text:
\section{Introduction}
The~BCS theory\cite{bcs} proved extremely effective as a~theory of superconductivity.
The~key idea of this theory---the~attractive interaction which binds electrons into
Cooper pairs---has its essential origins in earlier conclusions drawn
Fr\"ohlich\cite{fr:1}, who performed a~unitary transformation~$U$ of
the~electron-phonon Hamiltonian~$H_{e-ph}$. $U$~was adjusted so as to eliminate
the~electron-phonon interaction as far as possible and replace it by an~effective
interaction~$V_{eff}$ between electrons dressed in the phonon field.
$V_{eff}$~proved to be attractive for 1-electron energies close to~$\varepsilon_F$.
A~reduced form of~$V_{eff}$ was subsequently used by~Bardeen, Cooper and~Schrieffer
in their theory\cite{bcs}.

It is worth emphasizing that Fr\"ohlich's~transformation is not~strictly unitary,
because the~less significant terms of the~resulting expansion of~$UH_{e-ph}U^{-1}$
were discarded. Correctness of the~remaining terms, included into the~Hamiltonian of
a~superconductor, was confirmed by the~success of the~BCS theory.

Unfortunately, BCS~theory proved incapable of explaining superconductivity in
type-II superconductors, heavy fermions and high-$T_c$ superconductors~(HTSC).
The~search for an~alternative theory of superconductivity proceeds in various
directions and one of them exploits the~idea of extending BCS~theory by adding
to the~BCS Hamiltonian~$H_{BCS}$ further interactions. Rickayzen\cite{rick}
suggested incorporation of a~4-fermion interaction, motivating such choice by
analogies between theory of superconductivity and nuclear theory, where such
interactions had been considered. This idea was also mentioned
by~Volovik\cite{volovik}, but remained in the~realm of theoretical concepts until
the~late nineties, when Ma\'ckowiak and one of
authors~(PT)\cite{mt:3,mt:4,mt:2,mt:1,tm,t:1,t:2} (MT) proposed a
Hamiltonian~$H_{MT}=H_{BCS}+W+V_{MT}$, where
\begin{equation}
	H_{BCS}=T+V_{BCS}=\sum_{\we{k}\sigma}\xi_{\we{k}}n_{\we{k}\sigma}
	-|\Lambda|^{-1}\sum_{\we{k}\we{k'}}G_{\we{k}\we{k'}}a_{\we{k}+}^\ast
	a_{-\we{k}-}^\ast a_{-\we{k'}-}a_{\we{k'}+},
\label{H_BCS}
\end{equation}
\begin{equation}
	W=\sum_{\we{k}}\gamma_{\we{k}}n_{\we{k}+}n_{\we{k}-},
\label{W}
\end{equation}
\begin{equation}
	V_{MT}=|\Lambda|^{-1}\sum_{\we{k},\we{k'}}g_{\we{k}\we{k'}}
	a_{\we{k}-}^\ast a_{\we{k}+}^\ast a_{-\we{k}-}^\ast a_{-\we{k}+}^\ast
	a_{-\we{k'}+}a_{-\we{k'}-}a_{\we{k'}+}a_{\we{k'}-},
\label{V}
\end{equation}
$|\Lambda|$ in Eqs.~\eqref{H_BCS}--\eqref{V} denotes the system's volume,
whereas~$g_{\we{k}\we{k'}}$, $G_{\we{k}\we{k'}}$ are bounded functions
and~$\gamma_{\we{k}}$, $g_{\we{k}\we{k'}}$, $G_{\we{k}\we{k'}}$ are invariant under
time reversal~$\we{k}\to -\we{k}$ or~$\we{k'}\to-\we{k'}$.

Considerations which led to the~introduction of~$W$ were founded on the~analysis of
the~HTSC normal state. There are grounds to believe that this is not a~normal
Fermi~liquid state. The interaction~$W$, first added to~$H_{BCS}$
by~Czerwonko\cite{cz:1,cz:2}, guaranteed normal-state behaviour characteristic of
the~so-called statistical spin liquid, considered earlier by~Spa\l ek
and~\mbox{W\'ojcik}\cite{spawoj:1, spawoj:2}.

$W$ is a~2-electron interaction. This raises the~question, whether it can be
obtained by a~reduction procedure (different than the BCS~one) of the~interaction
derived by~Fr\"ohlich. It can be easily verified that this is impossible. More
precisely, for the~unique possible reduction of~momenta, the~coupling vanishes,
meaning that the~nature of~$W$ is not~phononic.

Introduction of the~4-fermion interaction~$V_{MT}$ was justified in
Ref.~\onlinecite{mt:3} by its possible role as an~attraction between pairs in~HTSC,
mediated by phonons or other quanta\cite{mt:1}. A~conjecture put forward in
Ref.~\onlinecite{mt:3} suggested also that~$V_{MT}$ could be expected to arise as
one of the~higher-order terms of Fr\"ohlich's~expansion of~$UH_{e-ph}U^{-1}$.
An~alternative justification was given in Ref.~\onlinecite{mt:1}, where~$V_{MT}$
was viewed as a~BCS-type interaction between quasi-particles of a~free gas
represented by~$W$ written in the~form~$\sum_{\we{k}}\gamma_{\we{k}}c_{\we{k}}^\ast
c_{\we{k}}$, with~$c_{\we{k}}=a_{\we{k}+}a_{\we{k}-}$. Both of these ideas
essentially exploit the~concept of phonon-type mediation of interactions.
The~significance of this mediation in HTSC was stressed
by~Wysoki\'nski~\cite{wysok:1}.

The~question of the~form of higher-order terms of Fr\"ohlich's expansion is
interesting itself, not only as providing a possible explanation of
the~MT extension, but, first of all, because these terms could throw some light on
further possible extensions of~$H_{BCS}$ related to effects of electron-lattice
interaction. Since Fermi-liquid theory will remain the~foundation of our formalism,
we shall focus our interest on effective electron interactions.

It is worth noting that the~possible presence of fermion quadruples in superconductors
and superfluids was considered in a~number of papers. Schneider and~Keller\cite{schn:1}
measured the~various characteristics of some cuprates and Chevrel-phase superconductors,
especially concentrating on the~relation between the~critical temperature and zero
temperature condensate density. They noticed that the~experimental data for
e.g.,~$\mathrm{Y}\mathrm{Ba}_2 \mathrm{Cu}_3 \mathrm{O}_{6.602}$ point to similarities
with the~behaviour of a~dilute Bose~gas. As a~result they suggested Bose condensation of
weakly interacting fermion pairs as a~mechanism of transition from normal to
superconducting state. Bunkov~et~al.\cite{bunkov} pointed to presence of fermion
quadruples in~$\mbox{}^3\mathrm{He}$. Their work was devoted to the~problem of
influence of spatial disorder on the~order parameter in
superfluid~$\mbox{}^3\mathrm{He}$. By resorting to the~work of~Volovik\cite{volovik}, they suggested that unusual spectra of~$\mbox{}^3\mathrm{He}$ in aerogel could be
explained by a~process in which impurities tend to destroy the~anisotropic
correlations of the~order parameter, while correlations of higher symmetry can survive
(e.g.,~four-particle correlations). Recently Schneider~et~al.\cite{schn:2} discovered
of~half-$h/2e$ magnetic flux in SQUIDs  fabricated of
bicrystalline~$\mathrm{Y}\mathrm{Ba}_2 \mathrm{Cu}_3 \mathrm{O}_{7-\delta}$ films.
This situation corresponds to the~presence of fermion quadruples in the~system.
Based on this observation, Aligia~et~al.\cite{aligia} investigated a~model of interface
between two superconductors, based on a~one~dimensional boson lattice model and
proposed formation of quartets of electrons.

In Section~\ref{2} higher order terms of the~expansion of
Fr\"ohlich's~transformation~$UH_{e-ph}U^{-1}$ are discussed qualitatively in order to
exhibit the~emerging structure. Owing to the~complexity of this procedure, only terms
proportional to the~third power of the~electron-phonon coupling are found in
Section~\ref{3}. The~resulting extended Fr\"ohlich Hamiltonian is transformed in the
next section by a~second Fr\"ohlich-type transformation, which produces 4-electron
terms. These are discussed in detail in Section~\ref{5}. In particular, a~reduction
similar to the~BCS one is performed, which yields an~interaction
of the~form~$V_{MT}$ in Eq.~\eqref{V}. The~expression for~$g_{\we{k}\we{k'}}$ is
derived. Detailed analysis of this expression is performed in the~next section, in
particular, we showed, by applying some approximations, that this expression is
negative i.e.,~$V_{MT}$ is attractive. The~original Fr\"ohlich's~transformation and
the~resulting terms, up to 6th power of the~coupling, are commented in
Section~\ref{6}. The final section contains a~discussion, summary and open questions.
\section{\label{2}Higher order terms of the Fr\"ohlich's transformation}
Following Fr\"ohlich\cite{fr:1} (see Appendix~\ref{a} for details), let us consider
the~electron-phonon Hamiltonian:
\begin{equation}
	H_{e-ph}=H_0+H_{int}=\sum_{\we{k}\sigma}\varepsilon_{\we{k}}
	a_{\we{k}\sigma}^\ast a_{\we{k}\sigma}+
	\sum_{\we{w}}\omega_{\we{w}}b_{\we{w}}^\ast b_{\we{w}}+
	i\sum_{\we{w}}D_{\we{w}}\left(
  	b_{\we{w}}\rho_{\we{w}}^\ast -b_{\we{w}}^\ast \rho_{\we{w}}\right),
\label{H_e-ph}
\end{equation}
where
\begin{equation}
  \rho_{\we{w}}=\sum_{\we{k}\sigma}a_{\we{k}-\we{w}\sigma}^\ast
  a_{\we{k}\sigma},
\label{rho}
\end{equation}
and~$a_{\we{k}\sigma}$ ($b_{\we{k}}$) are fermion (boson) operators.
The~coupling~$D_{\we{w}}$ will be assumed small and~$\hslash\equiv 1$.

Since the~interaction is spin-independent, the spin index will be suppressed.
Summation over electron momenta will include summation over spins.

Fr\"ohlich performed a~unitary transformation of~$H_{e-ph}$ in order to eliminate
(as far as possible) the~interaction term. The~transformed Hamiltonian is
\begin{equation}
	H=\mathrm{e}^{S^\ast}H_{\mathrm{e-ph}}\mathrm{e}^S=H_{\mathrm{e-ph}}-
	[S,H_{\mathrm{e-ph}}]+
        \frac{1}{2}[S,[S,H_{\mathrm{e-ph}}]]+\ldots\, ,
\label{h_trans}
\end{equation}
where
\begin{equation}
	S=\sum_{\we{q}}S_{\we{q}}=\sum_{\we{q}}\left( \gamma_{\we{q}}^\ast
	b_{\we{q}}^\ast -\gamma_{\we{q}} b_{\we{q}}\right)=-S^\ast,
\label{S}
\end{equation}
\begin{equation}
	\gamma_{\we{q}}=\sum_{\we{k}}\phi(\we{k},\we{q})a_{\we{k}}^\ast
	a_{\we{k}-\we{q}},
\label{gamma}
\end{equation}
and the~unknown function~$\phi(\we{k},\we{q}):
\mathbb{R}^3\times\mathbb{R}^3\to\mathbb{C}^1$ is adjusted to achieve the~cancellation.

Subsequently, a~term which is a~combination of products, each with $f$~fermion operators
and $b$~boson operators will be written as~$(f,b)$. Clearly, $f$~will always be even.
For example, $H_0$~consists of terms~$(2,0)$ and~$(0,2)$.

The~rhs of Eq.~\eqref{h_trans} expresses in terms of commutators~$[(f_1,b_1),(f_2,b_2)]$.
One easily finds that
\begin{equation}
	[(f_1,b_1),(f_2,b_2)]=[f_1,f_2]b_1 b_2 + f_2 f_1 [b_1,b_2]=[f_1,f_2]b_2 b_1 +
	f_1 f_2 [b_1,b_2].
\label{com}
\end{equation}
The necessary commutators~$[f_1,f_2]$, $[b_1,b_2]$ are given in Appendix~\ref{b}.

According to Eq.~\eqref{h_trans}, the~transformation can be performed, given commutators
of the~form occurring in Eq.~\eqref{com} with the~first argument equal~$S$. The latter
is a~$(2,1)$~expression, hence
\begin{equation}
  [S,(f,b)]=[(2,1),(f,b)]=(f,b+1)+(f+2,b-1),
\label{com_s}
\end{equation}
by virtue of Eqs.~\eqref{com},~\eqref{com_4f}. Clearly~$(f,b-1)=0$ for~$b=0$.

Based on this ground, Fr\"ohlich obtained the~transformed Hamiltonian
(see Appendix~\ref{a}):
\begin{equation*}
	H_F=\sum_{\we{k}}\varepsilon_{\we{k}}n_{\we{k}}-
	\frac{1}{2} \sum_{\we{k}\we{q}\we{w}}
	\frac{D_{\we{w}}^2 \left( 1+\Delta\left(\we{k},\we{w}\right)\right)
	\left( 1-\Delta(\we{q},\we{w})\right)}{\varepsilon_{\we{q}-\we{w}}-
	\varepsilon_{\we{q}}+\omega_{\we{w}}}\left( a_{\we{k}}^\ast a_{\we{k}-
	\we{w}}a_{\we{q}-\we{w}}^\ast a_{\we{q}} +c.c.\right).
\end{equation*}
The second term represents an~effective interaction between electrons dressed in
the~phonon field. If~$\varepsilon_{\we{q}-\we{w}}-\varepsilon_{\we{q}}+
\omega_{\we{w}}>0$, this interaction is attractive.

Omission of higher terms in Eq.~\eqref{h_trans} results in violation of unitarity.
The~question thus arises whether partial inclusion of these terms (first of all
those of 3rd~order in the~coupling) could improve agreement between theory and
experiment.

Following the~general rule for the~action of~$S$ in consecutive orders, expressed
by Eq.~\eqref{com_s}, one easily finds the~form of subsequent terms:
\begin{equation}
	\left\{
		\begin{aligned} (2,0) \\ (0,2) \end{aligned}
	\right\} \xrightarrow{S} (2,1) \xrightarrow{S}
	\left\{
		\begin{aligned} (4,0) \\ (2,2) \end{aligned}
	\right\} \xrightarrow{S} (4,1) \xrightarrow{S}
	\left\{
		\begin{aligned} (6,0)\\(4,2) \end{aligned}
	\right\} \xrightarrow{S} (6,1) \xrightarrow{S}
	\left\{
		\begin{aligned} (8,0) \\ (6,2) \end{aligned}
	\right\}.
\label{sgraf}
\end{equation}
In each consecutive step one obtains terms proportional to the~next power of
the~coupling.

