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\def\gtrsim{\gtrsim}

\title{Renormalization theory of Feynman amplitudes on configuration spaces}

\author{Nikolay M. Nikolov}

\begin{document}

\maketitle

\thispagestyle{empty}

\vspace{-0.8cm}

\begin{center}
\scriptsize
Institute for Nuclear Research and Nuclear Energy, \\
Tsarigradsko Chaussee 72, BG-1784 Sofia, Bulgaria \\
mitov@inrne.bas.bg
\end{center}

\vspace{0cm}

\begin{abstract}
In a previous paper
``\textit{Anomalies in Quantum Field Theory and Cohomologies of Configuration Spaces}''
(arXiv:0903.0187) we presented a new method for renormalization in
Euclidean configuration spaces based on certain renormalization maps.
This approach
is aimed to serve for
developing an algebraic algorithm
for computing the Gell--Mann--Low  renormalization group action.
In the present work we introduce a modification of the theory of renormalization maps
for the case of Minkowski space and we give the way how it is combined within
the
causal perturbation theory.
\end{abstract}%

\tableofcontents

\Section{Introduction}{Intr}

The causal approach to perturbative quantum field theory (QFT)
originates in the work of Stuckelberg and Bogolubov
and was fully developed by
(and named after) Epstein and Glaser (\cite{EG}, see also \cite{S2}, \cite{DF}).
In this method
the
renormalization is done for the
products of fields (time--ordered, or retarded).
This facilitates the generalization of
the
perturbation theory on manifolds
but still it has the disadvantage of being
rather complicated technically,
especially for concrete calculations.

In paper \cite{N1} (see also its review \cite{N2}) we have found an equivalent construction
to the Epstein--Glaser procedure, which is entirely
set up
in terms of
renormalization of Feynman amplitudes (integrals of functions).
This approach then has the additional advantage
of being independent of concrete models
of quantum fields like
the
$\varphi^4$--theory or
quantum electrodynamics etc.
In this way we get rid of the
technical difficulties present in
a particular
model, in other words, we separate them
from the renormalization problem.
Furthermore, our reformulation of the renormalization problem
makes possible to give a geometric characterization for the
renormalization ambiguity.
Our main goal there was to use this geometric analysis in order to derive
an algebraic algorithm for determining the Gell--Mann--Low  renormalization group action,
i.e., the action of the one parameter group ${\mathbb R}^+$ on the space of coupling constants,
which is induced by the scaling transformations (in terms of formal diffeomorphisms).
In particular, we are interested in algebraic algorithms for calculating the perturbative expansions
of $\beta$--functions and anomalous dimensions.

In the present paper we introduce a Minkowski space version
of the theory of renormalization maps developed in \cite{N1}
and we also combine this theory
with
the
causal perturbation theory.

\Section{Axiomatic properties of time--ordered products}{se2}

Let us briefly recall some basic facts from
the
causal perturbation theory (\cite{EG}).
In this approach one constructs
time--ordered products of fields
$$
T_n \bigl(\Theta_1 ({\mathrm{x}}_1) \cdots \Theta_n ({\mathrm{x}}_n)\bigr),
$$
where $\Theta_j({\mathrm{x}})$ are local free field polynomials like
$\varphi({\mathrm{x}})$, $\NPR{\varphi^2} ({\mathrm{x}})$, $\NPR{\partial^{\mu} \varphi \, \partial_{\mu} \varphi} ({\mathrm{x}})$, $\dots$
(i.e., $\Theta_j ({\mathrm{x}})$ are composite fields of free fields).
The main axioms for these time--ordered products are the following:

\begin{itemize}
\item[$(T_0)$]
\textit{Domain}: $T_n \bigl(\Theta_1 ({\mathrm{x}}_1) \cdots \Theta_n ({\mathrm{x}}_n)\bigr)$ are operator valued distributions
acting on \textit{invariant} domain that
contains the domain of the Wightman fields;
\item[$(T_1)$]
\textit{Permutation symmetry}:
$$
T_n \bigl(\Theta_1 ({\mathrm{x}}_1) \cdots \Theta_n ({\mathrm{x}}_n)\bigr)
\, = \,
(-1)^{\varepsilon} \,
T_n \bigl(\Theta_{\sigma_1} ({\mathrm{x}}_{\sigma_1}) \cdots \Theta_{\sigma_n} ({\mathrm{x}}_{\sigma_n})\bigr)
$$
for every permutation $(\sigma_1,\dots,\sigma_n)$ of $(1,\dots,n)$,
where $\varepsilon$ $(=$
$\varepsilon \bigl(\sigma;$ $\Theta_1,$ $\dots,$ $\Theta_n\bigr))$
is the \textit{fermionic} parity of the permutation $\sigma$
for the given set of local Wick polynomials $\Theta_1,$ $\dots,$ $\Theta_n$.
\item[$(T_2)$]
\textit{Covariance}:
\begin{eqnarray}
&& \hspace{-15pt}
U_g T_n \bigl(\Theta_1 ({\mathrm{x}}_1) \cdots \Theta_n ({\mathrm{x}}_n)\bigr) U_g^{-1}
\nonumber \\ && \hspace{-15pt} = \,
T_n \Bigl(\bigl((\pi(g)^{-1}\Theta_1\bigr) (g{\mathrm{x}}_1) \cdots \bigl(\pi(g)^{-1}\Theta_n\bigr) (g{\mathrm{x}}_n)\Bigr)
\nonumber
\end{eqnarray}
for every Poincar\'e transformation $g$.\footnote{%
The translations act trivially on the fields, i.e.,
$(\pi(g)\Theta)({\mathrm{x}})$ $=$ $\Theta({\mathrm{x}})$.}
\item[$(T_3)$]
\textit{Causality}:
\begin{eqnarray}
&& \hspace{-15pt}
T_{m+n} \bigl(\Theta_1 ({\mathrm{x}}_1) \cdots \Theta_n ({\mathrm{x}}_n)\bigr)
\nonumber \\ && \hspace{-15pt} = \
T_m \bigl(\Theta_1 ({\mathrm{x}}_1) \cdots \Theta_m ({\mathrm{x}}_m)\bigr)
\,
T_n \bigl(\Theta_{m+1} ({\mathrm{x}}_{m+1}) \cdots \Theta_{m+n} ({\mathrm{x}}_{m+n})\bigr)
\nonumber
\end{eqnarray}
in the domain\footnote{%
The relation ${\mathrm{x}} \gtrsim {\mathrm{y}}$ means that ${\mathrm{x}} \neq {\mathrm{y}}$ and ${\mathrm{x}}$ is not contained in the past of ${\mathrm{y}}$,
i.e., ${\mathrm{x}}$ $\notin$ ${\mathrm{y}}-\overline{V}_+$,
where $\overline{V}_+$ is the closure of the open future light--cone $V_+$.}
${\mathrm{x}}_j \gtrsim {\mathrm{x}}_{m+k}$ for $j=1,\dots,m$, $k=1,\dots,n$.
In particular, $T_1 \bigl((\Theta ({\mathrm{x}})\bigr)$ $=$ $\Theta ({\mathrm{x}})$.
\item[$(T_4)$]
\textit{Causal Wick expansion}:
\begin{eqnarray}\label{WTF}
\hspace{-25pt}
T_n \bigl(\Theta_1 ({\mathrm{x}}_1) \cdots \Theta_n ({\mathrm{x}}_n)\bigr)
\, = && \hspace{-15pt}
\mathop{\sum}\limits_{\Mbf{r}_1,\dots,\Mbf{r}_n}
\frac{1}{\Mbf{r}_1! \cdots \Mbf{r}_n!} \,
\langle 0 | \hspace{1pt}
T_n \bigl(\Theta_1^{(\Mbf{r}_1)} ({\mathrm{x}}_1) \cdots \Theta_n^{(\Mbf{r}_n)} ({\mathrm{x}}_n)\bigr)
| 0 \rangle
\nonumber \\ && \hspace{-15pt} \times \,
\NPR{\Phi^{\Mbf{r}_1} ({\mathrm{x}}_1) \cdots \Phi^{\Mbf{r}_n} ({\mathrm{x}}_n)} \,,
\end{eqnarray}
where we are using the same ``superquadri--index'' notations like in the Epstein--Glaser paper
\cite{EG}
(their notation $\NPR{\mathrm{A} ({\mathrm{x}})^{\mbf{\mathrm{r}}}}\,$)\footnote{%
The right hand side is well defined due to the zeroth theorem of Epstein--Glaser (\cite{EG}).}.
\item[$(T_5)$]
\textit{Scaling degree}:
there is a grading on the space of composite fields provided by the
so called scaling dimension $\text{dim }\Theta$ and
\begin{eqnarray}
&
\text{The Steinmann scaling degree of }
\langle 0 | \hspace{1pt}
T_n \bigl(\Theta_1 ({\mathrm{x}}_1) \cdots \Theta_n ({\mathrm{x}}_n)\bigr)
| 0 \rangle
& \nonumber \\ & \leqslant \,
\text{dim }\Theta_1  + \cdots + \text{dim }\Theta_n \,.
& \nonumber
\end{eqnarray}
\item[$(T_6)$]
\textit{Unitarity}:
\begin{eqnarray}
&& \hspace{-15pt}
\hspace{-25pt}
T_n \bigl(\Theta_1 ({\mathrm{x}}_1)^{+} \cdots \Theta_n ({\mathrm{x}}_n)^{+}\bigr)^{+}
\, = \,
\mathop{\sum}\limits_{m \, \geqslant \, 0} \,
\mathop{\sum}
\limits_{\text{\scriptsize
\(\begin{array}{c}
\text{ordered partitions}
\\
(S_1,\dots,S_m)
\text{ of}
\\
\{1,\dots,n\}
\end{array}\)}}
(-1)^{n+m} \ \,
T_{S_1} \cdots  T_{S_m} \,,
\nonumber
\end{eqnarray}
where
$T_{S_j}$ $:=$ $T_{|S_j|} \Bigl(\mathop{\prod}\limits_{k \, \in \, S_j} \Theta_j ({\mathrm{x}}_j)\Bigr)$
and $(\cdot)^+$ stands for the Hermitian conjugation.
\end{itemize}

There are some further requirements to the time--ordered products,
like the action Ward identity.
We shall not consider the latter
at this stage but shall make some comments on it in
Sect.~\ref{se6.2}.
Let us also point out that we would like to have the time--ordered products
$T_n \bigl(\Theta_1 ({\mathrm{x}}_1)$ $\cdots$ $\Theta_n ({\mathrm{x}}_n)\bigr)$
constructed for every set of composite fields
$\Theta_1 ({\mathrm{x}}),$ $\dots,$ $\Theta_n ({\mathrm{x}})$,
and this also includes the multilinearity condition
mentioned
in \cite[Sect.~4.1]{S2}.

\Section{Sketch of the construction of time--ordered products
by re\-n\-o\-rmalization maps}{se2a}

\noindent
In this section we shall draw in sketch our ideas
for constructing time--ordered products.
We shall explain them more precisely in the subsequent sections.

So, our idea for a construction of time--ordered products,
which is alternative to the old Epstein--Glaser procedure,
is to use
directly
the
the causal Wick expansion
(\ref{WTF}).
In other words, we would like to set
\begin{eqnarray}\label{WTFM}
\hspace{-0pt}
T_n \bigl(\Theta_1 ({\mathrm{x}}_1) \cdots \Theta_n ({\mathrm{x}}_n)\bigr)
= \! && \hspace{-15pt}
\mathop{\sum}\limits_{\Mbf{r}_1,\dots,\Mbf{r}_n}
\frac{1}{\Mbf{r}_1! \cdots \Mbf{r}_n!} \,
R_n \Bigl(
\langle 0 |
\Theta_1^{(\Mbf{r}_1)} ({\mathrm{x}}_1) \cdots \Theta_n^{(\Mbf{r}_n)} ({\mathrm{x}}_n)
| 0 \rangle
\Bigr)
\nonumber \\ && \hspace{-15pt} \times \,
\NPR{\Phi^{\Mbf{r}_1} ({\mathrm{x}}_1) \cdots \Phi^{\Mbf{r}_n} ({\mathrm{x}}_n)} \ ,
\end{eqnarray}
where $R_n$ are suitable linear maps
$$
\left\{\hspace{-2pt}
\begin{array}{c}
\text{algebra {${\mathrsfs{O}\hspace{-1pt}}_n$} of non globally defined} \\
\text{$n$--point {\it complex analytic} functions}
\end{array}
\hspace{-2pt}\right\}
\hspace{5pt}
\raisebox{-3pt}{$\mathop{\text{\Huge $\to$}}\limits^{\raisebox{3pt}{\large $R_n$}}$}
\hspace{5pt}
\left\{\hspace{-2pt}
\begin{array}{c}
\text{space of globally} \\
\text{defined distributions}
\end{array}
\hspace{-2pt}\right\} .
$$
The problem then is what are the properties of the maps $R_n$, which will ensure
the properties $(T_0)$--$(T_6)$.
In \cite{N1} we have introduced such maps
for Euclidean Green functions
and called them renormalization maps.
In Sect.~\ref{se5} we shall give a modification of this theory on Minkowski space
and shall show that they provide the main properties of
the
time--ordered products.

