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Junius Institute Launches Digitization Project with Grant from Davenant Trust
March 17, 2014 | By Jordan Ballor
Grand Rapids, MI (March 17, 2014)—The Junius Institute's first digitization initiative of early modern sources gained important support today when the Davenant Trust announced a grant to create digital editions of the collected works of one of the most significant Reformed theologians of the sixteenth century.
Girolamo Zanchi (1516-1590) was a third generation reformer, an exile from Italy who taught in the Reformation cities of Strasbourg and Heidelberg. His collected works, covering 8 total volumes and nearly 4,000 pages, were published after his death and exercised a major influence on the development of Reformed thought into the seventeenth century and beyond. The Davenant Trust is sponsoring the digitization of Zanchi's Opera omnia, providing a significant beginning to the larger initiative to digitize the significant rare book holdings at the Calvin College campus.
"We're excited to begin our project to create electronic editions of these major sources with the complete works of Zanchi," said Todd Rester, director of the Junius Institute for Digital Reformation Research at Calvin Theological Seminary. "Getting high-quality digital versions of these documents into the hands and onto the hard drives of scholars is the first step in moving the next generation of scholarship forward."
This is one of the first major grants made by the newly-formed Davenant Trust, an institution formed to advance the scholarship and reception of Reformed theology in the academy and church today. "We are very excited about this opportunity to partner with the Junius Institute," said Dr. W. Bradford Littlejohn, president of the Davenant Trust. "Their important work in the retrieval of Reformation sources dovetails perfectly with Davenant Trust's vision of Protestant resourcement and we expect many fruitful collaborations in future. Zanchi's work in particular should be a cornerstone of any such resourcement, and we are eager to help bring it into wider circulation."
The digitization of Zanchi's collected works will feature all of the significant contributions from the pen of this noteworthy theologian. And yet these thousands of pages of published text represent just a small foretaste of a larger, multi-phase project to digitize the world-class collection of rare books at the H. Henry Meeter Center for Calvin Studies and Hekman Library of Calvin College and Seminary. While the Junius Institute will continue to raise funds to eventually make available all of the rare book holdings, this initial phase will cover roughly 60 volumes and 60,000 pages of printed text.
Girolamo Zanchi (1516-1590)
Dr. Richard Muller, a senior fellow at the Junius Institute and the P.J. Zondervan Professor of Historical Theology at Calvin Theological Seminary, noted that Zanchi's contributions to the development and maturation of the Reformed tradition are often under-appreciated. "Zanchi was one of the key figures in the later generations of the Reformation. As a younger contemporary of John Calvin, Heinrich Bullinger, and Peter Martyr Vermigli, Zanchi stands as a major contributor in a variety of areas, especially in his philosophical treatises, exegesis, and doctrinal and scholastic works," said Muller.
Zanchi's works, like all of the project's digital editions, will be released under a Creative Commons license, allowing wide use and distribution for academic and scholarly purposes. The digitization initiative will also create a collection of source texts that can be used for a variety of future digital research purposes and projects.
Jordan J. Ballor
Junius Institute for Digital Reformation Research at Calvin Theological Seminary
About the Junius Institute for Digital Reformation Research
The Junius Institute for Digital Reformation Research at Calvin Theological Seminary (www.juniusinstitute.org) seeks to further the advancement of studies in early modern (ca. 16th to 18th century) theology and interconnected disciplines through the use of digital research tools, skills, and sources; to foster the presentation, preservation, and public use of primary and secondary sources within the public domain; and to encourage via educational and curricular means the study of the documents themselves, their content, as well as the technical skills required to interpret and analyze these materials.
About the Davenant Trust
The Davenant Trust (www.davenanttrust.org) sponsors Christian resourcement in scholarship. The Trust's aim is to revitalize contemporary Reformed and evangelical discourse by sponsoring scholarly endeavors at the intersection of the church and academy: the proliferation of the digital archive, the retrieval and translation of classic texts, the development and support of Christian study centers, and the sponsorship of individual scholars engaged in historical work for the sake of the contemporary church.
Posted in Announcements | Tagged davenant trust, digitization, girolamo zanchi, hekman library, meeter center
A Whir, Click, and Rustle
December 17, 2013 | By Jordan Ballor
Todd Rester, Junius Institute Director
Todd Rester, the director of the Junius Institute, introduces the institute and our new digitization initiative over at the Calvin Seminary website. Click through for the whole thing, but here's a snippet:
How do you make rare 16th century theological treasures available to students and scholars in the twenty first century? The innovative answer at the Junius Institute for Digital Reformation Research (www.juniusinstitute.org) sounds like the whir of a robot, the click of a camera, and the rustle of 1,500 pages an hour.
The Junius Institute for Digital Reformation Research is a new institute at Calvin Theological Seminary. This institute is dedicated to the preservation and presentation of rare works from the Reformation and post-Reformation era to students and scholars at home and abroad. The hum of these mechanized sounds emanate from an automated book scanner designed by one of the doctoral candidates at Calvin Theological Seminary, Todd Rester, who is also the director of the Junius Institute. "We have an opportunity to present and preserve the rare book holdings of the Hekman Library and the Meeter Center for Calvin Studies in a cost-effective and digital way to make a global impact."
Todd will be posting more details soon on our proposed plan to digitize a first selection of items from the Hekman Library rare book collection and the holdings of the Meeter Center. But in the meantime, you can also view our digitization project page to see some of the very first works we have digitized.
For more information on the development of the Junius Institute, and ways to support our work, please read this letter from our director.
Posted in Announcements | Tagged calvin theological seminary, digitization, hekman library, meeter center
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text:
\section{Introduction}
\setcounter{equation}{0}
Neutral atoms and molecules interact through the long range, attractive van der Waals potential  which has a decay as $-R^{-6}$
for large separation $R$. The quantum origin  of this force was first recognized by London \cite{London}.
Lieb and Thirring \cite{LT} supply a non-perturbative proof valid for very general charge configurations. If one goes beyond
the static Coulomb interaction in quantizing the Maxwell field, then the action is   no longer instantaneous but travels with the speed of light
between
atoms. In a now  very famous paper \cite{CP} Casimir and Polder establish  that thereby   the effective
interaction  potential  decays somewhat faster,  namely   as $-R^{-7}$, which is known as  the retarded van der Waals potential. For
two hydrogen atoms, the cross over between $R^{-6}$ and $R^{-7}$ sets  in at roughly 100 Bohr radii. There are both direct and indirect measurements \cite{Ex} which confirm the theoretical  prediction.

The starting point of Casimir and Polder  is nonrelativistic QED for two atoms separated by  a distance $R$. Within dipole approximation they expand  to fourth order  in the coupling  to the Maxwell field and obtain a prefactor  of $-R^{-7}$ which is proportional to the square of the electric dipole moment
 of a single atom.
  Later on alternative routes and simplified derivations were proposed. For an extensive discussion we refer to the book by Milonni \cite{Milonni},
  see also the monograph by  Margenau and Kestner \cite{MK} and  the lecture notes of Martin and B\"unzli \cite{MB}. Feinberg and Sucher \cite{FS0, FS} reconsider the issue  by employing  a dispersion-theoretic
  approach. Their prefactor  turns out  to be  quadratic in the electric and magnetic dipole moment of a single atom. Somewhat later  Boyer \cite{Boyer}
   rederived the  same prefactor using quantum zero-point energy and semiclassical expressions for the level shifts due to the presence of the atoms.
In our note we stick to nonrelativistic QED, no dipole approximation and no assumption  on  small coupling, and expand in $1/R$. We use the path integral formulation, in which  the subtraction of the ground state energy at $R=\infty$ is particularly transparent.
As in previous studies  the strength of the retarded van der Waals potential is quadratic  in the electric and magnetic dipole moments, but with  modified  coefficients
as compared   to \cite{FS, Boyer}.

In mathematical physics there has been  a revived interest in nonrelativistic QED \cite{Spohn}. It is conceivable that some parts of the argument can be elevated to a rigorous proof. In our paper we mostly ignore this line of research, but will provide   a more detailed  discussion in the conclusions.

\section{Hamiltonian and van der Waals potential}
\setcounter{equation}{0}

We consider a single hydrogen atom with an infinitely heavy nucleus located at the origin. The nucleus has charge $e$, $e>0$,
the electron has charge $-e$. We will use units in which $\hbar=1, c=1$, and the bare mass of the electron $m=1$.
In Section \ref{DL} we will restore the proper physical units.
 Let $x,p$ be position and momentum of the electron. Then the nonrelativistic QED hamiltonian for this system reads
\begin{align}
H=\frac{1}{2}\big(p-eA(x)\big)^2-e^2V(x)+H_{\mathrm{f}}.\label{1PL}
\end{align}
For $H$ to make sense the electron is assumed to have a prescribed  charge distribution $\varphi$ with the following properties: $\varphi$ is  normalized,
$\int \mathrm{d}x\, \varphi(x)=1$, rotation invariant,
$ \varphi(x)=\varphi_{\mathrm{rad}}(|x|)$,  of rapid decrease, and its Fourier transform,  $\hat{\varphi}$,  is   real. Then
$V$ is the smeared Coulomb potential
\begin{align}
V(x)=\int\mathrm{d}k\, |\hat{\varphi}(k)|^2|k|^{-2} \mathrm{e}^{-\mathrm{i} k\cdot x}.
\end{align}
$A(x)$ is the quantized  vector potential and $H_{\mathrm{f}}$  is the field energy. These are defined through a two-component Bose field
$a(k, \lambda), k\in \mathbb{R}^3, \lambda=1,2,$ with commutation relation
\begin{align}
[a(k, \lambda), a(k', \lambda')^*]= \delta_{\lambda\lambda'}\delta(k-k').
\end{align}
Explicitly
\begin{align}
H_{\mathrm{f}}=\sum_{\lambda=1,2}\int_{\mathbb{R}^3}\mathrm{d}k\, \omega(k)a(k, \lambda)^*a(k, \lambda)
\end{align}
with dispersion  relation
\begin{align}
\omega(k)=|k|
\end{align}
and
\begin{align}
A(x)=\sum_{\lambda=1,2}\int_{\mathbb{R}^3}\mathrm{d}k\, \frac{\hat{\varphi}(k)}{\sqrt{2\omega(k)}}\varepsilon(k, \lambda)\big(\mathrm{e}^{\mathrm{i} k\cdot x}a(k, \lambda)+\mathrm{e}^{-\mathrm{i} k\cdot x}a(k, \lambda)^*\big)
\end{align}
with the standard dreibein $\varepsilon(k,1), \varepsilon(k, 2), \hat{k}=k/|k|$. Thus the Hilbert space for $H$ is
\begin{align}
\mathcal{H}=L^2(\mathbb{R}^3_x)\otimes \mathfrak{F},
\end{align}
where $\mathfrak{F}$ is the bosonic Fock space over $
L^2(\mathbb{R}^3)\otimes \mathbb{C}^2$.
From the quantization  of the classical system of charges coupled to
the Maxwell field it follows that for the smearing of $A(x)$ and of
$V$ the same charge distribution has to be used. We refer to
\cite{Spohn} for details.
As proved by Griesemer, Lieb,  and Loss \cite{BFS, LL, GLL}, $H$ has a unique
ground state, denoted here  by $\psi$, with ground state energy $E$, $H\psi=E\psi$.

The asymptotic strength $\kappa $ of the van der Waals potential  depends on the properties of a single hydrogen atom only through its electric and magnetic
dipole moment, $\alpha_{\mathrm{E}}$ and $\alpha_{\mathrm{M}}$. They
are defined  through the energy, $W$,  of our   system for weak external uniform electric and  magnetic fields according to
\begin{align}
W=-\frac{1}{2} \alpha_{\mathrm{E}} E_{\mathrm{ex}}^2-\frac{1}{2} \alpha_{\mathrm{M}} B_{\mathrm{ex}}^2.
\end{align}
To say,  $H$ is perturbed by $e E_{\mathrm{ex}}\cdot x$ and the vector potential  is perturbed by $\frac{1}{2} B_{\mathrm{ex}}\wedge x$.
Then by second order perturbation theory  it follows that
\begin{align}
\alpha_{\mathrm{E}}&=2 \big( \tfrac{1}{3} \langle \psi, x\cdot (H-E)^{-1} x\psi\rangle\big)e^2,
\nonumber \\
\alpha_{\mathrm{M}}&= -\frac{1}{4} \big( \tfrac{1}{3} \langle \psi, x^2\psi\rangle\big)e^2.
\end{align}
As a convention, $\langle\cdot, \cdot \rangle$ denotes always the inner
product on the respective Hilbert space.

To investigate  the van der Waals potential we consider  two hydrogen atoms, one
located at $0$ and the other at $r=(0,0,R), R\ge 0$. It will be  convenient to
define the position of the second electron relative to $r$. Then $x_1,
x_2+r$ are positions  and $p_1, p_2$ the momenta of the two
electrons. The two-electron hamiltonian reads
\begin{align}
H_R=&\frac{1}{2}\big(p_1-eA(x_1)\big)^2-e^2V(x_1)+\frac{1}{2}\big(p_2-eA(x_2+r)\big)^2-e^2V(x_2)\nonumber \\
&+H_{\mathrm{f}}+e^2 V_R(x_1, x_2)\label{FullHami}
\end{align}
with the interaction potential
\begin{align}
V_R(x_1, x_2)&= -V(x_1-r)-V(x_2+r)+V(r)+V(r+x_2-x_1)\nonumber \\
&=\int_{\mathbb{R}^3}\mathrm{d}k\, |\hat{\varphi}(k)|^2\mathrm{e}^{\mathrm{i} k\cdot r}
|k|^{-2}
(1-\mathrm{e}^{-\mathrm{i} k\cdot x_1})(1-\mathrm{e}^{\mathrm{i} k\cdot x_2}).
\end{align}
$H_R$  acts on the Hilbert space $L^2(\mathbb{R}^3_{x_1})\otimes
L^2(\mathbb{R}^3_{x_2})\otimes \mathfrak{F}$. $H_R$ has a unique ground
state with energy $E(r)$.

