text:
\section{The standard two-dimensional
effective theory with complex
Hamiltonian}

The phenomenological characteristics of a resonance or a decaying
particle are
its energy $E_R$ (resonance energy) or
relativistically its mass $m$, and its
width $\Gamma=\hbar/\tau_R$ where $\tau_R$ is its lifetime
(in its rest frame).
These two real numbers are combined into a complex energy
$z_R=E_R-i{\gamma\over2}$ $\bigl($(or relativistically into
a complex value
$s_R=(m-i{\gamma\over2})^2$ of the Mandelstam variable $s$,
which is the
(energy)$^2$ in the rest frame of the decaying state or the
(total energy)$^2$ in
the center of mass frame of the decay products (often not
$m$ but $M$ defined by
$s_R=M^2-iM\gamma$ is called the mass of the resonance)$\bigr)$.
This complex energy
$z_R$ or complex mass-squared $s_R$ can be  defined as the position
of the first
order (resonance) pole in the second sheet of the analytically
continued
$S$-matrix, in the same way as stationary states are given by
bound state poles of
the $S$-matrix.  Empirically, stability or the value of the
lifetime does not appear
to be a criterion for elementarity.  Stable particles
are not qualitatively different
from quasistable particles, but only quantitatively
different by a zero or very
small value of $\Gamma$.   (A particle decays if it can
decay and it is stable
if selection rules for some quantum numbers prevent
it from decaying.)
Therefore both stable and quasistable states should be
described on the same
footing, as is the case if one defines them by $S$-matrix poles.

Since a stationary state characterized by
a real energy value $E_s$ is
described not only by a bound state pole on the negative real axis but also
by an eigenvector of the (self-adjoint, semi-bounded)
Hamiltonian $H$ with
eigenvalue $E_s$, a ``pure" decaying state should
also be described by an
``eigenvector" of $H$ but with complex eigenvalue $z_R$.
In the standard,
Hilbert space formulation of quantum mechanics such vectors do not exist.
But since it is of  practical importance for the  phenomenological
description of
experiments to have a vector space  description, this
deficiency of the Hilbert space quantum mechanics
has not prevented the
practitioners  from using eigenvectors with complex energy in
``phenomenological",
``effective" theories of decay.$^{1)}$
The Lee-Oehme-Yang theory for the time evolution of the
two-resonance neutral Kaon system is the most celebrated
example$^{2)}$ of this.

The Lee-Ohme-Yang theory is usually justified as the Weisskopf-Wigner
approximation applied to the two Kaon states $|K^0\rangle,|\bar
K^0\rangle$, in which the neutral $K^0$-system is prepared in an
inelastic scattering experiment by strong interaction (e.g.
$\pi^-p\to\Lambda K^0$).

One then introduces the two linear combinations $|K_1\rangle$ and
$|K_2\rangle$ (leaving aside the CP violating Hamiltonian) ---
or the $|K_S\rangle$ and
$|K_L\rangle$
(if one includes CP violation in the Hamiltonian):
\poczn
\begin{eqnarray}
&|K_S\rangle={N_S\over\sqrt{2}}
\left((1+\epsilon_S)|K^0\rangle+(1-\epsilon_S)|\bar
K^0\rangle\right)\approx|K_1\rangle\phantom{0}
{\rm for}\ \epsilon_S\to0\\
&|K_L\rangle={N_L\over\sqrt{2}}
\bigl((1+\epsilon_L)|K^0\rangle-(1-\epsilon_L)|\bar
K^0\rangle\bigr)\approx|K_2\rangle\phantom{0}
{\rm for}\ \epsilon_L\to0
\end{eqnarray}
\konn
The $|K_1\rangle$ and $|K_2\rangle$
(or $|K_S\rangle$ and $|K_L\rangle$)
are defined
to be eigenstates of the complex
``effective" Hamiltonian $H=M-{i\over2}\Gamma$.
These vectors, and not the $|K^0\rangle,|\bar K^0\rangle$,
are
assumed to represent the particle states,
with the justification that ``since we
should properly reserve the name ``particle"
for an object with unique
lifetime,
$K_1$, and $K_2$ are the true particles".$^{3)}$
This effective Hamiltonian we
call $H$ if the CP  violating term $H_{sw}$ is left aside,
and we call it
\begin{equation}
\relax{\rm I\kern-.18em H}=H+H_{sw}=H_0+H_w+H_{sw}=H_0+\relax{\rm I\kern-.18em H}_{int}\,\,\,{\rm with}\,\,\,
[H_{sw},CP]\not=0
\end{equation}
if {\bf s}ome {\bf w}eak additional
CP-violating interaction Hamiltonian $H_{sw}$ (not
necessarily superweak) is included.
$H$ is assumed to be normal
(i.e. $M,\Gamma$
are hermitian and commute, which excludes Jordan blocks),
and thus it is diagonalizable with
eigenvalues given by
\poczn
\begin{equation}
H|K_1^{0\thinspace-}\rangle=
(m_1-i{\gamma_1\over2})|K^{0-}_1\rangle\quad;\quad
H|K^{0-}_2\rangle=(m_2-i{\gamma_2\over2})K^{0-}_2\rangle\ ,
\end{equation}
and similarly for ${\relax{\rm I\kern-.18em H}}$
\begin{equation}
{\relax{\rm I\kern-.18em H}}|K_S^{\enspace-}\rangle=(m_S-i{\gamma_S\over2})|K_S^{\enspace-}
\rangle\quad;\quad
{\relax{\rm I\kern-.18em H}}|K_L^-\rangle=
(m_L-i{\gamma_L\over2})|K_L^{\enspace-}\rangle\ .
\end{equation}
The time evolution for $t\geq0$ is given by
\stepcounter{numrow}
\setcounter{equation}{0}
\begin{eqnarray}
e^{-iHt}|K^{0\enspace-}_1\rangle=e^{-im_1t}
e^{-{\gamma_1\over2}t}|K_1^{0\enspace-}\rangle\\
e^{-iHt}|K^{0\enspace-}_2\rangle=e^{-im_2}
e^{-{\gamma_2\over2}t}|K_2^{0\enspace-}\rangle\nonumber
\end{eqnarray}
or, if \relax{\rm I\kern-.18em H}\  is the Hamiltonian, by
\begin{eqnarray}
e^{-i{\relax{\rm I\kern-.18em H}}t}|K_S^{\enspace-}\rangle=
e^{-i(m_S-{i\gamma_S\over2})t}|K_S^{\enspace-}\rangle\\
e^{-i{\relax{\rm I\kern-.18em H}}t}|K_L^{\enspace-}\rangle=
e^{-i(m_L-{i\gamma_L\over2})t}|K_L^{\enspace-}\rangle\nonumber
\end{eqnarray}
\konn
The entire effective theory of the neutral
$K$-meson-system then takes place in
this 2-dimensional space ${\cal H}_2$ spanned
by the eigenvectors (3).  An
arbitrary coherent mixture (superposition) in the
$K^0$-beam is given by
\begin{equation}
\phi^{eff}(0)=|K_S^-\rangle a_S+|K_L^{\enspace-}\rangle
a_L\propto|K_L^-\rangle+\rho|K_S^-\rangle
\end{equation}
It has the time evolution:
\begin{equation}
\phi^{eff}(t)=e^{-i\relax{\rm I\kern-.18em H}
t}\phi^{eff}=e^{-im_St}
e^{-{i\gamma_S\over2}t}|K_S^{\enspace-}\rangle
a_S+e^{-im_Lt}e^{-{i\gamma_2\over2}t}
|K_L^{\enspace-}\rangle a_L
\end{equation}
(or a corresponding expansion in terms of $K_1,K_2$, with
$a_1, a_2$, if (3a) and (4a)
hold).  The $|K_S\rangle$ and $|K_L\rangle$
in the coherent (pure) beam
state (5) are  conventionally
expressed in terms of $|K^0\rangle$ and
$|\bar K^0\rangle$ by (1),
though the $|K^0\rangle,|\bar K^0\rangle$ and the
$|K_S^{\enspace-}\rangle,\ |K_L^{\enspace-}\rangle$
belong to different Hamiltonians
$H_0$ and
$\relax{\rm I\kern-.18em H}$ (or
$H_0$ and
$H$ for $H_{sw}=0$), respectively.
The $|K^0\rangle$  and $|\bar K^0\rangle$ should, therefore,
span a space different from
${\cal H}_2$.   We annotate this difference between these
two kinds of vectors by the superscript
$^-$ in
$|K_2^{\enspace-}\rangle$ etc.
We shall discuss this notation in more detail in section 3
below.   An incoherent mixture,
which one usually encounters experimentally in the initial
$K^0$ beam$^{4)}$ (or which arises from a pure state if one
uses a theory based on a Liouville
equation for the neutral $K$-system$^{5)}$),
is described by a density matrix
or a statistical operator in the space ${\cal H}_2$.

The quantity that one always considers$^{6)}$ is the
instantaneous decay rate of the $K^0$
state at time $t$ into the detected decay channel $c$.
This transition rate is proportional to
$2\pi\rho_c|<c|\relax{\rm I\kern-.18em H}_{int}|\phi(t)\rangle|^2$, where $c=\pi^+\pi^-$ or
$\pi^0\pi^0$ or any other decay channel.
$\rho_c$ is the phase space factor (density of states in
channel
$c$) and
$|c\rangle$ is the eigenvector of the interaction-free
Hamiltonian $H_0$.
The amplitude of this rate is according to (6) given by
\begin{eqnarray}
\langle\pi\pi|\relax{\rm I\kern-.18em H}_{wt}|\phi^{eff}(t)\rangle&=&
\langle\pi\pi|\relax{\rm I\kern-.18em H}_{int}|K_S^-\rangle a_Se^{-im_St}
e^{-{\gamma_L\over2}t}\nonumber\\
&&+\langle\pi\pi|\relax{\rm I\kern-.18em H}_{in}|K_L^-\rangle a_Le^{-im_St}
e^{-{\gamma_L\over2}t}
\end{eqnarray}
One usually considers the ratio
\begin{equation}
R(t)\equiv{|\langle\pi\pi|\relax{\rm I\kern-.18em H}_{int}|\phi^{eff}(t)
\rangle|^2
\over|\langle\pi\pi|\relax{\rm I\kern-.18em H}_{int}|K_S\rangle|^2}
\end{equation}
for which one obtains from (7)
\begin{equation}
R(t)=|a_S|^2e^{-\gamma_St}+|a_L\eta|^2e^{-\gamma_Lt}
+2|a_S||a_L||\eta|e^{-(\gamma_L+\gamma_S){t\over2}}
\cos(\Delta mt+\varphi)
\end{equation}
where
\[
\Delta m=m_L-m_S
\]
and where
\begin{eqnarray}
\eta&=
{\langle\pi\pi|\relax{\rm I\kern-.18em H}_{int}|K_L\rangle\over\langle\pi\pi|\relax{\rm I\kern-.18em H}_{int}
|K_S\rangle}=|\eta|e^{-i\varphi}\nonumber\\
\varphi&=arg\  a_S-arg\ (\eta a_L)
\end{eqnarray}
If $|\phi^{eff}\rangle$ is
$|K^0\rangle$, then
$a_S=a_L={1\over\sqrt{2}}$.  (If $\phi^{eff}$ is a coherent
mixture behind a regenerator then
$a_S=\rho\ ; a_L=1$.)
For large values of $t\approx20\tau_S=20{1\over\gamma_S}$
($\tau_S=0.893\pm0.001\times10^{-10}\sec$) only the
second term in $R(t=20\tau_S)$ of (9)
does not have a factor of
$e^{-{1\over2}\gamma_St}\approx10^{-5}$ or smaller,
so $R(t)$ is given
by
\begin{equation}
R(t=20\tau_S)={1\over2}|\eta|^2e^{-\gamma_Lt}\enspace,
\quad\gamma_L/\gamma_S\approx1.72\times10^{-3}\ .
\end{equation}
If in (2)
$H_{sw}$ would be zero and if in (1a)
$|K_L\rangle=|K_2\rangle$ with
$CP|K_2\rangle=-|K_2\rangle$, then (due to
$CP|\pi\pi\rangle=+|\pi\pi\rangle)$
$\langle\pi\pi|\relax{\rm I\kern-.18em H}_{int}
|K_L\rangle\rightarrow\langle\pi\pi|H_w|K_2\rangle=0$, i.e.
$\eta$ in (11) should
be zero.
Experimentally, however,  one observes
$R(t=$ large) $\not=0$ (Princeton effect$^{7)}$).
This is explained by the existence of an $H_{sw}$
with the properties of (2), and by the
decaying particle states
$|K_L\rangle$
and
$|K_S\rangle$
not being the CP eigenstates
$|K_2\rangle$ and $|K_1\rangle$ respectively.

