text:
\section{\label{Introduction}Introduction}

In papers \cite{CQG} and \cite{Bonachon} we have begun a unified study of
decoherence in open and closed systems, with or without dissipation. In this
paper we continue this study focused in some important details of a common
formalism on this subject. As considered in \cite{CQG}, \cite{OmnesP1}, and
\cite{OmnesP2} decoherence is a particular case of one of the phenomenon of
quantum mechanics: irreversibility. Decoherence is just an example of an
irreversible process.

The problem of irreversibility is that when a quantum state $\rho(t)$ follows
a unitary evolution given by the time-operator $\mathcal{U}(t)=e^{-i\frac
{H}{\hbar}t}$, the unitary nature of this evolution prevents the state to
reach equilibrium when $t\rightarrow\infty$. Therefore, if the non-unitary
evolution towards equilibrium is to be accounted for, a further element must
be added to the unitary evolution. From the most general viewpoint, this
element consists in the splitting of the maximal information about the system
into both a relevant part and an irrelevant part: whereas the irrelevant part
is disregarded, the relevant part is retained and its evolution may reach a
final equilibrium situation. It is important to observe that from the more
general point of view, when we speak about the relevant part and the
irrelevant one we are referring to part of the information of the system and
it does not imply a separation into groups of particles that belong to the
system. This last case would only be a particular way of separating the
maximal system information. From our perspective since the split into relevant
and irrelevant part can be performed in many ways, with no privileged
decomposition, there is no need of an unequivocal criterion for deciding where
to place the cut between \textquotedblleft the\textquotedblright\ system and
\textquotedblleft the\textquotedblright\ environment. In paper \cite{EPSA} we
argue that decoherence is a relative phenomenon, better understood from a
closed-system perspective according to which the split of a closed quantum
system into an open subsystem and its environment is just a way of selecting a
particular space of relevant observables of the whole closed system.

In operators language: the maximal information about the system is given by
the space of all potentially possible observables $\mathcal{O}$, i.e. self
adjoint operators in a Hilbert space. The splitting of this maximal
information into both a relevant part and an irrelevant one is done by
choosing the observed part of the system and ignore the rest. Then, usually we
select a particular subspace $\mathcal{O}_{R},$ of the space $\mathcal{O},$ as
the set that gets \ the relevant information. Moreover, we want to emphasize
that choosing to observe a part of the system does not necessarily imply a
loss of dimensionality.

We have already mentioned in \cite{CQG}\ that, to explain decoherence, in all
its possible versions, it is necessary to choose a space of relevant
observables. In this paper we will develop two examples:

\begin{enumerate}
\item In the \textit{Self Induced Decoherence} (SID) (see \cite{CQG},
\cite{Bonachon}, \cite{SID}, and \cite{CastaOrdo}) approach the choice of
$O_{R}\in\mathcal{O}_{R}$ corresponds to the van Hove observables $O_{VH
\in\mathcal{O}_{VH}$ (as it is defined below in eq. (\ref{vH})). This choice
of the relevant observables removes the non relevant observables from the
space $\mathcal{O}$. This choice does not imply that we necessarily ignore the
information about some particles, or that we only observe one subsystem. But
we can just ignore the information of some observables. In this particular
example there is no reduction of the \textquotedblleft size\textquotedblrigh
\ of $\mathcal{O}_{R}=\mathcal{O}_{VH}$, precisely
\begin{equation}
\dim(\mathcal{O})=\dim(\mathcal{O}_{VH}) \label{INT-b-01
\end{equation}
because $\mathcal{O}_{VH}$ is a dense space \cite{vH}. In this paper we
consider the case of systems with continuous spectrum, for the discrete case
(particles with spin) see \cite{Discreto}.

\item Another choice of $O_{R}\in\mathcal{O}_{R}$ can be the
\textit{Environment Induced Decoherence} (EID) choice (see \cite{CQG},
\cite{JPZ}, \cite{EID}), where the space of observables is decomposed in
$\mathcal{O}=\mathcal{O}_{S}\otimes\mathcal{O}_{E}$ and the relevant
observables are
\begin{equation}
O_{R}=O_{S}\otimes I_{E} \label{INT-b-02
\end{equation}
where $O_{S}$ only gets information from a subspace $\mathcal{O}_{S}$ and
$I_{E}$ is the unit operator of the correspondent space $\mathcal{O}_{E}$. In
cases like this, many authors call $S$ the factor space of the \textit{system}
and $E$ the factor space of the \textit{environment}. EID is a formalism with
many choices since we can define different $S$ and $E$.
\end{enumerate}

The expectation values $\left\langle O_{R}\right\rangle _{\rho(t)}=Tr\left(
\rho(t)O_{R}\right)  $ of the observables $O_{R}\in\mathcal{O}_{R}$ in the
state $\rho(t)$ express the relevant information about the system. Of course,
the decision about which observables are to be considered as relevant depends
on the particular purposes of each situation; but without this decision
irreversible evolutions cannot be described.

Based on these ideas the phenomenon of decoherence can be expressed in a
general way leading to a \textit{General Theoretical Framework for
Decoherence} (GTFD) that was presented in a previous paper (see \cite{CQG} and
\cite{Studies}). According to this general framework, the phenomenon of
decoherence can be explained in four general steps:

\begin{enumerate}
\item \textbf{First step.} The space $\mathcal{O}_{R}$ of relevant observables
is defined.

\item \textbf{Second step.} The expectation value $\langle O_{R}\rangle
_{\rho(t)}$, for any $O_{R}\in\mathcal{O}_{R}$, is obtained. This step can be
formulated in two different but equivalent ways:

\begin{itemize}
\item $\langle O_{R}\rangle_{\rho(t)}$ is computed as the expectation value of
$O_{R}$ in the unitarily evolving state $\rho(t)$.

\item A coarse-grained state $\rho_{G}(t)$ is defined with a non-unitary
evolution. The quantum system state and the coarse-grained state are not equal
and they evolve in a different way because we only consider the relevant
observables (see \cite{Mackey} for details). The coarse-grained state is a
state such that if we compute the mean value of a relevant observable using
the quantum system state, then this value must be the same as the mean value
obtained from the same observable using the coarse-grained state, i.e.
\begin{equation}
\langle O_{R}\rangle_{\rho(t)}=\langle O_{R}\rangle_{\rho_{G}(t)}\text{ for
any }O_{R}\in\mathcal{O}_{R}. \label{CG-01
\end{equation}

\end{itemize}

\item \textbf{Third step.} In many cases (see paper \cite{libromax}) it is
proved that $\langle O_{R}\rangle_{\rho(t)}=\langle O_{R}\rangle_{\rho_{G
(t)}$ reaches a final equilibrium value $\langle O_{R}\rangle_{\rho_{\ast}}$:
\begin{equation}
\lim_{t\rightarrow\infty}\langle O_{R}\rangle_{\rho(t)}=\lim_{t\rightarrow
\infty}\langle O_{R}\rangle_{\rho_{G}(t)}=\langle O_{R}\rangle_{\rho_{\ast}}
\label{INT-01
\end{equation}

This also means that the coarse-grained state $\rho_{G}(t)$ evolves towards a
final equilibrium state:
\begin{equation}
\lim_{t\rightarrow\infty}\langle O_{R}\rangle_{\rho_{G}(t)}=\langle
O_{R}\rangle_{\rho_{G\ast}} \label{INT-02
\end{equation}
This is possible because in an infinite dimensional Poincare system time is
infinite. The final equilibrium state $\rho_{G\ast}$ is obviously diagonal in
its own eigenbasis, which turns out to be the final preferred basis. But, from
eqs. (\ref{INT-01}) or (\ref{INT-02}) we cannot say that $\lim_{t\rightarrow
\infty}\rho(t)=\rho_{\ast}$ or $\lim_{t\rightarrow\infty}\rho_{G
(t)=\rho_{G\ast}$. But rigorously the unitarily evolving quantum state
$\rho(t)$ of the whole system only has a \textit{weak convergence} (see
\cite{Mackey}) or \textit{weak limit}, symbolized as:
\begin{equation}
W-\lim_{t\rightarrow\infty}\rho(t)=\rho_{\ast} \label{INT-03
\end{equation}
This formula is simply another way to formulate the equation (\ref{INT-02}).
As a consequence, the coarse-grained state $\rho_{G}(t)$ also has a weak
limit, as follows from eq.(\ref{INT-02}):
\begin{equation}
W-\lim_{t\rightarrow\infty}\rho_{G}(t)=\rho_{G\ast} \label{INT-04
\end{equation}
The meaning of \ eqs. (\ref{INT-03}) and (\ref{INT-04}) is that although the
off-diagonal terms of $\rho(t)$ never vanish through the unitary evolution,
the system reaches equilibrium \textit{from an observational point of view},
that is, from the viewpoint given by any relevant observable $O_{R
\in\mathcal{O}_{R}$.

\item \textbf{Fourth step. }Also a \textit{moving preferred basis}
$\{|j(t)\rangle_{P}\}$ must be defined as we will see in section I.B. This
basis is the eigen basis of certain state $\rho_{P}(t)$ such tha
\begin{equation}
\lim_{t\rightarrow\infty}\langle O_{R}\rangle_{(\rho_{R}(t)-\rho_{P
(t))}=0,\ \ \ \ \ \ \ \ \ \ \ \forall O_{R}\in\mathcal{O}_{R
\end{equation}
The characteristic time for this limit is the $t_{D}$, the \textit{decoherence
time} (see \cite{Bonachon} for details)
\end{enumerate}

Then our \textit{General Theoretical Framework for Decoherence} (GTFD) is
introduced but we are unable to use it until we will specify the
$\mathcal{O}_{R}$\ for each particular case. Only then we will find the
$\rho_{G}$\ evolution for EID and SID.

In this paper we show how the GTFD improves the understanding of approaches
EID, SID and decoherence in general. In Section II we will present a general
theoretical formalism for decoherence. Section
\ref{EID as particular case of the GTFD} is devoted to EID and Section
\ref{SID as particular case of the GTFD} to SID. Section
\ref{Physical relevance of EID and SID obsevables} deals with the physical
relevance of EID and SID and contains what would be the main conclusion of the
paper. Section \ref{Characteristic times} deals with the characteristic times.
We present our conclusion in Section \ref{Conclusion}.

\subsection{\label{Coarse-graining}Coarse-grained process}

We will call \textit{coarse-graining} to a process such that we can select a
part of the information of the system under study and only consider the
physical quantities that correspond to the selected information. Thus,
considering only one part of the complete system, we can reduce, in some
cases, the number of degrees of freedom that we use in the description. In
quantum mechanics such a process leads, to the elimination of some components
of the state. This process can be understood as the projection of the Hilbert
space, associated with the complete system, on a smaller subsystem that
contains relevant observables. If in the complete system the state operator is
$\rho(t)$ and $\mathcal{O}$ is the space of all possible observable then,
there is a subspace of relevant observable $\mathcal{O}_{R}$ observables that
contains those that provide the physical information.

Thus, we have explained the first step of section 1. Precisely, we define the
\textit{coarse-grained state} $\rho_{G}(t)$, associated with the relevant
subsystem, to a state $\rho_{G}(t)$ such that it would satisfy the equation
(\ref{CG-01}). The space where the operators act are specified case by case
(see e.g. eq. (\ref{EIDgen-02}) and (\ref{d})). This $\rho_{G}(t)$
\footnote{See the mathematical definition in eq. (\ref{EIDgen-b-06}).} would
contain the maximal possible information that can be obtained from the
observables of $\mathcal{O}_{R}$. Is important to observe that the condition
$\langle O_{R}\rangle_{\rho(t)}=\langle O_{R}\rangle_{\rho_{G}(t)}$ does not
imply that $\rho(t)=\rho_{G}(t)$.