In the Born-Oppenheimer approximation $E(r)$ is the effective
potential  between the two neutral hydrogen atoms in their ground state. Thus the issue at
hand is to investigate $E(r)$ for large $R$. By  rotation invariance $E(r)$ depends only on $|r|=R$ and
we also write $E(r)=E(R)$.
 For $R\to \infty$ the two atoms become independent and  one
can show that $E(R)$ converges to $2E$. The Casimir-Polder result  is
that, for small $e$,
\begin{align}
\lim_{R\to \infty}R^7(E(R)-2E)=-\frac{23}{4\pi}\big(\frac{1}{2\pi}\big)^2
\big(\frac{1}{2}\alpha_{\mathrm{E, at}}\big)^2 .\label{CP}
\end{align}
The factor $(1/2\pi)^2$ results from our use of the Lorentz-Heaviside
units.
We remark that Casimir and Polder omit in their definition of $\alpha_{\mathrm{E}}$ the factor $2$, which accounts for the extra $1/2$.
 $\alpha_{\mathrm{E, at}}$ is the dipole moment of a decoupled hydrogen atom.
It is defined through
\begin{align}
\alpha_{\mathrm{E}, \mathrm{at}}= 2 \big(\tfrac{1}{3} \langle\psi_{\mathrm{at}},
x\cdot (H_{\mathrm{at}}-E_{\mathrm{at}})^{-1} x\psi_{\mathrm{at}}\rangle \big)e^2
\label{Prefactor}
\end{align}
with
\begin{align}
H_{\mathrm{at}}=\frac{1}{2}p^2-e^2 V(x)
\end{align}
and $\psi_{\mathrm{at}}$   the ground state of the hydrogen atom,
$H_{\mathrm{at}}\psi_{\mathrm{at}}=E_{\mathrm{at}}\psi_{\mathrm{at}}$.

 In our  set-up, the natural dimensionless coupling constant is the Sommerfeld fine-structure constant
 \begin{align}
 \alpha=\frac{e^2}{4\pi \hbar c}.
 \end{align}
 The energy unit is set by the ionization energy of the hydrogen atom,
 which is  $\alpha^2 m c^2$ and  the length unit is the Bohr radius
 $r_{\mathrm{B}}=\hbar/\alpha m c$. Anticipating  a decay as $R^{-7}$,
 the dimensionless coupling strength, $\kappa$,  is defined through
 \begin{align}
 E(R)-2E =-\kappa(\alpha, \lambda_{\mathrm{c}}\Lambda) \alpha m c^2 (R/r_{\mathrm{B}})^{-7}
 \end{align}
 valid for large $R$. $\kappa$ depends on $\alpha$ and on the ultraviolet cutoff $\Lambda$ in units of the inverse Compton wave length $\lambda_{\mathrm{c}}=\hbar / mc$. For this  interpretation the form factor $\hat{\varphi}$ is chosen as  $\hat{\varphi}(k)=\hat{\varphi}_1(\Lambda^{-1}k)$,
 where $\hat{\varphi}_1$ rapidly interpolates between  $\hat{\varphi}_1=(2\pi)^{-3/2}$ for $|k|\le 1-\delta$ and $\hat{\varphi}_1=0$ for $|k|\ge 1$.

The goal of our note is to obtain an exact expression for the strength $\kappa$. Readers not so much interested in the details  of the computation
 may skip ahead  to Section \ref{DL} where the result  is discussed.

\section{Path integration}\setcounter{equation}{0}

As noted by Feynman \cite{Feynman}, in the functional integral
representation
 of $\mathrm{e}^{-tH_R}, t\ge 0$, the interaction with the radiation field
 is linear in $A$. Therefore one can carry out the Gaussian
 integration over the fluctuating photon field. This form will be particularly
 convenient for the Born-Oppenheimer energy $E(R)-2E$. After such a detour we
 will return to operators. Our notation is formal, but rigorous
 versions
are available \cite{Hiroshima1, Hiroshima2}.

We denote by $q_j(t)\in \mathbb{R}^3$ the path of electron $j$. In case of
a single electron we omit the index $j$. For the ground state energy
of  the two-electron system  one obtains
\begin{align}
E(r)=&-\lim_{T\to \infty}\frac{1}{2T}\log
\int\Big[\Pi\mathrm{d}q_1(\cdot)\Big]\int\Big[\Pi\mathrm{d}q_2(\cdot)\Big]
\nonumber \\
&\exp\Big[-\int_{-T}^T\mathrm{d}t\Big(\sum_{j=1,2}\big(\tfrac{1}{2}\dot{q}_j(t)^2-e^2V(q_j(t))
\big)
+e^2 V_R(q_1(t), q_2(t))
\Big)\nonumber \\
&-\sum_{j=1,2}\tfrac{1}{2}e^2\int_{-T}^T\mathrm{d}s\int_{-T}^T\mathrm{d}t\,
\dot{q}_j(s)\cdot W_0\big(q_j(s)-q_j(t), s-t\big)\dot{q}_j(t)\nonumber
\\
&- e^2\int_{-T}^T\mathrm{d}s\int_{-T}^T\mathrm{d}t\,
\dot{q}_1 (s)\cdot W_R\big(q_1(s)-q_2(t), s-t\big)\dot{q}_2(t)
\Big].
\end{align}
Here $\int\big[\Pi\mathrm{d}q_j(\cdot)\big]$ is the ``sum  over all
paths"  and $W_R$ is the photon propagator,
 \begin{align}
W_R(q,t)=\int_{\mathbb{R}^3}\mathrm{d}k\,
|\hat{\varphi}(k)|^2\frac{1}{2\omega(k)}
({\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l-| \hat{k}\rangle\langle\hat{k}|)\, \mathrm{e}^{-\omega(k)|t|}\mathrm{e}^{\mathrm{i} k\cdot
  r}\mathrm{e}^{-\mathrm{i} k\cdot q}\label{Propagator}
\end{align}
as a  $3\times 3$ matrix.
Here ${\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l $ is the unit matrix and   $|\hat{k}\rangle\langle \hat{k}|$ the orthogonal
projection onto $\hat{k}$, $\hat{k}=k/|k|$.

Correspondingly for a single electron
\begin{align}
E=&-\lim_{T\to \infty}\frac{1}{2T}\log \int
\Big[\Pi \mathrm{d}q(\cdot)\Big]
\exp\Big[
-\int_{-T}^T \mathrm{d}t\, \big(\tfrac{1}{2}\dot{q}(t)^2-e^2 V(q(t))\big)
\nonumber \\
&-\tfrac{1}{2}e^2\int_{-T}^T\mathrm{d}s\int_{-T}^T\mathrm{d}t\,
\dot{q}(s)\cdot W_0\big(q(s)-q(t), s-t\big)\dot{q}(t)
\Big].\label{SingleEnergy}
\end{align}
Therefore
\begin{align}
E(R)-2E=&-\lim_{T\to \infty}\frac{1}{2T}\log \mathbb{E}\times \mathbb{E} \Big[
\exp\Big[
-e^2\int_{-T}^T\mathrm{d}t\, V_R(q_1(t), q_2(t))
\nonumber \\
&-e^2\int_{-T}^T\mathrm{d}s\int_{-T}^T\mathrm{d}t\,
\dot{q}_1(s)\cdot W_R\big(q_1(s)-q_2(t), s-t\big)\dot{q}_2(t)
\Big]
\Big].\label{PathBindingEr}
\end{align}
Here $q_1(t)$ and $q_2(t)$  are two independent copies  of the ground state process $q(t)$ with a  path measure as written in (\ref{SingleEnergy}).
$q(t)$ is stationary and the
distribution of $q(t)$ at a fixed time is the electronic density
computed from  the ground state    $\psi$ of $H$.
The average with respect to the ground state process is denoted by $\mathbb{E}[\cdot]$ and the average over the two independent copies by $\mathbb{E}\times \mathbb{E}[\cdot]$

We note that in the  expression (\ref{PathBindingEr}) the only $R$
dependence sits in $V_R$ and $W_R$, which in a certain sense are
small. Thus it is  natural to use the cumulant expansion.
Denoting the exponent by $X_T(R)$, one arrives at
\begin{align}
-\frac{1}{2T}\log \mathbb{E}\times \mathbb{E}[ \mathrm{e}^{- e^2 X_T(R)}]
&=-\frac{1}{2T}\big(
e^2 C_1(R,T)+\frac{1}{2}e^4 C_2(R, T)+\cdots\label{2ndCum}
\big),\nonumber\\
C_1(R, T)&= -\mathbb{E}\times \mathbb{E}[X_T(R)],\nonumber \\[1ex]
C_2(R,T)&=\mathbb{E}\times \mathbb{E} [X_T(R)^2] -\mathbb{E}\times \mathbb{E} [X_T(R)]^2.
\end{align}
As we will see below $\mathbb{E}\times \mathbb{E} [ X_T(R)^2]/2T=\mathcal{O}(R^{-6})$, not using the cancellation between  $V_R$  and $W_R$ terms.
Since we are heading for a decay as  $R^{-7}$    for large $R$,  it should
 suffice to stop the expansion at the second cumulant $C_2(R, T)$.
{\it  This will be our main assumption.}
$\mathbb{E}\times \mathbb{E}[ X_T(R)]/2T$ is exponentially small. The large $R$ behavior of the
second cumulant will be investigated  in detail in the following section.

We note that the expectation  under the $k$-integral always factorizes with respect to
$q_1, q_2$. Using their independence we will need to compute
only a few expectations for the ground state process of a single electron. They are listed
now for  later convenience and proved in Appendix \ref{(i)to(v)}. We set $\varepsilon=\varepsilon(k, \lambda)$,
$\varepsilon_j=\varepsilon(k_j, \lambda_j)$ for either $\lambda_j=1$
or $\lambda_j=2$ and $\theta(t)$ the step function, $\theta(t)=-1$ for $t\le 0$, $\theta(t)=1$ for $t> 0$.
\begin{itemize}
\item[{\rm (i)}] $\displaystyle  \mathbb{E}[ \mathrm{e}^{\mathrm{i} k\cdot q(t)}]=\langle \psi,
  \mathrm{e}^{\mathrm{i} k\cdot x}\psi\rangle$,
\item[{\rm (ii)}] $\displaystyle \mathbb{E}[ \mathrm{e}^{\mathrm{i} k\cdot q(t)}(\varepsilon(k,
  \lambda)\cdot \dot{q}(t))]=-\langle \psi, \mathrm{e}^{\mathrm{i} k\cdot
  x}(H-E)(\varepsilon(k, \lambda)\cdot x)\psi\rangle=0 ,$
\item[{\rm (iii)}] $\displaystyle \mathbb{E}[ \mathrm{e}^{-\mathrm{i} k_1\cdot q(s)}\mathrm{e}^{\mathrm{i}
  k_2\cdot q(t)}]=\langle \psi, \mathrm{e}^{-\mathrm{i} k_1\cdot
  x}\mathrm{e}^{-|s-t|(H-E)}\mathrm{e}^{\mathrm{i} k_2\cdot x}\psi\rangle$,
\item[{\rm (iv)}] $ \displaystyle  \mathbb{E} [(\varepsilon_1\cdot
  \dot{q}(s))\mathrm{e}^{-\mathrm{i} k_1\cdot q(s)}(1-\mathrm{e}^{-\mathrm{i} k_2\cdot q(t)})]\vspace{3mm}
\\=
\theta(t-s)\langle \psi, (\varepsilon_1\cdot x)(H-E)\mathrm{e}^{-\mathrm{i}
  k_1\cdot x} \mathrm{e}^{-|s-t|(H-E)}(1-\mathrm{e}^{-\mathrm{i} k_2\cdot x})\psi\rangle,
$
 \item[{\rm (v)}]$\displaystyle
 \mathbb{E} [(\varepsilon_1\cdot \dot{q}(s))\mathrm{e}^{-\mathrm{i} k_1\cdot q(s)}\mathrm{e}^{-\mathrm{i}
  k_2\cdot q(t)}(\varepsilon_2\cdot \dot{q}(t))] \vspace{3mm}
  \\
 =-\langle\psi,  (\varepsilon_1\cdot x )(H-E)\mathrm{e}^{-\mathrm{i} k_1\cdot
  x}\mathrm{e}^{-|s-t|(H-E)}\mathrm{e}^{-\mathrm{i} k_2\cdot x} (H-E)(\varepsilon_2\cdot
x)\psi\rangle\vspace{3mm}\\
 +\delta(t-s)(\varepsilon_1\cdot \varepsilon_2)\langle \psi, \mathrm{e}^{-\mathrm{i} k_1\cdot x}
\mathrm{e}^{-\mathrm{i} k_2\cdot x}\psi\rangle. \label{(v)}
$
\end{itemize}
 In (v) the second term arises because  locally $q(t)$ is
like a standard Brownian motion  for  which  $\mathrm{d}q(s)\otimes
\mathrm{d}q(t)={\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l \delta(s-t)\mathrm{d}s\mathrm{d}t.
$

