\magnification 1200
\centerline {\bf Macroscopic Quantum Electrodynamics of a Plasma
Model:}
\vskip 0.3cm
\centerline {\bf Derivation of the Vlasov Kinetics}
\vskip 1cm
\centerline {\bf by Geoffrey L. Sewell}
\vskip 0.5cm
\centerline {\bf Department of Physics, Queen Mary and Westfield
College, London E1 4NS}
\vskip 1cm
\centerline {\bf Abstract.}
\vskip 0.3cm\noindent
We derive the large-scale Vlasov kinetics of a plasma model from
its underlying quantum electrodynamics. The model comprises a
system of non-relativistic electrons, coupled to a quantised
electromagnetic field and to a passive, positively charged,
neutralising background.
\vskip 1cm
\centerline {\bf 1. Introduction.}
\vskip 0.3cm\noindent
This note is concerned with the extraction of the large-scale
kinetics of a plasma model from its underlying quantum
electrodynamics. The model consists of a system of
non-relativistic electrons, coupled to a quantised
electromagnetic field and to a passive, neutralising, positively
charged background. Its specific form is obtained by coupling the
purely electrostatic plasma of [1,2] to a radiation field. Our
reason for favouring a quantum, rather than classical, model is
simply that quantum mechanics is quintessential both to the
stability of matter under electromagnetic interactions [3,4] and
to the very specific properties of the plasmas that arise in
condensed matter physics [5].
\vskip 0.2cm\noindent
Our principal result is that the macroscopic dynamics of this
quantum model corresponds to a {\it classical} Vlasov-cum-Maxwell
hydro-electrodynamics. Its new feature, at least at the level of
mathematical physics, is the incorporation of the magnetic and
transverse electric fields into the theory: these are certainly
of cardinal importance for plasma physics (cf. [6]).
\vskip 0.2cm\noindent
We shall present our material as follows. In Section 2, we shall
formulate our model as an interacting quantum system of $N$
electrons and a radiation field in a box of side $L,$ which
provides a background of uniformly distributed, neutralising,
positive charge. In Section 3, we shall reformulate it on a
'macroscopic' scale, with length unit $L,$ and then
simplify it by a short distance cut-off. The
resultant model reduces to a mean field theoretic form. In
Section 4, we shall present our principal results, namely
Theorems 4.1 and 4.2, which assert that the large scale dynamics
of the model conforms to the classical Vlasov-cum-Maxwell
kinetics in a limit where $N$ tends to infinity and the density,
$N/L^{3},$ remains fixed and finite. There, we shall observe
that, as in [2], this kinetics supports non-equilibrium phase
transitions from deterministic (Eulerian) to stochastic flows.
Section 5 will be devoted to obtaining key estimates, which we
shall employ in Section 6 to extend the Vlasov methodology of
[7-10] and [1,2] and thereby to prove the above-mentioned
theorems. We shall conclude in Section 7 with some brief comments
about outstanding problems.
\vskip 0.5cm
\centerline {\bf 2. The Model.}
\vskip 0.3cm\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 provides a background of passive, uniformly distributed,
neutralising positive charge. 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 vectors in ${\bf
R}^{3}$ 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 $({\Omega}^{(L)})^{N}.$ For notational simplicity, we take the
particles to be spinless; though, in view of the stability
properties established in [4], the electron spin would cause no
difficulties. The radiation is formulated in terms of a
transversely gauged vector potential, $A,$ and the transverse
electric field, $F,$ which are both Hermitian distribution-valued
operators in a Fock-Hilbert space, ${\cal H}_{rad}^{(L)},$ and
are defined by the following conditions.
\vskip 0.2cm\noindent
(1) $A$ and $F$ satisfy the canonical commutation relations
$$[A_{\mu}(X),F_{\nu}(X^{\prime})]=
i{\hbar}D_{{\mu}{\nu}}^{(L)}(X-X^{\prime}),
\eqno(2.3)$$
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
We 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.
\vskip 0.2cm\noindent
We assume that the interactions are the standard electromagnetic
ones, and thus that the Hamiltonian of the model takes the
following form (cf [11]).
$$H^{(N,L)}={\sum}_{j=1}^{N}
{1\over 2m}(P_{j}+{\epsilon}(A_{\kappa}(X_{j}))^{2}+
{\epsilon}^{2}{\sum}_{k,l(>k)=1}^{N}U^{(L)}(X_{k}-X_{l})+$$
$${1\over 2}{\int}_{{\Omega}^{(L)}}:(F(X))^{2}+
c^{2}({\nabla}^{(L)}{\times}A(X))^{2}:dX,\eqno(2.4)$$
where the replacement of $A$ by $A_{\kappa}$ in the first term
represents the cut-off obtained by removing from $A$ its Fourier
components whose wave-vectors have magnitude greater than
${\kappa}:={\hbar}/mc,$ and where $U^{(L)},$ the two-body Coulomb
interaction between electrons in the presence of the neutralising
background is given by (cf. [12])
$$U^{(L)}(X)=L^{-3}{\sum}_{Q{\in}
(2{\pi}L^{-1}{\bf Z})^{3}{\backslash}{\lbrace}0{\rbrace}}
{{\exp}(iQ.X)\over Q^{2}}.\eqno(2.5)$$
Our aim 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 0.5cm\noindent
\centerline {\bf 3. The Rescaled Description.}
\vskip 0.3cm\noindent
We take our macroscopic description of the model to be a
'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
$${\hbar}_{N}={{\hbar}\over mL^{2}{\omega}_{p}}
{\equiv}{{\hbar}\over m{\omega}_{p}}
({n_{0}\over N})^{2/3},\eqno(3.1)$$
and the speed of light is
$$c_{0}={c\over L{\omega}_{p}}{\equiv}{c\over {\omega}_{p}}
({n_{0}\over N})^{1/3}.\eqno(3.2)$$
\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
${\Omega}:={\Omega}^{(1)}{\equiv}({\bf R}/{\bf Z})^{3}.$ Thus,
we define ${\cal H}^{(N)}$ to be ${\cal H}^{(N,1)}$ 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,
$(a,f),$ by the following formulae.