\subsection{The \textit{coarse-grained state} as a projection in the complete
state}

As a consequence of the definition (\ref{CG-01}), a coarse-graining usually
implies a projection whose action is to eliminate some components of the state
vector corresponding to the thinner description. If this idea is generalized,
coarse-graining can be conceived as a projection that defines the properties
of the relevant observables and also as a consequence the space of states. In
this subsection we will prove that the coarse-grained state $\rho_{G}(t)$ can
be conceived as the projection of the complete state $\rho(t)$\ on the
relevant observables subspace $\mathcal{O}_{R}$. Let us use the notation
$\left\langle O\right\rangle _{\rho}=(\rho|O)$ inspired in the algebraic
formalism which was initiated by the Brussels school in \cite{Antoniou}. Let
the basis of $\mathcal{O}_{R}$\textit{\ }be\textit{\ }$\left\{  \left\vert
O_{R}^{\alpha}\right)  \right\}  $, where $\alpha$ is, e.g., a continuous
index that identifies the basis elements and let us define a projector on
$\mathcal{O}_{R}$\ a
\begin{equation}
\pi=\int|O_{R}^{\alpha})(\rho^{\alpha}|d\alpha\label{CGP-01
\end{equation}
Let $\mathcal{O}_{R}^{\prime}$ be a space of the linear functional on
$\mathcal{O}_{R}$, the states are a linear combination of functionals $\left(
\rho^{\alpha}\right\vert $, satisfying\footnote{If we were working in a finite
dimensional space $\mathcal{O}$, we could choose $\alpha=\left(  i,j\right)
$, $\beta=\left(  k,l\right)  $, $\left\vert O_{R}^{\alpha}\right)
=\left\vert i\right\rangle \left\langle j\right\vert $, $\left(  \rho^{\beta
}\right\vert =\left\vert k\right\rangle \left\langle l\right\vert $ so
$\left(  \rho^{\beta}|O_{R}^{\alpha}\right)  =Tr\left(  \left\vert
i\right\rangle \left\langle j|k\right\rangle \left\langle l\right\vert
\right)  =\delta_{jk}\delta_{il}$.}
\begin{equation}
(\rho^{\alpha}|O_{R}^{\beta})=\delta(\alpha-\beta) \label{CGP-02
\end{equation}
It is clear that $\pi$ is a projector, because $\pi^{2}=\pi$. Then we can
define
\begin{equation}
(\rho_{G}|=(\rho|\pi\label{CGP-03
\end{equation}
Therefore,
\begin{align}
(\rho_{G}|O_{R}^{\beta})  &  =(\rho|\pi|O_{R}^{\beta})=(\rho|\int
|O_{R}^{\alpha})(\rho^{\alpha}|O_{R}^{\beta})d\alpha\nonumber\\
&  =(\rho|\int|O_{R}^{\alpha})\delta(\alpha-\beta)d\alpha=(\rho|O_{R}^{\beta})
\label{CGP-04
\end{align}
i.e. eq. (\ref{CG-01}) since $\langle O\rangle_{\rho}=(\rho|O)$. Thus, making
linear combinations of the $|O_{R}^{\alpha})$, we obtain
\begin{equation}
\langle O_{R}\rangle_{\rho}=(\rho|O_{R})=(\rho|\pi|O_{R})=(\rho_{G
|O_{R})=\langle O_{R}\rangle_{\rho_{G}}\text{ \ \ \ }\forall O_{R
\in\mathcal{O}_{R} \label{CGP-05
\end{equation}
i.e. equation (\ref{CG-01}). This demonstration is also valid for discrete
spectra \cite{Discreto}.

It is interesting to remark, that the lost of irrelevant information, just
described, is somehow, similar to the one that appears in thermodynamics. In a
classical mechanical system we know the position and velocity of all its
particles. From a thermodynamical point of view this information is excessive
and cannot be handled. Moreover we are just interested in some macroscopic
magnitudes and their relation through thermodynamical equations. Then we must
introduce a coarse graining to eliminate the excess of information i.e. the
position and velocity of all the molecules.

\subsection{The evolution of the \textit{coarse-grained state}\ and its limit
as a projection of the complete state}

The just defined $\rho_{G}(t)$ is the result of the projection of the state
$\rho(t)$ onto the space $\mathcal{O}_{R}$ of relevant observables (see eq.
(\ref{CGP-03})). Now we prove that the final state $\rho_{G\ast}$ of $\rho
_{G}(t)$ is the result of the projection of the final state $\rho_{\ast}$ of
$\rho(t)$ onto $\mathcal{O}_{R}$. In fact we have
\begin{equation}
(\rho_{G}(t)|O_{R})=(\rho(t)|\pi|O_{R})=\int(\rho(t)|O_{R}^{\alpha
)(\rho^{\alpha}|O_{R})d\alpha\label{CGP-06
\end{equation}
So, using equation (\ref{INT-03}) (if this limit exists, as in the case of SID
and EID)
\begin{align}
\lim_{t\rightarrow\infty}(\rho_{G}(t)|O_{R})  &  =\lim_{t\rightarrow\infty
}(\rho(t)|\pi|O_{R})=\lim_{t\rightarrow\infty}\int(\rho(t)|O_{R}^{\alpha
})(\rho^{\alpha}|O_{R})d\alpha=\nonumber\\
&  =\int(\rho_{\ast}|O_{R}^{\alpha})(\rho^{\alpha}|O_{R})d\alpha=(\rho_{\ast
}|\pi|O_{R}^{\alpha})=(\rho_{G\ast}|O_{R}) \label{CGP-07
\end{align}
where we have defined
\begin{equation}
(\rho_{G\ast}|=(\rho_{\ast}|\pi\label{CGP-08
\end{equation}
This limit only exists for concrete examples, for example SID and EID cases.
From eq.(\ref{CGP-07}) we obtain
\begin{equation}
W-\lim_{t\rightarrow\infty}(\rho_{G}(t)|=(\rho_{G\ast}| \label{CGP-09
\end{equation}

\subsection{\label{The master equation as a projection Liouville equation}The
master equation as a projection of Liouville equation}

As we said in Section \ref{Introduction}, the second step of GTFD can be
formulated computing the expectation value of $O_{R}$ in the unitarily
evolving state $\rho(t)$ e.g. $\langle O_{R}\rangle_{\rho(t)}$\ or computing
the expectation value of $O_{R}$ in the non-unitarily evolving state $\rho
_{G}(t)$ e.g. $\langle O_{R}\rangle_{\rho_{G}(t)}$, where $\rho_{G
(t)$\ follows a non-unitary evolution governed by a master equation. In this
subsection we show how the master equation can be written as a projected
Liouville equation. In fact, let us consider the equation
\begin{equation}
i\hbar\frac{d}{dt}|\rho)=\left[  H,\rho\right]  =L|\rho) \label{ME-01
\end{equation}
where $L$ is the Liouville \textquotedblleft superoperator\textquotedblright.
Let us project this equation a

\begin{equation}
i\hbar\frac{d}{dt}\pi|\rho)=\pi L|\rho) \label{ME-02
\end{equation}
where $L$ is the Liouville \textquotedblleft superoperator\textquotedblrigh
\ (see definition in \cite{Cos}). Now in general $[\pi,L]\neq0$, in fact we defin

\begin{equation}
\lbrack\pi,L]=N \label{ME-05
\end{equation}
S
\begin{equation}
i\hbar\frac{d}{dt}\pi|\rho)=L\pi|\rho)+N|\rho) \label{ME-06
\end{equation}
As $\pi|\rho)=|\rho_{G})$, then we hav
\begin{equation}
i\hbar\frac{d}{dt}|\rho_{G})=L|\rho_{G})+N|\rho) \label{ME-03
\end{equation}

This is the general form of a master equation. Clearly\ (\ref{ME-03})\ is the
Liouville equation with a extra term that in general, transforms the unitary
evolution of the coarse-grained state $|\rho_{G})$\ in a non-unitary evolution.

For practical purposes the master equation will be presented in a more
intuitive way. Precisely: if the projector $\pi$\ is known, we only need the
operator $N=[\pi,L]$\ and we can use it in the equation (\ref{ME-06}). Of
course in this case we can define $\pi=P$ and $Q=I-P$ and we can write the
last equation as the system
\begin{equation}
i\hbar\frac{d}{dt}P|\rho)=PLP|\rho)+PLQ|\rho) \label{ME-07
\end{equation
\begin{equation}
i\hbar\frac{d}{dt}Q|\rho)=QLP|\rho)+QLQ|\rho) \label{ME-08
\end{equation}
and solve this system by well-known methods (e. g. the Nakayima Zwanzig method
\cite{Nakayima}) that yield a non unitary evolution and finally they lead us
to the eq. (\ref{CGP-09}).

\section{\label{EID as particular case of the GTFD}EID as particular case of
the GTFD}

In paper \cite{CQG} we shown how the three first steps of the GTFD fit
perfectly with EID.\ In EID a system $S$ (usually a small system of
macroscopic nature) and an environment $E$ (usually a big system of
microscopic nature \footnote{In fact, decoherence is one of the steps of the
classical limit for macroscopic systems.}) are defined (in more or less
arbitrary way) and the closed system becomes $U=E\cup S.$ Then we have the
observable subspaces $\mathcal{O}_{E}$ and $\mathcal{O}_{S}$ and the operator
\begin{equation}
\mathcal{O}_{U}=\mathcal{O}_{S}\otimes\mathcal{O}_{E} \label{EIDgen-01
\end{equation}
where the relevant observables $O_{R}$ rea
\begin{equation}
O_{R}=O_{S}\otimes I_{E} \label{EIDgen-02
\end{equation}
As $U=E\cup S$ the corresponding spaces are $\mathcal{H}_{U}=\mathcal{H
_{S}\otimes\mathcal{H}_{E}.$ Let $\{|i\rangle\}$ be the basis of
$\mathcal{H}_{S},$ let $\{|\alpha\rangle\}$ be the basis of $\mathcal{H}_{E}$,
therefore $\{|i,\alpha\rangle\}$ is the basis of $\mathcal{H}_{U},$ Under
these conditions as we are only interested in the relevant information that
\ the observable $O_{R}$ sees, i. e. in the mean value
\begin{equation}
\langle O_{R}\rangle_{\rho}=\sum_{ij\alpha\beta}\rho_{i\alpha,j\beta
}O_{S\text{ }ij}\delta_{\alpha\beta}=\sum_{ij}\left(  \sum_{\alpha
\rho_{i\alpha,j\alpha}\right)  O_{ij}=\langle O_{S}\rangle_{\rho_{S}}
\label{EIDgen-03
\end{equation}
where
\begin{equation}
\rho_{S}=Tr_{E}\rho, \label{EIDgen-04
\end{equation}
In many cases it can be proved that this $\rho_{S}(t)$ evolves in a non
unitary way and it reaches equilibrium \cite{libromax}.

\subsection{The EID projector}

Let $\{|ij\alpha\beta)=|i,\alpha\rangle\langle j,\beta|\}$ the basis of
$\mathcal{H}_{U}\otimes\mathcal{H}_{U},$ then the EID projector read
\begin{equation}
P_{S}=\frac{1}{\sqrt{n}}\sum_{ij\alpha\beta}|ij\alpha\alpha)(ij\beta\beta|
\label{EIDP-01
\end{equation}
In fact the generic state of $\mathcal{H}_{U}\otimes\mathcal{H}_{U}$ i
\begin{equation}
(\rho|=\sum_{ij\alpha\beta}\rho_{ij\alpha\beta}(ij\alpha\beta|
\label{EIDgen-b-01
\end{equation}
the
\begin{equation}
(\rho|P_{S}=\frac{1}{\sqrt{n}}\sum_{ij\alpha\beta hk\gamma\delta
\rho_{hk\gamma\delta}(hk\gamma\delta|ij\alpha\alpha)(ij\beta\beta|=\frac
{1}{\sqrt{n}}\sum_{ij\delta}\rho_{ij\delta\delta}(ij|\sum_{\beta}(\beta
\beta|=\frac{1}{\sqrt{n}}\sum_{ij\delta}\rho_{ij\delta\delta}(ij|
\label{EIDgen-b-02
\end{equation}
since $\sum_{\beta}(\beta\beta|=\sum_{\beta}|\beta\rangle\langle\beta|=1$, and
$\rho_{S,ij}=\sum_{\delta}\rho_{ij\delta\delta}$ $=(Tr_{S}\rho)_{ij}$ finall
\begin{equation}
(\rho|P_{S}=\frac{1}{\sqrt{n}}\sum_{ij\delta}\rho_{S,ij}(ij|
\label{EIDgen-b-03
\end{equation}
On the other han
\begin{equation}
P_{S}^{2}=\frac{1}{n}\sum_{ij\alpha\beta i^{\prime}j^{\prime}\alpha^{\prime
}\beta}|ij\alpha\alpha)(ij\beta\beta|i^{\prime}j^{\prime}\alpha^{\prime
\alpha^{\prime})(i^{\prime}j^{\prime}\beta^{\prime}\beta^{\prime}
\acute{
=P_{S}=\frac{1}{n}\sum_{ij\alpha\beta^{\prime}}|ij\alpha\alpha)(ij\beta
\beta^{\prime}|\sum_{\alpha^{\prime}\beta}\delta_{\alpha^{\prime}\beta}=P_{S}
\label{EIDgen-b-04
\end{equation}
so $P_{S}$ is a projector.