From the~view-point of Fermi~liquid theory, the~terms representing effective
inter-electron interactions are most interesting. According to Eq.~\eqref{sgraf},
new terms~$(6,0)$ (proportional to~$D_{\we{w}}^4$), describing 3-electron
interactions and~$(8,0)$ (proportional to~$D_{\we{w}}^6$), representing 4-electron
interactions, appear. Attractive interactions of this type leading to the~formation
of fermion triples and quadruples, could affect the~behaviour of a~superconductor.
However, the~total spin of an~electron triple is nonzero, so such clusters are
unstable, as they are not invariant under time inversion. So far, there has been
no experimental evidence of such objects.

Most electrons in a~superconductor below~$T_c$ are paired, so the 4-electron
interaction between Cooper~pairs can be expected to prevail. Furthermore, quadruples
with a~total spin equal zero and appropriate 1-electron momenta are stable under
time inversion. On the~other hand, under Fr\"ohlich's~conditions for convergence of
series \eqref{h_trans}, the~effect of the~terms~$(6,0)$,~$(8,0)$ is weaker.

Evaluation of these terms is a~complicated procedure. Before doing this, let us
first examine the~3rd~order corrections.
\section{\label{3}Third order of the transformation}
Let us consider the~effect of the~first higher orders discarded by Fr\"ohlich, i.e.,
terms proportional to~$D_{\we{w}}^3$. Then the~corrected Hamiltonian takes the~form
\begin{equation}
	H'=H_0-\left(\left[S,H_0\right]-H_{int}\right)+
	\left(\frac{1}{2}\left[S,\left[S,H_0\right]\right] -
	\left[ S,H_{int}\right]\right)
	-\left(\frac{1}{6}\left[S,\left[ S, \left[ S,H_0\right]\right]\right]
	-\frac{1}{2}\left[S,\left[S,H_{int}\right]\right]\right)+\ldots\, .
\label{trans_3rd_order}
\end{equation}
The~additional two terms in the~last bracket are equal, explicitly,
\begin{equation}
\begin{split}
	& \frac{1}{2}\left[S,\left[ S,H_{int}\right]\right]-
	\frac{1}{6}\left[S,\left[ S, \left[ S, H_0 \right]\right] \right]
	=\sum_{\we{q}\we{k}}A_{\we{k}\we{q}}b_{\we{q}}^\ast
	n_{\we{k}}\gamma_{\we{q}}^\ast\\
	& + \sum_{\we{q}\we{k}\we{w}\we{k'}}b_{\we{w}}^\ast \left\{
	B_{\we{k}\we{q}\we{w}\we{k'}} a_{\we{k}-\we{w}}^\ast a_{\we{k}-\we{q}}
	a_{\we{k'}-\we{q}}^\ast a_{\we{k'}}
	+ C_{\we{k}\we{q}\we{w}\we{k'}}a_{\we{k'}}^\ast a_{\we{k'}-\we{q}}
	a_{\we{k}-\we{q}}^\ast a_{\we{k}+ \we{w}}\right\}+c.c.:=H_e,\\
\end{split}
\label{3order}
\end{equation}
where
\begin{equation}
	2A_{\we{k}\we{q}}=iD_{\we{q}}\phi^\ast(\we{k},\we{q})+
	iD_{\we{q}}\phi(\we{k}+\we{q},\we{q})
	+\frac{1}{3}\left(\varepsilon_{\we{k}-\we{q}}-\varepsilon_{\we{k}}+
	\omega_{\we{q}} \right) |\phi(\we{k},\we{q})|^2-\frac{1}{3}
	\left( \varepsilon_{\we{k}}-\varepsilon_{\we{k}+\we{q}}
	+\omega_{\we{q}}\right)|\phi(\we{k}+\we{q},\we{q})|^2 +c.c.,
\label{A}
\end{equation}
\begin{equation}
\begin{split}
	& 2B_{\we{k}\we{q}\we{w}\we{k'}} = iD_{\we{q}}\Bigl\{\phi^\ast(\we{k'},\we{q})
	\phi^\ast(\we{k-\we{q}},\we{w})-\phi^\ast(\we{k'},\we{q})
	\phi^\ast(\we{k},\we{w})
	+\phi^\ast(\we{k},\we{w})\phi(\we{k},\we{q}) -\phi^\ast(\we{k}-
	\we{q},\we{w})\phi(\we{k}-\we{w},\we{q})\Bigr\}\\
        &+\frac{1}{3}\phi^\ast(\we{k'},\we{q})\Bigl\{ \phi(\we{k},\we{q})
	\phi^\ast(\we{k},\we{w})\left(\varepsilon_{\we{k}}-\varepsilon_{\we{k}-
	\we{q}}+\varepsilon_{\we{k'}-\we{q}}-\varepsilon_{\we{k'}}\right)
	+\phi(\we{k}-\we{w},\we{q})\phi^\ast(\we{k}-\we{q},\we{w})
	\left(\varepsilon_{\we{k}-\we{w}-\we{q}}-\varepsilon_{\we{k}-\we{w}}-
	\varepsilon_{\we{k'}-\we{q}}+\varepsilon_{\we{k'}}\right)\Bigr\},\\
\end{split}
\label{B}
\end{equation}
\begin{equation}
\begin{split}
	2C_{\we{k}\we{q}\we{w}\we{k'}}&=
	iD_{\we{q}}\Bigl\{ \phi(\we{k'},\we{q})\phi^\ast(\we{k}+\we{w}
	-\we{q},\we{w})-\phi(\we{k'},\we{q})\phi^\ast(\we{k}+\we{w},\we{w})\\
	& - \phi^\ast(\we{k}+\we{w}-\we{q},\we{w})\phi^\ast(\we{k}+\we{w},\we{q})+
	\phi^\ast(\we{k}+\we{w},\we{w})\phi^\ast(\we{k},\we{q})\Bigr\}\\
	&+\frac{1}{3}\phi(\we{k'},\we{q})\Bigl\{ \phi^\ast(\we{k}+\we{w},\we{q})
	\phi^\ast(\we{k}+\we{w}-\we{q},\we{w}) \left(
	\varepsilon_{\we{k'}-\we{q}}-\varepsilon_{\we{k'}}-
	\varepsilon_{\we{k}+\we{w}-\we{q}}+\varepsilon_{\we{k}+\we{w}}\right)\\
	& +\phi^\ast(\we{k},\we{q})\phi^\ast(\we{k}+\we{w},\we{w})
	\left( \varepsilon_{\we{k}-\we{q}}-\varepsilon_{\we{k}}-
	\varepsilon_{\we{k'}-\we{q}}+\varepsilon_{\we{k'}}\right)\Bigr\},\\
\label{C}
\end{split}
\end{equation}
It can be seen that a~new phonon index~$\we{w}$ has appeared in Eq.~\eqref{3order}.
It results from the~commutator of~$S$ with the~terms~$(4,0)$, so the~harmonic
approximation has not been violated.

Substitution of the~expression~\eqref{3order} into~$H'$ yields
\begin{equation*}
	H'=H_a+H_b+H_c+H_d+H_e,
\end{equation*}
where
\begin{equation*}
	H_a=\frac{1}{2}\sum_{\we{k}}\varepsilon_{\we{k}} n_{\we{k}}+
	\frac{1}{2}\sum_{\we{q}}\omega_{\we{q}}b_{\we{q}}^\ast b_{\we{q}}+c.c.,
\end{equation*}
\begin{equation}
	H_b=\sum_{\we{q}\we{k}}b_{\we{q}}^\ast b_{\we{q}}n_{\we{k}}
	\Bigl\{ iD_{\we{q}}\left(\phi(\we{k} +\we{q},\we{q})-
	\phi(\we{k},\we{q})\right)
	 -\frac{1}{2}\left(\varepsilon_{\we{k}}-\varepsilon_{\we{k}+\we{q}}+
	 \omega_{\we{q}}\right) |\phi(\we{k}+\we{q},\we{q})|^2+\frac{1}{2}
	 \left(\varepsilon_{\we{k}-\we{q}} -\varepsilon_{\we{k}}+
	 \omega_{\we{q}}\right)|\phi(\we{k},\we{q})|^2 \Bigr\} +c.c.,
\label{H_b}
\end{equation}
\begin{equation}
	H_c=\sum_{\we{q}\we{k}}b_{\we{q}}\Bigl( iD_{\we{q}}+\left(
	\varepsilon_{\we{k}-\we{q}}-\varepsilon_{\we{k}}+\omega_{\we{q}}
	\right)\phi(\we{k},\we{q})\Bigr)
	a_{\we{k}}^\ast a_{\we{k}- \we{q}}+c.c.,
\label{H_c}
\end{equation}
\begin{equation}
	H_d=\sum_{\we{q}\we{k}\we{k'}}\left(iD_{\we{q}}\phi(\we{k},\we{q})+
	\frac{1}{2}(\varepsilon_{\we{k}	-\we{q}}-\varepsilon_{\we{k}}+
	\omega_{\we{q}})\phi(\we{k},\we{q})\phi^\ast(\we{k'},\we{q})\right)
	a_{\we{k}}^\ast a_{\we{k}-\we{q}}a_{\we{k'}-\we{q}}^\ast
	a_{\we{k'}}+c.c..
\label{H_d}
\end{equation}
$H_b$ is linear in~$b_{\we{q}}^\ast b_{\we{q}}$ but differs from~$H_a$, $H_c$~is part
of~$H_{int}$ which had been excluded from the~transformation to avoid convergence
problems, $H_d$~is the~2-electron interaction obtained, whereas~$H_e$ contains terms
of the~form~$(4,1)$ which represent the~obtained correction to Fr\"ohlich's~Hamiltonian.

The~function~$\phi(\we{k},\we{q})$ remains in the~form~\eqref{phi}, which guarantees
minimization of the~contribution~$H_c$.
\section{\label{4}The second transformation}
The~structure of~$H'$ bears similarities to that of~$H_{e-ph}$. Both~$H'$
and~$H_{e-ph}$ contain terms describing the~interaction of electrons with phonons:
the~counterpart of~$H_{int}$ in~$H_{e-ph}$ is~$H_e$ in~$H'$. To estimate the~effect
of~$H_e$, let us repeatedly apply Fr\"ohlich's~method and perform a~second unitary
transformation adjusted to eliminate~$H_e$ as far as possible. The form of~$H_e$
suggests to take
\begin{equation*}
	S'=\sum_{\we{u}}\left(\eta_{\we{u}}^\ast b_{\we{u}}^\ast -
	\eta_{\we{u}}b_{\we{u}}+\xi_{\we{u}}^\ast
	b_{\we{u}}^\ast -\xi_{\we{u}}b_{\we{u}}+
	\zeta_{\we{u}}^\ast b_{\we{u}}^\ast -\zeta_{\we{u}}b_{\we{u}}\right),
\end{equation*}
where
\begin{equation}
	\eta_{\we{u}}^\ast=\sum_{\we{l}\we{m}}
	\psi^\ast(\we{l},\we{m},\we{u})a_{\we{l}}^\ast
	a_{\we{l}}a_{\we{m}-\we{u}}^\ast a_{\we{m}},
\label{eta}
\end{equation}
\begin{equation}
	\xi_{\we{u}}^\ast=\sum_{\we{l}\we{m}\we{t}}
	\chi^\ast(\we{l},\we{m},\we{t},\we{u})
	a_{\we{l}-\we{u}}^\ast a_{\we{l}-\we{t}}a_{\we{m}-\we{t}}^\ast a_{\we{m}},
\label{xi}
\end{equation}
\begin{equation}
	\zeta_{\we{u}}^\ast=\sum_{\we{l}\we{m}\we{t}}
	\varphi^\ast(\we{l},\we{m},\we{t},\we{u})
	a_{\we{l}}^\ast a_{\we{l}-\we{t}}a_{\we{m}-\we{t}}^\ast a_{\we{m}+\we{u}}.
\label{zeta}
\end{equation}
The~explicit form of the~functions~$\psi$, $\chi$, $\varphi$ will be found below.