$$x_{j}:=L^{-1}VX_{j}V^{-1}; \ p_{j}
:=(mL{\omega}_{p})^{-1}VP_{j}V^{-1}
{\equiv}-i{\hbar}_{N}{\nabla}_{x_{j}},\eqno(3.3)$$
where ${\nabla}$ (or ${\nabla}_{x}$) is the gradient operator in
${\Omega},$ and
$$a(x):={{\epsilon}\over mL{\omega}_{p}}VA(Lx)V^{-1}; \ f(x):=
{{\epsilon}\over mL{\omega}_{p}^{2}}VF(Lx)V^{-1}.\eqno(3.4)$$
Likewise, we define the rescaled Hamiltonian, representing the
macroscopic description, to be
$$H^{(N)}=(mL^{2}{\omega}_{p}^{2})^{-1}VH^{(N,L)}V^{-1}$$
$$={1\over 2}{\sum}_{j=1}^{N}
(p_{j}+a_{\kappa}(x_{j}))^{2}+
N^{-1}{\sum}_{j,k(>j)=1}^{N}U(x_{j}-x_{k})+$$
$${N\over 2}{\int}_{\Omega}:(f(x))^{2}+
c_{0}^{2}({\nabla}{\times}a(x))^{2}:dx,\eqno(3.5)$$
where
$$U(x)=
{\sum}_{q{\in}(2{\pi}{\bf Z})^{3}{\backslash}{\lbrace}0{\rbrace}}
{{\exp}(iq.x)\over q^{2}}\eqno(3.6)$$
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
$$b={\nabla}{\times}a.\eqno(3.7)$$
\vskip 0.2cm\noindent
Further, by equns. (2.1)-(2.3) and
(3.4), the fields $a$ and $f$ satisfy the CCR
$$[a_{\mu}(x),f_{\nu}(x^{\prime})]=iN^{-1}{\hbar}_{N}
D_{{\mu}{\nu}}(x-x^{\prime}),\eqno(3.8)$$
where $D{\equiv}D^{(1)}$ is the transverse part of the product
of the unit tensor in ${\bf R}^{3}$ and the Dirac distribution
in ${\Omega}.$
\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. (3.6), we replace
$U$ and $a_{\kappa}$ by $U_{g}$ and $a_{g}^{(N)},$ respectively,
where
$$U_{g}=g*U; \ a_{g}^{(N)}=g^{(N)}*a\eqno(3.9)$$
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
$N{\rightarrow}{\infty}.$ We define
$$b_{g}^{(N)}=g^{(N)}*b; \ f_{g}^{(N)}=g^{(N)}*f.
\eqno(3.10)$$
\vskip 0.2cm\noindent
{\bf Note.} The regularisation of the interactions
by convolution with $g$ (or $g^{(N)}$) is radically 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
$$H_{g}^{(N)}={1\over 2}{\sum}_{j=1}^{N}v_{j}^{2}+
N^{-1}{\sum}_{j,k(>j)=1}^{N}U_{g}(x_{j}-x_{k})+Nh_{rad},
\eqno(3.11)$$
where
$$v_{j}=p_{j}+a_{g}^{(N)}(x_{j})\eqno(3.12)$$
is the velocity of the j'th particle and
$$h_{rad}={1\over 2}
{\int}:(f(x))^{2}+c_{0}^{2}(b(x))^{2}:dx\eqno(3.13)$$
is the radiative energy, as measured in units of $N.$ Note that
$H_{g}^{(N)}$ is expressed in terms of gauge invariant operators
only.
\vskip 0.2cm\noindent
{\bf Fixing of $c_{0}.$} The physical demand that the particle
speeds of ${\Sigma}_{g}^{(N)}$ cannot exceed the speed of light
for the model implies that $c_{0}$ must 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, {\it even when passing
to the limit $N{\rightarrow}{\infty}.$} This is appropriate for
a treatment of ${\Sigma}_{g}^{(N)},$ where, on the one hand,
$N>>1,$ so that this finite system is 'close' to its hydro-
thermodynamic limit; while, on the other hand, $N$ is small
enough to ensure that $c_{0},$ as given by equn. (3.2), is at
least of the order of unity. One may easily check that both these
requirements can be fulfilled in realistic situations, e.g. with
$L=1$ cm. and $N{\simeq}10^{13}.$
\vskip 0.2cm\noindent
To express the fields $b, \ f$ 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'
$$b({\phi}):={\int}_{\Omega}dxb(x).{\phi}(x); \
f({\psi}):={\int}_{\Omega}f(x).{\psi}(x) \ {\forall}{\phi},
\ {\psi}{\in}{\cal D}_{tr}.\eqno(3.14)$$
Thus, $b, \ f$ are maps from ${\cal D}_{tr}$ into the self-
adjoint operators in ${\cal H},$ and the CCR (3.8) yields the
following one for these distributions.
$$[b({\phi}),f({\psi})]=iN^{-1}{\hbar}_{N}
({\phi},{\nabla}{\times}{\psi}) \
{\forall}{\phi},{\psi}{\in}{\cal D}_{tr},\eqno(3.15)$$
where $(.,.)$ denotes the $L^{2}$ inner product. Further, by
equns. (3.8), (3.12), and (3.15), the only other non-zero
commutators between the positions, momenta and fields of the
model are the following.
$$[x_{j,{\mu}},v_{k,{\nu}}]=i{\hbar}_{N}{\delta}_{jk}{\delta}_
{{\mu}{\nu}}I; \
[v_{j,{\mu}},v_{k,{\nu}}]=i{\hbar}_{N}
{\epsilon}_{{\mu}{\nu}{\sigma}}b_{g,{\sigma}}^{(N)}(x_{j})
{\delta}_{jk};$$
$$and \
[v_{j},f({\psi})]=i{\hbar}_{N}N^{-1}{\int}dxg^{(N)}(x){\psi}(x),
\eqno(3.16)$$
where ${\epsilon}$ is the alternate tensor, i.e.,
${\epsilon}_{{\mu}{\nu}{\sigma}}=1 \ (resp. \ -1)$ if
$({\mu},{\nu},{\sigma})$ is an even (resp. odd) permutation of
$(1,2,3),$ and is otherwise zero.
\vskip 0.2cm\noindent
Thus, the algebra generated by the operators
${\lbrace}x_{j},v_{j},b({\phi}),f({\psi}){\vert}j=1,. \ .,N; \
{\phi},{\psi}{\in}{\cal D}_{tr}{\rbrace}$ is closed w.r.t.
commutation. We take the Hermitian elements of this algebra to
be the observables of ${\Sigma}_{g}^{(N)}$ and the density
matrices in ${\cal H}^{(N)}$ to represent its states, so that the
expectation value of an observable, $A,$ in the state,
${\rho}^{(N)},$ is ${\rho}^{(N)}(A):=Tr({\rho}^{(N)}(A)).$
\vskip 0.2cm\noindent
The dynamics of ${\Sigma}_{g}^{(N)},$ in the Schr\"odinger
representation, is given by the unitary transformations
${\rho}^{(N)}{\rightarrow}{\rho}_{t}^{(N)}$ of its states, with
$${\rho}_{t}^{(N)}={\exp}(-iH_{g}^{(N)}t/{\hbar}_{N}){\rho}^{(N)}
{\exp}(iH_{g}^{(N)}t/{\hbar}_{N}) \ {\forall}t{\in}{\bf R}.
