\section{The string effective action}

\setcounter{equation}{0}

\section{Introduction}

The issue of coordinate transformations (also referred to as field redefinition
or reparametrisation invariance) in quantum field theory has been in the past
\cite{s1,s2,s3,s4,s5,s6,s7,s8}, and still is\cite{s9,s10,s11,s12,s13,s14,s15,s16,s17}
a subject of renewed interest.
The so-called ''equivalence theorem'' ensures, in principle \cite{ss1,ss2},  that the elements of
the $S$-matrix
remain the same after a field redefinition. However,
it is certainly not clear what becomes of the
renormalisability of a theory when a coordinate transformation is carried out.
We should mention that the reparametrisation invariance of the effective action
has benn discussed in \cite{toms2}.
\par
{} For instance,
it is well-known that the Lagrangian density for a complex scalar field
as given by
\begin{eqnarray}
{\cal L} &=& \partial_\mu \Phi^\star \partial^\mu \Phi
-m^2\left(\Phi^\star\Phi\right)
-\lambda\left(\Phi^\star\Phi\right)^2
\,\,\,\,\,\,.
\end{eqnarray}
is a renormalisable theory. In Cartesian coordinates where
\begin{equation}
\Phi={1\over\sqrt{2}} \left(\chi_1+i\chi_2\right)
\end{equation}
this model
describes two massive real fields $\chi_1$ and $\chi_2$ with a quartic interaction
as seen from the Lagrangian
\begin{eqnarray}
{\cal L} &=& {1\over 2}\partial_\mu \chi_1 \partial^\mu \chi_1
+ {1\over 2}\partial_\mu \chi_2 \partial^\mu \chi_2
 -{m^2\over 2}\left(\chi_1^2+\chi_2^2\right)
- {\lambda\over 4}\left(\chi_1^2+\chi_2^2\right)^2\,\,\,\,\,\,.
\label{cartesian}
\end{eqnarray}

\par
Let us now perform a field redefinition and
parametrise the complex scalar field $\Phi$
as
\begin{equation}
\Phi={1\over \sqrt{2}}\,\rho\,e^{i\theta/v}\,\,\,\,\,,
\end{equation}
where $\rho$ and $\theta$ are the polar coordinates (real fields)
and $v$ is a dimensionful constant. The Lagrangian density
becomes
\begin{eqnarray}
{\cal L} &=& {1\over 2}\partial_\mu \rho \partial^\mu \rho
+{1\over 2}{\rho^2\over v^2}\,\partial_\mu \theta \partial^\mu \theta -{m^2\over 2}\,\rho^2
-{\lambda\over 4}\,\rho^4  \,\,\,\,\,\,.
\label{polar-coord}
\end{eqnarray}
The first thing to notice in this parametrisation
is that the kinetic term for the real field $\theta$ is not of the standard form.
Furthermore, it is not clear how one can see that the spectrum of the theory
contains two massive fields (as in Cartesian coordinates).
\par
If one insists on treating the Lagrangian (\ref{polar-coord}) as a conventional field theory,
then a change of variables is necessary. {}For instance, we could
change $\rho$ to
\begin{equation}
\rho=v\,e^{\psi/v}\,\,\,\,\,,
\end{equation}
where $\psi$ is the new field.
The Lagrangian (\ref{polar-coord}) becomes then
\begin{eqnarray}
{\cal L} &=& e^{2\psi/v}\left[
{1\over 2}\partial_\mu \psi \partial^\mu \psi
+{1\over 2}\partial_\mu \theta \partial^\mu \theta\right]
-{m^2v^2\over 2}\,e^{2\psi/v} -
{\lambda v^4\over 4}\, e^{4\psi/v} \,\,\,\,\,\,.
\label{polar-coord-1}
\end{eqnarray}
Expanding the exponential, $e^{2\psi/v}= 1 + 2\psi/v + 2\psi^2/v^2 + \dots$,
leads to a standard kinetic terms for the two real fields $\psi$ and
$\theta$. In this parametrisation, the field $\theta$ is massless
while all the mass is appropriated by the field $\psi$.

\par
The important feature of the Lagrangians (\ref{polar-coord}) and (\ref{polar-coord-1})
is that their
interaction parts are non-polynomial in nature. That is,  they involve
derivatives of the fields. Therefore, the usual power-counting argument
of renormalisability does not apply here.

\par
The  aim of this note is to address the issue of the renormalisability
of theories like the one in (\ref{polar-coord}) when a change of variables
is carried out.
Our strategy is to treat the type of Lagrangians in (\ref{polar-coord}) and (\ref{polar-coord-1})
as a four-dimensional
non-linear sigma model supplemented with a potential term and study
their renormalisability. We use, for this purpose, the background field
method which we will review in the next section.

\section{The covariant background field expansion}

The non-linear sigma model is defined as follows: Let ${\Sigma}$ denote
the four-dimensional spacetime with coordinates $x^\mu$ and derivative $\partial_\mu$
and let ${\cal M}$ be a Riemannian manifold (the target space) with
metric $g_{ij}$. The field of the sigma model $\phi^i(x)$ is
a map from ${\Sigma}$ to ${\cal M}$. The Lagrangian for
the non-linear sigma model is
\begin{eqnarray}
{L}\left(\phi\right)= {1\over 2}
g_{ij}\left(\phi\right)\partial_\mu \phi^i \partial^\mu \phi^j
\,\,\,\,.
\label{sigma}
\end{eqnarray}
The field $\phi^i(x)$ labels the coordinates of the target space ${\cal M}$.

\par

The background field expansion method \cite{bf1,bf2,bf3,bf4,bf5,bf6} consists in splitting
the field $\phi^i(x)$ of the non-linear sigma model as
$\phi^i(x)=\varphi^i(x)+\pi^i(x)$, where $\varphi^i$ is the background field (a classical field)
and $\pi^i$ is the quantum fluctuation around this background field.
The quantum field $\pi^i=\phi^i-\varphi^i$, being a difference between
two coordinates of the target space, does not lead to a covariant expansion
of the non-linear sigma model. In order to respect the geometric nature of
the non-linear sigma model we expand the action in terms of the
quantun field $\xi^i$ instead. This field transforms as a vector on the target space and is
defined as follows:
\par
Let $\sigma^i(x,s)$ be
the unique geodesic joining  the two target space points $\varphi^i(x)$ and
$\varphi^i(x)+\pi^i(x)$. The affine parameter $s\in \left[0\,,\,1\right]$
parametrises this geodesic and we have the
interpolating conditions
\begin{equation}
\sigma^i(x,s=0)=\varphi^i(x) \,\,\,\,\,\,\,\,\,\,\,{\rm and}\,\,\,\,\,\,\,\,\,\,\,
\sigma^i(x,s=1)=\phi^i(x)=\varphi^i(x)+\pi^i(x)
\,\,\,\,\,.
\end{equation}
The geodesic equation is given by
\begin{equation}
{{\rm d}^2\sigma^i\over {\rm d}^2s}+\Gamma^i_{jk}\left(\sigma\right){{\rm d}\sigma^j\over {\rm d}s}
{{\rm d}\sigma^k\over {\rm d}s}=0\,\,\,\,,
\end{equation}
where $\Gamma^i_{jk}\left(\sigma\right)$ are the Christoffel symbols corresponding to the target space metric
$g_{ij}\left(\sigma\right)$. Let $\xi^i_s(x,s)$ denote the tangent vector to the geodesic $\sigma^i(x,s)$.
In other words,
\begin{equation}
\xi^i_s(x,s)={{\rm d}\sigma^i\over {\rm d}s}\,\,\,\,\,.
\end{equation}
The quantum field $\xi^i$ that will enter in the covariant expansion is defined
as the tangent vector to the geodesic at the point $\varphi^i(x)$. That is,
\begin{equation}
\xi^i\left(x\right)=\left.\xi^i_s\left(x,s\right)\right|_{s=0}\,\,\,\,\,\,.
\end{equation}
Since $\xi^i(x)$ transforms as a vector on the target space, the expansion of the
action in terms of this field will be automatically covariant.

\par
In order to obtain the covariant expansion, we start by extending the Lagrangian as
\begin{equation}
L\left(s\right)={1\over 2}g_{ij}\left(\sigma\left(s\right)\right)\partial_\mu\sigma^i\left(s\right)
\partial^\mu\sigma^j\left(s\right)\,\,\,\,\,
\end{equation}
so that $L\left(\phi\right)=L\left(s=1\right)$. We then expand $L(s)$ in powers
of $s$ around $s=0$. We obtain
\begin{eqnarray}
L\left(\varphi+\pi\right)=L\left(s=1\right)
&=&\left.\sum_{n=0}^\infty {1\over n!}{{\rm d}^nL\left(s\right)
\over
{\rm d}s^n}\right|_{s=0}
=\left.\sum_{n=0}^\infty {1\over n!}\left(\nabla_s\right)^nL\left(s\right)\right|_{s=0}\,\,\,\,\,,
\label{expansion}
\end{eqnarray}
where we have used the fact that $L\left(\sigma(x,s)\right)$ is a scalar to get the last equality.
Here $\nabla_s$ is the covariant derivative along the curve $\sigma^i(x,s)$.
Its acts on an arbitrary tensor $T^i_j$ as
\begin{equation}
\nabla_s T^i_j={{\rm d}\over {\rm d}s}T^i_j +\Gamma^i_{kl}\left(\sigma\right)\xi_s^k T^l_j
-\Gamma^l_{kj}\left(\sigma\right)\xi_s^k T^i_l \,\,\,\,.
\end{equation}
Of course, the tensor $T^i_j$ could have more indices.
If $T^i_j$ is a tensor function of $\sigma^i(x,s)$ only then
\begin{equation}
\nabla_s T^i_j\left(\sigma\right)=\xi^k_s\nabla_k T^i_j\left(\sigma\right)
=\xi^k_s\left[{\partial \over \partial \sigma^k}T^i_j+\Gamma^i_{kl}\left(\sigma\right) T^l_j
-\Gamma^l_{kj}\left(\sigma\right)T^i_l\right]\,\,\,\,\,.
\label{tensor}
\end{equation}
Here $\nabla_i$ is the usual covariant derivative with respect to $\Gamma^i_{jk}$.

\par
The expansion (\ref{expansion}) is evaluated using (\ref{tensor}) together with the formulae
\begin{eqnarray}
\nabla_s\partial_\mu\sigma^i &=& \nabla_\mu{{\rm d}\sigma^i\over{\rm d}s}=\nabla_\mu \xi^i_s
\equiv \partial_\mu \xi^i_s +\Gamma^i_{jk}\left(\sigma\right)\partial_\mu \sigma^j\xi^k_s\,\,\,\,,
\nonumber \\
\nabla_s g_{ij}\left(\sigma\right) &=& 0\,\,\,\,,
\nonumber \\
\nabla_s {{\rm d}\sigma^i\over{\rm d}s} &=& \nabla_s \xi^i_s=0\,\,\,\,,
\nonumber \\
\nabla_s \nabla_\mu \xi^i_s &=& R^i_{jkl}\xi^j_s\xi^k_s\partial_\mu\sigma^l
\,\,\,\,.
\end{eqnarray}
Here $R^i_{jkl}$ is the Riemann tensor{\footnote {Our convention is
$R^i_{jkl}=\partial_k\Gamma^i_{jl} +\Gamma^i_{km}\Gamma^m_{jl}-\left(k\leftrightarrow l\right)$.}}.
\par
The first few terms in the expansion of the Lagrangian $L\left(\phi\right)$ around
the background $\varphi^i$ are
\begin{eqnarray}
L\left(\phi\right) &=&
{1\over 2}g_{ij}\left(\varphi\right)\partial_\mu\varphi^i
\partial^\mu\varphi^j+g_{ij}\left(\varphi\right)\partial_\mu\varphi^i\nabla^\mu\xi^j
+{1\over 2}g_{ij}\left(\varphi\right)\nabla_\mu\xi^i
\nabla^\mu\xi^j
\nonumber \\
&+& R_{iklj}\left(\varphi\right)\partial_\mu\varphi^i\partial^\mu\varphi^j
\xi^k\xi^l
+\dots\,\,\,\,\,\,\,\,\,,
\label{expansion-1}
\end{eqnarray}
where
\begin{equation}
\nabla_\mu\xi^i=\partial_\mu\xi^i +\Gamma^i_{jk}\left(\varphi\right)\partial_\mu\varphi^j\xi^k\,\,\,\,.
\end{equation}
and $R_{iklj}=g_{im}R^m_{klj}$.
\par
If our Lagrangian contains a potential term like
\begin{equation}
L\left(\phi\right) =
{1\over 2}g_{ij}\left(\phi\right)\partial_\mu\phi^i\partial^\mu\phi^j-V\left(\phi\right)\,\,\,\,\,
\end{equation}
then the expansion of the potential is simply
\begin{equation}
V\left(\phi\right)=V\left(\varphi\right)+\sum_{n=1}^{\infty}{1\over n!}
\nabla_{j_{1}}\dots\nabla_{j_{n}}V\left(\varphi\right)\xi^{j_{1}}\dots\xi^{j_{n}}\,\,\,\,\,\,.
\label{expan-V}
\end{equation}

