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\title{Absence of Debye Screening in the Quantum Coulomb System}
\author{David C. Brydges,
\thanks{Research partially supported by NSF grant DMS 9102584}
Georg Keller
\thanks{Supported by the Swiss NSF}
\\Department of Mathematics \\
University of Virginia \\ Charlottesville, VA 22903}
%\\ Max Planck Institut f\"ur Physik und Astrophysik\\
% Werner-Heisenberg-Institut f\"ur Physik\\
% F\"ohringer Ring 6\\
% Postfach 401 212, 8000 M\"unchen 40\\
% Germany }
\date{\today}
\maketitle
\begin{abstract}
We present an approximation to the quantum Coulomb plasma at
equilibrium which captures the power-law violations of Debye
screening which have been reported in recent papers. The objectives
are: (1) to produce a simpler model which we will study in
forthcoming papers, (2) to develop a strategy by which the absence
of screening can be proven for the low density quantum Coulomb
plasma itself.
\end{abstract}
\section{The Classical Coulomb Gas}\label{intro}
The partition function for a (charge symmetric) classical Coulomb gas
in three dimensions is
\begin{equation}
\labeq{partitionfunction}
Z = \sum \frac{\tilde{z}^N}{N!} \int d^N p\, d^N\xi \, e^{-\beta H}
\end{equation}
where $\xi = (x,e)$ and $d\xi$ unites an integral over $x \in
{\rm container}$ with a sum over charges $e = \pm 1$.
\begin{equation}
\labeq{hamiltonian}
H = \sum \frac{p^2_i}{2m}
+ \frac{1}{2} \int \rho v_l \rho + i\int \phi_{\mbox{ext}}\rho
\end{equation}
where we define the charge density observable by
\begin{equation}
\rho(x) = \sum e_i \delta(x-x_i)
\end{equation}
so that
\begin{equation}
\int \rho v_l \rho = \sum e_i e_j v_l(x_i - x_j).
\end{equation} $i\phi_{{\rm ext}}$ is an external field. We put in
the strange factor $i$ because it will lead to simpler expressions
when we explain the sine-Gordon transformation. $v_l(x-y) = $``${1
\over r}$''. We have put the quotes around the $1/r$ because it is
necessary to place a cutoff on the singularity of the Coulomb
potential at short distances in order to have a stable interaction.
This cutoff will be characterised by a length $l$. In particular
$v_l(0) = \frac{1}{l}$. Having enforced a cutoff the self energies of
the particles are finite and we have included them in the interaction
energy. The natural choice for this length $l$ is the thermal
wavelength which is the size of the typical one particle wavefunction
in a corresponding quantum ideal gas
\begin{equation}
l= \sqrt{\frac{\beta \hbar^2}{m}}
\end{equation}
since it is the Pauli exclusion principle and quantum mechanics that
give rise to a stable system which we are approximating classically.
The other lengths which naturally arise are $\beta$ and the Debye
length
\begin{equation}
l_D = \frac{1}{\sqrt{2z\beta}}
\end{equation}
where
\begin{equation}
z = \tilde{z} \int dp e^{-\frac{\beta}{2m}p^2} e^{-\frac{\beta}{2l}}.
\end{equation}
$z$ is the physical activity in the sense that the expectation of the
density of particles is asymptotic to $2z$ as $z \rightarrow 0$. The
factor $e^{-\beta/l}$ accounts for the inclusion of the self--energies
in the interaction.
For this system the following theorem has been proved
\cite{BrydgesFederbush80}, \cite{BrydgesFederbush81},
\cite{Imbrie}, \cite{Imbrie83}.
\begin{thm}
For
\begin{equation}
zl^3 \ll e^{-\frac{\beta}{2l}},\hspace{.5cm} zl_D^3 \gg 1,
\end{equation}
all charge - charge correlations decay exponentially, i.e., there
are constants $C_1$ and $L>0$ such that
\begin{equation}
|\langle\rho(x) \rho(y)\rangle| \le C_1 e^{\frac{-|x-y|}{L}}
\end{equation}
and higher truncated charge correlations decay exponentially as
the length of the shortest tree on the positions of the observables.
\label{thm1}
\end{thm}
Also $L \simeq l_D$ when $zl^3$ and $zl_D^3$
are as in the theorem.