\eqno(3.17)$$
\vskip 0.2cm\noindent
The time-derivatives of the observables are determined, in
the Heisenberg picture, by the action on them of the
derivation
$${\Lambda}_{g}^{(N)}=
{i\over {\hbar}_{N}}[H_{g}^{(N)},.] \ .\eqno(3.18)$$
In particular, we see from equations (3.11), (3.15) and (3.16)
that this action is given by
$${\Lambda}_{g}^{(N)}x_{j}=v_{j};
\ {\Lambda}_{g}^{(N)}v_{j}=-f_{g}^{(N)}(x_{j})-
(v_{j}{\times}b_{g}^{(N)}(x_{j}))_{sym}
+N^{-1}{\sum}_{k{\neq}j}{\nabla}U_{g}(x_{j}-x_{k})
\eqno(3.19)$$
and
$${\Lambda}_{g}^{(N)}b=-{\nabla}{\times}f; \
{\Lambda}_{g}^{(N)}f=
c_{0}^{2}{\nabla}{\times}b_{g}+
N^{-1}{\sum}_{k=1}^{N}(v_{k}.D_{g}^{(N)}(x-x_{k}))_{sym},
\eqno(3.20)$$
where
$$D_{g}^{(N)}=g^{(N)}*D,\eqno(3.21)$$
$(v.D_{g}^{(N)})_{\mu}:={\sum}_{\nu}v_{\nu}D_{g,{\mu}{\nu}}^{(
N)}$
and the subscript $'sym'$ denotes symmetrised product. Thus,
the last term in equn. (3.20) represents the transverse part of
the regularised current density.
\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. (3.1), (3.15) and (3.16) that
the effective Planck constant, ${\hbar}_{N},$ vanishes and that
the observables $x_{j}, \ p_{j}, \ b, \ f$ all intercommute in
the limit $N{\rightarrow}{\infty};$ while, on the other hand,
the last terms in equns. (3.19) and (3.20) are of typical mean
field theoretic form, namely arithmetic means of $N$ copies of
single-particle observables.
\vskip 0.2cm\noindent
The following definitions are designed to accommodate the
anticipated classical structures of the macroscopic dynamics of
the model.
\vskip 0.2cm\noindent
{\bf Definition 3.1.} (1) We define the 'classical single-
particle phase space' $K:={\Omega}{\times}{\bf R}^{3}$ and its
dual ${\hat K}:=(2{\pi}{\bf Z})^{3}{\times}{\bf R}^{3}.$
\vskip 0.2cm\noindent
(2) We define ${\cal P}(K)$ to be the space of probability
measures on $K,$ with the vague topology.
\vskip 0.2cm\noindent
(3) We define the Weyl maps, ${\lbrace}W^{(N,n)}:{\hat
K}^{n}{\times}({\cal D}_{tr})^{2}{\rightarrow}
{\cal U}({\cal H}^{(N)}){\vert}n=0,. \ .,N{\rbrace},$
by the formula
$$W^{(N,n)}({\xi}_{1},{\eta}_{1};. \ .;{\xi}_{n},{\eta}_{n}
;{\phi},{\psi})=$$
$${\exp}({i\over 2}(b({\phi})+f({\psi})))
({\Pi}_{j=1}^{n}{\exp}({1\over 2}{\eta}_{j}.v_{j})
{\exp}(i{\xi}_{j}.x_{j})
{\exp}({1\over 2}{\eta}_{j}.v_{j}))
{\exp}({i\over 2}(b({\phi})+f({\psi}))).\eqno(3.22)$$
\vskip 0.2cm\noindent
(4) We define the quantum characteristic functions
${\lbrace}C_{t}^{(N,n)}:{\hat K}^{n}{\times}({\cal
D}_{tr})^{2}{\rightarrow}{\bf C}{\vert}n=0,. \ .,N{\rbrace}$ by
the formula
$$C_{t}^{(N,n)}={\rho}_{t}^{(N)}{\circ}W^{(N,n)}.\eqno(3.23)$$
\vskip 0.2cm\noindent
Thus, the characteristic functions
${\lbrace}C_{t}^{(N,n)}{\vert}n=0,1,. \
.,N{\rbrace}$ faithfully represent the state
${\rho}_{t}^{(N)}.$
\vskip 0.5cm
\centerline {\bf 4. The Classical Vlasov Limit.}
\vskip 0.3cm\noindent
In order to treat the dynamics of ${\Sigma}_{g}^{(N)}$ in the
limit $N{\rightarrow}{\infty},$ we shall need to formulate the
evolution of the family of systems
${\lbrace}{\Sigma}_{g}^{(N)}{\vert}N{\in}{\bf N}{\rbrace}.$ We
assume the following initial conditions.
\vskip 0.2cm\noindent
{\it (I.1). The expectation value of the energy per particle of
${\Sigma}_{g}^{(N)},$ in its initial state, ${\rho}^{(N)},$ is
bounded, uniformly w.r.t. $N,$ i.e.,
$${\rho}^{(N)}(H_{g}^{(N)})<{\gamma}N \ {\forall}N{\in}{\bf N},
\eqno(4.1)$$
where ${\gamma}$ is a finite constant. Since the system is
conservative, this inequality is equivalent to}
$${\rho}_{t}^{(N)}(H_{g}^{(N)})<{\gamma}N \ {\forall}N{\in}{\bf
N}, \ t{\in}{\bf R}.\eqno(4.1)^{\prime}$$
This energy bound for ${\Sigma}_{g}^{(N)}$ corresponds to one
proportional to $N^{5/3}$ for the original system
${\Sigma}^{(N,L)},$ and is chosen to represent the situation
where the latter 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 and
$L-$independent. For then, both the particle kinetic energy and the
electromagnetic field energy are proportional to $N^{5/3}.$
\vskip 0.2cm\noindent
{\it (I.2) The characteristic functions $C_{0}^{(N,n)}$
factorise, in the limit $N{\rightarrow}{\infty},$ according to
the formula}
$${\lim}_{N\to\infty}[C_{0}^{(N,n)}({\xi}_{1},{\eta}_{1};. \
.,{\xi}_{n},{\eta}_{n};{\phi},{\psi})-({\Pi}_{j=1}^{n}
C_{0}^{(N,1)}({\xi}_{j},{\eta}_{j};0,0))
C_{0}^{(N,0)}({\phi},{\psi})]=0.\eqno(4.2)$$
This condition represents the situation where ${\Sigma}^{(N,L)}$
is prepared in a pure phase, carrying correlations only of short
range, which scale down to zero range for
${\Sigma}_{g}^{(N)}$ in the limit $N{\rightarrow}{\infty}.$
\vskip 0.2cm\noindent
{\it (I.3) The initial state of the radiation field is
