$ and $V$ to be the canonical isometry of
${\cal H}^{(N,L)}$ onto ${\cal H}^{(N)},$ corresponding to the
mapping $X{\rightarrow}x:=X/L$ of ${\Omega}^{(L)}$ onto
${\Omega}.$ We then define the particle positions and momenta,
$(x_{j},p_{j}),$ and the radiation field, $(a,f),$ to the
macroscopic description, by the following formulae.
$$x_{j}:=L^{-1}VX_{j}V^{-1}; \ p_{j}
:=(mL{\omega}_{p})^{-1}VP_{j}V^{-1}
{\equiv}-i{\hbar}_{N}{\nabla}_{x_{j}}\eqno(2.9)$$
where ${\nabla}$ (or ${\nabla}_{x}$) is the gradient operator in
${\Omega},$
$$a(x):={{\epsilon}\over mL{\omega}_{p}}VA(Lx)V^{-1}; \ f(x):=
{{\epsilon}\over mL{\omega}_{p}^{2}}VF(Lx)V^{-1}\eqno(2.10)$$
and
$$H^{(N)}=(mL^{2}{\omega}_{p})^{-1}VH^{(N,L)}V^{-1}$$
$$={1\over 2}{\sum}_{j=1}^{N}
(p_{j}-a_{\kappa}(x_{j}))^{2}+
N^{-1}{\sum}_{j,k(>j)=1}^{N}U(x_{j}-x_{k})+$$
$${N\over 2}{\int}_{\Omega}dx:(f(x)^{2}+
c_{0}^{2}({\nabla}{\times}a(x))^{2}):\eqno(2.11)$$
where
$$U(x)=
{\sum}_{q{\in}(2{\pi}{\bf Z})^{3}{\backslash}{\lbrace}0{\rbrace}}
{\exp}(iq.x)/q^{2}\eqno(2.12)$$
and $a_{\kappa}$ is the regularised version of $a,$ obtained by
discarding the Fourier components of this field with wave-numbers
greater than
${\kappa}_{N}={\kappa}L{\equiv}{\kappa}(N/n_{0})^{1/3}.$ The
magnetic field vector is
$$b={\nabla}{\times}a\eqno(2.13)$$
\vskip 0.2cm\noindent
Thus, in the rescaled description, the system corresponds to a
model ${\Sigma}^{(N)},$ whose observables and normal states are
the self-adjoint operators and density matrices, respectively,
in ${\cal H}^{(N)},$ and whose dynamics is governed by the
Hamiltonian $H^{(N)}.$ Further, by equns. (2.1), (2.2), (2.5) and
(2.10), the fields $a$ and $f$ satisfy the CCR
$$[a_{\mu}(x),f_{\nu}(x^{\prime})]=iN^{-1}{\hbar}_{N}
D_{{\mu}{\nu}}(x-x^{\prime})\eqno(2.14)$$
where $D{\equiv}D^{(1)}$ is the transverse part of the product
of the unit tensor and the Dirac distribution in ${\Omega}.$
\vskip 0.2cm\noindent
To express the fields $a, \ f$ as distribution-valued
operators, we introduce the Schwartz space ${\cal D}_{tr}$ of
infinitely differentiable, divergence-free vector fields in
${\Omega},$ whose Fourier transforms are fast-decreasing
functions on $(2{\pi}{\bf Z})^{3},$ and we define the 'smeared
fields'
$$a({\phi}):={\int}_{\Omega}dxa(x).{\phi}(x); \
f({\psi}):={\int}_{\Omega}f(x).{\psi}(x) \ {\forall}{\phi},
\ {\psi}{\in}{\cal D}_{tr}\eqno(2.15)$$
Thus, $a, \ f$ are maps from ${\cal D}_{tr}$ into the self-
adjoint operators in ${\cal H},$ and the CCR (2.14) may be re-
expressed as
$$[a({\phi}),f({\psi})]=
iN^{-1}{\hbar}_{N}{\int}_{\Omega}dx{\phi}(x).{\psi}(x)
\ {\forall}{\phi},{\psi}{\in}{\cal D}_{tr}\eqno(2.14^{\prime})$$
\vskip 0.2cm\noindent
{\bf Note on $c_{0}.$} In view of the definition (2.8) of
$c_{0},$ the physical demand that the particle speeds in
${\Sigma}^{(N,L)}$ cannot exceed $c$ signifies that $c_{0}$ must
always be at least of the order of unity. In order to meet this
demand, we shall treat $c_{0},$ rather than $c,$ as a constant
parameter when we pass to the limit $N{\rightarrow}{\infty}.$ In
physical terms, this is tantamount to assuming that, in the
finite model, ${\Sigma}^{(N,L)},$ under consideration, the length
$c{\omega}_{p}^{-1}$ is at least of the order of $L$ and also
that $N>>1.$ One may easily check that these requirements may
both be fulfilled in realistic situations [Se1].