At this point we shall only describe
the algebras of regular functions on which we apply the renormalization maps.
Thus, this is a sequence of algebras ${\mathrsfs{O}\hspace{-1pt}}_2$, ${\mathrsfs{O}\hspace{-1pt}}_3$, $\dots$, ${\mathrsfs{O}\hspace{-1pt}}_n$, $\dots$
etc., where ${\mathrsfs{O}\hspace{-1pt}}_n$ is an algebra of translation
invariant complex analytic functions of $n$ vector arguments
belonging to the \textit{symmetrized extended $n$--point backward tube}.\footnote{%
The
symmetrized extended $n$--point backward tube
is the domain of analyticity of the $n$--point Wightman functions, which consists
of the orbit of the $n$--point backward tube
$$
\Bigl\{({\mathrm{x}}_1,\dots,{\mathrm{x}}_n) \, \in \, M^{\times n} + i M^{\times n} \, : \,
{\mathrm{x}}_k-{\mathrm{x}}_{k+1} \, \in \, M - i V_+ \text{ for } k = 1,\dots,n \Bigr\}
$$
($V_+$ being the open future light--cone)
under the action
of the complexified Lorentz group and the permutation group \cite[IV.5]{J}.}
These functions one can think of as coming from
\textit{analytically extended} Wightman functions
of composite fields.
More precisely, we assume that the
algebra ${\mathrsfs{O}\hspace{-1pt}}_n$ is linearly spanned by all finite
linear combinations of products of the form
$$
G \, = \,
\mathop{\prod}
\limits_{1 \, \leqslant \, j \, < \, k \, \leqslant \, n} \,
G_{jk} \bigl({\mathrm{x}}_j-{\mathrm{x}}_k\bigr)
, \quad
$$
where
${\mathrm{x}}_k$ $=$ $(x_k^0,\dots,x_k^{D-1})$ are vectors belonging, in general, to the complexified
Min\-ko\-w\-s\-ki space $M+iM$ $\equiv$ ${\mathbb R}^{D-1,1}$ $+$ $i{\mathbb R}^{D-1,1}$
and
the functions $G_{jk}({\mathrm{x}})$ belong to the
algebra ${\mathrsfs{O}\hspace{-1pt}}_2$.
The latter algebra, ${\mathrsfs{O}\hspace{-1pt}}_2$, is supposed to be a subalgebra of
the algebra of analytic functions
on the extended backward tube in $M+iM$.
One can think of the algebra ${\mathrsfs{O}\hspace{-1pt}}_2$ as an
algebra containing the propagators of the theory.
In this way the algebras ${\mathrsfs{O}\hspace{-1pt}}_n$ for $n>2$ are
entirely determined by the algebra ${\mathrsfs{O}\hspace{-1pt}}_2$.
To retain the permutation symmetry on the algebras ${\mathrsfs{O}\hspace{-1pt}}_n$
it is also convenient to continue to consider them
as algebras of regular functions on Euclidean configuration spaces like in \cite{N1}
but now we assume in addition that they also posses certain analytic continuations
mentioned above.

We retain
the technical assumption for the algebra ${\mathrsfs{O}\hspace{-1pt}}_2$
(and thus for all other ${\mathrsfs{O}\hspace{-1pt}}_n$) that it is closed
with respect to multiplication of its elements by
polynomials as well as with respect to applying derivatives.

We
conclude
this section
with a more suitable form of the
causal Wick expansion
\begin{eqnarray}\label{WTFN}
&& \hspace{-15pt}
T_n
\bigl(
\Theta_1 ({\mathrm{x}}_1) \cdots \Theta_n({\mathrm{x}}_n)
\bigr)
\nonumber \\
&& \hspace{-15pt} = \,
R_n \Biggl(
\mathop{\prod}\limits_{1 \, \leqslant \, j \, < \, k \, \leqslant \, n}
\exp \Biggl(
\mathop{\sum}\limits_{\mathrm{r},\mathrm{s}} C_{\mathrm{r},\mathrm{s}} \bigl({\mathrm{x}}_j-{\mathrm{x}}_k\bigr)
\frac{\partial}{\partial \varphi_{\mathrm{r}} ({\mathrm{x}}_j)} \frac{\partial}{\partial \varphi_{\mathrm{s}} ({\mathrm{x}}_k)}
\Biggr)
\Biggr)
\nonumber \\ && \hspace{-15pt} \hspace{13pt} \times \,
:\!\Theta_1 ({\mathrm{x}}_1) \cdots \Theta_n({\mathrm{x}}_n)\!: \,,
\end{eqnarray}
where\
\(
\varphi_{\mathrm{r}} ({\mathrm{x}}) :=
\partial_{{\mathrm{x}}}^{\mathrm{r}} \varphi ({\mathrm{x}})
, \
C_{\mathrm{r},\mathrm{s}} \bigl({\mathrm{x}}_1-{\mathrm{x}}_2\bigr) :=
\partial_{{\mathrm{x}}_1}^{\mathrm{r}} \partial_{{\mathrm{x}}_2}^{\mathrm{s}}
\langle 0 | \varphi({\mathrm{x}}_1) \varphi({\mathrm{x}}_2) | 0 \rangle
\)\
and $\partial_{{\mathrm{x}}}^{\mathrm{r}}$ $=$
$\Bigl(\frac{\partial}{\partial x^0}\Bigr)^{r_0}$ $\cdots$ $\Bigl(\frac{\partial}{\partial x^{D-1}}\Bigr)^{r_{D-1}}$.
The formula (\ref{WTFN}) is understood in the following way.
The product
\begin{equation}\label{DOP}
\mathop{\prod}\limits_{1 \, \leqslant \, j \, < \, k \, \leqslant \, n}
\exp \Biggl(
\mathop{\sum}\limits_{\mathrm{r},\mathrm{s}} C_{\mathrm{r},\mathrm{s}} \bigl({\mathrm{x}}_j-{\mathrm{x}}_k\bigr)
\frac{\partial}{\partial \varphi_{\mathrm{r}} ({\mathrm{x}}_j)} \frac{\partial}{\partial \varphi_{\mathrm{r}} ({\mathrm{x}}_k)}
\Biggr)
\end{equation}
in the argument of $R_n$ is considered, after expanding the exponents,
as a differential operator in $\frac{\partial}{\partial \varphi_{\mathrm{r}} ({\mathrm{x}}_j)}$,
which acts on
$:\!\!\Theta_1 ({\mathrm{x}}_1)$ $\cdots$ $\Theta_n({\mathrm{x}}_n)\!\!:$,
where every $\Theta_j ({\mathrm{x}}_j)$ is considered as a polynomial in
$\{\varphi_{\mathrm{r}} ({\mathrm{x}}_j)\}_{\mathrm{r}}$
for $j=1,\dots,n$.
The coefficients of so obtained differential operator (\ref{DOP})
are elements of the algebra ${\mathrsfs{O}\hspace{-1pt}}_n$ on which we then apply the renormalization map $R_n$.

Let us mention that the formula (\ref{WTFN}) is known in the literature also in the form
that uses variational derivatives:
\begin{eqnarray}
&& \hspace{-15pt}
\hspace{-20pt}
T_n
\bigl(
\Theta_1 ({\mathrm{x}}_1) \cdots \Theta_n({\mathrm{x}}_n)
\bigr)
\nonumber \\
&& \hspace{-15pt}
\hspace{-20pt}
= \,
R_n \Biggl( \hspace{-2pt}
\exp \Biggl( \hspace{-1pt} \frac{1}{2}
\int \mathrm{d}{\mathrm{z}} \, \mathrm{d}{\mathrm{w}} \,
C \bigl({\mathrm{z}}-{\mathrm{w}}\bigr) \,
\frac{\delta}{\delta \varphi ({\mathrm{z}})} \frac{\delta}{\delta \varphi ({\mathrm{w}})}
\hspace{-1pt} \Biggr)
\hspace{-2pt} \Biggr)
:\!\Theta_1 ({\mathrm{x}}_1) \cdots \Theta_n({\mathrm{x}}_n)\!: \,,
\nonumber
\end{eqnarray}
where $C({\mathrm{x}}-{\mathrm{y}})= \langle 0 | \varphi ({\mathrm{x}}) \varphi ({\mathrm{y}}) | 0 \rangle$.

\Section{More precise formulation of time--ordered products}{se3}

Let $F$ be the linear span of the generating fields $\varphi_A$ in our theory.
In other words, $F$ is a vector space equipped with a basis $\varphi_A$
that is in one--to--one correspondence with the basic Wightman fields\footnote{%
With a slight abuse of the notations we shall use the one and the same letters
$\varphi_A$ in the two cases.}
$\varphi_A ({\mathrm{x}})$.
In particular, $F$ is some representation of the Lorentz group,
which in general is a sum of irreducible representations.
The space $F$ is ${\mathbb Z}_2$--graded (bosons and fermions).
We work with \textit{complex} fields.
Thus, in what follows $F$ and all the vector spaces and algebras
will be considered over the field of complex numbers ${\mathbb C}$.

The space of the derivative fields is
$$
\FSP^{(1)} \, = \, {\mathbb C} [\partial] \otimes F \,,
$$
which we shall also write as
$\FSP^{(1)} = {\mathbb C} [\partial] F$.
Here, $\partial$ $:=$ $(\partial_0,$ $\dots,$ $\partial_{D-1})$ $\equiv$
$\Bigl(\frac{\partial}{\partial x^0},$ $\cdots,$ $\frac{\partial}{\partial {\mathrm{x}}^{D-1}}\Bigr)$
and so, $\FSP^{(1)}$ is just the linear span of all derivative fields of the form
$$
\bigl(\partial^{\mathrm{r}} \varphi_A\bigr) ({\mathrm{x}}) \, \equiv \, \varphi_{A,\mathrm{r}} ({\mathrm{x}})
\, := \,
\bigl(\partial_0^{r_0} \cdots \partial_{D-1}^{r_{D-1}} \varphi_A\bigr) ({\mathrm{x}}),
$$
where $\mathrm{r}$ $=$ $(r_0,\dots,r_{D-1})$ is a multiindex.
The space of all composite fields is then introduced as
the free graded--commutative algebra over $\FSP^{(1)}$, i.e.,
$$
\mathcal{F}_{}
\, = \,
{\mathbb C} \bigl[\FSP^{(1)}\bigr] \,.
$$
In this way, $\mathcal{F}_{}$ is also a differential algebra
with even derivatives $\partial_0,$ $\dots,$ $\partial_{D-1}$ (with respect to the ${\mathbb Z}_2$--grading).

Thus, we assumed that the algebra $\mathcal{F}_{}$ consists of
``\textit{off--shell}'' fields $\Theta$,
i.e. it is not factorized by any ideal at the first step.
Later, the elements $\Theta$ $\in$ $\mathcal{F}_{}$ will
be mapped to the space of composite fields $\Theta ({\mathrm{x}})$ of free fields
and their derivatives, which act on a (dense domain in a) Hilbert space.
Let us also point out that the product in the algebra $\mathcal{F}_{}$
does not correspond to the product in the operator algebra
of quantum fields, but it rather corresponds to the pointwise normal product.\footnote{%
Our notation $\mathcal{F}_{}$ should correspond to $\mathcal{P}^{\oplus}$
from \cite{S2}.}

For the axiom $(T_5)$ we also need a grading ``$\text{dim}$'' on $\mathcal{F}_{}$
in such a way that it becomes a graded commutative differential
algebra with derivatives $\{\partial_{\mu}\}_{\mu \, = \, 0}^{D-1}$ of degree~$1$.
The latter implies that the grading function on $\mathcal{F}_{}$ is completely determined
by its restriction on the space $F$: for instance,
$\text{dim} \, (\varphi_A \partial_{\mu} \varphi_B)$ $=$
$1$ $+$ $\text{dim} \, \varphi_A$ $+$ $\text{dim} \, \varphi_B$.
We assume that the ${\mathbb Z}_2$--grading function on $\mathcal{F}_{}$ is expressed
by the grading function $\text{dim}$ as
$2 \, \text{dim}$ $(\text{mod} \, 2)$.
We also require that
\begin{eqnarray}\label{COMP}
&
\text{The Steinmann scaling degree of }
\langle 0 | \hspace{1pt}
\varphi_A ({\mathrm{x}}_1) \varphi_B ({\mathrm{x}}_2)\bigr)
| 0 \rangle
& \nonumber \\ &
\leqslant \, \text{dim} \, \varphi_A + \text{dim} \, \varphi_B \,.
&
\end{eqnarray}

Next, let us consider for every positive integer $j \in {\mathbb N} \equiv \{1,2,\dots\}$,
a copy of $\mathcal{F}_{}$ denoted by $\mathcal{F}_{ \{j\}}$ together with an
identification
\begin{equation}\label{IDE}
\mathcal{F}_{} \cong \mathcal{F}_{ \{j\}} : \Theta \to (\Theta)_j \,.
\end{equation}
The elements $(\Theta)_j$ of the space $\mathcal{F}_{ \{j\}}$
will further play the role of local fields $\Theta ({\mathrm{x}}_j)$ in the variable ${\mathrm{x}}_j$.
Finally, let us introduce the spaces of all polylocal (off--shell) fields
$\mathcal{F}_{ S}$:
$$
\mathcal{F}_{ S} \, = \, \bigotimes_{j \, \in \, S} \, \mathcal{F}_{ \{j\}}
$$
for every finite nonempty subset $S \subset {\mathbb N}$.
Thus, $\mathcal{F}_{ S}$ is an algebra,\footnote{%
Again the product in this algebra $\mathcal{F}_{ S}$
does not correspond to the product in the operator algebra
of quantum fields.}
which is the tensor product of the \textit{graded} differential algebras
$\{\mathcal{F}_{ \{j\}}\}_{j \, \in \, S}$.
The algebras $\mathcal{F}_{ S}$ naturally form an \textit{inductive} system
(i.e., if $S' \subseteq S''$ then $\mathcal{F}_{ S'}$ $\hookrightarrow$ $\mathcal{F}_{ S''}$)
and we introduce also the inductive limit
$$
\mathcal{F}_{ {\mathbb N}} \, = \, \mathop{\lim}\limits_{\raisebox{5pt}{$\longrightarrow$}} \, \mathcal{F}_{ S}
\, \equiv \, \mathop{\bigcup}\limits_{S \hspace{-4pt} \mathop{\subset}\limits_{\text{finite}} \hspace{-4pt} {\mathbb N}} \, \mathcal{F}_{ S}\,.
$$
(Thus, we identify $\mathcal{F}_{ \{j\}}$ and $\mathcal{F}_{ S}$
with subalgebras in $\mathcal{F}_{ {\mathbb N}}$.)