The~interaction~$H_e$ is 3rd~order in~$D_{\we{q}}$, therefore bearing in mind
Fr\"ohlich's approach, we shall restrict the expansion
of~$\exp[S'^\ast]H'\exp[S']=\hat{H}$ to terms which are 6th~order in~$D_{\we{q}}$.
These terms are
\begin{equation}
	\hat{H}=H'-[S',H']+\frac{1}{2}[S',[S',H_a]] +\ldots \, .
\label{hatH}
\end{equation}
To evaluate the~rhs, one needs the~commutators
\begin{equation*}
\begin{split}
	\left[S',H_a\right]&=
	\sum_{\we{u}\we{k}\we{l}}b_{\we{u}}^\ast
	\psi^\ast(\we{l},\we{k},\we{u})
	\left( \varepsilon_{\we{k}}-\varepsilon_{\we{k}-\we{u}}-\omega_{\we{u}}\right)
	a_{\we{l}}^\ast a_{\we{l}} a_{\we{k}-\we{u}}^\ast a_{\we{k}}\\
	& + \sum_{\we{u}\we{k}\we{l}\we{t}}b_{\we{u}}^\ast
	\chi^\ast(\we{l},\we{k},\we{t},\we{u})
	\left(\varepsilon_{\we{k}}-\varepsilon_{\we{k}-\we{t}}+
	\varepsilon_{\we{l}-\we{t}}-\varepsilon_{\we{l}-\we{u}}-\omega_{\we{u}}\right)
	a_{\we{l}-\we{u}}^\ast a_{\we{l}-\we{t}}a_{\we{k}-\we{t}}^\ast a_{\we{k}}\\
	& + \sum_{\we{u}\we{k}\we{l}\we{t}}b_{\we{u}}^\ast
	\varphi^\ast(\we{l},\we{k},\we{t},\we{u})
	\left( \varepsilon_{\we{k}+\we{u}}-\varepsilon_{\we{k}-\we{t}}+
	\varepsilon_{\we{l}-\we{t}}-\varepsilon_{\we{l}}-\omega_{\we{u}}\right)
	a_{\we{l}}^\ast a_{\we{l}-\we{t}}a_{\we{k}-\we{t}}^\ast
	a_{\we{k}+\we{u}}+c.c.,\\
\end{split}
\end{equation*}
\begin{equation*}
	\left[S',H_b\right]=-\sum_{\we{q}\we{k}}b_{\we{q}}^\ast n_{\we{k}}
	\left(\eta_{\we{q}}^\ast+\xi_{\we{q}}^\ast
	+ \zeta_{\we{q}}^\ast\right)
	\left( D_{\we{k}\we{q}}+D_{\we{k}\we{q}}^\ast\right)+c.c.,
\end{equation*}
where~$D_{\we{k}\we{q}}$ is defined by~$H_b=\sum_{\we{q}}b_{\we{q}}^\ast b_{\we{q}}
n_{\we{k}}D_{\we{k}\we{q}}+c.c.$ in Eq.~\eqref{H_b},
\begin{equation*}
\begin{split}
	&\left[S',H_c\right]=-\sum_{\we{q}\we{k}}E_{\we{k},\we{q}}
	a_{\we{k}}^\ast a_{\we{k}-\we{q}}\left( \eta_{\we{q}}^\ast+\xi_{\we{q}}^\ast+
	\zeta_{\we{q}}^\ast\right)\\
        &+\sum_{\we{q}\we{k}\we{l}}E_{\we{k},\we{q}}b_{\we{q}}^\ast
	b_{\we{q}}\Bigl\{ \psi^\ast(\we{l},\we{k},\we{q})
	n_{\we{l}}\left(n_{\we{k}-\we{q}}-n_{\we{k}}\right)+
	\bigl( \psi^\ast(\we{k},\we{l},\we{q})-\psi^\ast(\we{k}-\we{q},\we{l},\we{q})
	\bigr) a_{\we{k}}^\ast a_{\we{k}-\we{q}} a_{\we{l}-\we{q}}^\ast
	a_{\we{l}}\Bigr\}\\
	&+\sum_{\we{q}\we{k}\we{l}\we{t}}b_{\we{q}}^\ast b_{\we{q}}
	\Bigl\{ \bigl( E_{\we{k},\we{q}}\chi^\ast(\we{l},\we{k},\we{t},\we{q})-
	E_{\we{k}-\we{t},\we{q}}\chi^\ast(\we{l},\we{k}-\we{q},\we{t},\we{q})
	\bigr) a_{\we{l}-\we{q}}^\ast a_{\we{l}-\we{t}}a_{\we{k}-\we{t}}^\ast
	a_{\we{k}-\we{q}}\\
	&+ \bigl( E_{\we{k}+\we{q}-\we{t},\we{q}}\chi^\ast(\we{k}+\we{q},\we{l},
	\we{t},\we{q})-E_{\we{k},\we{q}}\chi^\ast(\we{k},\we{l},\we{t},\we{q})
	\bigr) a_{\we{k}}^\ast a_{\we{k}-\we{t}}a_{\we{l}-\we{t}}^\ast a_{\we{l}}\\
	&+ \bigl( E_{\we{k}+\we{q}+\we{t},\we{q}}\varphi^\ast(\we{l},\we{k}+\we{t},
	\we{t},\we{q}) - E_{\we{k},\we{q}}\varphi^\ast(\we{l},\we{k}+\we{t}-\we{q},
	\we{t},\we{q})\bigr) a_{\we{l}}^\ast a_{\we{l}-\we{t}}a_{\we{k}}^\ast
	a_{\we{k}+\we{t}}\\
	&+ \bigl( E_{\we{k},\we{q}}\varphi^\ast(\we{k}+\we{t},\we{l},\we{t},\we{q})-
	E_{\we{k}+\we{t},\we{q}}\varphi^\ast(\we{k}+\we{t}-\we{q},\we{l},\we{t},\we{q})
	\bigr) a_{\we{k}+\we{t}}^\ast a_{\we{k}-\we{q}}a_{\we{l}-\we{t}}^\ast
	a_{\we{l}+\we{q}}\Bigr\}+c.c.,\\
\end{split}
\end{equation*}
where~$E_{\we{k},\we{q}}$ is defined by~$H_c=\sum_{\we{q}\we{k}}b_{\we{q}}a_{\we{k}}^\ast
a_{\we{k}-\we{q}} E_{\we{k},\we{q}}+c.c.$ in Eq.~\eqref{H_c},
\begin{equation*}
\begin{split}
  	\left[S', H_d \right]& =
	\sum_{\we{u}\we{q}\we{k}\we{k'}}b_{\we{u}}^\ast
	\left(F_{\we{k},\we{k'},\we{q}} +F_{\we{k'},\we{k},\we{q}}^\ast\right)
	\Bigl\{ a_{\we{k}}^\ast a_{\we{k}-\we{q}} \left[ \eta_{\we{u}}^\ast,a_{\we{k'}-
	\we{q}}^\ast a_{\we{k'}}\right]+
	\left[\eta_{\we{u}}^\ast,a_{\we{k}}^\ast a_{\we{k}-\we{q}}\right]a_{\we{k'}-
	\we{q}}^\ast a_{\we{k'}}\\
	&+a_{\we{k}}^\ast a_{\we{k}-\we{q}}\left[\xi_{\we{u}}^\ast,a_{\we{k'}-
	\we{q}}^\ast a_{\we{k'}}\right]+\left[\xi_{\we{u}}^\ast,a_{\we{k}}^\ast
	a_{\we{k}-\we{q}}\right]a_{\we{k'} -\we{q}}^\ast a_{\we{k'}
	+a_{\we{k}}^\ast a_{\we{k}-\we{q}}\left[\zeta_{\we{u}}^\ast,a_{\we{k'}
        -\we{q}}^\ast a_{\we{k'}}\right]+\left[\zeta_{\we{u}}^\ast,a_{\we{k}}^\ast
	a_{\we{k}-\we{q}} \right]a_{\we{k'}-\we{q}}^\ast a_{\we{k'}}\Bigr\}+c.c.,\\
\end{split}
\end{equation*}
where~$F_{\we{k},\we{k'},\we{q}}$ is defined by~$H_d=\sum_{\we{q}\we{k}\we{k'}}F_{\we{k},
\we{k'},\we{q}}a_{\we{k}}^\ast a_{\we{k}-\we{q}}a_{\we{k'}-\we{q}}^\ast a_{\we{k'}}+
c.c.$ in Eq.~\eqref{H_d},
\begin{equation*}
\begin{split}
	&\left[S',H_e\right]=
	-\sum_{\we{q}\we{k}}A_{\we{k},\we{q}}b_{\we{q}}^\ast b_{\we{q}}
	\Bigl\{ \left[ \eta_{\we{q}},n_{\we{k}}\gamma_{\we{q}}^\ast\right]+
	\left[\xi_{\we{q}},n_{\we{k}}\gamma_{\we{q}}^\ast\right]
	+\left[\zeta_{\we{q}},n_{\we{k}}\gamma_{\we{q}}^\ast\right]\Bigr\}\\
	&-\sum_{\we{q}\we{w}\we{k}\we{k'}}B_{\we{k},\we{q},\we{w},\we{k'}}
	b_{\we{w}}^\ast b_{\we{w}}\Bigl\{ \left[\eta_{\we{w}},a_{\we{k}-\we{w}}^\ast
	a_{\we{k}-\we{q}}a_{\we{k'}-\we{q}}^\ast a_{\we{k'}}\right]
  	+\left[\xi_{\we{w}},a_{\we{k}-\we{w}}^\ast a_{\we{k}-\we{q}}a_{\we{k'}-
	\we{q}}^\ast a_{\we{k'}}\right]
  	+\left[\zeta_{\we{w}},a_{\we{k}-\we{w}}^\ast a_{\we{k}-\we{q}}a_{\we{k'}-
	\we{q}}^\ast a_{\we{k'}} \right] \Bigr\} \\
	& - \sum_{\we{q}\we{w}\we{k}\we{k'}}C_{\we{k},\we{q},\we{w},\we{k'}}
	b_{\we{w}}^\ast b_{\we{w}}\Bigl\{ \left[\eta_{\we{w}},a_{\we{k'}}^\ast
	a_{\we{k'}-\we{q}}a_{\we{k}- \we{q}}^\ast a_{\we{k}+\we{w}}\right]
   	+ \left[\xi_{\we{w}},a_{\we{k'}}^\ast a_{\we{k'}-\we{q}}a_{\we{k}-\we{q}}^\ast
	a_{\we{k}+\we{w}}\right] +\left[\zeta_{\we{w}},a_{\we{k'}}^\ast a_{\we{k'}-
	\we{q}}a_{\we{k}-\we{q}}^\ast a_{\we{k}+\we{w}} \right]\Bigr\}\\
   	&-\sum_{\we{q}\we{k}}A_{\we{k},\we{q}}\Bigl\{ \eta_{\we{q}}
	n_{\we{k}}\gamma_{\we{q}}^\ast+\xi_{\we{q}} n_{\we{k}} \gamma_{\we{q}}^\ast+
	\zeta_{\we{q}}n_{\we{k}}\gamma_{\we{q}}^\ast\Bigr\} \\
	& - \sum_{\we{q}\we{w}\we{k}\we{k'}}B_{\we{k},\we{q},\we{w},\we{k'}}
	\Bigl\{ \eta_{\we{w}}a_{\we{k}-\we{w}}^\ast a_{\we{k}-\we{q}}a_{\we{k'}-
	\we{q}}^\ast a_{\we{k'}}
	+\xi_{\we{w}}a_{\we{k}- \we{w}}^\ast a_{\we{k}-\we{q}}a_{\we{k'}-\we{q}}^\ast
	a_{\we{k'}}+\zeta_{\we{w}}a_{\we{k}-\we{w}}^\ast a_{\we{k}-\we{q}}a_{\we{k'}-
	\we{q}}^\ast a_{\we{k'}}\Bigr\} \\
	& - \sum_{\we{q}\we{w}\we{k}\we{k'}}
	C_{\we{k},\we{q},\we{w},\we{k'}}\Bigl\{ \eta_{\we{w}} a_{\we{k'}}^\ast
	a_{\we{k'}-\we{q}}a_{\we{k}-\we{q}}^\ast a_{\we{k}+\we{w}}
	+ \xi_{\we{w}} a_{\we{k'}}^\ast a_{\we{k'}-\we{q}}a_{\we{k}-\we{q}}^\ast
	a_{\we{k}+\we{w}}+\zeta_{\we{w}} a_{\we{k'}}^\ast a_{\we{k'}-\we{q}}
	a_{\we{k}-\we{q}}^\ast a_{\we{k}+\we{w}}\Bigr\}+c.c.,\\
\end{split}
\end{equation*}
where~$A_{\we{k},\we{q}}$,~$B_{\we{k},\we{q},\we{w},\we{k'}}$,~$C_{\we{k},\we{q},\we{w},
\we{k'}}$ are given, respectively by Eqs.~\eqref{A}, \eqref{B} i \eqref{C},
\begin{equation}
\begin{split}
	&\left[S',\left[S',H_a\right]\right]=
	-\sum_{\we{q}\we{k}\we{k'}}G_{\we{k},\we{k'},\we{q}}
	b_{\we{q}}^\ast b_{\we{q}} \Bigl\{ \left[\eta_{\we{q}},a_{\we{k'}}^\ast
	a_{\we{k'}}a_{\we{k}-\we{q}}^\ast a_{\we{k}}\right]+\left[\xi_{\we{q}},
	a_{\we{k'}}^\ast a_{\we{k'}}a_{\we{k}-\we{q}}^\ast a_{\we{k}}\right]
	+\left[\zeta_{\we{q}},a_{\we{k'}}^\ast a_{\we{k'}}a_{\we{k}-\we{q}}^\ast
	a_{\we{k}} \right] \Bigr\} \\
	& - \sum_{\we{q}\we{k}\we{k'}\we{t}}H_{\we{k},\we{k'},\we{t},\we{q}}
	b_{\we{q}}^\ast b_{\we{q}} \Bigl\{ \left[\eta_{\we{q}},a_{\we{k'}-\we{q}}^\ast
	a_{\we{k'}- \we{t}}a_{\we{k}-\we{t}}^\ast a_{\we{k}}\right]+
	\left[\xi_{\we{q}},a_{\we{k'}-\we{q}}^\ast a_{\we{k'}-\we{t}}a_{\we{k}-
	\we{t}}^\ast a_{\we{k}}\right]
	+\left[\zeta_{\we{q}},a_{\we{k'}-\we{q}}^\ast a_{\we{k'}-\we{t}}
	a_{\we{k}-\we{t}}^\ast a_{\we{k}}\right]\Bigr\}\\
	& - \sum_{\we{q}\we{k}\we{k'}\we{t}}I_{\we{k},\we{k'},\we{t},\we{q}}
	b_{\we{q}}^\ast b_{\we{q}}\Bigl\{
	\left[\eta_{\we{q}},a_{\we{k'}}^\ast a_{\we{k'}-\we{t}}a_{\we{k}-\we{t}}^\ast
	a_{\we{k}+\we{q}}\right]
	+\left[\xi_{\we{q}},a_{\we{k'}}^\ast a_{\we{k'}-\we{t}}a_{\we{k}-\we{t}}^\ast
	a_{\we{k}+\we{q}} \right]+
	\left[\zeta_{\we{q}},a_{\we{k'}}^\ast a_{\we{k'}-\we{t}}a_{\we{k}-\we{t}}^\ast
	a_{\we{k}+\we{q}}\right]\Bigr\}\\
	&-\sum_{\we{q}\we{k}\we{k'}}G_{\we{k},\we{k'},\we{q}} \Bigl\{
	\eta_{\we{q}} n_{\we{k'}}a_{\we{k}-\we{q}}^\ast a_{\we{k}}+\xi_{\we{q}}
	n_{\we{k'}}a_{\we{k}-\we{q}}^\ast a_{\we{k}} +\zeta_{\we{q}} n_{\we{k'}}
	a_{\we{k}-\we{q}}^\ast a_{\we{k}}\Bigr\}\\
	&-\sum_{\we{q}\we{k}\we{k'}\we{t}}H_{\we{k},\we{k'},\we{t},\we{q}} \Bigl\{
	\eta_{\we{q}}a_{\we{k'}-\we{q}}^\ast a_{\we{k'}-\we{t}}a_{\we{k}-\we{t}}^\ast
	a_{\we{k}}+\xi_{\we{q}}a_{\we{k'}-\we{q}}^\ast a_{\we{k'}-\we{t}}a_{\we{k}-
	\we{t}}^\ast a_{\we{k}}+\zeta_{\we{q}}a_{\we{k'}-\we{q}}^\ast a_{\we{k'}-\we{t}}
	a_{\we{k}-\we{t}}^\ast a_{\we{k}}\Bigr\}\\
	&-\sum_{\we{q}\we{k}\we{k'}\we{t}}I_{\we{k},\we{k'},\we{t},\we{q}} \Bigl\{
	\eta_{\we{q}} a_{\we{k'}}^\ast a_{\we{k'}-\we{t}}a_{\we{k}-\we{t}}^\ast
	a_{\we{k}+\we{q}}+\xi_{\we{q}} a_{\we{k'}}^\ast a_{\we{k'}-\we{t}}
	a_{\we{k}-\we{t}}^\ast a_{\we{k}+\we{q}}+\zeta_{\we{q}}
	a_{\we{k'}}^\ast a_{\we{k'}-\we{t}}a_{\we{k}-\we{t}}^\ast a_{\we{k}+\we{q}}
	\Bigr\}+c.c.,\\
\end{split}
\label{S'S'Ha}
\end{equation}
where
\begin{equation}
 	G_{\we{k},\we{k'},\we{q}}=\psi^\ast(\we{k'},\we{k},\we{q})
	\left(\varepsilon_{\we{k}}-\varepsilon_{\we{k}-\we{q}}-
	\omega_{\we{q}}\right),
\label{G}
\end{equation}
\begin{equation}
	H_{\we{k},\we{k'},\we{t},\we{q}}=
	\chi^\ast(\we{k'},\we{k},\we{t},\we{q})
	\left(\varepsilon_{\we{k}}-\varepsilon_{\we{k}-\we{t}}+
	\varepsilon_{\we{k'}-\we{t}}-\varepsilon_{\we{k'}-\we{q}}-\omega_{\we{q}}
	\right),
\label{H}
\end{equation}
\begin{equation}
	I_{\we{k},\we{k'},\we{t},\we{q}}=
	\varphi^\ast(\we{k'},\we{k},\we{t},\we{q})
	\left(\varepsilon_{\we{k}+\we{q}}-\varepsilon_{\we{k}-\we{t}}+
	\varepsilon_{\we{k'}-\we{t}}-\varepsilon_{\we{k'}}-\omega_{\we{q}}
	\right).
\label{I}
\end{equation}
The~commutators in the~terms containing phonon operators have not been evaluated, as we
are interested first of all in expressions containing exclusively electron operators.