\par
Notice that the term $g_{ij}(\varphi)\partial_\mu\xi^i\partial^\mu\xi^j$
needed for the determination of the propagator for the quantum field
$\xi^i$ is not of the standard form due the the presence of the non-constant
metric $g_{ij}(\varphi)$. The remedy to this is to define a new quantum field
$\xi^a$ as
\begin{equation}
\xi^a=e^a_i\xi^i\,\,\,\,\,\, {\rm or} \,\,\,\,\,\,
\xi^i=E^i_a\xi^a\,\,\,\,\,,
\end{equation}
where we have introduced the vielbiens $e^a_i$ such that
\begin{equation}
g_{ij}=\eta_{ab}\,e^a_ie^b_j\,\,\,\,\,.
\end{equation}
The constat matrix $\eta_{ab}$ is invertible and $E^a_i$ is the inverse
of $e^i_a$. That is,
\begin{eqnarray}
e^a_iE^i_b=\delta^a_b\,\,\,\,\,\,,\,\,\,\,\,\, E^i_a e^a_j=\delta^i_j
\,\,\,\,\,\,.
\end{eqnarray}
With this field redefinition the expansions (\ref{expansion-1}) becomes
\begin{eqnarray}
L\left(\phi\right) &=&
{1\over 2}g_{ij}\left(\varphi\right)\partial_\mu\varphi^i\partial^\mu\varphi^j
+\eta_{ab}\,e^a_i\partial_\mu\varphi^i{\cal D}^\mu\xi^b
+{1\over 2}\eta_{ab}\,{\cal D}_\mu\xi^a {\cal D}^\mu\xi^b
\nonumber \\
&+& R_{iabj}\left(\varphi\right)\partial_\mu\varphi^i\partial^\mu\varphi^j\xi^a\xi^b
 +\dots\,\,\,\,\,\,\,\,\,,
\end{eqnarray}
where
\begin{eqnarray}
{\cal D}_\mu\xi^a &=& \partial_\mu\xi^a+\omega^a_{ib}\,\partial_\mu\varphi^i\xi^b\,\,\,\,,
\nonumber \\
\omega^a_{ib} &=& e^a_j\left(\partial_iE^j_b+\Gamma^j_{ik}E^k_b\right)
=e^a_j\nabla_iE^j_b\,\,\,\,\,
\end{eqnarray}
and $R_{iabj}=R_{iklj}E^k_aE^l_b$. The propagator is now computed
from the term $\eta_{ab}\,\partial_\mu \xi^a\partial^\mu\xi^b$ which has the standard
form.
\par
Similarly, the expansion (\ref{expan-V}) of the potential is
\begin{eqnarray}
V\left(\phi\right) &=& V\left(\varphi\right)
+\sum_{n=1}^{\infty}{1\over n!}V_{{a_{1}}\dots{a_n}}\left(\varphi\right)
\xi^{a_{1}}\dots\xi^{a_{n}}
\nonumber \\
V_{{a_1}\dots{a_n}}\left(\varphi\right) &\equiv&
E^{j_{1}}_{a_{1}}\dots E^{j_{n}}_{a_{n}}
\nabla_{j_{1}}\dots\nabla_{j_{n}}V\left(\varphi\right)
\,\,\,\,\,\,.
\end{eqnarray}
It is clear that $V_{{a_1}\dots{a_n}}$ is symmetric under the exchange of any two
indices. It is convenient to write this as
\begin{eqnarray}
V_{{a_1}\dots{a_n}} =D_{a_{1}}D_{a_{2}}\dots D_{a_{n}}V\,\,\,\,,
\label{V_{...}}
\end{eqnarray}
where the derivative $D_a$ acts as
\begin{equation}
D_aX_{bc}=E^i_a\left(\partial_iX_{bc}-\omega^d_{ib}X_{dc}-\omega^d_{ic}X_{bd}\right)\,\,\,
\end{equation}
on an arbitrary tensor $X_{bc}$.

\section{Field redefinition and sigma model}

We start with a field theory as described by the Lagrangian
\begin{eqnarray}
{\cal L} = {1\over 2}
\eta_{ab} \partial_\mu f^a \partial^\mu f^b
-V\left(f\right)
\,\,\,\,,
\label{renorma}
\end{eqnarray}
where $\eta_{ab}$ is a constant metric and $f^a(x)$ is a set of fields.
In the special case of the complex scalar
theory in (\ref{cartesian}), we have  $f^a=\left(\chi_1\,,\,\chi_2\right)$
and $\eta_{ab}={\rm diag}\left(1\,,\,1\right)$.
In four dimensions, this theory is renormalisable in the usual
power-counting sense (Dyson criterion) if the potential $V\left(f\right)$
is at most quartic in the fields $f^a$.
\par
Let now $\phi^i$ denote another set of fields which parametrise the same theory
as the fields $f^a$. In other words, we have made a change of variables from
the field $f^a$ to the fields $\phi^i$. We may therefore write
\begin{equation}
f^a=f^a\left(\phi^i\right)\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\
\phi^i=\phi^i\left(f^a\right)\,\,\,\,\,,
\end{equation}
where we have assumed that the change of variables is invertible.
Under this field redefinition,  the Lagrangian (\ref{renorma}) becomes
\begin{eqnarray}
{L}= {1\over 2}
g_{ij}\left(\phi\right)\partial_\mu \phi^i \partial^\mu \phi^j
-V\left(\phi\right)
\,\,\,\,,
\end{eqnarray}
where the metric $g_{ij}$ is given by
\begin{eqnarray}
g_{ij}=\eta_{ab}\partial_if^a\partial_jf^b\,\,\,\,\,.
\end{eqnarray}
Here $\partial_a={\partial\over\partial f^a}$ and $\partial_i={\partial\over \partial\phi^i}$ and the
range of the indices $a,b,..$ is the same as the range of the indices
$i,j,\dots\,\,\,$.

\par
{}From the relations $\partial_i\phi^j=\delta^j_i$ and $\partial_a f^b=\delta^b_a$
together with the chain rule, we deduce that
\begin{equation}
\partial_if^a\partial_a\phi^j=\delta^j_i\,\,\,\,\,\,,\,\,\,\,\,\,
\partial_a\phi^i\partial_i f^b=\delta^b_a\,\,\,\,.
\label{e-E}
\end{equation}
Hence we could identify the vielbeins $e^a_i$ and their inverse $E^i_a$
with
\begin{equation}
e^a_i=\partial_i f^a\,\,\,\,\,\,,\,\,\,\,\, E^i_a=\partial_a\phi^i\,\,\,\,.
\label{vielbeins}
\end{equation}
The inverse of the metric is $g^{ij}=\eta^{ab}\partial_a\phi^i\partial_a\phi^j$,
where $\eta^{ab}$ is the inverse of $\eta_{ab}$.
The Christoffel connection is then given by
\begin{equation}
\Gamma^i_{jk}={1\over 2}g^{il}\left(\partial_jg_{lk}+ \partial_kg_{lj}-\partial_lg_{jk}\right)
=\partial_j\partial_kf^a\partial_a\phi^i\,\,\,\,\,
\end{equation}
and all the components of the Riemann tensor $R^i_{jkl}$ vanish.
\par
Using the chain rule and the relations in (\ref{e-E}),
we find that the spin connection,
$\omega^a_{ib} = e^a_j\left(\partial_iE^j_b+\Gamma^j_{ik}E^k_b\right)$, vanishes. Indeed,
\begin{eqnarray}
\omega^a_{ib} &=& \partial_i\partial_jf^a\partial_b\phi^j+
\partial_jf^a\partial_if^c\partial_c\partial_b\phi^j
\nonumber \\
&=& \partial_i\left(\partial_jf^a\partial_b\phi^j\right) - \partial_jf^a\partial_c\partial_b\phi^j\partial_if^c +
\partial_jf^a\partial_if^c\partial_c\partial_b\phi^j = 0
\,\,\,\,\,.
\label{omega}
\end{eqnarray}
We also need the expression of the tensor $V_{a_{1}\dots a_{n}}$ as defined in (\ref{V_{...}}). This is
found to be
\begin{eqnarray}
V_{ab\dots c}=\partial_a\partial_b\dots\partial_c\,V\,\,\,\,\,.
\end{eqnarray}
If we assume that {\underline{ $V$ is at most quartic in the fieds}}, then
the background field expansion, \underline{{\it to all order}} in the quantum field $\xi^a$,
of the Lagrangian yields
\begin{eqnarray}
L &=&
{1\over 2}
g_{ij}\left(\varphi\right)\partial_\mu \varphi^i \partial^\mu \varphi^j
-V\left(\varphi\right)
\nonumber \\
&+& \eta_{ab}\,e^a_i\partial_\mu\varphi^i\partial^\mu\xi^b
+{1\over 2}\eta_{ab}\,\partial_\mu\xi^a \partial^\mu\xi^b
\nonumber \\
&-&\partial_aV\xi^a -{1\over 2}\partial_a\partial_bV\xi^a\xi^b -{1\over 6}\partial_a\partial_b\partial_c V\xi^a\xi^b\xi^c
-{1\over 24}\partial_a\partial_b\partial_c\partial_d V\xi^a\xi^b\xi^c\xi^d
\,\,\,\,\,.
\label{all-orders}
\end{eqnarray}
We notice that the resulting expansion is at most quartic in the quantum
field $\xi^a$. It is then clear that the theory is renormalisable in
the usual power-counting sense. We will illustrate this by an example
in the following section.

\section{An example: the interacting complex scalar field in polar coordinates}

In the notation of the previous sections, we have
$f^a=\left(\chi_1\,,\,\chi_2\right)$ and the constant
metric $\eta_{ab}={\rm diag}\left(1\,,\,1\right)$.
The new set of fields are $\phi^i=\left(\rho\,,\,\theta\right)$
and we have made the change of variables
\begin{eqnarray}
\chi_1=\rho\cos\left(\theta/v\right)\,\,\,\,\,,\,\,\,\,\,
\chi_2=\rho\sin\left(\theta/v\right)\,\,\,\,.
\label{chi}
\end{eqnarray}
The Lagrangian (\ref{polar-coord}) is a non-linear sigma model with a
metric $g_{ij}$ given by
\begin{equation}
g_{ij}= \left(
\begin{array}{cc}
1 & 0   \\
0 & {\rho^2/v^2} \end{array}
\right)\,\,\,\,\,.
\end{equation}
The non-vanishing components
of its Christoffel symbols are
\begin{eqnarray}
\Gamma^1_{22}=-{\rho\over v^2}\,\,\,\,\,\,,\,\,\,\,\,
\Gamma^2_{12}={1\over \rho}\,\,\,\,\,.
\end{eqnarray}

\par
Using (\ref{vielbeins}) and (\ref{chi}), the vielbeins $e^a_i$ and their inverses $E^i_a$ are{\footnote{
As matrices, $e^a_i$ and $E^i_a$ should be read as $e_{ia}$ and $E_{ai}$, respectively.}}
\begin{equation}
e^a_i= \left(
\begin{array}{cc}
\cos(\theta/v) & \sin(\theta/v)   \\
-{\rho\over v}\,\sin(\theta/v) & {\rho\over v}\,\cos(\theta/v) \end{array}
\right)
\,\,\,\,,\,\,\,\,
E^i_a= \left(
\begin{array}{cc}
\cos(\theta/v) &-{v\over \rho}\, \sin(\theta/v)   \\
\sin(\theta/v) & {v\over \rho}\,\cos(\theta/v) \end{array}
\right)
\,\,\,\,\,.
\end{equation}
Acoording to (\ref{omega}), all the components of the spin connection
$\omega^a_{ib}$ vanish.

\par
The background field expansion, {\it to all order} in the quantum field $\xi^a$,
of the Lagrangian (\ref{polar-coord}) yields
\begin{eqnarray}
L\left(\varphi\,,\,\xi\right) &=&
{1\over 2}\partial_\mu \rho \partial^\mu \rho
+{1\over 2}{\rho^2\over v^2}\,\partial_\mu \theta \partial^\mu \theta
-\left({m^2\over 2}\,\rho^2 +
{\lambda\over 4}\,\rho^4 \right)
\nonumber \\
&+& \eta_{ab}\,e^a_i\partial_\mu\varphi^i\partial^\mu\xi^b
+{1\over 2}\eta_{ab}\,\partial_\mu\xi^a \partial^\mu\xi^b
\nonumber \\
&-&V_a\xi^a -{1\over 2}V_{ab}\xi^a\xi^b -{1\over 6}V_{abc}\xi^a\xi^b\xi^c
-{1\over 24}V_{abcd}\xi^a\xi^b\xi^c\xi^d
\,\,\,\,\,.
\end{eqnarray}
Here we have used $\varphi^i=\left(\rho\,,\,\theta\right)$ to denote also
the background fields.

\par
The non-vanishing components of the symmetric tensor $V_{{a_1}\dots{a_n}}$ are
\begin{eqnarray}
V_1 &=&\rho\left(m^2+\lambda\rho^2\right)\cos(\theta/v)\,\,\,\,\,\,\,\,,\,\,\,\,\,\,
\nonumber \\
V_2 &=& \rho\left(m^2+\lambda\rho^2\right)\sin(\theta/v)\,\,\,\,\,\,\,\,,
\nonumber \\
V_{11} &=& m^2 + \lambda\rho^2\left(1+2\cos(\theta/v)^2\right)
\,\,\,\,\,\,\,\,,\,\,\,\,\,\,
\nonumber \\
V_{12} &=& 2\lambda\rho^2\cos(\theta/v)\sin(\theta/v)
\,\,\,\,\,,
\nonumber \\
V_{22}  &=& m^2 + \lambda\rho^2\left(1+2\sin(\theta/v)^2\right)
\,\,\,\,\,\,\,\,,\,\,\,\,\,\,
\nonumber \\
V_{111} &=& 6\lambda \rho\cos(\theta/v)\,\,\,\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\,\,
\nonumber \\
V_{112} &=& 2\lambda \rho\sin(\theta/v)\,\,\,\,,
\nonumber \\
V_{122} &=& 2\lambda \rho\cos(\theta/v)\,\,\,\,,
\nonumber \\
V_{222} &=& 6\lambda \rho\sin(\theta/v)\,\,\,\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\,\,
\nonumber \\
V_{1111} &=& 6\lambda \,\,\,\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\,\,
\nonumber \\
V_{1122} &=& 2\lambda
\,\,\,\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\,\,
\nonumber \\
V_{2222} &=& 6\lambda \,\,\,\,\,.
\end{eqnarray}
Let us now see what happens at the one-loop level in perturbation theory.

\noindent
\vskip 1.0cm
\noindent
{\bf\Large{One-loop renormalisation}:}
\vskip 1.0cm

\par
The generating functional for connected Green's functions,
$W\left[J\right]$,  is defined in the usual way by
\begin{eqnarray}
Z\left[J\right]=e^{iW\left[J\right]}=
N\int\left[{\rm d}\xi\right]e^{i\left[S\left(\varphi\,,\xi\right)
+J_a\xi^a\right]}\,\,\,\,\,,
\end{eqnarray}
where $N$ is normalising factor and
\begin{equation}
\left[{\rm d}\xi\right]=\prod_x\prod_{a=1}^{D}
\sqrt{g\left(x\right)}\,{\rm d}\xi^a\,\,\,\,
\end{equation}
is the coordinate independent measure.