macroscopically coherent, in that its fluctuations reduce to
zero, on the ${\Sigma}_{g}^{(N)}$ scale, in the limit
$N{\rightarrow}{\infty},$ i.e.,
$${\lim}_{N\to\infty}C_{0}^{(N,0)}({\phi},{\psi})=
{\exp}i(b_{0}({\phi})+f_{0}({\psi})),\eqno(4.3)$$
where $b_{0}$ 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 Theorem 4.1.} {\it (1) Under the assumption (I.1), the
quantum characteristic functions $C_{.}^{(N,n)}$ tend pointwise
and subsequentially, as $N{\rightarrow}{\infty},$ to classical
ones, $C_{.}^{(n)},$ the Fourier transforms of probability
measures
$M_{.}^{(n)}$ on $K^{n}{\times}({\cal D}_{tr}^{\prime})^{2},$
i.e.,
$${\lim}_{N\to\infty}C_{t}^{(N,n)}({\xi}_{1},{\eta}_{1};. \
.;{\xi}_{n},{\eta}_{n};{\phi},{\psi})=$$
$${\int}dM_{t}^{(n)}(x_{1},. \ .,x_{n};v_{1},. \
.,v_{n};b,f)
{\exp}i[{\sum}_{j=1}^{n}({\xi}_{j}.x_{j}+{\eta}_{j}.v_{j})+
b({\phi})+f({\psi})].\eqno(4.4)$$
\vskip 0.2cm\noindent
(2) The set ${\lbrace}M_{t}^{(n)}{\vert}n{\in}{\bf N}{\rbrace}$
canonically defines a probability measure $M_{t}$ on $K^{\bf
N}{\otimes}({\cal D}_{tr}^{\prime})^{2},$ that is symmetric
w.r.t. the $K-$components and satisfies the conditions
$${\int}dM_{t}v_{j}^{2}{\leq}{\gamma}_{1} \ {\forall}j{\in}
{\bf N}, \ t{\in}{\bf R}\eqno(4.5)$$
and
$${\int}dM_{t}(b({\phi}))^{2}{\leq}
c_{0}^{-2}{\gamma}_{1}{\Vert}{\phi}{\Vert}^{2};
\ {\int}dM_{t}(f({\psi}))^{2}<
{\gamma}_{1}{\Vert}{\psi}{\Vert}^{2}
\ {\forall}{\phi},{\psi}{\in}{\cal D}_{tr}, \ t{\in}
{\bf R},\eqno(4.6)$$
where ${\gamma}_{1}$ is a finite constant.}
\vskip 0.3cm\noindent
{\bf Note.} Here, $x_{j}, \ v_{j}, \ b$ and $f$ denote classical
quantities, whereas previously they represented quantum
observables. In the theory that follows, it should be
clear, in each specific context, whether they are to be
interpreted as classical or quantum variables.
\vskip 0.3cm\noindent
{\bf Theorem 4.2.} {\it Under the initial conditions (I.1)-(I.4),
\vskip 0.2cm\noindent
(1) the convergence of the characteristic functions in Theorem
4.1 becomes fully sequential;
\vskip 0.2cm\noindent
(2) $M_{t}$ takes the form
$$M_{t}=m_{t}^{{\otimes}{\bf N}}{\otimes}{\delta}_{b_{t},f_{t}}
\ {\forall}t{\in}{\bf R},\eqno(4.7)$$
where $m_{t}$ is a probability measure on $K$, which satisfies
the condition
$${\int}dm_{t}v^{2}{\leq}{\gamma}_{1} \ {\forall}t{\in}
{\bf R}\eqno(4.8)$$
and ${\delta}_{b_{t},f_{t}}$ is the Dirac measure on $({\cal
D}_{tr}^{\prime})^{2},$ with support
at a point $(b_{t},f_{t});$ and
\vskip 0.2cm\noindent
(3) $(m_{t},b_{t},f_{t})$ is the unique solution of the following
classical Vlasov-Maxwell equations, subject to the
condition (4.8):-
$${d\over dt}{\int}dm_{t}h={\int}dm_{t}(v.{\nabla}_{x}h-
(e_{g,t}+v{\times}b_{g,t}).{\nabla}_{v}h) \
{\forall}h{\in}C_{0}^{(1,1)}(K),\eqno(4.9)$$
$${{\partial}b_{t}\over {\partial}t}
=-{\nabla}{\times}f_{t}\eqno(4.10)$$
and
$${{\partial}f_{t}\over {\partial}t}
=c_{0}^{2}{\nabla}{\times}b_{t}+
{\int}dm_{t}*v.D_{g},\eqno(4.11)$$
where $*$ denotes convulution w.r.t. the position variable $x$
only,
$$e_{g,t}=f_{g,t}+
{\int}dm_{t}*{\nabla}U_{g}\eqno(4.12)$$
and
$$b_{g,t}=g*b_{t}; \ f_{g,t}=g*f_{t}; \ D_{g}=g*D.\eqno(4.13)$$
Thus, these equations define a one-parameter group
${\lbrace}{\tau}_{t}{\vert}t{\in}{\bf R}{\rbrace}$ of
transformations of the space ${\cal P}(K){\times}({\cal
D}_{tr}^{\prime})^{2},$ according
to the formula}
$${\tau}_{t}(m_{0},b_{0},f_{0})=(m_{t},b_{t},f_{t}).\eqno(4.14)$$
\vskip 0.3cm\noindent
{\bf Note on Hydrodynamical Phase Transitions.}
\vskip 0.2cm\noindent
In a previous work [2], we showed that the electrostatic model
supports transitions from Eulerian deterministic to stochastic
flow when the initial conditions are sufficiently non-uniform.
The present model must exhibit these same transitions, since the
Vlasov-Maxwell equations (4.9)-(4.13) harbour the solutions of
those of the electrostatic model, with $b=f=0.$ The interesting
question, of course, concerns what other hydrodynamical phase
transitions the present model may support.
\vskip 0.5cm\noindent
\centerline {\bf 5. Key Estimates.}
\vskip 0.3cm\noindent
{\bf Lemma 5.1} {\it Under the assumption (I.1), there is a
finite constant,
${\gamma}_{1},$ such that
$${\rho}_{t}^{(N)}(v_{j}^{2})<{\gamma}_{1}, \ {\forall}
t{\in}{\bf R}, \ N{\in}{\bf N}, \ j=1,. \ .,N\eqno(5.1)$$
and
$${\rho}_{t}^{(N)}(f({\phi})^{2}+c_{0}^{2}b({\phi})^{2})
<{\gamma}_{1}{\Vert}{\phi}{\Vert}^{2}+N^{-1}{\hbar}_{N}c_{0}
{\Vert}{\phi}{\Vert} \
{\Vert}{\nabla}{\times}{\phi}{\Vert} \
{\forall}t{\in}{\bf R}, \ N{\in}{\bf N}, \
{\phi}{\in}{\cal D}_{tr},\eqno(5.2)$$
where ${\Vert}.{\Vert}$ is the $L^{2}$ norm.}
\vskip 0.3cm\noindent
{\bf Proof.} Since $U_{g}$ is bounded, it follows from equns.