The two complex parameters,
$\eta_{+-}=|\eta_{+-}|e^{i\varphi+-}$, given by (10) for
$\langle\pi\pi|=\langle\pi^+\pi^-|$, and $\eta_{00}$ given by (10) for
$\langle\pi\pi|=\langle\pi^0\pi^0|$,
are the observable quantities in terms of which one
usually expresses the experimental data ascribed to CP violation.
The latest experimental
data, which may$^{8	)}$ or which may not$^{9)}$
indicate direct CP violation
($\langle\pi\pi|\relax{\rm I\kern-.18em H}_{int}|K_2^{\enspace-}\rangle\not=0,\
\eta_{+-}\not=\eta_{00})$,  give
the following values for the CP-violation parameters:
\begin{equation}
|\eta_{+-}|=(2.269\pm0.023)10^{-3},\enspace
\phi_{+-}=44.3^{\circ}\pm0.8^\circ\ ;\enspace
(\eta_{00}\approx\eta_{+-})\ .
\end{equation}
Inserting this into (11) we obtain the following experimental
value for $R(t)$:
\begin{equation}
R(t=20\tau_s)\approx{1\over2}
|2.27\cdot10^{-3}|^2\cdot 0.966\approx{1\over2}
|2.23\cdot10^{-3}|^2
\end{equation}
This number we shall use for the phenomenological
analysis in section 4.
\setcounter{equation}{0}
\section{
The Rigged Hilbert Space Formulation of Quantum Theory}

Vectors with the properties (1.4) and (1.3) have of course no place in the
Hilbert space of the standard quantum theory,
where all vectors have a unitary
time evolution and where (self adjoint semi-bounded) Hamiltonians cannot have
complex eigenvalues.  If, on the other hand,
one forces the neutral $K$ vectors
into the Hilbert space then one derives all kind of
mathematical consequences for which there exists no experimental evidence,
like deviations
from the exponential decay law, and vacuum regeneration of $K_S$ from
$K_L$.$^{10)}$
The resolution
of these incongruities is, of course, that the two-dimensional space
${\cal H}_2$ of the decaying neutral
$K$-meson system can not be contained in
the Hilbert space.
But we shall show that ${\cal H}_2$ is contained in the dual space
$\Phi^\times$ of a rigged Hilbert space
(or Gelfand triplet) $\Phi\subset{\cal
H}\subset\Phi^\times$.

A theory of resonance scattering and decay
has been developed over the past two
decades$^{11,12)}$ which uses the rigged Hilbert space (RHS)
for\-mu\-la\-tion$^{13)}$
of quantum mechanics.

Whereas in the Hilbert space the solutions of the
Schr\"odinger equation can have only unitary
time evolution and the eigenvalues of self-adjoint
operators can only be real, the RHS
formulation allows for a greater variety of
solutions and for new initial and boundary
conditions with a preferred direction of time (irreversibility).
In this formulation,
heuristic motions like Dirac kets $|E\rangle$, Gamow's
(exponentially decaying ``state")
vectors $|E-i{\Gamma\over2}\rangle$,
and Peierls purely outgoing boundary conditions,$^{1)}$
can be given a unified and mathematically meaningful foundation.
Vectors with the properties
(1.3) and (1.4) are well defined in the RHS formulation as
generalized eigenvectors (cf
Appendix (A6)) of a self-adjoint Hamiltonian
(semi-bounded essentially-self-adjoint operator
$H$ in an infinite dimensional space),
and the time evolution of these vectors is given by an
irreversible semigroup generated by the Hamiltonian.
These new vectors, which can also be obtained
from a resonance pole of the analytically continued
$S$-matrix and consequently have a Breit-Wigner
energy distribution, have been called Gamow vectors.

Since this paper addresses a physics problem, we do not give here the
precise mathematical definition of the RHS in general and of the particular
spaces $\Phi,\ \Phi_+,\ \Phi_-$ and their topological duals (space of
continuous antilinear functionals)
$\Phi^\times,\ \Phi_+^\times,\
\Phi^\times_-$, which we shall use in this paper.
In the appendix A we give a brief and casual
description of the RHS.
For the calculations in this paper we adopt the modus operandi of a
physicist and do not worry about the precise mathematical definitions of
${\cal H}$ or $\Phi$ or
$\Phi_+,\ \Phi_-,\ \Phi^\times_+$ etc.  All these spaces are different
topological completions of the same pre-Hilbert space $\Psi$ ($i.e.$ a linear
space $\Psi$ with ``scalar product" denoted by $(\psi,F)$ or
by
$\langle\psi|F\rangle$).
Here we use the algebraic space $\Psi$ and some additional
rules
which can be justified by the mathematics of the RHS.
These additional rules include
the well known rules of the Dirac bra-and ket formalism,
some mathematical properties of the
Gamow vectors
(e.g. those given by (1.3) and (1.4)) and in particular some basis
vector representations
which were not part of the Dirac formalism
(e.g. the so called ``complex spectral representation" of Appendix B).
These rules can
only be justified by the full RHS theory.
We shall simply introduce these rules as needed
while referring the reader to the
literature$^{11,14,15)}$ for their justification.

However, since the distinction between the spaces
$\Phi_+$ and $\Phi_-$ will be of physical
importance we need to explain some of their mathematical differences.
This is done using
their mathematical ``realizations".
One often says in mathematics that an abstract (linear
topological) space is ``realized" by a function space if
there exists a correspondence between
each vector $\phi\in\Phi$ and an element (or elements)
of the space of functions (probably in
mathematics functions are more ``real" than vectors).
The basis vector expansions of
Appendix B are examples of mathematical ``realizations".
In this manner the space
$\Phi$ is represented or ``realized" by the standard test function space
(Schwartz space).  The space
$\Phi_+$ (and
$\Phi_-$) is ``realized" by the subspace of Hardy class functions,
where the $+$ ($-$)
refers to analyticity in the upper (lower) half plane of the
second sheet of the complex
energy surface.$^{14,15)}$

In physics the abstact mathematical objects are realized by physical objects.
Thus a
physicists' ``realization" of the linear spaces
$\Phi_+,\ \Phi_-$ and their duals are by quantum
physical objects like states and observables.
The standard quantum theory uses the same Hilbert
space for both states and observables.

In contrast, distinct initial-boundary conditions for state vectors
$\phi^+$ ($e.g.$, in-states
$\phi^+$ of a scattering experiment) and observables
$|\psi^-\rangle\langle\psi^-|$ (e.g., so-called out-states $\psi^-$ of a
scattering experiment) lead to
two different rigged Hilbert spaces$^{12)}$:
\begin{eqnarray}
&\phi^+\in\Phi_-\subset{\cal
H}\subset\Phi^\times_-
&\mbox{for\ in-states\  of\ a\  scattering\
experiment which}\nonumber\\
&&
\mbox{are\ prepared\  by\  a\ preparation\ apparatus,}\nonumber\\
&&\mbox{e.g.\  an\  accelerator}\\
&\psi^-\in\Phi_+\subset{\cal
H}\subset\Phi^\times_+
&\mbox{for\  observables\  or\
out-states, which\  are}\nonumber\\
&&\mbox{measured by a\ registration\  apparatus,}\nonumber\\
&&\mbox{e.g.\  a\
detector}
\end{eqnarray}
The Hilbert space ${\cal H}$ is the same in both RHS's (1) and (2).
However $\Phi_+$ and
$\Phi_-$ are different
but they have more than the zero vector in common,
$\Phi_-\cap\Phi_+\not=\{\emptyset\}$;
in fact $\Phi_-\cap\Phi_+$ is in general infinite dimensional.
To use different mathematical spaces for states
(in-states) and observables (so-called out-``states")
is one of the new features of the RHS
formulation of quantum mechanics.

In (1), $\Phi_-$ describes the possible state vectors experimentally given by
the preparation apparatus (e.g. $\phi^{\rm in}$ or
$\phi^+$ of a scattering experiment) and in (2) $\Phi_+$  describes the
possible observables (e.g., $|\psi^{\rm out}\rangle\langle\psi^{\rm out}|$ or
$|\psi^-\rangle\langle\psi^-|$ of a scattering experiment) experimentally
specified by the detector.
The $|F^\pm\rangle\in\Phi^\times_\mp$ represent quantities connected
with the microphysical
system (e.g. ``scattering states $|E^\pm\rangle$ or decaying states
$|E-i{\Gamma\over2}^\pm\rangle$.
The superscripts for the vectors $\phi^+\in\Phi_-$ and
$\psi^-\in\Phi_+$, and the superscripts in
$|E^\pm\rangle\in\Phi^\times$ for the
eigenkets of the Hamiltonian $H=H_0+H_{int}$
refer to the standard notation of
scattering theory where $\phi^+$ represent
the in-states and $\psi^-$ represent the
out-observables (also called out-states).  The vectors $|E^\pm\rangle$
are related by the
Lippman-Schwinger equations to the eigenkets $|E\rangle$ of $H_0$, and the
superscripts of the eigenkets
$|E-i{\Gamma\over2}^-\rangle\in\Phi^\times_+$ and
$|E+i{\Gamma\over2}^+\rangle\in\Phi^\times_-$ of the
(essentially self-adjoint,
semibounded) Hamiltonian $H$ with complex eigenvalue
($E\mp i{\Gamma\over2})$ are an
extension of the labels in $|E^\mp\rangle$.
(The antithetical subscripts for the spaces have their
origin in the mathematicians notation for
the Hardy class functions.)
In this paper we shall only use the
Gamow vectors
$|E-i{\Gamma\over2}^-\rangle$ which have the property (1.4)
describing exponentially
decaying microphysical objects.
\setcounter{equation}{0}
\section{The Complex 2-dimensional Hamiltonian of the neutral K-system
as a Truncation of the exact Hamiltonian in the RHS.}

The neutral $K$ meson is produced in an inelastic scattering experiment.
We discuss here the case
that this inelastic scattering process produces pure $K^0$ states, as e.g.:
in the reaction:$^{4)}$
\begin{equation}
\pi^- p\to\Lambda K^0\to\Lambda\pi\pi\ .
\end{equation}
The principles of a scattering experiment applied to
this process are depicted in Fig. 1.  The scattering experiment consists of a
preparation part and a registration part.$^{12,15,16)}$  A meson beam
$\pi^-$ (i.e.
in the $II_3y=1,-1,0$ state), which is prepared
as a $\phi^{\rm in}$
before the interaction with the
target
$B$, evolves in the interaction region as a
$\phi^+\in\Phi_-$.  Due to the interaction it makes a transition into the
prepared state of the $K^0$ flavor,
described according to (2.1) by a
$\phi^+_{y=1}\in\Phi_-$.  Thus the
part of the experiment,
which prepares the state $\phi^+_{y=1}$,
consists of the apparatus for the preparation of the
$\pi^-$--beam and the strong interaction with the baryon-target
$B$ (changing the target state from
$p$ to
$\Lambda$).  The registration part
of the experiment
determines the so-called out-``state" $\psi^-$ which is
registered outside the interaction region as e.g. either
$\psi^{\rm out}=|\pi^+\pi^-\rangle$ or $\psi^{\rm out}=|\pi^0\pi^0\rangle$.
Its principal component is the $\pi^+\pi^-$ and/or $\pi^0\pi^0$ detector.
According to (2.2) the out-state, which is actually an
observable $|\psi^-\rangle\langle\psi^-|$,  fulfills
$\psi^-\in\Phi_+$.
In the conventional formulation of scattering theory in the Hilbert
space,
the in-state
$\phi^+$ (and $\phi^{\rm in}$) as well as
the out-observable
$\psi^-$ (and $\psi^{\rm out}$)
can be any vector of the Hilbert space ${\cal H}$.
In reality the $\phi^+$ and $\psi^-$
are subject to different conditions, namely, initial conditions for $\phi^+$
and final conditions for $\psi^-$.  This is described in the new RHS quantum
theory by using different mathematical conditions, $\phi^+\in\Phi_-$
and
$\psi^-\in\Phi_+$,
as shown in
equation (2.1) and (2.2), respectively.