\par
The Feynman propagator is computed from the free part of the action
\begin{equation}
L_{\rm {free}}\left(\xi\right)={1\over 2}\eta_{ab}\partial_\mu\xi^a\partial^\mu\xi^b
- {1\over 2}m^2\eta_{ab}\xi^a\xi^b\,\,\,\,\,.
\end{equation}
The mass term comes from $V_{11}$ and $V_{22}$. This propagator
is given by
\begin{eqnarray}
\Delta^{ab}_{{\rm F}}\left(x-y\right)=
{\eta^{ab}}\,{1\over(2\pi)^4}\int{e^{-ik.(x-y)}\over k^2-m^2}\, {\rm d}^4k\,\,\,
\end{eqnarray}
and satisfies
\begin{equation}
\eta_{ac}\left(\partial_\mu\partial^u+m^2\right)\Delta^{cb}_{\rm {F}}
\left(x\right)=-\delta^a_b\delta^4\left(x\right)
\,\,\,\,\,.
\end{equation}

\par
The loop expansion in terms of Feynman graphs is generated using
the Dyson-Wick perturbation theory. This is obtained from
\begin{eqnarray}
e^{iW\left(\varphi\right)}=<0|\exp\left[i\int{\rm d}^4x\,L_{{\rm int}}
\left(\varphi\,,\,\xi\right)\right]|0>
\,\,\,\,\,,
\end{eqnarray}
where
\begin{equation}
L_{{\rm int}}
\left(\varphi\,,\,\xi\right)=L
\left(\varphi\,,\,\xi\right)-L_{{\rm free}}
\left(\xi\right)\,\,\,\,.
\end{equation}
We then expand $\exp\left[i\int{\rm d}^4x\,L_{{\rm int}}
\left(\varphi\,,\,\xi\right)\right]$
and calculate the vacuum graphs by considering all the
possible Wick contractions involving the quantum field $\xi^a(x)$.
The background field $\varphi$ is treated as an external field.

\par
We use dimensional regularisation to isolate the divergences
in Feynman integrals. The dimension of spacetime is assumed
to be $d=4-\epsilon$. The original Lagrangian (\ref{polar-coord})
is  extended to $d$ dimensions as
\begin{eqnarray}
L =
{1\over 2}\partial_\mu \rho \partial^\mu \rho
+{1\over 2}{\rho^2\over v^2}\,\partial_\mu \theta \partial^\mu \theta
-\left({m^2\over 2}\,\rho^2 +
{\mu^{4-d}\lambda\over 4}\,\rho^4 \right)\,\,\,\,\,,
\label{L-mu}
\end{eqnarray}
where $\mu$ is an arbitrary mass parameter. In $d$ dimensions
$\rho$ has a mass dimension equal to ${1\over 2}d-1$ while
$\theta$ has a mass dimension equal to $1$. The coupling constant
$\lambda$ is kept dimensionless in $d$ dimensions.

\par
The first divergent diagram  is shown in figure 1 and contributes
\begin{figure}[thp]
\begin{center}
\includegraphics[scale=0.8]{complex-scalar-graph-1.eps}
\end{center}
\caption[]{One loop divergent diagram with one vertex ${\widetilde V}_{ab}$.}
\label{graph1}
\end{figure}

\begin{eqnarray}
I_1=\left(i\right)^2\times\left(-{1\over 2}\right)
{\widetilde{V}}_{ab}\eta^{ab}\mu^{4-d}\int{{\rm d}^dp\over(2\pi)^d}\,{1\over p^2-m^2}\,\,\,\,,
\end{eqnarray}
where ${\widetilde{V}}_{11}=V_{11}-m^2$, ${\widetilde{V}}_{22}=V_{22}-m^2$
and ${\widetilde{V}}_{12}=V_{12}$.
The divergent part of the integral is (see for example \cite{ryder})
\begin{equation}
I_1=-2\lambda\rho^2\left({im^2\over 8\pi^2\epsilon}+\,\,\,{\rm finite}\right)\,\,\,\,.
\end{equation}

\par
The second divergent graph
is drawn in figure 2 and gives
\begin{figure}[thp]
\begin{center}
\includegraphics[scale=0.8]{complex-scalar-graph-2.eps}
\end{center}
\caption[]{One loop divergent diagram with two vertices ${\widetilde V}_{ab}$ and ${\widetilde V}_{cd}$.}
\label{graph2}
\end{figure}

\begin{eqnarray}
I_2 &=& {\left(i\right)^4\over 2}\times
\left(-{1\over 2}\right)^2
{\widetilde V}_{ab}{\widetilde V}_{cd}\left(\eta^{ac}\eta^{bd}+\eta^{ad}\eta^{bc}\right)
\nonumber \\
 &\times&
\left(\mu^2\right)^{4-d}
\int{{\rm d}^dp\over(2\pi)^d}\,{1\over \left(p^2-m^2\right)}{1\over \left[\left(p-q\right)^2-m^2\right]}\,\,\,\,.
\end{eqnarray}
The infinite part of the integral is extracted (see for example \cite{ryder}) and we find
\begin{eqnarray}
I_2={5\over 2}\lambda^2\rho^4\left({i\mu^\epsilon\over 8\pi^2\epsilon} +\,\,\,{\rm finite}\right)\,\,\,\,.
\end{eqnarray}

\par
Green's functions are then rendered finite by adding to the
Lagrangian (\ref{L-mu}) the counter-term Lagrangian
\begin{eqnarray}
L_{{\rm CT}} =
{A\over 2}\partial_\mu \rho \partial^\mu \rho
+{B\over 2}{\rho^2\over v^2}\,\partial_\mu \theta \partial^\mu \theta
-\left({C\over 2 }\,\rho^2 +
{D\mu^{\epsilon}\lambda\over 4}\,\rho^4 \right)\,\,\,\,\,.
\end{eqnarray}
This leads to the bare Lagrangian
\begin{eqnarray}
L_{\rm B} =L+L_{{\rm CT}}=
{1\over 2}\partial_\mu \rho_{\rm B} \partial^\mu \rho_{\rm B}
+{1\over 2}{\rho^2_{\rm B}\over v^2_{\rm B}}\,\partial_\mu \theta_{\rm B} \partial^\mu \theta_{\rm B}
-\left({m^2_{\rm B}\over 2}\,\rho^2_{\rm B} +
{\lambda_{\rm B}\over 4}\,\rho_{\rm B}^4 \right)\,\,\,\,\,.
\end{eqnarray}
The bare quantities are defined as
\begin{eqnarray}
\rho_{\rm B} &=&\sqrt{Z_{\rho}}\,\rho \,\,\,\,\,\,,\,\,\,\,\,\,
Z_\rho=1+A\,\,\,\,\,\,,
\nonumber \\
m_{\rm B} &=&{Z_m}m \,\,\,\,\,\,,\,\,\,\,\,\,
Z_{m}^2={m^2+C\over m^2(1+A)}\,\,\,\,\,\,,
\nonumber \\
\lambda_{\rm B} &=&\mu^\epsilon{Z_\lambda}\lambda \,\,\,\,\,\,,\,\,\,\,\,\,
Z_\lambda={1+D\over (1+A)^2}\,\,\,\,\,\,,
\nonumber \\
\theta_{\rm B} &=&\sqrt{Z_{\theta}}\,\theta
\,\,\,\,\,\,,\,\,\,\,\,\, v_{\rm B} ={Z_v}v
\,\,\,\,\,\,,\,\,\,\,\,\,{Z_\rho Z_\theta\over Z_v^2}=1+B\,\,\,\,.
\end{eqnarray}

\par
At the one loop level, we have
\begin{equation}
C=-{\lambda m^2\over 2\pi^2\epsilon}\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\,
D={5\lambda \over 4\pi^2\epsilon}
\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\,
A=B=0
\,\,\,\,\,\,.
\end{equation}
This leads to $Z_\rho=Z_\theta=Z_v=1$. The one-loop result is precisely
the one found in the literature (see for example \cite{srednicki}).

\par
In conclusion, we have shown that a power-counting renormalisable scalar field theory
maintains this property in another reparametrisation of the fields.
The question of field redefinitions is much more important in theories involving gauge
fields. This is due to the fact that in some cases (like the Higgs model)
one could work with gauge invariant
variables which would eliminate the Faddeev-Popov ghosts. It is then crucial
to see what happens to the theory under such change of variables.
Some progress has already been made in this direction \cite{toms1,sonoda1,sonoda2,masson}.

\smallskip
\smallskip
\smallskip
\bigskip

\noindent $\underline{\hbox{\bf Acknowledgments}}$: I am very greatful to
David J. Toms for answering some questions regarding this work.
I would like also to thank Ian Jack and John Gracey for correspondence.

\smallskip
\smallskip


\section{Introduction}\label{intro}

Time evolutions of open quantum systems are not appearing as unitary maps.
Phenomenological equations describing non-equilibrium processes,
like diffusion, radiative decay, radiative excitation, etc., have no
terms describing memory-effects, they are Markovian.
These equations show the arrow of time, one cannot follow evolutions indefinitely backwards,
so they are to be described with \textbf{semigroups} of maps $\Tt^t$, acting on density matrices.

Basic dynamics is reversible.
In the quest for answering the question of
how irreversibility can emerge one has to perform limiting processes.
The methods of deriving $\Tt^t$ from basic unitary dynamics, see \cite{D74,D76,D76b}, using weak coupling or singular coupling limits,
give rise to semigroups of \textbf{completely positive} maps.

The generally possible structure of generators of such quantum dynamical processes
has been completely characterized for norm-continuous  semigroups,
yielding the ``GKS-Lindblad-equation'', \cite{GKS76,L76}.
Further attempts to study strongly continuous evolutions by E.B.Davies, \cite{D77,D77b,D80},
led, with some special assumptions, to a ``standard form'' of generators.
A.S.Holevo extended this study further, \cite{K95,H96,H97}.
He defines a generator in the Heisenberg picture as a quadratic form on the Hilbert space.
Crucial results of investigations in this sense
are presented by F.Fagnola and R.Rebolledo   in \cite{FR06}, sect. 3.

\par
In this paper we rework their methods. We
define a \textbf{generalized standard form}
for each generator of a QDS (quantum dynamical semigroup which is strongly continuous)
in the Schr\"{o}dinger picture and from a purely functional analytic point of view.
It allows for a complete characterization in case
the QDS is unambiguously defined by its ``matrix normal'' action on the matrix units
in some basis of the Hilbert space.
Two new examples of a QDS are presented, showing all the special features of the present analysis.
One is as simple and clear as possible, the other one models a natural process
observed in modern quantum physics.

\section{Setup and main statements}\label{setup}

We use the known fact that each strongly continuous semigroup on a Banach space
has a generator $\L$,
a closed operator with dense domain of definition $\DD(\L)$ which is
the set of all those Banach space elements $\rho$ for which the limit $t\rightarrow 0$
in the  formula defining $\L$,
\beq\label{generator}
\L(\rho)= \lim_{t\rightarrow 0}\frac{1}{t} \left( \Tt^t(\rho)- \rho \right),
\eeq
exists in norm. (In \cite{D76b} such a semigroup is denoted as  ``$c_0$ semigroup''.)
Now a QDS acts on three levels:  Hilbert space, density matrices, ``super operators'' acting on density matrices.
The goal is to represent the generator $\L$ by way of decomposing its action
into actions described by Hilbert space operators. Under certain assumptions stated below, this can be done by considering
\begin{itemize}
  \item  a separable Hilbert space $\HH$,
  \item  the Banach space of trace class operators $\mathfrak{T}(\HH)$ with the trace norm,
  \item  a strongly continuous semigroup of trace preserving completely positive
  maps $\Tt^t :$ $\mathfrak{T}(\HH)\rightarrow\mathfrak{T}(\HH)$ with generator $\L$.
\end{itemize}

\begin{define}{ \textbf{\textrm{ Generalized standard form}}}\label{gksldh}
\newline
Consider $L_k$ and $M$, linear operators on $\HH$ with dense domains of definition, and with the properties that
$M$ is  closed and generates a semigroup on $\HH$, and each $L_k$ is relatively bounded with respect to $M$.
Consider, moreover, ``generalized operators''
$L_k^\dag$ and $M^\dag$, linear operators mapping $\HH$ into the conjugate algebraic dual of $\DD(M)$,
such that for all $\psi\in\HH$ and for all $\eta\in\DD(M)$,
\beq\label{genop1}
\langle \eta|L_k^\dag|\psi\rangle := \langle \psi|L_k|\eta\rangle^\ast,  \quad \quad
\langle \eta|M^\dag|\psi\rangle   := \langle \psi|M|\eta\rangle^\ast.
\eeq
Then the generator $\L$ of a trace preserving QDS on $\mathfrak{T}(\HH)$ is in generalized standard form if
\beq\label{gensform}
\L(\rho) = \sum_k L_k\tdt \rho\tdt L_k^\dag - M\tdt\rho - \rho\tdt M^\dag,
\eeq
\beq
  \sum_k L_k^\dag L_k = M+M^\dag. \label{formequal}
\eeq
Equation (\ref{gensform}) holds for  $\rho$ of finite rank
which can be represented as finite sums of dyadic products
$\sum_n |\phi_n\rangle\langle\psi_n|$ with $\{\phi_n,\psi_n\}\subset\DD(M)$.
The domain $\DD(\L)$ is the closure of this set $\DD_0$
in graph norm $\|\rho\|_\L=\|\rho\|_1 +\|\L(\rho)\|_1$.
Equation (\ref{formequal}) holds for the noted entities as quadratic forms with $\DD(M)$ as the common form domain.
\end{define}
In sloppy words, the operators $M^\dag$ and $L_k^\dag$ have their domain of definition at the left hand side.
In Section \ref{discussion} we discuss these notations and the relation to Davies' investigations.