The elements of $\mathcal{F}_{ {\mathbb N}}$ will be mapped later either to
sums of
normal products
$:\hspace{-3pt}\Theta_1 ({\mathrm{x}}_1)$ $\cdots$ $\Theta_n ({\mathrm{x}}_n)\hspace{-3pt}:$,
or to time--ordered products
$T \bigl(\Theta_1 ({\mathrm{x}}_1)$ $\cdots$ $\Theta_n ({\mathrm{x}}_n)\bigr)$.

So, the algebra $\mathcal{F}_{ {\mathbb N}}$ is the
free (graded--)commuta\-ti\-ve algebra with generators
$\{\varphi_{A,\mathrm{r},j}\}_{A,\,\mathrm{r},\,j}$
(corresponding to the fields
$(\partial^{\mathrm{r}}\varphi_A)({\mathrm{x}}_j)$).
This algebra has various structures on it
(several products, derivatives, coproducts, structure of a jet algebra, \dots)
and it is very interesting to find the
interplay
between them.
We shall not investigate here the possible algebraic structures on $\mathcal{F}_{ {\mathbb N}}$
but we shall mention only one, which will be used further.
This is the permutation symmetry:
for every bijection $\sigma : {\mathbb N} \cong {\mathbb N}$ there is a natural action
$\sigma : \mathcal{F}_{ {\mathbb N}} \to \mathcal{F}_{ {\mathbb N}}$, which is a representation
of the (infinite) permutation group $\mathrsfs{S} ({\mathbb N})$ $\ni$ $\sigma$.
To write this action explicitly, note that the algebra $\mathcal{F}_{ {\mathbb N}}$
is linearly spanned by products of the form
\begin{equation}\label{MF1}
\mathop{\prod}\limits_{j \, \in \, {\mathbb N}} \, \Theta_j
\end{equation}
where $\Theta_j \in \mathcal{F}_{ \{j\}}$ (the $j$-th copy of $\mathcal{F}_{}$)
and only finitely many $\Theta_j$ are different from 1.
Then the action $\sigma : \mathcal{F}_{ {\mathbb N}} \to \mathcal{F}_{ {\mathbb N}}$ is defined by the formula
$$
\sigma \Biggl(\mathop{\prod}\limits_{j \, \in \, {\mathbb N}} \, \Theta_j\Biggr)
\, = \,
\mathop{\prod}\limits_{j \, \in \, {\mathbb N}} \,
\Theta_{\sigma(j)}
\,.
$$
(In particular, in the purely bosonic case the above action of $\mathrsfs{S}({\mathbb N})$ is trivial.)

In order to introduce the linear maps from $\mathcal{F}_{ {\mathbb N}}$
into the space of operator valued distributions,
which give the normal and time--ordered products
let us denote for every finite $S \subset {\mathbb N}$:
$$
\Dd'_S \, := \, \Dd' \bigl(M_S\bigr) \,,
$$
where
$$
M_S \, := \, M^S \bigl/ M
$$
is the space of $S$--point configurations $\bigl({\mathrm{x}}_j\bigr)_{j \, \in \, S}$ of vectors in $M$
modulo translations
(the elements of $M_S$ will be denoted by
$$
\bigl[{\mathrm{x}}_j\bigr]_{j \, \in \, S} \, := \,
\bigl({\mathrm{x}}_j\bigr)_{j \, \in \, S} \text{ mod } M \,).
$$
Thus, $\Dd'_S$ is
the space of all translation invariant ``$S$--point'' distributions
over the Minkow\-s\-ki space $M$ (i.e., distributions whose vector arguments ${\mathrm{x}}_j$
are indexed by elements $j \in S$).
Also, let us denote
$$
\widehat{\Dd}\hspace{0.75pt}'\hspace{-3.75pt}_S \, := \, \Dd' \bigl(M^S, Op (\mathfrak{D})\bigr)
$$
the space of all $S$--point operator valued distributions acting on an invariant dense domain
$\mathfrak{D}$ in a Hilbert space.
Let us introduce again the inductive limits
$$
\Dd'_{{\mathbb N}} \, = \,
\mathop{\bigcup}\limits_{S \hspace{-4pt} \mathop{\subset}\limits_{\text{finite}} \hspace{-4pt} {\mathbb N}} \, \Dd'_S
,\quad
\widehat{\Dd}\hspace{0.75pt}'\hspace{-3.75pt}_{{\mathbb N}} \, = \,
\mathop{\bigcup}\limits_{S \hspace{-4pt} \mathop{\subset}\limits_{\text{finite}} \hspace{-4pt} {\mathbb N}} \, \widehat{\Dd}\hspace{0.75pt}'\hspace{-3.75pt}_S \,,
$$
and the actions of the permutation group $\mathrsfs{S} ({\mathbb N})$ $(\ni \sigma)$
on the spaces $\Dd'_{{\mathbb N}}$ and $\widehat{\Dd}\hspace{0.75pt}'\hspace{-3.75pt}_{{\mathbb N}}$, which are given by the formula
$$
(\sigma f) ({\mathrm{x}}_1,{\mathrm{x}}_2,\dots) \, = \, f\bigl({\mathrm{x}}_{\sigma^{-1}(1)},{\mathrm{x}}_{\sigma^{-1}(2)},\dots\bigr) \,.
$$

Now, in a Wightman theory of a system of free fields $\{\varphi_A\}$ we have linear maps
$$
N : \mathcal{F}_{ {\mathbb N}} \to \widehat{\Dd}\hspace{0.75pt}'\hspace{-3.75pt}_{{\mathbb N}}
\, , \quad
T : \mathcal{F}_{ {\mathbb N}} \to \widehat{\Dd}\hspace{0.75pt}'\hspace{-3.75pt}_{{\mathbb N}}
\, .
$$
The first of them is the normal product:
$$
N \Biggl(\mathop{\prod}\limits_{j \, \in \, {\mathbb N}} \, \Theta_j\Biggr)
\, := \
\NPR{\Theta_1 ({\mathrm{x}}_1) \, \Theta_2 ({\mathrm{x}}_2) \cdots}
$$
and it has a canonical realization so that
we shall not characterize it here axiomatically.
The second map $T$ gives the time ordered products:
$$
T \Biggl(\mathop{\prod}\limits_{j \, \in \, {\mathbb N}} \, \Theta_j\Biggr)
\, := \
T \bigl(\Theta_1 ({\mathrm{x}}_1) \, \Theta_2 ({\mathrm{x}}_2) \cdots\bigr)
$$
and it does not have a canonical construction
but is defined up to
a
renormalization ambiguity.

It is easy to
reformulate
the axiomatic properties $(T_0)$--$(T_6)$ for
the time ordered products in terms of the maps $N$ and $T$.
In particular,
both maps, $N$ and $T$, have the permutation symmetry:
$$
\sigma \circ N \, = \, N \circ \sigma
\, , \quad
\sigma \circ T \, = \, T \circ \sigma
$$
for $\sigma \in \mathrsfs{S} ({\mathbb N})$.

The causality reads
\begin{equation}\label{CAUS}
T\bigl(\Theta_{S'} \,\Theta_{S''}\bigr)\vrestr{12pt}{\mathfrak{C}_{S;S'}}
\, = \,
T\bigl(\Theta_{S'}\bigr) \, T\bigl(\Theta_{S''}\bigr)\vrestr{12pt}{\mathfrak{C}_{S;S'}}
\end{equation}
where $S = S' \, \dot{\cup} \, S''$ (disjoint union),
$\Theta_{S'}$ $\in$ $\mathcal{F}_{S'}$ and $\Theta_{S''}$ $\in$ $\mathcal{F}_{S''}$,
and $\mathfrak{C}_{S;S'}$ is the open region:
$$
\mathfrak{C}_{S;S'} \, := \,
\Bigl\{
\bigl[{\mathrm{x}}_j\bigr]_{j \, \in \, S} \, : \,
{\mathrm{x}}_{j'} \, \gtrsim \, {\mathrm{x}}_{j''} \text{\, for\, }
j' \, \in \, S' \text{\, and\, } j'' \, \in \, S''
\Bigr\} \,
$$
(the relation ${\mathrm{x}}_{j'} \gtrsim {\mathrm{x}}_{j''}$ stands for ${\mathrm{x}}_{j''}$ $\notin$ ${\mathrm{x}}_{j'}-\overline{V}_+$).

The unitarity reads
\begin{eqnarray}\label{UNI}
&& \hspace{-15pt}
T \bigl(\Theta_S^{*}\bigr)^{+}
\, = \,
\mathop{\sum}\limits_{m \, \geqslant \, 0} \,
\mathop{\sum}
\limits_{\text{\scriptsize
\(\begin{array}{c}
\text{ordered partitions}
\\
(S_1,\dots,S_m)
\text{ of } S
\end{array}\)}}
(-1)^{|S|+m} \ \,
T \bigl(\Theta_{S_1}\bigr) \cdots  T \bigl(\Theta_{S_m}\bigr) \qquad
\end{eqnarray}
where $\Theta_{S_k}$ $\in$ $\mathcal{F}_{ S_k}$,
$\Theta_S$ $=$ $\Theta_{S_1}$ $\cdots$ $\Theta_{S_m}$
and $\Theta$ $\mapsto$ $\Theta^{*}$
is the involutive anti--automorphism of the algebra $\mathcal{F}_{ {\mathbb N}}$
generated by the antilinear involution on the space $F$ that represents the field conjugation.

\Section{Causal Wick expansion and causality}{se4}

In order to formulate the
causal Wick expansion
in terms of the notations of Sect.~\ref{se3}
let us first introduce
for every finite subset $S$ $\subset$ ${\mathbb N}$
the algebra ${\mathrsfs{O}\hspace{-1pt}}_S$, which is the linear span of all
\begin{equation}\label{GSF}
G \, = \,
\mathop{\prod}
\limits_{\mathop{}\limits^{j,\, k \, \in \, S}_{j \, < \, k}} \,
G_{jk} \bigl({\mathrm{x}}_j-{\mathrm{x}}_k\bigr)
, \quad
G_{jk} ({\mathrm{x}}) \in {\mathrsfs{O}\hspace{-1pt}}_2.
\end{equation}
The system of algebras $\{{\mathrsfs{O}\hspace{-1pt}}_S\}_S$ is again an inductive system and we set
$$
{\mathrsfs{O}\hspace{-1pt}}_{{\mathbb N}} \, = \,
\mathop{\bigcup}\limits_{S \hspace{-4pt} \mathop{\subset}\limits_{\text{finite}} \hspace{-4pt} {\mathbb N}} \, {\mathrsfs{O}\hspace{-1pt}}_S
\, = \,
\mathop{\bigcup}\limits_{n \, = \, 1}^{\infty} \, {\mathrsfs{O}\hspace{-1pt}}_n \,.
$$
As in the above cases of such inductive limits we have also a natural action
$$
\sigma : {\mathrsfs{O}\hspace{-1pt}}_{{\mathbb N}} \to {\mathrsfs{O}\hspace{-1pt}}_{{\mathbb N}}
$$
of $\mathrsfs{S} ({\mathbb N})$ $\ni$ $\sigma$.