The first cumulant can be dealt with immediately.
Using (ii) in the above list one obtains
\begin{align}
\mathbb{E}\times \mathbb{E}\big[ \dot{q}_1(s)\cdot W_R\big(q_1(s)-q_2(t), s-t\big)\dot{q}_2(t)\big]=0.
\end{align}
For the potential term it holds
\begin{align}
&\mathbb{E}\times \mathbb{E}[ V_R(q_1(t), q_2(t))]
\nonumber \\
&=\int_{\mathbb{R}^3}\mathrm{d}k\,
|\hat{\varphi}(k)|^2\omega(k)^{-2}\mathrm{e}^{\mathrm{i} k\cdot r}
\langle \psi, (1-\mathrm{e}^{-\mathrm{i} k\cdot x})\psi\rangle\langle \psi, (1-\mathrm{e}^{\mathrm{i} k\cdot
  x})\psi\rangle
  \nonumber \\
&=\int_{\mathbb{R}^3}\mathrm{d}x\int_{\mathbb{R}^3}\mathrm{d}y\big((\varrho-\delta)*\varphi\big)(x)
(4\pi |x+r-y|)^{-1}\big((\varrho-\delta)*\varphi\big)(y).
\end{align}
Here $\varrho(x)$ is the electron density for $\psi$, which is known  to
have an exponential decay \cite{Gr}. $\delta$ is the Dirac delta
and $*$ denotes convolution. Since $\int_{\mathbb{R}^3}\mathrm{d}x\,
(\varrho(x)-\delta(x))=0$ and since $\varrho$ decays rapidly, by
Newton's theorem  it holds that there are suitable constants $c_1,
c_2$ such that
\begin{align}
\big|\mathbb{E}\times \mathbb{E}[ V_R(q_1(t), q_2(t))]\big| \le c_1 \mathrm{e}^{-c_2R}.
\end{align}

\section{The second cumulant}\setcounter{equation}{0}
\label{Computation}

The second cumulant consists of three terms. They are all
proportional to $2T$ by the stationarity of $q_1(t), q_2(t)$.
We could take the limit $T\to \infty$ first, but the symmetric version
is more convenient. All expectations are written in terms of  $H$ and
its ground state $\psi$. For notational symplicity we replace  $H-E$ by
$H$,
 hence $H\psi=0$. For inverses as $\langle \phi_1,
 H^{-n}\phi_2\rangle$
we make sure that either $\langle \phi_1, \psi\rangle=0$ or $\langle
\phi_2, \psi\rangle=0$. Note that the ground state is nondegenerate \cite{Hiroshima1}.
But $\langle \phi_1, H^{-n}\phi_2\rangle$ could still be
infinite.
If $H$ is replaced by $H_{\mathrm{at}}$, then $H_{\mathrm{at}}$ has a  spectral gap  and therefore  an inverse on the
orthogonal  complement of $\psi_{\mathrm{at}}$.

We set
\begin{align}
C_2(R, T)=I_{VV}+2I_{VW}+I_{WW}\label{Dec2ndCum}
\end{align}
and compute each term separately.
The second step is a partial time-integration through which one can
understand how the $R^{-6}$ decay from the interaction potential is
canceled.
 In a final step we collect terms according to their number of
 time-integrations and discuss their $R$-dependence.

\subsection{Expectations}
  a) $I_{VV}$. Setting $\hat{\varphi}(k_j)=\hat{\varphi}_j,
  \omega(k_j)=\omega_j$,
we have
\begin{align}
I_{VV}=&\int\mathrm{d}t_1\mathrm{d}t_2\int\mathrm{d}k_1\mathrm{d}k_2\,
|\hat{\varphi}_1|^2|\hat{\varphi}_2|^2(|k_1|^2|k_2|^2)^{-1}\mathrm{e}^{\mathrm{i}
  (k_1+k_2)\cdot r}
  \nonumber\\
&\times \Big\{
\big|
\mathbb{E}[(1-\mathrm{e}^{-\mathrm{i} k_1\cdot q_1(t_1)})(1-\mathrm{e}^{-\mathrm{i} k_2\cdot q_1(t_2)})]
\big|^2
\nonumber\\
&-\big|
\mathbb{E}[ (1-\mathrm{e}^{-\mathrm{i} k_1\cdot q_1(t_1)})]\mathbb{E}[(1-\mathrm{e}^{-\mathrm{i} k_2\cdot q_1(t_2)})]
\big|^2
\Big\}
\nonumber \\
=&\int\mathrm{d}t_1\mathrm{d}t_2\int\mathrm{d}k_1\mathrm{d}k_2\,
|\hat{\varphi}_1|^2|\hat{\varphi}_2|^2(|k_1|^2 |k_2|^2)^{-1}\mathrm{e}^{\mathrm{i}
  (k_1+k_2)\cdot r}
  \nonumber\\
&\times \Big\{
\big|
\langle\psi,  (1-\mathrm{e}^{-\mathrm{i} k_1\cdot x})\mathrm{e}^{-|t_1-t_2|H}(1-\mathrm{e}^{-\mathrm{i} k_2\cdot x})\psi\rangle
\big|^2
\nonumber\\
&-\big|
\langle\psi, (1-\mathrm{e}^{-\mathrm{i} k_1\cdot x})\psi\rangle\langle\psi, (1-\mathrm{e}^{-\mathrm{i} k_2\cdot x})\psi\rangle
\big|^2
\Big\}. \label{IVV}
\end{align}

Note that  the integrand in (\ref{IVV}) decays to zero, since
\begin{align}
\mathrm{e}^{-|t_1-t_2|H}\to |\psi \rangle\langle \psi |
\end{align}
as $|t_1-t_2|\to \infty$.\medskip\\\medskip\\
b) $I_{VW}$. Setting $\varepsilon_j=\varepsilon(k_j, \lambda_j)$ and noting
\begin{align}
\sum_{\lambda=1,2}| \varepsilon(k, \lambda)\rangle\langle \varepsilon(k, \lambda)|={\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l -| \hat{k}\rangle \langle \hat{k}|, \label{Polarization1}
\end{align}
 one has
\begin{align}
I_{VW}=&\int\mathrm{d} t_1\mathrm{d} t_2 \mathrm{d} t_3\sum_{\lambda_1=1,2} \int \mathrm{d} k_1\mathrm{d} k_2 |\hat{\vphi}_1|^2|\hat{\vphi}_2|^2 (2\omega_1 |k_2|^2)^{-1}
\mathrm{e}^{\mathrm{i} (k_1+k_2)\cdot r} \mathrm{e}^{-\omega_1|t_1-t_2|}
\nonumber\\
&\times \mathbb{E}[ (\varepsilon_1\cdot \dot{q}_1(t_1))\mathrm{e}^{-\mathrm{i} k_1\cdot q_1(t_1)}(1-\mathrm{e}^{-\mathrm{i} k_2\cdot q_1(t_3)})]
\nonumber\\
&\times  \mathbb{E}[  (\varepsilon_1\cdot \dot{q}_2(t_2))\mathrm{e}^{\mathrm{i} k_1\cdot q_2(t_2)}(1-\mathrm{e}^{\mathrm{i} k_2\cdot q_2(t_3)})]
\nonumber \\
=&\int\mathrm{d} t_1\mathrm{d} t_2 \mathrm{d} t_3\sum_{\lambda_1=1,2} \int \mathrm{d} k_1\mathrm{d} k_2 |\hat{\vphi}_1|^2|\hat{\vphi}_2|^2 (2\omega_1 |k_2|^2)^{-1}
\mathrm{e}^{\mathrm{i} (k_1+k_2)\cdot r} \mathrm{e}^{-\omega_1|t_1-t_2|}
\nonumber \\
&\times \theta(t_1-t_3)\langle\psi,  (\varepsilon_1\cdot x) H \mathrm{e}^{-\mathrm{i} k_1\cdot x}\mathrm{e}^{-|t_1-t_3|H}(1-\mathrm{e}^{-\mathrm{i} k_2\cdot x})\psi\rangle
\nonumber \\
&\times \theta(t_2-t_3)\langle\psi,  (\varepsilon_1\cdot x) H \mathrm{e}^{\mathrm{i} k_1\cdot x}\mathrm{e}^{-|t_2-t_3|H}(1-\mathrm{e}^{\mathrm{i} k_2\cdot x})\psi\rangle.
 \end{align}
As proved in Appendix \ref{GrEx}, it holds
\begin{align}
\langle\psi,  (\varepsilon_1\cdot x)H \mathrm{e}^{-\mathrm{i} k_1\cdot x}\psi\rangle=\mathrm{i} \langle \psi, (\varepsilon_1\cdot (p-eA(x)))\mathrm{e}^{-\mathrm{i} k_1\cdot x}\psi\rangle=0.
\end{align}
Thus no truncation  of the expectation is needed.
\medskip\\
\medskip\\
c) $I_{WW}$. One has
\begin{align}
I_{WW}
=&\int \mathrm{d} t_1\mathrm{d} t_2\mathrm{d} t_3\mathrm{d} t_4 \sum_{\lambda_1, \lambda_2}\int \mathrm{d} k_1\mathrm{d} k_2 |\hat{\vphi}_1|^2 |\hat{\vphi}_2|^2(2\omega_12\omega_2)^{-1}
\mathrm{e}^{\mathrm{i} (k_1+k_2)\cdot r}
\nonumber \\
&\times \mathrm{e}^{-\omega_1|t_1-t_2|}\mathrm{e}^{-\omega_2|t_3-t_4|}
 \mathbb{E}\big[ (\varepsilon_1 \cdot \dot{q}_1(t_1)) \mathrm{e}^{-\mathrm{i} k_1\cdot q_1(t_1)}(\varepsilon_2 \cdot \dot{q}_1(t_3))\mathrm{e}^{-\mathrm{i} k_2\cdot q_1(t_3)}\big]
 \nonumber \\
 &\times \mathbb{E}\big[ (\varepsilon_1 \cdot \dot{q}_2(t_2)) \mathrm{e}^{\mathrm{i} k_1\cdot q_2(t_2)}(\varepsilon_2 \cdot \dot{q}_2(t_4))\mathrm{e}^{\mathrm{i} k_2\cdot q_2(t_4)}\big]
 \nonumber \\
 =&\int \mathrm{d} t_1\mathrm{d} t_2\mathrm{d} t_3\mathrm{d} t_4 \sum_{\lambda_1, \lambda_2}\int \mathrm{d} k_1\mathrm{d} k_2 |\hat{\vphi}_1|^2 |\hat{\vphi}_2|^2(2\omega_12\omega_2)^{-1}
\mathrm{e}^{\mathrm{i} (k_1+k_2)\cdot r}
\nonumber \\
&\times \mathrm{e}^{-\omega_1|t_1-t_2|}\mathrm{e}^{-\omega_2|t_3-t_4|}
\nonumber \\
&\times\big(
 -\langle\psi,  (\varepsilon_1 \cdot x) H  \mathrm{e}^{-\mathrm{i} k_1\cdot x} \mathrm{e}^{-|t_1-t_3|H}\mathrm{e}^{-\mathrm{i} k_2\cdot x }H  (\varepsilon_2 \cdot x)\psi\rangle
 \nonumber \\
 &+ \delta(t_1-t_3)(\varepsilon_1\cdot \varepsilon_2) \langle \psi, \mathrm{e}^{-\mathrm{i} (k_1+k_2)\cdot x}\psi\rangle
 \big)
 \nonumber \\
 &\times \big(
- \langle\psi,  (\varepsilon_1 \cdot x) H  \mathrm{e}^{\mathrm{i} k_1\cdot x} \mathrm{e}^{-|t_2-t_4|H}\mathrm{e}^{\mathrm{i} k_2\cdot x }H  (\varepsilon_2 \cdot x)\psi\rangle
\nonumber \\
 &+ \delta(t_2-t_4)(\varepsilon_1\cdot \varepsilon_2) \langle \psi, \mathrm{e}^{\mathrm{i} (k_1+k_2)\cdot x}\psi\rangle
 \big).
  \end{align}