(3.11), (3.13) and (4.1)$^{\prime},$ together with the Fermi
statistics of the electrons, that the expectation values of
$v_{j}^{2}$ and
$h_{rad},$ for the state ${\rho}_{t}^{(N)},$
are both majorised by some finite constant, ${\gamma}_{1}.$ This
establishes the estimate (5.1) and the inequality
$${\rho}_{t}^{(N)}(h_{rad})<{\gamma}_{1}.\eqno(5.3)$$
To obtain the estimate (5.2), we Fourier analyse the
test function ${\phi}$ and the fields $b$ and $f,$ noting that,
for each wave-vector $k \ ({\in}(2{\pi}{\bf
Z})^{3}{\backslash}{\lbrace}0{\rbrace}),$ we may choose two unit
vectors $u_{k,-1}, \ u_{k,1},$ such that the orthonormal vectors
$(k^{(1)} \ (:=k/{\vert}k{\vert}),u_{k,-1},u_{k,1})$ form a
right-handed triad in ${\bf R}^{3},$ i.e.,
$$k^{(1)}{\times}u_{k,s}=-su_{k,-s}; \ and \
u_{k,-1}{\times}u_{k,1}=k^{(1)}.\eqno(5.4)$$
Thus, in view of the CCR (3.15), the Fourier decompositions of
${\phi}, \ f$ and $b$ take the form
$${\phi}(x)={\sum}_{k,s}{\hat {\phi}}_{k,s}u_{k,s}{\exp}(ik.x)
,\eqno(5.5)$$
$$f(x)={\sum}_{k,s}
({1\over 2}N^{-1}{\hbar}_{N}c_{0}{\vert}k{\vert})^{1\over 2}
({\alpha}_{k,s}+{\alpha}_{-k,s}^{\star})u_{k,s}{\exp}(ik.x)
\eqno(5.6)$$
and
$$c_{0}b(x)=-{\sum}_{k,s}
({1\over 2}N^{-1}{\hbar}_{N}c_{0}{\vert}k{\vert})^{1\over 2}
({\alpha}_{k,s}-{\alpha}_{-k,s}^{\star})su_{k,-s}
{\exp}(ik.x),\eqno(5.7)$$
where $k$ and $s$ run over $(2{\pi}{\bf
Z})^{3}{\backslash}{\lbrace}0{\rbrace}$ and
${\lbrace}-1,1{\rbrace},$ respectively, and the
${\alpha}^{\star}$'s and ${\alpha}$'s
are creation and annihilation operators satisfying the CCR
$$[{\alpha}_{k,s},{\alpha}_{k^{\prime},s^{\prime}}^{\star}]=
{\delta}_{kk^{\prime}}{\delta}_{ss^{\prime}}; \
[{\alpha}_{k,s},{\alpha}_{k^{\prime},s^{\prime}}]=0.\eqno(5.8)$$
Hence,
$$f({\phi})={\sum}_{k,s}
({1\over 2}N^{-1}{\hbar}_{N}c_{0}{\vert}k{\vert})^{1\over 2}
({\alpha}_{k,s}+{\alpha}_{-k,s}^{\star}){\hat {\phi}}_{-k,s}
\eqno(5.9)$$
and
$$c_{0}b({\phi})=-i{\sum}_{k,s}
({1\over 2}N^{-1}{\hbar}_{N}c_{0}{\vert}k{\vert})^{1\over 2}
({\alpha}_{k,s}-{\alpha}_{-k,s}^{\star})s{\hat {\phi}}_{-k,-s}.
\eqno(5.10)$$
It follows from these formulae and the Schwartz inequality that
$$f({\phi})^{2}+c_{0}^{2}b({\phi})^{2}-
(:f({\phi})^{2}+c_{0}^{2}b({\phi})^{2}:)=
N^{-1}{\hbar}_{N}{\sum}_{k,s}c_{0}{\vert}k{\vert} \
{\vert}{\hat {\phi}}_{k,s}{\vert}^{2}{\leq}$$
$$N^{-1}{\hbar}_{N}{\sum}_{k,s}c_{0}{\Vert}{\phi} \
{\Vert} \ {\Vert}{\nabla}{\times}{\phi}{\Vert},\eqno(5.11)$$
and also, by equns. (3.13) and (5.3), that
$${\rho}_{t}^{(N)}(:f({\phi})^{2}+c_{0}^{2}b({\phi})^{2}:){\leq}
{\rho}_{t}^{(N)}(h_{rad}){\Vert}{\phi}{\Vert}^{2}.\eqno(5.12)$$
The estimate (5.2) follows immediately from these last two
inequalities and equn. (5.3).
\vskip 0.3cm\noindent
{\bf Definition 5.2.} (1) For each $R{\in}{\bf N},$ we define
${\cal D}_{tr}^{(R)}$ to be the subspace of ${\cal D}_{tr}$ whose
Fourier coefficients vanish outside the ball of radius $2{\pi}R;$
and, for ${\phi}{\in}{\cal D}_{tr},$ we define ${\phi}^{(R)}$ to
be the element of ${\cal D}_{tr}^{(R)}$ whose Fourier
coefficients coincide with those of ${\phi}$ inside this ball.
\vskip 0.2cm\noindent
(2) We define the truncated characteristic functions
$C_{t}^{(N,n,R)}:{\hat K}^{n}{\times}({\cal
D}_{tr})^{2}{\rightarrow}{\bf C}$ by the formula
$$C_{t}^{(N,n,R)}({\xi}_{1},{\eta}_{1};. \
.;{\xi}_{n},{\eta}_{n}:{\phi},{\psi}){\equiv}
C_{t}^{(N,n)}({\xi}_{1},{\eta}_{1};. \
.;{\xi}_{n},{\eta}_{n}:{\phi}^{(R)},{\psi}^{(R)}).\eqno(5.13)$$
\vskip 0.3cm\noindent
{\bf Lemma 5.3.} {\it $C_{.}^{(N,n,R)}$ converges pointwise to
$C_{.}^{(N,n)}$ as $R{\rightarrow}{\infty},$ the convergence
being uniform w.r.t. $N, \ t$ and the ${\xi}$'s and ${\eta}$'s.}
\vskip 0.3cm\noindent
{\bf Proof.} By equn. (3.22),
$$W^{(N,n)}({\xi}_{1},{\eta}_{1};. \ .;{\xi}_{n},{\eta}_{n};
2{\phi},2{\psi}){\equiv}$$
$$W^{(N,0)}({\phi},{\psi})
W^{(N,n)}({\xi}_{1},{\eta}_{1};. \ .;{\xi}_{n},{\eta}_{n};0,0)
W^{(N,0)}({\phi},{\psi}).\eqno(5.14)$$
Further, by the CCR (3.15) and equn. (3.22),
$$W^{(N,0)}({\phi},{\psi})-
W^{(N,0)}({\phi}^{(R)},{\psi}^{(R)})=
W^{(N,0)}({\phi},{\psi})(I+F^{(N,R)}+G^{(N,R)}),\eqno(5.15)$$
where
$$F^{(N,R)}=W^{(N,0)}({\phi}-{\phi}^{(R)},
{\psi}-{\psi}^{(R)})-I\eqno(5.16)$$
and
$$G^{(N,R)}=W^{(N,0)}(({\phi}-{\phi}^{(R)}),
({\psi}-{\psi}^{(R)}))
[{\exp}({i\over 2}N^{-1}{\hbar}_{N}
(({\psi}^{(R)},{\phi})
-({\phi}^{(R)},{\psi})))-1],\eqno(5.17)$$
$(.,.)$ being the $L^{2}$ inner product.