The system of renormalization maps $R_S$ $:$ ${\mathrsfs{O}\hspace{-1pt}}_S$ $\to$ $R_S$
(to be defined in Sect.~\ref{se5})
will be consistent with the above inductive limits and thus, they will
induce a linear map
\begin{equation}\label{RMAN}
R : {\mathrsfs{O}\hspace{-1pt}}_{{\mathbb N}} \to \Dd'_{{\mathbb N}}
\ ,\qquad
R \bigl({\mathrsfs{O}\hspace{-1pt}}_S \bigr) \, \subseteq \, \Dd'_S
\,.
\end{equation}

Now, the
causal Wick expansion
(\ref{WTFN})
has also more compact form in the above notations,
\begin{equation}\label{WTE1}
T\bigl(\Theta_S\bigr) \, = \,
R \bigl(\widehat{G}_S\bigr) N\bigl(\Theta_S\bigr)
\,,
\end{equation}
where $\Theta_S \in \mathcal{F}_S$,
and $\widehat{G}_S$ is an operator
\begin{equation}\label{gh1}
\widehat{G}_S : {\mathrsfs{O}\hspace{-1pt}}_S \otimes \mathcal{F}_S \to {\mathrsfs{O}\hspace{-1pt}}_S \otimes \mathcal{F}_S
\end{equation}
defined by
\begin{equation}\label{gh2}
\widehat{G}_S \, = \, \mathop{\prod}\limits_{\mathop{}\limits^{j,\, k \, \in \, S}_{j \, < \, k}}
\widehat{G}_{jk}
,\quad
\widehat{G}_{jk} \, = \,
\exp \Biggl(
\mathop{\sum}\limits_{A,B,\mathrm{r},\mathrm{s}} C_{A,\mathrm{r};B,\mathrm{s}} \bigl({\mathrm{x}}_j-{\mathrm{x}}_k\bigr)
 \otimes
\frac{\partial}{\partial \varphi_{A,\mathrm{r},j}} \frac{\partial}{\partial \varphi_{B,\mathrm{s},k}}
\Biggr)
\end{equation}
and
\(
C_{A,B,\mathrm{r},\mathrm{s}} \bigl({\mathrm{x}}_1-{\mathrm{x}}_2\bigr) :=
\partial_{{\mathrm{x}}_1}^{\mathrm{r}} \partial_{{\mathrm{x}}_2}^{\mathrm{s}}
\langle 0 | \varphi_A({\mathrm{x}}_1) \varphi_B({\mathrm{x}}_2) | 0 \rangle
\in {\mathrsfs{O}\hspace{-1pt}}_2
\).
Let us note that the derivations $\frac{\partial}{\partial \varphi_{A,\mathrm{r},j}}$
entering in Eq.~(\ref{gh2}) are the canonical graded derivations
on the algebra $\mathcal{F}_{ {\mathbb N}}$ regarded as a free graded commutative algebra
with generators $\{\varphi_{A,\mathrm{r},j}\}_{A,\mathrm{r},j}$ (these are the ``vertical derivations'').
We also note that equation (\ref{WTE1}) is more correct to write as
\begin{equation}\label{WTENN1}
T \bigl(\Theta_S\bigr) \, = \, \text{mult} \circ \bigl(R \otimes N\bigr) \bigl(\widehat{G}_S \Theta_S\bigr) \,,
\end{equation}
where $\text{mult}$ $:$ $\Dd'_{{\mathbb N}} \otimes \widehat{\Dd}\hspace{0.75pt}'\hspace{-3.75pt}_{{\mathbb N} \, (\text{norm})}$ $\to$ $\widehat{\Dd}\hspace{0.75pt}'\hspace{-3.75pt}_{{\mathbb N}}$
is the operation of multiplication between the distributions in the space $\Dd'_{{\mathbb N}}$
and the operator--valued distributions in the subspace $\widehat{\Dd}\hspace{0.75pt}'\hspace{-3.75pt}_{{\mathbb N} \, (\text{norm})}$ $\subset$ $\widehat{\Dd}\hspace{0.75pt}'\hspace{-3.75pt}_{{\mathbb N}}$
spanned by the normal products of local fields of different arguments
(this multiplication exists due to the zeroth theorem of Epstein and Glaser \cite{EG}).

\medskp

\begin{Remark}\label{RR3}
The maps $\widehat{G}_S$ are consistent with the inductive limit and generate a linear map
$\widehat{G}$ $:$ ${\mathrsfs{O}\hspace{-1pt}}_{{\mathbb N}} \otimes \mathcal{F}_{ {\mathbb N}}$ $\to$ ${\mathrsfs{O}\hspace{-1pt}}_{{\mathbb N}} \otimes \mathcal{F}_{ {\mathbb N}}$.
Then Eq.~(\ref{WTENN1}) reads
$$
T \, = \, \text{mult} \circ \bigl(R \otimes N\bigr) \circ \widehat{G} \,.
$$
We shall work with $\widehat{G}_S$ because of the combinatorics related to the causality.
\end{Remark}

\medskp

We have also a
Wick expansion formula
for ordinary products.
In order to formulate it we need additional notations.
Since the functions belonging to ${\mathrsfs{O}\hspace{-1pt}}_S$ are analytic functions
on certain $S$--point tube domains we have another linear maps
${\mathrsfs{O}\hspace{-1pt}}_{S}$ $\to$ $\Dd'_{S}$, the boundary values of analytic functions.
In order to define them we introduce the notion
of an \textit{ordered} set $\VE{S}$
that is a set $S$ $=$ $\{j_1,\dots,j_n\}$ equipped with a total order
$j_1$ $\prec$ $\cdots$ $\prec$ $j_n$ on it.
We shall also write it as
$$
\VE{S} \, = \, \La j_1,\dots,j_n \Ra \,
$$
and $S$ will be called a body of $\VE{S}$.
If $S \subset {\mathbb N}$ we shall consider the order $\prec$ on $S$
as an \textit{independent} structure
on it, which may not coincide with the order $<$ induced by ${\mathbb N}$.
For every ordered set $\VE{S}$
$=$ $\La j_1,\dots,j_n \Ra$
we have
a standard backward\footnote{%
Since we use in (\ref{GSF}) differences of a type left minus right coordinate
then it follows that the boundary values appear
with respect to backward tubes instead of forward.}
tube domain associated to $\VE{S}$,
$$
\mathcal{T}_{\VE{S}}
\, := \Bigl\{[{\mathrm{x}}_j]_{j \, \in \, S} \, \in \, M_S + i M_S \, : \,
{\mathrm{x}}_{j_k}-{\mathrm{x}}_{j_{k+1}} \, \in \, M - i V_+ \text{ for } k = 1,\dots,n \Bigr\}
$$
($V_+$ being the open forward light--cone in $M$)
and then we define a boundary value map
$$
\text{b.v.}_{\VE{S}} : {\mathrsfs{O}\hspace{-1pt}}_S \to \Dd'_S
$$
with respect to this tube $\mathcal{T}_{\VE{S}}$.
Thus, the linear maps $\text{b.v.}_{\VE{S}}$ will produce the Wightman functions in the theory.

Now, the
Wick formula
for ordinary products is
\begin{equation}\label{WTEW}
N\bigl(\Theta_{j_1}\bigr) \cdots N\bigl(\Theta_{j_1}\bigr)
\, = \,
\text{b.v.}_{\VE{S}} \bigl(\widehat{G}_S\bigr) \ N \bigl(\Theta_S\bigr)
\end{equation}
($\VE{S}$ $=$ $\La j_1,\dots,j_n \Ra$).
Again Eq.~(\ref{WTEW})
is understood as
\begin{equation}\label{WTEW1}
N\bigl(\Theta_{j_1}\bigr) \cdots N\bigl(\Theta_{j_1}\bigr)
\, = \,
\text{mult} \circ
\bigl(\text{b.v.}_{\VE{S}} \otimes N\bigr) \bigl(\widehat{G}_S \Theta_S\bigr) \,.
\end{equation}

Equation (\ref{WTEW1}) shows that the right hand side of Eq.~(\ref{WTENN1})
always defines an operator $\mathcal{F}_{ {\mathbb N}}$ $\to$ $\widehat{\Dd}\hspace{0.75pt}'\hspace{-3.75pt}_{{\mathbb N}}$
since the right hand sides of Eqs.~(\ref{WTEW1}) and (\ref{WTENN1})
can be written as:
$$
\bigl(\text{b.v.}_{\VE{S}} \otimes \text{\it id}\bigr) \circ
\bigl(\text{\it id} \otimes N\bigr) \bigl(\widehat{G}_S \Theta_S\bigr)
\quad \text{and} \quad
\bigl(R \otimes \text{\it id}\bigr) \circ
\bigl(\text{\it id} \otimes N\bigr) \bigl(\widehat{G}_S \Theta_S\bigr)
\,,
$$
respectively, and the map $\text{b.v.}_{\VE{S}}$ is an \textit{injection}.
Hence, our idea to construct time--ordered products by
Eq.~(\ref{WTFN}) or Eq.~(\ref{WTFM})
is correct even if we start with off--shell fields:
the corresponding new formula is (\ref{WTENN1}) and it will
preserve
the kernel in the passage from off--shell fields to on--shell fields.

An important consequence of Eq. (\ref{WTEW}) is
\begin{equation}\label{WTE2}
N \bigl(\Theta_{S'}\bigr) N \bigl(\Theta_{S''}\bigr) \, = \,
\text{b.v.}_{\VE{S}} \bigl(\widehat{G}_{S',\,S''}\bigr) \ N \bigl(\Theta_S\bigr)
\,,
\end{equation}
where for a disjoint union $S$ $=$ $S' \, \dot{\cup} \, S''$
of ordered sets $\VE{S}'$ and $\VE{S}''$ we equip $S$ with an
order $S'$ $\prec$ $S''$ and introduce the splittings
$\Theta_S$ $=$ $\Theta_{S'}\Theta_{S''}$ and
\begin{eqnarray}
&& \hspace{-15pt} \hspace{100pt}
\widehat{G}_S \, = \, \widehat{G}_{S'} \, \widehat{G}_{S''} \, \widehat{G}_{S',\,S''} \,,
\nonumber \\ && \hspace{-15pt}
\widehat{G}_{S'} \, = \, \mathop{\prod}\limits_{\mathop{}\limits^{j,\, k \, \in \, S'}_{j \, \prec \, k}}
\widehat{G}_{jk}
\, , \quad
\widehat{G}_{S''} \, = \, \mathop{\prod}\limits_{\mathop{}\limits^{j,\, k \, \in \, S''}_{j \, \prec \, k}}
\widehat{G}_{jk}
\, , \quad
\widehat{G}_{S',\,S''} \, = \, \mathop{\prod}\limits_{\mathop{}\limits^{j \, \in \, S'}_{j \, \in \, S''}}
\widehat{G}_{jk}
\,.
\nonumber
\end{eqnarray}

The causality relation (\ref{CAUS}) will follow now from Eq. (\ref{WTE2})
if $R$ satisfies the following equation:
$$
R \bigl(\widehat{G}_S\bigr) \vrestr{12pt}{\mathfrak{C}_{S;S'}} \, = \,
R \bigl(\widehat{G}_{S'}\bigr) \, R \bigl(\widehat{G}_{S''}\bigr) \, \text{b.v.}_{\VE{S}} \bigl(\widehat{G}_{S',\,S''}\bigr)
\vrestr{12pt}{\mathfrak{C}_{S;S'}}
\,.
$$
This will be implied by the recursive axiomatic property $(r4)$ of the renormalization maps $R$
introduced in the next section.
Before we pass to the definition of renormalization maps we shall show
how Eq.~(\ref{WTE2}) implies (\ref{CAUS}).
{\samepage
On $\mathfrak{C}_{S;S'}$ we have:
\begin{eqnarray}
T\bigl(\Theta_S\bigr)
\, = && \hspace{-15pt}
R \bigl(\widehat{G}_S\bigr) N\bigl(\Theta_S\bigr)
\, = \,
R \bigl(\widehat{G}_{S'}\bigr) R \bigl(\widehat{G}_{S''}\bigr) \, \text{b.v.}_{\VE{S}} \bigl(\widehat{G}_{S',\,S''}\bigr)
\, N\bigl(\Theta_S\bigr)
\nonumber \\
\, = && \hspace{-15pt}
R \bigl(\widehat{G}_{S'}\bigr) R \bigl(\widehat{G}_{S''}\bigr)
N \bigl(\Theta_{S'}\bigr) N \bigl(\Theta_{S''}\bigr)
\, = \,
T\bigl(\Theta_{S'}\bigr) T\bigl(\Theta_{S''}\bigr)
\nonumber
\end{eqnarray}
($\Theta_S := \Theta_{S'} \Theta_{S''}$).}

\Section{Theory of renormalization maps on Minkowski space}{se5}

Let us now consider the modification of the theory of renormalization maps
on the Minkowski space.
We introduced above an improved version of them
replacing the whole system $\{R_S\}_S$ with a single linear map
$$
R : {\mathrsfs{O}\hspace{-1pt}}_{{\mathbb N}} \to \Dd'_{{\mathbb N}} \,
$$
(and so, $R_S$ $\equiv$ $R\vrestr{10pt}{{\mathrsfs{O}\hspace{-1pt}}_S}$).
In order to make sure that this be possible
we need to assume that the system
of renormalization maps $R_S : {\mathrsfs{O}\hspace{-1pt}}_S \to \Dd'_S$ is
consistent with the inductive limits.
This is then equivalent to the enhanced version of the requirement $(r1)$,
which we discussed in Remarks 2.1--2.3 of \cite{N1}.

Thus, the modified axiomatic requirements on the map $R$ are the following.
The conditions $(r1)$--$(r3)$ remains essentially the same
on the Minkowski and the Euclidean space:

\medskip

$(r1)$ \textit{Permutation symmetry.}
For every $\sigma \in \mathrsfs{S} ({\mathbb N})$ we require
$\sigma \circ R$ $=$ $R \circ \sigma$.