\vskip 0.2cm\noindent
We now regularise the interactions by replacing $U$ and
$a_{\kappa}$ by their respective convolutions with a positive,
${\cal D}-$ class, $L-$independent function $g,$ whose integral
over ${\Omega}$ is unity. Thus, in view of the above
specification of $a_{\kappa},$ following equn. (2.12), we replace
$U$ and $a_{\kappa}$ by $U_{g}$ and $a_{g}^{(N)},$ respectively,
where
$$U_{g}=g*U; \ a_{g}^{(N)}=g^{(N)}*a\eqno(2.16)$$
and $g^{(N)}$ is the truncated form of $g$ obtained by removal
of its Fourier components of wave-vector lying outside the ball
of radius ${\kappa}(N/n_{0})^{1/3}.$ Evidently, $g^{(N)}$
converges to $g$ in the ${\cal D}$ topology, as
$N{\rightarrow}{\infty}.$
\vskip 0.2cm\noindent
{\bf Note.} This regularisation is quite different from that
involved in the definition of $A_{\kappa},$ since
it corresponds to a {\it macroscopic} cut-off, at distance
proportional to $L,$ when referred back to ${\Sigma}^{(N,L)}.$
\vskip 0.2cm\noindent
It follows from our specifications that the Hamiltonian of
the modified model, ${\Sigma}_{g}^{(N)},$ is
$$H_{g}^{(N)}={1\over 2}{\sum}_{j=1}^{N}v_{j}^{2}+
N^{-1}{\sum}_{j,k(>j)=1}^{N}U_{g}(x_{j}-x_{k})+Nh_{rad}
\eqno(2.17)$$
where
$$v_{j}=p_{j}-a_{g}^{(N)}(x_{j})\eqno(2.18)$$
is the velocity of the j'th particle, and
$$h_{rad}={1\over
2}{\int}dx:((f(x))^{2}+({\nabla}{\times}a(x))^{2}):
\eqno(2.19)$$
is the radiative energy, as measured in units of $N.$
\vskip 0.2cm\noindent
The algebraic structure of ${\Sigma}_{g}^{(N)}$ is governed by
the commutation relations between its position, velocity and
field observables. In fact, it follows easily from our
definitions that the only non-zero commutators between these
operators are those given by equn. (2.14) and the following ones.
$$[x_{j,{\mu}},v_{k,{\nu}}]=i{\hbar}_{N}{\delta}_{jk}{\delta}_
{{\mu}{\nu}}I; \
[v_{j,{\mu}},v_{k,{\nu}}]=i{\hbar}_{N}{\epsilon}_{{\mu}{\nu}{\
sigma}}b_{g,{\sigma}}^{(N)}(x_{j}){\delta}_{jk};$$
$$and \
[v_{j},f({\psi})]=-i{\hbar}_{N}N^{-1}{\int}dxg^{(N)}(x){\psi}(x)
\eqno(2.20)$$
where ${\epsilon}$ is the alternate tensor, i.e.,
${\epsilon}_{{\mu}{\nu}{\sigma}}=1 \ (resp. \ -1)$ if
$({\mu},{\nu},{\sigma})$ is an even (resp. odd) permutaion of
$(1,2,3),$ and is otherwise zero.
\vskip 0.2cm\noindent
The time-derivatives of the observables are determined by the
action on them of the derivation
$${\Lambda}_{g}^{(N)}=
{i\over {\hbar}_{N}}[H_{g}^{(N)},.]\eqno(2.21)$$
In particular, we see from equations (2.14), (2.17) and (2.20)
that this action is given by
$${\Lambda}_{g}^{(N)}x_{j}=v_{j};
\ {\Lambda}_{g}^{(N)}v_{j}=f_{g}^{(N)}(x_{j})-
(v_{j}{\times}b_{g}^{(N)}(x_{j}))_{sym}
-N^{-1}{\sum}_{k{\neq}j}{\nabla}U_{g}(x_{j}-x_{k})
\eqno(2.22)$$
and
$${\Lambda}_{g}^{(N)}a(x)=f(x); \ {\Lambda}_{g}^{(N)}f(x)=
c_{0}^{2}{\Delta}a(x)+N^{-1}{\Sigma}_{k=1}^{N}
(v_{k}g^{(N)}(x-x_{k}))_{sym}\eqno(2.23)$$
where
$$b_{g}^{(N)}=g^{(N)}*b \ and \ f_{g}^{(N)}=g^{(N)}*f
\eqno(2.24)$$
are the regularised magnetic and transverse electric fields,
respectively and the subscript $(sym)$ denotes symmetrised
product.
\vskip 0.2cm\noindent
{\bf Comment.} The model ${\Sigma}_{g}^{(N)},$ which carries the
macrodynamics of the original one, ${\Sigma}^{(N,L)},$ exhibits
the hallmarks of a {\it classical mean field theory.} For, on the
one hand, it follows from equns. (2.7), (2.14) and (2.20) that
the effective Planck constant, ${\hbar}_{N},$ vanishes, and that
the observables $x_{j}, \ p_{j}, \ a, \ f$ all intercommute in
the limit $N{\rightarrow}{\infty}:$ while, on the other hand, the
last terms in the formulae (2.22) and (2.23) are of typical mean
field theoretic form, namely $N^{-1}{\sum}_{k=1}^{N}Q_{k},$ with
the $Q_{k}$'s copies of the same single-particle observable.