\section{Discussion}\label{discussion}
In \cite{BrydgesFederbush81}, p.428, it was claimed that screening of
observables in the sense of exponential decay as in the theorem above
will not hold for the quantum plasma. The argument was strengthened
by some lower bounds (but only on time-dependent observables) given by
Brydges and Seiler \cite{BrydgesSeiler}. Since then Alastuey and
Martin \cite{AlastueyMartin88}, \cite{AlastueyMartin89} have made
detailed calculations which state that within perturbation theory (the
Wigner- Kirkwood expansion) screening is destroyed by effects due to
diagrams with power-law decay at order $\hbar^4$ and higher. They
show, for example, that for NaCl ions in water at room temperature
there will be screening out to about 60 Debye lengths at which point
there is a cross-over to a power law tail. According to their
analysis the typical power-law is $r^{-6}$ but it can be higher
depending on the correlation and the system. This violation of
screening has nothing to do with statistics. It is similar in
mechanism to Van der Waals forces, but it occurs
\cite{AlastueyMartin89} even for one component plasmas in which there
are no atoms or molecules. Similar comments appear in a paper by
Ashcroft et al. \cite{AshcroftMaggs}.
In this paper we exhibit an appproximation to the quantum Coulomb
plasma that captures the mechanism by which quantum fluctuations
destroy the screening. The present paper will motivate conclusions
which we will obtain by a complete mathematical analysis of this model
to appear shortly \cite{BrydgesKeller2}. The approximations we
present are a possible strategy by which the conclusions of Alastuey
and Martin could be established nonperturbatively, but this appears to
be an unreasonably lengthy enterprise at the moment.
To motivate the choice of our model we first review some aspects of
the proof of screening in the classical case. We introduce the
Gaussian measure $d\mu_{\frac{1}{\beta} v_l}(\phi)$ on functions
$\phi(x)$ which by definition satisfies
\begin{equation}
e^{-\frac{\beta}{2} \int \rho v_l \rho} =
\int d\mu_{\frac{1}{\beta} v_l}(\phi) e^{-i \beta \int
\rho \phi}
\labeq{10}
\end{equation}
If $l$ were zero, no cutoff, then formally
\begin{equation}
d\mu_{\frac{1}{\beta} v_0}(\phi) = D[\phi] e^{-\frac{\beta}{2} \int
(\nabla \phi)^2}.
\end{equation}
By substituting \refeq{10} into the partition functon
\refeq{partitionfunction} and interchanging the integral over $d \mu$
with the $\sum$ and $\int dp \, d\xi$ we are led to the well known
sine--Gordon representation of the partition function. This
represents the interacting gas as a superposition over all external
fields of ideal gas partition functions for particles in external
fields,
\begin{equation}
\labeq{sinegordon}
Z = \int d\mu_{\frac{1}{\beta} v_l}(\phi) Z_{{\rm ideal}}(i \phi +
i\phi_{{\rm ext}})
\end{equation}
\begin{eqnarray}
Z_{{\rm ideal}}(i \phi) &=& e^{\tilde{z} \int dp \, d\xi \, e^{-\beta h(i
\phi)}}\\
&=& e^{2ze^{\frac{\beta}{2l}}\int dx\,\cos{\beta \phi(x)}}
\end{eqnarray}
where
\begin{equation}
h(i \phi) = \frac{ p^2}{2m} + i e \phi.
\end{equation}
It is tempting to make the approximation $\cos{\beta \phi} \simeq 1 -
\frac{1}{2} \beta^2 \phi^2$ but this is not quite right in cases where
$\beta/l \gg 1$. Instead the first step in the proof of Theorem
\ref{thm1} is to integrate out fluctuations of the field on all scales
up to the Debye length $l_D$. Under the hypotheses of Theorem
\ref{thm1} this is done (exactly) by a Mayer expansion which is
convergent because the hypotheses say that the plasma inside a Debye
sphere is close to an ideal gas. The result is that the short
distance cutoff $l$ in the Gaussian measure $d\mu_{\frac{1}{\beta}
v_l}$ in \refeq{sinegordon} is replaced by $l_D$ while the exponent
$2ze^{\frac{\beta}{2l}}\int dx\,\cos{\beta \phi(x)}$ becomes a
convergent series of non-local monomials in $\exp{[i e \beta
\phi(x)}]$ but this series is still dominated by the leading term
which is local and has the form $2z\int dx\,\cos{\beta \phi(x)}$. In
other words, the effect of a renormalization group transformation is,
to a controllable approximation, to replace $v_l$ by
\begin{equation}
v \equiv v_{l_D}
\end{equation}
in the measure and to drop the constant $e^{\frac{\beta}{2l}}$. In
fact there are also renormalizations of parameters, e.g. the dominant
term is prefaced by a constant which tends to one as $zl^3
e^{\frac{\beta}{2l}} \longrightarrow 0$, $zl_D^3 \longrightarrow
\infty$ but we shall pretend these are not there throughout this
paper.