\medskip

$(r2)$ \textit{Preservation of the filtrations.}
The scaling degree does not increase
$\text{\rm sc.{\hspace{1pt}}d.} \, R (G)$ $\leqslant$ $\text{\rm sc.{\hspace{1pt}}d.} \, G$.

\medskip

$(r3)$ \textit{Commutativity with multiplication by polynomials.}
If $p$ is a polynomial on $M_S$ ($S \subset {\mathbb N}$) then
$R (p \hspace{1pt} G)$ $=$ $p \, R (G)$.

\medskip

\begin{Remark}\label{Rm}
Property $(r3)$ might look artificial from physical point of view
and in fact it is not necessary for the construction of the time--ordered products.
In paper \cite{N1} this property  plays a crucial role for the reduction of
the cohomological analysis of the renormalization group
to de Rham cohomologies of configuration spaces.
We considered in $(r3)$ only polynomials
since we wish to work algebraically.
But if we work on manifolds
then it is natural to require commutativity
between the renormalization maps and multiplication
by everywhere smooth functions,
i.e., $R (p \hspace{1pt} G)$ $=$ $p \, R (G)$ for
$p \in \mathcal{C}^{\infty} \bigl(M_S\bigr)$.
Then the latter property becomes very natural
from geometric point of view since it allows us to make localization
(i.e., to use localization techniques like partition of unity).
\end{Remark}

\medskp

The last condition $(r4)$ needs more essential modification

\medskip

$(r4)$
\textit{Causality.}
For every disjoint union $S$ $=$ $S' \, \dot{\cup} \, S''$ we have
\begin{equation}\label{RCAUS}
R \bigl(G_S\bigr) \vrestr{12pt}{\mathfrak{C}_{S;S'}} \, = \,
R \bigl(G_{S'}\bigr) \, R \bigl(G_{S''}\bigr) \, \text{b.v.}_{\VE{S}} \bigl(G_{S',\,S''}\bigr)
\vrestr{12pt}{\mathfrak{C}_{S;S'}}
\end{equation}
for every $G_S \in {\mathrsfs{O}\hspace{-1pt}}_S$ of the form (\ref{GSF}),
where we consider some order on the sets
$S'$ and $S''$ and equip $S$ with the order induced by
$S'$ $\prec$ $S''$; finally, as above, we introduce the splitting
\begin{eqnarray}\label{GSPL}
&& \hspace{-15pt} \hspace{100pt}
G_S \, = \, G_{S'} \, G_{S''} \, G_{S',\,S''} \,,
\nonumber \\ && \hspace{-15pt}
G_{S'} \, = \, \mathop{\prod}\limits_{\mathop{}\limits^{j,\, k \, \in \, S'}_{j \, \prec \, k}}
G_{jk}
\, , \quad
G_{S''} \, = \, \mathop{\prod}\limits_{\mathop{}\limits^{j,\, k \, \in \, S''}_{j \, \prec \, k}}
G_{jk}
\, , \quad
G_{S',\,S''} \, = \, \mathop{\prod}\limits_{\mathop{}\limits^{j \, \in \, S'}_{j \, \in \, S''}}
G_{jk}
\,. \qquad
\end{eqnarray}
The right hand side of Eq.~(\ref{RCAUS}) is correct due to the following lemma.

\medskip

\begin{Lemma}\label{L1}
The product
$R \bigl(G_{S'}\bigr) R \bigl(G_{S''}\bigr) \text{b.v.}_{\VE{S}} \bigl(G_{S',\,S''}\bigr)$
exists on
$M_S$.
\end{Lemma}

\medskip

\noindent
\textit{Proof.}
We have to show that the sums of the wave front sets
over\footnote{%
We shall specify in the wave front also the base on which the distribution is
considered since we are using inductive systems of spaces $\Dd'_S$
$( \, := \Dd' (M_S),$ $M_S := M^S / M)$
and
a distribution from $\Dd'_S$ can be also considered in $\Dd'_{S'}$ for $S \subseteq S'$.}
$M_S$
\begin{eqnarray}\label{WF0}
&
\WF{M_S}{R \bigl(G_{S'}\bigr)}+\WF{M_S}{R \bigl(G_{S''}\bigr)},
& \nonumber \\ &
\WF{M_S}{R \bigl(G_{S'}\bigr)}+\WF{M_S}{\text{b.v.}_{\VE{S}} \bigl(G_{S',\,S''}\bigr)},
& \nonumber \\ &
\WF{M_S}{R \bigl(G_{S''}\bigr)}+\WF{M_S}{\text{b.v.}_{\VE{S}} \bigl(G_{S',\,S''}\bigr)},
& \nonumber \\ &
\WF{M_S}{R \bigl(G_{S'}\bigr)}+\WF{M_S}{R \bigl(G_{S''}\bigr)}
+\WF{M_S}{\text{b.v.}_{\VE{S}} \bigl(G_{S',\,S''}\bigr)}
& \quad
\end{eqnarray}
do not intersect the zero section of the cotangent bundle $T^* \bigl(M_S\bigr)$.
This is clear for the first three sums and for the last we first use
a general statement about the wave front of boundary values of analytic functions,
which implies
$$
\text{b.v.}_{(-)} \bigl(G_{jk}\bigr) \, \subseteq \, M \oplus V_-
$$
where $\text{b.v.}_{(-)} \bigl(G_{jk}\bigr)$ is the boundary value of $G_{jk} ({\mathrm{x}})$
in the backward tube $M+iV_-$, $V_-$ $=$ $-V_+$ (the opened backward light--cone).
It then follows that
\begin{equation}\label{WF1}
\WF{M_S}{\text{b.v.}_{\VE{S}} \bigl(G_{S',\,S''}\bigr)}
\, \subseteq \,
\mathop{\sum}\limits_{\mathop{}\limits^{j \, \in \, S'}_{k \, \in \, S''}}
\bigl(\pi^S_{\{j,k\}}\bigr)^*
\Bigl(M \oplus V_-\Bigr)\,,
\end{equation}
where $\bigl(\pi^S_{S'}\bigr)^*$ $:$ $T^* \bigl(M_{S'}\bigr)$ $\to$ $T^* \bigl(M_S\bigr)$
is defined for every $S' \subseteq S$ as the pull-back of
the natural projection $\pi^S_{S'}$ $:$ $M_S$ $\to$ $M_{S'}$ $:$
$[{\mathrm{x}}_j]_{j \, \in \, S}$ $\mapsto$ $[{\mathrm{x}}_j]_{j \, \in \, S'}$.
Let us combine Eq.~(\ref{WF1}) with
\begin{eqnarray}\label{WF2}
\WF{M_S}{R \bigl(G_{S'}\bigr)}
\, \subseteq && \hspace{-15pt}
\bigl(\pi^S_{S'}\bigr)^*
\Bigl(\WF{M_{S'}}{R \bigl(G_{S'}\bigr)}\Bigr)
\,, \quad
\nonumber \\
\WF{M_S}{R \bigl(G_{S''}\bigr)}
\, \subseteq && \hspace{-15pt}
\bigl(\pi^S_{S''}\bigr)^*
\Bigl(\WF{M_{S''}}{R \bigl(G_{S''}\bigr)}\Bigr)
\end{eqnarray}
and use the splitting
\begin{equation}\label{splt}
M_S \cong M_{S'} \hspace{1pt} \oplus \, M_{S''} \hspace{1pt} \oplus \, M : [{\mathrm{x}}_j]_{j \, \in \, S}
\hspace{1pt} \mapsto \hspace{1pt}
[{\mathrm{x}}_j]_{j \, \in \, S'} \hspace{1pt} \oplus \, [{\mathrm{x}}_j]_{j \, \in \, S''} \hspace{1pt} \oplus \,
\bigl({\mathrm{x}}_{\max \, S'} - {\mathrm{x}}_{\min \, S''}\bigr)
.
\end{equation}
Then
the fibers in the tangent bundle $T \bigl(M_S\bigr)$ spits according to (\ref{splt}),
which then implies a decomposition of the cotangent bundle.
Let $pr_3$ $:$ $T^* \bigl(M_S\bigr)$ $\to$ $T^* \bigl(M_S\bigr)$
be the projection that corresponds to the third summand in Eq.~(\ref{splt})
(i.e., the projection onto
the annihilator of the tangent spaces of the first two summands in Eq.~(\ref{splt})).
it follows that $pr_3$ maps the right hand sides in Eq.~(\ref{WF2}) to the zero section,
and on the other hand, $pr_3$ maps the right hand side of Eq.~(\ref{WF1}) to $M$ $\oplus$ $V_-$.
So, the fourth sum in Eq.~(\ref{WF0}) also does not intersect the zero section
of $T^* \bigl(M_S\bigr)$.$\quad\Box$

\bigskip

This completes the list of modified axiomatic conditions for the renormalization map
$R$
on the Minkowski space.
Let us add one more natural assumption

\medskip

$(r5)$ \textit{Lorentz invariance.}
The map $R$ $:$ ${\mathrsfs{O}\hspace{-1pt}}_{{\mathbb N}} \to \Dd'_{{\mathbb N}}$ intertwines the natural actions
of the Lorentz group on ${\mathrsfs{O}\hspace{-1pt}}_{{\mathbb N}}$ and $\Dd'_{{\mathbb N}}$.

\medskip

As in the Euclidean case, the above invariance is ensured later by the construction
of $R$.

Now, the construction of $R$ follows the same scheme like in the Euclidean case.
First, since the the sets $\mathfrak{C}_{S;S'}$ form an open covering of $M_S \bigr\backslash\hspace{1pt}\{\Mbf{0}\}$
we have again inductively defined secondary renormalization maps
$$
{\mathop{\RMA}\limits^{\text{\tiny $\bullet$}}}_S : {\mathrsfs{O}\hspace{-1pt}}_S \to \Dd'_{temp} \bigl(M_S \bigr\backslash\hspace{1pt}\{\Mbf{0}\}\bigr) \,.
$$
Hence, what remains to do is to compose ${\mathop{\RMA}\limits^{\text{\tiny $\bullet$}}}_S$
$$
{\mathop{\RMA}\limits^{\text{\tiny $\bullet$}}}_S \circ P_S \, =: \, R_S
$$
with a primary renormalization map\footnote{%
The systems of linear $P_S$ and ${\mathop{\RMA}\limits^{\text{\tiny $\bullet$}}}_S$ do not form inductive systems.
This is because the system of the intermediate spaces $\Dd'_{temp} \bigl(M_S \bigr\backslash\hspace{1pt}\{\Mbf{0}\}\bigr)$
is not an inductive system.}
$$
P_S : \Dd'_{temp} \bigl(M_S \bigr\backslash\hspace{1pt}\{\Mbf{0}\}\bigr) \to \Dd'_S \,.
$$
Following  \cite{N1}, the linear maps $P_S$ can be constructed
by means of a larger
system of linear maps $\mathcal{P}_N$ $:$ $\Dd'_{temp} \bigl({\mathbb R}^N \bigr\backslash\hspace{1pt}\{\Mbf{0}\}\bigr)$ $\to$ $\Dd' ({\mathbb R}^N)$,
so that $P_S$ $\cong$ $\mathcal{P}_{D(|S|-1)}$.
The axiomatic properties for $\mathcal{P}_N$: $(p1)$--$(p5)$ from \cite{N1}
together with $(p6)$ (\cite[Remark 2.2]{N1}),
remain almost the same
except
the requirement of Euclidean invariance in $(p3)$, which have to be replaced
by the permutation symmetry and Lorentz invariance for $P_S$.\footnote{%
It is more convenient to exclude from the initial requirements for $\mathcal{P}_N$ in
\cite{N1} the property $(p3)$ and just at the end ensure it in the above modified form
for $P_n$ $\cong$ $\mathcal{P}_{D(n-1)}$ by using semi-simplicity of the Lorentz group.}

\medskip

This completes the construction of the renormalization map $R$ on the Minkowski space.

\medskp

\begin{Remark}\label{RM2}
Condition $(r4)$ can be generalized also for arbitrary
\textit{ordered} $S$--partitions.
Theorem 2.9 in \cite{N1}
about changes of renormalization will remain valid in the same form on Minkowski space.
Its proof however should be modified and we have to use now ordered partitions
in order to apply the recursion according to generalized property $(r4)$.
\end{Remark}

\medskp

At the end of this section let us summarize what we have proven
for the construction of time--ordered products.