\vskip 0.3cm\noindent
{\bf 2.3. Dynamics of ${\Sigma}_{g}^{(N)}$.}
\vskip 0.2cm\noindent
We shall now formulate the evolution of this system from an
initial, possibly pure, state, represented by a density matrix,
${\rho}_{0}^{(N)},$ in ${\cal H}^{(N)}.$ The evolute of this
state at time $t$ is thus
$${\rho}_{t}^{(N)}={\exp}(-iH_{g}^{(N)}t/{\hbar}_{N})
{\rho}_{0}^{(N)}{\exp}(iH_{g}^{(N)}t/{\hbar}_{N})
\eqno(2.25)$$
\vskip 0.2cm\noindent
The following definition provides a phase space for the
anticipated classical limit of the macroscopic dynamics.
\vskip 0.2cm\noindent
{\bf Definition 2.1.} We define $K$ to be the classical, one-
particle phase space ${\Omega}{\times}{\bf R}^{3}$ and ${\hat K}$
to be its dual, $(2{\pi}{\bf Z})^{3}{\times}{\bf R}^{3}.$
\vskip 0.2cm\noindent
We now represent the state ${\rho}_{t}^{(N)},$ by a family of
quantum characteristic functions (QCF's)
$${\lbrace}C_{t}^{(N,n)}:{\hat K}^{n}{\times}({\cal
D}_{tr})^{2}{\rightarrow}{\bf C}{\vert}n=0,1,. \ .,N{\rbrace},$$
as defined by the formula
$$C_{t}^{(N,n)}({\xi}_{1},{\eta}_{1};. \ .;{\xi}_{n},{\eta}_{n};
{\phi},{\psi})=$$
$${\langle}{\rho}_{t}^{(N)};
{\exp}({i\over 2}(a({\phi})+f({\psi}))
({\Pi}_{j=1}^{n} {\exp}({1\over 2}{\eta}_{j}.v_{j})
{\exp}(i{\xi}_{j}.x_{j})
{\exp}({1\over 2}{\eta}_{j}.v_{j}))
{\exp}({i\over 2}(a({\phi})+f({\psi})){\rangle}$$ \
$${\forall}({\xi}_{j},{\eta}_{j}){\in}{\hat K}, \ j=1,. \ .,n \
{\phi},{\psi}{\in}{\cal D}_{tr}\eqno(2.26)$$
where ${\langle}{\rho};A{\rangle}{\equiv}Tr({\rho}A).$ Thus, but
for the non-commutativity of the observables, $C_{t}^{(N,n)}$
would be the characteristic function of a classical probability
measure.
\vskip 0.2cm\noindent
We assume the following initial conditions.
\vskip 0.2cm\noindent
$(I.1)$ The expectation value of the energy per particle of
${\Sigma}^{(N)}$ is bounded, uniformly w.r.t. $N,$ i.e.
$${\rho}_{0}^{(N)}(H_{g}^{(N)})<{\gamma}N, \
{\forall}N{\in}{\bf N}\eqno(2.27)$$
where ${\gamma}$ is a constant. This energy bound corresponds to
one proportional to $N^{5/3}$ for ${\Sigma}^{(N,L)},$ and is
chosen to represent the situation where the latter system is
prepared in a state where the charge density and current
densities are of the form ${\sigma}(X/L)$ and $Lu(X/L),$
respectively, with ${\sigma}$ and $u$ smooth. For then, both the
particle kinetic energy and the electromagnetic field energy are
proportional to $N^{5/3}.$
\vskip 0.2cm\noindent
{\bf Note.} Since ${\Sigma}_{g}^{(N)}$ is a conservative system,
the condition (2.27) remains valid if ${\rho}_{0}^{(N)}$ is
replaced by ${\rho}_{t}^{(N)}.$ Hence, as this state is invariant
w.r.t. permutations of the coordinates $x_{j},$ it follows easily
from equns. (2.17)-(2.19) that we can find a finite constant
${\gamma}_{1},$ such that
$${\langle}{\rho}_{t}^{(N)};v_{j}^{2}{\rangle}<{\gamma}_{1}; \
{\langle}{\rho}_{t}^{(N)};
(a({\nabla}{\times}{\phi}))^{2}{\rangle}<
{\gamma}_{1}{\Vert}{\phi}{\Vert}_{2}^{2}; \ and \
{\langle}{\rho}_{t}^{(N)};
(f({\psi}))^{2}{\rangle}<
{\gamma}_{1}{\Vert}{\psi}{\Vert}_{2}^{2};$$
$${\forall}N{\in}{\bf N}, \ t{\in}{\bf R}, \ j=1,. \ .,n, \
{\phi},{\psi}{\in}{\cal D}_{tr}\eqno(2.28)$$
where ${\Vert}.{\Vert}_{2}$ is the $L^{2}$ norm.