Given the~transformed Hamiltonian~$\hat{H}$, we are now in position to minimize
the~effect of~$H_e$ by imposing, similarly as~Fr\"ohlich, the~condition~$H_e-\left[S',
H_a\right]=0$. This leads to equations for~$\psi$, $\chi$, $\varphi$, which determine these functions uniquely, viz,
\begin{equation}
	\psi^\ast(\we{k'},\we{k},\we{q})=
	\frac{\phi^\ast(\we{k},\we{q})A_{\we{k'},\we{q}}}{\varepsilon_{\we{k}}
	-\varepsilon_{\we{k}-\we{q}}-\omega_{\we{q}}}.
\label{psi}
\end{equation}
After substituting~$\phi$ and~$A$ given by Eqs.~\eqref{phi},~\eqref{A}, one obtains
\begin{equation}
	\psi^\ast(\we{k'},\we{k},\we{q})=
	\frac{2iD_{\we{q}}^3\left( 1-\Delta(\we{k},\we{q})\right)}{3\left(
  	\varepsilon_{\we{k}-\we{q}}-\varepsilon_{\we{k}}+\omega_{\we{q}}\right)^2}
	\left( \frac{1-\Delta(\we{k'},\we{q})}{\varepsilon_{\we{k'}-\we{q}}-
	\varepsilon_{\we{k'}}+\omega_{\we{q}}}-\frac{1-\Delta(\we{k'}+\we{q},
  	\we{q})}{\varepsilon_{\we{k'}}-\varepsilon_{\we{k'}+\we{q}}+
	\omega_{\we{q}}}\right).
\label{psi_exp}
\end{equation}
Introduction of additional functions in order to preserve convergence is not necessary
here, as~$\Delta$ already guarantees this property.

As for~$\chi$ and~$\varphi$, additional functions preserving convergence are
indispensable, viz,
\begin{equation}
	\chi^\ast(\we{k'},\we{k},\we{t},\we{q})=
	\frac{B_{\we{k'},\we{t},\we{q},\we{k}}}{\varepsilon_{\we{k}}-
	\varepsilon_{\we{k}-\we{t}}+\varepsilon_{\we{k'}-\we{t}}-\varepsilon_{\we{k'}-
	\we{q}}-\omega_{\we{q}}} \left( 1-\tilde{\Delta}(\we{k},\we{k'},\we{t},
	\we{q})\right),
\label{chi}
\end{equation}
where
\begin{equation}
	\tilde{\Delta}(\we{k},\we{k'},\we{t},\we{q})=\begin{cases}
	        1 \text{,} & \, \text{if} \quad \left|\varepsilon_{\we{k}}-
		\varepsilon_{\we{k}-\we{t}}+\varepsilon_{\we{k'}-\we{t}}-
		\varepsilon_{\we{k'}-\we{q}}-\omega_{\we{q}}\right| <
		\tilde{\Gamma}_{\we{q}} \\
       	        0 \text{,} & \, \text{if} \quad \left|\varepsilon_{\we{k}}-
		\varepsilon_{\we{k}-\we{t}}+\varepsilon_{\we{k'}-\we{t}}-
		\varepsilon_{\we{k'}-\we{q}}- \omega_{\we{q}}\right| \ge
		\tilde{\Gamma}_{\we{q}} \\
	      \end{cases}.
\label{Delta_tilde}
\end{equation}
Taking into account Eqs.~\eqref{phi},~\eqref{B}, one obtains
\begin{equation}
\begin{split}
	&\chi^\ast(\we{k'},\we{k},\we{t},\we{q})=\frac{iD_{\we{t}}^2
	D_{\we{q}}}{6\left(\varepsilon_{\we{k}}-\varepsilon_{\we{k}-\we{t}}+
	\varepsilon_{\we{k'}-\we{t}}-\varepsilon_{\we{k'}-\we{q}}-
	\omega_{\we{q}} \right)}
	\left( \frac{\left( 1-\Delta(\we{k},\we{t}) \right)\left(1-\Delta(\we{k'},
	\we{q})\right)}{\left( \varepsilon_{\we{k}-\we{t}}-\varepsilon_{\we{k}}+
	\omega_{\we{t}}\right)\left(
	\varepsilon_{\we{k'}-\we{q}}-\varepsilon_{\we{k'}}+\omega_{\we{q}}\right)}\left(
	2+\Delta(\we{k'},\we{t})\right)\right.\\
	&-\frac{\left( 1-\Delta(\we{k},\we{t}) \right)\left(1-\Delta(\we{k'}-\we{t},
	\we{q})\right)}{\left(
	\varepsilon_{\we{k}-\we{t}}-\varepsilon_{\we{k}}+\omega_{\we{t}}\right)\left(
	\varepsilon_{\we{k'}-
	\we{t}-\we{q}}-\varepsilon_{\we{k'}-\we{t}}+\omega_{\we{q}}\right)}\left( 2+
	\Delta(\we{k'}-\we{q},\we{t})\right)
	+\frac{\left( 1-\Delta(\we{k'},\we{t}) \right)\left(1-\Delta(\we{k'},\we{q})
	\right)}{\left(
	\varepsilon_{\we{k'}-\we{t}}-\varepsilon_{\we{k'}}+\omega_{\we{t}}\right)\left(
	\varepsilon_{\we{k'}
	-\we{q}}-\varepsilon_{\we{k'}}+\omega_{\we{q}}\right)}\left( 4-\Delta(\we{k},
	\we{t})\right)\\
	&\left.-\frac{\left( 1-\Delta(\we{k'}-\we{q},\we{t}) \right)\left(1-
	\Delta(\we{k'}-\we{t},\we{q})
	\right)}{\left( \varepsilon_{\we{k'}-\we{q}-\we{t}}-\varepsilon_{\we{k'}-\we{q}}+
	\omega_{\we{t}}
	\right)\left( \varepsilon_{\we{k'}-\we{t}-\we{q}}-\varepsilon_{\we{k'}-\we{t}}+
	\omega_{\we{q}}
	\right)}\left( 4-\Delta(\we{k},\we{t})\right)\right)\left( 1-
	\tilde{\Delta}(\we{k},\we{k'},\we{t},\we{q})\right),\\
\end{split}
\label{chi_exp}
\end{equation}
\begin{equation}
	\varphi^\ast(\we{k'},\we{k},\we{t},\we{q})=\frac{C_{\we{k},\we{t},\we{q},
	\we{k'}}}{\varepsilon_{\we{k}+\we{q}}-\varepsilon_{\we{k}-\we{t}}+
	\varepsilon_{\we{k'}-\we{t}} -\varepsilon_{\we{k'}}-\omega_{\we{q}}}\left( 1-
	\hat{\Delta}(\we{k},\we{k'},\we{t},\we{q}) \right),
\label{varphi}
\end{equation}
where
\begin{equation}
	\hat{\Delta}(\we{k},\we{k'},\we{t},\we{q})=\begin{cases}
	        1 \text{,} & \, \text{if} \quad \left| \varepsilon_{\we{k}+\we{q}}-
		\varepsilon_{\we{k}-\we{t}}+\varepsilon_{\we{k'}-\we{t}}-
		\varepsilon_{\we{k'}}-\omega_{\we{q}}\right| < \hat{\Gamma}_{\we{q}}\\
	        0 \text{,} & \, \text{if} \quad \left|\varepsilon_{\we{k}+\we{q}}-
		\varepsilon_{\we{k}-\we{t}}+\varepsilon_{\we{k'}-\we{t}}-
		\varepsilon_{\we{k'}}-\omega_{\we{q}}\right| \ge \hat{\Gamma}_{\we{q}}\\
	      \end{cases},
\label{Delta_hat}
\end{equation}
and substitution of~$\phi$ and~$C_{\we{k},\we{t},\we{q},\we{k'}}$ from
Eqs.~\eqref{phi},~\eqref{C} yields
\begin{equation}
\begin{split}
	& \varphi^\ast(\we{k'},\we{k},\we{t},\we{q}) = \frac{iD_{\we{t}}^2
	D_{\we{q}}}{6\left( \varepsilon_{\we{k}+\we{q}}-\varepsilon_{\we{k}-\we{t}}+
	\varepsilon_{\we{k'}-\we{t}}- \varepsilon_{\we{k'}}-\omega_{\we{q}}
	\right)}
	\left( \frac{\left( 1-\Delta(\we{k'},\we{t})\right)\left(1-\Delta(\we{k}+
	\we{q}-\we{t},\we{q})\right)}{\left(\varepsilon_{\we{k'}-\we{t}}-
	\varepsilon_{\we{k'}}+\omega_{\we{t}}\right)
	\left(\varepsilon_{\we{k}-\we{t}}-\varepsilon_{\we{k}+\we{q}-\we{t}}+
	\omega_{\we{q}}\right)}\left( 2+\Delta(\we{k}+\we{q},\we{t})\right)\right.\\
	&-\frac{\left( 1-\Delta(\we{k'},\we{t})\right)\left(1-\Delta(\we{k}+\we{q},
	\we{q})\right)}{\left( \varepsilon_{\we{k'}-\we{t}}-\varepsilon_{\we{k'}}+
	\omega_{\we{t}}\right)\left(\varepsilon_{\we{k}}-
	\varepsilon_{\we{k}+\we{q}}+\omega_{\we{q}}\right)}\left( 2+\Delta(\we{k},
	\we{t})\right)
	+\frac{\left(1-\Delta(\we{k}+\we{q},\we{t})\right)\left(1-\Delta(\we{k}+\we{q}-
	\we{t}, \we{q})\right)}{\left(\varepsilon_{\we{k}+\we{q}-\we{t}}-
	\varepsilon_{\we{k}+\we{q}}+\omega_{\we{t}}\right)\left(\varepsilon_{\we{k}-
	\we{t}}-\varepsilon_{\we{k}+\we{q}-\we{t}}+\omega_{\we{q}}\right)}\left( 4-
	\Delta(\we{k'},\we{t})\right)\\
	&\left.-\frac{\left( 1-\Delta(\we{k},\we{t})\right)\left(1-\Delta(\we{k}+\we{q},
	\we{q})\right)}{\left(\varepsilon_{\we{k}-\we{t}}-\varepsilon_{\we{k}}+
	\omega_{\we{t}}\right)\left( \varepsilon_{\we{k}}-\varepsilon_{\we{k}+\we{q}}+
	\omega_{\we{q}}\right)}\left( 4-\Delta(\we{k'},\we{t})\right)\right)\left( 1-
	\hat{\Delta}(\we{k},\we{k'},\we{t},\we{q})\right),\\
\end{split}
\label{varphi_exp}
\end{equation}
Having established~$\psi$, $\chi$, $\varphi$, let us average~$\hat{H}$ over the~phonon
vacuum:
\begin{equation}
\begin{split}
	\hat{H}_{\mathrm{av}}&=\frac{1}{2}\sum_{\we{k}}\varepsilon_{\we{k}}n_{\we{k}}+
	\sum_{\we{q}\we{k}\we{k'}} F_{\we{k},\we{k'},\we{q}}a_{\we{k}}^\ast a_{\we{k}-
	\we{q}}a_{\we{k'}-\we{q}}^\ast a_{\we{k'}}+
	\sum_{\we{q}\we{k}}E_{\we{k},\we{q}}a_{\we{k}}^\ast a_{\we{k}-\we{q}}
	\left(\eta_{\we{q}}^\ast+ \xi_{\we{q}}^\ast+\zeta_{\we{q}}^\ast\right)\\
	&+\sum_{\we{q}\we{k}\we{k'}}\left( A_{\we{k'},\we{q}}\phi^\ast(\we{k},\we{q})-
	\frac{1}{2} G_{\we{k},\we{k'},\we{q}}\right)
	\Bigl\{ \eta_{\we{q}}n_{\we{k'}}a_{\we{k}-\we{q}}^\ast a_{\we{k}}+
	\xi_{\we{q}}n_{\we{k'}}a_{\we{k}-\we{q}}^\ast a_{\we{k}}+\zeta_{\we{q}}
	n_{\we{k'}}a_{\we{k}- \we{q}}^\ast a_{\we{k}}\Bigr\}\\
	&+\sum_{\we{q}\we{k}\we{k'}\we{t}}\left( B_{\we{k'},\we{t},\we{q},\we{k}}-
	\frac{1}{2}H_{\we{k},\we{k'},\we{t},\we{q}}\right)\Bigl\{\eta_{\we{q}}
	a_{\we{k'}-\we{q}}^\ast a_{\we{k'}-\we{t}}a_{\we{k}
	-\we{t}}^\ast a_{\we{k}}+\xi_{\we{q}}a_{\we{k'}-\we{q}}^\ast a_{\we{k'}-
	\we{t}}a_{\we{k}-\we{t}}^\ast a_{\we{k}}
	+ \zeta_{\we{q}}a_{\we{k'}-\we{q}}^\ast a_{\we{k'}-\we{t}}a_{\we{k}-
	\we{t}}^\ast a_{\we{k}} \Bigr\} \\
	& + \sum_{\we{q}\we{k}\we{k'}\we{t}}\left(
	C_{\we{k},\we{t},\we{q},\we{k'}}-\frac{1}{2}I_{\we{k},\we{k'},\we{t},
	\we{q}}\right)\Bigl\{\eta_{\we{q}}a_{\we{k'}}^\ast a_{\we{k'}-\we{t}}a_{\we{k}-
	\we{t}}^\ast a_{\we{k}+\we{q}}
	+ \xi_{\we{q}}a_{\we{k'}}^\ast a_{\we{k'}-\we{t}}a_{\we{k}-\we{t}}^\ast
	a_{\we{k}+\we{q}}+\zeta_{\we{q}}a_{\we{k'}}^\ast a_{\we{k'}-\we{t}}a_{\we{k}-
	\we{t}}^\ast a_{\we{k}+\we{q}}\Bigr\}+c.c.,\\
\end{split}
\label{H_av}
\end{equation}