We now consider the state of the meson at times (in the rest frame of the
$K^0$) $t\geq0$.  Here $t=0$ is the time before which the preparation
of the neutral
$K$-meson state (or in general of the $\phi^+$ state) is
completed and after which the registration of
$\psi^{\rm out}\leftarrow\psi^-$ begins.
The time
$t$ is the proper time of the Kaon which in the actual experiment
is measured by the
distance $d$ from the target position (or from the exit
face of the regenerator in
regeneration experiments when
$\phi^+\sim f\phi^+_{y=1}+\bar f\phi^+_{y=-1}$)
to the decay vertex ($t=d m_K/cp$,
where $p$ is the component of the Kaon momentum
along the beam line).
To each decay vertex we associate a microphysical
exponentially decaying
state of the neutral $K$-system,
which is represented by a Gamow vector.
This exponentially decaying Gamow state is a component
of $\phi^+(t)$  that
has dynamically evolved in time from the original,
prepared state
$\phi^+=\phi^+(t=0)$ by the exact Hamiltonian $H$.
(The Hamiltonian $H$
contains all interactions,
including the one responsible for the Kaon decay.)
Since $\phi^+$ is according to (B2)
a superposition of functionals the time evolution
operator is given by the semigroup $e^{iH^\times t}$; thus:$^{17)}$
\begin{equation}
\phi^+(t)=e^{-iH^\times t}\phi^+\phantom{00000}\mbox{or}\phantom{00000}
\phi^+(t)=e^{-i{\relax{\rm I\kern-.18em H}}^\times t}\phi^+\phantom{00000}t\geq0\ .
\end{equation}

We first want to discuss a theory in which the CP violating
$H_{sw}$ is assumed
to be zero and for which the exact Hamiltonian is denoted by $H$.
Then
[$H$,\ CP]=0 and we choose $H$ and CP as the
complete system of commuting
observables (c.s.c.o.). We ignore all other observables for the
neutral $K$ system except for the
energy operator
$H$, the weak-interaction-free energy operator $H_0$,
the hypercharge $\hat Y$ and the discrete symmetries like CP.
(This means we are
always working in the $K$-meson rest frame where we have
($\vec p_K, j_3,\ j^\pi)=(\vec 0,\ 0,\ 0^-$) which we do not write ).
In place of the
c.s.c.o.
$H$, CP, one can also consider the c.s.c.o.  $H_0,\ \hat Y$.
Besides these two, there is still
another c.s.c.o.:
$H_0$ and CP which we however do not want to consider.
The operators $H$ and
$\hat Y$ do not form a c.s.c.o.

Now we shall make use of some exact results in Appendix B.
We want to contrast the two basis systems
given by the nuclear spectral theorem (B1)
and by the complex basis vector expansion (B2) for the
c.s.c.o. H, CP.  The system of basis vectors is denoted respectively by
\poczn
\begin{equation}
|E,\ cp^-
\rangle\phantom{00000}\mbox{for}
\phantom{00000}0\leq E<\infty,
\end{equation}
and by
\begin{equation}
|z_i,\ cp^-\rangle;\quad\mbox{for}
\quad z_i=z_{K_1},z_{K_2};
\quad\mbox{and}\quad|\omega,
cp;^+\rangle\quad{\rm for}
\qquad
-\infty_{II}<\omega\leq0\ .
\end{equation}
\konn
And we also shall consider the
eigenvectors of the c.s.c.o. $H_0,Y$ which we
denote by
\begin{equation}
|E,y\rangle\qquad{\rm for}\qquad 0\leq E<\infty
\end{equation}
The {\it missing} superscript $^-$ in (4)
indicates that $|E,y\rangle$ are eigenvectors of $H_0$
and not of
$H$  (this latter notation is not in agreement with the
standard notation of scattering theory
where $|E\rangle$ would denote the eigenvector of
$H_{00}= H_0-H_{\rm strong}$).

The vectors
\begin{equation}
|E=m_K,\quad y=\pm1\rangle\qquad {\rm correspond\  to\  the\
usual}\qquad |K^0\rangle,\ |\bar K^0\rangle\ .
\end{equation}
while the vectors
\begin{equation}
|z_1,cp=+,^-\rangle;\quad|z_{2,}
cp=-,^-\rangle\qquad
\mbox{\rm correspond\  to\  the\  usual}\qquad
|K_1\rangle;\enspace |K_2\rangle
\end{equation}
Therefore we also use the notation:
\addtocounter{equation}{-1}
\poczn
\begin{equation}
|K_{i,}^{\enspace-}\rangle=|z_i,
cp=-(-1)^i,^{\enspace-}\rangle,\quad z_i=m_{K_i}-i
{\gamma_i\over2}\phantom{00000}i=1,2
\end{equation}
\konn
(up to an arbitrary ``normalization" constant,
cf. (18) below and footnote 20).
This notation also explains why already in equations
(1.3) and (1.4) we employed the
unusual notation
$|K_{1}^{\enspace-}\rangle$ and $|K_{2}^{\enspace-}\rangle$
with a superscript $^-$,  in contrast to the
usual notation $|K_{1}\rangle$ and $|K_2\rangle$.
The $|K_{i}^{\enspace-}\rangle$ are the exact
generalized eigenvectors of the self-adjoint,
semi-bounded Hamiltonian $H$ with complex
eigenvalues $z_i=m_i-i{\gamma_i\over2}$, i.e.
they are the Gamow kets of the neutral
Kaons system. The Gamow kets fulfill {\it exactly }
the (exponential) time evolution equation
(1.4), with $H$ being the infinite dimensional self-adjoint
Hamiltonian operator and not a complex
two dimensional matrix $H^{eff}$.  But (1.4) (and also (1.3))
is to be understood as a functional
equation over the space $\Phi_+$:
\begin{eqnarray}
&\langle e^{iHt}\psi^-|z^-_i\rangle\equiv
\langle\psi^-|e^{-iH^\times t}
|z^-_i\rangle=
e^{-i(m_i-i{\gamma_i\over2})t}
\langle\psi^-|z_i^-\rangle\nonumber\\
&{\rm for\  all}\
\psi^-\in\Phi_+\phantom{00000}
{\rm and}\phantom{00000}t\geq0.
\end{eqnarray}
This means one can only form Dirac bra-ket of (1.4)
with a $\psi^-$ that is an element of the
infinite dimensional space $\Phi_+$ and not in general with
$\phi^+\in\Phi$, cf. footnote 17.

That the vectors $\psi^-$ in (7) can only be of the space $\Phi_+$
($i.e.$ representing
the registered decay products, e.g.
$\psi^{\rm out}=\pi^+\pi^-$) is not a
restriction on physics since we are interested
in decay probabilities or
transition rates into observed out-``states" and
not in arbitrary matrix elements.
That $t\geq0$
in (7), i.e. that the time evolution of the
Gamow vectors is given by a semigroup, not a unitary
group, is derived using the mathematics of the RHS$^{7)}$
and is the  appropriate
restriction for decay processes (arrow of time, microphysical irreversibility).

Using the c.s.c.o. $H, CP$,  we have two choices for the basis vector
expansion:\\[3pt]
1. Dirac basis vector expansion (B1):
\begin{equation}
\phi^+=\sum_{cp=\pm1}\int^\infty_0dE\quad |E,{\rm cp}^+
\rangle b_{\rm cp}(E)
\end{equation}
where the expansion coefficients (energy wave function)
\begin{equation}
b_{cp}(E)=\langle^+E,cp|\phi^+\rangle\in {\cal S}\enspace
\mbox{(Schwartz\ space)}.
\end{equation}
If $\phi^+$ is a state of the neutral Kaon system
then $b_{cp}$(E) are presumed to be peaked
at $E={m_{K_1}}$ for $cp=1$,  and at $E={m_{K_2}}$ for $cp=-$1.  These
functions
may in the
neighborhood of the energy value  $E={m_{K_i}}$
come close to being Breit-Wigner amplitudes (with
widths $\gamma_i$).

The expansion (8) has its
{\it analogue} in Hilbert space quantum mechanics.
Hilbert space quantum mechanics amounts
to the assumption that
the energy wave functions fulfill the conditions
\begin{equation}
b_{cp}(E)\in L^2\qquad {\rm and}\qquad
Eb_{cp}(E)\in L^2\qquad\mbox{(Lebesgue
square integrable)}.
\end{equation}
Then
$b_{cp}(E)$ cannot be a
Breit-Wigner amplitude. Further the assumption (8) with
(9) or (10) causes the well known problems of
deviations from the exponential decay law in
general$^{18)}$ and  some additional problems
specifically for the neutral $K$-system.$^{10)}$

If the prepared state $\phi^+$ is a pure strangeness =$+1$ state
$\phi^+=\phi^+_{y=1}$,
which would usually be denoted by $\phi^+_{y=1}=|K^0\rangle$,
then for that $\phi^+_{y=1}$ the wave functions in (8) would fulfill:$^{19)}$
\begin{equation}
b_{cp=-1}(E)=b_{cp=+1}(E)\equiv{b(E)\over\sqrt{2}}
\end{equation}
This means, a prepared state vector with a definite hypercharge
$y=+1$ or $-1$ has the
Dirac basis vector expansion:
\begin{equation}
\phi^+_{y}=\int^\infty_0dE\ (|E,\
cp=+1,^+\rangle+\mbox{sign}(y)\ |E,cp=-1,^+\rangle)\
{b(E)\over\sqrt{2}}
\end{equation}
If one defines
\begin{equation}
``|K^{0}_{1,2}\rangle"=\int^\infty_0dE|E,cp=+1,-1,^+\rangle b(E)
\end{equation}
and uses the notation
\begin{equation}
\phi^+_{y=1}=|K^{0}\rangle\enspace \qquad  \phi^+_{y=-1}=|\bar
K^{0}\rangle,
\end{equation}
then (12) is the standard expression
\begin{equation}
|K^{0}\rangle={1\over\sqrt{2}}(``|K^{0}_1\rangle"+``|K^{0}_2\rangle")
\enspace\qquad |\bar
K^{0}\rangle=
{1\over\sqrt{2}}
(``|K^{0}_1\rangle-``|K_2^{0}\rangle")
\end{equation}
However
the vectors
`` $|K_{1,2}^{0}\rangle"$
do not give the Lee-Oehme-Yang theory, because they cannot have the
properties (1.4a) and
(1.3a) that one demands of the  usual $|K_{1,2}^{0}\rangle$. Indeed, if one
defines
$|K^{0}_{1,2}\rangle$ by (13), and imposes the precise conditions (10)
given by the
Hilbert space formulation, then one arrives at all kinds of ``CP-violation
problems of the exact
(Hilbert space) quantum theory."$^{10)}$

We shall not use Hilbert space quantum theory
here and thus not make the assumption (10).  Instead we will use
the conditions (2.1), which in terms of the wave function
$b_{cp}(E)=\langle^+E,cp|\phi^+\rangle$ means $b_{cp}(E)\in{\cal
S}\cap{\cal H}_-$ (Hardy class
function from below$^{14)}$).  Then one can also use the Dirac basis vector
expansion
(B1) to obtain (8) for $\phi^+\in\Phi_-$.  But as an alternative to (B1)
one has a second
choice:\\[3pt]
2.  The complex basis vector expansion (B2):
\begin{equation}
\phi^+=|K_1^{0\enspace-}\rangle b_1+|K_2^{0\enspace-}\rangle
b_{-1}+F^-_{cp=+1}+F^-_{cp=-1}
\end{equation}
Here we have defined
\begin{equation}
F^-_{cp}=\int^{-\infty_{II}}_0dE\quad|E,cp^-\rangle b_{cp}(E)
\end{equation}
which we call the background term.  And we have defined the (differently
normalized$^{20)}$) Gamow vectors
\begin{equation}
|K^{0\thinspace-}_i\rangle=|z_i,cp=-(-1)^i,^-\rangle\sqrt{2\pi\gamma_i}\quad
;\quad
z_i=m_i-i{\gamma_i\over2}
\end{equation}
or the microphysical state operator
\begin{equation}
|K_i^{0\enspace-}\rangle\langle^+K^{0}_i|=|z_i,^-\rangle2\pi\gamma_i\langle^
+z_i|
\end{equation}
Since  $|K_i^{0\enspace-}\rangle\in\Phi^\times_+$,
the vectors defined by the r.h.s. of (17)
must also be in
$\Phi^\times_-:\ F^-_{cp}\in\Phi^\times_+$, because
$\phi^+\in\Phi_-$.