\subsection{\label{The coarse-grained state in EID}The \textit{coarse-grained
state} in EID}

To obtain the coarse-grained state in EID we must project the complete state
on space $\mathcal{O}_{S}^{^{\prime}}$ . So:
\begin{equation}
\left(  \rho_{G}\right\vert =\left(  \rho\right\vert P_{S} \label{EIDgen-b-05
\end{equation}
Note that the dimension of the space that contains $\left(  \rho
_{G}\right\vert $ is iqual that the dimension of the space that
contains$\left(  \rho\right\vert $ but this does not happen with $\left(
\rho_{G}\right\vert $ and $\left(  \rho_{S}\right\vert $ because $\rho
_{S}=Tr_{E}\rho$. If we want to recover $\rho_{G}$\ starting from $\rho_{S}$,
we have: from the second step of GTFD that $\langle O_{R}\rangle_{\rho
(t)}=\langle O_{R}\rangle_{\rho_{G}(t)}$, and frpm (\ref{EIDgen-03}) that
$\langle O_{R}\rangle_{\rho}=\langle O_{S}\rangle_{\rho_{S}}$\ where
$O_{R}=O_{S}\otimes I_{E}$. Then we can define
\begin{equation}
\rho_{G}=\frac{\rho_{S}\otimes I_{E}}{Tr\left(  I_{E}\right)  }
\label{EIDgen-b-06
\end{equation}
The
\begin{align}
\langle O_{R}\rangle_{\rho(t)}  &  =\langle O_{R}\rangle_{\rho_{G}(t)}=\langle
O_{S}\otimes I_{E}\rangle_{\rho_{G}(t)}=Tr\left(  \left(  O_{S}\otimes
I_{E}\right)  \left(  \frac{\rho_{S}\otimes I_{E}}{Tr\left(  I_{E}\right)
}\right)  \right) \nonumber\\
&  =\frac{Tr\left(  O_{S}\rho_{S}\right)  Tr\left(  I_{E}\right)  }{Tr\left(
I_{E}\right)  }=Tr\left(  O_{S}\rho_{S}\right)  =\left\langle O_{S
\right\rangle _{\rho_{S}(t)} \label{EIDgen-b-07
\end{align}
If we want to find the final coarse-grained state we can proceed proving, case
by case, in each system or example that
\begin{equation}
\lim_{t\rightarrow\infty}(\rho(t)|O_{R})=(\rho_{\ast}|O_{R}),\text{
\ \ }\forall O_{R}\in\mathcal{O}_{R}\text{ or }W-\lim_{t\rightarrow\infty
}(\rho(t)|=(\rho_{\ast}| \label{EIDgen-b-08
\end{equation}
the
\begin{equation}
\lim_{t\rightarrow\infty}(\rho(t)|\pi|O_{R})=(\rho_{\ast}|\pi|O_{R}),\text{
thus }\lim_{t\rightarrow\infty}(\rho_{G}(t)|O_{R})=(\rho_{G\ast}|O_{R}),\text{
} \label{EIDCG-01
\end{equation}
\ \

We can use eq. (\ref{EIDCG-01}) and now we have enough equations to find
$(\rho_{G\ast}|O_{R})$ and therefore to find all the relevant coordinates of
$(\rho_{G\ast}|.$ So for any $O_{R}\in\mathcal{O}_{R}$ we hav
\begin{equation}
W-\lim_{t\rightarrow\infty}(\rho_{G}(t)|=(\rho_{R\ast}| \label{EIDgen-b-09
\end{equation}
The characteristic time of this evolution is $t_{R}$ that can be computed
using the poles technique. The decoherence time $t_{D}<t_{R}$ can also be
computed with the same technique. The two times can be also computed
case-by-case in several models \cite{Bonachon} \cite{polesPRA}.

\subsection{\label{EID Environment}Comments: EID Dissipative Environment}

The intuitive explanation of EID is \textit{dissipation}. EID would be, in
principle, a dissipative formalism, since in many models the microscopic $S$
gives its energy to the macroscopic $E$ where this energy is stored. The
kinetic energy of$\ S$ becomes zero and $S$ reaches equilibrium and classical
motion stops in the macroscopic-collective variables of$\ S$. Decoherence is
produced before equilibrium and it is proved that, for macroscopic systems,
the decoherence time is a small fraction of the relaxation time (see
\cite{CQG} and \cite{Bonachon})

I.- \textbf{A trivial example:}

This trivial example will become quite persuasive when we compare it with the
SID analog.

Let us consider a (small) stone $S$ and a (big) poll $E.$ The stone (which
initially has all the energy) falls into the motionless pool, creating big
waves of big wave length and low frequency in the water. The evolution makes
that waves would become smaller and smaller and their frequencies grow, ending
in microscopic (thermal) waves, while the stone stops its motion and reaches
equilibrium\footnote{Following the laws of the thermodynamic, the total energy
is conserved, but the mechanical energy is \textquotedblleft
degraded\textquotedblright\ in heat.}. The stone has dissipated its energy
into the pool. Essentially, in this example we see that
\textit{big-low-frequency-macroscopic} waves end in
\textit{small-high-frequency-microscopic} waves where the energy is dissipated.

Then, essentially we have two processes:

a.- Macro to Micro dissipation. The energy of the macroscopic waves
\textquotedblleft dissipates\textquotedblright\ into those of microscopic word.

b.- Evolution of the motion from low frequencies to high frequencies. The
macroscopic wave has low frequency while the microscopic one has high frequency.

We do not say that classical dissipation leads to quantum dissipation, but
this is a good analogy to understand the phenomena.

II.-\textbf{ For more general (non-trivial) example} (see \cite{Z}, 3.2, page
48). In Zwanzig's general formalism of the master equation we have relevant
channels (corresponding to relevant observables) and irrelevant channels
(corresponding to irrelevant observables) and the information goes to deeper
and deeper spaces of irrelevant channels. So information is dissipated in this case.

\section{\label{SID as particular case of the GTFD}SID as particular case of
the GTFD}

In SID approach the game is played in the complete set of commuting
observables (CSCO) that contains the Hamiltonian $H$ of the closed system $U$
and the constants of motion $C_{i}$ such that $\left[  H,C_{i}\right]  =0$.
The corresponding basis is $\{|\omega,c_{i}\rangle\}$, being these states
stationary, an
\begin{equation}
H=\int\omega\sum_{i}|\omega,c_{i}\rangle\langle\omega,c_{i}|d\omega
\label{SID-b-01
\end{equation}
We will see that in this case we can directly obtain a state equilibrium limit
$\rho(t)\rightarrow\rho_{\ast}$. Then, as we will see, all the characters of
the play: state, energy, etc. are constants of the motion, and therefore there
is no energy transfer and no dissipation in the $\{|\omega,c_{i}\rangle\}$
context (SID is not a dissipative formalism). This is the main difference with EID.

Nevertheless point \textquotedblleft b\textquotedblright\ of Section
\ref{EID Environment}.I allows us to see a crucial resemblance with EID:

The observables are (for simplicity we forget the $c_{i}$ indices
\begin{equation}
O=\int\widetilde{O}(\omega,\omega^{\prime})|\omega,\omega^{\prime})d\omega
d\omega^{\prime} \label{SID-b-02
\end{equation}
where $|\omega,\omega^{\prime})=|\omega\rangle\langle\omega^{\prime}|$ and
$\ \widetilde{O}(\omega,\omega^{\prime})$ is any kernel or distribution. The
relevant observables are those obtained by the van Hove choice \cite{vH}
\footnote{The non rigorous $\delta(\omega-\omega^{\prime})$ will soon
disappear from this text. In fact the formalism below is precisely a way \ to
eliminate this $\delta(\omega-\omega^{\prime})$ \ \ We will use this heuristic
object \textquotedblleft$\delta(\omega-\omega^{\prime})$\textquotedblrigh
\ just to give some examples below.}
\begin{equation}
\ \widetilde{O}(\omega,\omega^{\prime})=O(\omega)\delta(\omega-\omega^{\prime
})+O(\omega,\omega^{\prime}) \label{vH
\end{equation}
where $O(\omega,\omega^{\prime})$ is a regular function, precisely
$O(\omega,\omega^{\prime})\in\mathbb{L}^{2}(\omega-\omega^{\prime})$. Then we
define
\begin{equation}
O_{R}=\int O(\omega)|\omega)d\omega+\int O(\omega,\omega^{\prime
)|\omega,\omega^{\prime})d\omega d\omega^{\prime} \label{d
\end{equation}
where $|\omega)=|\omega\rangle\langle\omega|$, $|\omega,\omega^{\prime
})=|\omega\rangle\langle\omega^{\prime}|$\ and the states rea
\begin{equation}
\rho_{R}=\int\rho(\omega)(\omega|d\omega+\int\rho(\omega,\omega^{\prime
})(\omega,\omega^{\prime}|d\omega d\omega^{\prime} \label{e
\end{equation}
where $(\omega|,$ $(\omega,\omega^{\prime}|$ is the cobasis of $|\omega),$
$|\omega,\omega^{\prime}),$ where $\rho(\omega,\omega^{\prime})$ is also a
regular function, i. e. $\rho(\omega,\omega^{\prime})\in\mathbb{L}^{1
(\omega-\omega^{\prime})$. and
\begin{equation}
\rho_{R}(\omega)=\rho_{R}^{\ast}(\omega),\text{ }\rho_{R}(\omega)\geq0,\text{
}\int\rho_{R}(\omega)d\omega=1 \label{g
\end{equation}
Then

\begin{equation}
\langle O_{R}\rangle_{\rho_{R}(t)}=(\rho_{R}|O_{R})=\int\rho(\omega
)O(\omega)d\omega+\int\int\rho(\omega^{\prime},\omega)O(\omega,\omega^{\prime
})\exp\left[  -i\frac{(\omega-\omega^{\prime})}{\hbar}t\right]  d\omega
d\omega^{\prime} \label{0
\end{equation}
and
\begin{equation}
\lim_{t\rightarrow\infty}\langle O_{R}\rangle_{\rho_{R}(t)}=\lim
_{t\rightarrow\infty}(\rho_{R}|O_{R})=\int\rho(\omega)O(\omega)d\omega
\end{equation}
since $\rho(\omega^{\prime},\omega)O(\omega,\omega^{\prime})\in\mathbb{L
^{1}(\omega-\omega^{\prime}).$

In the particular case $O=H$ (a particular van Hove observables) equation
(\ref{d}) reads
\begin{equation}
H=\int\omega|\omega)d\omega\label{SID-b-03
\end{equation}
an
\begin{equation}
\langle H\rangle_{\rho(t)}=\int\rho(\omega)\omega d\omega\label{SID-b-04
\end{equation}
Therefore the energy of the system remains constant in time and it is only
concentrated in the diagonal terms $\rho(\omega)$. Thus, there is no energy
transfer$.$ Anyhow the van Hove observables see the motion in the states
$\rho(t)$ and therefore there is no quantum equilibrium at the initial stage.