\vskip 0.2cm\noindent
$(I.2)$ The characteristic functions $C_{0}^{(N,n)}$ factorise,
in the limit $N{\rightarrow}{\infty},$ according to the formula
$${\lim}_{N\to\infty}[C_{0}^{(N,n)}({\xi}_{1},{\eta}_{1};. \
.;{\xi}_{n},{\eta}_{n};{\phi},{\psi})-({\Pi}_{j=1}^{n}
C_{0}^{(N,1)}({\xi}_{j},{\eta}_{j};0,0))
C_{0}^{(N,0)}({\phi},{\psi})]=0\eqno(2.29)$$
This condition represents the situation where ${\Sigma}^{(N,L)}$
is prepared in a pure phase, carrying correlations only of short
range correlations, which scale down to zero range for
${\Sigma}_{g}^{(N)}$ in the limit $N{\rightarrow}{\infty}.$
\vskip 0.2cm\noindent
$(I.3)$ The initial state of the radiation field is
macroscopically coherent, in that it fluctuations reducing to
zero on the ${\Sigma}_{g}^{(N)}$ scale in the limit
$N{\rightarrow}{\infty}.$ Hence,
$${\lim}_{N\to\infty}C_{0}^{(N,0)}({\phi},{\psi})=
{\exp}i(a_{0}({\phi})+f_{0}({\psi}))\eqno(2.30)$$
where $a_{0}$ and $f_{0}$ are classical fields.
\vskip 0.2cm\noindent
$(I.4)$ These latter fields are continuous functions on $X.$
\vskip 0.3cm\noindent
{\bf 2.4. The Vlasov Dynamics.}
\vskip 0.2cm\noindent
The following theorem represents our main result, concerning the
large-scale classical electrodynamics of the quantum plasma
model.
\vskip 0.2cm\noindent
{\bf Theorem 2.2} {\it (1) Under the assumption (I.1-4), and for
each $n{\in}{\bf N}, \ C_{t}^{(N,n)}$ converges pointwise, as
$N{\rightarrow}{\infty},$ to the characteristic functions of a
classical probability measure} $M_{t}^{(n)}$ {\it on}
$K^{(n)}{\times}({\cal D}_{tr}^{\prime})^{2},$ {\it i.e.,}
$${\lim}_{N\to\infty}C_{t}^{(N,n)}({\xi}_{1},{\eta}_{1};. \
.;{\xi}_{n},{\eta}_{n};{\phi},{\psi})=$$
$${\int}dM_{t}^{(n)}(x_{1},v_{1};. \ .;x_{n},v_{n};a,f)
{\exp}i[{\sum}_{j=1}^{n}({\xi}_{j}.v_{j}+{\eta}_{j}.x_{j})+
a({\phi})+f({\psi})]\eqno(2.31)$$
{\it Furthermore, the set ${\lbrace}M_{t}^{(n)}{\vert}n{\in}{\bf
N}{\rbrace}$ are invariant under permutations of the
$(x_{j},v_{j})'$s and canonically defines a probability measure
$M_{t}$ on the space $K^{n}{\times}({\cal D}_{tr}^{\prime})^{2}.$
\vskip 0.2cm\noindent
(2) $M_{t}$ factorises into the form $m_{t}^{{\otimes}{\bf
N}}{\otimes}{\delta}_{a_{t},f_{t}},$ where $m_{t}$ is a
probability measure on $K$ and ${\delta}_{a_{t},f_{t}}$ is the
Dirac measure on $({\cal D}_{tr}^{\prime})^{2})$ with support at
classical fields $(a_{t},f_{t}).$
\vskip 0.2cm\noindent
(3) $(m_{t},a_{t},f_{t})$ evolves according to the following
Vlasov-Maxwell equations, and these have a unique global
solution.}
$${d\over dt}{\int}dm_{t}h=$$
$${\int}dm_{t}{\lbrack}v.{\nabla}_{x}h+
{\lbrace}f_{g,t}+v{\times}b_{g,t}-{\int}dm_{t}(x^{\prime},v)
{\nabla}(U_{g}(x-x^{\prime}){\rbrace}.{\nabla}_{v}h{\rbrack}
\ {\forall}h{\in}{\cal C}_{0}^{(1)}(K)\eqno(2.32)$$
{\it and}
$$c_{0}^{2}{\Delta}a_{t}-
{{\partial}^{2}a_{t}\over {\partial}t^{2}}=
{\int}dm_{t}(x^{\prime},v)vg(x-x^{\prime})\eqno(2.33)$$
{\it where}
$$f_{t}={{\partial}a_{t}\over {\partial}t}; \
b_{t}={\nabla}{\times}a_{t}\eqno(2.34)$$
{\it and}
$$b_{g,t}=g*a_{g,t}; \ f_{g,t}=g*f_{t}\eqno(2.35)$$
\vskip 0.2cm\noindent
The proof of this theorem, which we shall provide elsewhere
[Se3], is based on a generalisation of the methods employed in
the purely electrostatic case. The main steps in this proof are
the following.