To prove the existence of such a generalized standard form we
need the assumption that there is some basis,
in which the matrix elements of density operators are differentiable functions of time.
We, therefore, consider:
\begin{itemize}
\item  a basis $\{e_k\}\subset\HH$ and the dense set  of finite linear combinations of these basis elements
  \beq\label{de}
  \DD_e:=\lspan\{e_k\},
  \eeq
  \item  the set of finite linear combinations of matrix units in this basis
  \beq\label{desquared}
  \DD_e^2 := \lspan\{|e_k\rangle\langle e_\ell|\}\subset\mathfrak{T}(\HH).
  \eeq
\end{itemize}

\begin{define}{ \textrm{\textbf{ Matrix normal QDS}}}\label{decomposable}
\newline
A QDS on $\mathfrak{T}(\HH)$ is called
``matrix normal'' if there exists a basis $\{e_k\}$ of $\HH$ such that $\DD_e^2$ as defined in  (\ref{desquared})
is a core for its generator $\L$.
\end{define}
The operator $\L$ is then completely defined by its action on each $\rho$ in the core $\DD_e^2$.
It is hence \emph{decomposable} into its actions on the matrix units $|e_k\rangle\langle e_\ell|$ in the basis $\{e_k\}$.
It is, for our intentions, not sufficient to assume nothing but that $\DD_e^2$ is just contained in the domain of definition of $\L$.
In Section \ref{discussion} we explain this remark and give an example.

This formal assumption on decomposability with respect to matrix elements
is now related to the description of generators $\L$ in the generalized standard form,
using operators on the underlying Hilbert space $\HH$,
without a priori reference to a basis.
The main result of the present investigation is:
\begin{thm}{ \textbf{\textrm{ Matrix normality is equivalent to the existence of a generalized standard form.}}}\label{gksldhthm}
\newline
There exists a generalized standard form for the generator $\L$
of a trace preserving QDS if and only if it is matrix normal.
In this case  $\DD_e\subset\DD(M)$.
\end{thm}

\section{Proofs and further details}\label{procedure}

\subsection{From decomposability to the standard form.}\label{adctostandard}

Before we come to the proof of Theorem \ref{gksldhthm}, some preparation is necessary. We start by stating the following
\begin{lem}\label{lprop}

The generator $\L$ of a matrix normal trace preserving QDS has the properties:

\begin{enumerate}
  \item $\forall\phi\in\DD_e: \quad \langle\phi\,|\L(|\phi\rangle\langle\phi|)\,|\phi\rangle\leq 0$, \label{litem1}
  \item $\forall\phi\in\DD_e,\forall\psi\textrm{ with } \langle\phi|\psi\rangle=0 :\quad
           \langle\psi\,|\L(|\phi\rangle\langle\phi|)\,|\psi\rangle\geq 0$,       \label{litem2}
  \item $\forall\{\phi_k\in\DD_e,\psi_k\} \textrm{ with } \sum_k\langle\phi_k|\psi_k\rangle=0 :\quad
           \sum_{k,\ell}\langle\psi_k|\,\L(|\phi_k\rangle\langle\phi_\ell|)\,|\psi_\ell\rangle\geq 0$. \label{litem3}
\end{enumerate}
\end{lem}
\begin{proof}
Item (\ref{litem1}) is a consequence of preservation of both positivity and trace by transformation of $|\phi\rangle\langle\phi|$ by $\Tt^t$.
Starting at $t=0$ with $\rho=|\phi\rangle\langle\phi|$,
some new positive diagonal elements  may appear in $\Tt^t(|\phi\rangle\langle\phi|)$.
Since  the sum of all the diagonal elements is constant, the original single element,
which was ``$1$'', when $\|\phi\|=1$, has to diminish.
Item (\ref{litem2}) concerns the newly appearing diagonal elements as $t$ increases,
starting from zero, and follows again from positivity of $\Tt^t$.
Item (\ref{litem3}) follows from complete positivity of $\Tt^t$. It is item (\ref{litem2})
      applied to the generator  $\hat{\L}$ of the semigroup  $\hat{\Tt^t} =  \Tt^t \otimes \textrm{Id}$,
      acting on density matrices on $\HH\bigotimes\ell^2$,
      where $\textrm{Id}$ is the identity-map,
      with $\hat{\phi}=\sum_k \phi_k\otimes f_k$,  $\hat{\psi}=\sum_k \psi_k\otimes f_k$,
      where $\{f_k\}$ is an orthonormal basis of $\ell^2$.
\end{proof}
The subsequent construction follows Holevo \cite{H97};
(see also, for the case of norm continuous semigroups, \cite{F99};
the method, according to F.Fagnola, has already been used earlier by A.Frigerio):

\begin{define}{ \textbf{The generator of contraction in Hilbert space, M}}\label{mcontractor}
\newline
For a matrix normal QDS
choose some vector $\chi\in\DD_e$ with $\|\chi\|=1$.
On $\DD(M):=\{\psi\,:\,|\psi\rangle\langle\chi|\in \DD(\L) \}$ an operator $M$
is defined by
\beq\label{em}
M\psi := -\L(|\psi\rangle\langle\chi|)\chi +
\half \langle\chi|\,\L(|\chi\rangle\langle\chi|)\,|\chi\rangle\, \psi.
\eeq
\end{define}
The naming is explained in Proposition \ref{contractor}.
Note that $\DD_e\subset \DD(M)$, which will be used in Equation (\ref{expandermap}).

\begin{proposition}{ \textbf{The expander in the generator of the QDS}}\label{expander}
\newline
For a matrix normal QDS
the map
\beq\label{expandermap}
\L_+ :\,\,\rho\mapsto \L(\rho)+M\rho+\rho M^\dag
\eeq
is a completely positive map from $\DD_e^2$ into $\mathfrak{T}(\HH)$.
\end{proposition}
\begin{proof}
Each $\sigma\in\DD_e^2$  can be expanded as a linear combination of at most four positive matrices of finite rank.
Each positive matrix $\rho\in\DD_e^2$  can be represented as $\rho=\sum_{n=1}^N  |e_n\rangle\langle e_n|$
for some elements $e_n\in\DD_e$, and there exists
$$
\Tr\{M\rho+\rho M^\dag\} = \sum_{n=1}^N \langle e_n|(M+M^\dag)|e_n\rangle <\infty.
$$
Now $\L$ generates a semigroup which preserves the trace, so for all $\rho\in\DD(\L)$
\beq\label{ltrace}
\Tr\{\L(\rho)\}=\left[\frac d {dt}\Tr\{\Tt^t(\rho)\}\right]_{t=0}=0,
\eeq
and this implies, for all $\rho\in\DD_e^2$
\beq
\Tr\{\L_+(\rho)\}=\Tr\{M\rho + \rho M^\dag\},  \label{lplusone}
\eeq
and $\L_+$ is well defined on $\DD_e^2$.

The criterion for complete positivity states that
$\sum_{k,\ell} \langle\psi_k|\,\L_+(|\phi_k\rangle\langle\phi_\ell|)\,|\psi_\ell\rangle$
is non-negative for all
$
N$ and $ \{(\phi_k,\psi_k), k=\,1 \ldots N,\,\,\phi_k\in\DD_e\}.
$
To check whether this criterion is fulfilled,
we insert (\ref{em}) into (\ref{expandermap}).
We get
\beqa
\langle\psi_k|\,\L_+(|\phi_k\rangle\langle\phi_\ell|)\,|\psi_\ell\rangle = & & \nonumber\\
 \langle\psi_k|\,\L(|\phi_k\rangle\langle\phi_\ell|)\,|\psi_\ell\rangle
&+& \langle\chi|\,\L(|\chi\rangle\langle\chi|)\,|\chi\rangle \cdot \langle\psi_k|\phi_k\rangle\cdot\langle\phi_\ell|\psi_\ell\rangle\nonumber\\
- \langle\psi_k|\,\L(|\phi_k\rangle\langle\chi|)\,|\chi\rangle \cdot\langle\phi_\ell|\psi_\ell\rangle
&-& \langle\chi|\,\L(|\chi\rangle\langle\phi_\ell|)\,|\psi_\ell\rangle \cdot \langle\psi_k|\phi_k\rangle. \label{hol}
\eeqa
Defining
\beq\label{double}
\phi_{N+k}:=- \langle\psi_k|\phi_k\rangle\cdot \chi,\,\qquad \psi_{N+k}:=\chi,
\eeq
we continue the reformulation of (\ref{hol}), writing it as
\beq
\sum_{i=0}^1 \,\sum_{j=0}^1\,\, \langle\psi_{k+iN}|\,\L(|\phi_{k+iN}\rangle\langle\phi_{\ell+jN}|)\,|\psi_{\ell+jN}\rangle .
\eeq
Summation over all indices gives
\beq
\sum_{k=1}^N \sum_{\ell=1}^N    \langle\psi_k|\,\L_+(|\phi_k\rangle\langle\phi_\ell|)\,|\psi_\ell\rangle
= \sum_{k=1}^{2N} \sum_{\ell=1}^{2N}    \langle\psi_k|\,\L(|\phi_k\rangle\langle\phi_\ell|)\,|\psi_\ell\rangle \geq 0.
\eeq
The inequality at the end holds because of  the property stated in Lemma (\ref{lprop}.\ref{litem3}) of $\L$
and $\sum_{k=1}^{2N}\langle\phi_k|\psi_k\rangle = 0$, which follows from (\ref{double}).
Complete positivity of $\L_+$ is thus established.
\end{proof}

With these preparations we now come to the proof of the algebraic part of Theorem \ref{gksldhthm}:
\begin{proof}{ \textbf{Algebraic skeleton of the extended standard form:}}
\newline
Due to Proposition \ref{expander}, Theorem \ref{krausthm} and Corollary \ref{krausdomain},
given in the Appendix in Section \ref{unboundedcp},
we have for all $ \rho \in \DD_e^2$ that $
\L(\rho)$ can be decomposed as
$$
\L(\rho)=\L_+(\rho)-M\rho-\rho M^\dag.
$$
Moreover, there exist generalized Kraus Operators $L_k$, acting on $\DD_e$, such that
$$
\L_+(\rho)=\sum_k L_k\tdt\rho\tdt L_k^\dag.
$$

Proof of Equation (\ref{formequal}):
\newline
Operations which are standard procedures in dealing with ordinary operators
have to be carefully adapted to the case where generalized operators appear.
Consider $\rho=|\phi\rangle\langle\phi| \in \DD_e^2$ with $\|\phi\|=1$.
So $\L(\rho)\in\mathfrak{T}(\HH)$ and $\Tt^t(\rho)$ is differentiable:
$$
0= \frac d{dt} \Tr\left(\Tt^t(\rho)\right)|_{t=0} = \Tr\left(\L(\rho)\right)=
\sum_\alpha\langle\alpha|\left(\sum_k L_k\rho L_k^\dag -M\rho-\rho M^\dag\right)|\alpha\rangle=
$$
$$
=\sum_\alpha \left( \sum_k\langle\alpha|L_k|\phi\rangle\langle\phi|L_k^\dag|\alpha\rangle
-\langle\alpha|M|\phi\rangle\langle\phi|\alpha\rangle- \langle\alpha|\phi\rangle\langle\phi|M^\dag|\alpha\rangle\right),
$$
for any basis $\alpha$. We choose a basis with $|\alpha=1\rangle=|\phi\rangle$,
so $\langle\alpha|\phi\rangle=\delta_{\alpha,1}$, and
$$
\Tr\left(\L(\rho)\right)=\sum_\alpha\left( \sum_k |\langle\alpha|L_k|\phi\rangle|^2
-(\langle\alpha|M|\phi\rangle\ + \langle\phi|M^\dag|\alpha\rangle)\delta_{\alpha,1} \right)=
$$
$$
=\sum_\alpha \left(\sum_k |\langle\alpha|L_k|\phi\rangle|^2 \right)  -\langle\phi|(M + M^\dag)|\phi\rangle.
$$
Since $\Tr\left(\L(\rho)\right)=0$ and the expectation value of $M$ is finite, the first term with the double sum is also finite. This sum over non-negative numbers, however, can be rearranged as follows:
$$
\sum_\alpha \sum_k |\langle\alpha|L_k|\phi\rangle|^2 =
\sum_k \sum_\alpha \langle\phi| L_k^\dag  |\alpha\rangle \langle\alpha|  L_k|\phi\rangle =
 \langle\phi|\sum_k L_k^\dag L_k|\phi\rangle.
$$
Therefore, the formula
$$
\langle\phi|\left(\sum_k L_k^\dag L_k -M-M^\dag  \right)|\phi\rangle=0
$$
holds for each vector $\phi\in\DD_e$, and Equation (\ref{formequal}) is proven
to be valid on the restricted form domain $\DD_e$.
\end{proof}

Having built the algebraic structure, we consider analytic-topological features.
From Equation (\ref{formequal})
on its skeleton domain it follows immediately that each $L_k$ is bounded relative to $M$, that is
$$
\|L_k\phi\|^2=2 \textrm{Re} \langle\phi|M|\phi\rangle \leq 2\|\phi\|\cdot\|M\phi\| < (\|\phi\|+\|M\phi\|)^2.
$$
The $L_k$ can therefore be uniquely extended to the entire domain $\DD(M)$. The super operator $\L_+$ and the validity of Equation (\ref{gensform}) can be extended to $\DD_0$,
and  Equation (\ref{formequal}) holds on the whole form domain.

In order to finish the proof of one half of Theorem \ref{gksldhthm}, our last task is to show
that $M$ is closed and generates a contraction semigroup.
\begin{proposition}{ \textbf{ M is closed.}}
\newline
The operator $M$, the mover-and-contractor presented in Definition \ref{mcontractor}, is closed.
\end{proposition}
\begin{proof}

We consider a sequence $\psi_n \in \DD(M)$, with $\psi_n\rightarrow\psi$ and $M\psi_n\rightarrow\eta$.
With $\psi_{m,n}:=\psi_m-\psi_n$,
using   Equation (\ref{lplusone}), we have
$$
0\leq\Tr\{\L_+(|\psi_{m,n}\rangle\langle \psi_{m,n}|)\}=
\Tr\{M|\psi_{m,n}\rangle\langle\psi_{m,n}|  + |\psi_{m,n}\rangle\langle\psi_{m,n}|M^\dag\}$$
$$\leq 2\cdot\|\psi_m-\psi_n\|\cdot\|M(\psi_m-\psi_n)\| \rightarrow 0
$$
as $m,n\,\rightarrow\infty$.
All following limits of operators are now existing in trace norm.