\medskip

\begin{Theorem}\label{T2}
Let $R : {\mathrsfs{O}\hspace{-1pt}}_{{\mathbb N}} \to \Dd'_{{\mathbb N}}$
be a renormalization map that satisfies
the requirements $(r1)$--$(r5)$ listed above.
Let us define a linear map $T$ $:$ $\mathcal{F}_{ {\mathbb N}}$ $\to$ $\widehat{\Dd}\hspace{0.75pt}'\hspace{-3.75pt}_{{\mathbb N}}$ by
Eq.~$($\ref{WTE1}$)$ $($or, Eq. $($\ref{WTENN1}$)$$)$.
Then the map $T$ is well defined and satisfies the properties
$(T_0)$--$(T_5)$ listed in Sect.~\ref{se2}
$($but possibly without the unitarity~$(T_6)$$)$.
\end{Theorem}

\medskip

\noindent
\textit{Final remarks on the proof.}
We have already argued why $(T_0)$--$(T_4)$ are satisfied
and we would like to discuss here condition $(T_5)$,
the scaling degree.
This is ensured by the fact that the operators $\widehat{G}_{jk}$ (\ref{gh2}),
and hence, $\widehat{G}_S$ (\ref{gh1}),
preserve the filtration on ${\mathrsfs{O}\hspace{-1pt}}_{{\mathbb N}} \otimes \mathcal{F}_{ {\mathbb N}}$
(this filtration is the induced one from the scaling degree on ${\mathrsfs{O}\hspace{-1pt}}_{{\mathbb N}}$
and the grading on $\mathcal{F}_{ {\mathbb N}}$: $\leqslant \text{\rm sc.{\hspace{1pt}}d.} \otimes \text{dim}$).
This is simply because the operators
$C_{A,\mathrm{r};B,\mathrm{s}} \bigl({\mathrm{x}}_j-{\mathrm{x}}_k\bigr)$
$\frac{\partial}{\partial \varphi_{A,\mathrm{r},j}}$ $\frac{\partial}{\partial \varphi_{B,\mathrm{s},k}}$
in the arguments of the exponent in
$\widehat{G}_{jk}$ have this property:
the derivatives
$\frac{\partial}{\partial \varphi_{A,\mathrm{r},j}}$ $\frac{\partial}{\partial \varphi_{B,\mathrm{s},k}}$
decreases the filtration by
$\text{dim} \, \varphi_{A,\mathrm{r},j}$ $+$ $\text{dim} \, \varphi_{B,\mathrm{s},k}$,
while $C_{A,\mathrm{r};B,\mathrm{s}} \bigl({\mathrm{x}}_j-{\mathrm{x}}_k\bigr)$
($=$ $\partial_{{\mathrm{x}}_j}^{\mathrm{r}} \partial_{{\mathrm{x}}_k}^{\mathrm{s}} \langle 0 | \varphi_A({\mathrm{x}}_j)$ $\varphi_B({\mathrm{x}}_k) | 0 \rangle$)
increases the filtration with the same value.
Note that at this point we also use condition (\ref{COMP})
on the grading function $\text{dim}$,
which implies that
the scaling degree of
$C_{A,\mathrm{r};B,\mathrm{s}} \bigl({\mathrm{x}}_j-{\mathrm{x}}_k\bigr)$
is less than or equal to
$\text{dim} \, \varphi_{A,\mathrm{r},j}$ $+$ $\text{dim} \, \varphi_{B,\mathrm{s},k}$.

\Section{Unitarity}{se6.1}

As we mentioned in Theorem \ref{T2}
we do not know
whether the properties $(r1)$--$(r5)$ imply the unitarity
condition $(T_6)$ (Eq.~(\ref{UNI}))
for the time--ordered products.
What we can only say now is that a sufficient condition for this would be
the following additional property on the renormalization map $R$:
\begin{eqnarray}\label{er6}
\overline{R \bigl(G_S^{\hspace{1pt}\raisebox{1pt}{\scriptsize $\CNj$}}\bigr)}
\, = \,
\mathop{\sum}\limits_{m \, \geqslant \, 0} \,
\mathop{\sum}
\limits_{\text{\scriptsize
\(\begin{array}{c}
\text{ordered partitions}
\\
(S_1,\dots,S_m)
\text{ of } S
\end{array}\)}}
&& \hspace{-15pt}
(-1)^{|S|+m} \ \,
R\bigl(G_{S_1}\bigr) \cdots \bigl(G_{S_m}\bigr)
\nonumber \\ && \hspace{-15pt}
\times \
\text{b.v.}_{\VE{S}_{S_1,\dots,S_m}} \Bigl(G_{\{S_1,\dots,S_m\}}\Bigr)
\end{eqnarray}
($G_S \in {\mathrsfs{O}\hspace{-1pt}}_S$ has the form (\ref{GSF})).
Here:
the $\star$--operation on $G_S^{\hspace{1pt}\raisebox{1pt}{\scriptsize $\CNj$}}$ is the ``CPT--operation'',\footnote{%
In particular, this assumes that ${\mathrsfs{O}\hspace{-1pt}}_S$ is $\star$--invariant,
which thus should be added to the axiomatic requirements on ${\mathrsfs{O}\hspace{-1pt}}_S$ (and enough, for ${\mathrsfs{O}\hspace{-1pt}}_2$).
The operation $\star$ is an anti--automorphism of ${\mathrsfs{O}\hspace{-1pt}}_{{\mathbb N}}$.}
$$
G_S^{\hspace{1pt}\raisebox{1pt}{\scriptsize $\CNj$}} ({\mathrm{x}}_{j_1},\dots,{\mathrm{x}}_{j_n}) \, := \,
\overline{G_S (-{\mathrm{x}}_{j_1},\dots,-{\mathrm{x}}_{j_n})}
\qquad (S \, = \, \{j_1,\dots,j_n\})
\,;
$$
the ordered sets $\VE{S}_{S_1,\dots,S_m}$ defining the boundary values in Eq.~(\ref{er6})
is obtained by any orders on $S_k$ and
$S_1$ $\prec$ $\cdots$ $\prec$ $S_m$;
the notation $G_{\{S_1,\dots,S_m\}}$ follows the conventions from \cite[see Eq. (2.14)]{N1},
i.e.,
$$
G_{\{S_1,\dots,S_m\}} \, = \,
\mathop{\prod}\limits_{1 \, \leqslant \, a \, < \, b \, \leqslant \, m} \,
\mathop{\prod}\limits_{\mathop{}\limits^{j \, \in \, S_a}_{k \, \in \, S_b}} \,
G_{jk} ({\mathrm{x}}_j-{\mathrm{x}}_k) \,
$$
(which generalizes the notation $G_{S',\,S''}$ (\ref{GSPL})).
Note that the product of the distributions in the right hand side of
Eq.~(\ref{er6}) exists, which can be proven in the same way as Lemma~\ref{L1}.
The unitarity condition (\ref{UNI}) can be derived from Eq. (\ref{er6})
if we use a generalization of Eq.~(\ref{WTE2}) that is
\begin{equation}\label{WTE3}
N \bigl(\Theta_{S_1}\bigr) \cdots  N \bigl(\Theta_{S_m}\bigr) \, = \,
\text{b.v.}_{\VE{S}_{S_1,\dots,S_m}} \Bigl(\widehat{G}_{\{S_1,\dots,S_m\}}\Bigr) \,
N \bigl(\Theta_S\bigr)
\end{equation}
($\Theta_{S_k}$ $\in$ $\mathcal{F}_{ S_k}$, $\Theta_S$ $=$ $\Theta_{S_1}$ $\cdots$ $\Theta_{S_m}$).

The next problem is to find an additional condition on the primary renormalization maps,
which would imply Eq.~(\ref{er6}).

Nevertheless, it is interesting to study the renormalization group even without
the unitarity condition.
In fact, in the Euclidean case the counterpart of the unitarity is just the fact
that the renormalization maps transform real functions to real distributions,
which is thus satisfied by definition.
Let us also point out that even if we construct time--ordered products, which do not obey
the unitarity then there is a simple, purely algebraic way to pass to a new
system of time--ordered products that satisfy all $(T_0)$--$(T_6)$ (\cite{EG}).

\Section{Action Ward identity}{se6.2}

This is the condition
\begin{equation}\label{AWI}
T\bigl(\partial_{x_k^{\mu}}\Theta\bigr) \, = \,
\partial_{x_k^{\mu}} T\bigl(\Theta\bigr)
\end{equation}
($\Theta \in \mathcal{F}_{ {\mathbb N}}$, $k \in {\mathbb N}$, $\mu=0,\dots,D-1$).
In fact, since we are working only with translation invariant distributions
we have the translation invariance~of~$T$:
\begin{equation}\label{TRI}
\mathop{\sum}\limits_{k \, \in \, {\mathbb N}} \,
T\bigl(\partial_{x_k^{\mu}}\Theta\bigr) \, = \,
\mathop{\sum}\limits_{k \, \in \, {\mathbb N}} \,
\partial_{x_k^{\mu}} T\bigl(\Theta\bigr)
\,.
\end{equation}
Condition (\ref{AWI}) has been established in the off-shell (functional) approach
(\cite{DF})
but it has simple obstructions in the operator formalism.
For instance, for a free scalar field of mass $m$
the field equations together with the
causal Wick expansion
implies that
$$
\bigl(\Box_{{\mathrm{x}}_1} - m^2\bigr) \, T\bigl(\varphi ({\mathrm{x}}_1) \varphi({\mathrm{x}}_2)\bigr)
= \delta({\mathrm{x}}_1-{\mathrm{x}}_2) \neq
T\bigl(\bigl(\Box_{{\mathrm{x}}_1} - m^2\bigr) \varphi  ({\mathrm{x}}_1) \varphi({\mathrm{x}}_2)\bigr) = 0 \,.
$$

A sufficient condition for Eq.~(\ref{AWI}) would be to require
a similar additional condition on the renormalization map $R$:
\begin{equation}\label{RAWI}
R\bigl(\partial_{x_k^{\mu}}G\bigr) \, = \,
\partial_{x_k^{\mu}} R\bigl(G\bigr)
\end{equation}
($G \in {\mathrsfs{O}\hspace{-1pt}}_{{\mathbb N}}$, $k \in {\mathbb N}$, $\mu=0,\dots,D-1$).
But now this contradicts to the requirement $(r3)$
that is also equivalent to the identities
\begin{equation}\label{RX}
R\bigl(x_k^{\mu} \, G\bigr) \, = \,
x_k^{\mu} \, R\bigl(G\bigr)
\,.
\end{equation}
The reason is that the two conditions, (\ref{RAWI}) and (\ref{RX}),
would imply that the renormalization map commute with the action
of all linear partial differential operators, which is not possible.
Of course,
one can ask why not to use
condition (\ref{RX}) instead of (\ref{RAWI})?
The proof of Lemma 2.7 in \cite{N1} shows
that in the case of (\ref{RAWI})
the recursion there goes in the wrong direction
(see also Remark~\ref{Rm}).
And in fact, there are again simple obstructions for Eq.~(\ref{RAWI}):
the function $({\mathrm{x}}^2)^{-\frac{D}{2}+1}$
(on $D$--dimensional Minkowski space)
has a unique extension\footnote{%
since its scaling degree is less than $D$}
$R \Bigl(({\mathrm{x}}^2)^{-\frac{D}{2}+1}\Bigr)$ and this is the Green function for
the D'Alembert operator;
hence,
$$
\Box_{{\mathrm{x}}} \, R \Bigl(({\mathrm{x}}^2)^{-\frac{D}{2}+1}\Bigr) \, = \, \delta ({\mathrm{x}}) \, \neq \,
R \Bigl(\Box_{{\mathrm{x}}} \, ({\mathrm{x}}^2)^{-\frac{D}{2}+1}\Bigr) \, = \, 0 \,.
$$

\Section{Renormalizing perturbative Euclidean field theory}{se6.3}

The Euclidean version of our ``theory of renormalization maps''
is easier but to formulate perturbative Euclidean QFT in the spirit
of algebraic perturbative QFT on Minkowski space is more difficult.
The problem is what would replace the time--ordered products in the Euclidean case.
Of course, we can use again the formula
$T\bigl(\Theta_S\bigr)$ $=$ $R \bigl(\widehat{G}_S\bigr)$ $N\bigl(\Theta_S\bigr)$
(Eq.~(\ref{WTE1})) for a definition of the map $T$.
The problem then is that in the Euclidean case there are no
pointwise
Wick products!
In other words, $N\bigl(\Theta_S\bigr)$ do not generally exist as
densely defined operators on the Euclidean Fock space.
They can be only represented as quadratic forms.
For instance, one may see this problem even for Wick squares:
if $\varphi_E ({\mathrm{x}})$ is the free Euclidean field of mass $m$ can we define
$\NPR{\varphi_E^2} ({\mathrm{x}})$ as an operator on the Euclidean Fock space?
If so,
then
$$
\langle 0 | \NPR{\varphi_E^2} ({\mathrm{x}}) \NPR{\varphi_E^2} ({\mathrm{y}}) | 0 \rangle \, = \, G_E ({\mathrm{x}}-{\mathrm{y}})^2 \,.
$$
But in contrast to the Minkowski case $G_E ({\mathrm{x}}-{\mathrm{y}})^2$ does not exists
(while the square or product of any Wightman functions
do exit).

A possible
way-out
is to work only off-shell (\cite{S1,K}) and hope that we can construct a positive functional
on the algebra of renormalized Euclidean time--ordered products.
However, it is unlikely that the latter is possible.
The reason is that such a positive functional will extend the vacuum functional on the initial
Euclidean field algebra (which is in fact, affiliated with a von Neumann commutative algebra)
and hence, it will be possible to represent the time--ordered products
on the Euclidean space, which already exist as quadratic forms there, also as operators.