  \subsection{Partial time integration}
  The next step is a partial time integration for $I_{VW} $ and $I_{WW}$. For the integrand of $I_{VW}$ we use the identity
  \begin{align}
 & \theta(s-t) \langle \psi, (\varepsilon_1\cdot x) H \mathrm{e}^{-\mathrm{i} k_1\cdot x} \mathrm{e}^{-|s-t|H} (1-\mathrm{e}^{-\mathrm{i} k_2\cdot x})\psi\rangle
  \nonumber \\
  &=-\frac{\partial }{\partial s} \langle \psi, ( \varepsilon_1\cdot x) H \mathrm{e}^{-\mathrm{i} k_1\cdot x} H^{-1} \mathrm{e}^{-|s-t|H} (1-\mathrm{e}^{-\mathrm{i} k_2 \cdot x})\psi\rangle
  \end{align}
and for the integrand of $I_{WW}$ the identity
\begin{align}
&-\langle \psi, (\varepsilon_1\cdot x) H \mathrm{e}^{-\mathrm{i} k_1\cdot x }\mathrm{e}^{-|s-t|H} \mathrm{e}^{-\mathrm{i} k_2\cdot x} H (\varepsilon_2\cdot x)\psi\rangle
\nonumber \\
&=\frac{\partial}{\partial s}\frac{\partial}{\partial t} \langle \psi, (\varepsilon_1\cdot x) H \mathrm{e}^{-\mathrm{i} k_1\cdot x }H^{-2} \mathrm{e}^{-|s-t|H} \mathrm{e}^{-\mathrm{i} k_2\cdot x} H (\varepsilon_2\cdot x)\psi\rangle
\nonumber \\
& -2 \delta(s-t)\langle \psi, (\varepsilon_1\cdot x) H \mathrm{e}^{-\mathrm{i} k_1\cdot x}H^{-1} \mathrm{e}^{-\mathrm{i} k_2\cdot x}H (\varepsilon_2 \cdot x)\psi\rangle.
\end{align}
We insert these expectations in $I_{VW}, I_{WW}$, integrate partially in time, and use
\begin{align}
\frac{\partial}{\partial s}\frac{\partial}{\partial t}\mathrm{e}^{-\omega|s-t|}=-\omega^2\mathrm{e}^{-\omega|s-t|}+2\omega\delta(s-t).
\end{align}
The boundary terms vanish. For $I_{VW} $ one obtains
\begin{align}
I_{VW}=&\int\mathrm{d} t_1\mathrm{d} t_2\mathrm{d} t_3 \sum_{\lambda_1}\int\mathrm{d} k_1\mathrm{d} k_2 |\hat{\vphi}_1|^2|\hat{\vphi}_2|^2 (2\omega_1|k_2|^2)^{-1} \mathrm{e}^{\mathrm{i} (k_1+k_2)\cdot r}
\nonumber \\
&\times \big(-\omega_1^2\mathrm{e}^{-\omega_1|t_1-t_2|}+2\omega_1 \delta(t_1-t_2)\big)\nonumber \\
&\times \langle \psi, (\varepsilon_1\cdot x) H \mathrm{e}^{-\mathrm{i} k_1\cdot x}H^{-1}\mathrm{e}^{-|t_1-t_3|H}(1-\mathrm{e}^{-\mathrm{i} k_2\cdot x})\psi\rangle
\nonumber \\
&\times \langle \psi, (\varepsilon_1\cdot x) H \mathrm{e}^{\mathrm{i} k_1\cdot x} H^{-1} \mathrm{e}^{-|t_2-t_3|H} (1-\mathrm{e}^{\mathrm{i} k_2\cdot x})\psi\rangle.
\end{align}
Finally  $I_{WW}$  is given by
\begin{align}
I_{WW}=&\int \mathrm{d} t_1\mathrm{d} t_2\mathrm{d} t_3\mathrm{d} t_4 \sum_{\lambda_1, \lambda_2}\int \mathrm{d} k_1\mathrm{d} k_2 |\hat{\vphi}_1|^2 |\hat{\vphi}_2|^2(2\omega_12\omega_2)^{-1}
\mathrm{e}^{\mathrm{i} (k_1+k_2)\cdot r}
\nonumber \\
&\times\Big\{ \big(-\omega_1^2 \mathrm{e}^{-\omega_1|t_1-t_2|}+2\omega_1 \delta(t_1-t_2)\big)\big(-\omega_2^2 \mathrm{e}^{-\omega_2|t_3-t_4|}+2\omega_2 \delta(t_3-t_4)\big)
\nonumber \\
&
 \times \langle\psi,  (\varepsilon_1 \cdot x) H  \mathrm{e}^{-\mathrm{i} k_1\cdot x} H^{-2} \mathrm{e}^{-|t_1-t_3|H}\mathrm{e}^{-\mathrm{i} k_2\cdot x }H  (\varepsilon_2 \cdot x)\psi\rangle
 \nonumber \\
  &  \times \langle\psi,  (\varepsilon_1 \cdot x) H  \mathrm{e}^{\mathrm{i} k_1\cdot x} H^{-2}\mathrm{e}^{-|t_2-t_4|H}\mathrm{e}^{\mathrm{i} k_2\cdot x }H  (\varepsilon_2 \cdot x)\psi\rangle\
  \nonumber \\
 &+\theta(t_1-t_2)\omega_1 \theta(t_3-t_4)\omega_2 \mathrm{e}^{-\omega_1|t_1-t_2|} \mathrm{e}^{-\omega_2|t_3-t_4|}
 \nonumber \\
 &\times \Big(
 \langle\psi,  (\varepsilon_1 \cdot x) H  \mathrm{e}^{-\mathrm{i} k_1\cdot x}H^{-2} \mathrm{e}^{-|t_1-t_3|H}\mathrm{e}^{-\mathrm{i} k_2\cdot x }H  (\varepsilon_2 \cdot x)\psi\rangle
 \nonumber \\
 &\times\delta(t_2-t_4)B_+(k_1, \lambda_1; k_2, \lambda_2)
 \nonumber \\
 & + \langle\psi,  (\varepsilon_1 \cdot x) H  \mathrm{e}^{\mathrm{i} k_1\cdot x} H^{-2}\mathrm{e}^{-|t_2-t_4|H}\mathrm{e}^{\mathrm{i} k_2\cdot x }H  (\varepsilon_2 \cdot x)\psi\rangle\
 \nonumber \\
 & \times \delta(t_1-t_3)B_-(k_1, \lambda_1; k_2, \lambda_2)
 \Big)
 \nonumber \\
 &+\mathrm{e}^{-\omega_1|t_1-t_2|}\mathrm{e}^{-\omega_2|t_3-t_4|}\delta(t_1-t_3)\delta(t_2-t_4)
 \nonumber \\
 &\times B_-(k_1, \lambda_1; k_2, \lambda_2)B_+(k_1, \lambda_1; k_2, \lambda_2) \Big\},
\end{align}
where
\begin{align}
&B_-(k_1, \lambda_1; k_2, \lambda_2)
\nonumber \\
&=(\varepsilon_1\cdot \varepsilon_2)\langle \psi, \mathrm{e}^{-\mathrm{i}(k_1+k_2)\cdot x}\psi\rangle-2\langle \psi, (\varepsilon_1\cdot x) H \mathrm{e}^{-\mathrm{i} k_1\cdot x} H^{-1}\mathrm{e}^{-\mathrm{i} k_2\cdot x}H (\varepsilon_2\cdot x)\psi\rangle,
\\
&B_+(k_1, \lambda_1; k_2, \lambda_2)
\nonumber \\
&=(\varepsilon_1\cdot \varepsilon_2)\langle \psi, \mathrm{e}^{\mathrm{i}(k_1+k_2)\cdot x}\psi\rangle-2\langle \psi, (\varepsilon_1\cdot x) H \mathrm{e}^{\mathrm{i} k_1\cdot x} H^{-1}\mathrm{e}^{\mathrm{i} k_2\cdot x}H (\varepsilon_2\cdot x)\psi\rangle
\end{align}

We use (\ref{Dec2ndCum}) and  collect the terms of $C_2(R, T)$ according to the number of their time-integrations, divide by $1/2T$, and take the limit as $T\to \infty$. To prepare
 for the limit $R\to \infty$,
we rescale the momentum integration as $k_j\leadsto k_j/R, j=1,2$, and introduce
the unit vector $\hat{n}=r/R=(0,0,1)$. Note that $\varepsilon(k/R, \lambda)=\varepsilon(k, \lambda)$.
 The two-time, three-time, and four-time integrations are treated separately.

\subsection{Two-time integrations}\label{TwoTime}
The sum of all terms involving  two-time integrations  is denoted by $I_2(R)$. One has
\begin{align}
I_2(R)=&\int\mathrm{d} k_1\mathrm{d} k_2 |\hat{\vphi}(k_1/R)|^2|\hat{\vphi}(k_2/R)|^2 \mathrm{e}^{\mathrm{i} (k_1+k_2)\cdot \hat{n}}
\nonumber \\
& \times \Big[ \int \mathrm{d} t\Big\{
R^{-2} (|k_1|^2 |k_2|^2)^{-1} \Big(
\big|
\langle\psi,  (1-\mathrm{e}^{-\mathrm{i} k_1\cdot x/R})\mathrm{e}^{-|t|H}(1-\mathrm{e}^{-\mathrm{i} k_2\cdot x/R})\psi\rangle
\big|^2
\nonumber \\
& -\big|
\langle\psi, (1-\mathrm{e}^{-\mathrm{i} k_1\cdot x/R})\psi\rangle\langle\psi, (1-\mathrm{e}^{-\mathrm{i} k\cdot x_2/R})\psi\rangle
\big|^2
\Big)
\nonumber \\
&+2\sum_{\lambda_1} R^{-4} |k_2|^{-2} \langle \psi, (\varepsilon_1\cdot x) H \mathrm{e}^{-\mathrm{i} k_1\cdot x/R}H^{-1}\mathrm{e}^{-|t|H} (1-\mathrm{e}^{-\mathrm{i} k_2\cdot x/R})\psi\rangle
\nonumber \\
&\times \langle \psi, (\varepsilon_1\cdot x) H \mathrm{e}^{\mathrm{i} k_1\cdot x/R} H^{-1}\mathrm{e}^{-|t|H} (1-\mathrm{e}^{\mathrm{i} k_2\cdot x/R})\psi\rangle
\nonumber \\
&+\sum_{\lambda_1, \lambda_2}R^{-6} \langle \psi, (\varepsilon_1\cdot x) H\mathrm{e}^{-\mathrm{i} k_1\cdot x/R}H^{-2} \mathrm{e}^{-|t|H} \mathrm{e}^{-\mathrm{i} k_2\cdot x/R} H(\varepsilon_2\cdot x)
\psi\rangle
\nonumber \\
&\times \langle \psi, (\varepsilon_1\cdot x) H\mathrm{e}^{\mathrm{i} k_1\cdot x/R}H^{-2} \mathrm{e}^{-|t|H} \mathrm{e}^{\mathrm{i} k_2\cdot x/R} H(\varepsilon_2\cdot x)
\psi\rangle
\Big\}
\nonumber \\
&+\sum_{\lambda_1, \lambda_2}R^{-3} (2\omega_1\omega_2 (\omega_1+\omega_2))^{-1}
\nonumber \\
&\times  B_-(k_1/R, \lambda_1; k_2/R, \lambda_2)B_+(k_1/R, \lambda_1; k_2/R, \lambda_2)
\Big]\label{TwoTime1}
\nonumber \\
=& I_{2,1}(R)+I_{2,2}(R).
\end{align}

We consider the sum, $I_{2,1}(R)$,  of the first three terms and expand in $1/R$,  which yields  expectations of the form $\langle \psi, (a\cdot x) \mathrm{e}^{-|t|H} (b\cdot x)\psi\rangle$ with $a,b\in \mathbb{R}^3$. By rotation invariance
\begin{align}
\langle \psi, (a\cdot x) \mathrm{e}^{-|t|H}(b\cdot x) \psi\rangle=(a\cdot b)\tfrac{1}{3}\langle \psi, x\cdot \mathrm{e}^{-|t|H}x\psi\rangle.
\end{align}
Using
\begin{align}
\sum_{\lambda_1, \lambda_2}(\varepsilon_1\cdot \varepsilon_2)^2=1+(\hat{k}_1\cdot \hat{k}_2)^2,\ \ \sum_{\lambda_1} (\varepsilon_1\cdot \hat{k}_2)^2=1-(\hat{k}_1\cdot \hat{k}_2)^2,
\end{align}
 one arrives at the lowest order
\begin{align}
R^{-6}\big(\tfrac{1}{3}\langle \psi, x\cdot \mathrm{e}^{-|t|H}x\psi\rangle\big)^2\big((\hat{k}_1\cdot \hat{k}_2)^2+2-2(\hat{k}_1\cdot \hat{k}_2)^2+1+(\hat{k}_1\cdot \hat{k}_2)^2\big). \label{Cal1}
\end{align}
The nonsmooth contributions, containing $(\hat{k}_1\cdot
\hat{k}_2)^2$, are canceled exactly, while  the smooth part, at this
order, is given by
\begin{align}
&I_{2,1}(R)
\nonumber \\
&=3\int\mathrm{d} t\int \mathrm{d} k_1\mathrm{d} k_2 |\hat{\vphi}(k_1/R)|^2|\hat{\vphi}(k_2/R)|^2\mathrm{e}^{\mathrm{i} (k_1+k_2)\cdot \hat{n}} R^{-6} \big(\tfrac{1}{3}\langle \psi, x\cdot \mathrm{e}^{-|t|H}x\psi\rangle\big)^2,
\end{align}
which inherits the rapid decay in $R$ from $\varphi$.  At the next order one picks up the quadratic contributions  $k_1^2, k_1\cdot k_2, k_2^2$ with coefficients integrable in $t$. By power counting  one arrives at
$I_{2, 1}(R)\simeq R^{-8}$ as $R\to \infty$.