According to the theory that underlies (B2), the eigenvalues
\[
z_i=m_i-i{\gamma_i\over2}
\]
in (18) and (19) are the complex energies of $K^{0}_i(i=1,2)$. This means
they are the positions
of the resonance poles for the two resonances $K^{0}_1$ and $K^{0}_2$.
(e.g. of the $S$-matrix
for the scattering process with resonance formation:$^{21)}$
\begin{equation}
\pi\pi\to(K^{0},\bar K^{0})=(K^{0}_1,K_2^{0})\to\pi\pi
\end{equation}
or any other scattering processes in which $K$-resonance formation occurs).
This value is
identical with the generalized eigenvalue of $H$.

The expansion coefficients -- i.e. the coordinates of the vector $\phi^+$
along the basis
vectors (3a) and (3b) -- are according to (B2) given by
\begin{eqnarray}
&&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
b_{cp=+1,}=\langle^+K_1^{0}|\phi^+\rangle\quad,\quad
b_{cp=-1}^{\phantom{00000}}=\langle^+K^{0}_2|\phi^+\rangle=
\sqrt{2\pi\gamma_2}\langle^+z_2|\phi^+\rangle\\
&&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
b_{cp}(E_{II})=\langle^-E_{II},cp|\phi^+\rangle,\quad
\langle^-E_{II},
cp|\phi^+\rangle=S_{cp}(E_{II})\langle^+E_{II}cp|\phi^+\rangle,
\end{eqnarray}
where $S(E)$ is the $S$-matrix
analytically continued
to values $E_{II}$ at the negative
real axis on the 2nd sheet (of e.g. the scattering process (20) if we
consider, as we shall,
the decays $K{^0}\to\pi\pi$).

The representation (16) is the special case of (B2) if there are no bound
states (i.e.
no neutral K's with a mass below the $\pi^+\pi^-$ threshold) and if there
are only two
resonance states with $j^\pi=0^-$.  Under these assumptions the representation (16) is
therefore exact (like the spectral theorem) for every $\phi^+\in\Phi_-$.
Whereas the vectors $|K^{0\enspace-}_i\rangle$ are well known basis vectors,
the ``background
integrals" (17) (integrals along the negative real axis of the second sheet)
depend upon the dynamics $H$ or $S$, and upon the prepared state $\phi^+$
and are not well known.$^{23)}$

The best situation is obtained if one can prepare a state $\phi^+$ such
that the background
term is very small.  Then one has from (16) for any pure (i.e. coherent
``mixture" or superposition) neutral Kaon state:
\begin{equation}
\phi^+\approx|K^{0\thinspace-}_1\rangle b_1,+|K^{0\thinspace-}_2\rangle b_{-1}
\end{equation}
This means that for values of
$|b_{cp}(E)|\ll|b_{cp}/\sqrt{\gamma_i}|$
any prepared pure neutral
Kaon state is -approximately- a superposition of
$|K_1^{0\enspace-}\rangle$ and
$|K_2^{0\enspace-}\rangle$.

We now shall take for the prepared state $\phi^+$ a hypercharge eigenstate
as produced e.g.
by the elastic scattering process (1)$^{24)}$
(Gell-Mann and Pais hypothesis) which we call
$\phi^+_{y=+1}$ (corresponding to $|K^{0}\rangle$) or $\phi^+_{y=-1}$
(corresponding
to $|\bar K^{0}\rangle$).   Then
\begin{eqnarray}
&&\phi^+_y=|z_1, cp=+1^-\rangle\tilde b^y_1+|z_2,cp=-1^-\rangle\tilde b^y_{-1}
\nonumber\\
&&+\int^{-\infty_{II}}_0dE\bigl(|E,cp=+1^{\enspace-}\rangle
b^y_1(E)+|E,cp=-1,^-\rangle b^y_{-1}(E)\bigr)
\end{eqnarray}
and using the same arguments that led to
(11),$^{25)}$  one obtains:
\begin{equation}
{1\over\sqrt{2\pi\gamma_1}}
\tilde b^{y=\pm1}_1=\pm b_{-1}^{y=\pm1}\equiv{b\over\sqrt{2}},\qquad
\pm b^{y=\pm1}_1(E)=
b_{-1}^{y=\pm1}(E)\equiv {b(E)\over\sqrt{2}}.
\end{equation}
For a hypercharge eigenstate the approximation (23) then becomes:
\begin{equation}
\phi^+_{y=\pm1}\approx(|K_1^-\rangle\pm|K_2^{\enspace-}\rangle){b\over\sqrt{2}}
\end{equation}
This is the analogue of the standard expressions (15), except that the
$|K_i^{\enspace-}\rangle$
in (26)
are the Gamow vectors with the properly (1.3) that have the exact time
evolution (7) or (1.4).

We now obtain the time evolution (2) of the prepared state $\phi^+_y$ for
$t>0$ using the
representation (24), since for $t\geq0$ the evolution of the Gamow vectors
is known from (7)
\begin{eqnarray}
\phi^+_{y=\pm1}(t)&=&
e^{-iH^\times t}\phi_{y=\pm1}=
{b\over\sqrt{2}}
(e^{-im_1t}e^{-{\gamma_1\over2}t}|K_1^{\enspace-}\rangle\pm
e^{-im_2t}e^{-{\gamma_2\over2}t}|K_2^{\enspace-}\rangle)\nonumber\\
&\!\!\!\!\!+&\!\!\!\!\!\int^{-\infty_{II}}_0
dE\bigl(\pm|E,cp=+1^-\rangle+|E,cp=
-1^-\rangle\bigr)e^{-iEt}{b(E)\over\sqrt{2}},
\end{eqnarray}
Note that according to (27), $e^{-iH^\times t}$ cannot transform from the
background term
$(F_{+1}^{\enspace-}+F^{\enspace-}_{-1})$
to
$|K^{\enspace-}_1\rangle$ or
$|K^{\enspace-}_2\rangle$ and it cannot transform
from
$|K_2^{\enspace-}\rangle$ to $|K_1^{\enspace-}\rangle$.  This means our
theory does not predict
anything that could be interpreted as  ``vacuum regeneration of $K_1$ (or
$K_S$) from
$K_L$."$^{10)}$

The approximation of a prepared state for very small background
term,
$|b(E)\sqrt{\gamma_{i}}/b|\ll1$, given by (26), has the time evolution:
\begin{equation}
\phi^+_{y=\pm}(t)\equiv
e^{-iH^\times
t}\phi^+_{y=\pm1}={b\over\sqrt{2}}(e^{-im_1t}e^{-{\gamma_1\over2}t}|K_1^-
\rangle\pm
e^{-im_2t}e^{-{\gamma_2\over2}t}|K_2^{\enspace-}\rangle)
\end{equation}
This means that the RHS quantum theory reproduces to a certain extent the
Lee-Oehme-Yang theory, and if $\phi^+$ can be prepared such that the
background term
$|b_{cp}(E_{II})|$ is negligibly small, then the Lee-Oehme-Yang theory
emerges as its
approximation.  Vice versa, the worthiness of the Lee-Oehme-Yang theory can
be taken as a
measure of how small the background term
$F_{cp}^{\enspace-}\in\Phi^\times_+$ in (16) and (24) can be made if the
$K^0$-state
$\phi^+_y\in\Phi_-$ is suitably prepared.

Thus the ``complex spectral" resolution (B2) of the RHS formulation of
quantum mechanics chooses
the basis system in $\Phi^\times_+$ such that the two dimensional space
${\cal H}_2$ of the
standard neutral Kaon model is spanned by the generalized basis vectors $|K_1^{\enspace-}\rangle,
|K_2^{\enspace-}\rangle\in\Phi^\times_+$.  One does not have to make any
special assumptions
about a complex effective Hamiltonian.  The Hamiltonian $H$ (precisely its
closure $H^\dagger$)
is just required to have the standard properties:  It is a self-adjoint
operator, bounded from
below.

The two dimensional matrix $H^{eff}$ emerges as the matrix of this infinite
dimensional
operator $H$ in the two dimensional subspace ${\cal
H}_2\subset\Phi^\times_+$.  Thus the RHS
formulation justifies the effective Lee-Oehme-Yang theory -- in contrast to
the exact Hilbert space
formulation, which contradicts it.$^{10)}$  The RHS formulation also gives
some additional
results, such as the semigroup $(t\geq0)$ time evolution of the Kaon decay,
(27), (28).
Furthermore,
for the
Kaon system in particular, it predicts the background term, which, however
small, must be
different from zero (because $\phi^+\in\Phi_-$ cannot be a superposiiton of
two vector
$|K_1^{\enspace-}\rangle, |K_2^{\enspace-}\rangle\in\Phi^\times_+$).

The existence of this background term in (27) has some significant
consequences, which may or
may not be of practical importance since $|{\sqrt{\gamma} b(E)\over b}|\ll1$
could be too
small to be observed.
However, since in the case of CP violation and direct CP-violation one is
talking of effects of
the order of $10^{-3}$ or 10$^{-6}$ and since one also discusses CPT
violations and violations of
microphysical quantum coherence,$^{26,5)}$ which are orders of magnitude
smaller
than $10^{-3}$,
a
discussion of possible effects from the background terms is warranted.
\setcounter{equation}{0}
\section{Long-time $2\pi$ decays of neutral $K$ without CP
violating Hamiltonian}