$\hat{H}_{\mathrm{av}}$ contains the~free-electron term, the~2-electron
Fr\"ohlich~interaction and terms representing 3-electron and 4-electron interactions.
The~3-electron terms were generated by the~second transformation of~$H_c$.
The~source of these terms is thus the~non-transformed part of the~original
interaction, discarded by~BCS. If higher-order corrections to BCS~theory are of
interest, all terms arising from that part, i.e., the~3-electron ones, can be therefore
neglected. The~same conclusion was drawn above on the grounds of unstability.
\section{\label{5}4-fermion interactions}
The~Hamiltonian~$\hat{H}_{\mathrm{av}}$ contains several terms representing 4-fermion
interactions. Using Eqs.~\eqref{G}, \eqref{psi}, \eqref{chi}, \eqref{varphi}, one
finds
\begin{equation*}
  	A_{\we{k'},\we{q}}\phi^\ast(\we{k},\we{q})-\frac{1}{2}G_{\we{k},\we{k'},\we{q}}
	=\frac{1}{2}G_{\we{k},\we{k'},\we{q}}=\frac{1}{2}\phi^\ast(\we{k},\we{q})
	A_{\we{k'},\we{q}}.
\end{equation*}
Additionally, taking into account Eqs.~\eqref{H},~\eqref{I}, we get
\begin{equation*}
	B_{\we{k'},\we{t},\we{q},\we{k}}-\frac{1}{2}H_{\we{k},\we{k'},\we{t},\we{q}}
	=\frac{1}{2} H_{\we{k},\we{k'},\we{t},\we{q}}=
	\frac{1}{2}B_{\we{k'},\we{t},\we{q},\we{t}}
	\left( 1-\tilde{\Delta}(\we{k},\we{k'},\we{t},\we{q}) \right),
\end{equation*}
\begin{equation*}
	C_{\we{k},\we{t},\we{q},\we{k'}}-\frac{1}{2}I_{\we{k},\we{k'},\we{t},\we{q}}
	=\frac{1}{2}I_{\we{k},\we{k'},\we{t},\we{q}}=
	\frac{1}{2}C_{\we{k},\we{t},\we{q},\we{k'}}\left( 1-\hat{\Delta}(\we{k},
	\we{k'},\we{t},\we{q})\right).
\end{equation*}
In terms
of~$G_{\we{k},\we{k'},\we{q}}$,~$H_{\we{k},\we{k'},\we{t},\we{q}}$,~$I_{\we{k},
\we{k'},\we{t},\we{q}}$ the~4-fermion interactions present in~$\hat{H}_{\mathrm{av}}$
express as
\begin{equation}
	H_4^1=\sum_{\we{q}\we{k}\we{k'}\we{l}\we{m}}\frac{1}{2}G_{\we{k},\we{k'},
	\we{q}}\psi(\we{l},\we{m},\we{q})a_{\we{m}}^\ast a_{\we{m}-\we{q}}
	n_{\we{l}}n_{\we{k'}}a_{\we{k}-\we{q}}^\ast a_{\we{k}}+c.c.,
\label{H41}
\end{equation}
\begin{equation}
	H_4^2=\sum_{\we{q}\we{k}\we{k'}\we{l}\we{m}\we{t}}\frac{1}{2}G_{\we{k},\we{k'},
	\we{q}}\chi(\we{l},\we{m},\we{t},\we{q})a_{\we{m}}^\ast a_{\we{m}-\we{t}}
	a_{\we{l}-\we{t}}^\ast a_{\we{l}-\we{q}} n_{\we{k'}} a_{\we{k}-\we{q}}^\ast
	a_{\we{k}}+c.c.,
\label{H42}
\end{equation}
\begin{equation}
	H_4^3=\sum_{\we{q}\we{k}\we{k'}\we{l}\we{m}\we{t}}\frac{1}{2}G_{\we{k},\we{k'},
	\we{q}}\varphi(\we{l},\we{m},\we{t},\we{q})a_{\we{m}+\we{q}}^\ast a_{\we{m}-
	\we{t}}a_{\we{l}-\we{t}}^\ast a_{\we{l}}n_{\we{k'}}a_{\we{k}-\we{q}}^\ast
	a_{\we{k}}+c.c.,
\label{H43}
\end{equation}
\begin{equation}
	H_4^4=\sum_{\we{q}\we{k}\we{k'}\we{l}\we{m}\we{t}}\frac{1}{2}H_{\we{k},\we{k'},
	\we{t},\we{q}}\psi(\we{l},\we{m},\we{q})a_{\we{m}}^\ast a_{\we{m}-\we{q}}
	n_{\we{l}}a_{\we{k'}-\we{q}}^\ast a_{\we{k'}-\we{t}}a_{\we{k}-\we{t}}^\ast
	a_{\we{k}}+c.c.,
\label{H44}
\end{equation}
\begin{equation}
	H_4^5=\sum_{\we{q}\we{k}\we{k'}\we{l}\we{m}\we{t}\we{w}}\frac{1}{2}H_{\we{k},
	\we{k'},\we{t},\we{q}} \chi(\we{l},\we{m},\we{w},\we{q})a_{\we{m}}^\ast
	a_{\we{m}-\we{w}}a_{\we{l}-\we{w}}^\ast a_{\we{l}-\we{q}} a_{\we{k'}-\we{q}}^\ast
	a_{\we{k'}-\we{t}}a_{\we{k}-\we{t}}^\ast a_{\we{k}}+c.c.,
\label{H45}
\end{equation}
\begin{equation}
	H_4^6=\sum_{\we{q}\we{k}\we{k'}\we{l}\we{m}\we{t}\we{w}}\frac{1}{2}H_{\we{k},
	\we{k'},\we{t},\we{q}} \varphi(\we{l},\we{m},\we{w},\we{q})a_{\we{m}+\we{q}}^\ast
	a_{\we{m}-\we{w}}a_{\we{l}-\we{w}}^\ast a_{\we{l}} a_{\we{k'}-\we{q}}^\ast
	a_{\we{k'}-\we{t}}a_{\we{k}-\we{t}}^\ast a_{\we{k}}+c.c.,
\label{H46}
\end{equation}
\begin{equation}
	H_4^7=\sum_{\we{q}\we{k}\we{k'}\we{l}\we{m}\we{t}}\frac{1}{2}I_{\we{k},\we{k'},
	\we{t},\we{q}} \psi(\we{l},\we{m},\we{q})a_{\we{m}}^\ast a_{\we{m}-\we{q}}
	n_{\we{l}}a_{\we{k'}}^\ast a_{\we{k'}-\we{t}} a_{\we{k}-\we{t}}^\ast a_{\we{k}+
	\we{q}}+c.c.,
\label{H47}
\end{equation}
\begin{equation}
	H_4^8=\sum_{\we{q}\we{k}\we{k'}\we{l}\we{m}\we{t}\we{w}}\frac{1}{2}I_{\we{k},
	\we{k'},\we{t},\we{q}} \chi(\we{l},\we{m},\we{w},\we{q})a_{\we{m}}^\ast
	a_{\we{m}-\we{w}}a_{\we{l}-\we{w}}^\ast a_{\we{l}-\we{q}}
	a_{\we{k'}}^\ast a_{\we{k'}-\we{t}}a_{\we{k}-\we{t}}^\ast a_{\we{k}+\we{q}}+c.c.,
\label{H48}
\end{equation}
\begin{equation}
	H_4^9=\sum_{\we{q}\we{k}\we{k'}\we{l}\we{m}\we{t}\we{w}}\frac{1}{2}I_{\we{k},
	\we{k'},\we{t},\we{q}} \varphi(\we{l},\we{m},\we{w},\we{q})a_{\we{m}+\we{q}}^\ast
	a_{\we{m}-\we{w}}a_{\we{l}-\we{w}}^\ast a_{\we{l}} a_{\we{k'}}^\ast
	a_{\we{k'}-\we{t}}a_{\we{k}-\we{t}}^\ast a_{\we{k}+\we{q}}+c.c..
\label{H49}
\end{equation}

Five of these interactions, viz,~$H_4^1$, $H_4^2$, $H_4^3$, $H_4^4$, $H_4^7$, contain
the~operator~$n_{\we{k}}$, therefore they are not reducible to the 4-fermion MT
potential~\eqref{V}. This potential is particularly interesting not only for its
possible relevance to the~physics of superconductors, but also because
the~thermodynamics of the~Hamiltonian~$H_{BCS}+W+V_{MT}$ is exactly solvable
(Brankov~et~al.\cite{brankov}).

Let us consider the~8-fold product of operators in~$H_4^5$ with explicit spin indices:
\begin{equation*}
	a_{\we{m}\sigma''}^\ast a_{\we{m}-\we{w}\sigma''}a_{\we{l}-\we{w}\sigma'''}^\ast
	a_{\we{l}-\we{q}\sigma'''} a_{\we{k'}-\we{q}\sigma}^\ast
	a_{\we{k'}-\we{t}\sigma}a_{\we{k}-\we{t}\sigma'}^\ast
	a_{\we{k}\sigma'}.
\end{equation*}
With respect to 1-fermion momenta these are nine possible reductions to
the~form~\eqref{V}, and each of them allows two or four possibilities related to spin
reduction. The~values assumed by each momentum index are collected in
Table~\ref{tab1}. In four cases~$\we{q}=0$ and the~coupling vanishes,
since~$\chi(\we{l},\we{m},\we{w},0)=0$, for all~$\we{l}$, $\we{m}$, $\we{w}$.
As a~consequence, $H_4^5$~assumes the~reduced form
\begin{equation}
\begin{split}
        H^5_{4(red)}&=\sum_{\we{k}\ne\we{m}}
        \Bigl\{ 2H_{\we{k},2\we{k}+\we{m},\we{k}+\we{m},2\we{k}+2\we{m}}
        \, \chi(2\we{m}+\we{k},\we{m},\we{k}+\we{m},2\we{k}+2\we{m})\\
	& - H_{\we{k},\we{m},\we{k}+\we{m},2\we{m}}\bigl(\chi(2\we{m}+\we{k},\we{m},
	\we{k}+\we{m},2\we{m})+
	\chi(2\we{m}-\we{k},\we{m},\we{m}-\we{k},2\we{m})\bigr)\\
	& - \chi(\we{k},\we{m},\we{k}+\we{m},2\we{k})\left( H_{\we{k},2\we{k}+\we{m},
	\we{k}+\we{m},2\we{k}} + H_{\we{k},2\we{k}-\we{m},\we{k}-\we{m},2\we{k}}
	\right)\Bigr\}
   	a_{\we{m}-}^\ast  a_{\we{m}+}^\ast a_{-\we{m}-}^\ast  a_{-\we{m}+}^\ast
	a_{-\we{k}+}a_{-\we{k}-} a_{\we{k}+}a_{\we{k}-}+ c.c..\\
\end{split}
\label{red1}
\end{equation}
Terms with~$\we{k}=\we{m}$ have been excluded, similarly as in~$V_{BCS}$. Their
contribution is accounted for by a~shift of 1-fermion energies. The~coefficients on
the~rhs result after performing summation over spins.

\begin{table}
\begin{tabular}{|c|c|c|c|c|c|l|}
	\hline
	$H_4^5$&$\we{l}$&$\we{k'}$&$\we{q}$&$\we{t}$&$\we{w}$&spins\\\hline
	1&$2\we{m}+\we{k}$&$2\we{k}+\we{m}$&$2\we{k}+2\we{m}$&$\we{k}+\we{m}$&$\we{k}+
	\we{m}$&$\sigma''=-\sigma'''\text{,}\sigma=-\sigma'$\\\hline
	2&$2\we{m}+\we{k}$&$\we{m}$&$2\we{m}$&$\we{k}+\we{m}$&$\we{k}+\we{m}$&$\sigma''=-
	\sigma'''=-\sigma=\sigma'$\\\hline
	3&$2\we{m}-\we{k}$&$\we{m}$&$2\we{m}$&$\we{k}+\we{m}$&$\we{m}-\we{k}$&$\sigma''=-
	\sigma'''=\sigma=-\sigma'$\\\hline
	4&$\we{k}$&$2\we{k}+\we{m}$&$2\we{k}$&$\we{k}+\we{m}$&$\we{k}+\we{m}$&$\sigma''=-
	\sigma=-\sigma'''=\sigma'$\\\hline
	5&$\we{k}$&$\we{m}$&$0$&$\we{k}+\we{m}$&$\we{k}+\we{m}$&$\sigma''=-\sigma\text{,
	}\sigma'''=-\sigma'$\\\hline
	6&$-\we{k}$&$\we{m}$&$0$&$\we{k}+\we{m}$&$\we{m}-\we{k}$&$\sigma''=-\sigma=
	\sigma'''=-\sigma'$\\\hline
	7&$\we{k}$&$2\we{k}-\we{m}$&$2\we{k}$&$\we{k}-\we{m}$&$\we{k}+\we{m}$&$\sigma''=-
	\sigma'=\sigma=-\sigma'''$\\\hline
	8&$\we{k}$&$-\we{m}$&$0$&$\we{k}-\we{m}$&$\we{k}+\we{m}$&$\sigma''=-\sigma'=-
	\sigma=\sigma'''$\\\hline
	9&$-\we{k}$&$-\we{m}$&$0$&$\we{k}-\we{m}$&$\we{m}-\we{k}$&$\sigma''=-\sigma'
	\text{,}\sigma'''=-\sigma$\\\hline
\end{tabular}
\caption{Reductions of $H_4^5$}
\label{tab1}
\end{table}