For any $\gamma\in\DD_e$, any $z\in\C$, we have
$$
0\leq\L_+(|\gamma+z \psi_{m,n}\rangle\langle\gamma+z \psi_{m,n}|)=$$
$$=\L_+(|\gamma\rangle\langle\gamma|)+z\L_+(|\psi_{m,n}\rangle\langle\gamma|)+
z^\ast\L_+(|\gamma\rangle\langle\psi_{m,n}|)+|z|^2\L_+(|\psi_{m,n}\rangle\langle\psi_{m,n}|).
$$

The first term is constant and the last one vanishes in the limit $m,n\rightarrow\infty$ for arbitrary $|z|$.
The two terms in between, depending linearly on $z$ and being negative for some argument of $z$, have to remain bounded
in relation to $\L_+(|\gamma\rangle\langle\gamma|)$, independently of $z$, hence each one
has to go to zero, especially $\L_+(|\psi_{m,n}\rangle\langle\gamma|)\rightarrow 0$.

It follows that
$\L_+(|\psi_n\rangle\langle\gamma|)$ is a Cauchy sequence, and therefore also
$\L(|\psi_n\rangle\langle\gamma|)$ converges. Since $\L$ is closed,  $|\psi\rangle\langle\gamma|\in\DD(\L)$
for each $\gamma\in\DD_e$
and hence $\psi\in\DD(M)$ and $M\psi=\eta$, by definition.
\end{proof}

\begin{proposition}{ \textbf{ M generates a semigroup on the Hilbert space.}}\label{contractor}
\newline
The operator $M$ is maximal accretive.
It generates the contractive semigroup $e^{-tM}$ on the Hilbert space $\HH$.
\end{proposition}

\begin{proof}
$M$ is accretive ($-M$ is dissipative), i.e. $\textrm{Re}\langle\phi|M|\phi\rangle\geq 0$
$\forall\phi\in\DD(M)$, as is seen in Equation \ref{formequal}.
It generates a semigroup iff it is maximal accretive.\par
Suppose now that it is not maximal and that there exists an accretive extension $\bar{M}$,
being maximal and hence generating a semigroup.
In its domain $\DD(\bar{M})$, considered as a Hilbert space with norm (squared) $\|\phi\|_{\bar{M}}^2=\|\bar{M}\phi\|^2+\|\phi\|^2$,
there exists a non-empty subspace $\bar{\DD}$ orthogonal to $\DD(M)$.
Extend further the operators $L_k$ from acting on $\DD(M)$ to acting on  $\DD(\bar{M})$ by $L_k\phi=0$ for all $\phi\in\bar{\DD}$.
Then the inequality $\sum_k L_k^\dag L_k \leq\bar{M}+\bar{M}^\dag$ holds.
Therefore, under these conditions there exists a contractive semigroup on $\mathfrak{T}(\HH)$
with generator $\bar{\LL}$ such that
$$
 \bar{\LL}(|\phi\rangle\langle\psi|)=
\sum_k |L_k\phi\rangle\langle L_k\psi| -|M\phi\rangle\langle\psi| - |\phi\rangle\langle M\psi|,\qquad\forall (\phi,\psi)\subset\DD(\bar{M}),
$$
as proven in \cite[Section 3]{F99}. However, such an $\bar{\LL}$ would be an extension of $\LL$,
a contradiction to the maximal dissipativity of generators of semigroups.
\end{proof}

The properties of $M$ and of the $L_k$ are in precise correlation
to the structure of generators of QMS as defined in \cite{F99} and \cite{FR06}.

\subsection{Deriving decomposability from the standard form}\label{standardtoadc}

We  assume the existence of a generator $\L$ in generalized standard form
and we divide it, as above, into two pieces.
One of it is the completely positive map $\L_+$, the other one shall now be investigated in more detail.
\begin{define}\label{msuperdef}
On the Banach space $\Tf(\HH)$ we consider the operator
              \beq\label{msuper}
              \M:\,\,\rho\mapsto M\rho + \rho M^\dag,
              \eeq
with domain $\DD_0$, as defined in Definition \ref{gksldh}
as a subset of $\DD(\L)$, namely
\beq\label{dmdefine}
\DD_0 := \{\rho = \sum_{n=1}^N   |\phi_n \rangle\langle\psi_n|,\, \{\phi_n ,\psi_n\}\in\DD(M),\,N<\infty\}\label{rhodec}.
\eeq
\end{define}
It follows that also $\L_+=\L+\M$ is well defined on $\DD_0$.
Note that the expander $\L_+$ is dominated by $\M$. Namely, as in the proof of Proposition \ref{expander}, Equation (\ref{ltrace}), we have the general condition that
$\L$ generates a semigroup which preserves the trace, hence for all $\rho\in\DD(\L)$
\beq
\Tr\{\L(\rho)\}=\left[\frac d {dt}\Tr\{\Tt^t(\rho)\}\right]_{t=0}=0,
\eeq
implying
$$
0\leq \Tr\{\L_+(\rho)\}=\Tr\{\M(\rho)\}\leq \Tr\{|\M(\rho)|\} =\|\M(\rho)\|_1,\qquad \forall\rho\in\DD_0,\,\,\rho\geq 0.
$$
In order to take also non-positive matrices into account we look at general matrix units.
\begin{lem}\label{schwarztype}
For $(\phi,\psi)\subset\DD(M)$ we have
$$
\|\L_+( |\phi \rangle\langle\psi|)\|_1\leq  \left(\|\L_+( |\phi \rangle\langle\phi|)\|_1 \right)^{1/2}\cdot
\left(\|\L_+( |\psi \rangle\langle\psi|)\|_1 \right)^{1/2}.
$$
\end{lem}
\begin{proof}
We use generalized Kraus operators, see Section \ref{unboundedcp}. In case their number is infinite
we shall in the following first consider summations over finite $k\in\{0,\ldots, K\}$
and then in the end look at $K\rightarrow\infty$.
Moreover, the triangle inequality for the trace-norm
and the Schwarz inequality for the inner product in $\ell^2$ shall be used. We have
\beqa
\|\L_+( |\phi \rangle\langle\psi|)\|_1=  \|\sum_k |L_k \phi \rangle\langle L_k\psi| \|_1
\leq   \sum_k \| |L_k \phi \rangle\langle L_k\psi|\|_1 = \sum_k \|L_k \phi\|\cdot \| L_k\psi\| \nonumber\\
\leq  \left( \sum_k \|L_k \phi\|^2\right)^{1/2}\cdot\left(\sum_k \| L_k\psi\|^2 \right)^{1/2}
=\left(\|\L_+( |\phi \rangle\langle\phi|)\|_1 \right)^{1/2}\cdot
\left(\|\L_+( |\psi \rangle\langle\psi|)\|_1 \right)^{1/2}.  \nonumber
\eeqa
\end{proof}
For a general matrix unit we establish  approximations.
\begin{lem}\label{approx}
Consider two pairs,  $\{(\phi,\psi),\, (e,f)\}\subset\DD(M)$
which lie close to each other in the graph of $M$,
$$(\|\phi-e\|,\|M\phi-Me\|,\|\psi-f\|,\|M\psi-Mf\|)\subset[0,\eps].$$
Then for actions of super operators the following approximations hold
$$
\| \M( |\phi\rangle\langle\psi|)- \M( |e \rangle\langle f|)\|_1 \leq \eps\cdot (\|\phi\|+\|M\phi\|+\|\psi\|+\|M\psi\|+2\eps),
$$
and
$$
\| \L_+( |\phi\rangle\langle\psi|)- \L_+( |e \rangle\langle f|)\|_1 \leq \eps\cdot (\|\phi\|+\|M\phi\|+\|\psi\|+\|M\psi\|+2\eps).
$$
\end{lem}
\begin{proof}
\beqa
&&\| \M( |\phi\rangle\langle\psi|)- \M( |e \rangle\langle f|)\|_1 =
\| \M( |\phi-e\rangle\langle\psi|) + \M( |e \rangle\langle\psi- f|)\|_1 \nonumber\\
 &\leq& \|M\phi-M e\|\tdt\|\psi\| + \|\phi-e\|\tdt\|M\psi\| +\|Me\|\tdt\|\psi- f\|+\|e\|\tdt\|M\psi- M f\| \nonumber\\
&\leq& \eps\cdot (\|\psi\|+\|M\psi\|+\|\phi\|+\eps+\|M\phi\|+\eps).
\nonumber
\eeqa
For the action of $\L_+$ we use Lemma \ref{schwarztype} and domination by $\M$ when acting on positive $\rho$. Therefore, we have
\beqa
&&\| \L_+( |\phi\rangle\langle\psi|)- \L_+( |e \rangle\langle f|)\|_1 =
\| \L_+( |\phi-e\rangle\langle\psi|) + \L_+( |e \rangle\langle\psi- f|)\|_1 \nonumber\\
&\leq&  \left(\|\L_+( |\phi -e\rangle\langle\phi-e|)\|_1 \right)^{1/2}\cdot
\left(\|\L_+( |\psi \rangle\langle\psi|)\|_1 \right)^{1/2} \nonumber\\
&& \qquad\qquad + \left(\|\L_+( |e\rangle\langle e|)\|_1 \right)^{1/2}\cdot
\left(\|\L_+( |\psi -f \rangle\langle\psi -f|)\|_1 \right)^{1/2}\nonumber\\
&\leq&  \left(\|\M( |\phi -e\rangle\langle\phi-e|)\|_1 \right)^{1/2}\cdot
\left(\|\M( |\psi \rangle\langle\psi|)\|_1 \right)^{1/2}     \nonumber\\
&& \qquad\qquad + \left(\|\M( |e\rangle\langle e|)\|_1 \right)^{1/2}\cdot
\left(\|\M( |\psi -f \rangle\langle\psi -f|)\|_1 \right)^{1/2}\nonumber\\
&\leq&  2\eps\cdot \left[ ( \|M\psi \|\tdt\|\psi\| )^{1/2}
 +  (\|M e\|\tdt\|e\| )^{1/2} \right]      \nonumber\\&\leq&
 \eps\cdot (\|M\psi\|+\|\psi\|+\|M\phi\|+\|\phi\|+2\eps).
\nonumber
\eeqa
Here we used the inequality between the geometric and arithmetic mean to transform the penultimate line.
\end{proof}

\begin{proposition}\label{easypart}
For a generator $\L$ in generalized standard form
there exists a basis $\{e_k\}$ such that $\DD_e$ is a core for $M$,
and $\DD_e^2$ is a core for $\L$.
\end{proposition}
\begin{proof}
The graph $\{(\phi,M\phi),\,\phi\in \DD(M)\}$
of the closed operator $M$
forms a complete Hilbert space with the graph norm $\|(\phi,M\phi)\|^2=\|\phi\|^2+\|M\phi\|^2$.
In this Hilbert space there exists a basis $\{(\phi_k,M\phi_k)\}$.
Since $\DD(M)$ is dense in $\HH$, the $\{\phi_k\}$ make a total set, and by the Gram Schmidt orthogonalization
one can construct out of them a basis $\{e_k\}$ of $\HH$.
Both the Gram Schmidt procedure and its reverse, representing each $\phi_k$
by the $\{e_k\}$, involve only finite linear combinations,
so the subspace $\DD_e$ is a core for $M$.

Now  a general $\rho\in\DD_0$ is of finite rank and thus a finite sum of matrix units,
$$
\rho=\sum_{n=1}^N |\phi_n\rangle\langle\psi_n|.
$$
For any $\eps>0$ there are vectors $e_n,\,f_n$ in $\DD_e$ close to $\phi_n,\,\psi_n$ in graph norm.
Due to these approximations in graph norm Lemma \ref{approx} can be applied. Combining both inequalities stated there, yields
$$
\| \L( |\phi_n\rangle\langle\psi_n|)- \L( |e_n \rangle\langle f_n|)\|_1 \leq 2\eps\cdot \left(\|\phi_n\|+\|M\phi_n\|+\|\psi_n\|+\|M\psi_n\|+2\eps\right).
$$
Then, by the triangle inequality and by summation of all these inequalities, we arrive at
\beqa
&&\|\L( \rho) - \L( \sum_n|e_n \rangle\langle f_n|)\|_1 \leq
\sum_n\| \L( |\phi_n\rangle\langle\psi_n|)- \L( |e_n \rangle\langle f_n|)\|_1 \nonumber\\
&&\leq N\tdt\eps\cdot (\max_n(\|\phi_n\|+\|M\phi_n\|+\|\psi_n\|+\|M\psi_n\|)+2\eps).
\eeqa

Now $\DD(\L)$ is the closure of the set of operators with finite rank in $\DD(\L)$,
so it follows that $\DD_e^2$ is a core for $\L$.
\end{proof}

\section{Discussion}\label{discussion}

QDS can be analyzed on three levels: the underlying Hilbert space, the space of Hilbert space operators,
and the ``super-operators'' acting on density operators.
On the ``highest level'', the level of super-operators, we
refer to general abstract theorems concerning strongly continuous semigroups
in Banach spaces.
The existence of a generator $\L$, under the condition that $\Tt^t$ is strongly continuous and trace preserving,
is the presupposition and basis for all our definitions, propositions and theorems.
On the intermediate level we discuss the form in which the generator can be represented
using operators on the Hilbert space. This problem is,
as is shown in this paper, related to the appearance of the QDS
on the lowest level.
On this level we may see, in principle, the evolution equation as a set of
coupled differential equations for
matrix elements in certain bases of $\HH$.