The quantity that is measured in the neutral $K$ experiments$^{27)}$ is the
instantaneous decay
rate of the $K^{0}$'s  into $\pi\pi$, ($\pi^+\pi^-$ or
$\pi^0\pi^0$).  In analogy to the instantaneous transition rate of
$\phi^{eff}(t)$ into
$\pi\pi$, which is given by (1.8) with (1.7) and (1.5), we take for the
instantaneous transition
rate of the prepared state $\phi^+(t)$ into $\pi\pi$ the
matrix element
${\hbox{$<\pi\pi|H_w|\phi_{y=1}^+(t)\rangle$}}$,
where
$H_w$ is the CP conserving interaction Hamiltonian.
The difference
between the standard phenomenology and the model that we want to consider
in this section
is the following:  Our Hamiltonian is
\begin{equation}
H=H_0+H_w\qquad{\rm with}\qquad[H,CP]=0\qquad[H_w,CP]=0
\end{equation}
and our $K^0$ state ($\phi^+_{y=1}$ prepared in the inelastic scattering
process (3.1)) is
given by (3.27), which evolves with a preferred time direction, $t\geq0$.
In the effective
theory summarized in section 1, the Hamiltonian is $\relax{\rm I\kern-.18em H}$ given by (1.2),
but the time evolving
state vector  is given by (1.6) (and though it is only used for the forward
time direction there
is no theoretical reason that $\phi^{eff}(t)$ could not also evolve
backward in time).  For our
model we define the ratio
$R(t)$ by replacing in (1.8)
$\relax{\rm I\kern-.18em H}_{in}$ by $H_w,\ \phi^{eff}$ by $\phi^+$ and
$<\pi\pi|\relax{\rm I\kern-.18em H}_{int}|K_S\rangle$ by $<\pi\pi|H_w|K_1^{\enspace-}\rangle$, since
$|K_S^{\enspace-}\rangle\to|K_1^{\enspace-}\rangle$ for an $H$ given by
(1).  Thus the
normalized instantaneous
rate as a function of $t$ (proper
time in the $K^0$ rest frame) is given by:$^{28)}$
\begin{equation}
R(t)=\Big|{<\pi\pi|H_w\big|
\phi_{y=+1}^{+}(t)>\over<\pi\pi|H_w|K_1^{\enspace-}>}\Big|^2
\end{equation}
We calculate $R(t)$ using our theoretical result (3.27) and obtain by
(inserting (3.27) into
(2)):
\begin{equation}
R(t)=\Big|e^{{-\gamma_1\over2}t}
e^{-im_1t}{b\over\sqrt{2}}+
\int^{-\infty_{II}}_0d
E{<\pi\pi|H_w|E,cp=+1^{\enspace-}>\over<\pi\pi|H_w|K_1^{\enspace-}>}
{b(E)\over\sqrt{2}}
e^{-iEt}\Big|^2,
\end{equation}
since $<\pi\pi|H_w|K_2^{\enspace-}\rangle=0$, and
$<\pi\pi|H_w|E,cp=-1,^{\phantom{0}-}\rangle=0$  due to (1).
In the same way as for the
$R(t)$ of
(1.9), the first term vanishes for large values of
$t\approx20{1\over\gamma_s}$ ($e^{-{1\over2}\gamma_st}\approx10^{-5})$ so that
\begin{eqnarray}
&&R(t=20\tau_s)
=\Big|{1\over\sqrt{2}<\pi\pi|H_w|
K_1^{\enspace-}>}\Big|\nonumber\\
&&\times\int_0^{-\infty_{II}}
dE<\pi\pi|H_w|E, cp=+1^{\enspace-}>b(E)e^{-iEt}\Big|^2
\end{eqnarray}

Theoretically, not much can be said at this stage about the integral on the
r.h.s. of (4).
We write it using (3.22) as:
\begin{eqnarray}
&&\hspace{-25pt}I(t)\nonumber\\
&&\hspace{-25pt}=-\int^0_{-\infty_{II}}
dE<\pi\pi|H_w|E,cp=+1^{-}>
S_{cp=+1}(E)<^+E,cp=+1|\phi^+>e^{-iEt}\nonumber\\
&&\hspace{-25pt}=-\int^0_{-\infty_{II}}
dE<\pi\pi|H_w|E,cp=+1^+><^+E,cp=+1|\phi^+>e^{-iEt}
\end{eqnarray}
Then (3) can be written as
\begin{equation}
R(t)=\Big|e^{-{\gamma_1\over2}t}e^{-im_1t}{b\over\sqrt{2}}+
{I(t)\over<\pi\pi|H_w|K_1^{\enspace-}>\sqrt{2}}\Big|^2,
\end{equation}
The integral $I(t)$ describes that part of the transition
$\phi^+\to\pi\pi$ which does not go through $K_1^{\enspace-}$ resonance
formation.
We know that $<^+E|\phi^+\rangle\in S\cap{\cal H}_-$ (Hardy class function
of the lower half plane,
second sheet).
If we also knew that $\langle\pi\pi|H_w|E^-\rangle S(E)\in S\cap{\cal
H}_-$, then one could
prove (using the Riemann-Lebesgue lemma for Hardy class functions)
that
$I(t)$
also
decreases
for increasing $t$, but it decreases
less than an
exponential $e^{-\gamma t}$.
Even though we do not have this information
let us
assume that the non-resonant background $I(t)$ will survive the exponential
$e^{-\gamma_1t}$:
\begin{equation}
|I(t)|^2\geq\  {\rm const.}\  e^{-\gamma_1t}\qquad{\rm for}\qquad
t\geq20{1\over\gamma_1}\ .
\end{equation}
The magnitude of
$\langle^+E_{II},cp=1|\phi^+\rangle$
(which can be calculated from \mbox{$\langle^+E|\phi^+\rangle$}
on the positive real
axis i.e. at physical
values of $E$ because of its Hardy class property$^{14)}$)
depends upon $\phi^+$ i.e. upon the preparation of the $K^0$
state.
Since experimentally the $\pi^+\pi^-$ are selected
such that their invariant mass is near the center of a
$K^0$-Breit-Wigner energy distribution, the magnitude of the non-resonance
contributions
\mbox{$|\langle^+E_{II}|\phi^+\rangle|$} to the observed $R(t)$ will be small.
But we need
only a small
contribution
on the r.h.s. of (4) in order
to explain the experimental value (1.13) for $R(20\tau_s)$.  Mathematically
\mbox{$\langle^+E_{II}|\phi^+\rangle$} must be different from zero,
because \mbox{$\phi^+\in\Phi_-$}.
But its magnitude
could be arbitrarily small, too small to account even for the small value
(1.13)
of
$R(20\tau_1)$.  Thus the question is not whether such a term
\mbox{$|I(t)/\langle\pi\pi(H_w|K^{\enspace-}_1\rangle|$}, which decreases
more slowly in time
than the
exponential $e^{{-\gamma_1}t}$,
exists
but
whether this term in (6) has
the right magnitude to explain the value (1.13).

Since we
have no theoretical prediction, we shall use the second term on the r.h.s.
of (6) as
phenomenological parameter.

The phenomenological value that we obtain for
this background integral from the values (1.12),$^{29)}$ and
(1.13) is:
\begin{equation}
\Big|{I(t=20\tau_1)\over\langle\pi\pi|H_w|K_1^{\enspace-}\rangle}\Big|
\approx2.23\cdot10
^{-3}\ .
\end{equation}
This means that if the non-resonant contributions to the transitions of the
prepared state
$\phi^+$ into $\pi\pi$ are about 2.23$\cdot10^{-3}$ of the $K_1$-resonance
term, then these
contributions can explain the Princeton effect without the assumption of a
CP-violating
Hamiltonian.  Theoretically
$|{I(t)\over\langle\pi\pi|H_w|K_1^{\enspace-}\rangle}|$
could of course be much
smaller than
$10^{-3}$, in fact it can be arbitrarily small as long as it is not equal
to zero and still
fulfill the mathematical conditions that $\phi^+\in\Phi_-$.

To display the distinction between the way the effective Lee-Oehme-Yang
theory explains the
existence of a long time $K^0\to\pi\pi$ decay mode and the way the exact
theory in the RHS
explain this effect, we compare the state vector for $K$-short and $K$-long
in both theories.
According to (1.1) we have (except for some normalization):
\poczn
\begin{eqnarray}
|K_S^{\enspace-}>&=&|K_1>+\epsilon_S|K_2>\\
|K_L^{\enspace-}>&=&\epsilon_L|K_1>+|K_2>
\end{eqnarray}
\konn
Here $|K_S^{\enspace-}>$ and $|K_L^{\enspace-}>$, and not
$|K_1>$ and
$|K_2>$ are assumed to be the Gamow
vectors,  and $\epsilon_L\approx\epsilon_S\approx\epsilon\approx10^{-3}$
are very small.  In
analogy to this we define
\poczn
\begin{eqnarray}
\phi^+_\sigma&=&|K_1^{\enspace-}>+{1\over b}F_{cp=-1}^{-}\\
\phi^+_\lambda&=&{1\over b}F^-_{cp=1}+|K_2^{\phantom{0}-}>
\end{eqnarray}
\konn
where $|K_{1,2}^{\phantom{000}-}>$
are the Gamow vectors and $F_{cp}^{\enspace-}$ are the vectors (3.17) in
the expansion (3.16).  Then
$\phi^+_y$ of (3.24), (3.25) can be written as
\begin{equation}
\phi^+_{y=\pm1}=
\phi^+_\sigma{b\over\sqrt{2}}
\pm\phi_\lambda^+{b\over\sqrt{2}}\ .
\end{equation}
This formula is the analogue of (1.5) (with $a_S=\pm a_L={b\over\sqrt{2}}$)
of the effective
theory.

Comparing (10) with (11) we see that
\begin{center}
\begin{minipage}{0.8\textwidth}
$\phi^+_\sigma$ are, like the $|K_S>$,
mostly
$cp=+1$ vectors with  a  small $cp=-1$ contribution.
\end{minipage}\\
\begin{minipage}{0.8\textwidth}
$\phi^+_\lambda$ are, like the $|K_L>$,
mostly
$cp=-1$ vectors with  a  small $cp=+1$ contribution.
\end{minipage}
\end{center}
But whereas in (9) the small admixtures of the opposite parity is given by
$\epsilon|K_2>\in{\cal H}_2$ and
$\epsilon|K_1>\in{\cal H}_2$ respectively, the opposite parity admixtures
in (10),
${1\over b}F^-_{cp=-1}$ and
${1\over b} F^-_{cp=+1}$ respectively, are not
elements of ${\cal H}_2$.  (Recall that ${\cal H}_2$
$\subset\Phi^\times_+$ is the space spanned by the vectors
$|K_1^{\enspace-}>,|K_2^{\enspace-}>)$.  Since in the effective theory one
has only the space
${\cal H}_2$, one had to postulate an effective Hamiltonian $\relax{\rm I\kern-.18em H}$ with
$[\relax{\rm I\kern-.18em H}, CP]\not=0$ in order to
obtain (9b) and
\begin{equation}
<\pi\pi|\relax{\rm I\kern-.18em H}|K_L^{\enspace-}>=
\epsilon<\pi\pi|\relax{\rm I\kern-.18em H}|K_1^{\enspace-}>+<\pi\pi|\relax{\rm I\kern-.18em H}|K_2^{\enspace-}>\not=0
\end{equation}
In the infinite dimensional RHS formulation one has other vectors outside
of ${\cal H}_2$ which have
the opposite parity so that
\begin{equation}
<\pi\pi|H|\phi_\lambda^+>={1\over b}<\pi\pi|H|F^-_{cp=+1}>\not=0
\end{equation}
even though $[H,CP]=0$.  This term gives the $\pi\pi$ transitions for large
$t\approx20\gamma_s$
when the amplitude $<\pi\pi|H|\phi^+_\sigma>$ is suppressed by a factor of
$e^{-\gamma_1t/2}$.  Thus the analogue of
$\epsilon<\pi\pi|\relax{\rm I\kern-.18em H}|K_1^{\enspace-}>$ is in the RHS
formulation given by ${1\over b}<\pi\pi|H|F^{\enspace-}_{cp=+1}>$ which has
nothing to do with CP
violation.
In the exact RHS theory with non-CP-violating H there can, however, not be
an anologue of the
direct CP-violation amplitude $<\pi\pi|\relax{\rm I\kern-.18em H}|K_2^{\enspace-}>$.

