text:
\global\long\def\fps#1#2{#1\![[#2]]}

\global\long\def\poisson#1#2{\left\lfloor #1,#2\right\rceil }

\global\long\def\bld#1{\boldsymbol{#1}}

\begin{fmffile}{DimRegPos01}

\def\FGH{\parbox{10mm}{
\begin{center}
\begin{fmfgraph}(20,15)
\fmfbottom{g,h} \fmftop{f}
\fmfdot{f,g,h}
\end{fmfgraph}
\end{center}}}

\def\FGoneHone{\parbox{10mm}{
\begin{center}
\begin{fmfgraph}(20,15)
\fmfbottom{g,h} \fmftop{f}
\fmfdot{f,g,h}
\fmf{plain}{g,h}
\end{fmfgraph}
\end{center}}}

\def\FoneGHone{\parbox{10mm}{
\begin{center}
\begin{fmfgraph}(20,15)
\fmfbottom{g,h} \fmftop{f}
\fmfdot{f,g,h}
\fmf{plain}{f,h}
\end{fmfgraph}
\end{center}}}

\def\FoneGoneH{\parbox{10mm}{
\begin{center}
\begin{fmfgraph}(20,15)
\fmfbottom{g,h} \fmftop{f}
\fmfdot{f,g,h}
\fmf{plain}{f,g}
\end{fmfgraph}
\end{center}}}

\def\FGtwoHtwo{\parbox{10mm}{
\begin{center}
\begin{fmfgraph}(20,15)
\fmfbottom{g,h} \fmftop{f}
\fmfdot{f,g,h}
\fmf{plain,right=.5}{g,h}
\fmf{plain,left=.5}{g,h}
\end{fmfgraph}
\end{center}}}

\def\FoneGoneHtwo{\parbox{10mm}{
\begin{center}
\begin{fmfgraph}(20,15)
\fmfbottom{g,h} \fmftop{f}
\fmfdot{f,g,h}
\fmf{plain}{f,h}
\fmf{plain}{g,h}
\end{fmfgraph}
\end{center}}}

\def\FoneGoneoneHone{\parbox{10mm}{
\begin{center}
\begin{fmfgraph}(20,15)
\fmfbottom{g,h} \fmftop{f}
\fmfdot{f,g,h}
\fmf{plain}{f,g}
\fmf{plain}{g,h}
\end{fmfgraph}
\end{center}}}

\def\FtwoGHtwo{\parbox{10mm}{
\begin{center}
\begin{fmfgraph}(20,15)
\fmfbottom{g,h} \fmftop{f}
\fmfdot{f,g,h}
\fmf{plain,right=.5}{f,h}
\fmf{plain,left=.5}{f,h}
\end{fmfgraph}
\end{center}}}

\def\FtwoGoneHone{\parbox{10mm}{
\begin{center}
\begin{fmfgraph}(20,15)
\fmfbottom{g,h} \fmftop{f}
\fmfdot{f,g,h}
\fmf{plain}{f,g}
\fmf{plain}{f,h}
\end{fmfgraph}
\end{center}}}

\def\FtwoGtwoH{\parbox{10mm}{
\begin{center}
\begin{fmfgraph}(20,15)
\fmfbottom{g,h} \fmftop{f}
\fmfdot{f,g,h}
\fmf{plain,right=.5}{f,g}
\fmf{plain,left=.5}{f,g}
\end{fmfgraph}
\end{center}}}

\def\FthreeGoneHtwoF{\parbox{35mm}{
\begin{center}
\begin{fmfgraph*}(40,30)
\fmfbottom{g,h} \fmftop{f}
\fmfdot{f,g,h}
\fmflabel{$F^{(5)}$}{f}
\fmflabel{$G^{(4)}$}{g}
\fmflabel{$H^{(3)}$}{h}
\fmf{plain,right=.3}{f,g}
\fmf{plain}{f,g}
\fmf{plain,left=.3}{f,g}
\fmf{plain,right=.3}{g,h}
\fmf{plain,right=.3}{h,f}
\fmf{plain,left=.3}{h,f}
\end{fmfgraph*}
\end{center}}}

\def\LineX{\parbox{15mm}{
\begin{center}
\begin{fmfgraph*}(30,20)
\fmfleft{v} \fmfright{w}
\fmfdot{v,w}
\fmf{plain,label=$e$, l.side=left, l.dist=1mm}{v,w}
\fmfv{label=$v$,l.angle=-90}{v}
\fmfv{label=$w$,l.angle=-90}{w}
\end{fmfgraph*}
\end{center}}}

\chapter{\label{cha:DimReg-PositionSpace}Dimensional Regularization in Position
Space}

The work of Bollini and Giambiagi on dimensional regularization in
position space, mentioned previously, focused on the Fourier transform
of this regularization method between momentum space and position
space \citep{BolliniGiambiagi1996}. In contrast to their work, our
analysis will be formulated exclusively in position space, and a direct
translation to momentum space will generally not be possible. However,
an advantage of the method presented here is that all expressions
will depend smoothly on the mass parameter $m^{2}$, which makes it
possible to apply the covariant framework of \citep{Hollands2004},
see also \citep{DuetschFredenhagen2004}. We will analyze the graph
structure of the time-ordered product in the first section and use
this in the second section to construct for any graph a unique dimensionally
regularized amplitude. We will define the dimensionally regularized
time-ordered product and the corresponding scattering matrix. This
dimensionally regularized $\Sm$-matrix will then be used as an example
in the solution of the Epstein-Glaser recursion, and to establish
the relation to Connes-Kreimer theory of renormalization in the last
chapter.