For the second summand, $I_{2,2}(R)$, we use that to  leading order in $1/R$, \begin{align}
B_{\pm}(k_1/R, \lambda_1; k_2/R, \lambda_2)= R^{-2} |k_1| |k_2| (\varepsilon_1\cdot\hat{ k}_2)   (\varepsilon_2\cdot \hat{k}_1)(\tfrac{1}{3}\langle \psi, x^2\psi\rangle)+\mathcal{O}(R^{-4}),
\end{align}
as in shown in Appendix \ref{GrEx}.
Using that $|\hat{\vphi}(0)|^2=(2\pi)^{-3}$ one arrives at
\begin{align}
&I_{2,2}(R)
\nonumber \\
&=  R^{-7}\big(\tfrac{1}{3} \langle \psi, x^2 \psi\rangle\big)^2(2\pi)^{-6}\frac{1}{2} \int \mathrm{d} k_1 \mathrm{d} k_2  (\omega_1\omega_2(\omega_1+\omega_2))^{-1}|k_1|^2 |k_2|^2
\nonumber \\
&\times \mathrm{e}^{\mathrm{i} (k_1+k_2)\cdot \hat{n}}
 (1-(\hat{k}_1\cdot
\hat{k}_2)^2)^2
 +\mathcal{O}(R^{-9})
\nonumber \\
&= \frac{128}{\pi}\big(\frac{1}{2\pi}\big)^2  e^{-4 } \alpha_{\mathrm{M}}^2 R^{-7}
+\mathcal{O}(R^{-9}).
\end{align}
The evaluation  of the numerical coefficient is discussed in Appendix \ref{NumC}.
  Altogether
  \begin{align}
  \lim_{R\to \infty} R^7I_2(R)= \frac{128}{\pi}\big(\frac{1}{2\pi}\big)^2 e^{-4} \alpha_{\mathrm{M}}^2. \label{I2}
  \end{align}
\subsection{Three-time integrations}
The sum of all terms involving three-time integrations is denoted by $I_3(R)$. One obtains
\begin{align}
I_3(R)=&\int\mathrm{d} t_1\mathrm{d} t_2 \int \mathrm{d} k_1\mathrm{d} k_2 |\hat{\vphi}(k_1/R)|^2|\hat{\vphi}(k_2/R)|^2 \mathrm{e}^{\mathrm{i}(k_1+k_2)\cdot \hat{n}}
\nonumber \\
&\times \Big\{
-R^{-5} \omega_1 |k_2|^{-2} \mathrm{e}^{-\omega_1|t_1-t_2|/R}
\nonumber \\
&\times\Big( \sum_{\lambda_1}  \langle \psi, (\varepsilon_1\cdot x) H \mathrm{e}^{-\mathrm{i} k_1\cdot x/R}H^{-1}
\mathrm{e}^{-|t_1|H} (1-\mathrm{e}^{-\mathrm{i} k_2\cdot x/R})\psi\rangle
\nonumber \\
&\times  \langle \psi, (\varepsilon_1\cdot x) H \mathrm{e}^{\mathrm{i} k_1\cdot x/R}H^{-1}
\mathrm{e}^{-|t_2|H} (1-\mathrm{e}^{\mathrm{i} k_2\cdot x/R})\psi\rangle\Big)
\nonumber \\
&-R^{-7}\big(
\tfrac{1}{2}\omega_1 \mathrm{e}^{-\omega_1|t_1-t_2|/R}+\tfrac{1}{2}\omega_2 \mathrm{e}^{-\omega_2|t_1-t_2|/R}\big) \nonumber \\
&\times \Big(\sum_{\lambda_1, \lambda_2}
\langle \psi, (\varepsilon_1\cdot x) H \mathrm{e}^{-\mathrm{i} k_1\cdot x/R}H^{-2} \mathrm{e}^{-|t_1|H}\mathrm{e}^{-\mathrm{i} k_2\cdot x/R} H(\varepsilon_2\cdot x)\psi\rangle
\nonumber \\
&\times
\langle \psi, (\varepsilon_1\cdot x) H \mathrm{e}^{\mathrm{i} k_1\cdot x/R}H^{-2} \mathrm{e}^{-|t_2|H}\mathrm{e}^{\mathrm{i} k_2\cdot x/R} H(\varepsilon_2\cdot x)\psi\rangle\Big)\nonumber
\end{align}

\begin{align}
&+R^{-6}\tfrac{1}{4} \theta(t_1) \mathrm{e}^{-\omega_1 |t_1|/R} \theta(t_2)\mathrm{e}^{-\omega_2 |t_2|/R}
\nonumber \\
&\times\Big(\sum_{\lambda_1, \lambda_2} \big(
\langle\psi,  (\varepsilon_1 \cdot x) H  \mathrm{e}^{-\mathrm{i} k_1\cdot x/R}H^{-2} \mathrm{e}^{-|t_1-t_2|H}\mathrm{e}^{-\mathrm{i} k_2\cdot x/R }H  (\varepsilon_2 \cdot x)\psi\rangle
\nonumber \\
&\times B_+(k_1, \lambda_1; k_2, \lambda_2)
\nonumber \\
&+\langle\psi,  (\varepsilon_1 \cdot x) H  \mathrm{e}^{\mathrm{i} k_1\cdot x/R}H^{-2} \mathrm{e}^{-|t_1-t_2|H}\mathrm{e}^{\mathrm{i} k_2\cdot x/R }H  (\varepsilon_2 \cdot x)\psi\rangle
\nonumber \\
&\times B_-(k_1, \lambda_1; k_2, \lambda_2)
 \big)\Big)
\Big\}\nonumber \\
=& I_{3,1}(R)+I_{3,2}(R).
\end{align}

We expand the integrand in $1/R$ which yields
\begin{align}
&\int\mathrm{d} t_1\mathrm{d} t_2 \int \mathrm{d} k_1\mathrm{d} k_2 |\hat{\vphi}(k_1/R)|^2|\hat{\vphi}(k_2/R)|^2 \mathrm{e}^{\mathrm{i}(k_1+k_2)\cdot \hat{n}}
\nonumber \\
&\times \Big\{
-R^{-7} \omega_1  \mathrm{e}^{-\omega_1|t_1-t_2|/R} (1-(\hat{k}_1\cdot\hat{k_2})^2)
\nonumber \\
&\times \tfrac{1}{3}\langle \psi, x\cdot
\mathrm{e}^{-|t_1|H} x\psi\rangle \tfrac{1}{3}\langle \psi, x\cdot
\mathrm{e}^{-|t_2|H} x\psi\rangle
\nonumber \\
&-R^{-7}\big(
\tfrac{1}{2}\omega_1 \mathrm{e}^{-\omega_1|t_1-t_2|/R}+\tfrac{1}{2}\omega_2 \mathrm{e}^{-\omega_2|t_1-t_2|/R}\big) (1+(\hat{k}_1\cdot \hat{k}_2)^2)
\nonumber \\
&\times \tfrac{1}{3}\langle \psi, x\cdot
\mathrm{e}^{-|t_1|H} x\psi\rangle \tfrac{1}{3}\langle \psi, x\cdot
\mathrm{e}^{-|t_2|H} x\psi\rangle
\nonumber \\
&+R^{-6} \tfrac{1}{4}\theta(t_1) \mathrm{e}^{-\omega_1 |t_1|/R} \theta(t_2)\mathrm{e}^{-\omega_2 |t_2|/R}
\nonumber \\
&\times \Big(\sum_{\lambda_1, \lambda_2}
\tfrac{1}{3}\langle \psi, x\cdot
\mathrm{e}^{-|t_1-t_2|H} x\psi\rangle
\nonumber \\
&\times (\varepsilon_1\cdot \varepsilon_2)\big(
 B_+(k_1/R, \lambda_1; k_2/R, \lambda_2)
+ B_-(k_1/R, \lambda_1; k_2/R, \lambda_2)
 \big)\Big)
\Big\}\label{ThreeT1}
\nonumber \\
=&-R^{-7}\int\mathrm{d} t_1\mathrm{d} t_2 \int\mathrm{d} k_1\mathrm{d} k_2 |\hat{\vphi}(k_1/R)|^2|\hat{\vphi}(k_2/R)|^2
\mathrm{e}^{\mathrm{i} (k_1+k_2)\cdot \hat{n}} \omega_1 \mathrm{e}^{-\omega_1|t_1-t_2|/R}
\nonumber \\
&\times \tfrac{1}{3}\langle \psi, x\cdot
\mathrm{e}^{-|t_1|H} x\psi\rangle \tfrac{1}{3}\langle \psi, x\cdot
\mathrm{e}^{-|t_2|H} x\psi\rangle
\nonumber \\
&+ R^{-6}\int\mathrm{d} t_1\mathrm{d} t_2 \int\mathrm{d} k_1\mathrm{d} k_2 |\hat{\vphi}(k_1/R)|^2|\hat{\vphi}(k_2/R)|^2\Big(
\sum_{\lambda_1, \lambda_2}
\nonumber \\
&\times \mathrm{e}^{\mathrm{i} (k_1+k_2)\cdot \hat{n}}\tfrac{1}{4}\theta(t_1)\mathrm{e}^{-\omega_1|t_1|/R} \theta(t_2) \mathrm{e}^{-\omega_2 |t_2|/R}
 \tfrac{1}{3}\langle \psi, x\cdot
\mathrm{e}^{-|t_1-t_2|H} x\psi\rangle
\nonumber \\
&\times (\varepsilon_1\cdot \varepsilon_2)\big(
 B_+(k_1/R, \lambda_1; k_2/R, \lambda_2)
+ B_-(k_1/R, \lambda_1; k_2/R,  \lambda_2)
 \big)\Big).
\end{align}
For the first summand the nonsmooth contributions are canceled
 exactly, as in Section \ref{TwoTime}, while the smooth contribtions decay rapidly
 since $\varphi$ does so. Its  next order  picks up an extra factor $R^{-2}$.  Thus
$I_{3,1}(R)\simeq R^{-9} $ for large $R$. For the second summand we perform explicitly the time integration with the result
\begin{align}
I_{3,2}(R)=&R^{-6} \int \mathrm{d} k_1\mathrm{d} k_2  |\hat{\vphi}(k_1/R)|^2|\hat{\vphi}(k_2/R)|^2\Big(
\mathrm{e}^{\mathrm{i} (k_1+k_2)\cdot \hat{n}}
\nonumber \\
&\times  R (\omega_1+\omega_2)^{-1}\tfrac{1}{3}
\langle \psi, x\cdot \big(H (H+R^{-1}\omega_1)^{-1} (H+R^{-1}\omega_2)^{-1}\big)
x\psi\rangle
\nonumber \\
&\times\sum_{\lambda_1, \lambda_2}
 (\varepsilon_1\cdot \varepsilon_2)\big(
 B_+(k_1/R, \lambda_1; k_2/R, \lambda_2)
+ B_-(k_1/R, \lambda_1; k_2/R, \lambda_2)\big)\Big).
\end{align}
Since $B_{\pm}=R^{-2} |k_1| |k_2|(\varepsilon_1\cdot \hat{k}_2) (\varepsilon_2\cdot \hat{k}_1)(\tfrac{1}{3}\langle \psi, x^2\psi\rangle)+\mathcal{O}(R^{-4})$, as shown in  Appendix \ref{GrEx}, we conclude that
\begin{align}
I_{3,2}(R)=&-R^{-7}\big(\tfrac{1}{3}\langle \psi, x^2\psi\big\rangle\big)\big(\tfrac{1}{3}\langle \psi, x\cdot H^{-1}x\psi\rangle\big)
\nonumber \\
&\times 2 (2\pi)^{-6} \int \mathrm{d} k_1 \mathrm{d}  k_2\,  \mathrm{e}^{\mathrm{i} (k_1+k_2)\cdot \hat{n}} (\omega_1\omega_2(\omega_1+\omega_2))^{-1}
\nonumber \\
&\times |k_1|^2 |k_2|^2 (\hat{k}_1\cdot \hat{k}_2)
(1-(\hat{k}_1\cdot \hat{k}_2)^2)
  +\mathcal{O}(R^{-9})
\nonumber \\
=&\frac{52}{\pi}\big(\frac{1}{2\pi}\big)^2 e^{-4}\alpha_{\mathrm{E}}\alpha_{\mathrm{M}} R^{-7}+\mathcal{O}(R^{-9}).
\end{align}
  The evaluation of the numerical coefficient is discussed  in Appendix \ref{NumC}. Altogether one has
  \begin{align}
  \lim_{R\to \infty} R^7 I_3(R)=\frac{52}{\pi}\big(\frac{1}{2\pi}\big)^2  e^{-4 } \alpha_{\mathrm{E}}\alpha_{\mathrm{M}}. \label{I3Lim}
    \end{align}

\subsection{Four-time integrations}
There is only a single term with four-time integrations, namely
\begin{align}
I_4(R)=&  \int\mathrm{d} t_1\mathrm{d} t_2\mathrm{d} t_3 \sum_{\lambda_1, \lambda_2}\int\mathrm{d} k_1 \mathrm{d} k_2 |\hat{\vphi}_1|^2|\hat{\vphi}_2|^2
 \mathrm{e}^{\mathrm{i} (k_1+k_2)\cdot r}
 \nonumber \\
 &\times \frac{1}{4}\omega_1\omega_2 \mathrm{e}^{-\omega_1|t_1-t_2+t_3|}\mathrm{e}^{-\omega_2 |t_3|}
 \nonumber \\
 &\times \langle \psi, (\varepsilon_1\cdot x) H \mathrm{e}^{-\mathrm{i} k_1\cdot x}H^{-2}\mathrm{e}^{-|t_1|H} \mathrm{e}^{-\mathrm{i} k_2\cdot x} H (\varepsilon_2\cdot x)\psi\rangle
 \nonumber \\
 &\times \langle \psi, (\varepsilon_1\cdot x) H \mathrm{e}^{\mathrm{i} k_1\cdot x}H^{-2}\mathrm{e}^{-|t_2|H} \mathrm{e}^{\mathrm{i} k_2\cdot x} H (\varepsilon_2\cdot x)\psi\rangle.
 \end{align}
We rescale $k_j \leadsto k_j/R$ as before and in addition $t_3\leadsto t_3R$. Then the exponentially decaying  terms are
\begin{align}
\mathrm{e}^{-\omega_1|t_3+(t_1-t_2)/R|}\mathrm{e}^{-\omega_2 |t_3|}\simeq \mathrm{e}^{-\omega_1 |t_3|} \mathrm{e}^{-\omega_2 |t_3|}
\end{align}
for large $R$. Expanding in $1/R$ yields
\begin{align}
I_4(R)=  &R^{-7}  \big(\tfrac{1}{3} \langle \psi, x\cdot H^{-1}x\psi\rangle\big)^2 2 \int_0^{\infty}\mathrm{d} t \int \mathrm{d} k_1 \mathrm{d} k_2 |\hat{\vphi}(k_1/R)|^2|\hat{\vphi}(k_2/R)|^2
 \nonumber \\
 &\times \mathrm{e}^{\mathrm{i} (k_1+k_2)\cdot \hat{n}}
  \omega_1\omega_2 \mathrm{e}^{-t (\omega_1+\omega_2)} (1+(\hat{k}_1\cdot \hat{k}_2)^2)  +\mathcal{O}(R^{-9})
  \nonumber \\
= &R^{-7}  (2\pi)^{-6}\big(\tfrac{1}{3} \langle \psi, x\cdot H^{-1}x\psi\rangle\big)^2  2\int \mathrm{d} k_1 \mathrm{d} k_2
   \nonumber \\
 &\times \mathrm{e}^{\mathrm{i} (k_1+k_2)\cdot \hat{n}} \frac{\omega_1\omega_2}{\omega_1+\omega_2} (1+(\hat{k}_1\cdot \hat{k}_2)^2)+\mathcal{O}(R^{-9})
 \nonumber \\
 =& R^{-7} \big(\frac{1}{2\pi}\big)^2 e^{-4}\frac{23}{8\pi} \alpha_{\mathrm{E}}^2+\mathcal{O}(R^{-9}). \label{I4}
  \end{align}
  The prefactor of $R^{-7}$ agrees with  the one computed already by
  Casimir and Polder.