The theoretical CP problems that the Princeton effect caused
for the Lee-Oehme-Yang theory
clearly had its origin in the confinement of the theory to the
two-dimensional space ${\cal
H}_2$.  This confinement to ${\cal H}_2$ is an ad hoc assumption which
cannot be justified in an
infinite dimensional space of states, except as the zeroth order of
perturbation theory:   If
one considers
$H_0=H-H_w$ as
the exact Hamiltonian and $H_w$ as
a perturbation of $H$ and if one takes the eigenvectors $|K_1^{\enspace-}>$ and
$|K_2^{\enspace-}>$ of $H$ as the zeroth order eigenvectors of $H_0$, then
one knows from
perturbation theory that the higher order eigenvectors of $H_0$ are
generally not linear
combination of the zeroth order eigenvectors.  The exact eigenvectors
$|K^0>, |\bar K^{0}>$
of $H_0$ are
then given by
the highest ($\infty$) order of perturbation theory.  They
therefore cannot in general be linear combinations of the
zeroth eigenvectors
$|K_1^{\enspace-}>$ and
$|K_2^{\enspace-}>$ only.  But this is exactly what would be required in
(1.5) (for
$a_S=a_L={1\over\sqrt{2}}$ or
$\phi^{eff}=|K^0>$).  Though the complex basis vector expansion (B2) was
the origin of the idea
that the Princeton effect can be explained without a CP-violating
Hamiltonian, it
is not really needed in order to ask the question why the prepared state
$|K^{0}>$ should be
expandable (with
an accuracy of $10^{-3}$)
only in terms of vectors of the two-dimensional ${\cal H}_2$.
The existence of a finite ($>2$) or infinite number of linearly independent
$(\hbox{\rm basis-)}$ vectors -- irrespective of what definition of
convergence one uses -- is
already sufficient to see the problem.  The RHS is needed to
explain the existence of the eigenvectors (1.3a) with the property (1.4a) for $t\geq0$, and
to justify the inclusion of  these eigenvectors in a
complete basis system for the prepared state vectors
$\phi^+_{y=1}\in\Phi_-$,
(where
$\phi^+_{y=1}$ corresponds to $|K^0>$).
For these properties one needs the mathematical (topological) completion of
the linear
scalar product space $\Psi$ (Appendix A).
\setcounter{equation}{0}
\section{The Standard Phenomenological Description of CP Violation
as a Truncation of the
Exact Theory in the RHS}

Even if the $\pi\pi$ decay of the long lived neutral $K$ meson state can be
explained without a CP
violating interaction Hamiltonian, it does not mean that there is no CP
violation of the
Hamiltonian.  As was already mentioned above, the background term in (3.24)
-- though it needs to
be there -- may be so small ($|\sqrt{\gamma_i}b(E)/b|\ll10^{-3}$) that it cannot
account for the
Princeton effect.  Also, the observed time dependence of the transition
amplitude (1.8), in
particular the interference term in (1.9), may be such that it cannot be
explained by the
background integrals in (4.3) or (3.27).
Therefore, we want to
apply now the same exact theory of sections 2 and 3 to the
CP-violating Hamiltonian
\[
\relax{\rm I\kern-.18em H}=H+H_{sw}=H_0+H_w+H_{sw}\quad;\quad[\relax{\rm I\kern-.18em H},\ CP]\not=0 \ .
\]
Then, in place of the eigenvectors $|K^-_1\rangle,|K^-_2\rangle$ of $H$, we
use the
eigenvectors
$|K^-_S\rangle,|K_L^-\rangle$ of $\relax{\rm I\kern-.18em H}$ in the complex basis vector
expansion (B2) of the K-meson
state vector $\phi^+_y$:
\begin{eqnarray}
\phi^+&=&{1\over\sqrt{2}}(|K_S^{\enspace-}>b_S+|K_L^{\enspace-}>b_L)
\nonumber\\
&+&\sum_\beta\int^{-\infty_{II}}_0dE|E,\beta^->b_\beta(E)/\sqrt{2}\ .
\end{eqnarray}
In here, $|E,\beta^->$ are the generalized eigenvectors of $\relax{\rm I\kern-.18em H}$, and
$\beta$ are the degeneracy
quantum numbers (where now $\beta\not= cp$).  The complex basis vector
expansion (1) is, as
before, very general and exact, under the assumption that
$|K_S^{\enspace-}>$ and
$|K_L^{\enspace-}>$ are the only
Gamow vectors with the right quantum numbers for the neutral $K$-system.
The complex
expansion coefficients $b_S, b_L$ and
$b_\beta(E)$
$=<E,\beta|\phi^+>$
depend again predominantly upon the experimental conditions for the
preparation of
$\phi^+$.
Again for mathematical reasons
$b_\beta(E)\in{\cal H}_-^2\cap S$ cannot be exactly zero but it could be
arbitrarily small
(mathematics provides no information about the order of magnitude
involved).  Thus if
$|{\sqrt{\gamma_i}b(E)\over b_{S,L}}|\ll10^{-3}$, then, as far as the Princeton
effect (which is of
order
$10^{-3}$) is concerned, every pure neutral-$K$ state obtained from (1) is
adequately approximated
as
\begin{equation}
\phi^+\approx(|K_S^{\enspace-}>b_S+|K_L^{\enspace-}>b_L)
{1\over\sqrt{2}}\sim|K_S^{\enspace-}>\rho+|K_L^{\enspace-}>\ .
\end{equation}
Since the time evolution of the Gamow vectors $|K_S^{\enspace-}>$ and
$|K_L^{\enspace-}>$ is derived (not assumed) to be given by (1.4b) (with
the additional result
that
$t\geq0$), we have obtained
in the RHS an exact theory which contains
the standard phenomenological description of the neutral $K$-system with
CP-violation as an approximation.  No new physics has been developed,
but the standard phenomenological description has been given an exact
theoretical foundation.
The background terms in (1) may play (an observable) role in other experimental
investigations.$^{26)}$

Since the basis vectors on the r.h.s. of (1) are generalized eigenvectors
of ${\relax{\rm I\kern-.18em H}}$, the
time evolution operator $e^{-i\relax{\rm I\kern-.18em H}^\times t}$ is diagonal:
\begin{eqnarray}
\phi^+(t)&=&e^{-i\relax{\rm I\kern-.18em H}^\times t}\phi^+={1\over\sqrt{2}}
(e^{-im_St}e^{-{\gamma_S\over2}t}|K_S^{\enspace-}>b_S+e^{-im_Lt}e^{-{\gamma_
L\over2}t}|K_L^{\enspace-}>b_L)\nonumber\\
&+&\sum_\beta\int^{-\infty_{II}}_0dEe^{-iEt}|E,\beta^{\enspace-}>b^\beta(E)/
\sqrt{2};\phantom{0000}
t\geq0
\end{eqnarray}
In particular $|K_L^{\enspace-}>$ cannot evolve by its own Hamiltonian
$\relax{\rm I\kern-.18em H}$ (i.e. without
additional interaction with a regenerator) into $|K_S^{\enspace-}>$ or
vice-versa (i.e.
there is no ``vacuum regeneration of $K_S$ from $K_L$"), and neither can $K_S$ be regenerated
due to
$e^{-i\relax{\rm I\kern-.18em H} t}$ from the background term
\[
F_\beta^-=\int^{-\infty_{II}}_0dE|E,\beta^{\enspace-}>b^\beta(E)/\sqrt{2}\ .
\]
The Gamow vectors $|K_S^{\enspace-}>$ and $|K_L^{\enspace-}>$  evolve
(as a consequence of their definition from the resonance poles)
irreversibly
and obey the exact exponential decay law (1.4b):
\begin{eqnarray}
e^{-i\relax{\rm I\kern-.18em H}^\times
t}|K_S^{\enspace-}>&=&e^{-im_St}e^{-{\gamma_S\over2}t}|K_S^{\enspace-}>\qquad
,\qquad t\geq0\nonumber\\
e^{-i\relax{\rm I\kern-.18em H}^\times
t}|K_L^{\enspace-}>&=&e^{-im_Lt}e^{-{\gamma_L\over2}t}|K_L^{\enspace-}>
\qquad,\qquad t\geq0
\end{eqnarray}
There is no additional term on the r.h.s. of (4), in contrast to exact
infinite dimensional
theories in the Hilbert space.$^{10)}$  Also, $|K_S^{\enspace-}>$ and
$|K_L^{\enspace-}>$ cannot be
expressed in terms of
$\phi^+_{y=+1}$ and
$\phi^+_{y=-1}$, or any other finite or infinite
superposition of
$\phi^+_y\in\Phi_-$, due to the terms $F_\beta^-\in\Phi^\times_+$ on the
r.h.s. of (1).  The
time evolution in (3) is irreversible,
$t\geq0$, and $\phi^+(t)\in\Phi^\times_+$.  This means that it can
only be evaluated as a functional $<\psi^-|\phi^+(t)>$ at $\psi^-\in\Phi_+$
(which
represent  observed decay products
like $\psi^-=|\pi\pi>$).  In particular the
functional $\phi^+(t)$ cannot be evaluated
at $\phi^+\in\Phi_-$ i.e. the quantity
$\hbox{$<\phi^+_{y^{\prime}}|\phi^+_y(t)>$}$=\break\hfil
$<\phi^+_{y'}|e^{-iIH^\times
t}|\phi^+_y>$,  which would represent a vacuum regeneration
amplitude of $\bar K^0(y^{\prime}=-1)$ from
$K^0(y=+1)$, and vice versa, makes no sense in our theory.  These kind of
quantities have also no
observable meaning since no experiment can measure the probability for a
``transition" from
an in-state
$\phi^+_{y=1}$ into another in-state
$\phi^+_{y=-1}$.
\setcounter{equation}{0}
\section{Summary and Conclusions}

The purpose of this paper was to use the neutral $K$-system of two
interfering resonances to
test some aspects of the RHS-quantum theory of microphysical
irreversibility.  We limited our
investigation to the hypothesis that the two decaying $K$-states are ordinary,
first order, S-matrix-pole resonances (by the choice of $(B2)$) since the
standard theory with
complex effective Hamiltonian makes the same hypothesis (by the choice of a
diagonalizable complex Hamiltonian matrix rather than a Jordan block).
Then we saw that the effective Lee-Oehne-Yang theory is a subtheory of the
exact theory in
the RHS.
It must be emphasized that this is not the case for the exact theory in
Hilbert space
because the Hilbert space theory
does not allow for a complex basis vector expansion.  As a bonus, we saw
that the
remainder of the
exact theory, which is always ignored in the two dimensional effective
subtheory, leads to a non-zero $2\pi$ decay rate of the neutral $K$-system for
large time even if we choose a CP conserving Hamiltonian.  This may or may
not be of practical
significance since at this stage nothing can be said about its magnitude.

Many more experimental properties are known about the instantaneous
transition rate
$|<\pi\pi|H_{int}|\phi^+(t)>|^2$ than have been used in our discussion in
this paper.$^{27)}$
To
make adequate use of these properties the background terms
$<\pi\pi|H_w|F^{\enspace-}_{cp}(t)>$ need to be investigated further
and more of its characteristics needs to be known than just the property that it
decreases slower than exponentially in time.  Of particular interest is the
transition rate at
instances around
$t=12\tau_s$ where it has been fitted$^{27)}$ to the interference term
$\cos(\Delta
mt+\varphi)$ of (1.9); a result which, in that form, can probably not be
obtained from the
background term.  These questions will have to be discussed in a subsequent
paper.