This procedure has been also applied to~$H_4^6$, $H_4^8$ $H_4^9$. The~corresponding
values of 1-fermion momenta are given in Tables~\ref{tab2}, \ref{tab3}, \ref{tab4}.
The~additional remark~$\we{k}\to\we{k}-2\we{m}$ (or similar), indicates that
a~translation of one momentum index is necessary after reduction.

\begin{table}
\begin{tabular}{|c|c|c|c|c|c|l|c|}
	\hline
	$H_4^6$&$\we{l}$&$\we{k'}$&$\we{q}$&$\we{t}$&$\we{w}$&spins&\\\hline
	1&$-\we{k}$&$-\we{m}$&$-2\we{k}-2\we{m}$&-$\we{k}-\we{m}$&$\we{k}+
	\we{m}$&$\sigma''=-\sigma'''\text{,}\sigma=-\sigma'$&$\we{m}\to\we{m}-
	2\we{k}$\\\hline
	2&$\we{k}$&$-\we{m}$&$-2\we{m}$&$\we{k}-\we{m}$&$\we{k}+\we{m}$&$\sigma''=-
	\sigma'''=-\sigma=\sigma'$&\\\hline
	3&$-\we{k}$&$-\we{m}$&$-2\we{m}$&$\we{k}-\we{m}$&$\we{m}-\we{k}$&$\sigma''=-
	\sigma'''=\sigma=-\sigma'$&\\\hline
	4&$-\we{k}$&$4\we{k}+\we{m}$&$2\we{k}$&$3\we{k}+\we{m}$&$\we{k}+\we{m}$&$
	\sigma''=-\sigma=-\sigma'''=\sigma'$&$\we{m}\to\we{m}-2\we{k}$\\\hline
	5&$\we{k}$&$\we{m}$&$0$&$\we{k}+\we{m}$&$\we{k}+\we{m}$&$\sigma''=-\sigma\text{,
	}\sigma'''=-\sigma'$&\\\hline
	6&$-\we{k}$&$\we{m}$&$0$&$\we{k}+\we{m}$&$\we{m}-\we{k}$&$\sigma''=-\sigma=
	\sigma'''=-\sigma'$&\\\hline
	7&$-\we{k}$&$-\we{m}$&$2\we{k}$&$-\we{k}-\we{m}$&$\we{k}+\we{m}$&$\sigma''=-
	\sigma'=\sigma=-\sigma'''$&$\we{m}\to\we{m}-2\we{k}$\\\hline
	8&$\we{k}$&$-\we{m}$&$0$&$\we{k}-\we{m}$&$\we{k}+\we{m}$&$\sigma''=-\sigma'=-
	\sigma=\sigma'''$&\\\hline
	9&$-\we{k}$&$-\we{m}$&$0$&$\we{k}-\we{m}$&$\we{m}-\we{k}$&$\sigma''=-\sigma'
	\text{,}\sigma'''=-\sigma$&\\\hline
\end{tabular}
\caption{Reductions of $H_4^6$}
\label{tab2}
\end{table}

\begin{table}
\begin{tabular}{|c|c|c|c|c|c|l|c|}
	\hline
	$H_4^8$&$\we{l}$&$\we{k'}$&$\we{q}$&$\we{t}$&$\we{w}$&spins&\\\hline
	1&$-\we{k}$&$-\we{m}$&$-2\we{k}-2\we{m}$&$\we{k}+\we{m}$&$-\we{k}-\we{m}$&$
	\sigma''=-\sigma'''\text{,}\sigma=-\sigma'$&$\we{k}\to\we{k}-2\we{m}$\\\hline
	2&$\we{k}+4\we{m}$&$-\we{m}$&$2\we{m}$&$\we{k}+\we{m}$&$\we{k}+3\we{m}$&$
	\sigma''=-\sigma'''=-\sigma=\sigma'$&$\we{k}\to\we{k}-2\we{m}$\\\hline
	3&$-\we{k}$&$-\we{m}$&$2\we{m}$&$\we{k}+\we{m}$&$-\we{m}-\we{k}$&$\sigma''=-
	\sigma'''=\sigma=-\sigma'$&$\we{k}\to\we{k}-2\we{m}$\\\hline
	4&$-\we{k}$&$\we{m}$&$-2\we{k}$&$\we{k}+\we{m}$&$-\we{k}+\we{m}$&$\sigma''=-
	\sigma=\sigma'=-\sigma'''$&\\\hline
	5&$\we{k}$&$\we{m}$&$0$&$\we{k}+\we{m}$&$\we{k}+\we{m}$&$\sigma''=-\sigma\text{,
	}\sigma'''=-\sigma'$&\\\hline
	6&$-\we{k}$&$\we{m}$&$0$&$\we{k}+\we{m}$&$\we{m}-\we{k}$&$\sigma''=-\sigma=-
	\sigma'=\sigma'''$&\\\hline
	7&$-\we{k}$&$-\we{m}$&$-2\we{k}$&$\we{k}-\we{m}$&$-\we{k}+\we{m}$&$\sigma''=-
	\sigma'=\sigma=-\sigma'''$&\\\hline
	8&$\we{k}$&$-\we{m}$&$0$&$\we{k}-\we{m}$&$\we{k}+\we{m}$&$\sigma''=-\sigma'=-
	\sigma=\sigma'''$&\\\hline
	9&$-\we{k}$&$-\we{m}$&$0$&$\we{k}-\we{m}$&$\we{m}-\we{k}$&$\sigma''=-\sigma'
	\text{,}\sigma'''=-\sigma$&\\\hline
\end{tabular}
\caption{Reductions of $H_4^8$}
\label{tab3}
\end{table}

\begin{table}
\begin{tabular}{|c|c|c|c|c|c|l|l|}
\hline
	$H_4^9$&$\we{l}$&$\we{k'}$&$\we{q}$&$\we{t}$&$\we{w}$&spins &\\\hline
	1&$-\frac{1}{3}\we{k}+\frac{2}{3}\we{m}$&$\frac{2}{3}\we{k}-\frac{1}{3}\we{m}$&$-
	\frac{2}{3}\we{k}-\frac{2}{3}\we{m}$&$\frac{1}{3}\we{k}+\frac{1}{3}
	\we{m}$&$\frac{1}{3}\we{k}+\frac{1}{3}\we{m}$&$\sigma''=-\sigma'''\text{,}\sigma
	=-\sigma'$&$\we{k}\to3\we{k}+2\we{m}\text{,}\we{m}\to\we{m}-2\we{k}$\\\hline
	2&$\we{k}-2\we{m}$&$\we{m}$&$-2\we{m}$&$\we{k}-\we{m}$&$\we{k}-\we{m}$&$\sigma''=-
	\sigma'''=-\sigma=\sigma'$&$\we{k}\to\we{k}+2\we{m}$\\\hline
	3&$2\we{m}-\we{k}$&$\we{m}$&$-2\we{m}$&$\we{k}-\we{m}$&$3\we{m}-\we{k}$&$\sigma''
	=-\sigma'''=\sigma=-\sigma'$&$\we{k}\to\we{k}+2\we{m}$\\\hline
	4&$\we{k}$&$-2\we{k}+\we{m}$&$-2\we{k}$&$-\we{k}+\we{m}$&$-\we{k}+\we{m}$&$
	\sigma''=-\sigma=\sigma'=-\sigma'''$&$\we{m}\to\we{m}+2\we{k}$\\\hline
	5&$\we{k}$&$\we{m}$&$0$&$\we{k}+\we{m}$&$\we{k}+\we{m}$&$\sigma''=-\sigma\text{,
	}\sigma'''=-\sigma'$&\\\hline
	6&$-\we{k}$&$\we{m}$&$0$&$\we{k}+\we{m}$&$\we{m}-\we{k}$&$\sigma''=-\sigma=-
	\sigma'=\sigma'''$&\\\hline
	7&$\we{k}$&$2\we{k}-\we{m}$&$-2\we{k}$&$3\we{k}-\we{m}$&$-\we{k}+\we{m}$&$
	\sigma''=-\sigma'=\sigma=-\sigma'''$&$\we{m}\to\we{m}+2\we{k}$\\\hline
	8&$\we{k}$&$-\we{m}$&$0$&$\we{k}-\we{m}$&$\we{k}+\we{m}$&$\sigma''=-\sigma'=-
	\sigma=\sigma'''$&\\\hline
	9&$-\we{k}$&$-\we{m}$&$0$&$\we{k}-\we{m}$&$\we{m}-\we{k}$&$\sigma''=-\sigma'
	\text{,}\sigma'''=-\sigma$&\\\hline
\end{tabular}
\caption{Reductions of $H_4^9$}
\label{tab4}
\end{table}

After reduction, $H_4^6$, $H_4^8$, $H_4^9$~take the~following forms:
\begin{equation}
\begin{split}
	& H^6_{4(red)}=\sum_{\we{k}\ne\we{m}} \Bigl\{ 2H_{\we{k},2\we{k}+\we{m},\we{k}+
	\we{m},2\we{k}+2\we{m}} \, \varphi(-\we{k},-2\we{k}-\we{m},-\we{k}-\we{m},2\we{k}
	+2\we{m})\\
	&-H_{\we{k},\we{m},\we{k}+\we{m},2\we{m}}\bigl( \varphi(\we{k},-\we{m},\we{k}-
	\we{m},2\we{m})+ \varphi(-\we{k},-\we{m},-\we{m}-\we{k},2\we{m})\bigr)\\
	&-\varphi(-\we{k},\we{m}-2\we{k},-\we{k}+\we{m},2\we{k})\bigl( H_{\we{k},2\we{k}+
	\we{m},\we{k}+\we{m},2\we{k}}+H_{\we{k},2\we{k}-\we{m},\we{k}-\we{m},2\we{k}}
	\bigr) \Bigr\}
	a_{\we{m}-}^\ast  a_{\we{m}+}^\ast  a_{-\we{m}-}^\ast  a_{-\we{m}+}^\ast
	a_{-\we{k}+}a_{-\we{k}-} a_{\we{k}+} a_{\we{k}-}+c.c.,\\
\end{split}
\label{red2}
\end{equation}
\begin{equation}
\begin{split}
	H^8_{4(red)}&=\sum_{\we{k}\ne\we{m}} \Bigl\{ 2I_{-\we{k}-2\we{m},-\we{m},-\we{k}-
	\we{m},2\we{k}+ 2\we{m}} \, \chi(\we{k}+2\we{m},\we{m},\we{k}+\we{m},2\we{k}+
	2\we{m})\\
	&-I_{\we{k}-2\we{m},-\we{m},\we{k}-\we{m},2\we{m}}\bigl(\chi(\we{k}+2\we{m},
	\we{m},\we{k}+\we{m},2\we{m})+\chi(-\we{k}+2\we{m},\we{m},\we{m}-\we{k},2\we{m})
	\bigr)\\
	&-\chi(\we{k},\we{m},\we{k}+\we{m},2\we{k})\bigl( I_{-\we{k},\we{m},-\we{k}+
	\we{m},2\we{k}}+ I_{-\we{k},-\we{m},-\we{k}-\we{m},2\we{k}}\bigr)\Bigr\}
	a_{\we{m}-}^\ast  a_{\we{m}+}^\ast  a_{-\we{m}-}^\ast  a_{-\we{m}+}^\ast
	a_{-\we{k}+}a_{-\we{k}-} a_{\we{k}+} a_{\we{k}-} + c.c.,\\
\end{split}
\label{red3}
\end{equation}
\begin{equation}
\begin{split}
	& H^9_{4(red)}=\sum_{\we{k}\ne\we{m}} \Bigl\{ 2I_{-\we{k}-2\we{m},-\we{m},-\we{k}-
	\we{m},2\we{k}+	2\we{m}} \, \varphi(-\we{k},-2\we{k}-\we{m},-\we{k}-\we{m},
	2\we{k}+2\we{m})\\
	&-I_{\we{k}-2\we{m},-\we{m},\we{k}-\we{m},2\we{m}}\bigl( \varphi(\we{k},-\we{m},
	\we{k}-\we{m},2\we{m}) + \varphi(-\we{k},-\we{m},-\we{k}-\we{m},2\we{m})\bigr)\\
	&-\varphi(-\we{k},-2\we{k}+\we{m},-\we{k}+\we{m},2\we{k})\bigl( I_{-\we{k},
	\we{m},-\we{k}+\we{m},2\we{k}}+I_{-\we{k},-\we{m},-\we{k}-\we{m},2\we{k}}
	\bigr)\Bigr\}
	\times a_{\we{m}-}^\ast  a_{\we{m}+}^\ast  a_{-\we{m}-}^\ast  a_{-\we{m}+}^\ast
	a_{-\we{k}+} a_{-\we{k}-} a_{\we{k}+} a_{\we{k}-} + c.c..\\
\end{split}
\label{red4}
\end{equation}
Collecting all terms in Eqs.~\eqref{red1}--\eqref{red4}, one obtains a~$V_{MT}$
interaction as in Eq.~\eqref{V} with
\begin{equation}
        g_{\we{m}\we{k}}=-4\omega_{2\we{k}+2\we{m}}
	\Lambda_{\we{k}\we{m}}\Lambda_{\we{m}\we{k}}^\ast+
	2\Theta_{\we{k}\we{m}}+2\Theta_{\we{m}\we{k}}^\ast,
\label{gmk}
\end{equation}
where
\begin{equation}
        \Lambda_{\we{k}\we{m}}=\chi^\ast(2\we{k}+\we{m},\we{k},\we{k}+\we{m},2\we{k}+
	2\we{m}) + \varphi^\ast(-\we{m},-\we{k}-2\we{m},-\we{k}-\we{m},2\we{k}+2\we{m}),
\label{Lambda}
\end{equation}
\begin{equation}
\begin{split}
	& \Theta_{\we{k}\we{m}} =\omega_{\we{m}}\bigl( \chi^\ast(\we{m},\we{k},\we{k}+
	\we{m},2\we{m})+\varphi^\ast(-\we{m},\we{k}-2\we{m},\we{k}-\we{m},2\we{m})
	\bigr) \\
       &\times \Bigl( \chi(2\we{m}+\we{k},\we{m},\we{k}+\we{m},2\we{m})+\chi(2\we{m}-
       \we{k},\we{m},\we{m}-\we{k},2\we{m})
       +\varphi(\we{k},-\we{m},\we{k}-\we{m},2\we{m})+\varphi(-\we{k},-\we{m},-\we{k}-
       \we{m},2\we{m})\Bigr).\\
\end{split}
\label{Theta}
\end{equation}
\section{4-fermion interaction coupling}
The~most important question about the~4-fermion interaction is positive- or
negative-valuedness of~$g_{\we{m}\we{k}}$. In particular, it would be of interest to
determine the~domain of~$g_{\we{m}\we{k}}$ in momentum space where it is attractive.
Unfortunately, the~form of~$g_{\we{m}\we{k}}$ given by Eq.~\eqref{gmk} is extremely
complicated, so this problem cannot be resolved in general.