  \subsection{Sum of all terms}
    We add the limits listed in (\ref{I2}), (\ref{I3Lim}),  and (\ref{I4}), which yields
  \begin{align}
 \kappa(e)= \lim_{R\to \infty} R^7 \Big(\lim_{T\to \infty} \frac{1}{2T}e^4 \frac{1}{2}C_2(R,T)\Big)=\big(\frac{1}{2\pi}\big)^2 \Big(\frac{23}{16\pi}\alpha_{\mathrm{E}}^2
  +\frac{26}{\pi}\alpha_{\mathrm{E}}\alpha_{\mathrm{M}}
  +\frac{64}{\pi}\alpha_{\mathrm{M}}^2\Big).\label{SumAll}
    \end{align}

   \subsection{Comparison with previous results}

Magnetic contributions  to the  $-R^{-7}$ decay were first considered by Feinberg and Sucher \cite{FS0, FS} and by Boyer \cite{Boyer} .
They find that $\alpha^2_{\mathrm{E,at }}$ and $\alpha_{\mathrm{M, at}}^2$ have the same coefficient, namely  $23/16\pi$, while  the one of $\alpha_{\mathrm{E, at}}
\alpha_{\mathrm{M, at}}$ is $7/8\pi$. This raises  the issue on  the origin for the discrepancy.

The $\alpha_{\mathrm{E, at}}^2$ term  can be most easily obtained through the dipole approximated hamiltonian
\begin{align}
H_{\mathrm{dip}}=&\frac{1}{2} (p_1-eA(0))^2-e^2 V(x_1)+\frac{1}{2} (p_2-e A(r))^2-e^2 V(x_2)+H_{\mathrm{f}}
\nonumber \\ &+ e^2 V_R(x_1, x_2). \label{Dipole}
\end{align}
   One shifts $p_1$ by $eA(0)$ and  $p_2$ by $eA(r)$ through the unitary
   $U=\exp[-\mathrm{i} (x_1 \cdot A(0) +x_2 \cdot A(r))]$.  Then
  \begin{align}
  U H_{\mathrm{dip}} U^{-1} =& \frac{1}{2}p_1^2-e^2 V(x_1)+\frac{1}{2}p_2^2-e^2 V(x_2)+H_{\mathrm{f}}
  \nonumber \\
  &-e x_1\cdot E_{\perp}(0)-e x_2\cdot E_{\perp}(r)+e^2 V_R(x_1, x_2)
  \nonumber \\
  &+e^2 \int \mathrm{d}k\, |\hat{\varphi}(k)|^2 \Big(\frac{1}{3} x_1^2+\frac{1}{3}x_2^2+\mathrm{e}^{\mathrm{i} k\cdot r}\big(
  x_1\cdot x_2-|k|^{-2} (x_1\cdot k)(x_2\cdot k)
  \big)\Big),\label{UniTra}
  \end{align}
   where $E_{\perp}(r)$ is the quantized transverse electric field.
   Note that the long range part of $V_R$ is cancelled. From the $4$-th order  perturbation in $-e x_1\cdot E_{\perp}(0)-ex_2 \cdot E_{\perp}(r)$
    one obtains the Casimir-Polder result.

    To include magnetic effects one proceeds to the next order of  the multipole expansion and defines
    \begin{align}
    H_{\mathrm{mul}}=&\frac{1}{2} \big(p_1-eA(0)-e x_1\cdot \nabla_r A(0)\big)^2-e^2 V(x_1)
    \nonumber \\
    &+\frac{1}{2} \big(p_2-e A(r)-e x_2\cdot \nabla_r A(r)  \big)^2-e^2 V(x_2)
\nonumber \\ &+H_{\mathrm{f}}+ e^2 V_R(x_1, x_2),
  \end{align}
    which,
   as before, is unitarily transformed  to
   \begin{align}
  U H_{\mathrm{mul}}U^{-1}=&\frac{1}{2} \big(p_1-e x_1\cdot \nabla_r A(0)\big)^2-e^2 V(x_1)
    \nonumber \\
    &+\frac{1}{2} \big(p_2-e x_2\cdot \nabla_r A(r)  \big)^2-e^2 V(x_2)
\nonumber \\ &+H_{\mathrm{f}}
-e x_1\cdot E_{\perp}(0)-e x_2\cdot E_{\perp}(r)+e^2 V_R^{\mathrm{cor}}(x_1, x_2), \label{UniTra2}
   \end{align}
   where $V_R^{\mathrm{cor}}$ is the interaction potential  from (\ref{UniTra}). Note that $\nabla_x\wedge (x\cdot \nabla_r A(r))=B(r)=\nabla_r\wedge A(r)$.
   Thus  the hamiltonian (\ref{UniTra2}) clearly displays the fluctuating  electric  and magnetic  fields. Expanding in the terms proportional to
   $\nabla_r A(0), \\
   \nabla_r A(r), E_{\perp}(0), E_{\perp}(r)$ to $4$-th order  leads to an energy with a large $R$ asymptotics  in agreement  with (\ref{SumAll}).
   In  spirit  the authors of \cite{FS0, FS, Boyer} make  a further gauge transformation through  the unitary
   $
   \exp[ -\mathrm{i} (x_1\cdot \nabla_r (x_1\cdot A(0))+x_2 \cdot \nabla_r(x_2\cdot A(r)))].
   $
   Thereby $x_1\cdot \nabla_rA(0)$ is transformed  to $\frac{1}{2} x_1\wedge B(0)$ and correspondingly  for  $x_2\cdot \nabla_r A(0)$.
   In addition there are terms coming from the shifting  of $E_{\perp}$. In the $4$-th order perturbation only  the former terms  are  taken into account. More
   precisely
    the term
   $\langle \psi_{\mathrm{at}}\otimes \Omega, (x_1\wedge B(0))^2 P H^{-1}  P (x_2\wedge B(r))^2\psi_{\mathrm{at}}\otimes \Omega\rangle$, which
   yields indeed the integrand (\ref{SS1}) and thus $23/16\pi$ for  the prefactor. Here $P={\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l-|\psi \rangle\langle \psi|$ and $\Omega$  is the  Fock vacuum.
    For the cross-term they use
   $
   \langle \psi_{\mathrm{at}}\otimes \Omega, x_1\cdot E_{\perp}(0) P H^{-1} P  x_1\cdot E_{\perp}(0) P H^{-1} P  (x_2\wedge B(r))^2\psi_{\mathrm{at}}\otimes \Omega\rangle$,
    which  yields the integral (\ref{SS2}) with the term $(1-(\hat{k}_1\cdot \hat{k}_2)^2)$ omitted   and  thus $7/8\pi$ for the prefactor of $\alpha_{\mathrm{E, at}} \alpha_{\mathrm{M, at}}$.

   \section{The  dimensionless strength}\setcounter{equation}{0}
\label{DL}

   We restore the physical units in $H$ of (\ref{1PL}). Then
   \begin{align}
   H=\frac{1}{2m} \big(p-\tfrac{e}{c} A(x)\big)^2-e^2 V(x)+H_{\mathrm{f}}
   \end{align}
   with $p=-\mathrm{i} \hbar \nabla_x$,
   \begin{align}
   H_{\mathrm{f}}=\sum_{\lambda=1,2} \int_{\mathbb{R}^3} \mathrm{d} k\, \hbar c  |k| a(k, \lambda)^* a(k, \lambda),
   \end{align}
   and
   \begin{align}
    A(x)=\sum_{\lambda=1,2} \int_{\mathbb{R}^3} \mathrm{d} k \sqrt{c/2|k|} \hat{\varphi}(k)\varepsilon(k, \lambda) \big(\mathrm{e}^{\mathrm{i} k\cdot x}a(k, \lambda)+\mathrm{e}^{-\mathrm{i} k\cdot x}a(k, \lambda)^*\big) .
    \end{align}
   $H$ is   transformed to atomic units through the canonical transformation $U$ defined as
   \begin{align}
   U^* a(k, \lambda)U&=(\alpha^{-2} \lambda_{\mathrm{c}})^{3/2} a(\alpha^{-2} \lambda_{\mathrm{c}} k, \lambda),
   \nonumber \\
   U^* x U& =\alpha^{-1}r_{\mathrm{B}}x, \ \ \ U^* pU =\alpha r _{\mathrm{B}}^{-1} p.
   \end{align}
   Then
   \begin{align}
   U^* H U&= \alpha^2 m c^2 \Big(
   \frac{1}{2} \big(
   -\mathrm{i} \nabla_x - \sqrt{4\pi} \alpha^{3/2} \tilde{A}(\alpha x)
   \big)^2   -  \tilde{V}(x) +\tilde{H}_{\mathrm{f}}\Big)
   \nonumber \\
   &= \alpha^2 m c^2 \tilde{H},
   \end{align}
   where
   \begin{align}
   \tilde{H}_{\mathrm{f}}= \sum_{\lambda=1,2} \int_{\mathbb{R}^3} \mathrm{d} k\,   |k| a(k, \lambda)^* a(k, \lambda) \label{THf}
   \end{align}
    and
    \begin{align}
    \tilde{V}(x)&=4\pi \int_{\mathbb{R}^3}\mathrm{d} k\,   \hat{\varphi}(\alpha^2 \lambda_{\mathrm{c}}^{-1} k)^2 |k|^{-2} \mathrm{e}^{-\mathrm{i} k\cdot x},\label{TV}
    \\
         \tilde{A}(x)&=\sum_{\lambda=1,2}\int_{\mathbb{R}^3} \mathrm{d} k\,    \frac{\hat{\varphi}(\alpha^2 \lambda_{\mathrm{c}}^{-1} k)     }{\sqrt{2|k|}}\varepsilon(k, \lambda) \big(\mathrm{e}^{\mathrm{i} k\cdot x}a(k, \lambda)+\mathrm{e}^{-\mathrm{i} k\cdot x}a(k, \lambda)^*\big) .    \label{TA}
          \end{align}
   Energy is now in units of $\alpha^2 m c^2$  and distances are in units of the Bohr radius $r_{\mathrm{B}}$.

   From (\ref{THf}), (\ref{TA}) the photon propagator in atomic units is obtained as

   \begin{align}
\tilde{W}_R(q,t)=4\pi
\int_{\mathbb{R}^3}\mathrm{d}k\,
|\hat{\varphi}(\alpha^2 \lambda_{\mathrm{c}}^{-1}k)|^2\frac{1}{2|k|}
({\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l-| \hat{k}\rangle\langle\hat{k}|)\, \mathrm{e}^{-\alpha^{-1} |k||t|}\mathrm{e}^{\mathrm{i} k\cdot
  r}\mathrm{e}^{-\mathrm{i} k\cdot q}.
   \end{align}
   Comparing with (\ref{Propagator}) this amounts to replacing  $e^2$ by $4\pi$ and $t$ by $t/\alpha$. One merely has  to follow through
    this change in the computation of Section \ref{Computation}. The final result is
    \begin{align}
    E(R)-2E=-\kappa (\alpha, \lambda_{\mathrm{c}} \Lambda) \alpha m c^2 (R/r_{\mathrm{B}})^{-7}
    \end{align}
    for large $R$ with  the strength $\kappa$ obtained as
    \begin{align}
    \kappa (\alpha, \lambda_{\mathrm{c}}\Lambda)= \frac{4}{\pi} \Big(\frac{23}{16}\tilde{\alpha}_{\mathrm{E}}^2
  +26\tilde{\alpha}_{\mathrm{E}}\tilde{\alpha}_{\mathrm{M}}
  +64\tilde{\alpha}_{\mathrm{M}}^2\Big)    .
    \end{align}
    Here,  the dimensionless electric and magnetic dipole  moments are
    \begin{align}
    \tilde{\alpha}_{\mathrm{E}}=2 \big(\tfrac{1}{3} \langle \psi, x\cdot (\tilde{H}-E)^{-1} x\psi\rangle\big)
    \end{align}
    and
    \begin{align}
    \tilde{\alpha}_{\mathrm{M}} =-\alpha^2 \frac{1}{4} \big(\tfrac{1}{3} \langle \psi, x^2\psi\rangle\big)
    \end{align}
    with $\psi$  the ground state of $\tilde{H}$, $\tilde{H}\psi=E\psi$. The ultraviolet cut-off is implemented by  replacing $\hat{\varphi}(k)$ by
     the form factor $\hat{\varphi}_1(k/ \Lambda )$: $\hat{\varphi}_1$ decreases rapidly at $|k|=1$ from $(2\pi)^{-3/2}$ to $0$.

    Following Bethe \cite{Bethe} a physically reasonable choice  is $\lambda_{\mathrm{c}} \Lambda=1$.  One would like to remove the ultraviolet cut-off
    through $\Lambda\to \infty$.
    But  this limit  is not well understood. In any case, the bare mass $m$ would  have to  substituted by the renormalized mass.  For fixed $\Lambda,\
      \lambda_{\mathrm{c}}\Lambda =1$, in the limit $\alpha\to 0$,
    $\tilde{H}$ decouples from the radiation field and $\kappa(\alpha,
    \lambda_{\mathrm{c}} \Lambda)$ tends to the strength obtained by  Casimir  and Polder. Then $\tilde{\alpha}_{\mathrm{E}}\simeq 9/4$ and $\tilde{\alpha}_{\mathrm{M}}\simeq -\alpha^2$ with  quality for the strict Coulomb potential.
      For a systematic  expansion in $\alpha$ one would need the correction of
    order $\alpha$ to $\tilde{\alpha}_{\mathrm{E}}$ and to  the bare mass $m$.