{\bf Choice of units of length:} Set $l_D = 1.$ With this choice of
units the hypotheses of Theorem \ref{thm1} imply that $\beta \ll 1$
and
\begin{equation}
2z = {1 \over \beta}.
\end{equation}
Having removed all scales up to $l_D = 1$ the next step in the proof
of Theorem \ref{thm1} is to control the approximation
\footnote{actually $e^{{1 \over \beta} \cos{\beta \phi(x)}} \approx
\sum_n e^{-\frac{\beta}{2}[\phi - {2\pi \over \beta} n ]^2}$}
\begin{equation}
\cos{\beta \phi} \simeq 1 - \frac{1}{2} \beta^2 \phi^2 + O(\beta^4).
\end{equation}
Within this approximation the partition
function becomes, up to constants which cancel in correlations, a
Gaussian integral
\begin{eqnarray}
Z &\simeq& \int
d\mu_{\frac{1}{\beta}v} \,
e^{-\frac{1}{2} \beta\int (\phi + \phi_{{\rm ext}})^2}\\
&\equiv&
\left [
\int d\mu_{\frac{1}{\beta}v} \,
e^{-\frac{1}{2} \beta\int \phi^2}\right ]
e^{- {1 \over 2} \beta \int \phi_{{\rm ext}}[1 - u]
\phi_{{\rm ext}}}
\end{eqnarray}
where $u$ is the {\it exponentially decaying} kernel of
\begin{equation}
u \equiv (v^{-1} + 1)^{-1} \labeq{v_m}
\end{equation}
in terms of which one can compute correlations of charge observables
by functional derivatives with respect to $\phi_{{\rm ext}}$ and
obtain the results of Debye--H\"uckel theory, in particular
exponential decay of correlations.
\section{The Quantum coulomb Gas}\label{qcg}
Now we turn to the analogous representation in the quantum case.
For simplicity we discuss the case of Boltzmann statistics, but
there are similar representations for Fermion and Bose statistics.
This simplification is reasonable since we are discussing a regime
in which the gas is very far from degenerate, $l \ll $ interparticle
distance by the hypotheses of the theorem.
The correct way to take into account the failure of commutativity
$[p,x] \neq 0$ is to replace $\int dp \, d\xi$ by the trace over the one
particle Hilbert space and use time-ordered exponentials in
$Z_{{\rm ideal}}(i \phi)$ so that
\begin{equation}
\labeq{quantumidealgas}
Z_{{\rm ideal}}(i \phi) = e^{\tilde{z}
Tr( e^{-\int^{\beta}_0 d\tau h(i \phi)} ) }
\end{equation}
where $\phi$ is now an imaginary-time dependent external field
$\phi(\tau,x)$ which is integrated over using the Gaussian measure
$d\mu_{v_l \otimes I}$ whose covariance is $v_l(x-y)\delta(\tau -
\sigma)$. With these substitutions in \refeq{sinegordon} the
sine--Gordon representation \refeq{sinegordon} is still valid.
To understand this, set $\phi_{{\rm ext}} = 0$, and consider $\sum
\frac{\tilde{z}^N}{N!} Tr_Ne^{-\beta H}$ where $H$ is the many body
quantum Hamiltonian obtained from \refeq{hamiltonian} by $p
\rightarrow \frac{\hbar}{i} \frac{\partial}{\partial x}$ and $Tr_N$ is
the $N$ body trace. Then
\begin{equation}
Tr_Ne^{-\beta H} =Tr_N\lim_{n \rightarrow \infty} \prod_1^n \left [
e^{-\frac{\beta}{n}H_0} e^{-\frac{\beta}{2n} \int \rho v_l \rho} \right].