What is new in our presentation, in comparison to Davies' definition of a ``Standard Form''?
We use $M^\dag$ and $L^\dag$ at places where Davies puts - formally - the adjoint operators.
In the present context, the adjoint $M^\ast$ exists with a dense domain of definition, but its range is in general not large enough;
the operators $L_k$ are in general not closable, and the $L_k^\ast$  not densely defined.
These features can be explicitly seen in the examples illustrated in Section \ref{examples}.
Looking closely into Davies' papers, one may observe that he just writes  $M^\ast$ and $L^\ast$,
without really using these adjoint operators, when giving a precise sense to the formal notations.
Being honest, we admit that our ``generalized standard form'' is therefore not really new.
But his kind of notation may lead to some misunderstandings, which we try to avoid.
To characterize $M^\dag$ and $L^\dag$ more precisely,
one may note that these operators map $\HH$ into the dual space of $\HH_M$,
in the Gelfand triple
$\HH_M \subset \HH\subset \HH_M^\ast$,
where $\HH_M$ is
the domain of definition of $M$ equipped with the norm $\|\phi\|_M$, $\|\phi\|_M^2 = \|\phi\|^2+ \|M\phi\|^2$.

\textbf{Remark on the notation:}
      Our notation indicates that $\L(\rho)$ denotes  a trace class operator defined on the whole Hilbert space $\HH$.
      The restriction onto a domain concerns the set of density matrices, not of Hilbert space vectors.
      We choose nevertheless to write $\L$ in a way which is similar to Davies' expression of a standard form,
      and which looks like a formula in matrix analysis.
      This is a matter of taste; if one doesn't like the $M^\dag$ and the $L^\dag$,
      one can write, instead of (\ref{gensform}) and (\ref{formequal})
\beq\label{alternform}
\L(\rho) = \sum_k  L_k (L_k \rho^\ast)^\ast - M\tdt\rho - (M\tdt\rho^\ast)^\ast,
\eeq
with
\beq
\sum_k \| L_k \phi\|^2 =
\langle \phi | M \phi\rangle+\langle M \phi |\phi\rangle,\qquad\forall \phi\in\DD(M). \label{alterformequal}
\eeq

\textbf{Remark on the domains:} To construct the operators appearing in the generalized standard form
      it is not sufficient to assume that $\DD_e^2$ is contained in the domain of definition of $\L$.
      As a well-known counter example consider an induced unitary evolution $\Tt^t(\rho)=U_t\rho U_t^\ast$, where the generator
      of the group $U_t$ is a Laplacian on a finite interval $[a,b]$ with Dirichlet boundary conditions.
      With a basis $\{e_k\}$, where the limits $x\rightarrow a$ and  $x\rightarrow b$ of each function $e_k(x)$
      and of its derivative is $0$, the matrix elements give no information on boundary conditions. They define an
      hermitian operator with nonzero defect indices.

From the point of view of a functional analyst some important problems still demand further studies:
One is the question under which conditions a strongly continuous QDS is matrix normal
in an appropriate basis.
Another problem is how to reverse the implications, to
give a functional analytic characterization of the generators, appearing in the form
given in Definition \ref{gksldh},
without the a priori assumption that the semigroup is trace preserving.
This problem appears in the literature, starting essentially with \cite{C91}, continuing with
many more investigations, e.g. \cite{CF98},  \cite{FPM12},
as finding conditions for the generated semigroup to be ``conservative''.
Last but not least, one would like to have a ``standard non-standard form''
for generators which are not matrix normal.
We have examples, one from Davies and one from Holevo,
of such non-standard generators, \cite{D77,K95,H96}.
Both are built as extensions of generators in standard form,
but generating semigroups which do not preserve the trace.
The partition $\L=\L_+-\M$, defined here with Equations (\ref{gensform},\ref{formequal},\ref{expander},\ref{msuper})
is extended to $\L=\L_n+\L_+-\M$, including a non-standard completely positive map $\L_n$
which is not to be represented with generalized Kraus operators. Is this the ultimate most general form? A question that still needs to be answered.

\section{Examples}\label{examples}

\subsection{Translation, followed by drop out}\label{dropout}

The following simple example shows all the special features of the present analysis.
(A similar example is presented in \cite{A02}, sect. 2)
\par
Consider the Hilbert space
$$
\HH=\LL^2(\R_+)\oplus\C \,\, =\,\left\{ \left( \begin{array}{c}
                                                  \psi(x)\\
                                                   a
                                                \end{array}
\right)\right\}$$
and define $$|\omega\rangle :=  \left( \begin{array}{c}
                                                  0 \\
                                                  1
                                                \end{array}
\right),\quad \Omega := |\omega\rangle\langle\omega|.
$$
Moreover,
$$
\Tt^t(\rho)= \Tr(P_t \rho)\Omega + S_t \rho S_t^\ast
$$
with
$$
S_t|\omega\rangle=|\omega\rangle, \quad (S_t\psi)(x)=\psi(x+t),
\quad P_t |\omega\rangle=0,\quad  (P_t\psi)(x)=\chi_{[0,t]}(x)\psi(x).
$$
The generator is given by
$$\L(\rho)=L\rho\,L^\dag +\partial \rho+\rho\partial^\dag ,$$
where $\partial|\omega\rangle=0$ and $
(\partial\psi)(x)=\psi'(x)$ without boundary conditions, and
$$
 L         \left( \begin{array}{c}
                                                   \psi(x)\\
                                                  a
                                                \end{array}
\right) =  \left( \begin{array}{c}
                                                 0 \\
                                                 \psi(0)
                                                \end{array}
\right),\qquad \DD(L) \supset \DD(\partial)
$$
Note that $L$ -- formally $L=|\omega\rangle\langle\delta(x)|$ -- is not closable,
so $L^\ast$ is not a densely defined operator. As in Equation (\ref{genop1}), $L^\dag$   is a generalized operator,
a linear map from  $\HH$, the space of ket-vectors, to a space
of unbounded linear functionals on the Hilbert space of bra-vectors.
Formally $L^\dag=|\delta(x)\rangle\langle\omega|$.

The dual semigroup acts as
$$\Tt^{t\ast}(A)= \langle\omega|A|\omega\rangle  P_t +  S_t^\ast A S_t.
$$
It does not map $\CC (\HH)$, the space of compact operators, into $\CC (\HH)$. It, therefore, does not fulfill the condition
stated in Theorems 3.4 and 4.1 in \cite{D77b}, a requirement necessary in Davies' approach to prove
that the generator can appear in standard form.
It is not even strongly continuous and has no generator in the usual sense, namely
as an operator mapping $\BB(\HH)$ to $\BB(\HH)$.

One could define a generalized ``standard'' form for the dual generator,
mapping $\BB(\HH)$ to the set of quadratic forms $\mathfrak{Q}(\HH)$, by
$$
\L'(A)=L^\dag A\,L + \partial^\dag A+A\partial.
$$
It involves a completely positive mapping of $A$ to the quadratic form $L^\dag A\,L$.
A general theory of such extended Kraus operators is given in the Appendix
in Definition \ref{genops} and Theorem \ref{krausthm}.
Another characterization has been stated by Holevo in \cite{H97}.
He considered the generator as a bilinear functional, a ``form'', acting onto a pair
which consists of an observable and a density matrix.

We note that this example is not far from representing
a physical process as described in the following subsection.

\subsection{Quantum reflection with sticking probability}\label{sticking}

We present a simple model for a process observed in modern physics, see \cite{ZB83,BD06}.
Consider a hydrogen atom that moves freely in half space and is reflected from the surface of ultracold helium, the plane which is the boundary to the other half space. With some probability, however, it remains bounded to this surface
and creates there a ``ripplon''.\par
To describe this situation in a mathematical model, we consider the Fock space $\F$ of ripplons,
the space\footnote{
What we denote here by $\LL^2$ is the usual Hilbert space $L^2$, since the letter $L$ is reserved for operators.}
 $\LL^2(\R^2, dxdy)$ of wave functions for the hydrogen bounded to the surface,
and the space of wave functions in the half space, $\LL^2(\R^2, dxdy)\otimes \LL^2(\R_+,dz)$.
The total Hilbert space, therefore, is
\beqa
\HH&=& \F\otimes[\LL^2(\R^2, dx dy)\oplus \LL^2(\R^2, dx dy)\otimes \LL^2(\R_+,dz)]\nonumber\\
&=& \F\otimes \LL^2(\R^2, dx dy)\otimes (\C\oplus \LL^2(\R_+,dz)).
\eeqa
The free movement yields the Hamiltonian, here forming the imaginary part of the ``mover-and-contractor'' $M$,
$$
M=i(H_R +H_2 + H_z).
$$
$H_R$ is the self-adjoint Hamiltonian for the free ripplons.
$H_2$ is also self-adjoint and is the negative Laplacian on the plane, representing
both the free movement of the hydrogen atom when it is bounded to the surface,
and the parallel part of free movement in the half plane.
The evolution in the direction orthogonal to the boundary is modeled by $H_z=-\frac{d^2}{dz^2}$
with boundary conditions $\psi'(0)=w\psi(0)$, $w\in\C$, $\textrm{Im}(w)>0$.
This part is not self-adjoint, so $M+M^\dag=i(H_z-H_z^\dag)$. Moreover, the imaginary part of $w$ is related to the probability of sticking.
The transfer of the hydrogen atom into the layer at the surface is modeled by $L_z$ acting on
$\C\oplus\LL^2(\R_+,dz)$ as
$$
L_z|\psi\rangle=\left(2\,\textrm{Im}(w)\right)^{1/2}|\omega\rangle\psi(0),
$$
and $L_z|\omega\rangle=0$,
where $|\omega\rangle$ is a unit vector in the Hilbert space $\C$.
To complete the physics, there is an isometry from $\F\otimes \LL^2(\R^2)\otimes\C$ to $\F\otimes \LL^2(\R^2)\otimes\C$, namely the scattering matrix $S$.
It creates a ripplon with angular momentum $\kappa$
and reduces the momentum of the sticking hydrogen from $k$ to $k-\kappa$.
These vectors of momenta are all in the plane and $\kappa$ has the same direction as $k$. The absolute value of $\kappa$ depends on $|k|$ and is consistent
with the conservation of total energy.
On the total Hilbert space the generalized Kraus operator $L=S\tdt L_z$ is then acting.

From the perspective of mathematical physics, one would like to improve this model
and also to derive it from Schr\"{o}dinger quantum mechanics. Our conjecture is that this task can be fulfilled by using a strong coupling limit.

\section{Appendix: Generalized Kraus operators for unbounded completely positive maps}\label{unboundedcp}

\subsection{ Generalization of ``completely positive map''}

K. Kraus, \cite{K71} gave a representation of the Stinespring theorem
in the form
$$\Q(W)=\sum_k K_k\tdt W \tdt K_k^\ast,
$$
where $\Q$ is an ``operation'', a completely positive bounded map from $\mathfrak{T}(\HH)$ to $\mathfrak{T}(\HH)$.
It has been observed several times, e.g. in \cite{SG05}, that
in case of a finite dimensional Hilbert space $\HH$ this representation can be deduced from
the Choi-Jamiolkowski dualism of maps and states.

We give here a generalization to infinite dimensions and
densely defined unbounded completely positive maps.
Moreover, we consider the case, where the target of the map is not a space of operators
but the set of quadratic forms, $\mathfrak{Q}(\HH)$.

\begin{define}\textbf{Generalized Operators:}\label{genops}

Consider two bases, $\{e_n\}$ and $\{f_n\}$ in the Hilbert space of infinite dimension.
The operators $K$, used in the following, are linear mappings from $\DD_e$
into the algebraic dual of
$\DD_f$. Their adjoints, $K^\dag$, are operators from $\DD_f$
into the algebraic dual of $\DD_e$, related to $K$ by
\beq
\langle e|K^\dag|f\rangle= \langle f|K|e\rangle^\ast,
\eeq
for $e\in\DD_e$, $f\in\DD_f$.
\end{define}
In the sense of Holevo's investigations
these operators can also be considered as bilinear forms on $\lspan\{|f\rangle\langle e|\}$ and $\lspan\{|e\rangle\langle f|\}$ respectively.
In another sense one may characterize $M^\dag$ and $L^\dag$ more precisely
as acting  in the Gelfand triple
$\HH_M \subset \HH\subset \HH_M^\ast$.
These operators map there $\HH$ into the dual space of $\HH_M$, which is
the domain of definition of $M$ equipped with the norm $\|\phi\|_M$, $\|\phi\|_M^2 = \|\phi\|^2+ \|M\phi\|^2$.

\begin{thm}\textbf{Generalized Kraus-Operators}\label{krausthm}

If $\Q$ is a completely positive map from $\DD_e^2$ into $\mathfrak{Q}(\HH)$ with $\DD_f$ as common form domain,
there exists a countable set of ``generalized Kraus operators'' $K_\alpha$
such that
\beq
\Q(\rho) = \sum_\alpha  K_\alpha \rho K_\alpha^\dag.
\eeq
If, moreover,  each quadratic form $\Q(\rho)$ can be
associated with an operator $\breve{\Q}(\rho)$ in $\BB(\HH)$,
such that $\langle g|\Q(\rho)|f\rangle=\langle g|\breve{\Q}(\rho)f\rangle$ holds for all $f\in\DD_f$ and for all $g\in\DD_f$,
each $K_\alpha$ is an operator mapping $\DD_e$ into $\HH$.
\end{thm}

Note that there exist moreover several considerations of ``Completely Positive Maps''
in a different context, see e.g. \cite{CTU11,P12}.