Hence, by equns. (3.22), (5.16) and (5.17), together with the
inequality ${\vert}1-{exp}(ix){\vert}{\leq}{\vert}x{\vert},$
whether $x$ is a real number or Hermitian operator,
$${\rho}_{t}^{(N)}(F^{(N,R){\star}}F^{(N,R)}){\leq}
2{\rho}_{t}^{(N)}((b({\phi}^{(R)}-{\phi}))^{2}+
(f({\psi}^{(R)}-{\psi}))^{2})\eqno(5.18)$$
and
$${\Vert}G^{(N,R)}{\Vert}{\leq}{1\over 2}N^{-1}{\hbar}_{N}
({\Vert}{\phi}^{(R)}-{\phi}{\Vert} \
{\Vert}{\psi}{\Vert}+{\Vert}{\psi}^{(R)}-{\psi}{\Vert} \
{\Vert}{\phi}{\Vert}),\eqno(5.19)$$
this latter estimate being an upper bound of the modulus of the
exponent in equn. (5.17). It follows from the last two
inequalities and Def. 5.2 that
both ${\rho}_{t}^{(N)}(F^{(N,R){\star}}F^{(N,R)})$ and
${\Vert}G^{(N,R)}{\Vert}$ tend to zero, uniformly w.r.t. $N$ and
all other arguments, as $R{\rightarrow}{\infty};$ and by equns.
(3.23) and (5.13)-(5.15), together with the uniform boundedness
of $W^{(N,n)},$ this implies the required result.
\vskip 0.5cm\noindent
\centerline {\bf 6. Proof of Theorems.}
\vskip 0.3cm\noindent
{\bf Proof of Theorem 4.1.} This is based on an extension of the
method employed in [9] and [1] to obtain the corresponding
result in the absence of the quantum field $(b,f).$
\vskip 0.2cm\noindent
(1) We start by noting that $C_{t}^{(N,n,R)},$ as defined in Def.
5.2(2), may be canonically identified with the restriction of
$C_{t}^{(N,n)}$ to ${\hat K}^{n}{\times}({\cal
D}_{tr}^{(R)})^{2}.$ We next observe that, by equns. (3.17),
(3.18), (3.22) and (3.23),
$${d\over dt}C_{t}^{(N,n)}={\rho}_{t}^{(N)}
{\circ}{\Lambda}_{g}^{(N)}W^{(N,n)},\eqno(6.1)$$
the r.h.s. of this equation being well-defined, by Lemma 5.1 and the
fact that, by equns. (3.11), (3.15), (3.16) and (3.18),
${\Lambda}_{g}^{(N)}W^{(N,n)}$ is of the form $W^{(N,n)}{\cal
Q},$ where ${\cal Q}$ is a linear combination of velocities,
smeared fields and bounded operators.
\vskip 0.2cm\noindent
It follows now from equns. (3.22), (3.23) and (6.1), as
restricted to ${\hat K}^{n}{\times}({\cal D}_{tr}^{(R)})^{2},$
together with the commutation relations (3.15) and (3.16), that
the derivatives of $C_{.}^{(N,n,R)}$ w.r.t. the continuous
variables $t,{\eta}_{j}$ and the Fourier coefficients of
${\phi}^{(R)},{\psi}^{(R)},$ of which there are but a finite
number, are bounded, uniformly on the compacts. Consequently (cf.
[9]), it may be inferred from the Arzela-Ascoli theorem that
$C_{.}^{(N,n,R)}$ converges subsequentially and pointwise, over
the full range of these arguments, as $N{\rightarrow}{\infty}.$
Hence, by Lemma 5.1, the same is true for $C_{.}^{(N,n)}.$ We
denote its limit by $C_{.}^{(n)}.$
\vskip 0.2cm\noindent
In view of the asymptotic commutativity of the observables
$x_{j},v_{j},b({\phi}),f({\psi}),$ as $N{\rightarrow}{\infty}$
(cf. equns. (3.15) and (3.16)), it is now a simple matter to
employ Bochner's theorem, as in [9], to prove that $C_{t}^{(n)}$
is indeed the characteristic function of a classical probability
measure $M_{t}^{(n)}.$
\vskip 0.2cm\noindent
(2) By Def. 3.1(4) and equn. (4.4),
${\lbrace}M_{t}^{(n)}{\vert}n{\in}{\bf N}{\rbrace}$ canonically
induces a probability measure $M_{t}$ on $K^{\bf
N}{\otimes}({\cal D}_{tr}^{\prime})^{2},$ according to the
specification that $M_{t}^{(n)}$ is the restriction of $M_{t}$
to $K^{n}{\otimes}({\cal D}_{tr}^{\prime})^{2}.$ The symmetry of
$M_{t}$ w.r.t. the $K-$components is ensured by the Fermi
statistics of the electrons.
\vskip 0.2cm\noindent
To establish the estimate (4.5), we employ equn. (4.4) for the
case where $n=1$ and ${\xi}_{1}, \ {\phi}, \ {\psi}$ and all but
the ${\mu}$'th component of ${\eta}_{1}$ vanish. On integrating
the resultant equation against ${\hat F}({\eta}_{1,{\mu}}),$
where ${\hat F}$ is the Fourier transform of a ${\cal D}-$class
function $F$ on ${\bf R},$ we see from Def. 3.1(4) that
$${\lim}_{N\to\infty}{\rho}_{t}^{(N)}(F(v_{1,{\mu}}))=
{\int}dM_{t}F(v_{1,{\mu}}).\eqno(6.2)$$
We now choose $F$ to be of the form
$$F(y)=y^{2}G_{R}(y),\eqno(6.3)$$
where $G_{R}$ and $1-G_{R}$ are positive valued functions, with
$G_{R}$ taking the value unity on the interval $[-R,R]$ and
vanishing outside $[-R-1,R+1].$ Thus, defining ${\chi}_{R}$ to
be the index function for $[-R,R],$It follows from these
specifications and equations (4.2), (6.2) and (6.3) that
$${\sum}_{{\mu}=1}^{3}{\int}dM_{t}v_{1,{\mu}}^{2}
{\chi}_{R}(v_{1,{\mu}}){\leq}
{\sum}_{{\mu}=1}^{3}{\int}dM_{t}v_{1,{\mu}}^{2}
G_{R}(v_{1,{\mu}})=$$
$${\lim}_{N\to\infty}{\sum}_{{\mu}=1}^{3}{\rho}_{t}^{(N)}
(v_{1,{\mu}}^{2}G_{R}(v_{1,{\mu}}){\leq}{\gamma}_{1}.$$
Since $R$ may chosen arbitrarily, this estimate implies the
validity of equn. (4.5) for $j=1,$ and hence, by symmetry, for
all
$j.$ The proof of the estimate (4.6) may be obtained analogously.