\end{equation}
We use the representation \refeq{10} for each factor of
$e^{-\frac{\beta}{2n} \int \rho v_l \rho}$, each requiring its own
auxiliary field $\phi_i(x)$, $i = 1, \ldots, n$. Then
\begin{eqnarray}
Tr_Ne^{-\beta H} &=& \lim_{n \rightarrow \infty} \int d^n \mu_{{n
\over \beta}v_l}(\phi) \,Tr_N
\prod_1^n \left [ e^{-\frac{\beta}{n}H_0} e^{-\frac{\beta}{n} i \int
\rho \phi_j} \right] \\
&=& \lim_{n \rightarrow \infty} \int d^n \mu_{{n \over
\beta}v_l}(\phi) \, Tr_N
e^{-\int_0^{\beta} d\tau \, \left [H_0 + i \int
\rho \phi\right ]}
\end{eqnarray}
where the exponential is time ordered and the collection of fields
$\phi_j(x)$ is united into one time dependent field $\phi(\tau,x)
\equiv \phi_j(x)$ when $\tau \in
[\frac{(j-1)\beta}{n},\frac{j\beta}{n})$.
Finally we note
that the trace over the many body Hilbert space factors into a product
of one body traces so that
\begin{equation}
\sum \frac{\tilde{z}^N}{N!} Tr_Ne^{-\beta H} = \int d \mu_{v_l\otimes
I}(\phi) \, Z_{{\rm ideal}}(i\phi + i\phi_{{\rm ext}})
\end{equation}
where we have put the external field back in. We write $\int d
\mu_{v_l \otimes I}$ but we mean $\lim_{n \rightarrow \infty} \int
d^n \mu(\phi)$.
Following \cite{Ginibre}
$Z_{{\rm ideal}}(i \phi)$ can be
written as a sum over all continuous closed paths
$X(\tau),\hspace{.25cm} \tau \in [0,\beta]$, using the Feynman- Kac
formula,
\begin{equation}
\labeq{ginibre}
Tr( e^{-\int^{\beta}_0 d\tau h(i \phi)} ) = \sum_e
\int dW^{\beta}(X) e^{-ie \int_0^\beta d\tau \phi(\tau, X(\tau))}.
\end{equation}
$dW$ is the Wiener measure associated with the kernel of
$\exp[t(\frac{\hbar^2}{2m}\Delta)]$.
The combination of \refeq{sinegordon} and \refeq{ginibre} is a
representation for the quantum partition function which appears in
\cite{FroehlichPark}. It is also derived and used in
\cite{AlastueyMartin89}.
Notice that there is a Goldstone mode:
\[
\phi(\tau,x) \rightarrow \phi(\tau,x) + f(\tau)
\]
where $f$ is any function such that $\int_0^\beta d\tau f(\tau) = 0$.
This will be the origin of the long range forces. The intuition is
that the Feynman-Kac formula represents the quantum gas as a classical
gas of closed charge loops with instantaneous Coulomb interactions.
Each loop represents the quantum uncertainty around a classical position.
This leads to a time-dependent dipole force superimposed on the
Coulomb force for the classical system. A dipole can polarise other
dipoles leading to induced dipole-dipole or multipole-multipole forces
which are power laws. The standard textbook discussions do not see
this effect because they make a static approximation which loses these
time dependences. The mechanism is very similar to the Van der Waals
forces, except that it takes place without any need for neutral
objects such as atoms or molecules.
To bypass some terrible technicalities we now alter the Wiener measure
to obtain a simple model which exhibits destruction of screening
by the same mechanism that we claim will occur in the complete model.
\section{The Semiquantum Simplification}\label{simplification}
We replace the integration $\int dW^{\beta}$ over all Wiener paths by
a new integration concentrated on just one kind of path which
oscillates about the initial point by a distance $O(l)$ (the size of
the wavepacket) in a random direction: let $d\sigma(\vec{e})$ be a
spherically symmetric measure on vectors $\vec{e}$. Then
\begin{equation}
dW^{\beta} \longrightarrow dx \, d\sigma(\vec{e}).
\labeq{path}
\end{equation}
The right hand side is a measure on paths because $(x,\vec{e})$ labels
the path:
\begin{eqnarray}
X(\tau) &=& x + l\vec{e}f(\tau)\\ \nonumber
f(\tau) &=& \sin{[{2 \pi \tau \over \beta}]}.