First, let us specify the~quantities occurring in Eq.~\eqref{gmk}. We assume that
$\varepsilon_{\we{k}}=a k^2$, $\omega_{\we{k}} = b k$, $\tilde{\Gamma}_{\we{k}} = c k$,
$\Gamma_{\we{k}}= d k$ and $\hat{\Gamma}_{\we{k}} = e k$, where $a$, $b$, $c$, $d$,
$e$ are real, positive constants.

Following Davydov\cite{davydov}, we apply some approximations in order to
estimate~$g_{\we{m}\we{k}}$. We assume that the~most significant contribution to
the~4-fermion~interaction (analogously as in the~BCS~theory) comes from 1-electron
momenta $\we{m}$, $\we{k}$ which satisfy the condition $m \approx k$.

The~case of exact equality is, of course, excluded (in accord with the~restriction
on summation in Eqs.~\eqref{red1} -- \eqref{red4}), so we use $g_{\we{k}\we{k}}$ only
as an~abbreviation for~$g_{\we{m}\we{k}}$ under our approximation $m \approx k$.

Thus, the~interaction coupling~$g_{\we{k}\we{k}}$ is given by
\begin{equation}
	g_{\we{k}\we{k}}=-4\omega_{4\we{k}}\Lambda_{\we{k}\we{k}}
    	\Lambda_{\we{k}\we{k}}^\ast + 2\Theta_{\we{k}\we{k}}+
    	2\Theta_{\we{k}\we{k}}^\ast.
\label{gkk}
\end{equation}
We take into account only vectors with almost compatible directions, otherwise
$g_{\we{k}\we{k}}$~vanishes.

We can thus rewrite Eq.~\eqref{gkk} in the~form:
\begin{equation}
	g_{\we{k}\we{k}}=-16\omega_{\we{k}}|\Lambda_{\we{k}\we{k}}|^2
  	+ 4 \Re \Theta_{\we{k}\we{k}},
\label{gkk1}
\end{equation}
where the~first term is always nonpositive and $\Re$ means real part. The~second
term requires detailed analysis. Detailed calculations are performed in Appendix~\ref{c}.
We have  found~$g_{\we{k}\we{k}}$ in all cases for different values of
the~constants~$a$, $b$, $c$, $d$, $e$. Most interesting is the case $b>c$,
$b>e$ and $b > d$, because the~essential part of initial Hamiltonian is transformed
under these conditions. Under a~further approximation, we find the form
of~$g_{\we{k}\we{k}}$ for~$k \approx k_F$, where $k_F$~is Fermi~momentum.
Additionally we put $c=d=e$ and $\varepsilon_{\we{k}_F} \gg \omega_{\we{k}_F}$
(this is at least true for metals). Under these assumptions
\begin{equation*}
	\Theta_{\we{k}_F \we{k}_F} \approx
	\frac{-D_{\we{k}_F}^6}{4\omega_{\we{k}_F}^5}, \qquad
	\Lambda_{\we{k}_F \we{k}_F}\approx
	\frac{iD_{\we{k}_F}^3}{12\omega_{\we{k}_F}^2 \varepsilon_{\we{k}_F}},
\end{equation*}
which implies
\begin{equation*}
	g_{\we{k}_F \we{k}_F} \approx \frac{-D_{\we{k}_F}^6}{\omega_{\we{k}_F}^5}
	\label{g_kF} < 0.
\end{equation*}

Now we can compare the~magnitudes of the~coupling constants of
4-fermion interaction and 2-fermion interaction (under the~same assumptions).
It is shown in Appendix~\ref{a} that $G_{\we{k}_F\we{k}_F}=
-D_{\we{k}_F}^2 / \omega_{\we{k}_F}$. Thus the~interaction
coupling for 4- and 2-fermion (BCS) interactions fulfils the~relation
$g_{\we{k}_F \we{k}_F}=G_{\we{k}_F\we{k}_F}^3 / \omega_{\we{k}_F}^2$. As a~consequence,
for a~strong pairing, there is a~significant contribution from 4-fermion interactions.

Moreover, the~4-fermion interaction coupling~$g_{\we{k}\we{k}}$ is
also negative in most cases without imposing the~approximations~$c=d=e$
and~$k\approx k_F$. This is shown in Appendix~\ref{c}.
\section{\label{6}4-fermion interactions in Fr\"ohlich's expansion}
As demonstrated in Section~\ref{3}, 4-fermion interaction terms appear in sixth order
of the~expansion of Fr\"ohlich's~original transformation. These terms are derived in
Appendix~\ref{b}. The~various resulting 4-fermion terms considerably outnumber those
obtained by applying a~second~transformation in Section~\ref{5}. Again a~reduction
procedure to~$V_{MT}$ is possible for all 4-fermion expressions~\eqref{fr_6ord}, which
do~not contain the~particle number operator and possess three phonon indices. However,
the~resulting couplings are also complicated functions and, therefore, will not be
examined in detail.

The~disadvantage of this method is appearance of 3-fermion interactions, which in
the~double transformation method gave a~small contribution. Here additional arguments
must be used to discard these terms.
\section{\label{7}Concluding remarks}
We have extended Fr\"ohlich's~transformation of~$H_{\mathrm{e-ph}}$ to higher order
terms. This has been done by performing a~second transform of the~first terms discarded
by~Fr\"ohlich. The~resulting interactions are of 3- and 4-fermion type. The~3-fermion
terms can be expected to be inessential in superconductivity because of their instability
under time inversion. The~4-fermion terms are, in general, reducible
to~$V_{\mathrm{MT}}$, a~BCS-type 4-fermion interaction. The~resulting 4-fermion coupling
is extremely complicated, but under reasonable approximations it~is negative-valued.
Moreover, for 1-fermion momenta in the~neighbourhood of Fermi~momentum, it~has the~simple
form~$-D_{\we{k}_F}^6 / \omega_{\we{k}_F}^5 = G_{\we{k}_F\we{k}_F}^3 /
\omega_{\we{k}_F}^2$, where $G_{\we{k}_F\we{k}_F}$ is the~2-fermion
coupling. This fact implies that 4-fermion interactions are significant
for systems with strong pairing and allows to estimate (relative to 2-fermion coupling
and phonon energy at Fermi~momentum) their magnitude.

As in BCS~theory, where the~relation between the~gap parameter~$\Delta$ and coupling
constant~$G$ of the BCS~interaction allows to estimate the~magnitude of~$G$, it can be
expected that detailed thermodynamics of~$H_{MT}$ will provide a~relation
between~$|g_{\we{m}\we{k}}|$ and other parameters of the~theory, thereby allowing to
estimate the~magnitude of~$g_{\we{m}\we{k}}$. Another emerging question is Cooper's
problem\cite{cooper} for a~bound quadruple in the~presence of~$V_{\mathrm{MT}}$
or~$V_{BCS}+V_{MT}$. Kamei and~Miyake deal with this question in a~recent work\cite{kamei}.

The~double transformation has unveiled the~structure of 3-, 4-, 5-fermion interactions.
Since they are proportional to 4th, 6th, 8th~power of~$D_{\we{w}}$, they are relatively
weak, so inclusion of these terms, apart from 4-fermion ones, appears unjustified at
present, although Schneider~et~al.\cite{schn:2} suggest existence of both quadruples and
sextets in some HTSC.

The~higher-order expansion terms of the~transformed~$H_{\mathrm{e-ph}}$, including
quadruple, sextet etc.\  interactions can be expected to reveal themselves in materials
with extremely strong pairing correlations between spin $1/2$ fermions, since
the~presence of strongly correlated pairs implies at least some kind of weak interaction
between them.

On the other hand, our extension of Fr\"ohlich's~transformation shows that if phonon
mediation exists in a~superconductor, the~4-fermion and 6-fermion interactions are always
present as supplementary to the~BCS~one. Unfortunately, the~Hamiltonian~$H_{\mathrm{BCS}}+
V_{\mathrm{MT}}$, although exactly solvable, leads to intricate mean-field equations.

Our results obtained can be generalized in many respects. First of all, the~process of
averaging over phonon vacuum could be replaced by averaging over the~phonon equilibrium
state, which could be justified at higher temperatures. This would already lead to
an~additional 1-fermion term in Fr\"ohlich's~Hamiltonian and modification of 2-fermion
and 3-fermion terms in our method. Another extension would result by going beyond
the~harmonic approximation and including all products of phonon operators. Method of
Bogoliubov\cite{bogol:1} could be applied to ``dangerous terms'' (divergent), omitted in
Fr\"ohlich's~method. These questions will be dealt with in further investigations.

redpajama_set_name: RedPajamaArXiv


1. Assimilate. Consciously disconnect your sense of identity from all groups of which you're a member to maximize potential for assimilation. When our social groups become our identity it is manifested in our behaviors that, consequentially, may force others to cling harder to theirs. Be a good representative but be yourself.
2. Home-stay: Know if you're an extrovert or extravert and make sure you're paired appropriately. Be candid when completing paperwork. The application is your chance to better try to specify your living situation abroad.
3. Academics: Pick out classes at your host institution early and maintain communication with professors and the program coordinator(s). Course structures – language, format, etc. change more than most Americans may be used to. Be flexible.
4. Read Everything: Pay close attention to and take advice given in all the documentation. For example, secure a power of attorney BEFORE you depart. You can never be certain of what will transpire in your absence and getting a power of attorney abroad may be extremely different and much more expensive. Or consider the pre-program language session. And figure out visa requirements. Information is out there but you have to act.


text:
Ramat Gilad Residents Prefer Their Mobile Homes to Luxury Homes
Nadav Shragai
Send me email alerts for new articles by Nadav Shragai
Last Shabbat marked five years since the murder of Gilad Zar, the security coordinator of the Samaria Regional Council. His sister, Michal Shoham, recalls that after the family completed the week-long mourning period, her father vowed on Gilad's grave that he would establish six settlements in his memory, one for each Hebrew letter of Gilad Zar's name.
Two outposts have already been established: Ramat Gilad - where Shoham lives with her husband, Shmuel, and their 11 children - and Havat Gilad. The Zar family members, who are scattered throughout Hebron and various West Bank outposts and hilltops, have no doubt that the other four will soon be built.
The hilltop of Ramat Gilad, near the Karnei Shomron settlement, practically touches the hilltop where Moshe Zar, the father of Michal and her six surviving siblings, built his home in 1979. Shoham said that she and the other residents of Ramat Gilad will defend their homes "the way a mother defends her child from a tiger's claws. For us, this place is Gilad's tombstone, a tombstone that speaks to me a lot more than the one in the cemetery. It's impossible to flip a switch and say, 'Now we'll get rid of them.'"
Gilad Zar, one of the founders of the settlement of Itamar, was shot dead by Palestinian terrorists on May 29, 2001 while driving between the West Bank settlements of Kedumim and Yitzhar. After his car overturned, gunmen shot him in the head at point-blank range. In addition to his parents and siblings, Zar, who was 41 when he died, is survived by his wife, Hagar, and eight children.
Shoham said that even though her family was joined at the outpost by only one other family in the first two years of living there, and even though they did not always have access to running water, she still prefers her current abode.
"The comfort that we didn't find in the fancy house in Karnei Shomron we found in a rickety caravan that was dragged here from an absorption site for Ethiopians. For three months we lived here without water," she said. "The Israel Defense Forces guarded the bringing of the first caravans here. There are pictures, so that no one can deny it. But afterward, the Civil Administration argued that the structures here are 'not authorized.'"
After that, the Ehud Barak government removed the caravan being used on Ramat Gilad as a synagogue, only to bring it back later, said Shoham, adding that the outpost was threatened with evacuation about 18 months ago.
"Today, heaven forbid, we face a similar threat, and I can promise that here there will be no hugs," she said, referring to last summer's disengagement from the Gaza Strip. "We will not go together to the buses."
Shoham and her family are not the first to take an interest in this particular hilltop. Ramat Gilad used to be a Jordanian military post, and there are still active minefields in its foothills. The outpost provides a good view of the surroundings, and Ariel Sharon, who used to be a close friend of Moshe Zar, used to bring some of his guests up there to take a look around.
Now seven families live in Ramat Gilad, including Michal Shoham and her family and two of the Shohams' married children, as well as a young couple from Tel Aviv and the daughter of the rabbi of Karnei Shomron. On the wall of the Shohams' living room hangs a picture of Tiferet Yisrael - an outpost one of their sons founded in Gush Katif.
Sharon the enemy
And what does Moshe Zar think about his friend Sharon's decision to evacuate Gush Katif? Zar and his wife think he is just like first-century historian Josephus Flavius.
"He too was a hero who was not afraid and who led the camp," said Zar. "But one day he turned his back and became an enemy, but not just any enemy - someone who is very familiar with all the strategies of his people, who knows the strongholds, the combat methods, and uses all this information to defeat his people, to fight them and destroy them. So too with Ariel Sharon, who built all the settlements and knows us and was part of us. He turned his back and took advantage of his close familiarity with us in order to destroy and expel."
Haaretz Contributor

redpajama_set_name: RedPajamaCommonCrawl


It is no secret that twentieth-century Britain was governed through a culture of secrecy, and secrecy was particularly endemic in military research and defence policy surrounding biological and chemical warfare. More generally, it is hard to exaggerate the role of secrecy in all past biological and chemical warfare programmes and several recent historical surveys of biological and chemical warfare research have emphasised that all state sponsored programmes, together with sub-state organised activities, were cloaked in utmost secrecy. Of these research programmes, Britain carried out one of the most significant in scale and scope in the twentieth century. Yet, partly because of the secrecy surrounding the programme, there is still little academic literature on its historical development. Equally, and despite secrecy being a pervasive feature of past and contemporary societies, social scientists and historians have paid relatively little scholarly attention to the nature, mechanics and effects of secrecy, particularly with regard to secrecy in relation to the production and governance of science and technology. Drawing on classical sociological writing on secrecy by Simmel, Merton and Shils this groundbreaking book by Brian Balmer draws on recently declassified documents to investigate significant episodes in the history of biological and chemical warfare. At the same time, it draws on more contemporary perspectives in science and technology studies that understand knowledge and social order as co-produced within heterogeneous networks of 'things and people' in order to develop a theoretical set of arguments about how the relationship between secrecy and science might be understood.