\vskip 0.3cm\noindent
{\bf Definition 6.1.} (1) We define ${\bf C}_{0,cyl}^{(n)}$
(resp. ${\bf C}_{0,cyl}^{(n,1)})$ to be the space of real-valued
functions on $K^{n}{\times}({\cal D}_{tr}^{\prime})^{2},$ whose
elements, $F^{(n)},$ are of the form
$$F^{(n)}(x_{1},v_{1},;. \ .;x_{n},v_{n};b,f)={\tilde F}^{(n)}
(x_{1},v_{1},;. \ .;x_{n},v_{n};b({\phi}_{1}),. \
.,b({\phi}_{m});f({\psi}_{1}),. \ .,f({\psi}_{m})),\eqno(6.4)$$
where ${\tilde F}^{(n)}$ is a continuous (resp. continuously
differentiable) function on $K^{n}{\times}{\bf R}^{2m},$ with
compact support, and the ${\phi}$'s and ${\psi}$'s are elements
of ${\cal D}_{tr},$ with $m<{\infty}.$
\vskip 0.2cm\noindent
(2) Under the canonical identification of ${\bf C}_{0,cyl}^{(n)}$
with a space of functions on $K^{\bf N}{\times}({\cal
D}_{tr}^{\prime})^{2},$ we define ${\bf C}_{0,cyl} \
(resp.\ {\bf C}_{0,cyl}^{1})$ to be ${\cup}_{n{\in}{\bf N}}{\bf
C}_{0,cyl}^{(n)} \ (resp. \ {\cup}_{n{\in}{\bf N}}{\bf
C}_{0,cyl}^{(n,1)}).$
\vskip 0.2cm\noindent
(3) For $F^{(n)}{\in}{\bf C}_{0,cyl}^{(n)},$ we define ${\hat F}$
to be the Fourier transform of ${\tilde F}^{(n)},$
i.e.,
$${\hat F}^{(n)}({\xi}_{1},{\eta}_{1};. \
.;{\xi}_{n},{\eta}_{n};q_{1}, \ .,q_{m};r_{1},. \ .,r_{m})=$$
$${\int}{\tilde F}^{(n)}(x_{1},v_{1},;. \ .;x_{n},v_{n};y_{1},.
\ .,y_{m};z_{1},. \ z_{m}){\times}$$
$${\exp}-i[{\sum}_{j=1}^{n}(x_{j}.{\xi}_{j}+v_{j}.{\eta}_{j})+
{\sum}_{k=1}^{m}(y_{j}q_{j}+z_{j}r_{j})]$$
$${\forall}{\xi}_{j}{\in}(2{\pi}{\bf Z})^{3},
\ {\eta}_{j}{\in}{\bf R}^{3}, \ q_{k},r_{k}{\in}{\bf R}; \
j=0,. \ .,n;k=1,. \ .,m.\eqno(6.5)$$
\vskip 0.2cm\noindent
(4) We define differentiation of the ${\bf C}_{0,cyl}^{(n,1)}-
$class functions w.r.t. the fields $b,f,$ by the formula
$${\tilde {{\partial}F^{(n)}\over {\partial}b}}={\sum}_{k=1}^{m}
{{\partial}{\tilde F^{(n)}}\over
{\partial}b({\phi}_{j})}{\phi}_{j};
\ {\tilde {{\partial}F^{(n)}\over
{\partial}f}}={\sum}_{k=1}^{m}
{{\partial}{\tilde F}^{(n)}\over
{\partial}f({\psi}_{k})}{\psi}_{k},\eqno(6.6)$$
where ${\tilde F}^{(n)}$ is the representative of $F^{(n)}$ given
by equn. (6.4).
\vskip 0.2cm\noindent
{\bf Definition 6.2.} (1) Let ${\cal D}_{tr}^{(N){\prime}}$ be
the subspace of ${\cal D}_{tr}^{\prime}$ consisting of elements
whose Fourier transforms have support in the ball of radius
${\kappa}_{N},$ and let $q{\rightarrow}q^{(N)}$ be the canonical
projection of ${\cal D}_{tr}^{\prime}$ onto ${\cal
D}_{tr}^{(N){\prime}}.$ We define ${\Sigma}_{g,c}^{(N)}$ to be
the classical system, whose phase space in $K^{N}{\times}({\cal
D}_{tr}^{(N){\prime}})^{2}$ and whose equations of motion for its
time-dependent phase points $(x_{1,t},v_{1,t};.
\ .;x_{N,t},v_{N,t};b_{t}^{(N)},f_{t}^{(N)})$ are
$${dx_{j,t}\over dt}=v_{j,t};
\ {dv_{j,t}\over dt}=-(f_{g,t}^{(N)}(x_{j,t})+
v_{j,t}{\times}b_{g,t}^{(N)}(x_{j,t}))
+N^{-1}{\sum}_{k{\neq}j}{\nabla}U_{g}(x_{j,t}-x_{k,t})
\eqno(6.7)$$
and
$${{\partial}b_{t}^{(N)}\over
{\partial}t}=-{\nabla}{\times}f_{t}^{(N)};
\ {{\partial}f_{t}^{(N)}(x)\over {\partial}t}=
c_{0}^{2}{\nabla}{\times}b_{t}^{(N)}(x)+
N^{-1}{\sum}_{k=1}^{N}v_{k,t}.D_{g}^{(N)}(x-x_{k,t}).\eqno(6.8)$$
These are equations of motion for a system of a finite number of
degrees
of freedom, as represented by the $x$'s and $v$'s and the Fourier
coefficients
of $b^{(N)}$ and $f^{(N)}$; and it follows from standard fixed
point methods
that they have a unique global solution. We define
$T_{t}^{(N)}$ to be the transformation of $K^{N}{\times}
({\cal D}_{tr}^{(N){\prime}})^{2}$ that takes $(x_{1,0},v_{1,0};.
\ .;x_{N,0},v_{N,0};b_{0}^{(N)},f_{0}^{(N)})$ to
$(x_{1,t},v_{1,t};. \
.;x_{N,t},v_{N,t};b_{t}^{(N)},f_{t}^{(N)}).$
\vskip 0.2cm\noindent
(2) We define $m_{t}^{(N)}$ is the probability measure on $K$
given by
$$m_{t}^{(N)}={\sum}_{j=1}^{N}{\delta}_{x_{j,t},v_{j,t}}
.\eqno(6.9)$$
\vskip 0.3cm\noindent
{\bf Proof of Theorem 4.2.} This is a consequence of Theorem 4.1
and the following three theorems, together with the observation
that equns. (4.5) and (4.7) imply equn. (4.8).