\section{Conclusions and Outlook}\setcounter{equation}{0}
The reader may wonder  how much  is still missing to a complete proof.
For  the full model, a central difficulty is that the ground state process $q(t)$ is not so well under control. Since $H$ has no spectral gap, the two-point correlation has a slow decay,
presumably as   $\mathbb{E}[ q(t)\cdot q(0)] \simeq |t|^{-4}$ for large $t$. This means that  higher order cumulants are difficult
to control. In  fact, we cannot even prove  that $\alpha_{\mathrm{E}}<\infty$.

 From the statistical mechanics point of view an interesting case  would be to replace the ground state process $q(t)$  by $q_{\mathrm{at}}(t)$, namely
  the one  governed by $H_{\mathrm{at}}$. This process is
  exponentially  mixing which should  help  in controlling  the error
  term.
  On the other hand, the double stochastic integral in the action
  causes extra difficulties.
  Unfortunately there is no obvious hamiltonian corresponding to this approximation.

  A further variant   would be  the dipole approximation of equation (\ref{Dipole}).
   In the path integral   (\ref{PathBindingEr})  this corresponds to replacing $W_R(x, t)$ by $W(0, t)$.
    The effective action is quadratic in $\dot{q}_j(t)$ and the partial time integration can be easily implemented.
    It results in a diagonal  term which  cancels the slowly decaying  part of $V_R$ and the remainder  action is given by
   $\int \mathrm{d} s \mathrm{d} t q(s)\cdot \ddot{W}_R(0, s-t) q(t)$. Since $\ddot{W}_R(0, t)\simeq t^{-4}$
     for large $t$, the difficulties
mentioned above remain. To have an exactly solvable  Gaussian  model,
on top one would have to use the  quadratic approximation for
$V$ and $V_R$.

For ground state properties a powerful method is the Feshbach
projection together with  a successive integration over high $k$-modes
of the  radiation
field \cite{BCFS, GH}. It would be interesting to understand whether this
technique could be  used  for  a  rigorous  control on the van der
Waals potential.

\begin{flushleft}
{\bf Acknowledgement:} We are grateful to P. Milonni for very useful
hints on the literature.
\end{flushleft}

redpajama_set_name: RedPajamaArXiv


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stlimc.org was taken down because of limited resources and serious spam problems. working to bring back a static version for archive/historical purposes when possible.
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According to state and federal court filings, an Arizona real estate developer used two nursing homes in Keokuk, Iowa to launder $3 million in Medicaid payments. While the developer was collecting this money, residents complained about neglect, including being left to lay in their own urine.
At one of the facilities, residents sometimes waited 30 or 40 minutes for their call lights to be answered by a staff member. Last fall, after a contractor was sued for allegedly installing a call-light system that was defective, an official from the facility said residents were put in great danger because of faulty call lights.
The developer has now been banned from managing or setting foot in either nursing home. The developer also owes $2 million to the people of Iowa, yet he lives in a home in Paradise Valley, Arizona that is worth $1.3 million. Paradise Valley is one of the wealthiest suburbs of Phoenix.
The Iowa attorney general sued the developer three years ago for $17.4 million. The attorney general alleged the developer's two nursing homes were stealing money from taxpayers through fake Medicaid claims.
The attorney general claims the developer has been committing Medicaid fraud since January 2010. The state alleges the developer promised the director of nursing at one facility that he would give her $204 per month to make payments on her new camper trailer if the facility passed inspection.
The facility passed inspection and the developer made the payments. However, he then filed a Medicaid report, claiming the money was spent on supportive care for residents. This triggered an increase in Medicaid payments to the facility. Just one month before this, the administrator of the nursing home said she was frustrated with cash shortfalls at the facility.
The developer is also being prosecuted in federal court – the Department of Justice accused him of bank fraud for obtaining federally backed loans for one health care facility and spending the money elsewhere.
He told an Iowa judge he could not pay a $2,000 court-ordered penalty because he lacked steady income or employment. He also said he was renting his house in Arizona using money loaned to him by his family. However, the developed was able to pay $1 million toward a $3 million settlement with the attorney general. Last year, a bank in Iowa said a $2.6 million debt with them had been paid.
If your elderly loved one suffered abuse or neglect at a nursing facility, you may be entitled to compensation. Contact a trusted attorney to discuss your legal options and learn how he or she can help you pursue justice.
The Wisconsin nursing home abuse attorneys at PKSD can review your situation in a free, no obligation legal consultation. You will not owe us for legal fees unless you receive compensation.


text:
API Documentation
=================

.. toctree::
   :maxdepth: 4

   rez
   rezplugins

redpajama_set_name: RedPajamaGithub


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Ideally situated in the heart of the Village close to the main swimming pool, enjoy this bar?s modern setting to recharge your batteries during the day and relax in the evening with family or friends.
Fancy a coffee after your round?


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text:
Q: Magento 2 Vue storefront [search] Can not connect the vue-storefront-api / ElasticSearch instance! While running npm run dev then after I am getting errors like
    [search] Can not connect the vue-storefront-api / ElasticSearch instance! Error: FetchError in request to ES: FetchError: request to http://localhost:8080/api/catalog/vue_storefront_catalog/attribute/_search?_source_include=activity%2Cattribute_code%2Cattribute_id%2Cdefault_frontend_label%2Cdefault_value%2Centity_type_id%2Cfrontend_input%2Cfrontend_label%2Cid%2Cis_user_defined%2Cis_visible_on_front%2Cis_visible%2Cis_comparable%2Coptions%2Ctier_prices&from=0&request=%7B%22query%22%3A%7B%22bool%22%3A%7B%22filter%22%3A%7B%22terms%22%3A%7B%22attribute_code%22%3A%5B%22color%22%2C%22erin_recommends%22%2C%22price%22%2C%22size%22%2C%22activity%22%2C%22final_price%22%2C%22id%22%2C%22image%22%2C%22name%22%2C%22new%22%2C%22original_price%22%2C%22original_price_incl_tax%22%2C%22price_incl_tax%22%2C%22product_links%22%2C%22sale%22%2C%22sku%22%2C%22special_price%22%2C%22special_price_incl_tax%22%2C%22special_from_date%22%2C%22special_to_date%22%2C%22status%22%2C%22tax_class_id%22%2C%22tier_prices%22%2C%22type_id%22%2C%22url_path%22%2C%22url_key%22%5D%7D%7D%7D%7D%7D&size=150&sort= failed, reason: connect ECONNREFUSED 127.0.0.1:8080
    at vsf-search-adapter-api-searchAdapter.server-bundle.js:155:39
    at processTicksAndRejections (internal/process/task_queues.js:88:5)
(node:11095) UnhandledPromiseRejectionWarning: Error: FetchError in request to ES: FetchError: request to http://localhost:8080/api/catalog/vue_storefront_catalog/attribute/_search?_source_include=activity%2Cattribute_code%2Cattribute_id%2Cdefault_frontend_label%2Cdefault_value%2Centity_type_id%2Cfrontend_input%2Cfrontend_label%2Cid%2Cis_user_defined%2Cis_visible_on_front%2Cis_visible%2Cis_comparable%2Coptions%2Ctier_prices&from=0&request=%7B%22query%22%3A%7B%22bool%22%3A%7B%22filter%22%3A%7B%22terms%22%3A%7B%22attribute_code%22%3A%5B%22color%22%2C%22erin_recommends%22%2C%22price%22%2C%22size%22%2C%22activity%22%2C%22final_price%22%2C%22id%22%2C%22image%22%2C%22name%22%2C%22new%22%2C%22original_price%22%2C%22original_price_incl_tax%22%2C%22price_incl_tax%22%2C%22product_links%22%2C%22sale%22%2C%22sku%22%2C%22special_price%22%2C%22special_price_incl_tax%22%2C%22special_from_date%22%2C%22special_to_date%22%2C%22status%22%2C%22tax_class_id%22%2C%22tier_prices%22%2C%22type_id%22%2C%22url_path%22%2C%22url_key%22%5D%7D%7D%7D%7D%7D&size=150&sort= failed, reason: connect ECONNREFUSED 127.0.0.1:8080
(node:11095) UnhandledPromiseRejectionWarning: Unhandled promise rejection. This error originated either by throwing inside of an async function without a catch block, or by rejecting a promise which was not handled with .catch(). (rejection id: 2646)

Does anyone have any idea how can I fix it?

A: It is possible you are running old elastic-search version in vuestorefront-api . generally vue-storefront-api provide two elasticsearch version based configuration. you can choose your compatible elatic-search version in docker-compose.yml file.
just replace this configuration
**es1:**
    container_name: elasticsearch
    build: **docker/elasticsearch/**
    ulimits:
      memlock:
        soft: -1
        hard: -1
    volumes:
      - **./docker/elasticsearch/config/elasticsearch.yml:/usr/share/elasticsearch/config/elasticsearch.yml:ro**
    ports:
      - '9200:9200'
      - '9300:9300'
    environment:
      - discovery.type=single-node
      - cluster.name=docker-cluster
      - bootstrap.memory_lock=true
      - "ES_JAVA_OPTS=-Xmx512m -Xms512m"

  kibana:
    build: docker/kibana/
    volumes:
      - ./docker/kibana/config/:/usr/share/kibana/config:ro
    ports:
      - '5601:5601'
    depends_on:
      - **es1**

with latest elastic search version
es7:
    container_name: elasticsearch
    build: docker/elasticsearch7/
    ulimits:
      memlock:
        soft: -1
        hard: -1
    volumes:
      - ./docker/elasticsearch7/config/elasticsearch.yml:/usr/share/elasticsearch/config/elasticsearch.yml:ro
    ports:
      - '9200:9200'
      - '9300:9300'
    environment:
      - discovery.type=single-node
      - cluster.name=docker-cluster
      - bootstrap.memory_lock=true
      - "ES_JAVA_OPTS=-Xmx512m -Xms512m"

  kibana:
    build: docker/kibana/
    volumes:
      - ./docker/kibana/config/:/usr/share/kibana/config:ro
    ports:
      - '5601:5601'
    depends_on:
      - es7

i hope this will solve your problem.

redpajama_set_name: RedPajamaStackExchange


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text:
Q: How to receive a file from c # via ajax I am trying to send a file through ajax and it is not possible to save it, someone to help me, I have already tried several serialized but not one that works. acontinuauacion I show you the client side code and the server side, it would be useful if you help me if I'm doing it right or there is another way with [webmethod] to do it

$("#btnCargar").on("click", function() {
  var filese = $("#fileComparendo")[0].files[0];
  var formData = new FormData();
  formData.append("archivo", filese);

  $.ajax({
    type: "POST",
    url: "<%=ResolveUrl("addcomp.aspx/prueba")%>",
    data: formData,
    contentType: false,
    processData: false,

    success: function(data) {
      alert(data);
      console.log("CallWM" + data.d);
      alert(data.d);
    },
    error: function(XMLHttpRequest, textStatus, errorThrown) {
      var err = eval("(" + XMLHttpRequest.responseText + ")");
      alert(err.Message)
      console.log("Ajax Error!");
    }
  });

});

    <input type="text" id="txtPrueba" class="form-contrav" />
    <input type="file" class="file-name" id="fileComparendo" />
    <input type="button" class="btn btn-danger" value="Cargar" id="btnCargar" />

[WebMethod]
public static  string prueba(HttpContext archivo)
{
   // var recipient = Request["Recipient"];
    var re =  HttpContext.Current.Request.Form["archivo"];
    archivo.Response.ContentType = "text/plain";

    string dirFullPath = HttpContext.Current.Server.MapPath("~/files/");
    string[] files;
    int numFiles;
    files = System.IO.Directory.GetFiles(dirFullPath);
    numFiles = files.Length;
    numFiles = numFiles + 1;
    string str_image = "";
    foreach (string s in archivo.Request.Files)
    {
        HttpPostedFile file = archivo.Request.Files[s];
        string fileName = file.FileName;
        string fileExtension = file.ContentType;

        if (!string.IsNullOrEmpty(fileName))
        {
            fileExtension = Path.GetExtension(fileName);
            str_image = "MyPHOTO_" + numFiles.ToString() + fileExtension;
            string pathToSave_100 = HttpContext.Current.Server.MapPath("~/files/") + str_image;
            file.SaveAs(pathToSave_100);
        }
    }
    archivo.Response.Write(str_image);
    return "hola";
}

A: [WebMethod] can only accept a content type of application/json, but you can use an .ashx: https://stackoverflow.com/a/26273565/1429439
Add a new Generic Handler called Upload.ashx, then replace ProcessRequest() with:
public void ProcessRequest(HttpContext archivo)
        {
            var re = HttpContext.Current.Request.Form["archivo"];
            archivo.Response.ContentType = "text/plain";

            string dirFullPath = HttpContext.Current.Server.MapPath("~/files/");
            string[] files;
            int numFiles;
            files = System.IO.Directory.GetFiles(dirFullPath);
            numFiles = files.Length;
            numFiles = numFiles + 1;
            string str_image = "";
            foreach (string s in archivo.Request.Files)
            {
                HttpPostedFile file = archivo.Request.Files[s];
                string fileName = file.FileName;
                string fileExtension = file.ContentType;

                if (!string.IsNullOrEmpty(fileName))
                {
                    fileExtension = System.IO.Path.GetExtension(fileName);
                    str_image = "MyPHOTO_" + numFiles.ToString() + fileExtension;
                    string pathToSave_100 = HttpContext.Current.Server.MapPath("~/files/") + str_image;
                    file.SaveAs(pathToSave_100);
                }
            }
            archivo.Response.Write(str_image);
        }