\labeq{f)}
\end{eqnarray}
We dont claim that this is a controllable approximation in the sense
that there is a physically natural parameter that can be driven to
some limit to obtain it, but it is one of the simplest ways to put a
little quantum mechanics into a classical model. We shall choose
\begin{equation}
d\sigma(\vec{e}) = {1 \over 2}[\delta(\vec{e}) + (2\pi)^{-{3 \over
2}} e^{-{ \|\vec{e}\|^2 \over 2 }}].
\end{equation}
This choice perhaps would look more natural if there were no delta
function: the delta function has the interpretation that half our
particles are truly classical whilst the other half are semiquantum.
The choice of proportions is not essential, indeed one could allow all
the particles to be semiquantum but the resulting model is harder to
analyse rigorously. We now make some more changes, but these ones, we
claim, are on a different footing from the last change. They are
attempts to extract an effective Lagrangian which, we believe, can be
justified by rigorous mathematics.
{\bf Approximation 1}
\begin{eqnarray}
\int_0^\beta d\tau \,\phi(\tau, X(\tau)) &=& \int_0^\beta d\tau
\,\phi(\tau, X(0))\nonumber\\ &+&
\int_0^\beta d\tau \int_{t=0}^{t=\tau} \,\nabla \phi(\tau, X(t))\cdot
dX(t)\labeq{dX}\\
&=& \phi([0,\beta], x) + \int_{t=0}^{t=\beta} \,\nabla \phi([t,\beta],
X(t))\cdot dX(t)\nonumber\\
&\approx& \phi([0,\beta],x) + \int_{t=0}^{t=\beta} \,\nabla \phi([t,\beta],
x)\cdot dX(t)
\end{eqnarray}
where $\nabla$ acts on the spatial variables and
\begin{equation}
\phi([t,\beta],x) \equiv \int_t^\beta d\tau \,\phi(\tau,x).
\end{equation}
The consequence of these
approximations is that the dependence of $Z_{{\rm ideal}}(i \phi +
i\phi_{{\rm ext}})$ on $\phi(\tau,x)$ is only through
\[
\phi_1(x) \equiv \frac{1}{\sqrt{\beta}} \phi([0,\beta],x)
\]
\[
\phi_2(x) \equiv \sqrt{\frac{2}{\beta}} \int_0^\beta d\tau \phi(\tau,x)
f(\tau).
\]
In fact, in terms of these fields we find by the integration by
parts $\int_{t=0}^{t=\beta} \,\nabla \phi([t,\beta],
x)\cdot dX(t) = \int_0^\beta dt \,\nabla \phi(t,x)\cdot [X(t)-x]$ that
\begin{equation}
\int_0^\beta d\tau \,\phi(\tau, X(\tau)) \approx \sqrt{\beta}
\phi_1(x) + \sqrt{{\beta \over 2}} l \vec{e}\cdot\nabla \phi_2(x).
\end{equation}
Since the fields $\phi_1$, $\phi_2$ are Gaussian and $\int d\mu_{v_l
\otimes I} \, \phi_i(x) \phi_j(y) = v_l(x-y) \delta_{ij}$, $\phi_1$,
$\phi_2$ are independently distributed according to the massless
Gaussian measure $d\mu_{v_l}$ encountered above in the classical model.
Thus the partition function becomes
\begin{eqnarray}
Z &=& \int d\mu_{v_l}(\phi_2) \int d\mu_{v_l}(\phi_1) \cdot {\rm
exp}[ 2 \tilde{z} \int dx \, \int d\sigma(\vec{e}) \,\nonumber \\
& & \cos{(\sqrt{\beta} \phi_1(x) +\sqrt{\beta} \phi_{{\rm ext}} +
\sqrt{\frac{\beta}{2}}l \vec{e}\cdot\nabla\phi_2)}].
\labeq{semiquantum}
\end{eqnarray}
Notice that if $d\sigma(\vec{e})$ is set to $\delta(\vec{e})$ we
revert to the classical Coulomb gas. If $\phi_1$ is set to zero then
by reversing the Sine-Gordon transformation we obtain the partition
function of a classical dipole gas with dipole moments $\vec{e}$
distributed according to $d\sigma$.