\vskip 0.3cm\noindent
{\bf Theorem 6.3.} {\it Under the assumption (I.1),
the probability measure $M_{t}$ satisfies the
Vlasov hierarchy
$${d\over dt}{\int}dM_{t}F=
{\int}dM_{t}{\cal L}F \ {\forall}F{\in}{\bf C}_{0,cyl}^{1}
,\eqno(6.10)$$
where the restriction ${\cal L}^{(n)}$ of $L$ to ${\bf
C}_{0,cyl}^{(n,1)}$ is given by}
$${\cal L}^{(n)}={\sum}_{k=1}^{n}[v_{k}.{\nabla}_{x_{k}}
-(f_{g}(x_{k})+(v_{k}{\times}b_{g}(x_{k})-({\nabla}
U_{g}(x_{k}-x_{n+1})).{\nabla}_{v_{k}})]$$
$$-({\nabla}{\times}f).{{\partial}\over {\partial}b}+
[c_{0}^{2}{\nabla}{\times}b
+(v_{k}.D_{g})(x-x_{k})].{{\partial}\over {\partial}f}
.\eqno(6.11)$$
\vskip 0.3cm\noindent
{\bf Theorem 6.4.} {\it (1) Assuming the estimate (4.8) and
initial condition (I.4), the Vlasov equations (4.9)-(4.13) have
a unique global solution.
\vskip 0.2cm\noindent
(2) This solution $(m_{t},b_{t},f_{t})$ is the limit, as
$N{\rightarrow}{\infty},$ of that
$(m_{t}^{(N)},a_{t}^{(N)},f_{t}^{(N)}),$ as specified in Def. 6.2
for the classical system ${\Sigma}_{g,c}^{(N)},$ subject to the
condition that the initial configuration of the latter system is
chosen so that $m_{0}$ is the limit of} $m_{0}^{(N)}.$
\vskip 0.3cm\noindent
{\bf Theorem 6.5.} {\it Assuming Theorem 4.1(2) and the
conditions (I.2) and (I.3), the Vlasov hierarchy (6.10) has the
unique solution
$$M_{t}=m_{t}^{{\otimes}{\bf N}}{\otimes}{\delta}_{b_{t},f_{t}}
\ {\forall}t{\in}{\bf R}, \ n{\in}{\bf N},\eqno(6.12)$$
where $(m_{t},b_{t},f_{t})$ is the solution of the Vlasov
equations (4.9)-(4.13).}
\vskip 0.3cm\noindent
{\bf Note.} It is the uniqueness of the solutions both of the latter
equations and of the hierarchy (6.10) that renders the
subsequential convergence of $C_{.}^{(N,n)}$ fully sequential.
\vskip 0.3cm\noindent
{\bf Proof of Theorem 6.3.} This is based on an extension of the
method employed in [1] for the purely electrostatic case. Thus,
we let $F^{(n)}$ be an element of ${\bf C}_{0,cyl}^{(n,1)}(K),$
for which ${\hat F}^{(n)}$ is of the Schwartz class ${\cal S},$
i.e. fast-decreasing and infinitely differentiable w.r.t. its
continuous arguments. On multiplying equn. (6.1) by ${\hat
F}^{(n)},$ integrating (or summing) over all arguments, and
passing to the limit $N{\rightarrow}{\infty},$ we infer from
equns. (3.15), (3.16), (3.18)-(3.20), Theorem 4.1 and Lemma 5.1,
after some straightforward, but rather tedious, manipulations,
that
$M_{t}$ satisfies the Vlasov hierarchy (6.10), as restricted to
the case where ${\hat F}^{(n)}$ is of class ${\cal S}.$ The
removal of this restriction is then achieved by continuity.
\vskip 0.3cm\noindent
{\bf Proof of Theorem 6.4.} This is based on an extension of the
method devised in [7,8] for the case where there is no
radiation
field $(b,f).$
\vskip 0.2cm\noindent
We start by solving equns. (4.10) and (4.11) for $(b_{t},f_{t})$
in terms of their initial values $(b_{0},f_{0})$ and the
measure $m_{.},$ by means of elementary Green function
techniques. Thus,
$$b_{t}={\dot
{\kappa}}_{t}*b_{0}-{\nabla}{\times}{\kappa}_{t}*f_{0}+
{\int}_{0}^{t}ds{\int}
dm_{s}*{\nabla}{\times}{\kappa}_{t-s}*v.D_{g}
\eqno(6.13)$$
and
$$f_{t}=c_{0}^{2}{\nabla}{\times}{\kappa}_{t}*b_{0}+
{\dot {\kappa}}_{t}*f_{0}-
{\int}_{0}^{t}ds{\int}dm_{s}*{\dot {\kappa}}_{t-s}*v.D_{g}
,\eqno(6.14)$$
where
$${\kappa}_{t}(x)=
{\sum}_{k{\in}(2{\pi}{\bf Z})^{3}}{\cos}(k.x)
{{\sin}(c_{0}{\vert}k{\vert}t)\over c_{0}{\vert}k{\vert}t}
\eqno(6.15)$$
and ${\dot {\kappa}_{t}}:={\partial}{\kappa}_{t}/{\partial}t.$
\vskip 0.2cm\noindent
We now consider equn. (4.9) as an equation of motion for
$m_{t},$ in which $e_{g,t},b_{g,t}$ are expressed in terms of
this measure via the equns. (4.12), (4.13), (6.13) and (6.14).
The auxilliary equations of motion for a particle, for whose
phase point $m_{t}$ is the probability measure, are then (cf.
[8])
$${dx_{t}\over dt}=v_{t}; \
{dv_{t}\over dt}=-(e_{g,t}(x_{t})+v_{t}{\times}b_{g,t}(x_{t}))
,\eqno(6.16)$$
i.e, defining
$${\kappa}_{g,t}=g*{\kappa}_{t},\eqno(6.17)$$
$${dx_{t}\over dt}=v_{t}; \
{dv_{t}\over dt}=
V_{1}(x_{t},v_{t}{\vert}t)+V_{2}(x_{t}{\vert}m_{t})+
{\int}_{0}^{t}dsV_{3}(x_{t},v_{t}{\vert}m_{s},t-s),
\eqno(6.18))$$
where
$$V_{1}(x,v{\vert}t)=
-[c_{0}^{2}{\nabla}{\times}{\kappa}_{g,t}*b_{0}](x)
+v{\times}[{\dot {\kappa}}_{g,t}*b_{0}-
{\nabla}{\times}{\kappa}_{g,t}*f_{0}](x),\eqno(6.19)$$
$$V_{2}(x{\vert}m)=[{\int}dm*{\nabla}U_{g}](x)\eqno(6.20)$$
and
$$V_{3}(x,v{\vert}m,t)
=[{\int}dm*{\dot {\kappa}}_{g,t}*v.D_{g}](x)
-v{\times}[{\int}dm*{\nabla}{\times}{\kappa}_{g,t}*v.D_{g}](x)
.\eqno(6.21)$$
It follows now from the ${\cal D}$ property of $g$ and
the assumed continuity of $(b_{0},f_{0})$ that
the $V$'s satisfy Lipschitz conditions of the form
$${\vert}V_{j}(x,v{\vert}.)-
V_{j}(x^{\prime},v^{\prime}{\vert}.){\vert}