Balsam pillows, maple syrup, spruce gum, custom-made rustic furniture — they're all products comprised of raw materials native to the Adirondacks. Other businesses, current or defunct, have similar roots, but occasionally in regional history we find homegrown livelihoods that seem an odd fit for the North Country. Among the unlikeliest of those is pearl harvesting — not in the St. Lawrence River or Lake Champlain, but in creeks and rivers of the Adirondacks and foothills.
The New York State Department of Environmental Conservation (DEC) has acquired 17 acres of woodland in the historic maritime hamlet of Setauket. The property will be added to Patriots Hollow State Forest.
Cold and flu season once again has sufferers scrambling for any kind of relief from all sorts of medicines. A little over a century ago, right here on Northern New York store shelves, next to cough drops by national companies like Smith Brothers and Luden's, was a local product made in Malone.
Sprucelets were created mainly from a raw material harvested in the Adirondacks: spruce gum. Like hops, blueberries, and maple syrup, the seasonal gathering and sale of spruce gum boosted the incomes of thousands of North Country folks seeking to make a dollar any way they could. Much of what they picked was sold to national gum companies, but some was used locally by entrepreneurs who established small factories and created many jobs.
An interactive and inclusive exhibit, Hunting the Whale: The Rise and Fall of a Southampton Industry adds new discoveries to the accumulation of documentation and artifacts collected over more than 100 years to illuminate Southampton Village's prominent role in the whaling industry at its mid-19th century height.
Whaling tools, maps, illustrations, archival images and text will be displayed with an eye toward making the exhibit accessible to audiences of varied interests and all ages. Among those whose roles will be highlighted are local indigenous people, slaves, servants, whaling captains, and the families that were sustained by the whaling industry.


Realization of a "metal origami" mural, on the occasion of the opening of the URBAN NATION museum, dedicated to urban contemporary art in Berlin.
The artist has invited his partner Snez (with whom she forms the duo Emes Crew), to make a "Mougeon" and pixo writing at the top of the wall.
Banksy, Shepard Fairey, Vhils, Icy and Sot, Marina Zumi, Jef Aérosol, Blek le Rat, Tilt, C215, Miss Van, JR, Seth Globepainter, Space invader, 1UP, etc.
Thanks to Yasha Young, museum director for her invitation and welcome and big up to all the team of Urban Nation !


Anecdote: Back in my college DJ days, I was on the air playing "Speak Like A Child," when a friend called me. He told me how he and some of his buddies were hanging around their apartment listening to our fine station. When they heard the Style Council over the airwaves, they said they came up with one conclusion: Karim Amir was on the air. They were taking the piss out of me, but as the lone 80's pop lover, I was used to it.
"Have You Ever Had It Blue?"
The Blow Monkeys, "Digging Your Scene"
Found this in the "related" column next to "Walls Come Tumbling Down." I was obsessed with this song when it came out. The 45 single had the song on both sides: one side was the American version, the other the UK version. Honestly, I don't remember the difference, but I played the hell out of both sides.
All the Jam-lovin' Mods in my circle of friends as a teen hated The Style Council with a misguided passion. At the time, I thought "Walls Come Tumbling Down" and "My Ever Changing Moods" were great pop songs, but kept that opinion secret from the close minded herd I ran with.
Oh, I'm sorry I gave NuBz18 credit for the Southern Belles quote in my recent RT journal entry instead of you. I must have had a brain-freeze or something.


This elegant roll-on PETAL Attar concentrate is steeped in the traditions of ancient Persia, where attars were used for anointing. Attar means "to smell sweet." A beautiful way to enjoy a scent pick-me-up anytime throughout the day, the PETAL Attar holds scent close to you without overpowering. Delightfully portable, ideal for travel, airline-compliant, PETAL Attar is an ideal companion on your amazing daily journey… from yoga, to lunch with dear friends, to the symphony, to wherever life takes you.


text:
Scientists Find Secret To 'Uncrushable' Bug's Strength
Looking like it was forged in apocalyptic fires, the diabolical ironclad beetle has a formidable reputation for being able to withstand being stabbed or run over by a car.
Now scientists have found a jigsaw-like mechanism in their exoskeletons that helps the little creature tolerate forces up to 39,000 times its own body weight.
Their discovery could have implications for engineering and robotics, experts say, and even challenge the position of cockroaches as models of insect indestructibility.
The beetle, which no longer has its ancestors' ability to fly away from predators, has developed crush-resistant exoskeletal forewings — called elytra — as a form of protection, according to the new study published in the journal Nature.
The beetle, which is roughly two centimetres (just under an inch) long, is an oak-dwelling fungivore primarily residing on the western coast of North America.
In spite of its ferocious name, it tends to scuttle into a hiding place or play dead when attacked, helped by its rough textured, craggy black body that gives it the appearance of a small rock.
"Beyond feigning death, this beetle has a remarkable ability to withstand crushing and piercing strikes from predators, and even the occasional automobile," the study said.
It said they are so tough that entomologists often bend the steel pins they use when they try to mount the beetles in display cases.
Scientists from the United States and Japan looked at the structure of the beetle's elytra, which allow it to withstand forces of up to 149 newtons — approximately 39,000 times its body weight.
They found was significantly more than other similar beetles could tolerate.
Facts about the diabolical ironclad beetle. Photo: AFP / John SAEKI
Using advanced imaging techniques, the researchers saw that the beetle has a series of interlocked jigsaw-shaped joints in its elytra and found that the shape of these blades and their laminated microstructure acted to toughen the exoskeleton.
They concluded that the strong yet flexible structure allows the beetles to hide under rocks or contort themselves into tight spaces in tree bark for shelter, bearing extra weight without damaging their internal organs.
The researchers then used 3D printing and simulations to investigate whether this geometric structure could be used as a mechanical fastener for joining different materials — such as plastics and metal — similar to that required in turbines in the aerospace industry.
The study found that these diabolical ironclad beetle-inspired designs provided enhanced strength and significantly increased toughness compared to a commonly used engineering joint.
Po-Yu Chen, in the Department of Materials Science and Engineering at Taiwan's National Tsing Hua University, said the weight the insect was able to withstand was roughly 10 times higher than the biting forces generated by potential predators.
"Cockroaches have a similar shape-changing ability, which has inspired the design of a compressible robot that can squeeze into, and move within, tight spaces. Such robots could be used to search for survivors in collapsed buildings after disasters," he said in a commentary in Nature.
Po-Yu Chen, who was not connected to the research, said more research was needed into the structure's potential engineering uses, but said the diabolical ironclad beetle "might now inspire new designs for compressible robots, or for armoured vehicles".
"And the authors' work demonstrates that you should never underestimate the capabilities of insects," he added.
Tags: Bugs, find, Scientists, Secret, Strength, Uncrushable
New approach could lead to designed plastics with specific properties
Nasa asteroid mission confident of success


Reduce, Reuse, and Recycle – We'd be entirely remiss if we didn't include a suggestion to reduce, reuse, and recycle everyday materials that are all around us. You can start small and still have a big impact on our planet. Bring reusable shopping bags every time you head to the grocery store; bring along a reusable water bottle instead of throwing out a single-use plastic bottle; try to use more reusable towels and paper products in your home, and shop for used furniture or clothing instead of buying new. Collectively, over an entire year, you and your family can make a big dent in the amount of waste you create each year.
Mind Your Plate – One of the biggest waste generators and contributors to climate change is food production and consumption. Consider the food items you regularly serve your family and make small changes to reduce the waste your household creates, as well as the waste created in producing your food. Try adopting a Meatless Monday plan in which you commit to not eating meat products all day on Mondays. Soon you won't even notice the switch, and you'll be able to add in other meatless days of the week. Try to buy and eat food products that aren't heavily processed and packaged in layers of plastic. Grow as much of your own food as you can, and shop locally for things like honey, vegetables, and herbs at community CSAs or farmers markets.
Walk, Bike, or Ride More – Especially if you live in Northern Colorado, there are many opportunities for alternative transportation that can make a big difference in the amount of driving you do. The less you use your car, the fewer greenhouse gases you emit. Consider taking the bus, riding your bike, or walking to your next outing. We're lucky to live in an area that has made alternative transportation types available – let's use them!
Consider Solar – Tax deductions for residential solar will be reduced by the end of 2019. As you make your green new year's resolutions, consider solar while you can still deduct the full 30% of the cost. Not only will you get a better deal on your system, but your solar panels will also go a long way to reducing your footprint and helping you live a greener life. Click here to learn more about the green benefits of installing solar.
As you implement your green new year's resolutions, contact Sandbox Solar to learn more about installing a solar system on your home or business. Sandbox Solar is a solar panel installation company located in Northern Colorado – serving all of Fort Collins, Loveland, Greeley, Denver, and beyond!. We will help you through the design, development, and installation of solar panels at your home or business, utilizing the highest-quality design tools and components. Click here to learn more about our services and how we can help you live a greener life in 2019!


text:
Learn How To Play A Sport
By: SupermanPosted on November 23, 2020 November 23, 2020
GCLUB Sports is an online community of sports games that allows you to take on the role of an active professional athlete. For every game you play, you get the opportunity to practice and sharpen your skills so you eventually become the next superstar athlete that gets signed by the professional leagues. The players of GCLUB sports are given a wide range of abilities, strengths and weaknesses that allow them to excel at their sport. They are not simply good enough to be the top athlete in the GCLUB community, they must also have the ability to perform under pressure.
The online sports games provide a fun and exciting environment for all age groups to enjoy. There are no age restrictions as to the participation of the players as the game rules are applicable to all ages. Most of the game modes are available in different languages including Spanish, Chinese and French. The players can easily switch between the various languages and play games in the language of their choice without any problem.
The different modes of games include shooting games, team competitions, bowling and many more. You can compete with other people from all around the world by signing up with your GCLUB account. It is simple to use, all you need to do is register with the website and then log in to the site and choose the game mode that suits you.
Online tournaments can be won and there are a lot of ways to earn prize money. This is one of the major reasons why so many people prefer to play this type of online game rather than joining a real life team sport. Learn more information about situs dominoqq terpercaya
The sport modes of GCLUB sports are designed to be fun, so much so that even people who are not familiar with the sport will be able to enjoy the game. The graphics are clear and the game physics are very realistic. The games are designed to be competitive and engaging, so no matter what level you are at, there is always something new to play. There are many categories of games available for beginners to advanced gamers.
These online sports games can help improve your sport knowledge and help you sharpen your skill. If you want to learn how to play a sport from the best, then this is probably the best way to go.
Categories: Game
How Is Online Games Different From Other Video Games?
Buying Rechargeable Batteries For Vaping


text:
Q: Vba selenium facing issues while finding element I've written a script using vba with selenium. Firstly it gets to the main page then placing the word "pump" in the search input and finally select them all by ticking in the checkbox. It is working fine until it comes to select the checkbox. However, upon reaching that line, it throws an error "element not visible". Apparently, it seems that I did nothing wrong. I would be very happy if somebody points out what I'm missing here:
Sub table_data()

Dim driver As New WebDriver

With driver
    .Start "chrome", "http://apps.tga.gov.au/Prod/devices"
    .get "/daen-entry.aspx"
    .Wait 3000
    .FindElementById("disclaimer-accept").Click
    .Wait 3000
    .FindElementById("medicine-name").SendKeys ("pump")
    .Wait 5000
    .FindElementById("medicines-check-all").Click   ''element not visible here
    .Wait 3000
End With
End Sub

Html elements for the section where element should be stored:

                                    <input type="checkbox" class="medicines-check-all">
                                    Devices found for 'pump...'

                                None selected

Here is the link:
"http://apps.tga.gov.au/Prod/devices/daen-entry.aspx"

A: According to provided HTML sample, "medicines-check-all" is not a value of ID attribute, but of class name attribute, so you should try
.FindElementByClassName("medicines-check-all").Click

instead of
.FindElementById("medicines-check-all").Click

A: Instead of clicking on checkbox, you can try with clicking on label as given below.
.FindElementById("medicines-header-text").Click

redpajama_set_name: RedPajamaStackExchange


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This is something I struggle with a lot when looking at the designs of systems or even the code of other systems. I keep having conversations along the lines of "oh we put that in there in case they want to do x, or we put this in there because one day me want to do z" and I can't tell you how many times X, Z was actually needed or actually needed how the programmers envisaged (and its even something I've been guilty of myself). This is also where a good business analyst and domain expert is worth their weight in gold as they can use their experience, judgement and knowledege of the business/business domain to identify the most likely future changes.
Its certainly something I've been guilty of, and probably something I'll continue to be guilty of in the future.
I find it hard to fight the urge to add functionality before its needed (specifically talking about code here), but I try to keep it in my mind and instead make a note about "I wanted to add X here" and if it becomes needed I can easily go back and put it in (or I can even talk to whoever makes the decision about functionality and tell them I thought X may be needed, if that becomes the case).