{\bf Approximation 2} This is the same step as discussed above for
the classical model in which the fluctuations on scales up to $l_D =
1$ are integrated out by a Mayer expansion, of which we keep only the
leading term. Thus $v_l$ becomes $v_1 \equiv v$ and $2 \tilde{z}$
becomes ${1 \over \beta}$.
\begin{eqnarray}
Z &\approx& \int d\mu_v(\phi_2) \int d\mu_v(\phi_1) \cdot {\rm exp}[
{1 \over \beta} \int dx \, \int d\sigma(\vec{e}) \,\nonumber \\ & &
\cos{(\sqrt{\beta} \phi_1(x) + \sqrt{\beta} \phi_{{\rm ext}} +
\sqrt{\frac{\beta}{2}}\vec{e}\cdot l\nabla\phi_2)}].
\labeq{mayer}
\end{eqnarray}
The integration over $d\sigma(\vec{e})$ can be performed explicitly
and the partition function becomes
\begin{eqnarray}
Z &=& \int d\mu_v(\phi_2) \int d\mu_v(\phi_1) \nonumber \\
& & \cdot {\rm exp}[
{1 \over \beta} \int dx \, w_2
\cos{(\sqrt{\beta} \phi_1(x) + \sqrt{\beta} \phi_{{\rm
ext}})}]\nonumber\\
w_2(x) &\equiv& {1 \over 2}[1 + e^{- \beta (l \nabla \phi_2(x))^2}].
\end{eqnarray}
The next approximation is of the same nature as the quadratic
approximation of the cosine used to prove Theorem \ref{thm1}. We
have by the hypotheses of Theorem \ref{thm1} that $\beta \ll 1$ so
that
{\bf Approximation 3}
\[ {1 \over \beta} \cos{(\sqrt{\beta} \phi_1 + \sqrt{\beta} \phi_{{\rm
ext}})} \simeq {1 \over \beta} - \frac{1}{2} (\phi_1 +
\phi_{{\rm ext}})^2.
\]
Now we can integrate out the $\phi_1 $ field: let
\begin{equation}
u_{w_2}(x,y) \equiv \mbox{kernel of the operator } [v^{-1} +
w_2]^{-1}.
\end{equation}
Then
\begin{eqnarray}
\lefteqn{ \int d\mu_v(\phi_1)\, e^{- \frac{1}{2} \int dx \, w_2
(\phi_1 + \phi_{{\rm ext}})^2} }\nonumber\\
&& = \left [\int d \mu_v(\phi_1) \, e^{- \frac{1}{2} \int dx \, w_2
\phi_1^2} \right]
e^{- \frac{1}{2} \int \phi_{{\rm ext}} [w_2 - u_{w_2}] \phi_{{\rm
ext}} }.
\end{eqnarray}
\section{Conclusions}\label{conclusions}
These approximations have led us to the following model of the quantum
Coulomb gas:
\begin{equation}
Z = \int d\mu_v(\phi_2) e^{{1 \over \beta} \int dx \, w_2}
\left [\int d \mu_v(\phi_1) \, e^{- \frac{1}{2} \int dx \, w_2
\phi_1^2} \right]
e^{- \frac{1}{2} \int \phi_{{\rm ext}} [w_2 - u_{w_2}] \phi_{{\rm
ext}} }.
\labeq{model}
\end{equation}
We will give a complete nonperturbative analysis of this model in a
forthcoming paper \cite{BrydgesKeller2}. Let expectations of static
charge densities be obtained by functional derivatives
\begin{equation}
\langle \prod_{i = 1}^r \rho(x_i) \rangle \equiv \left [ {1 \over
Z}\prod_{i = 1}^r {\delta \over \delta \phi_{{\rm
ext}}}(x_i) Z(\phi_{{\rm ext}})\right ]_{\phi_{{\rm ext}} = 0}.
\end{equation}
Here are the conclusions we expect:
\begin{enumerate}
\item The two point correlations decay exponentially:
\begin{equation}
\langle \rho(x_1)\rho(x_2) \rangle \sim {{\rm Const.} \over \beta }
u(x_1 - x_2)
\end{equation}
as $|x_1 - x_2| \longrightarrow \infty$. $u$ is given in
\refeq{v_m}.