\medskip

\noindent
\textbf{Acknowledgments.}
I am grateful to Professor R. Stora for his critical remarks to my paper \cite{N1},
which stimulated the present work.
I am also grateful to Professor I. Todorov for his comments.
This work was partially supported by the French--Bulgarian project Rila
under the contract Egide -- Rila N112 and by Bulgarian NSF grant DO 02--257.

\medskip

\newcommand{\Bibitem}[1]{\bibitem[#1]{#1}}
\newcommand{\BIbitem}[2]{\bibitem[#1]{#2}}
\addtocontents{toc}{\protect\vspace{-8pt}}
\addtocontents{toc}{\contentsline {section}{References}{\arabic{page}}}

redpajama_set_name: RedPajamaArXiv


Apple has a new patent (via AppleInsider) that could help explain leaked videos showing a front screen material that is incredibly durable, and resistant to bending, scratching and shattering. The screen is thought to be based on manufactured sapphire, a material that has a toughness rating above glass, but a new patent reveals how Apple might be stacking the odds in its favor with a process to make it even stronger.
The patent describes a way to mimic the same kind of strengthening effects used on glass (through chemical coatings, mostly, as exemplified by Corning's Gorilla Glass), but with an alternate "ion implanting" method that works on material like sapphire, with which the existing methods aren't necessarily going to work.
The process uses a similar technique to the one Apple currently employs to color its black bezels on iPhones, which works via implanted ions put directly into the display glass itself. The ions, when implanted in the way Apple describes, could tint the glass, so using them in areas where the company wants specifically to create a different-colored area would make the most sense, and would still provide some crack resistance, in the case of incidents like a dropped phone (phones often suffer the worst cracks when they fall on their corners).
As we near the launch of the next iPhone, one of the big questions on everyone's mind will be whether Apple does indeed employ sapphire in its new devices to make them extra durable. We'll be on hand to find out next Tuesday, when Apple unveils at least new phones, and likely a new wearable, too.


text:
Lockdown: Thirteen card players busted by police in Hackney social club
Published: 6:30 AM January 11, 2021
Stoke Newington Road - Credit: Google
Thirteen men playing cards in a social club, in breach of the coronavirus lockdown, were busted by police on Thursday night.
Police were called to the club in Stoke Newington Road at 10pm by witnesses who spotted several men going inside.
All of them were issued with fixed penalty notice fines, and Hackney council have been advised of the owner's actions.
Ch Insp Pete Shaw said: "My officers work incredibly hard, putting themselves at risk of catching this virus to keep those in Hackney and Tower Hamlets safe.
"It is very disappointing when officers have to deal with groups of people who appear to think that these rules do not apply to them.
"To make it very clear, anyone who is seen to be breaking the rules to hold or attend a house party or an unlicensed music event can expect to be fined by the police and the event shut down.
"We need to be doing everything we can as a community to try and keep ourselves, our family and friends as safe as possible and to reduce the burden on the NHS."
Hackney News

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Dress from Coast, Scarf from M&S, Bag from Accessorize, Belt from Acessorize, Wedges from Carvela at ASOS, Valentino glasses from TK Maxx. No time for links, sorry, but you'll find most of them on my pinterest!
You look great. I love the turquoise colour with the white. hope you're having a great holiday!


Connecting an entire nation with trails. Or creating a place for the neighbourhood kids to play and explore. We plan and design spaces for sport, leisure and outdoor activity. Shaping the landscape, building features to help people enjoy it, guiding visitors on their journeys, our team makes the most of the natural world, treading lightly, letting the great outdoors speak for itself.
We are an interdisciplinary office. The work above represents the efforts of colleagues and their perspectives and methodologies. Learn more about the design disciplines that come together sharing their perspectives and skills.


Our dance workshops are ideal for all ages as a fun and engaging way to develop confidence, communication, creativity, critical thinking and improve cultural awareness.
Warm up session which is relevant to the level of curriculum or area of study or interest.
Learn a piece of dance.
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I don't know how to word this post, so I'm not going to.
We're finally selling plots, oboy!1!!
If you don't speak "paint" you go to BKP and head east for about 30 seconds. If you don't see it by then, you have not in fact been heading east.
Here's the most fantastic minecraft picture you've ever seen, of the first floor. Take some extra time to notice the 2 random chunks that refused to render.
Never mind I couldn't find one of those either, so here's an amazing picture of me designing the town hall on one side, and talking about how it's not going to fit in CV on the other side.
All the shop plots are located in and under the two large dressers and the desk on the top floor. Across from the bed in the picture above. Note, the bed is now red, so don't even TRY to find a light blue one, it's NOT THERE.
Make sure to read our beautiful town rules before buying a plot, half of which were definitely not copy and pasted from other towns and villages. You can find these inside the town hall, the giant RED bed, on the top floor.
All home plots cost 20c regardless of where they are and, unless I'm a massive nub, they should all be the same size.. actually blame blue if they're not because she did it not me.
Shop plots vary between something like 12c and 60c.. depending on the size and location of the plot, with the majority either being 12 or 30, because rule 7.
Some people wanted to buy plots like a year and a half ago, but we said no, (even though we were at practically the same point we are now) so they get first pick. Those are for Unixnubmin, Grannykaf, LissaLaine and MCG8 in that order.
How to get a plot: message me in game, on the forums or via the CV discord. If it's a forums or discord message either put a screenshot of you and the plot you want, or put the plot name. You can find those by standing in the plot and typing "/rg i" because I was way too lazy to put 7000 signs for all the different plots. Also it would be pretty funny if you messaged BlueHiero in game instead of me xD (sadly I've been informed she has no clue how to give the plots, but it would still be funny).
you know I have calmed one, right?
(11-24-2018 01:30 PM)MCG8 Wrote: you know I have calmed one, right?
It's okay, Uni won't mind otherwise Monichu will shock him again.
I have my shop plot... will have to come stock it now.
I want a home plot please... Will have to come check them out too. Maybe in a day or two.
P.S. So I have to /msg Blue... ok will do.
(12-08-2018 12:26 AM)Aohariken Wrote: Woah. Just woah.
I still have to get there Delfy. I am ticking off my TODO list now..


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Due to varying commercial establishment sizes and unique project requests, please fill out our Free Quote Form below for exact commercial pricing. First Glance Solutions will contact you with detailed pricing information, or if further information is needed.
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text:
Q: Trying to create an Object from class but don't know what to put for parameter public Viewport view;

public Camera(Viewport newView)
        {
            view = newView;
            zoom = 2;
            rotation = 0;
        }

Sorry if this seems very basic but I can't figure it out for the life of me. I want to create an Object called 'camera' in my Game class, but I don't know what I need to input as the Viewport value.
Right now, in my Game class, I only have this:
Camera camera = new Camera(//don't know what goes here);

A: Your constructor of the "Camera" class needs an "Viewport" object. Not more not less!
Viewport vp = new Viewport();
 Camera cam = new Camera(vp);
I don't know the Viewport class... Possible is also...
a)
Viewport vp = new Viewport(param1, param2,...);
b)
Viewport vp = new Viewport() { Property1 = param1, Property2 = param2, ...};
c)
Or do it instant Camera cam = new Camera(new Viewport {Property1 = param1, Property2 = param2, ...}

A: You will need to pass it graphics.GraphicsDevice.Viewport as the parameter.
This assumes you are creating the instance of Camera in Game1.cs. Which you should be doing, since this effects the entire game.
Creating a new instance is redundant, and may not fit the target device.

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text:
Tripti Home
Tripti Joshi (Editor)
I am a Teacher who started creating online content for my students from 2016 so that they can get access to free knowledge online.
Ben 10: Destroy All Aliens
Rate This6.2
Director Victor Cook
Initial DVD release March 23, 2012 (USA)
Writer Marty Isenberg
6.2/10 IMDb
Genre Animation, Action, Family
Duration 1h 9m
Country Singapore United States India
Release date March 11, 2012 (2012-03-11) (Asia) March 23, 2012 (2012-03-23) (United States)
Based on Ben 10, by Man of Action
Directors Victor Cook, Dwayne McDuffie
Cast Tara Strong (Ben Tennyson, Upgrade), Meagan Smith (Gwen Tennyson), Paul Eiding (Grandpa Max), Steven Blum (Heatblast, Retaliator), Dee Bradley Baker (Stinkfly, Wildmutt), Richard McGonagle (Fourarms)
Characters Ben Tennyson (Tara Strong), Gwen Tennyson (Meagan Smith), Heatblast (Steven Blum), Stinkfly (Dee Bradley Baker), Grandpa Max (Paul Eiding)
Similar movies Tara Strong and Paul Eiding appear in Ben 10: Destroy All Aliens and Ben 10: Secret of the Omnitrix
Ben 10: Destroy All Aliens (originally titled Ben 10: Alien Dimensions) is a computer-animated science fiction action film that premiered on Cartoon Network Asia on March 11, 2012, and in the United States on Cartoon Network March 23, 2012, as part of "Ben 10 Week" which ran from March 19, 2012 – March 24, 2012. It was officially unveiled at San Diego Comic-Con International 2011. This is Cartoon Network Asia's first film in collaboration with Cartoon Network Studios. The movie would be Cartoon Network's first CG animated movie as well as the first time the Ben 10 universe would be interpreted in full CG. Tiny Island Productions, producer of Qubo's Shelldon, provided the CGI for the film. David Kwok, CEO of Tiny Island Productions, commented that having an opportunity to work on the film was amazing and challenging. This movie took place after the first Ben 10: Secret of the Omnitrix movie. It is dedicated to Dulce Lim Chen.
Zwiastun ben 10 destroy all aliens cartoon network
Critical reviews
Nielsen ratings
Taking place shortly after the events of the original Ben 10, Ben, Gwen and Grandpa Max spend their evenings fighting aliens on the streets of Bellwood. During a battle against a robotic tank, Ben as Upgrade and Gwen argue about the best way to defeat it, leading to Gwen using her extraordinarily strong pink-colored magic to cast a dismantling spell on the tank while Ben is attached to it. Unknown to the children, this leads to a malfunction in the Omnitrix.
Later, after a mishap in school where the Omnitrix accidentally teleports his homework, Ben is grounded by his parents and forced to stay home and do a history report. Ben attempts to work on the report before the Omnitrix teleports his laptop away as well. Shortly afterwards, Tetrax Shard arrives and tells Ben that Azmuth has asked to meet with him. They go to meet Azmuth but are attacked by an evil version of Waybig. Tetrax and Ben fight the alien, who is also teleported into the Omnitrix, though neither of them is aware of it. Tetrax decides to take Ben to search for Azmuth and activates a new function on the Omnitrix. Before he can explain this new ability, the ship is attacked and Ben is sucked out into the atmosphere, falling to the ground as Diamondhead. It is later revealed that the new ability prevents Ben from returning to human form when he times out, simply switching alien forms. Ben as Fourarms arrives at Stonehenge and is attacked by a weaponizing Galvanic Mechamorph who demands to know about the alien that attacked Ben. Ben tries to explain he knows nothing, but the Mechamorph only continues to attack and the Mechamorph's teleporting ability causes the pair to carry their battle across the globe, destroying part of Stonehenge and nearly damaging Christ the Redeemer in Rio de Janeiro and fighting at the Great Pyramids in Egypt before arriving back in Bellwood. Ben transforms into Grey Matter and manages to escape the Mechamorph.
Meanwhile, Gwen and Grandpa Max receive a call from Ben's parents Carl and Sandra informing them that Ben has escaped the house. They go in search of him, only to find Tetrax and discover Azmuth's ship. Looking at the ship's log, they discover that Azmuth disguised it as a semi truck to track down and fix the malfunctioning Omnitrix, but crashed and was destroyed by the evil Waybig. After escaping the Mechamorph, they soon encounter Ben back at his house and Tetrax restores Ben's Omnitrix to normal just as the Mechamorph catches up to them. Gwen sees how the Omnitrix reacts after Ben returns to human form and deduces what has happened just as she, Ben, Tetrax and the Mechamorph are transported into the Omnitrix itself.
Inside, Ben and the Mechamorph continue to battle before being attacked by a swarm of Stinkflies just as Gwen and Tetrax are ambushed by a herd of Wildmutt. The four catch up with each other and they realize that the evil Waybig is inside with them just as he resurfaces. The Mechamorph, who is revealed to be Azmuth's father, attacks the Waybig in revenge for killing his son while Gwen explains to Ben about the malfunction that she caused to the Omnitrix when she cast the dismantling spell on the robotic tank while Ben was attached to it. She uses Azmuth's father to reverse the transportation and return them to Earth. Unfortunately, the Omnitrix malfunctions again and transforms Ben's parents into Wildmutt and Heatblast, causing Gwen to realize that the evil Waybig is Azmuth himself. Leaving Grandpa Max and Tetrax to handle his parents, Ben and Gwen follow Azmuth's father and Azmuth to the city park where Ben transforms into Way Big to subdue Azmuth and hold off his father while he and Gwen try to reverse the transformation. Between the two Waybigs, they do a few hundred million dollars in property damage from smashing huge building after huge building (there are most likely no one in the buildings as the battle takes place in the morning before people arrive to work in them, which may be why they are not concerned about huge loss of life). They soon convince Azmuth's father that his son is the very alien that they are fighting and the three work together to convince Azmuth to regain control of himself, use his father's Mechamorph abilities to fix the Omnitrix and return Azmuth to normal. Azmuth reprimands Ben and Gwen for their errors and reveals his father is a Galvan wielding Mechamorph-type enhanced armor. Azmuth reveals that the Omnitrix was so filled with pink mana; Gwen asks what he means, to which he simply says that she will know eventually.
Returning to Ben's house, Azmuth restores Ben's parents to normal and repairs the Rust bucket, previously destroyed in the battle, before leaving to reconcile with his own father. Ben's parents, having forgotten that they grounded Ben, tell him to enjoy his fishing trip with Grandpa Max and Gwen. Ben declines, having learned to be more responsible, and stays home to finish his history report for school. The movie ends with Ben, Gwen, and Grandpa Max on another road trip where they encounter Doctor Animo and the three prepare for another exciting battle.
Only one short is based on Ben 10: Destroy All Aliens and it's named as Ben 10: Road Trip Rumble.
Tara Strong as Ben Tennyson, Upgrade, Sandra Tennyson, Computer Voice
Meagan Smith as Gwen Tennyson, Girl Student
Paul Eiding as Grandpa Max
Steven Blum as Heatblast, Mechamorph
David Fennoy as Tetrax Shard
Richard McGonagle as Four Arms, Teacher
Robert David Hall as Azmuth
Dee Bradley Baker as Stinkfly, Wildmutt, Cash Murray, Carl Tennyson, Vulpimancers, Lepidopterrans, Mutant Frog
Adam Wylie as JT
Richard Steven Horvitz as Grey Matter, Police Radio, Dispatcher
Jim Ward as Diamondhead
Fred Tatasciore as Way Big, Evil Way Big
Troy Baker as Azmuth's Father, Boy Student
Dwight Schultz as Dr. Animo
Nina Teo as Ben's Girl classmate
Xander Ching as Ben's Boy classmate
Victor Cook - Director, Supervising producer
Silas Hickey - Executive producer
Marty Isenberg - Writer
Collette Sunderman - Casting and Voice Director
Turner Broadcasting System Asia Pacific - Creative animation director
To promote the film, a toy line manufactured by Bandai has been announced to launch around the same time the film premieres. Cartoon Network Asia and Philippines held a contest for one male and one female to be "immortalized in Ben 10 history." The two winners would receive "the ultimate voice over role," assorted prizes (clothing, toys, etc.) and a VIP limo ride to walk on the "green carpet" at the Ben 10: Destroy All Aliens premiere. A similar competition was held in Malaysia and the Middle East. Filipino viewers heard the Philippine winners while the rest of Southeast Asia will hear the Malaysian kids. The winners of the contest in Philippines were later revealed to be Nina Teo and Xander Ching who are voicing two of Ben's classmates. Cartoon Network Africa held a contest to win Green Movie Tickets to the exclusive premiere of the film in Johannesburg on March 10, 2012.
The film was later released on iTunes and other electronic-based stores, such as the PlayStation Store.
Madman Entertainment released Destroy All Aliens in Region 4 on June 20, 2012. Warner Home Video released the film in Region 2 on October 1, 2012. The film was also released in Region 1 on April 16, 2013.
The film received positive reviews, as it currently holds a 83% rating on Rotten Tomatoes based on audience reviews. Emily Ashby of Commonsense Media gave Ben 10: Destroy All Aliens a 3 out of 5. Although criticizing on how much violence exists in the context of the story Ashby states that the story emphasizes the joy of family, even when that comes with rules a person does not like.
In the United States, Ben 10: Destroy All Aliens increased by double-digits across the board versus the same time period in 2011. The broadcast placed #1 in its time period in boys 2-11.
The film won the Best 3D Animated Program category at the Asian Television Awards 2012 as well as the Gold Award for Best Movie Campaign in the 2012 ProMax Awards. The award demonstrated the high level of animation talent on show in Asia and the talent available here, commented Sunny Saha from Turner International.
Ben 10: Destroy All Aliens Wikipedia
Ben 10: Destroy All Aliens IMDb Ben 10: Destroy All Aliens themoviedb.org
Similar TopicsDee Bradley Baker
Dwayne McDuffie
Marty Isenberg