Update the url in $.ajax() to point to Upload.ashx with:
url: "<%=ResolveUrl("Upload.ashx")%>",


text:
Canceled Flights: How To Handle Flight Cancellations As COVID-Linked Crew Shortages Leave Thousands Stranded
NEW YORK – An increase in cancellations during peak holiday travel season has left thousands of passengers across the country stranded in airport lounges as they attempt to return home.
Airlines officials said the recent spike in omicron COVID-19 cases has left flight crews immobilized and, as a result, they don't have enough people to make their scheduled flights.
Although such a situation is difficult to predict, travel experts say affected passengers still have options to reach their destination or, in the worst case, end their extended stay with as little hassle as possible.
"It's frustrating for travelers, but there's not much they can do about it," Brett Snyder, president of Cranky Concierge travel planning, told ABC News.
MORE: Omicron, storms disrupt air travel for 5th day in a row
Snyder said the most important thing for passengers to do is to be aware of their airline and the status of their flight. Affected airlines have updated their information to customers and airport agents as soon as they know a cancellation is imminent, but if any of them are at the top of their flight early, they have more time to determine its next steps, Snyder said.
"Speed is your friend," he said. "If you could skip the wait at the airport and make your arrangements earlier, you would be better off."
Scott Keyes, the founder of Scott's Cheap Flights, told ABC News that if a flight is canceled after you arrive at the airport, there are plenty of options for getting another flight quickly.
Keyes said it was important to get to the ticket agent queue quickly, as available seats are allocated on a first-come, first-served basis. He advised passengers to also contact their airline's customer service while in line, so they can arrange another flight.
Keyes said some stranded passengers may have success calling the airline's international call centers instead of the United States.
"You can call any of these alternative centers and connect with an agent who will handle your reservation the same way an American agent does," he told ABC News.
MORE: United States Should Consider COVID Vaccine Mandate for Domestic Air Travel, Says Fauci
Keyes noted that the current number of cancellations by airlines is around 5% of their flights, so there's a good chance a stranded passenger could get another flight to their destination soon, even if that includes connecting flights. He added that passenger cancellations, like those of people who need to be quarantined, also open the possibility of waiting stations.
Snyder said major airlines have agreements with each other and may offer passengers flights to destinations with another carrier.
"You can always get your money back and buy another ticket with that money," he said. "Just know what your airline's refund policies are, as some may not refund you for a few hours after cancellation."
In the worst-case scenario, where a passenger has to wait a long time for the next flight, travel experts say there are options to accommodate a passenger's extended stay.
Most airlines will offer a voucher for food and hotel, and others will request a voucher from a passenger whose flight is canceled. Keyes stressed that not all airlines will offer it voluntarily.
"It never hurts to ask. Make sure you make this request proactively," he said. "They are not required by federal law to house you, but most of the time they will be able to."
Keyes said some credit card companies offer travel protections for canceled flights that have been booked on the card.
"Being able to get reimbursed for that hotel room is more preferable than paying out of pocket," he said.
Snyder said air passengers should also seek other forms of credit and reimbursement from airlines. In the past, during massive cancellation events, airlines offered credit for a future flight or loyalty miles.
"We have seen airlines do this when they have abnormal collapses," he said.
Snyder and Keyes both said there would likely be more cancellations due to omicron-related staff shortages as the recent surge continues, and passengers preparing to fly for next month should be prepared for this to happen.
Both stressed that this is an uncertain time and that the best way to overcome any sudden cancellations is to remain calm and calm when speaking with airline agents or other staff.
"These agents have dealt with thousands and thousands of disgruntled and angry travelers and they have done so ruthlessly throughout the pandemic," Keyes said. "It's not their fault that the flight is canceled and if you can treat them with kindness, it will often go too far in their willingness to help you."
ABC News' Mina Kaji contributed to this report.
Wizz Air is looking for new crew members for EX-YU bases
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One more worry for Vikings fans: Sunday officials squad
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redpajama_set_name: RedPajamaCommonCrawl


Christmas lights. Christmas Day family hike. Christmas cookies. Christmas games. And Christmas tree photos. Just a few photos from my Christmas to yours. Okay, that doesn't really work, but you get what I mean. I hope you all had a wonderful Christmas and got lots of good presents, good food, and good time with loved ones.
Did you get any good presents? Any surprises? Any cute clothes? If you participated in The Great Christmas Exchange, how'd that go?
It really was. :) I hope you had a good one, too!
Merry Christmas! Looks like a wonderful time!
It really was a good time. :) Merry Christmas and a Happy New Year to you, too!
rock'em sock'em robots! CLASSIC! i hope your holiday was simply wonderful!
We actually got Rock 'em Sock 'em this Christmas from my uncle and it's definitely a favorite game of ours now. :) Hope you had a Merry Christmas and have a Happy New Year, friend!!!
These are such incredible photos (as always) AND AHHH THE FOGGY TREES + FAMILY + LIGHTS I love them.
Stop it yourself! Seriously, every time I read your compliments I start blushing or something. Seriously. :) But in all seriousness, thank you. From a talented photographer friend, that means a lot.
Looks like you had a wonderful Christmas!!! Your pictures are fabulous!
It was a wonderful Christmas--I hope yours was, too! And thanks! I love looking at everyone's Christmas photos so I thought I had to add mine, too. :) Happy New Year!!
Your photos are just beautiful!!! My goodness there is a DD party game!!!
Especially the one of your dog, too cute!


text:
Clip Time: ABC's "Revenge For Real: Murder In The Hamptons"
By Paul A. Johnson Mar 7, 2012, 1:30pm EST
video platformvideo managementvideo solutionsvideo player
(Best viewed in full screen mode.)
The first clip from ABC's oddball Revenge spinoff, "Revenge for Real: Murder in the Hamptons" is now available online and we have to admit that convicted murderer Danny Pelosi gives great soundbite. Some choice quotes from the 4-minute clip:
"I was a man with a tool belt. I gave her what the rich man couldn't."
"I thought I hit lotto."
"There is no motive to take a father away from his kids."
"Generosa did not kill Ted Ammon. She had him killed."
"I'm not guilty of murder. I'm guilty of being an idiot."
The program promises to follow up on new evidence provided by Mr. Pelosi that he says proves his innocence. What do you think they'll find?
Danny Pelosi, Convicted Hamptons Murderer, Claims Wife Wanted 'Revenge' on Multimillionaire Husband [ABC News]
Robert Downey Jr. is now the owner of the iconic East Hampton windmill cottage Keep in mind—it's never been an actual windmill, but rather serves as a copy of an original windmill in East Hampton.
Ira Rennert has to pay $213.2M for looting company funds that paid for his Sagaponack estate Valued at $248M and sitting at 73,000-square-feet, the property is one of the largest private homes in the United States.
Fly Over The Country's Largest Home Like Never Before


text:
ACC urges drivers to be sincere in the fight against corruption
HomeAYV NewsACC urges drivers to be sincere in the fight against corruption
Manager Kamara disclosed that Sierra Leone is losing huge amount of money as a result of corruption which is sometimes perpetuated by some drivers; who connive with public officers to deprive the state of much needed resources that would have been used to improve the social wellbeing of the people. He expressed his dissatisfaction on the attitude of some drivers who have chosen to remain lawless and put the good image of the driving profession in a negative spot light. The Regional Manager urged the executive of the union to do more in order to instill discipline and protect the interest of their members. Manager Kamara warned drivers to desist from offering bribery to public officers; as such acts constitute an offence and punishable by law. He implored them to be law abiding and ensure that their vehicles are road worthy and licenses up-to-date.
In his statement on the role and benefits of drivers in the fight against corruption, ACC's Senior Public Education Officer, David Kanekey Conteh said "Sierra Leone will not progress if we allow corruption to flourish." Conteh noted that the country should not be left in the hands of the corrupt as its effects are suffered by the majority of Sierra Leoneans. He notified the drivers of their roles in the fight against corruption such as: adherence to law, desist from obtaining fake licenses, demanding receipt after every payment at the Sierra Leone Road Safety Authority offices, abstaining from paying bribes and booking fees, report cases of soliciting to ACC and avoid reckless driving. Conteh highlighted some of the benefits they will gain if corruption is collectively kicked out of the country such as: impartial trial, construction of good roads, importation of quality spare parts, improve health facility, reduction in road fatality and better standard of living. He explained the various channels of reporting corruption and read out ACC toll free lines and those of the Regional Manager. The Senior Public Education Officer entreated the union executive to imbibe the values of transparency and accountability in the use of union funds.
ACC Public Education Officer, Abdulai Saccoh, dilating on the purpose of the meeting and provided an update on ACC's activities. He described the engagement as a demonstration of ACC's commitment to strengthen the already existing partnership in the anti-graft campaign. Saccoh stated that the meeting was intended to remind drivers of their roles in the anti-graft campaign, how to report, benefits of reporting graft and to solicit their support in the fight against corruption. "Corruption will not take place when there is no one to offer the bribe" he said. The Public Education Officer advised his audience to work within the ambit of the law as it is a respecter of no one.
Saccoh informed his audience of the planned launch of the Pay No Bribe campaign targeting: the SLP, Energy, Water Resources, Education and Health sectors; and which would be piloted in Bombali, Bo, Kenema and Western Area. He mentioned the conviction of a Traffic Warden Victorino Fayia Joseph who was sentenced to a thirty million Leones or a three year jail term and recent indictment of two officials of the Ministry of Agriculture, Forestry and Food Security (MAFFS) namely Mark Ivan Babatunde Dixon and Mohamed Alimamy Mansaray, both Store Keepers of that ministry for allegedly misappropriating 920 and 804 bags of fertilizers meant for farmers.
The meeting was chaired by ACC Senior Public Education Officer Al-Hassan Sesay; statements were also made by the Chairman of Drivers Union Bombali Alhaji Umaru S. Kamara and Chairman Truck Unit Sulaiman M. Suma. The presentation of education, information and communication materials formed the high point of the meeting.


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729 products Tags: Agitator Slurry Tank Mix Gold Agitator Slurry Tank Mix Mining Gold In this stage, the ore slurry flow to this tank, the chemicals add in this stage, of impeller preparing for next step that mix the ore slurry and reagents.


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Групова збагачувальна фабрика «Краснолиманська» — збудована за проектом «Дніпродіпрошахт» у 1960 році з виробничою потужністю 1600 тис. тон на рік і призначенням — збагачення газового вугілля для коксування. У 1978 році фабрика зазнала докорінної реконструкції за проектом «Дондіпрошахту» з спорудженням нового головного корпусу, вуглеприйому, сушильного відділення і т.ін. Виробнича потужність була підвищена до 2900 тис. тон. Нова технологія передбачала збагачення класу 13-150 мм у важкосередовищних сепараторах з магнетитовою суспензією, відсадку класу 0,5-13 мм, флотацію шламу у дві стадії — зважаючи на дуже важку його флотовність. Протягом певного часу фабрика випускала концентрати марки «Г» для коксування та енергетики і крупно-сортове вугілля для комунально-побутових потреб. В останні роки фабрика переробляє жирне коксівне вугілля місцевого видобутку (без привізного) і випускає лише концентрат на коксування, виділяючи деяку кількість крупного сорту на побутове споживання. При фабриці побудовано установку для виїмки і збагачення шламу, що міститься у мулонакопичувачі.

Місце знаходження: селище Родинське, Донецька обл., залізнична станція Легендарна.

Джерела

Збагачувальні фабрики України
Родинське
Підприємства, засновані 1960

redpajama_set_name: RedPajamaWikipedia


PROVIDENCE — Join Dr. Scott Hahn and the St. Paul Center at McVinney Auditorium, 43 Dave Gavitt Way, this Saturday, April 7, 2018 for a powerful morning presentation on two of the most pivotal aspects of Catholic life: Marriage and Mass.
Guests will also have the opportunity to learn more about the mission of the St. Paul Center.
Dr. Hahn is the bestselling author of more than 40 titles. He holds the Father Michael Scanlan Chair of Biblical Theology and New Evangelization at the Franciscan University of Steubenville, where he has taught since 1990. As founder and president of the St. Paul Center for Biblical Theology, he is editor-in-chief of Emmaus Road Publishing. He also serves as the McEssy Distinguished Visiting Professor of Biblical Theology at Mundelein Seminary. Dr. Hahn has six children and 15 grandchildren. For more information, visit ScottHahn.org.
Register today to reserve your ticket for this premiere event by visiting www.mcvinneyauditorium.com.


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on Jan 26, 2018 Jan 26, 2018
Dirty Old Town was written in 1949 by Ewan MacColl & is about Salford, Lancashire, where he grew up. It was originally written as an interlude to cover a set change in a play he had written, but gained a life of it's own as a folk standard. The first recorded version of the song was made by Ewan in 1952. This is the 1956 version made by Ewan with Peggy Seeger, Alan Lomax & the Ramblers.
The song is probably most associated with The Dubliners, who recorded a version in 1968. This is a great video of them performing it in 1976 in Finland.
The Pogues have often performed it too. One of the first gigs I went to was to see The Pogues on their Rum Sodomy & Lash tour. I was 16 & it was one of the best gigs of my life.
Dirty Old Town is such a wonderful song & it travels well.
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Farming at the Van Loveren Vineyards began in 1937 and nearly 80 years ago, the first wine was produced by Hennie Retief snr and today, his grandson Philip Retief continues in the tradition of producing quality award-winning wines. Van Loveren is a family business and harmony and love is an important part of their success. The farm itself is an international tourist attraction and boasts a bistro, MTB and hiking trails, and of course wine tasting and pairing. They have a long list of awards and accolades with several gold medals, Platter stars, and Double Gold awards.