We can follow the \textquotedblleft more general example\textquotedblright\ at
the end of the last section but now in the SID case. But upon a time there was
a myth that said that dissipation was necessary for the quantum states to
reach equilibrium and decoherence. Then as the states of \ a closed system
cannot dissipate, because they have no environment, they can neither decohere
nor reach equilibrium. The origin of this myth \ was a confusion between
classical objects and quantum states. In fact. to reach equilibrium a
classical object, e. g., needs friction to dissipate its kinetic energy in an
environment. But a quantum state is not a classical object. So today this myth
is dissipated (see \cite{libromax} page 93) but somehow the prejudice about
closed systems subsists. To be didactic let us consider a closed system. The
mean energy of a quantum state $\rho(t)$ in an arbitrary basis is$:
\[
(\rho(t)|H)=\sum_{ij}\rho_{ij}(t)H_{ji}=\sum_{i}\rho_{ii}(t)H_{ii}+\sum_{i\neq
j}\rho_{ij}(t)H_{ji
\]
where the first term of the r.h.s. would be the mean energy of the diagonal
terms $\rho_{ii}(t)$ and the second term the non diagonal ones $\rho_{ij}(t)$.
But in the energy eigenbasis this equation simply read
\[
(\rho(t)|H)=\sum_{i}\rho_{ii}\omega_{i
\]
where $\omega_{i}$ are the eigenvalues of $H.$ Namely in the Hamiltonian basis
the energy is concentrated in the constant diagonal terms \ and the variable
non diagonal terms do not contribute to the mean energy and therefore their
vanishing (according to SID) is irrelevant for the energy balance.

For all these reasons decoherence is clearly unrelated with dissipation, at
least in closed systems.

c.- SID decoherence is originated in the physical phenomenon of destructive
interference among the off diagonal terms of $\rho(t)$ or its mathematical
version: the Riemann-Lebesgue theorem (also illustrated by \cite{BH}).
Therefore SID is both physically and mathematically motivated. But nowadays
SID has not a direct experimental verification but it has indirect proves as
we will see. Also there is computational experiments as the Casati and Prosen
model \cite{CP} \cite{Cast}. Nevertheless there is a very long list of
physical theories that were introduced, adopted, and even popularized before
their experimental verification took place (e.g. Superstrings theory). Then
the essential requirement for a (provisional) theoretical physical formalism
is just that they would be soundly physically motivated.

\subsection{\label{The algebraic formalism}The algebraic formalism}

We can repeat this explanation in algebraic language \cite{CastaOrdo}: The
characteristic algebra $\mathcal{A}$ of the operators (see the complete
version in \cite{Ordoniez}) contains the space of the self-adjoints
observables $\mathcal{O}$ which in turn contains the minimal subalgebra
$\widetilde{\mathcal{A}\text{ }}$of the operators that commute with the
Hamiltonian $H$ (that we can consider as the typical \textquotedblright
diagonal\textquotedblright\ operators algebra). Then we have:
\begin{equation}
\widetilde{\mathcal{A}}\subset\mathcal{O\subset A} \label{SID-b-07
\end{equation}
Now we can make the quotien

\begin{equation}
\mathcal{A}/\tilde{\mathcal{A}}=\mathcal{V}_{nd} \label{SID-b-08
\end{equation}
where $\mathcal{V}_{nd}$ would represent the set of equivalence classes of
operators that do not commute with $H$ (the \textquotedblright non-diagonal
operators\textquotedblright). These equivalence classes read
\begin{equation}
\lbrack a]=a+\widetilde{\mathcal{A}},\qquad a\in\mathcal{A} \label{j
\end{equation}
So we can decompose $\mathcal{A}$ as:
\begin{equation}
\mathcal{A=}\widetilde{\mathcal{A}}+\mathcal{V}_{nd} \label{SID-b-09
\end{equation}
But eq. (\ref{j}) is not a direct sum, since we can add an arbitrary
$a\in\widetilde{\mathcal{A}}$ from the first term of the r. h. s. of the last
equation and substrate $a$ from the second term.

At this point we can ask ourselves which the observables are that really
matter in the case of SID under an evolution $e^{-iHt}$. Certainly the
observables that commute with $H$ which are contained in $\widetilde
{\mathcal{A}}$ (and correspond to diagonal matrices $\sim$\textquotedbllef
$\delta(\omega-\omega^{\prime})$\textquotedblright\ of eq. (\ref{vH})). The
observables that do not commute with $H$ correspond to the off-diagonal terms
contained in $\mathcal{V}_{nd}.$ These terms, must vanish when $t\rightarrow
\infty,$ so they must be endowed with mathematical properties adequated to
produce this limit. Riemann-Lebesgue theorem tells us that this fact takes
place if functions $O(\omega,\omega^{\prime})$, and therefore $\rho
(\omega,\omega^{\prime})$,\ are $\mathbb{L}_{1}$ , in such a way that, via the
Schwartz inequality the Riemann-Lebesgue theorem could be used as explained
above. Then we add this property to $\mathcal{V}_{nd}$. So we define a sub
algebra of $\mathcal{A}$ (that can be called a van Hove algebra \cite{vH}
since it is inspired in the works of this author) as

\begin{equation}
\mathcal{A}_{vh}\mathcal{=}\widetilde{\mathcal{A}}\oplus\mathcal{V}_{r
\subset\mathcal{A} \label{k
\end{equation}
where the vector space $\mathcal{V}_{r}$ is the space of operators of eq.
(\ref{d}) with $O(\omega)=0$ and $O(\omega,\omega^{\prime})\in\mathbb{L
_{2}(\omega-\omega^{\prime})$ as required under equation (\ref{vH}). Moreover
$\mathcal{O}_{R}\mathcal{=V}_{vhS},$ the space of self-adjoint operators of
$\mathcal{A}_{vh},$ which can be constructed in such a way to be dense in
$\mathcal{V}_{S}$ (because any distribution can be approximated by regular
functions). Therefore, essentially the introduced restriction is the minimal
possible coarse-graining. Now the $\oplus$ of equation (\ref{k}) is a direct
sum because $\widetilde{\mathcal{A}}$ contains the factor \textquotedbllef
$\delta(\omega-\omega^{\prime})$\textquotedblright\ and $\mathcal{V}_{r}$
contains just regular functions and a kernel cannot be both a distribution
$\delta$ and a regular function. Moreover, as our observables must be
self-adjoint the space of observables must be the just defined
\begin{equation}
\mathcal{O}_{R}\mathcal{=V}_{vhS}\mathcal{=}\widetilde{\mathcal{A}
\oplus\mathcal{V}_{rS}\subset\mathcal{V}_{S} \label{l
\end{equation}
where $\mathcal{V}_{rS}$ is the space of the self-adjoint operators of
$\mathcal{V}_{r}$. This decomposition corresponds to the one in eq. (\ref{d})
where $\mathcal{V}_{rS}$ only contains regular self-adjoint operators (namely
$O(\omega^{\prime},\omega)^{\ast}$=$O(\omega,\omega^{\prime})).$ Restriction
(\ref{l}) is just the choice (coarse-graining) of the relevant measurement
apparatuses for our problem, those that measure the diagonal terms in
$\widetilde{\mathcal{A}}$ and those that measure the non diagonal terms that
vanish when $t\rightarrow\infty$ in $\mathcal{V}_{rS}$\footnote{See
\cite{Ballentine} Section 8.2 (page 210) for the definition of these
observables.}. Under eq. (\ref{d}) we have called $|\omega)=|\omega
\rangle\langle\omega|$ the vectors of the basis of $\widetilde{\mathcal{A}}$
and $|\omega,\omega^{\prime})=|\omega\rangle\langle\omega^{\prime}|$ those of
$\mathcal{V}_{rS}$ . Then a generic observable of $\mathcal{O}_{R}$ reads as
in eq. (\ref{d})

The states must be considered as linear functionals over the space
$\mathcal{O}$ ($\mathcal{O}^{\prime}$ the dual of space $\mathcal{O}$)

\begin{equation}
\mathcal{O}_{R}^{\prime}\mathcal{=V}_{vhS}^{\prime}\mathcal{=}\widetilde
{\mathcal{A}}^{\prime}\oplus\mathcal{V}_{rS}^{\prime}\subset\mathcal{O
^{\prime} \label{ll
\end{equation}
Therefore the state reads as in eq. (\ref{e}). The space of these generalized
states (satisfying eq. (\ref{g})) is the convex space $\mathcal{S
_{R}\mathcal{\subset O}_{R}^{\prime}$. Now the mean value is given by eq.
(\ref{0}) and we can obtain the limits (\ref{EIDCG-01}) or (\ref{EIDgen-b-09
). \textit{This is the simple trick that allows us to deal with the
singularities (i.e. the \textquotedblleft}$\delta(\omega-\omega^{\prime
)$\textquotedblright) \textit{in a rigorous mathematical way and to obtain
correct physical results}. Essentially we have defined a new observable space
$\mathcal{O}_{R}$ (that contains the observables $O_{R}$ of eq. (\ref{d})) and
a space of states $\mathcal{S}_{R}$ that are adapted to solve our problem.

The algebraic approach has several applications in many chapters of physics.
The most important are ARQFT \cite{ARQFT} and Statistical Mechanics
\cite{Stat}. The approach presented here could be useful for defining
decoherence into these fields.

\subsection{\label{The projector into the space of regular functions}The
projector into the space of regular functions.}

Let us consider the rigged Hilbert space or Gel'fand triple
\begin{equation}
\Phi\subset\mathcal{H}\subset\Phi^{\prime
\end{equation}
where $\Phi$ is the test function space, $\mathcal{H}$ is a Hilbert space, and
$\Phi^{\prime}$ is the dual space of $\Phi$

Let
\begin{equation}
F\in\Phi^{\prime},\text{ \ }F:\Phi\rightarrow\mathbb{R,}\text{
F[\varphi]=x\in\mathbb{R} \label{SID-b-11
\end{equation}
be a functional or distribution on a space of test function $\Phi$ so
$\varphi\in\Phi$ [1]\footnote{More precisely $\Phi\subset\mathcal{H\subset
}\Phi^{\times}$, and $F:\Phi\rightarrow\mathbb{C}$ in the complex case, where
$\Phi^{\times}$ is the anti-dual space (see \cite{A.Bhom}).}$.$ A regular
function $f(x)\in\mathcal{H}$ can be used to define a generalized function (or
distribution) as a functional
\begin{equation}
F_{f}[\varphi]=\int f(x)\varphi(x)dx \label{1
\end{equation}
where $f(x)\in\mathcal{H}$, and $\varphi(x)\in\Phi$. Then, if $\left\{
e_{i}(x)\right\}  $\ is a basis of $\mathcal{H}$ we can decompose $f(x)$ and
$\varphi(x)$ a
\begin{equation}
f(x)=\sum_{i}f_{i}e_{i}(x),\varphi(x)=\sum_{i}\varphi_{i}e_{i}(x)
\label{SID-b-12
\end{equation}
Then we can also define a projector acting in a generalized function on the
space of regular functions $\mathcal{H}$ a
\begin{equation}
\pi F=\tilde{f}_{F}(x) \label{SID-b-13
\end{equation}
wher
\begin{equation}
\tilde{f}_{F}(x)=\sum_{i}F[e_{i}(x)]e_{i}(x) \label{2
\end{equation}
Then $\tilde{f}_{F}(x)$ is a $\mathcal{H}$ function if
\begin{equation}
\sum_{i}|F[e_{i}(x)]|^{2}<\infty\label{SID-b-14
\end{equation}
But in general it will not be the case and it may happen tha
\begin{equation}
\sum_{i}|F[e_{i}(x)]|^{2}\sim\infty\label{SID-b-15
\end{equation}
We will assume that we can approximate a distribution $F$ with a Hilbert space
function $f(x)$ (which can be written in a distribution form as $F_{f
[\varphi]$ as close as we wish). Then we can assume that the space of
functions of $\mathcal{H}$ is dense in the space of distributions
$\Phi^{\prime}$ in an adequate topology \cite{T}. This mathematical idea will
be enough for our physical purposes. Of course this fact must be demonstrated
case by case choosing mathematical structure with adequate properties.
Moreover we can study the problem using an algebra $\mathcal{A}$ and obtaining
the space $\mathcal{H}$ using the GNS theorem and its generalization (see
\cite{IC}) \ E. g. in paper \cite{AO} a detailed example can be found based in
the algebra $\mathcal{L(S(}\mathbb{R}^{+}))$(also see a detailed example in
\cite{CL})\}.