\vskip 0.2cm\noindent
(a) We extend the method of [NS] to infer part (1) of the
theorem, as restricted to a subsequence of $N'$s, from the
equations of motion for ${\Sigma}_{g}^{(N)},$
together with the commutation relations (2.14) and (2.20) and the
condition (2.28) (i.e. $(I.1)).$
\vskip 0.2cm\noindent
(b) We show that conditions (I.2,3) imply the specified
factorisation property of the resultant $M_{t}$ at $t=0.$
\vskip 0.2cm\noindent
(c) We derive a hierarchy of equations of motion for the measures
$M_{t}^{(n)}$ from the corresponding ones for the QCF's
$C_{t}^{(N,n)}$ by passing to the subsequential limit obtained
in (a).
\vskip 0.2cm\noindent
(d) We employ (b), (c) and condition (I.4) to generalise the
classical methods of Neuzert [Ne] and Spohn [Sp] so as to
establish the global existence and uniqueness of the solution of
the Vlasov-Maxwell equations (2.32)-(2.35), and to derive these
equations from the Vlasov hierarchy. The uniqueness of this
solution then serves to render the subsequential convergence of
(a) fully sequential.
\vskip 0.3cm\noindent
{\bf 2.5. Hydrodynamical Phase Transitions.}
\vskip 0.2cm\noindent
In a previous work [Se2], we showed that the electrostatic model
supports transitions from Eulerian deterministic to stochastic
flow when the initial conditions are sufficiently non-uniform.
The same is evidently true of the present model, since it may
readily be seen that the Vlasov-Maxwell equations (2.32)-(2.35)
have solutions for which $a$ and $f$ vanish and $m_{t}$ satisfies
precisely the same conditions that gave rise to those transitions
in the electrostatic case. The interesting question, of course,
concerns what other hydrodynamical phase transitions the present
model may support.
\vskip 0.5cm\noindent
\centerline {\bf 3. The Laser Model}
\vskip 0.3cm\noindent
{\bf 3.1. The Model.}
\vskip 0.2cm\noindent
This is a version [AS] of the Dicke-Hepp-Lieb model of matter
interacting with radiation. Specifically, it consists of $N$
identical two-level atoms, coupled to $n$ radiative modes by
dipolar interactions. Furthermore, each element of the model,
whether atom or mode, is an {\it open dissipative system}, the
atoms being coupled to pumps and sinks and the radiation to sinks
only.
\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,
completely positive (CP) contractions of ${\cal A},$ and ${\phi}$
is a normal state on this algebra. Here, the action of the
reservoirs on the system is incorporated into the structure of
this semigroup. We build
the model from its elements, consisting of the atoms and modes.
\vskip 0.2cm\noindent
{\bf 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
$${\sigma}_{x}^{2}={\sigma}_{y}^{2}={\sigma}_{z}^{2}=I; \
{\sigma}_{x}{\sigma}_{y}=-{\sigma}_{y}{\sigma}_{x}=i{\sigma}_{z},
\ etc.\eqno(3.1)$$
We define the spin raising and lowering operators
$${\sigma}_{\pm}={1\over 2}({\sigma}_{x}{\pm}i{\sigma}_{y})
\eqno(3.2)$$
${\lbrace}T_{at}(t){\vert}t{\in}{\bf R}_{+}{\rbrace}$ 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 given by the
formula
$$L_{at}{\sigma}_{\pm}=-({\gamma}_{\perp}{\mp}i{\epsilon})
{\sigma}_{\pm};
\ L_{at}{\sigma}_{z}=-{\gamma}_{\parallel}({\sigma}_{z}-{\eta}I);
\ L_{at}I=0\eqno(3.3)$$
where the ${\epsilon}(>0)$ is the energy difference
between the two eigenstates of the atom, the ${\gamma}'$s are
positive damping constants, and ${\eta}$ is a further constant
representing the terminal value of ${\sigma}_{z}$ for the
isolated atom. In fact, the values of the ${\gamma}'$s and
${\eta}$ depend on the outcome of the couplings of the atom to
its pump and sink, the latter constant being positive when the
pumping dominates. Further, the demands of complete positivity
impose the constraints [AS]
$${\gamma}_{\parallel}{\leq}2{\gamma}_{\perp}; \
-1<{\eta}<1\eqno(3.4)$$
on these parameters.