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Some friends of mine live in a mobile home park that is owned by the resident owners as a co-op. This could be a solution to affordable home ownership since even condominiums in the Bellingham area are usually well over $100,000. Buying a share in their mobile home park only costs $17,500. That's almost like owning the land under your home, but not quite. It's a share in the cooperatively owned park which allows one a space for their home, access to the central clubhouse and other benefits.
Each month members pay dues as well, but the dues are around $160 per month. That's quite reasonable compared to rent in most mobile home parks, let alone renting an apartment. The dues cover ongoing expenses such as water and sewer as well each resident's share of the property taxes that the park pays on all the land.
One also buys their mobile home unit which can be as low as $15,000. Most of the mobile homes are more than $15,000, but $25,000 - $60,000 is a likely range.
When one thinks about it, this is an affordable way to have home ownership, if ownership is desired. For instance, a mobile home for around $20,000 plus the park "buy in" share for $17,500 means buying a house for $37,500. For Bellingham, that's not bad.
Another affordable housing option in the Bellingham area is an organization called Kulshan Community Land Trust. They have some partial ownership schemes also, but I've noticed that the cheapest condominium, currently in their listing, is around $116,000!
That's Bellingham for you. They also work with owners who are buying houses, but these are usually even more expensive, tho I saw a house listed in their office window for $104,000.
Looks like mobile homes are less expensive, but Kulshan doesn't deal with those kind of properties. Mostly, I hear, because financing institutions basically overlook the mobile home option.
With lack of affordable housing such a big problem, I think mobile homes need to be taken more seriously. Problem is, urban zoning doesn't usually allow them. The park that my friends live in is near the city limits. It's still on the bus line tho. Mobile home parks aren't as dense as condominiums and density is good for environmental reasons, but they do tend to be denser than single family zones.
Pictured: One of my friend's neighbors has raised beds for gardening.
The park that my friends live in is quite nice with a small lake in the center, quite a bit of space between the units and a nice clubhouse.
Speaking of clubhouse, that brings up the topic of "community." People I know, that live in mobile home parks, seem to have lots of community. Talking to folks around the mailboxes, going to "Meet your neighbor" meals at the clubhouse and so forth.
My friends, who live in that park, think about it in terms of "intentional communities." They have interest in things like home gardening and sustainable communities. Mobile homes might be a good way to bring this about that is not super expensive.
But they are a good form of housing for mobile American society. Average American moves every 5 1/2 years? (maybe even more). We're nomadic. Maybe our houses need to move with us; like American Indian teepees. It's better than commuting, for instance if one owns a house in Everett, works for Boeing in Everett and then gets transferred down to the Renton Boeing facility.
Also mobile homes might do well for answering the question, "should New Orleans be rebuilt?" Make it mobile. Let it move around with the shifting wetlands. Put it where the changing river and coastline wants it, rather than locking it into one spot. Flexibility.
Mobiles might adapt better to floating also. A while back, I heard, on the Clark Howard Show (a radio financial affairs counsellor), a suggestion of making New Orleans homes float. This might solve much of the flooding problems. Radio said they do this in much of Holland. When the 50 year flood comes, the houses just float. Minimizes the damage.


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Reel Review: The Invisible Man (2020)
Tags: elisabeth moss leigh wannell oliver Jackson cohen The Invisible Man the invisible man 2020
Leigh Whannell's "The Invisible Man" is a must see, raw and cathartic cinematic experience that peers into the heart of female terror.
Whannell brilliantly breathes new life into this timely and pertinent re-adaptation of The Invisible Man that tackles many tough to discuss real-life issues with deft and sensitivity.
Leigh Whannell comes through in every way possible in his re-imagining of the Universal classic The Invisible Man. His full brilliance as not only a director but a writer is on display in the film that will surely become a horror classic in its own right.
The 2020 version of The Invisible Man is not an easy watch, as it deals with some extremely heavy and triggering subject matter, but it is a rewarding watch.
The film begins with Cecilia Kass, played by Mad Men and The Handmaid's Tale star Elisabeth Moss, as she escapes from the clutches of her abusive partner, Adrian Griffin. Adrian, played by The Haunting of Hill House's Oliver Jackson-Cohen, is essentially a genius and extremely wealthy as a result. Cecilia desperately flees in the night in order to attempt to start anew. She succeeds in getting away but not for long.
A couple of weeks after her liberation from Adrian, she learns that he has committed suicide and has left her five million dollars. This looks like it might be a fresh start for Cecilia — a reason for her to celebrate and a way for her to have an unfettered life where every aspect of her day to day isn't dictated by the controlling Adrian.
Her brief happiness is short lived, however, as she starts being stalked by a very much alive and invisible Adrian, who will absolutely stop at nothing to make Cecilia pay for leaving him.
In the midst of the terror of being pursued by an assailant that she cannot see, those around her start to doubt what she says and believe that her sanity is slipping after years of abuse at Adrian's hands.
Every actor in The Invisible Man is at the top of their respective game.
It's a stellar ensemble cast that is led by a gutsy Moss. Moss brings her trademark empathetic ability to the role of Cecilia, making the film an emotionally raw and involving experience. Cecilia's trauma is palpable. Every one of her actions is colored by her past with Adrian, as she tries desperately to shake off any influence he's had in her life. She is relearning how to be herself without him because he had such a vicious hold on her. Moss portrays Cecilia with the love and respect that the character deserves.
Opposite Moss's Cecilia is Jackson-Cohen's absolutely and deliciously despicable Adrian. For a character that the audience doesn't physically see for much of the movie, Cohen-Jackson capably brings style and flair to the intelligent megalomaniac. It takes a lot of talent to play someone that is so hateful and unsympathetic to their core, and he does so perfectly. It's a lovely turn after playing sweet and melancholic Luke in Mike Flanagan's The Haunting of Hill House, and it shows the depth of his range.
Supporting Moss and Jackson-Cohen are Aldis Hodge as James Lanier, a detective and friend of Cecilia's, Storm Reid as Sydney Lanier, James's teenage daughter, Harriet Dyer as Emily Kass, Cecilia's sister, and Michael Dorman as Tom Griffin, Adrian's lawyer brother.
Hodge and Reid are easily stand outs in the supporting cast and bring warmth and life to the film, acting as a safe harbor and support system for Cecilia. Hodge, Moss, and Reid have wonderful chemistry, which makes their scenes together a joy to watch. Sixteen-year-old Reid has quite the career ahead of her, and this film is another bit of proof of that. As a whole, the cast is amazingly solid, and there's not a weak link to be found.
Whannell has upgraded, so to speak, the idea of the invisible man himself, showing the audience that some of the most fearsome monsters are in reality human beings.
The film's commentary is not subtle because Whannell has drawn heavily on the real life struggles that women face in abusive relationships. He sheds light on the uncomfortable facets of abuse that aren't discussed as widely as they should be and thus creates an absolutely breathless and engrossing viewing experience. Whannell's vision is a bold one that doesn't shy away.
The Invisible Man puts gaslighting under a microscope and creates a modern allegory that is desperately needed.
Those who feared that the trailer shows most of the film can rest assured that nothing will be able to prepare them for the viewing experience that is The Invisible Man. Whannell doesn't show his full hand, and the film is better for it.
This is a masterclass is slow burn horror and tension.
Every twist is planned with care, and with each twist the tension is continually ratcheted up. There are many surprises in this film, and even if the audience manages to guess one, there's at least two more that will certainly stagger them.
The film's technical elements come together beautifully to tell this consuming story. The artful cinematography lends to the tautness and the terror that takes place on screen, which is paired with impressive special effects. The slick editing creates for a smooth viewing experience, paired beautifully with the blood pumping and triumphant score.
A lot of care went into crafting THE INVISIBLE MAN from top to bottom, and that should not go without mentioning.
It's a film that excels across the board, a rare viewing experience that deserves to be cherished and meditated upon. Missing out on this masterful and cathartic experience would be a shame, especially for those who are fans of horror and Whannell.
The Invisible Man is horror at its best. It is not one to be missed — and definitely isn't one to be forgotten.


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Lautodromo di Most è un circuito automobilistico situato vicino alla città di Most, nel nord-ovest della Repubblica Ceca.

Inaugurato nel 1983, ha ospitato eventi quali l'European Touring Car Cup, il FIA European Truck Racing Cup e l'European Le Mans Series; dal 2021 ospita inoltre il campionato mondiale Superbike.

Galleria d'immagini

Note

Altri progetti

Collegamenti esterni

redpajama_set_name: RedPajamaWikipedia


Forte Sales, Lettings & Property Management - WHAT NEXT FOR PROPERTY?
So, the decision is made. But how will this affect the property market and more importantly how will it affect Exeter and the surrounding area? Of course, the West Country is a very desirable area in which to live and work which will probably mean it is less affected compared to the rest of the UK. There is still a shortage of housing and interest rates remain low. Buyers are still registering their interest.
Demand for rental properties is still high and of course the University will bring its usual influx of students looking for quality accommodation in and around the city centre.
Our rental stock changes regularly as properties are let and our management portfolio is increasing as more and more Landlords recognise that a quality management service saves them time and money.
We watch and wait for the next step forward.