\bigskip

So $F[e_{i}(x)]$ can be approximated by a $f_{i}$ satisfying
\begin{equation}
\sum_{i}|f_{i}|^{2}<\infty\label{SID-b-16
\end{equation}
as close as we can and define a function
\begin{equation}
f(x)=\sum_{i}f_{i}e_{i}(x) \label{SID-b-17
\end{equation}
(this choice can be called a \textit{smoothing} process) and defines an
operator $\pi$ such that
\begin{equation}
\pi F=f \label{SID-b-18
\end{equation}

Now, from eq. (\ref{1}) we hav

\begin{equation}
F_{f}[\varphi]=\int f(x)\varphi(x)dx \label{SID-b-19
\end{equation}
Thu
\begin{equation}
F_{f}[e_{i}(x)]=\int f(x)e_{i}(x)dx=f_{i} \label{SID-b-20
\end{equation}
and from eq. (\ref{2}
\begin{equation}
\pi F_{f}[\varphi]=\sum_{i}f_{i}e_{i}(x)=f(x) \label{SID-b-21
\end{equation}
and we have that
\begin{equation}
\sum_{i}|f_{i}|^{2}<\infty\label{SID-b-22
\end{equation}
Then the projection of a $\mathcal{H}$ function is a $\mathcal{H}$ function
and $\pi^{2}=\pi,$ so $\pi$ is a projector and we have defined the projectio
\begin{equation}
\pi:\Phi^{\prime}\rightarrow\mathcal{H} \label{SID-b-23
\end{equation}

In a bra-ket language \ \{$e_{i}(x)\}$ becomes the basis \
$\vert
$e_{i}\rangle\}$ with cobasis is \{$\langle e_{i}|\}$ and the functional
$F[\varphi]$ is a bra $\langle F|.$ The
\begin{equation}
\langle e_{i}|e_{j}\rangle=\delta_{ij},\text{ }\pi
{\textstyle\sum_{i}}
|e_{i}\rangle\langle e_{i}|\text{ and therefore }\pi^{2}=\pi\label{SID-b-24
\end{equation}
and
\begin{equation}
\langle F|\pi=\langle f|\text{, \ \ }\pi
{\textstyle\sum_{i}}
\langle f|e_{i}\rangle\langle e_{i}| \label{3
\end{equation}
namely eq.(\ref{2}) where we have smoothed\ the $\langle F|e_{i}\rangle$ to
become the $\langle f|e_{i}\rangle.$

\subsection{The SID projector}

We can define the projector $\pi$ of SID such tha
\begin{equation}
\pi|O)=|O_{R})\text{ and }(\rho_{R}|=(\rho|\pi\label{4
\end{equation}
To begin with, we just stress that, intuitively, functions that oscillate with
infinite frequency can be associated with some kind of distributions. Then
these functions or distributions never reach equilibrium because they do not
suffer the destructive interference that would produce the factor $\exp\left[
-i\frac{(\omega-\omega^{\prime})}{\hbar}t\right]  $ in an infinite time.
Precisely these distributions are the ones that are not taken into account by
the van Hove observables.

From what we have explained in subsection
\ref{The projector into the space of regular functions},\ in this case the
projector $\pi$ reads
\begin{equation}
\pi:\mathcal{O}\rightarrow\mathcal{O}_{R}\mathcal{=V}_{vhS}\mathcal{=
\widetilde{\mathcal{A}}\oplus\mathcal{V}_{rS}\subset\mathcal{O},\text{
\ \ }\pi:\mathcal{O}^{\prime}\rightarrow\mathcal{O}_{R}^{\prime
\mathcal{=V}_{vhS}^{\prime}\mathcal{=}\widetilde{\mathcal{A}}^{\prime
\oplus\mathcal{V}_{rS}^{\prime}\subset\mathcal{O}^{\prime} \label{SID-b-25
\end{equation}
Moreover at the end of calculation we have seen that the decohered states
(namely the states that are candidates to become classical states when
$\hbar\rightarrow0$) only belong to space $\widetilde{\mathcal{A}}^{\prime}$
(with basis $\{(\omega|\},$ see eq. (\ref{e}))$.$ So the formalism yields the
definition of an important projector $\pi$ that projects the states over the
sub space $\mathcal{O}_{R}^{\prime}\mathcal{=V}_{vhS}^{\prime}$. We can call
$\pi$ the \textit{classical projector} because when $t\rightarrow\infty$ and
$\hbar\rightarrow0$, then $\mathcal{O}_{R}^{\prime}\mathcal{=V}_{vhS}^{\prime
}\rightarrow\widetilde{\mathcal{A}}^{\prime}$ so $\pi$ projects on the
\textquotedblright classical world\textquotedblright\ (see \cite{MCRL}).

Then according to the formalism of subsection
\ref{The projector into the space of regular functions}. we can define the
projector $\pi$ a
\begin{equation}
\pi|O)=|O_{R})=\int|\omega)(\omega|d\omega+\int\int_{\omega\neq\omega^{\prime
}}|\omega,\omega^{\prime})(\omega,\omega^{\prime}|d\omega d\omega^{\prime}
\label{proyd
\end{equation}
and we can say that if
$\vert
$O)$ and ($\rho|$ are generic operators or states the relevant ones will be
\begin{equation}
\pi|O)=|O_{R})=\int O(\omega)|\omega)d\omega+\int\int_{\omega\neq
\omega^{\prime}}O(\omega,\omega^{\prime})|\omega,\omega^{\prime})d\omega
d\omega^{\prime}\text{ and } \label{SID-b-26
\end{equation
\begin{equation}
(\rho_{R}|=(\rho|\pi=\int\rho(\omega)(\omega|d\omega+\int\int_{\omega
\neq\omega^{\prime}}\rho(\omega,\omega^{\prime})e^{-i(\omega-\omega^{\prime
})t}(\omega,\omega^{\prime}|d\omega d\omega^{\prime} \label{SID-b-27
\end{equation}
and since $(\omega|$, $(\omega,\omega^{\prime}|$\ is the cobasis of $|\omega
)$, $|\omega,\omega^{\prime})$, the product results
\begin{equation}
(\rho|O_{R})=(\rho|\pi|O)=(\rho_{R}|O)=\int\rho(\omega)O(\omega)d\omega
+\int\int\rho(\omega^{\prime},\omega)O(\omega,\omega^{\prime})d\omega
d\omega^{\prime} \label{SID-b-28
\end{equation}
Where we require that $O(\omega,\omega^{\prime})\in\mathbb{L}_{2
(\omega-\omega^{\prime}),$ then $\rho(\omega^{\prime},\omega)\in\mathbb{L
_{2}(\omega-\omega^{\prime})$, and from the Schwarz inequality $\rho
(\omega^{\prime},\omega)O(\omega,\omega^{\prime})\in\mathbb{L}_{1
(\omega-\omega^{\prime})$, and precisely this is the condition to use
Riemann-Lebesgue theorem.

\subsection{The \textit{coarse-grained state} in SID}

To obtain the coarse-grained state of SID we must project the complete state
on a van Hove space $\mathcal{O}_{VH}^{^{\prime}}$ . So with a similar
notation as the one of Section \ref{The coarse-grained state in EID}, we
have:
\begin{equation}
\left(  \rho_{G}\right\vert =\left(  \rho\right\vert P_{VH} \label{SID-b-29
\end{equation}
Note that, in this case, $\dim\left(  \rho_{G}\right\vert =\dim\left(
\rho\right\vert $ but, unlike EID, $\dim\left(  \rho_{G}\right\vert
=\dim\left(  \rho_{VH}\right\vert $ because $\rho_{VH}\neq Tr_{I}\rho$ where
$I$ is the unit operator. If we want to recover $\rho_{G}$\ starting from
$\rho_{VH}$, we need to do nothing because in this case $\rho_{G}=\rho_{VH}$.
From the second step of GTFD we have that $\langle O_{R}\rangle_{\rho
(t)}=\langle O_{R}\rangle_{\rho_{G}(t)}$, the
\begin{align}
\langle O_{R}\rangle_{\rho(t)}  &  =\langle O_{R}\rangle_{\rho_{G}(t)}=\langle
O_{VH}\rangle_{\rho_{G}(t)}=Tr\left(  O_{VH}\rho_{G}\right) \nonumber\\
&  =Tr\left(  O_{VH}\rho_{VH}\right)  =\left\langle O_{VH}\right\rangle
_{\rho_{VH}(t)} \label{SID-b-30
\end{align}
Let us now find the final coarse-grained state. In SID, using the
Riemann-Lebesgue theorem, it is proved tha
\begin{equation}
\lim_{t\rightarrow\infty}(\rho(t)|O_{R})=(\rho_{\ast}|O_{R}),\text{
\ \ }\forall O_{R}\in\mathcal{O}_{R}\text{ or }W-\lim_{t\rightarrow\infty
}(\rho(t)|=(\rho_{\ast}| \label{7
\end{equation}
The
\begin{equation}
\lim_{t\rightarrow\infty}(\rho(t)|\pi|O)=(\rho_{\ast}|\pi|O),\text{ thus
\lim_{t\rightarrow\infty}(\rho(t)|O_{R})=(\rho_{\ast}|O_{R}),\text{ }
\label{8
\end{equation}
\ \

From $(\rho_{\ast}|O_{R})$ of eq. (\ref{8}) we can deduce that, according to
the Riezs theorem, all the coordinates of $(\rho_{\ast}|$ e.g. in the finite
space we have $\dim\mathcal{O}_{R}=\dim\mathcal{O=}n$ we could take $n$
independent $\ |O_{R}^{i})$ $i=1,2,...n$ and since we have $n$ equations
$(\rho_{\ast}|O_{R})=$ \textit{certain known mean} \textit{value} we could
obtain all the coordinates of $(\rho_{\ast}|$ in space $\mathcal{O}$. So for
all $O_{R}\in\mathcal{O}_{R}$ we hav
\begin{equation}
W-\lim_{t\rightarrow\infty}(\rho(t)|=(\rho_{\ast}| \label{SID-b-31
\end{equation}

\section{\label{Physical relevance of EID and SID obsevables}Physical
relevance of EID and SID observables}

In the previous sections we have shown how the EID formalism fits perfectly in
the GTFD. The main concept in this framework is the coarse graining, as
explained in Section \ref{Coarse-graining}. But a question remains: if there
is a loss of information with physical relevance in a coarse graining
evolution. We have explained that the coarse graining is produced if we choose
a space of relevant observables $O_{S}$ of EID. All this is wellknown.

We will now consider the case of SID where the relevant observables are the
van Hove observables, of equation (\ref{d}), that belong to a space
$\mathcal{O}_{R}$. Then the corresponding states, of equation (\ref{e}),
belongs to a space $\mathcal{O}_{R}^{^{\prime}}$. Equations (\ref{d}) and
(\ref{e}) show that in SID a particular choice and their consequences are introduced

\begin{enumerate}
\item[i] $O(\omega,\omega^{\prime})$ is a regular function (i.e.
$O(\omega,\omega^{\prime})\in\mathbb{L}^{2}$ ) and not a generic distribution.
This makes $\mathcal{O}_{R}=\mathcal{O}_{VH}$. This is the restriction.