\item The higher correlations do not decay exponentially, for example:
\begin{eqnarray}
\lefteqn{ \langle \rho(x)\rho(-x) \rho(x + y)\rho(-x + y) \rangle
}\nonumber\\ && - \langle \rho(x)\rho(-x)\rangle \langle\rho(x +
y)\rho(-x + y) \rangle \nonumber \\ && \sim {\rm Const.} [{1 \over
\beta} u(-x) u(x)]^2 [\beta l^2]^2 {1 \over |y|^6}
\end{eqnarray}
as $y \longrightarrow \infty$ with $|x|$ large.
\end{enumerate}
{\bf Justification} Notice that in equation \refeq{model} the kernel
$u_{w_2}(x,y)$ decays uniformly in $w_2$ because $w_2(x)$ is smooth in
$x$ and $w_2(x) \ge {1 \over 2}$ for all \footnote{This is why we
decided to put a $\delta$ function into the measure $d\sigma$}
$\phi_2$. Therefore functional derivatives with respect to
$\phi_{{\rm ext}}$ are linked in pairs by exponentially decaying
propagators $u_{w_2}$. However the propagators $u_{w_2}$ depend on
the field $\nabla \phi_2$ through $w_2$. This field is distributed
according to a massless Gaussian measure $d\mu_v$ with a small
perturbation by the terms
\begin{eqnarray}
\left [\int d \mu_v(\phi_1) \, e^{- \frac{1}{2} \int dx \, w_2
\phi_1^2} \right] &\approx& e^{ \int [O(1) + O(1)\beta (l \nabla
\phi_2(x))^2] \, dx } \nonumber\\ e^{{1 \over \beta} \int dx \,
w_2} &\approx& e^{\int [{1 \over \beta} + O(1)(l\nabla
\phi_2))^2]\,dx }.
\end{eqnarray}
Since the perturbation is a function of $\nabla \phi_2$ it will not
make the measure massive.
{\bf Comments:} It is an artifact of this particular approximation and
charge symmetry that the action is separately even in $\phi_1$ and
$\phi_2$, which is the reason for the different types of decay.
Alastuey and Martin \cite{AlastueyMartin89} also found that there are
different decay rates for the correlations of two versus four
charge observables, but the differences were in the exponent of the
power law rather than the drastic exponential versus power law that we
obtain.
If in equation \refeq{semiquantum} the two fields $\phi_1,\phi_2$ were
the same field then that model would have exponential decay in all
correlations. This would be a classical system consisting of
particles which carry both a charge and a small dipole moment.
The quadratic approximation
$$\cos{(\sqrt{\beta} \phi_1 + \sqrt{\beta} \phi_{{\rm ext}} +
\sqrt{\frac{\beta}{2}}\vec{e}\cdot l\nabla\phi_2)} \simeq 1 -
\frac{\beta}{2} (\phi_1 + \phi_{{\rm ext}} +
\sqrt{\frac{1}{2}}\vec{e}\cdot l\nabla\phi_2)^2 $$ in equation
\refeq{mayer} would lose some higher order terms which are responsible
for the destruction of the screening.
For the full quantum Coulomb partition function \refeq{ginibre}
equation \refeq{dX} is replaced by:
\begin{eqnarray}
\int_0^\beta d\tau \,\phi(\tau, X(\tau))
&=& \phi([0,\beta], x) + \int_{t=0}^{t=\beta} \,\nabla \phi([t,\beta],
X(t))\cdot dX(t) + \nonumber\\
&&\int_0^\beta {dt \over \beta} \, l^2 \Delta
\phi([t,\beta], X(t)).
\end{eqnarray}
The extra term arises by the Ito calculus $dX(t)^2 = {dt \over \beta}
\, l^2$ and the $dX(t)$ integral is an Ito integral. This formula has
the good feature that the very singular field $\phi(\tau,x)$ which is
a white noise in its dependence on $\tau$ has been traded in for the
continuous field $\phi([t,\beta],x)$. It is possible that a
nonperturbative proof of the absence of screening for the quantum
Coulomb system can be constructed by integrating out the (massive)
time averaged field ${1 \over \beta} \phi([0,\beta],x)$ as we did
(approximately) in obtaining the model \refeq{model}.
\vspace{.5in}
\begin{center}
ACKNOWLEDGEMENT
\end{center}
We would like to thank Erhard Seiler and Philippe Martin for many
helpful conversations. To our knowledge the statement that Coulomb
quantum plasma have these particular power law corrections to
screening was first made by Paul Federbush.
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\end{document}