\vskip 0.2cm\noindent
We take ${\phi}_{at}$ to be the unique $T_{at}-$invariant state
on ${\cal A}_{at},$ and this is given by
$${\phi}_{at}({\sigma}_{z})={\eta}; \
{\phi}_{at}({\sigma}_{\pm})=0\eqno(3.5)$$
This state is faithful, and the condition that it carries an
inverted population is simply that ${\eta}>0.$
\vskip 0.2cm\noindent
{\bf The Matter.} We 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 (3.3) that the
action of the generator, $L_{mat},$ of $T_{mat}$ on the
observables of single spins is given by
$$L_{mat}{\sigma}_{\pm,r}=-({\gamma}_{\perp}{\mp}i{\epsilon})
{\sigma}_{\pm,r}; \ L_{mat}{\sigma}_{z,r}=
-{\gamma}_{\parallel}({\sigma}_{z,r}-{\eta}I);\
L_{mat}I=0\eqno(3.6)$$
\vskip 0.2cm\noindent
{\bf 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.
\vskip 0.2cm\noindent
We formulate ${\Sigma}_{rad}$ as a $W^{\star}-$
dynamical system $({\cal A}_{rad},T_{rad},{\phi}_{rad}),$ in the
following way. First, we represent the radiation modes by
creation and destruction operators
${\lbrace}a_{l}^{\star},a_{l}{\vert}l=0,1. \ .,n-1{\rbrace}$ in
a Hilbert space ${\cal H}_{rad},$ as defined by the standard
conditions that
\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
$$[a_{l},a_{m}^{\star}]_{-}={\delta}_{lm}I; \ [a_{l},a_{m}]_{-}=0
\eqno(3.7)$$
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
$$W(z)={\exp}i(z.a+(z.a)^{\star}), \ with \
z.a={\sum}_{l=0}^{n-1}z_{l}a_{l}\eqno(3.8)$$
Thus, by (3.7), $W$ satisfies the Weyl algebraic relation
$$W(z)W(z^{\prime})=W(z+z^{\prime}){\exp}
(iIm(z,z^{\prime})_{n})\eqno(3.9)$$
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 assume that the dynamical semigroup $T_{rad}$ is completely
positive, normal and quasi-free, and consequently is given by
Vanheuverszwijn's prescription [Vh], i.e.,
$$T_{rad}(t)[W(z)]=W({\xi}(t)z){\exp}(-{\theta}(t))\eqno(3.10)$$
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
$$({\xi}(t)z)_{l}=z_{l}{\exp}(-(i{\omega}_{l}+{\kappa}_{l})t)
\ for \ l=0,.. \ .,n-1\eqno(3.11)$$
and
$${\theta}(t)={1\over 2}({\Vert}z{\Vert}_{n}^{2}-
{\Vert}{\xi}(t)z{\Vert}_{n}^{2})
\eqno(3.12)$$
where ${\Vert}.{\Vert}_{n}$ is the ${\bf C}^{n}$ norm. The
generator of the semigroup $T_{rad}$ is
$$L_{rad}={\sum}_{l=0}^{n-1}
(i[{\omega}_{l}a_{l}^{\star}a_{l},.]_{-}
+2{\kappa}_{l}a_{l}^{\star}(.)a_{l}-
{\kappa}_{l}[a_{l}^{\star}a_{l},.]_{+})\eqno(3.13)$$
\vskip 0.2cm\noindent
{\bf The Composite System,} ${\Sigma},$ is obtained by coupling
${\Sigma}_{mat}$ to ${\Sigma}_{rad}$ by dipolar interactions,
represented by the partial Hamiltonian
$$H_{int}=iN^{-1/2}{\sum}_{r=1}^{N}{\sum}_{l=0}^{n-1}
{\lambda}_{l}({\sigma}_{-,r}a_{l}^{\star}
{\exp}(-2{\pi}ik_{l}r)-h.c.),\eqno(3.14)$$
where $k_{l}$ is the wave-number of the $l'$th mode and the
${\lambda}'$s are real-valued, $N-$independent coupling
constants.
\vskip 0.2cm\noindent
We now 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, ${\cal A}={\cal L}({\cal H}),$ where
${\cal H}={\cal H}_{mat}{\otimes}{\cal H}_{rad}.$
We 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
$${\phi}_{mat}{\otimes}{\phi}_{rad}\eqno(3.15)$$
\vskip 0.2cm\noindent
Finally, we construct the dynamical semigroup semigroup, $T,$ as
the perturbation of $T_{mat}{\otimes}T_{rad}$ corresponding to
the interaction Hamiltonian $H_{int}.$ In fact, the construction
of this semigroup has been achieved [AS] by a method based on
quantum stochastic processes [AFL, HP, Ku], which negotiates the
problems caused by the unboundedness of the putative generator
of $T,$ namely
$$L=L_{mat}+L_{rad}+i[H_{int},.]\eqno(3.16)$$
The net result of this construction is to obtain a dynamical
semigroup $T,$ such that
\vskip 0.2cm\noindent
(1) there is a norm-dense set, ${\cal S}_{0},$ of normal states
on ${\cal A},$ that is stable under the mapping
${\rho}{\rightarrow}{\rho}_{t}:={\rho}{\circ}T(t);$
\vskip 0.2cm\noindent
(2) these states are well defined on the algebra ${\cal F}({\cal
A}),$ of the affiliates of ${\cal A}$ given by the polynomials
in the operators $a_{l}^{\star},a_{l}$ and $W(z);$\footnote
*{Thus, if ${\rho}$ is the density matrix representing an ${\cal
S}_{0}-$class state and $Q{\in}{\cal F}({\cal A}),$ then the
operator ${\rho}Q$ is of trace class.} and
\vskip 0.2cm\noindent
(3) $${d\over dt}{\rho}_{t}(Q)={\rho}_{t}(LQ) \
{\forall}{\rho}{\in}{\cal S}_{0}, \
Q{\in}{\cal F}({\cal A})\eqno(3.17)$$
where $L$ is given by equn. (3.16).