\item[ii] $\rho(\omega,\omega^{\prime})$ is also a regular function since it
belongs to a $\mathcal{O}_{R}^{^{\prime}}=\mathcal{O}_{VH}^{^{\prime}}$ (and
therefore also $\rho(\omega,\omega^{\prime})\in\mathbb{L}^{2}$), this is the
consequence. Then, we must ask ourselves if the obtained spaces $\mathcal{O
_{VH}$ and $\mathcal{O}_{VH}^{^{\prime}}$ are generic enough to take into
account all physical reality. Below we give an argument to prove that it is so.
\end{enumerate}

Let us consider the Hamiltonian of the system
\begin{equation}
\left\vert H\right)  =\int_{0}^{\infty}\omega|\omega)\,d\omega\label{2-3-1-03
\end{equation}
and a particular observable $\left\vert Z\right)  =\left\vert z\right\rangle
\left\langle z\right\vert $ where $\left\langle z|z\right\rangle =1$, i.e.
$\left\vert Z\right)  $ a is projector. As it is a usual observable we have
that:
\begin{equation}
\left\vert Z\right)  =\int_{0}^{\infty}\int_{0}^{\infty}\tilde{Z
(\omega,\omega^{\prime})|\omega,\omega^{\prime})\,d\omega d\omega^{\prime}
\label{2-3-1-04
\end{equation}
without any loss of generality we can write this equation as
\begin{equation}
\left\vert Z\right)  =\int_{0}^{\infty}Z(\omega)|\omega)\,d\omega+\int
_{0}^{\infty}\int_{0}^{\infty}Z(\omega,\omega^{\prime})|\omega,\omega^{\prime
})\,d\omega d\omega^{\prime} \label{2-3-1-05
\end{equation}
where $Z(\omega)$ is a regular function and $Z(\omega,\omega^{\prime})$ is a
distribution (a \textquotedblleft$\delta(\omega-\omega^{\prime})
\textquotedblright\ is hidden in $|\omega)$); $Z(\omega)$ and $Z(\omega
,\omega^{\prime})$ represent the diagonal and non diagonal components of the
observable $\left\vert Z\right)  $ which, in principle, it is not a van Hove
observable. The non diagonal components can be written as
\begin{align}
\left\langle Z^{ND}\right\rangle _{\rho_{\omega^{\prime}\omega}}  &
=\left\langle z\right\vert \rho_{\omega^{\prime}\omega}\left\vert
z\right\rangle =\left\langle z|\omega^{\prime}\right\rangle \left\langle
\omega|z\right\rangle \nonumber\\
&  =\left\langle \omega|z\right\rangle \left\langle z|\omega^{\prime
}\right\rangle =Z(\omega,\omega^{\prime}) \label{2-3-1-09
\end{align}
The usual procedure to measure $Z(\omega,\omega^{\prime})$ is to divide the
plane $\left(  \omega,\omega^{\prime}\right)  $ in squares of area
$\Delta\omega\Delta\omega^{^{\prime}}$. For each one of these squares, i.e.
for the square of the center $\left(  \omega_{k},\omega_{l}\right)  $ a state
$\left(  \rho_{\omega_{k}\omega_{l}}\right\vert =\left\vert \omega
_{k}\right\rangle \left\langle \omega_{l}\right\vert $ can be prepared, and
then, after the repetitions of many measurements the mean value $Z(\omega
_{k},\omega_{l})=\left\langle z|\omega_{k}\right\rangle \left\langle
\omega_{l}|z\right\rangle $ is computed. Once $Z(\omega_{k},\omega_{l})$ is
chosen for each pair $\left(  \omega_{k},\omega_{l}\right)  $ a regular
function $f(\omega,\omega^{\prime})$ is defined such that it interpolates all
the measured values. With this function we define
\begin{equation}
\left\vert Z_{VH}\right)  =\int_{0}^{\infty}Z(\omega)|\omega)\,d\omega
+\int_{0}^{\infty}\int_{0}^{\infty}f(\omega,\omega^{\prime})|\omega
,\omega^{\prime})\,d\omega d\omega^{\prime} \label{2-3-1-10
\end{equation}
which is a van Hove function since $Z(\omega)$ and $f(\omega,\omega^{\prime})$
are regular functions. Of course $\left\vert Z_{VH}\right)  $ is not exactly
$\left\vert Z\right)  $, but the central point is that $\Delta\omega$ is
maximal accuracy of the energy measurement instruments then $\left\vert
Z_{VH}\right)  $ is indistinguishable of $\left\vert Z\right)  $ from the
experimental point of view. Then combining projectors, we can conclude that
for any observable (according to the decomposition spectral theorem) there is
a van Hove observable that is observationally indistinguishable from the
former. Thus the van Hove observables can give an account of reality. A
similar argument can be used in the case of states. As a consequence the
observables and states that do not belong to the van Hove spaces cannot be
characterized experimentally since they are beyond the measurement precision.
Then SID is able to describe the physical reality with the measurement
precision of nowadays.

Of course in EID, the criterion to neglect information is completely
different. All the information that is irrelevant for the proper system $S$ is neglected.

\section{\label{Characteristic times}Characteristic times}

After this consideration we must complete the subject defining the
characteristic times:

1.- In EID there is a moving preferred basis \footnote{In each example of EID
this prefered basis is defined unambiguously, a general definition can be
found in \cite{Bonachon}.} for the relevant subsystem and the off diagonal
terms vanish \ in this basis in a characteristic time, known as the
decoherence time $t_{D}$, that we will call the proper system decoherence time
in the moving pointer basis $t_{DS}$\footnote{See the discusion about $t_{DS}$
in paper \cite{Bonachon}.}. This is of course a quantum reasoning related with
a quantum state, $\rho_{S}(t)$ of EID.

We also have a time where the relevant proper subsystem stops its motion, at a
time $t_{R}$, that we will call the relaxation time of the proper subsystem
$t_{RS}$, In this case we are in the usual grounds.

We know that for macroscopic bodie
\[
t_{DS}\ll t_{RS
\]

2.- In SID there is a final pointer basis (the eigenbasis \ of $H)$ for the
closed system and the off diagonal terms vanish \ in this basis in a
characteristic time known as the relaxation time that we will call the proper
system decoherence time of the \textquotedblleft universe\textquotedblrigh
\ $t_{RU}.$ This time is studied in \cite{SID}, \cite{CastaOrdo} and
\cite{CasatiBonachon}. In SID there is also a moving preferred basis
introduced in \cite{CasatiBonachon} and the corresponding decoherence time
$t_{DU}$.

Of course we can prove that
\[
t_{DS}<t_{RS},\text{ }t_{US}<t_{DR
\]
and as proved in \cite{Bonachon} and \cite{CasatiBonachon
\[
t_{DS}<t_{DU},\text{ }t_{DS}<t_{RU
\]

This is only a general description about the characteristic times, a complete
study about this issue can be found in \cite{Bonachon}, \cite{polesPRA} and
\cite{Time}.

\section{\label{Conclusion}Conclusion}

\begin{enumerate}
\item Comparing EID and SID we can discuss the application of the these two
formalisms. In the case of EID its experimental consequence and its general
success to explain many physical phenomena is wellknown so it is useless to
list all its applications. This is not the case of the new arrived SID.
Besides we can list some important facts.

\begin{enumerate}
\item It explains the classical limit in the case of closed system as
cosmological systems (see paper \cite{Mott}, \cite{Gambini}) and other
interesting closed systems, like the Casati-Prosen model (\cite{CP},
\cite{Cast}.) It also gives a closed-system solution to the Mott problem
\cite{Mott}.

\item It may help to understand some formal aspects of quantum chaos
\cite{QCh}.

\item It explains the classical limit in the case of the Modal Hamiltonian
Interpretation of quantum mechanic \cite{LC}, a new member of the group of
modal interpretations that began with the work of van Frassen. This
interpretation satisfies all the Mermin desiderata \cite{CL1}.

\item That closed systems reach an equilibrium at Khalfin time was
experimentally proved in \cite{3A}.
\end{enumerate}

\item With this paper we have completed papers \cite{CQG}, \cite{Bonachon} and
we define the main common features of all decoherence formalisms, the choice
of relevant observables, for the case of EID and SID.

\item We have also shown that EID and SID are perfectly compatible with the GTFD.

\item We have proved that coarse graining appears both in EID and in SID
because some information has been neglected. In EID the neglected information
is the one that does not come from the proper system $S$. In SID it is the
information that cannot be obtained by any available physical device.
\end{enumerate}

redpajama_set_name: RedPajamaArXiv


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text:
Thurmon King's New Sackett Family Database
Prairie Rest Cemetery, Delavan, Tazewell County, Illinois, USA
Latitude: 40.36973, Longitude: -89.52971
All Burials
1. Lay, Richard E. d. 12 Jun 2007 Prairie Rest Cemetery, Delavan, Tazewell County, Illinois, USA I35559
2. Mangold, Franklin O. d. 5 Mar 1998 Prairie Rest Cemetery, Delavan, Tazewell County, Illinois, USA I65120
3. Nelson, Georgia M. bur. 19 Jun 2010 Prairie Rest Cemetery, Delavan, Tazewell County, Illinois, USA I34945
4. Sackett, Harold Alva d. 15 Mar 2000 Prairie Rest Cemetery, Delavan, Tazewell County, Illinois, USA I11163
5. Sackett, Jessie A. d. 27 Oct 2000 Prairie Rest Cemetery, Delavan, Tazewell County, Illinois, USA I11164
6. Sackett, Lulu Ursula bur. 8 Apr 1964 Prairie Rest Cemetery, Delavan, Tazewell County, Illinois, USA I11158
7. Sackett, Minnie L. d. 21 Aug 1982 Prairie Rest Cemetery, Delavan, Tazewell County, Illinois, USA I11168
8. Shannon, Praticia d. 31 Dec 2003 Prairie Rest Cemetery, Delavan, Tazewell County, Illinois, USA I65119
Maintained by Ted Smith for The Sackett Family Association.
The collected content at this site is © 1999-2021 The Sackett Family Association. All rights reserved.

redpajama_set_name: RedPajamaCommonCrawl


text:
Bone marrow-derived cells and biophysical stimulation for talar osteochondral lesions: A randomized controlled study
BACKGROUND: Osteochondral lesions of the talus (OLT) frequently occur after ankle sprains in young patients participating in sports activities. These injuries may lead to chronic pain, joint swelling, and finally osteoarthritis, therefore, surgical repair is frequently needed. A collagen scaffold seeded with bone marrow-derived cells (BMDCs) harvested from patient's iliac crest and implanted into the OLT through a single arthroscopic procedure has been recently proposed as an effective treatment option. Nevertheless, BMDCs, embedded in an inflammatory environment, tend to differentiate toward a fibroblast phenotype with a consequential loss of mechanical characteristics. Biophysical stimulation with pulsed electromagnetic fields (PEMFs) has been shown to promote anabolic chondrocyte activity, stimulate proteoglycan synthesis, and reduce the release of the most relevant pro-inflammatory cytokines. The aim of this randomized controlled trial was to evaluate the effects of PEMFs on clinical outcome in patients who underwent BMDCs transplantation for OLT. METHODS: Thirty patients affected by grade III and IV Outerbridge OLT underwent BMDCs transplantation. After surgery, patients were randomly assigned to either experimental group (PEMFs 4 hours per day for 60 days starting within 3 days after operation) or control group. Clinical outcome was evaluated with (American Orthopaedic Foot and Ankle Society) AOFAS score, Visual Analog Scale (VAS), and Short Form-36 (SF-36). RESULTS: Significantly higher AOFAS score was recorded in the experimental group both at 6 or 12 months follow-up. At 60 days and 6 and 12 months follow-up, significant lower pain was observed in the experimental group. No significant difference was found in SF-36 between groups.
Titolo: Bone marrow-derived cells and biophysical stimulation for talar osteochondral lesions: A randomized controlled study
Cadossi M.
BUDA, ROBERTO EMANUELE CESARE (Corresponding)
Ramponi L.
Sambri A.
Natali S.
Giannini S.
FOOT & ANKLE INTERNATIONAL
Abstract: BACKGROUND: Osteochondral lesions of the talus (OLT) frequently occur after ankle sprains in young patients participating in sports activities. These injuries may lead to chronic pain, joint swelling, and finally osteoarthritis, therefore, surgical repair is frequently needed. A collagen scaffold seeded with bone marrow-derived cells (BMDCs) harvested from patient's iliac crest and implanted into the OLT through a single arthroscopic procedure has been recently proposed as an effective treatment option. Nevertheless, BMDCs, embedded in an inflammatory environment, tend to differentiate toward a fibroblast phenotype with a consequential loss of mechanical characteristics. Biophysical stimulation with pulsed electromagnetic fields (PEMFs) has been shown to promote anabolic chondrocyte activity, stimulate proteoglycan synthesis, and reduce the release of the most relevant pro-inflammatory cytokines. The aim of this randomized controlled trial was to evaluate the effects of PEMFs on clinical outcome in patients who underwent BMDCs transplantation for OLT. METHODS: Thirty patients affected by grade III and IV Outerbridge OLT underwent BMDCs transplantation. After surgery, patients were randomly assigned to either experimental group (PEMFs 4 hours per day for 60 days starting within 3 days after operation) or control group. Clinical outcome was evaluated with (American Orthopaedic Foot and Ankle Society) AOFAS score, Visual Analog Scale (VAS), and Short Form-36 (SF-36). RESULTS: Significantly higher AOFAS score was recorded in the experimental group both at 6 or 12 months follow-up. At 60 days and 6 and 12 months follow-up, significant lower pain was observed in the experimental group. No significant difference was found in SF-36 between groups.
Bone Marrow–derived Cells and.pdf N/A Administrator Richiedi una copia


text:
Монгольское завоевание Анатолии произошло с 1241 по 1243 год. После поражения армии сельджуков Рума в битве при Кёсе-даге в 1243 году территория оказалась под контролем монголов вплоть до 1335 года.