redpajama_set_name: RedPajamaArXiv


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FILE - Chris Stapleton performs at the 50th annual CMA Awards in Nashville, Tenn., on Nov. 2, 2016. Stapleton will hit next month's Super Bowl stage to sing the national anthem, while R&B legend Babyface will perform "America the Beautiful."
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text:
Home » Records » Deepest Lakes
Deepest Lake in the World
& Deepest Lake in the United States
Satellite Image of Lake Baikal: Image by Geology.com using NASA Landsat data.
The World's Deepest Lake
Lake Baikal in southern Russia is the world's deepest lake. It is an estimated 5,387 feet deep (1,642 meters), and its bottom is approximately 3,893 feet (1,187 meters) below sea level. Lake Baikal is also the world's largest freshwater lake in terms of volume.
It is difficult to comprehend how a lake in the middle of the Asia could have a bottom that is nearly 4,000 feet below sea level. It is impossible for erosion to cut a channel that deep in the middle of a continent.
Lake Baikal is so deep because it is located in an active continental rift zone. The rift zone is widening at a rate of about 1 inch (2.5 centimeters) per year. As the rift grows wider, it also grows deeper through subsidence. So, Lake Baikal could grow wider and deeper in the future.
Related: World's Largest Lake
Lake Baikal map: Lake Baikal is located in southern Siberia near the city if Irkutsk. Map from the CIA Factbook.
Crater Lake: Panorama view of Crater Lake showing the steep crater wall that surrounds the lake and Wizard Island, a small volcano within the crater. Photo copyright iStockphoto / ziggymaj.
Deepest Lake in the United States:
The deepest lake in the United States is Crater Lake, a volcanic crater in southern Oregon. Its deepest measured depth is 1,949 feet (594 meters). It is the ninth-deepest lake in the world.
It is an amazing lake because no rivers flow into it or out of it. The water level in the lake is a balance between rainfall, groundwater flow and evaporation.
The lake is a volcanic crater that formed about 7600 years ago in the aftermath of one of the largest volcanic eruptions in recent geologic history. The explosive eruption ejected about 150 cubic kilometers of material, then the volcano collapsed into the empty magma chamber below to form a deep basin known as a caldera.
Deepest Lakes in the World
Baikal Siberia, Russia 5,387 ft (1,642 m)
Tanganyika Tanzania, D.R. Congo & Zambia 4,823 ft (1,470 m)
Caspian Sea Iran & Russia 3,363 ft (1025 m)
Vostok Antarctica 2,950 ft (900 m) minimums
O'Higgins-San Martin Chile & Argentina 2,742 ft (836 m)
Nyasa Mozambique, Tanzania & Malawi 2,316 ft (706 m)
Issyk Kul Kyrgyzstan 2,192 ft (668 m)
Great Slave Northwest Territories, Canada 2,015 ft (614 m)
Crater Lake Oregon, U.S.A. 1,949 ft (594 m)
Matano Indonesia 1,936 ft (590 m)
General Carrera Chile & Argentina 1,923 ft (586 m)
Hornindalsvatnet Norway 1,686 ft (514 m)
Quesnel Canada 1,660 ft (506 m)
Toba Indonesia 1,657 ft (505 m)
Sarez Tajikistan 1,657 ft (505 m)
Tahoe California & Nevada, U.S.A. 1,644 ft (501 m)
Argentino Argentina 1,640 ft (500 m)
Kivu D.R. Congo & Rwanda 1,575 ft (480 m)
Mjøsa Norway 1,535 ft (468 m)
Lake Chelan Washington, U.S.A. 1,486 ft (453 m)
Crater Lake bathymetry: Bathymetry image of Crater Lake by USGS. The deepest areas are in the northeast portion of the lake. Enlarge map.
[1] Crater Lake: Summary on the USGS Volcano Hazards Program website. Page last updated February 2015; last accessed August 2016.
[2] Facts about Crater Lake: Article on the Oregon Explorer Natural Resources Digital Library website, a collaboration of Oregon State University Libraries & Press and the Institute for Natural Resources. Last accessed August 2016.
[3] Morphometric Data for Lake Baikal: The INTAS Project 99-1669 Team, October 2002. Marc DeBatist of Ghent University, Ghent, Belgium. Last accessed August 2016.
Actual Lake Depths Will Vary
It is worth noting that estimated lake depths are just that - estimates. In fact, they are estimates of depths that change over time!
Searching online, a person may find several different depths listed for the same lake. Why is this?
The recorded depth of a lake may vary over time, depending on multiple factors.
Crater Lake, for example, does not have any streams or rivers flowing into or out of the lake. The water level is relatively constant because, remarkably, the amount of water coming into the lake (via rainfall and snowfall) generally equals the amount of water going out of the lake (via evaporation and seepage).
Since the depth of Crater Lake is directly influenced by the climate, it is easy to imagine how the water level would drop in a year of drought, or how the lake would become deeper in a year of record precipitation. These ideas can be applied to lakes fed and drained by rivers as well.
Another example of how the depth of a lake can change is with Lake Baikal, which is located over a continental rift. The rift is slowly getting wider and deeper each year, which means that the size of the lake is changing, too.
In addition to our planet changing over time, methods of measuring also change. Back in 1886, the depth of Crater Lake was estimated to be 608 meters - measured using a piano wire and lead weight. In 1959, the maximum depth was reported to be 589 meters with sonar measurement. And in July of 2000, 594 meters was the depth reached by a multibeam survey.
Three different depths were recorded at three different points in time with three different methods of measurement. Which is correct? They could all be accurate, or, none of them may be exactly right. There's no way to know with 100% certainty.
That is why it's important to keep in mind that these statistics are simply estimates, and the actual measurements are constantly changing, ever so slightly, even from one day to the next.
More Earth Extremes
Tallest Mountain
Largest Tsunami
Gifts That Rock
World*s Largest Diamonds
The Most Explosive Eruption
Largest Meteorite


Last updated April, 26 2019 at 12:17am.
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text:
Homogenes rubrogaster is a species of beetle in the family Cerambycidae. It was described by Napp and dos Santos in 1996.

References

Heteropsini
Beetles described in 1996

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1 The men of Ephraim were gathered together, and passed northward; and they said to Jephthah, "Why did you pass over to fight against the children of Ammon, and didn't call us to go with you? We will burn your house around you with fire!" 2 Jephthah said to them, "I and my people were at great strife with the children of Ammon; and when I called you, you didn't save me out of their hand. 3 When I saw that you didn't save me, I put my life in my hand, and passed over against the children of Ammon, and Yahweh delivered them into my hand. Why then have you come up to me this day, to fight against me?" 4 Then Jephthah gathered together all the men of Gilead, and fought with Ephraim; and the men of Gilead struck Ephraim, because they said, "You are fugitives of Ephraim, you Gileadites, in the midst of Ephraim, and in the midst of Manasseh." 5 The Gileadites took the fords of the Jordan against the Ephraimites. It was so, that when the fugitives of Ephraim said, "Let me go over," the men of Gilead said to him, "Are you an Ephraimite?" If he said, "No;" 6 then they said to him, "Now say 'Shibboleth;'" and he said "Sibboleth;" for he couldn't manage to pronounce it right: then they siezed him, and killed him at the fords of the Jordan. At that time, forty-two thousand of Ephraim fell. 7 Jephthah judged Israel six years. Then Jephthah the Gileadite died, and was buried in the cities of Gilead. 8 After him Ibzan of Bethlehem judged Israel. 9 He had thirty sons; and thirty daughters he sent abroad, and thirty daughters he brought in from abroad for his sons. He judged Israel seven years. 10 Ibzan died, and was buried at Bethlehem. 11 After him Elon the Zebulunite judged Israel; and he judged Israel ten years. 12 Elon the Zebulunite died, and was buried in Aijalon in the land of Zebulun. 13 After him Abdon the son of Hillel the Pirathonite judged Israel. 14 He had forty sons and thirty sons' sons, who rode on seventy donkey colts: and he judged Israel eight years. 15 Abdon the son of Hillel the Pirathonite died, and was buried in Pirathon in the land of Ephraim, in the hill country of the Amalekites.


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The Pittsburgh Steelers have had not had a real good whipping boy since the heyday of William Gay in 2009, and Antwon Blake has come close this season to bearing that level of animosity. And to be truthful, much of it had been merited based on his play. But it's also true that he has improved over the course of the season. And he has made some plays.
Blake's performance against the Bengals on Sunday was an improvement—not flawless, but pretty good, all things considered. An end zone interception didn't hurt matters, but he spent a lot of the game covering A.J. Green and Cincinnati scored a season-low 16 points.
The first play of the game was a fairly typical result for Blake against the Bengals, as the left outside cornerback lining up against bigger targets like Green and Marvin Jones. Andy Dalton regularly took advantage of his need to play off to hit short passes for medium gains, but the 5'9" corner defended them well.
The first pass of the game was a short pass to Jones with Blake set eight yards off. He broke off the throw and met Jones six yards out with a strong hit, wrapping by the waist and lifting up the receiver as he tried to advance forward, the safety coming in late to expedite his journey to the grass.
The Bengals threw a bunch formation at the Steelers off the left side later in the drive, with Blake playing off on what he diagnosed as a screen pass with two blockers set in front of Mohamed Sanu. Blake broke on the pass as soon as Dalton turned to his right, beating Green and wrapping up Sanu, to be joined by a trio of defenders for an old-fashioned gang tackle.
Jumping ahead to late in the first half, there is yet another of several examples of the Bengals taking what they can get with Blake going against their tall wide receivers. While he is clearly at a size disadvantage, he is better off playing off the line and coming up to the play rather than being outmuscled at the line to be beaten deep.
He makes the most of his assets to bring down players bigger than him, even if it doesn't always look well-coordinated. He is effective as long as he stays coordinated and his footwork holds up, which is not always the case, as on the second play of the third quarter as he slips out of his break to allow a five-yard first-down pass to Green.
Midway through the quarter, Blake was beaten down the field for his worst rep of the game, a third and two shot down the right sideline to Green, who used his size and speed to simply beat him down the field, though it seems the cornerback felt he pushed off.
Of course, later in the game, Blake recorded his second interception of the season, picking off Dalton on a lofted pass into the end zone, targeting Jones toward the back pylon. Set up from a bunch look, Blake was able to sit back and watch the play develop, enabling him to jump the route and nicely pluck the ball out of the air.
This time he showed instinct and urgency in his return, bringing the ball out to the 40-yard line until he inadvertently stepped out of bounds after hurdling a defender.
Late in the game, after the Bengals had taken the lead and the Steelers turned the ball over yet again, Blake was the first to make the run stop in the hopes of giving the offense one more chance to win the game. playing at the line with two tight ends set on his side, he was able to keep clean against his blocker on the edge to scrape off and bring the runner down after a yard.
Thought he did give up one deep shot, and his size enables taller receivers to get a free pass underneath, of which he Bengals are fully stocked, Blake had an overall solid game, and has generally been on an upward trajectory for most of the year. It's understandable why the coaching staff has him out there, even if most don't like it.