\vskip 0.2cm\noindent
{\bf Note.} This result serves to generalise the standard theory
of dynamical semigroups [GK, Li] to a class of unbounded
generators.
\vskip 0.3cm\noindent
{\bf 3.2. The Macroscopic Observables.}
\vskip 0.2cm\noindent
We take these to be the global intensive observables
$$s_{l}^{(N)}=N^{-1}{\sum}_{r=1}^{N}
{\sigma}_{-,r}{\exp}(-2{\pi}ik_{l}r); \ l=0,.. \
.,n-1\eqno(3.18)$$
and
$$p_{l}^{(N)}=N^{-1}{\sum}_{r=1}^{N}
{\sigma}_{z,r}{\exp}(-2{\pi}ik_{l}r); \ l=0,.. \
.,n-1\eqno(3.19)$$
together with the operators
$${\alpha}_{l}^{(N)}=N^{-1/2}a_{l}; \ l=0,.. \ .,n-1\eqno(3.20)$$
corresponding to a scaling of the number operators
$a_{l}^{\star}a_{l}$ in units of $N.$ 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 ${\cal M}^{(N)}.$ To simplify the
model, we choose
$$k_{l}={l\over n}\eqno(3.21)$$
so that ${\cal M}^{(N)}$ is a Lie algebra w.r.t. commutation.
Specifically, by (3.18)-(3.21), its non-zero Lie brackets
are the following ones, and their adjoints.
$$[s_{l}^{(N)},s_{m}^{(N){\star}}]=-N^{-1}p_{[l-m]}^{(N)}; \
[s_{l}^{(N)},p_{m}^{(N)}]=2N^{-1}s_{[l-m]}^{(N)};$$
$$[s_{l}^{(N){\star}},p_{m}^{(N)}]=
-2N^{-1}s_{[l+m]}^{(N){\star}}; \
[a_{l}^{(N)},a_{m}^{(N){\star}}]=N^{-1}I{\delta}_{lm}
\eqno(3.22)$$
where $[l{\pm}m]=l{\pm}m \ (mod \ n).$ Thus, the observables
${\cal M}^{(N)}$ become classical in the limit
$N{\rightarrow}{\infty}.$ Further, by (3.2), (3.3), (3.18) and
(3.19),
$${\Vert}s_{l}^{(N)}{\Vert}=1; \ {\Vert}p_{l}^{(N)}{\Vert}=1
\ for \ l=0,. \ .,n-1\eqno(3.23)$$
and
$$p_{0}^{(N){\star}}=p_{0}^{(N)}; \ and \ p_{l}^{(N){\star}}
=p_{n-l}^{(N)} \ for \ l=1,. \ .,n-1\eqno(3.24)$$
\vskip 0.2cm\noindent
By (3.14) and (3.18)-(3.20), the interaction Hamiltonian
$H_{int}$
is a function of the macro-observables only, i.e.,
$$H_{int}^{(N)}=iN{\sum}_{l=0}^{n-1}{\lambda}_{l}
({\alpha}_{l}^{(N){\star}}s_{l}^{(N)}-
{\alpha}_{l}^{(N)}s_{l}^{(N){\star}})\eqno(3.25)$$
\vskip 0.3cm\noindent
{\bf 3.3. The Macroscopic Dynamics.}
\vskip 0.2cm\noindent
Our objective will be to extract the dynamics of ${\cal M}^{(N)}$
from the microscopic equation of motion (3.17), 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
${\Sigma}, \ {\cal A}, \ T, \ L, \ H_{int}$ and ${\cal S}_{0}$
on this number by the superscript $(N).$
\vskip 0.2cm\noindent
We shall assume that the initial state, ${\rho}^{(N)},$ of
${\Sigma}^{(N)}$ lies in ${\cal S}_{0}^{(N)}$ and that the number
of photons it carries does not increase faster than $N,$ i.e.
that, for some finite constant $B,$
$${\rho}^{(N)}({\alpha}_{l}^{(N){\star}}{\alpha}_{l}^{(N)})**{\eta}_{1});$ and then the
radiation becomes chaotic by a mechanism precisely analogous to
that governing chaos in the famous Lorenz attractor.
\vskip 0.2cm\noindent
(4) [AS] More generally, in the multi-mode case, chaos can arise
either by the Ruelle-Takens [RT] or by the Landau mechanism [LL,
Section 27]. In the former case, it stems from a bifurcation of
the periodic orbit of (2) into a strange attractor: in the latter
one, it arises from the successive activation of a very large
number of modes.