Из-за нескольких мятежей против султана сельджуков Рума, в 1255 году монгольская орда легко преодолела центральную и восточную Анатолию. В Турции до сих пор можно найти следы культурного наследия монголов, в частности захоронения различных правителей, в том числе и сына ильхана Ирана Хулагу.

К концу 14 века большинство территории Анатолии находилось под контролем различных
бейликов вплоть до крушения династии Сельджукидов.
Туркменские бейлики также признавали себя вассалами монгольских ханов, оставаясь фактически самостоятельными правителями областей.
Они не чеканили монеты с изображением собственных правителей, вплоть до Османа I, выпустившего мелкие серебряные монеты, названные акче, по образцу монет Ильханов с собственным изображением. В соответствии с традициями ислама печать монет была прерогативой суверенов, тем самым Осман I заявил о своей независимости от монгольского ханства. Существует мнение, что османы продолжали в какой-то мере платить дань монгольским ильханам вплоть до 1335 года, то есть фактически полной независимости добились лишь после смерти Османа.

Ранняя история

В 12 столетии Византийской империи удалось повторно утвердить свою власть в восточной и северной части Анатолии. После завоевания Константинополя в 1204 году в ходе Четвертого крестового похода, были образованы Эпирский деспотат и Никейская Империя. Несколькими неделями раньше Алексей Комнин основал Трапезундскую империю.

Трабзон и Никея находились на границе с Монгольской Империей. Контроль над территориями поделили греки и сельджуки, с постоянным ослаблением влияния Византии.

Во время Чормагана, военные действия с турками-сельджуками не велись. Однако после смерти Ала ад-Дин Кей-Кубада I, во времена правления Гийас ад-Дина Кей-Хосрова II в 1236 году монгольская империя вторглась на территорию Великой Армении, находящейся под властью Конийского султаната.
После смерти Угэдея-хана в 1241 году, Кей-Хосров посчитал армию достаточно сильной для сопротивления армии Монгольской империи. Байджу призвал султана принять зависимость от Монголии, лично посетить Монголию и оставить в знак смирения заложников. После отказа выполнения условий, Байджу объявил войну. В ответ армия турок-сельджуков вторглась в Грузинское царство, часть Монгольской Империи.

Падение Эрзурума
В 1241 году армия Байджу подошла к городу Эрзурум. Перед осадой Байнджу предложил жителям сдаться, получив отказ Байджу осадил Эрзурум и после двухмесячной осады захватил его. Армянский хронист тех лет Киракос Гандзакеци, описывая это событие, отмечал, что:

татары [то есть монголы] разделили городские стены на участки между разными отрядами… воздвигли множество катапульт и разрушили стены. После этого они ворвались в город, без всякой пощады предали мечу жителей, ограбили их имущество и богатство, а город сожгли огнём.

Получив послание о падении города султан направил монгольскому военачальнику письмо, заявляя, что армия Байджу захватила лишь один из множества городов.

Однако с захватом Эрзурума Байджу обрел необходимый плацдарм для дальнейших военных действий в Анатолии, которые он возобновил весной 1243 г.

Битва при Кёсе-даге

Султан заключал союз со всеми странами, окружающими его. Царь Киликийского армянского государства из династии Хетумидов обещал военную помощь, однако до настоящего времени не дошли подтверждающие факты их участия. Кей-Хосров получил военную поддержку со стороны Трапезундской империи, султана Халеба, Никейской империи
В итоге, как сообщает Киракос Гандзакеци, султану удалось выставить «бесчисленное количество людей». Сельджукский хронист Ибн Биби приводит цифру — 70 тыс.
Сельджуки встретили своих противников, заняв выгодные позиции в горном ущелье Кёсе-даг, западнее Эрзинджана.
Однако Байджу сумел перехитрить султанских военачальников, использовав традиционную тактику кочевников с ложным отступлением и внезапной контратакой, в которой участвовали отборные части монголов, а также отряды грузинских и армянских князей. В результате находившиеся в засаде воины, по словам Гандзакеци:

встретив султанские войска, разбили и обратили их в бегство. Султан едва спасся и бежал. Монголы преследовали бегущих и беспощадно истребляли их.

Заключение мира

Султан Кей-Хосров послал Байнджу делегацию, для заключения мирного соглашения. В результате соглашения Конийский султанат был обложен данью, которую должен платить ежегодно в золоте, шелке, верблюдах и овцах. Территория, захваченная монголами, была присоединена к Монгольской империи. Таким образом, султанат был разделен на две части, султану были оставлены земли к западу от реки Кызыл-Ырмак, область к востоку от реки переходила под управление монгольского наместника.

После поражения союзника при Кёсе-даге никейский император Иоанн III Дука Ватац начал подготовку к наступлению монгольской армии. Ватац направил в Мунке-хан послов, с целью выиграть время. Однако Монгольская империя не принимала действий по захвату империи и не вмешивалась в раздел Константинополя, не препятствуя плану Иоанна Ватаца по захвату города, и не способствуя сохранению города в составе Латинской империи.

Примечания

Ссылки

 Словопедия Кольера
 Битва при Кёсе-даге

Монгольские завоевания
Войны XIII века

redpajama_set_name: RedPajamaWikipedia


BT TV App Quirk ?
I'm just curious if anyone else has experienced this little foible with the BT TV App.
When I launch the BT TV App on my Windows 10 PC, I am (of course) required to login, after which I click on "Let'sGo" and my browser opens up and shows me a choice of programming (the app dropping into the background).


As you wrap furniture in blankets, pack moving boxes and go through numerous things you've accumulated over the years, there are possibly two things going through your head.
a) you'll miss this place, after all – this has been your home for such a long time.
Usually, movers don't tend to bring (old) appliances with them when they're moving homes. If you are like the majority of movers, and you are leaving appliances like microwaves, refrigerators, water softeners, and water heaters behind, make sure the owner's manuals remain as well. No matter how simple they may seem to use, the new owners would probably instead follow the manual then guess and end up with a mess on their hands. Leaving owner's manuals is practical if the new owners need to order replacement parts, too.
In case you've got a stack of unused paint cans you used on your home recently, leave them to the new residents. We do understand wanting to save up, and all, but those cans will probably end up in the trash somewhere anyway. So, why not leave them for the new owners to use to touch up the scuff and scratches that come with moving (especially if it's an older home)? Store the cans where the new owners can easily find them (e.g., at the back of a closet, under a sink, in the garage, etc.) and where there are no temperature fluctuations. You can also let them know where you left them.
Unless your light fixtures have a sentimental value or you'd recently purchased them, then leave them to the new residents. After all, buyers expect to move into a house with attached objects; however, if you want to take them with you, notify the real estate agent, and replace the fixtures with other ones. Also, make sure you leave light bulbs, too, so the new owners don't get stuck in the dark.
We all love our gardens, don't we? Sure! But, it's a little weird transporting shrubs, trees, and other landscaping features when leaving a house. Unless agreed otherwise before the purchase, the new homeowner will expect to find all outdoor plantings on the property just as they were when they signed the contract. Plus, the greenery wouldn't even survive transplantation, so leaving them behind shouldn't change much.
If your (former) home comes with a portable wireless alarm system (the kind that syncs with your smartphone due to Wi-Fi-enabled cameras), you can bring it with you when you move. These are easy to rewire and relocate. However, if the house has hardwired alarms connected to your home's electrical system, leave them with the house. In case you have a security company monitoring your alarm system, let both the new owners and the company know that you are moving. That way, the new owners can switch it to their name, and you can discontinue the service.
Looking to sell your Home in Southern California? Don't waste time searching for a professional agent. Work with So Cal's 5 STAR Reputation Leader. Click Here to talk with us.


text:
Ubenimex (INN), also known more commonly as bestatin, is a competitive, reversible protease inhibitor. It is an inhibitor of arginyl aminopeptidase (aminopeptidase B), leukotriene A4 hydrolase (a zinc metalloprotease that displays both epoxide hydrolase and aminopeptidase activities), alanyl aminopeptidase (aminopeptidase M/N), leucyl/cystinyl aminopeptidase (oxytocinase/vasopressinase), and membrane dipeptidase (leukotriene D4 hydrolase). It is being studied for use in the treatment of acute myelocytic leukemia and lymphedema.
It is derived from Streptomyces olivoreticuli. Ubenimex has been found to inhibit the enzymatic degradation of oxytocin, vasopressin, enkephalins, and various other peptides and compounds.

See also
 Amastatin
 Pepstatin

References

External links
 The MEROPS online database for peptidases and their inhibitors: Bestatin

Enkephalinase inhibitors
Protease inhibitors
Experimental cancer drugs
Secondary amino acids


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Remember those tweets, back in December, indicating that Hillary Clinton would make a cameo on Broad City this season? Ilana Glazer and Abbi Jacobson, at a SXSW talk, revealed the episode would air this Wednesday. But don't take it as an endorsement, necessarily.
During an interview with them on Friday, Anne Fulenwider of Marie Claire asked the duo how they went about putting in the call.
No, but really—how did this all come about? "We had come up with this idea that Ilana's character would find the Hillary campaign in some way– would be a part of it," Abbi said.
That wouldn't necessarily require Hillary's presence, but at some point they decided, "Let's just ask her," Abbi recalled.
And also losing Michigan, Fulenwider pointed out. That crack elicited some groans, but not as many as one audience member got when she asked Abbi and Ilana why they chose to have Hillary on when they seem more like Berners. She expressed concern that the episode would influence people and asked whether it was trying to make a statement.
The show premieres 4/20, natch.


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The Middle: No Nagging... for a Short Time
You know how some kids habitually forget things? Like the whole "you'd forget your head if it wasn't attached" thing? I've never understood it. How can you not remember your coat or your bookbag? You use it every day! /soapbox Anyway, I thought that this episode was decent... every move was pretty much in-character and the actions followed what this series has deemed standard. I could have used a little more substance, but that's not the end of the world. It has only slight re-watching power, but I did find myself laughing quite a few times. I knew that we'd need to have an event that would set things back to status quo, but I also could have taken another episode with the kids running their own lives. You?
The Middle "The Sit Down" (S03E17): The family is running late in the morning, which is fairly normal for them. They're suffering from their normal hold-ups, like Axl refusing to get up or clean his room, and Brick and Sue forgetting to bring things home (specifically, a coffee pot and a coat, respectively). The next day, the
(ABC/RICHARD FOREMAN)
kids ask to have a meeting with the parents, claiming that Frankie and Mike are out of control... she nags constantly and he makes unreasonable punishments. [both true.] Frankie and Mike decide to let up, since the kids essentially want more responsibility. It starts off well, with Sue remembering her coat and Axl doing his homework, but nobody is around to break up an argument when Axl gets angry at Sue for bringing WeirdAshley to the house for wrestlerette practice. In fact, Mike and Frankie are doing things for themselves, like going bowling. [haha at the crack at the Duggars - let the kids police each other.] Sue loses her coat and Axl freaks out - wondering if this will blow their "sweet deal." [haha] Plus, Axl also has to read Of Mice and Men, so he has Brick read it and summarize while they drive around town to find the coat. [the idea was funny... but it didn't really come off that way in the end.] After a couple of false alarms, Sue thinks that her coat is at WeirdAshley's. While Sue is searching, Axl accidentally agrees to go to prom with her AGAIN. [this actually makes me angry... he better find a way to get out of this!] Brick realizes that he knows where the coat is - it's been at home the entire time. [the whole family really needs to remember to specifically ask Brick things in order to get accurate information!] The kids think that they're in the clear, but when a moth flies into Brick's ear, Axl calls Frankie for advice on which emergency room is best. [wow... what a way to freak someone out!] When they all meet up at the hospital, everyone gets upset, though a moth flying into brick's ear is a freak thing - nobody's fault. [...and that's that.]
by Amy K. Bredemeyer at 7:00 PM Topics: Amy K. Bredemeyer, television reviews, The Middle