\subsection{Proof of Theorem \ref{krausthm} and Corollaries on special cases}

We consider matrix units $E_{i,j}=|e_i\rangle\langle e_j|$, so that
\beq\label{supermatrix}
Q_{i,j;k,\ell}=\langle f_i| \Q( E_{k,\ell})|f_j\rangle.
\eeq
Since the Choi matrix $E=\sum_{i,j}E_{i,j}\otimes E_{i,j}$
in infinite dimensions is a quadratic form \cite{H11}, not an operator,
we use the approximating increasing sequence of bounded positive operators on $\HH\otimes\HH$
$$E_N=\sum_{i,j}^N E_{i,j}\otimes E_{i,j}=|\theta_N\rangle\langle\theta_N|, \qquad
\mathrm{with} \qquad|\theta_N\rangle=\sum_i^N |e_i\otimes e_i\rangle.$$
Next, we make use of complete positivity of $\Q$ and look at the map $\hat{\Q}=\Q\otimes\one$, which preserves positivity. It is, therefore, possible to form the positive square root of $\hat{\Q}(E_N)$
and define vectors
$$\Psi_{N;i,k}=\sqrt{\hat{\Q}(E_N)}|f_i\otimes e_k\rangle.$$
These vectors are unnormalized and change with $N$. But observe that, for $N\geq k$
$$\|\Psi_{N;i,k}\|^2=\langle f_i\otimes e_k|\hat{\Q}(E_N)|f_i\otimes e_k\rangle=
\langle f_i|\Q(E_{k,k})|f_i\rangle \leq \infty$$
and, for $N\geq k$ and $N\geq \ell$

\beqa
\langle\Psi_{N;i,k}|\Psi_{N;j,\ell}\rangle=\langle f_i\otimes e_k|\hat{\Q}(E_N)|f_j\otimes e_\ell\rangle &=& \nonumber \\
=\langle f_i\otimes e_k|\Q(E_{k,\ell})\otimes E_{k,\ell} |f_j\otimes e_\ell\rangle &=& \langle f_i|\Q(E_{k,\ell})|f_j\rangle.     \label{matrixT}
\eeqa

The vectors $\Psi$ will, most probably, go weakly to zero, as $N\rightarrow\infty$,
but their norms and inner products are  bounded, increasing in $N$ and remaining constant for large $N$.
So the internal ``geometry'' of the rhomboid which they form stays constant and can be represented with a fixed set of vectors. This can be done by mimicking the Gram-Schmidt procedure, generating an orthonormal set of vectors out of the $\Psi$-vectors for any finite $N$,
and mapping them onto a fixed basis.

\begin{proposition} \textbf{Construction of the $N$-independent vectors} \label{conphi}

A set of vectors $\Phi_{i,k}\in\HH\otimes\HH$ can be constructed
independently of $N$, such that
\beq \label{psiphi}
\langle\Phi_{i,k}|\Phi_{j,\ell}\rangle = \langle \Psi_{N;i,k}|\Psi_{N;j,\ell}\rangle, \qquad
\forall N\geq \max\{i,j,k,\ell\}.
\eeq
\end{proposition}

\begin{proof}
We first choose the ordering of indices for the basis $e_k$ in such a way, that $\Psi_{N;0,0}\neq 0$ and then define some order relation for the index-pairs $(i,k)$.
More precisely, we set $n=n(i,k):=\max\{i,k\}$, $\alpha=\alpha(i,k):=n^2-n+i-k$ and
denote $\Phi_\alpha=\Phi_{i,k}$.

Consider now an arbitrary basis $b_\alpha$ and make the Ansatz:
$$\Phi_\alpha = \sum_{\beta \leq\alpha} \gamma_{\alpha,\beta}\,b_\beta.$$
Note that $\|\Phi_0\|^2 \neq 0$,
hence $$\gamma_{0,0}:=\|\Phi_0\|>0.$$

Next, we proceed with nested inductions on $\alpha$ and $\beta$ with $\beta\leq\alpha$.
Having determined all $\gamma_{\alpha-1,\beta}$  for fixed $\alpha-1$, we go over
to fixed $\alpha$ and determine the sequence $\gamma_{\alpha,\beta}$, in increasing order of $\beta$. To this end, we solve Equation (\ref{psiphi}) and take for $\beta<\alpha$ the expansion
$$\langle\Phi_\beta|\Phi_\alpha\rangle =
\left[\sum_{\beta' <\beta} \gamma_{\beta,\beta'}^\ast\,\gamma_{\alpha,\beta'}\right] +
\gamma_{\beta,\beta}^\ast\,\gamma_{\alpha,\beta},
$$
into account, which determines $\gamma_{\alpha,\beta}$ in case $\gamma_{\beta,\beta}\neq 0$.

If $\gamma_{\beta,\beta}= 0$, the vector $\Phi_\beta$ is a linear combination of $\Phi_{\beta'}$
with $\beta'<\beta$. This already guarantees the validity of Equation (\ref{psiphi}).
In such a case we reject any use of $b_\beta$ and set $\gamma_{\alpha,\beta}=0$ for each $\alpha>\beta$,
to make sure that the induction on $\beta$ can be  finished at $\beta=\alpha$.
This procedure guarantees, at the end of induction on $\beta$, the  solvability of
$$\langle\Phi_\alpha|\Phi_\alpha\rangle =
\left[\sum_{\beta<\alpha} |\gamma_{\alpha,\beta}|^2\right] +
|\gamma_{\alpha,\alpha}|^2,
$$
enabling us to determine $\gamma_{\alpha,\alpha}\geq 0$
as the positive square root of $\|\Phi_\alpha\|^2-\sum |\gamma_{\alpha,\beta}|^2$.
\end{proof}

One may represent now the super matrix elements by combining Equations (\ref{supermatrix}, \ref{matrixT}, \ref{psiphi})
\beq\label{represent}
Q_{i,j;k,\ell}
=\langle\Phi_{i,k}|\Phi_{j,\ell}\rangle
=\sum_\alpha\langle\Phi_{i,k}|\alpha\rangle\langle\alpha|\Phi_{j,\ell}\rangle,
\eeq
where $|\alpha\rangle$ are some basis vectors in $\HH\otimes\HH$.
We shall consider the basis $|\alpha\rangle =b_\alpha$ with $\alpha =\alpha(j,\ell)$,
which was used in the proof of the Proposition \ref{conphi}.
Note that, due to the Gram-Schmidt-like procedure used to construct the $\Phi$-vectors, there appears for each $\Phi_{i,k}$
only a finite set of $\alpha$ in Equation (\ref{defkraus}).
So one can swap the two sums in Equation (\ref{finite}) following below.

Define now ``Kraus-operators'' $K_\alpha$ on $\DD_e$  by their matrix-elements
\beq  \label{defkraus}
\langle f_i|K_\alpha|e_k\rangle := \langle \Phi_{i,k}|\alpha\rangle.
\eeq
Then with (\ref{represent}), we have for matrices $\rho$ which are finite linear combinations of $E_{k,\ell}$ that
\beqa\label{finite}
[\Q(\rho)]_{i,j}&=&\sum_{k,\ell}Q_{i,j;k,\ell}\,\rho_{k,\ell}=
\sum_{k,\ell} \sum_\alpha\langle\Phi_{i,k}|\alpha\rangle \rho_{k,\ell}   \langle\alpha|\Phi_{j,\ell}\rangle =
\sum_\alpha \sum_{k,\ell}\ldots =\\
&=&\sum_\alpha\left( K_\alpha \rho K_\alpha^\dag\right)_{i,j}.
\eeqa
\begin{cor}\label{krausdomain}
If $\Q$ maps $\DD_e^2$ into $\BB(\HH)$, the Kraus Operators $K_\alpha$ map $\DD_e$ into $\HH$.
\end{cor}
\begin{proof}
Consider $\phi\in\DD_e$ and form $|\phi\rangle\langle\phi|$ which is in $\DD_e^2$ and is mapped by $\Q$
to an operator with finite norm. For each $\psi\in\DD_f$ which is dense in $\HH$ we have
$$
\|\Q(|\phi\rangle\langle\phi|)\|\cdot \|\psi\|^2  \geq \langle\psi| \Q(|\phi\rangle\langle\phi|) |\psi\rangle =
\sum_\alpha \langle\psi| K_\alpha |\phi\rangle\langle\phi| K^\dag_\alpha |\psi\rangle
= \sum_\alpha | \langle\psi| K_\alpha |\phi\rangle|^2.
$$
So $\|\Q(|\phi\rangle\langle\phi|)\| $ gives a bound for the supremum over all normed $\psi$ and thus an upper bound for the square of
the norm of each $K_\alpha|\phi\rangle$.
\end{proof}
However, note that the $K_\alpha^\dag$ are in general still ``generalized'' operators.
\begin{cor}
For trace preserving maps $\Q$ the $K_\alpha$ can be extended to bounded operators in $\BB(\HH)$,
and $K_\alpha^\dag =K_\alpha^\ast$.
\end{cor}
\begin{proof}
For arbitrary positive $\rho$ the formula holds for
a sequence of approximating positive $\rho_N\in\DD_e^2$ with finite rank. Moreover, in the limit $N\to\infty$
we have bounded convergence.

Note that the preservation of trace by $\Q$ implies
$$\sum_i \|\Phi_{i,k}\|^2, = 1,\quad\forall\, k\qquad\Rightarrow\qquad
\sum_i \langle\Phi_{i,k}|\Phi_{i,\ell}\rangle = 0,\quad\forall\, k,\ell,$$
therefore,
$$\sum_\alpha K_\alpha^\dag \cdot K_\alpha =\one,\qquad\Rightarrow\qquad
\quad\|K_\alpha\|\leq 1,\quad\forall\,\alpha.
$$
\end{proof}

\subsection{Acknowledgements}

We thank Franco Fagnola, who spotted an error in the first version
and gave to us important hints on relations to existing papers.

One of us (S.A.) has been supported through the FWF-project Nr. P22929-N16.

B.B. thanks V. Umanit\`{a} and the organizing committee of the
Workshop ``Quantum Markov Semigroups:
 Decoherence and empirical estimates'',
 Genova, 26-28 June 2013,
for invitation to this marvelous workshop.

We thank also Jan Schlemmer for important hints on methods.


\section{Introduction}

During the last few decades, new materials based on transition metal oxides (TMOs)
as the key parent component have surprised us with
their novel and unexpected behaviors. These include high temperature
superconductivity in the cuprates, giant magnetoresistance in
magnanites and
a plethora of fascinating new phenomena which has been recently reported on
oxide hetero-structures of TMOs and
devices \cite{Mannhart, Cen, Gozar}.
For example, an interface between
two insulators behaves as a metal \cite{Cen} which becomes
superconducting at sufficiently low temperatures, while an interface
between two antiferromagnets becomes ferromagnetic \cite{Millis}.
These new structures not only create a playground for unexpected
physical phenomena to be observed, but, in addition, they open up the
possibility for new applications based on radically different
foundations.   The complex, unusual, and as yet not fully
discovered or understood
behavior of small-gap TMOs can be manipulated in a variety of fundamentally
new applications \cite{Mottsolar}.

While in the late 80's, immediately after the discovery of the
cuprate superconductors, electronic structure calculations could
not predict the correct ground state of TMO based materials,
during the following decades significant progress has
been made which has restored some of the faith in the newly
developed \emph{ab initio} computational schemes.

The room temperature $M_1$ phase of VO$_2$, a
proto-typical  material\cite{Mott} in the  family of  TMOs, is  such a
small gap  system of correlated $d$-electrons.  Small-gap  TMOs have a
very rich, complex and  interesting phase diagram, where understanding
their electronic  structure and wave functions is  of high importance.
As a  result these  TMOs have been  widely studied  with sophisticated
\emph{ab initio} methods. A recent \emph{Letter} reported \cite{Eyert2}
that HSE06 non-local  range-separated hybrid density functional theory
(DFT) \cite{HSE} is  able to correctly describe the  ground state wave
functions of  VO$_2$, and produces a reasonable though  larger gap
than the  experimental one.  HSE  is a slower method  than traditional
(semi)local   DFT   functionals  but   much   faster  than   many-body
perturbation  methods, such  as the  GW-method or  the  dynamical mean
field  theory; thus, it  has been  assumed with  a growing  number of
followers
\cite{Heinemann-HSE, Zhuang-HSE, Gillen-HSE, Dutta-HSE}  that  HSE06 may
be  a very  practical method  to explore  the properties  of small-gap
TMOs.

Here  we   show  by  a  fully  dynamical   self-consistent  GW
(scGW) calculation  \cite{Hedin, Schilfgaarde}  that HSE06  does
not provide accurate wave functions and a semi-local DFT functional
is as good a starting  point for the  scGW calculation as  the
computationally demanding HSE06  method. Furthermore, we
demonstrate that even if the admixture  of  the Fock-exchange in
the hybrid  functional  is tuned  to reproduce the scGW
fundamental gap, still, the resulting wave functions deviate  from
those obtained by  scGW close  to the  Fermi-level.  We conclude
that (semi)local functionals  are as good starting points for the
scGW procedure as the hybrid functionals for small-gap TMOs.  The
former, however, is much less computationally demanding.

We  carried out  DFT  calculations on  the  $M_1$ phase  of VO$_2$  as
implemented  within  the  \textsc{VASP} package \cite{VASP, Kresse1,
  Kresse2}. We  used small  core projectors for  vanadium ions,  so we
explicitly included  $3s$ and $3p$  electrons as valence.  The valence
electron  states  were  expressed   as  linear  combinations  of  plane
waves.  We  found  that  the  plane wave  cutoff  of  400~eV  provided
convergent single particle levels. As we applied various functionals
and GW  schemes for calculating  the quasi-particle energies,  we used
the  experimental  geometry\cite{Anderson2}.
The Brillouin-zone integration was approximated
by a weighted sum on a special $k$-point set. We  found  that  the required
size  of  the Monkhorst-Pack  \cite{MP76}  $k$-point  set  depends
strongly  on  the existence  of  a gap. Convergent  charge
density could  be achieved with a $5\times5\times5$
Monkhorst-Pack $k$-point set
when there is a gap, while an $18\times18\times18$  $k$-point set was
required without a gap.  We applied 146 conduction bands in the
GW calculations.

The ground state electronic structure and wave functions were
calculated by the standard semi-local Perdew-Burke-Ernzerhof (PBE)
functional \cite{PBE} as well as non-local, range separated hybrid
functionals as proposed by Heyd-Scuseria-Ernzerhof (HSE)
\cite{HSE, HSE06}. The HSE functional for the exchange-correlation
part of the energy involves a parameter $\alpha$ which mixes the
contribution of the short-range parts of the Fock-exchange and the
PBE expression for the exchange energy \cite{HSE}. It also
involves a second parameter $\omega$ which defines what is meant
by the short and long ranged part of the Coulomb potential.