\vskip 0.2cm\noindent
Thus, the model exhibits optically quiescent, coherent and
chaotic phases.
\vskip 0.5cm
\centerline {\bf 4. Concluding Remarks}
\vskip 0.3cm\noindent
We have shown how the passage from the microscopic to the
macroscopic picture of both conservative and open quantum systems
can lead to rich electrodynamical structures, which support both
ordered and chaotic phases.
\vskip 0.2cm\noindent
Our treatment of both the plasma and the laser models was centred
on quantum characteristic functions (QCF's) of macroscopic
observables, first introduced in [NS]. This provides a natural
procedure for extracting classical phenomenological dynamical
laws from an underlying quantum dynamics, since, under rather
general conditions, these QCF's reduce to those of a classical
probability measure in a large-scale limit.
\vskip 0.2cm\noindent
We note that both of the models treated here reduce to
mean field theories, with the crucially simplifying feature that
their macroscopic variables evolve according to self-contained
dynamical laws. In the case of the plasma model, the mean field
theoretic character stems from the macroscopic cut-off introduced
in equn. (2.16). However, we conjecture (cf. [Se2]) that a proper
exploitation of the Heisenberg and Pauli principles could
effectively blunt the Coulomb singularity and thus remove the
need for the cut-off.
\vskip 0.2cm\noindent
In the case of the laser model, the self-contained character of
the macroscopic evolution arises from (a) the fixing of the
number $n$ of radiation modes, even in the limit
$N{\rightarrow}{\infty},$ and
(b) the stipulation that the wave-numbers satisfy the condition
(3.21), which ensures that the macro-observables ${\cal M}^{(N)}$
form a Lie algebra. The generalisation of the model to a more
realistic one, with $n=N,$ would evidently demand radically new
insights, since such a model does not appear to manifest any
clear separation between its microscopic and macroscopic
dynamics.
\vskip 0.5cm
\centerline {\bf References}
\vskip 0.2cm\noindent
[AFL] L. Accardi, A. Frigerio and J. T. Lewis: Publ. RIMS {\bf
18}, 97, 1982
\vskip 0.2cm\noindent
[AS] G. Alli and G. L. Sewell: J. Math. Phys. {\bf 36}, 5598,
1995
\vskip 0.2cm\noindent
[BP] D. Bohm and D. Pines: Phys. Rev. {\bf 82}, 625, 1951
\vskip 0.2cm\noindent
[BFS] V. Bach, J. Fr\"ohlich and I. M. Sigal: "Mathematical
Theory of Non-Relativistic Matter and Radiation", Preprint
\vskip 0.2cm\noindent
[Di] R. H. Dicke: Phys. Rev. {\bf 93}, 99, 1954
\vskip 0.2cm\noindent
[Gl] R. J. Glauber: Phys. Rev. {\bf 130}, 2529, 1963
\vskip 0.2cm\noindent
[GK] V. Gorini and A. Kossakowski: J. Math. Phys. {\bf 17}, 1298,
1976
\vskip 0.2cm\noindent
[HL] 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
[HP] R. Hudson and K. Parthasarathy: Commun. Math. Phys. {\bf
93}, 301, 1984
\vskip 0.2cm\noindent
[Ku] B. K\"ummerer: J. Funct. Anal. {\bf 63}, 139, 1985
\vskip 0.2cm\noindent
[Li] G. Lindblad: Commun. Math. Phys. {\bf 48}, 119, 1976
\vskip 0.2cm\noindent
[LL] L. D. Landau and E. M. Lifshitz: "Fluid Mechanics",
Pergamon, Oxford, New York, Toronto, Sydney, Paris, 1984
\vskip 0.2cm\noindent
[Ne] H. Neunzert: Fluid Dyn. Trans. {\bf 9}, 929, 1978
\vskip 0.2cm\noindent
[NS] H. Narnhofer and G. L. Sewell: Commun. Math. Phys. {\bf 79},
9, 1981
\vskip 0.2cm\noindent
[RT] D. Ruelle and F. Takens: Commun. Math. Phys. {\bf 20}, 167,
1971
\vskip 0.2cm\noindent
[Se1] G. L. Sewell: J. Math. Phys. {\bf 26}, 2324, 1985
\vskip 0.2cm\noindent
[Se2] G. L. Sewell: Helv. Phys. Acta {\bf 67}, 4, 1994
\vskip 0.2cm\noindent
[Se3] G. L. Sewell: In preparation
\vskip 0.2cm\noindent
[Sp] H. Spohn: Math. Meth. Appl. Sci. {\bf 3}, 445, 1981
\vskip 0.2cm\noindent
[Vh] P. Vanheuverzwijn: Ann. Inst. H. Poincare A {\bf 29}, 123,
1978; Erratum {\it ibid} {\bf 30}, 83, 1979
\end
**