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Castle of Tirana ready for the public; Veliaj invites also those who were against
By Oculus News Saturday, December 22, 2018 castle, Erion Veliaj, Tirana, toptani
Erion Veliaj (left), the wife of Edmond Toptani (middle) and Mirela Kumbaro (right) inaugurating the Old Castel of Tirana
Beginning from today, the Old Castle of Tirana is the city's newest cultural and commercial space, where many citizens and tourists can find souvenir shops, mosaics and traditional restaurants. After Skanderbeg Square and New Bazaar, thanks to the cooperation with the Toptani family about its rehabilitation, the old Castle of Tirana has turned to the citizens as a space where the old fashioned and the modern are intertwined, to serve as a culinary environment for traditional Albanian dishes, but also as an area for the development of artisanship, culture and art.
The head of the Municipality of Tirana, Erion Veliaj, accompanied by the Minister of Culture, Mirela Kumbaro, visited the rehabilitated Fortress facilities.
"I am convinced that this will be a success story, not just for all of us who are in Tirana and will enjoy this, but for many tourists. I'm sure the Castle will become soon a destination, will probably be one of the most visited places. If we continue in this spirit - there are some things we agree, how we can move forward - I am convinced that even the most septics, those who fought this project and did everything to stop this, should come to see now the Castle ..." Veliaj said.
Tirana Castle premises
As demonstrated in previous projects, the rehabilitation of the Tirana Castle area will bring a positive impact on the city's economy by opening up new jobs. Veliaj underlined that in these three years all the beautiful transformational projects launched by the Municipality of Tirana at the beginning were accompanied by debates. Anyhow, he said, the city is destined to grow.
"This city, in these 3-4 years that I've been trying to serve, is always surrounded by rage and debate about everything beautiful. I remember when we came here at first, they said: They are ripping the Castle, they are destroying it, the city is over, the towers are knocked, concrete. Now let's just explain it to them: the city is growing every year with 25-27 thousand people, needs to have places where people will live. We can not make ghettos, we cannot build informal constructions. Better to live in the small houses build in communist era where the garbage is collected, where the the address is functional, where there is water and energy." said Veliaj.
Tirana Old Castle Entrance
He stressed that this is also a moment to reflect, as Albanians will not have to waste time setting obstacles to each other, but to help and to be all together.
Old Castle of Tirana at the Entrance
"There is a moment to reflect on what we Albanians are concerned about, what time we spend, debates, why we put our hands on each other's throat, about what, for parties? In 2018 we are afraid of the party's ideal - this is red, Toptani is blue, these so, they too. Is it worth to behave as hysterical fans of the party, who is voted once in four years on who does the job better, nut I think it is time for us to join as 'Tirons', as Albanians, as Europeans to say: Is doing one thing well, we all must help, because by providing a space where there is an economy, moving money, employing people, paying taxes, the property gets value, the whole city gets better," said Veliaj.
Castle of Tirana ready for the public; Veliaj invites also those who were against Saturday, December 22, 2018 Rating: 5
toptani


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The Bachelor Red Wine Recap (Hometown Dates): 'You're not my person'
Tue Feb 23, 2021 at 10:39am ET
By Liz Long
During The Bachelor "Hometowns," Serena P. surprises Matt with some news. Pic credit: ABC
Monsters and Critics columnist Liz Long recaps The Bachelor, Season 25, Episode 7, with a little help from a glass of pinot (or two)…
Well friends, we are in the final homestretch. I know we've lost many along the way throughout Matt's journey for love, but, for those of you still here, we will do our due diligence to recap tonight's episode of "hometowns."
"Hometowns" is in quotes, of course, because they all take place at the Nemacolin (the real winner of the season). Tonight's highlights included a fall heard 'round Pennsylvania, the correct use of the word "literally," and a self-made exit.
Let's recap.
Michelle's "Hometown"
Michelle wins for cutest hometown. Not only did she match Matt in lavender, but she melted my icy heart when she introduced "Mr. James" to her students (via Zoom call). Oh yeah, and these were her parents:
The Bachelor's Michelle Young's adorable parents. Pic credit: ABC
Michelle tells them that she really can see herself with Matt and is completely ready for an engagement (and I genuinely believe her).
Why is she so ready? Well it's because Matt is creating gardens for inner city kids to grow their own food, of course. (And I love this is the thing she highlights about him.)
But her hometown felt a little bittersweet. While it was unbelievably cute, at the same time, I couldn't help but have a nagging sadness, believing he won't pick her.
Pic credit: @bachelortweettz/Twitter
Let's just move on to Rachael's hometown, shall we?
Rachael's "Hometown"
Rachael's date wins for "I risked my life …. literally" for you.
She takes him skydiving and we've seen this type of date many times. What we haven't seen, though, is a flat out nose dive into the Earth.
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Like, no joke, Rachael face-planted and didn't move for a solid minute.
It didn't help that her instructor looked like he was 12.
Pic credit: @EmilyHahahaha/Twitter
But the face-plant wasn't a total loss. Afterwards, Matt realized how much he is in love with Rachael, feeling like he almost lost her.
I'm still getting over the fact this might be the first time we've seen Rachael disheveled all season. I mean, she even looked good plummeting at 130 miles per hour. And by "disheveled," I mean a few pieces of grass in her hair and nose. No broken bones. No bruises. (Oddly, no ambulances?)
Rachael's before and after skydiving. Pic credit: ABC
Rachael's parents were tougher than Michelle's with her dad being the "biggest hurdle." In the end, he was supportive of the relationship, despite Matt not asking for the blessing like Rachael so badly wanted. (Foreshadowing of where his head is already at, perhaps?)
Bri's "Hometown"
Aside from the classic run/jump/leg-wrap hug that is seemingly reserved only for hometowns, Bri greets Matt with "Hey, remember when you almost killed me?"
Oh, Bri, if you only knew the near-death-experience Rachael just had.
Technicalities aside, Rachael takes Matt on a four-wheeling date to show just how "full circle" they've come. In other words, because their one and only other date involved four-wheeling, that is a full-circle moment for them.
Matt is excited to meet Bri's mom, who snidely(?) greets him with "I know what Matt's been doing … dating!"
"Yes, mom, I've been dating your daughter." Pic credit: ABC
But Matt manages to keep his ever-present respectful nature … so much so, that is it just me, or is he saying the same things with every single family? (You know, all the canned hometown stuff?)
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And, on cue, after departing from the parents, everyone is confessing their love for Matt.
However, with Bri, after she told Matt she loved him, Matt's response was "Thank you for sharing that with me." OMG, I'm so sorry, Bri.
At least you have your mom, who had this gem: "Worst case scenario, we'll be mending your broken heart together … but we'll survive." Now that's a mom.
Serena's "Hometown"
I honestly didn't know Serena was Canadian, but dangit, we're about to learn just how hardcore of a Canuck she is.
I'm going to quiz you on Canada, Matt! Pic credit: ABC
I guess she never said the word "doubt" in any other episode, because when she said it tonight, oh boy, you could tell she's Canadian!
After a nice tasting of Poutine, we meet Serena's family. Her mom asks her if "he's intellectually stimulating," and Serena–not yet wanting to admit she's not a fan of Matt's novel kissing style–tells mom how he checks off all her boxes … well … ALMOST.
It's just that … well … something is "off" about Serena. Of course, we all knew this during that tantric yoga date, when she couldn't even kiss him. But it takes Matt a good pow-wow session with Chris (oh, Chris) to figure things out.
Chris tells Matt, don't wait for a rose ceremony …. sounds like you need to do a "deep dive" now. (Just don't use those words around Rachael, guys.)
Lo and behold, Matt surprises a curly-haired Serena alone in her room, and he is about to be surprised himself!
Serena reveals, "You're just not my person, Matt." And Matt looks seriously dumbfounded. As in, wait a minute, that has never happened to me. Plus, I'm the Bachelor here.
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But Matt collects himself and even asks her if she'd walk him out. All's well, and they both seem to quickly recover, indicating this just wasn't meant to be from the start.
And then there were three. Pic credit: ABC
You know, this episode was somewhat refreshing. Fairly uneventful (aside from Rachael's splat moment) … but completely void of Mean Girl mentality. It would have almost been a good old-fashioned old-school Bachelor had we wrapped with a complete rose ceremony.
But, alas, Bri, Rachael, and Michelle will all have to wait for their roses another week (or two). Next week, we have The Women Tell All, which, honestly, I don't think I'm ready for.
In the meantime, I'm going to try and find where Serena got her camel-colored leather pants. And if you want some major spoilers regarding Matt and his final pick, click here. Til' next week, "Rose-Lovers."
The Bachelor airs on Mondays at 8/7c on ABC.
Liz Long
Liz is a reality TV junkie, reformed attorney, and mom to Kayla—an...read more
Tags Chris Harrison, Matt James, Rachael Kirkconnell, The Bachelor


It reduces pollution in the air we breathe, and creates no creosote build-up on fireplaces or chimney walls.
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Say goodbye to the many hours of buying, storing, chopping and hauling wood. Also once the fire is lit, no further effort is required.
A national survey indicates a savings of approximately 57% when gas logs are used instead of wood logs. Adding a fireplace can return 138% on investment at resale.
Note: Not all units are available in both glass options. Please refer to your manufacturer's catalog for glass offerings on specific units.


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Home » US hunger levels dip slightly, but millions still suffer
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US hunger levels dip slightly, but millions still suffer
09/06/17 5:00 PM By Bill Tomson
KEYWORDS children ERS food Food Insecurity household hunger usda
WASHINGTON, Sept. 6, 2017 - Millions of children are still going hungry or are not eating properly in the U.S., but the overall number of people suffering what USDA calls food insecurity continues to creep lower, according to a report released today by the department's Economic Research Service.
People in about 12.3 percent of U.S. households had trouble getting good food or didn't get enough to eat at some point last year, concluded to the survey-based report, "Household Food Security in the United States in 2016." That's down from 12.7 percent in 2015.
The category of food insecurity is broken into two subsets – low food security and very low food security. While low security is troubling because some people cannot get the quality and variety of food most Americans can afford, very low security means that members of a household have to skip meals or sometimes go without eating for days at a time.
While it's a positive sign that the level of food insecurity fell slightly in 2016, it's still above the pre-recession rate in 2007 of 11.1 percent, said Alisha Coleman-Jensen, one of the authors of the ERS report. The level peaked at 14.9 percent by 2011 and has been dropping gradually ever since.
Coleman-Jensen said it's difficult putting an accurate number on the number of children still going hungry. About 3.1 million households with children reported being food insecure during the year, but often those households have more than one child.
Roughly 6.5 million children live in homes where there is often not enough food to eat, she said, but she also stressed that it's usually the adults who may suffer the most.
"As food insecurity becomes more severe, adults may reduce their food intake while trying to insure children get enough to eat," she said. "Research has shown that parents will try to shield their children from food insecurity."
It's only in the worst cases that children are forced to skip meals or not eat for long periods of time. That describes children in about 300,000 homes last year, Coleman-Jensen said.
The ERS report showed that 4.9 percent of families in the U.S. experienced very low food security in 2016, little changed from 5 percent in 2015.
Also, while the level of overall food insecurity has dropped, that may not be true for all categories of households. For example, the report showed an uptick in the inability to buy enough food year-round in homes with a single, female head of the household.
The highest levels of food insecurity were found in 15 states: Alabama, Arkansas, Arizona, Indiana, Kentucky, Louisiana, Maine, Mississippi, North Carolina, New Mexico, Ohio, Oklahoma, Oregon, Texas and West Virginia. Food insecurity in those states was higher than the national average.
The charts provided in the ERS report did not track the prevalence of very low food security on the state level, but Coleman-Jensen said Hawaii and Delaware were the states with the least households in that category (3 percent), while Louisiana and Alabama had the most (7.7 percent).
Report shows hunger decreasing but not to pre-recession levels
Organic doing great, but feed grains? Most are still imported
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Marshall Matz specializes in agriculture at OFW Law in Washington, D.C., and in this opinion piece he shares his thoughts on where the U.S. ranks amongst countries globally.
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The Trump administration rolled out new rulemaking to replace an Obama administration effort after speaking with a major farm organization's members earlier in the week. Agri-Pulse's Phil Brasher and Spencer Chase discuss the week's events in today's video.
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