The  value of  the parameter  $\omega=0.2  a_0^{-1}$ is determined
to  give a  balanced description that  provides good accuracy and
speed  for both molecules and    solids \cite{HSE06}. The
Fock-exchange part  is calculated using the  short-range part of
the  Coulomb  interaction.   The  PBE \cite{PBE}  expression for
the  exchange energy functional is modified   to  use   the short
and long   range  parts  of   the  Coulomb interaction \cite{HSE}.
The choice  of $\omega$ and $\alpha$ may depend on the actual
system.  We fixed the parameter $\omega$ at $0.2 a_0^{-1}$ while
we varied the parameter $\alpha$. This parameter is often tuned to
agree with   experimental data,   such   as  the band gap
\cite{Moussa-HSE, HSE-TunedLaMo, TunedHSE-Pozun}   or   the
dielectric constant \cite{Marques1} of  a  given   crystal.
$\alpha$=0.25 corresponds  to the HSE06 functional   \cite{HSE06}
that  we  call now HSE-0.250.   We note   that   $\alpha$=0.25 was
rationalized in Ref.~\onlinecite{PEB} where  they showed that a
smaller  value is needed for systems with nearly-degenerate
ground-states. VO$_2$ may fall into this  category,  thus  we
applied $\alpha$=0.125 and $\alpha$=0.172 (HSE-0.125 and
HSE-0.172 functionals, respectively).

As HSE functionals  contain external  parameters they  are  not truly
\emph{ab initio} methods. Nevertheless, HSE DFT functionals may provide
a good  starting  point  for  many-body perturbation  methods,  such  as
the GW-method  which may  result in quasi-particle  energies and
wave functions  that are ideally \emph{independent}  from the starting
point.   We applied  several levels  of approximations  within  the GW
quasi-particle   scheme   as   implemented  in   VASP \cite{Shishkin1,
  Shishkin2,   Fuchs,  Shishkin3}.  (i) First, we  applied   the  simplest
single-shot GW  approach, i.e., the G$_0$W$_0$  approximation.
This means that we have used the  Kohn-Sham eigenvalues and
orbitals in G$_0$ and W$_0$. For W we took W$_0=\epsilon^{-1}V$,
where the dielectric matrix $\epsilon^{-1}_{{\bf G},{\bf G'}}({\bf
q},\omega)$, with ${\bf G}$ and ${\bf G}'$  denoting reciprocal
lattice vectors, were  calculated in the random  phase
approximation   and  the  self-energy  corrections  were evaluated
to first order  in the  difference between  the self-energy
$\Sigma$ and  the Kohn-Sham potential \cite{Louie, Sham}. (ii) As
a higher level  approximation,  we solved for  G self-consistently
within the  GW approximation     following the     procedure
described     in Ref.~\onlinecite{Shishkin3}. Typically,  four
iterations  in  G  were sufficient  to achieve convergence of
the self-consistent
quasi-particle  energies within  0.02~eV, and we denote this
procedure  by G$_4$W$_0$.  (iii) Last, we applied a fully
self-consistent  dynamical GW correction as proposed  by
van~Schilfgaarde \emph{et  al.} \cite{Schilfgaarde} and
implemented in  VASP \cite{Shishkin3}. In  this case, the  G and W
are updated  \emph{together}  with  the  wave  functions by means of
the following equation:
\begin{equation}
\label{eq:scgw}
\left[T + V + \Sigma(E_n)\right] |\psi_n\rangle = E_n |\psi_n\rangle ,
\end{equation}
where $T$ is the kinetic energy operator, $V$ is the electrostatic
potential, and, $\Sigma(E_n)$ is the energy-dependent Hermitian
part of the self-energy as calculated from the GW approximation.
$E_n$ and $\psi_n$ are the quasi-particle energies and wave
functions. Formally, Eq.~\ref{eq:scgw} looks like an ordinary
Kohn-Sham DFT equation. Thus, the resulting wave
functions and energies can be used to recalculate G and W of the
system, and, the (Hermitian) part of $\Sigma(E_n)$, until
self-consistency has been reached. We found that typically 13-18
iterations were sufficient to obtain self-consistent
quasi-particle energies \emph{and} wave functions.
\begin{table}
\begin{ruledtabular}
\caption{We list the direct gaps in the $M_1$ phase of VO$_2$
         calculated using different methods and compare with the
         results of other
         work and experimental values. Abbreviations of functionals
         are explained in the text. We note that
         PBE results in a metallic state (no gap).}
\begin{tabular}{ l r}
\label{tab:gaps}
 Source                         &   Gap          \\ \hline
 PBE (GGA)                       &   N/A       \\
 HSE-0.250                       &   1.01 eV       \\
 HSE-0.125                       &   0.26 eV        \\
 HSE-0.172                       &   0.55 eV        \\
 HSE-0.250+G$_0$W$_0$                   &   1.01 eV        \\
 HSE-0.125+G$_0$W$_0$                   &   0.69 eV        \\
 HSE-0.250+G$_4$W$_0$                   &   1.01 eV        \\
 HSE-0.125+G$_4$W$_0$                   &   0.69 eV        \\
 HSE-0.250+scGW                     &   0.54 eV        \\
 HSE-0.125+scGW                     &   0.54 eV        \\
 PBE+scGW                           &   0.54 eV        \\
 LDA+COHSEX+G$_0$W$_0$\footnotemark[1]        &   0.6  eV        \\
 LDA+empirical correction+G$_0$W$_0$\footnotemark[2] &   0.6  eV        \\
 Experiment\footnotemark[3]     &    $\sim$0.6 eV
\end{tabular}
\end{ruledtabular}
\footnotetext[1]{Ref.~\onlinecite{COHSEX} (with LDA lattice constant)}
\footnotetext[2]{Ref.~\onlinecite{LanyOxides} (empirical correction on d-orbitals)}
\footnotetext[3]{Ref.~\onlinecite{Epsilon2data}}
\end{table}

First, we discuss our results for the quasi-particle energies  close to
the  Fermi-level obtained with
 DFT and quasi-particle correction calculations
on the $M_1$ phase of VO$_2$.
The  $M_1$ phase  of  VO$_2$  has  a small  gap  of
$\sim$0.6~eV \cite{Epsilon2data}. PBE falsely predicts a metallic state
while HSE-0.250 (HSE06) yields too large a gap of 1.01~eV, in agreement
with a previous work \cite{Eyert2}. If  we apply HSE-0.125 then the gap
becomes too low at 0.26~eV. One  may assume that  HSE functionals are a
much better starting point for GW-calculation as they provide
a  gap, thus  G$_0$W$_0$ may  result  in good  results on  top of  HSE
functionals.  However,  G$_0$W$_0$  did  not improve  the  results  on
HSE-0.250.   The    calculated   gap   did   not   change,
(Table~\ref{tab:gaps})  which  might imply  that
HSE-0.250  produces   very  good  quasi-particle   energies  and  wave
functions.  However, the  G$_0$W$_0$  correction on  top of  HSE-0.125
gave   a   very   different   result, yielding a gap of   0.69~eV .
When G was
self-consistently updated,  the quasi-particle energies
did not change (see HSE-0.250+G$_4$W$_0$  and
HSE-0.125+G$_4$W$_0$ results  in Table~\ref{tab:gaps}).
We conclude that G$_0$W$_0$ and G$_4$W$_0$ corrections do not supply a
ground state in close agreement with experiment near the Fermi-level. A
more complete calculation is needed to approach the experimental situation.

We then applied a  fully dynamical self-consistent GW-method where the
wave functions were updated together with the G and W. The calculated
HSE-0.250+scGW  and HSE-0.125+scGW  band gaps  are the  same
(0.54~eV), which  is  quite close  to  the  experimental  one. In
addition,  the calculated  density  of  states   agrees  well  with  the
experimental photo-emission spectrum (see Fig.~\ref{fig:PES}) \cite{PES}.

A       previous      theoretical      study       indicated  that the
local-density-approximation  (LDA) can  be a  good starting  point for the
self-consistent  GW procedure for  VO$_2$ \cite{COHSEX}. Gatti~\emph{et
  al.} applied the self-consistent    GW-method in the static   COHSEX
approximation  first,  which  opened  a  band gap  from  the
metallic  solution. Then,  they applied  a fully  dynamical G$_0$W$_0$
correction  on  the  quasi-particle  energies \cite{COHSEX}.  Here,  we
applied a fully dynamical self-consistent GW on top of a semi-local PBE
functional. The calculated PBE+scGW band gap  agrees well
with HSE+scGW. From this, it is fairly clear that starting from the
computationally ``expensive'' hybrid
functional  wave   functions  is   not  advantageous   when  a
self-consistent GW calculation is required,  as seems to be the case
in many small-gap    systems,    such    as    the    TMOs \cite{LanyOxides}.
\begin{figure}
\epsfig{file=Fig1.eps,width=\figwidth}
\caption{(Color online) The photo-emission data taken from Ref.~\onlinecite{PES} is
  compared with the density of states (DOS) of the occupied states as calculated
  within scGW on top of different HSE functionals. The calculated data was smoothed using an exponentially weighted moving average
  with smoothing factor of 0.25. We averaged each data point with its seven
neighbors using a decreasing weight $(1-\alpha)^n$, where $n$ is the number of
points to the central point.}
\label{fig:PES}
\end{figure}

As the wave functions are updated in the scGW procedure, it is intriguing
to study the change in the wave functions due to the scGW correction. This
analysis was already carried out when starting from LDA wave
functions \cite{COHSEX}.  Now, we  analyze   the  case  of  HSE
functionals. To  show the  change  in wave  functions,  we present  the
projected density of states (PDOS) onto the spherical harmonics around
one unique vanadium atom close to the Fermi-level where the change is the most
significant. From these plots we  observed several interesting issues: (i) starting
with PBE  [c.f., Figs.~\ref{fig:wfs}(a) and (b)], the scGW procedure is  needed to open
the gap near  the Fermi-level so that the  contribution of d$_z^2$ and
(d$_{xz}$,d$_{yz}$)  will  be   significantly  smaller  than  that  of
d$_{xy}$ and  d$_{x^2-y^2}$ just below the  Fermi-level and \emph{vice
  versa}  just  above  the  Fermi-level, (ii)  the
unoccupied  wave  functions  with  energies of  $\sim$1~eV  above  the
Fermi-level in  PBE and PBE+scGW calculations are very similar, (iii)
HSE  naturally opens the gap [Figs.~\ref{fig:wfs}(c,e)],  but the
wave functions significantly differ from those
obtained with HSE+scGW
[c.f., Figs.~\ref{fig:wfs}(d) and (f)],  particularly,   at  energies  above  the
Fermi-level where  HSE+scGW yields similar  contributions from d$_z^2$
and d$_{x^2-y^2}$ orbitals at around 2~eV above the Fermi-level while
they      ``split''      in  the HSE     calculations. Apparently
[Figs.~\ref{fig:wfs}(b,d,f)],  the convergent  scGW wave  functions are
the same  \emph{regardless} of the  starting point. The  semi-local PBE
functional  provides relatively  good  wave functions for states
with energy 1~eV above  the
Fermi-level, \emph{unlike}  HSE-0.125 or HSE-0.250 functionals.
\begin{figure*}
\includegraphics[keepaspectratio,totalheight=5 in, width= 6 in]{Fig2.eps}
\caption{(Color online) The partial density of states (PDOS) near the
Fermi-level projected onto the
spherical harmonics around one of
the unique vanadium atoms (V1) in the $M_1$ phase of VO$_2$, for a variety of functionals.
The $\vec{x}$ axis is parallel to the $M_1$ "a" axis,
$\vec{y}$ is parallel to the $M_1$ "b" axis \cite{Anderson2},
and $\vec{z}$ is  perpendicular to those, using a right-hand rule.}
\label{fig:wfs}
\end{figure*}

Recently,  it has  been claimed  that  for this  material, the  hybrid
functional  HSE-0.250 gives a  good description  of the
ground  state  \cite{Eyert2}.  According  to  our  analysis,  this  is
questionable. Apart  from close vicinity to the  Fermi-level, the
semi-local  PBE wave  functions are  superior to  the  HSE-0.250 wave
functions in the VO$_2$ crystal. Given  the similar number  of iterations
required to  achieve convergence in the scGW procedure,  and the decreased
workload,  it seems  that using  the HSE  family of  functionals  as a
starting  point for  the  more accurate  scGW  approximation does  not
provide an improved calculation at all.

We further  note that HSE  functionals have been  used as a  basis for
investigating previously  unknown materials \cite{Zhuang-HSE},  and to
explore the  complicated     physics     of      phase     transitions
\cite{Dutta-HSE,Heinemann-HSE}.  In addition, the parameter
$\alpha$ has been tuned to agree with experimental data such as the band gap
\cite{Moussa-HSE, HSE-TunedLaMo, TunedHSE-Pozun, Marques1}.
In order to demonstrate the danger of fitting  the $\alpha$ parameter
to the band gap of $M_1$  phase of VO$_2$, we show that the ``tuned''
HSE functional does not produce the appropriate quasi-particle
energies and wave functions. We
found, by a simple linear interpolation of our HSE-0.250
and HSE-0.125 results, that  the HSE functional
with $\alpha$=0.172 yields about the same gap of
0.55~eV as that found by the scGW approach. Comparing
the    wave     functions    of    scGW     and    HSE-0.172    [c.f.,
  Figs.~\ref{fig:wfs}(f) and (g)], it  is clear that the tuned functional and scGW calculations
significantly disagree  for the wave functions with energies above  the
Fermi-level.  Since  the  character  of  the wave  functions  is  quite
important   for  several   properties  such   as   optical  excitations
\cite{Bechstedt-New,  Schleife2011, Mottsolar},  it  seems that  using
such tuned functionals  as a starting point  may be not appropriate
for VO$_2$, and this ``tuning'' has to be carefully checked in
small-gap TMOs.

In conclusion, we demonstrated that
the HSE-type of functionals
should be  applied with  great care on  small-gap TMOs
such as  the $M_1$ phase  of VO$_2$. We  found that
scGW  calculations  on top  of  semi-local  PBE  or the  non-local  HSE
functionals  provide  the  same  results  and they are
in agreement with  the  experimental
data. Furthermore, we have found that  the PBE wave functions
for states which have energy 1~eV  above the
Fermi-level are superior, thus,
the PBE provides at  least as good starting point as the HSE
functionals for detailed scGW calculations.

This work was supported in part by the U.S.\ National High Magnetic
Field Laboratory which is partially funded by the U.S.\ National
Science Foundation.