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Cartoon Time
People / Everyday Women in China
Telling China's Stories Through Elegant Illustrations
Editor: Wei Xuanyi
Li Min is an illustrator at China Daily. [China Women's News]
Li Min's success is not a coincidence but a result of a number of interwoven factors — the needs of the times, the support from her superiors, the understanding of editors and her own efforts.
"In the future, I hope my brushwork style will be more clear-cut. I will further dig into the traditional elements in Chinese culture and tell China's stories more vividly through illustrations," said Li Min, an illustrator at China Daily.
Her creations have won many domestically influential awards over the past several years.
Li has been quite busy recently: giving lectures at the Communication University of Zhejiang and China Jiliang University; offering an exhibition of her illustrations in Shenzhen, South China's Guangdong Province; and attending an exhibition at the invitation of her fellows from the Republic of Korea.
She said her original intention in accepting all these appointments was to present the charm of China Daily to more people.
Not long ago, the covers of China Daily's Global Edition began to be forwarded by microblog, Wechat and online portals, bringing a great deal of favorable comments and numerous fans.
The exquisite and beautiful stereotype-breaking newspaper cover illustrations were created by Li Min. She has worked in China Daily for nine years after she graduated from the Academy of Arts and Design of Tsinghua University.
Over that time, she has progressed in her job through continuous learning and bold attempts.
Zhang Geming, a professor at Tsinghua University, spoke highly of her illustrations, saying that the "elegant, fresh, lively and fashionable" illustrations give the newspaper vigor, and present it as a platform where illustrators could show their talents. Created by the illustrator's diligence and efforts, they are also a pleasant surprise to readers.
Many internet users give a thumbs-up to Li's creations, describing them as "design-oriented," "full of Chinese elements,""very nice," and "the pride of Chinese media."Li said the response has been beyond her expectations.
Li uses bright and eye-catching colors and smooth lines to produce strong visual impact. The traditional images of Chinese culture, such as pavilions, bamboo, pines, cranes and lotuses are used incisively and powerfully in her creations.
Since 2010, more than 300 of her illustrations have appeared on the front page of China Daily's overseas editions, which are published in more than 30 countries in Europe and Africa.
Li started to learn drawing at the age of about four and then received one-on-one tutorship from a teacher of the China Central Academy of Fine Arts during her early student years. Then she was enrolled by the school affiliated with the Central Academy of Craft Art (today's Academy of Arts and Design of Tsinghua University).
After seven years of studying visual communication design, she planned to further her education abroad. However, an internship provided by China Daily interrupted her plan.She chose to work for China Daily after the internship. In the first two years, drawing illustrations and cartoons for articles washer routine work, which made her dissatisfied with the job.
It wasn't long before the launch of China Daily Europe in 2011. Her superiors decided to use big illustration on the front page. It was an opportunity for Li to display her talent.
She took every mission seriously and kept improving her skills and urging herself to make innovations. She attended a six-month training program on gongbi painting organized by the Palace Museum and practiced tracing lines in her spare time.
She also frequently analyzed the illustrations on specialized websites. An illustration created by Victo Ngai, a New York based illustrator from Hong Kong Special Administraive, inspired her. Then she happened to realize that the characters can be created with more Chinese elements.
She said the priority of making illustration was the understanding of the content on the front page. Before drawing, she was briefed on the content by editors and communicates with the latter to better understand and digest it.
Li often gives lectures in universities and shares her experience in study and work. She encourages students to learn their majors well, since the knowledge they might consider boring may help them a lot in the future.
An illustration created by Li Min for China Daily's European Weekly [China Women's News]
(Source: China Women's News/Translated and edited by Women of China)
Ren's Love of CatsWomen of China2018-10-17Heart-warming Stories of Teachers2018-09-10Teacher Illustrates Changchun's Beauty in Book2018-08-30European Illustrator Advances Her New Book to Chinese Children2017-11-18Nostalgia for 80s Textbook Illustrations2017-08-16
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100 starts for Sam
Midfield general Sam Hutchinson began the New Year by posting a landmark appearance for Sheffield Wednesday.
Our New Year's Day clash with Birmingham City was the 29-year-old's 100th start for the club.
💯 The New Year's Day fixture at Hillsborough was Sam Hutchinson's 100th start for Sheffield Wednesday 💪#swfc pic.twitter.com/vGoT3mQlQH
— Sheffield Wednesday (@swfc) January 2, 2019
The tough-tackling midfield man joined Wednesday initially on loan in February 2014 and the switch from Chelsea was made permanent in July of that year.
Hutchinson made his debut in February 2014, coming on from the bench against Derby at Hillsborough and made his first start the following week in the Owls' 2-0 win away at Huddersfield.
During his S6 loan spell, Hutchinson's first goal came away at Bournemouth in an eventful 4-2 win, that later saw him sent off in the second half.
His permanent capture that summer was popular amongst Wednesdayites due to his wholehearted and full-blooded displays.
A regular under Stuart Gray and Carlos Carvalhal, Hutchinson became a mainstay in the side over the next three seasons; posting 20, 30 and 34 appearances in 14/15, 15/16 and 16/17.
And during the 2016/17 season, Hutchinson was voted as the August, October and December Player of the Month.
The ex-Chelsea man won the December award with an impressive 85% of the vote, the biggest landslide in the history of the swfc.co.uk award. It was in December that month that Hutchinson scored his second goal of the season in the South Yorkshire derby against Barnsley at Hillsborough.
The former Nottingham Forest and Vitesse Arnhem loanee has scooped a raft of awards with Wednesday including the Wise Old Owls, London Owls Player of the Year and Sheffield Star Player of the Year.
He was recalled to the side over the festive period, beginning with a substitute appearance in the 1-0 win over Preston at Hillsborough.
The fixtures with Middlesbrough, West Brom and Birmingham all saw the 29-year-old included in the starting XI, with the clash against the Blues his 100th start for the Owls.
Sam Hutchinson


Our city has the strength of nine centuries behind it. We are the birthplace of the Latvian national flag and we are a proud city. Proud but welcoming. And we are delighted to welcome the world's best orienteers as well as the participants of the OPEN race.
We have made sure that the WCUP2017 participants see the best our region has to offer – the courses planned through the most picturesque locations in Latvia. Thus making sure that not only will the event be memorable, but also unique and breathtakingly beautiful.
Cēsis has a long tradition of orienteering sport, with many of the best Latvian orienteers calling our city home, and this is the second time we have the honour of hosting an event of this calibre. I am sure you will appreciate the effort we have put into organising Orienteering World Cup 2017, and I am sure you will return here again to enjoy the nature around steep slopes and ravines of the ancient Gauja river valley and the rich atmosphere of the Cēsis old town.
Enjoy Cēsis, enjoy the competition and best of luck to all of you!


Placing a denture in water (or a denture cleanser solution) when it is not being worn helps the denture retain its shape, remain pliable and keeps it from drying out.
Dentures should never be placed in hot or boiling water, which could cause them to warp.
Denture adhesives are not a remedy for ill-fitting dentures, which may need to be relined or replaced to prevent oral sores from developing.
A company earns the ADA Seal of Acceptance by submitting scientific evidence that demonstrates the safety and efficacy of its product, which the ADA Council on Scientific Affairs evaluates according to objective requirements.
If chronic irritation (inflammation) exists beneath the denture bases (including but not limited to epulis fissuratum, oral ulcerations, or treatment-resistant Candida-related denture stomatitis). If denture adhesives are required to eat, or to retain the dentures socially (i.e., when the dentures will not remain in place by themselves), or when adhesives must be used more than once daily.
If the patient will not, or cannot, wear the removable prostheses.
If the denture has degraded sufficiently so that it is not stable in the mouth, no longer matches the other dentition, no longer fits well or if the denture itself or the prosthetic teeth are discolored, cracked, broken, or missing.
If there is a change in the teeth supporting a removable partial or overdenture.
If it has been more than 5 years since the denture was fabricated.
Denture cleansers and denture adhesives are products patients and/or their caregivers may consider.
A film of saliva is often all that is needed to hold dentures in place. Denture adhesives may be useful for those with dry mouth or xerostomia. Denture cream can help provide additional adhesion for well-fitting dentures, but may at best be only a temporary solution to the problem of an ill-fitting denture.10 A denture that fits poorly (i.e., feels loose or causes discomfort), may need to be relined or replaced.6, 7, 10 Ill-fitting dentures can contribute to the development of mouth sores or, with prolonged use, bone loss. Dental examinations and appropriate care will often help reduce or eliminate the need for denture adhesive products.
Follow the instructions provided with the denture adhesive. If the product does not come with instructions or the instructions are unclear, consult with a dental professional.
Do not use more adhesive than recommended.
Understand that some denture adhesives contain zinc, and that although they are safe to use in moderation as directed, if overused, they could contribute to harmful effects if overused.
Know that manufacturers may not always list their product ingredients.
Know that there are zinc-free denture adhesive products.
Stop using the denture adhesive and consult a physician if symptoms such as numbness or tingling sensations develop in the extremities.
Start with a small amount of adhesive—if the adhesive oozes off the denture into the mouth, that is likely too much adhesive.
Know that a 2.4-ounce tube of denture adhesive used by a consumer with upper and lower dentures should last seven to eight weeks.
Track the amount of denture adhesive used by marking on a calendar when a new tube is started, and when the tube is empty.
Consider speaking to their dentist to see that the dentures fit properly. Dentures can become ill-fitting as a person's gums change over time.
Denture cleansers with the ADA Seal have demonstrated that use of the product is both safe and effective, as evaluated according to objective requirements.
Substantiates the product does what it claims to do.
In at least one clinical study, subjects achieved a high degree of denture cleanliness (lack of visible stains, deposits or objectionable odor) using the product without any supervision.
Confirms that no hazards are expected when the product is used according to manufacturer's instructions.
Demonstration that the cleanser does not adversely affect the denture material or harm the oral tissues.
A company earns the ADA Seal for denture adhesives, or adherents, by producing scientific evidence demonstrating the safety and efficacy of its product, which is evaluated according to objective requirements.
Use of the product increases biting force.
Use of the product increases denture retention.
The adherent does not damage the denture.
American College of Prosthodontists. Facts and Figures. Accessed April 8, 2019.
Doundoulakis JH, Eckert SE, Lindquist CC, Jeffcoat MK. The implant-supported overdenture as an alternative to the complete mandibular denture. J Am Dent Assoc 2003;134(11):1455-8.
Aetna Inc. What Are Dentures? : Colgate 2001-2012. Accessed April 8, 2019.
Aetna Inc. Implant-Supported Denture (reviewed by the faculty of Columbia University College of Dental Medicine). Colgate 2013. Accessed April 8, 2019.
Rueggerberg F. Chapter 2: Dental Materials for Complete Dentures. Textbook of Complete Dentures. 6th ed: People's Medical Publishing House USA Ltd.; 2009. p. 7-23.
Felton D, Cooper L, Duqum I, et al. Evidence-based guidelines for the care and maintenance of complete dentures: a publication of the American College of Prosthodontists. J Am Dent Assoc 2011;142 Suppl 1:1s-20s.
Felton D, Cooper L, Duqum I, et al. Evidence-based guidelines for the care and maintenance of complete dentures: a publication of the American College of Prosthodontists. J Prosthodont 2011;20 Suppl 1:S1-s12.
Tuminelli FJ, Cooper LF, Campbell SD, et al. Position Statement: Frequency of Denture Replacement. American College of Prosthodontists 2015 (Reaffirmed 2018). Accessed April 8, 2019.
U.S. Food and Drug Administration. Denture Adhesives. 2018. Accessed April 8, 2019.
American Dental Association. News Releases: American Dental Association Statement on Zinc in Denture Cream. March 14, 2011. Accessed April 8, 2019.


Milwaukee, Wis. Dec. 16 - Hubert M. Schmidt, aged twenty years, said to be a janitor at Marquette university, and Thomas M. Stanton, aged twenty-five years, of Chicago, were found dead in a room on the second floor of the Plankinton house late last night. Schmit's throat was cut from ear to ear, and Stanton had a bullet hole in his right temple. The circumstances indicated that the case was a double suicide. When the bodies were found, Schmit was lying on the bed, and Stanton was sitting in a chair. Both were stark naked. They had come to the hotel early Sunday morning and registered from Chicago. Letters found in the pockets, however, indicated that Stanton alone was a Chicagoan, the letters being addressed to him at 1685 Grenshaw street, that city.
The discovery of the bodies was made by William O'Neil, a porter at the hotel. The men retired to their room immediately after coming to the hotel and had not left it. Late last night the night clerk, Gordon Long, became alarmed and sent O'Neil to investigate. When he reached the room on the second floor he found the door locked on the inside. It was forced and O'Neil stepped into the room. In an instant he leaped back, for on the bed was Schmit, his throat cut so deeply that the head was almost severed from the body. He lay in a pile of blood-soaked clothing, his feet almost hanging over the edge of the bed, while his head lay near the wail. Two razors lay on the bed beside him. In the chair, about four feet away, at the foot of the bed was the body of Stanton, the head dropping upon the shoulder. There was a bullet hole in his right temple and on the chair beside lay a thirty-two calliber Smith & Weston revolver with the five chambers full, one of the cartridges having been discharged. A slight cut was seen on the left side of the neck, evidently made with the second razor which was lying on the bed. No adequate cause has been discovered for the tragedy.


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HomeBizarreGoblins impregnate Gweru woman
Goblins impregnate Gweru woman
February 8, 2019 Staff Reporter Bizarre 0
A FAMILY from Mzila Village in Lower Gweru got a shocking report from their 19-year- old daughter when they quizzed her about who was responsible for her pregnancy and she claimed it was a goblin.
The woman, Simelokuhle Moyo, who lives with her grandmother reportedly claimed that she never slept with any man, but for the past few months always woke up with signs indicating that she had sex with someone during the night.
Moyo's grandmother, disturbed by what she had been told, invited other family members to interrogate the woman as her claims were mind-boggling.
When the family gathered, Moyo maintained her statement claiming that she saw the weird creature in her sleep and it was responsible for her pregnancy.
Her aunt said they were amazed by her claim. "Simelokuhle's statement have left us confused and in a dilemma. How can someone claim that she was impregnated by a goblin?
"She claims that she never had sex with a man, but we do not believe her story.
"We suspect that she just fabricated it after the boyfriend ran away or she slept with a number of men and she does not know who the real father is."
She further revealed that as a family they had agreed to consult a traditional healer to get to the bottom of the issue because the story gave their family a bad name in the community.
Simelokuhle told the same tale to B-Metro.
"I am five months pregnant, but I do not know who impregnated me because I do not remember sleeping with any man.
"It is unfortunate that my family thinks I am telling lies."
The Moyo family revealed that they were prepared to sell a cow and find out the truth from traditional healers and prophets.