\section{\label{sec:GraphStructureT}Graph structure of the Time-Ordered Product}

Before turning to the time-ordered product we want to introduce very
briefly the basic notions connected with the definition of a graph.
An oriented graph $\Gamma$ is a set of vertices $V(\Gamma)$ and
a set of edges $E(\Gamma)$ together with maps \[
\source,\target:E(\Gamma)\rightarrow V(\Gamma)\,,\]
which give source and target of an edge $e\in E(\Gamma)$, respectively.
Furthermore we give $\Gamma$ an \emph{orientation} by assigning to
any pair $\left(e,v\right)\in E(\Gamma)\times V(\Gamma)$ the value\[
\left(e:v\right):=\left\{ \begin{array}{cl}
+1 & \mbox{if }\target(e)=v\\
-1 & \mbox{if }\source(e)=v\\
0 & \mbox{otherwise.}\end{array}\right.\]
We call $e$ \emph{adjacent} \emph{to} $v$ if $\left(e:v\right)\neq0$.
A graph for which the orientation map $\left(e,v\right)\mapsto\left(e:v\right)$
is multi-valued we call \emph{tadpole}. However, the definition of
the time-ordered product in (\ref{eq:TimeOrderedProduct}) implies
that there are no tadpoles occurring in its graph expansion, i.e.,
Equation~(\ref{eq:TimeOrderedGraphStructure}) below; see the proof
of Proposition~\ref{pro:PartialAlgebraTimeProd} and the preceding
Remark~\ref{rem:TimeStar-and-Tadpoles}. In particular this implies
that we will only need to consider graphs for which\begin{equation}
\forall e\in E(\Gamma):\quad\source(e)\neq\target(e).\label{eq:NoTadpoleCondition}\end{equation}
Furthermore we remark that for scalar QFT the orientation of a given
(Feynman-) graph can be chosen freely, one speaks of an \emph{unoriented
graph} in this case. Let $\mathcal{G}$ denote the set of all unoriented
graphs $\Gamma$ for which the orientation map $\left(e,v\right)\mapsto\left(e:v\right)$
is single-valued, i.e. (\ref{eq:NoTadpoleCondition}) holds.

Consider the $n$-fold time-ordered product introduced in Section~\ref{sec:The-Renormalization-Problem}
as a map\[
\begin{array}{rccl}
\Time_{n}: & \fps{\mathcal{F}_{\loc}(\M)}{\hbar}^{\otimes n} & \rightarrow & \fps{\mathcal{F}(\M)}{\hbar}\\
 & F_{1}\otimes\cdots\otimes F_{n} & \mapsto & F_{1}\dT\cdots\dT F_{n}\,.\end{array}\]
It was defined with the help of a second order, symmetric functional
differential operator (\ref{eq:BiDifferentialTProduct}), which can
be written as \[
\Gamma_{H_{F}}(F\cdot G)=\left\langle H_{F},F^{\left(1\right)}\otimes G^{\left(1\right)}\right\rangle \,,\]
due to the absence of tadpoles. On the level of graphs this operation
is represented by drawing one line, $H_{F}$, between the interaction
vertices, $F$ and $G$. We can split the time-ordered product, $\Time_{n}$,
in a similar way into two parts \citep{FredenhagenGoettingen2009};
a differential operator, \[
\bld{\delta}^{\alpha}:F_{1}\otimes\cdots\otimes F_{n}\mapsto F_{1}^{\left(\alpha_{1}\right)}\otimes\cdots\otimes F_{n}^{\left(\alpha_{n}\right)}\,,\quad\alpha\in\mathbb{N}^{n}\,,\]
where $\alpha_{i}$ is the number of lines adjacent to the vertex
with interaction $F_{i}$,
\footnote{Since we do not want to restrict ourselves to any particular type
of interaction, the number of edges adjacent to a given vertex is
not fixed a priori.
} and a distribution,\[
S_{\Gamma}:F_{1}^{\left(\alpha_{1}\right)}\otimes\cdots\otimes F_{n}^{\left(\alpha_{n}\right)}\mapsto\left\langle S_{\Gamma},F_{1}^{\left(\alpha_{1}\right)}\otimes\cdots\otimes F_{n}^{\left(\alpha_{n}\right)}\right\rangle ,\]
containing the information as to which vertices of the graph $\Gamma$
are connected by a line. The $n$-fold time-ordered product can then
be written as\begin{equation}
F_{1}\dT\cdots\dT F_{n}=\sum_{\alpha\in\mathbb{N}^{n}}\sum_{\Gamma\in\mathcal{G}_{\alpha}}\frac{\hbar^{\left|E(\Gamma)\right|}}{\Sym(\Gamma)}\left\langle S_{\Gamma},\bld{\delta}^{\alpha}\left(F_{1}\otimes\cdots\otimes F_{n}\right)\right\rangle \,,\label{eq:TimeOrderedGraphStructure}\end{equation}
where $\mathcal{G}_{\alpha}$ is the set of (non-tadpole) graphs with
$n=\dim(\alpha)$ vertices and $\tfrac{\left|\alpha\right|}{2}$ lines
such that there are $\alpha_{i}$ lines joining at vertex $i$. $\Sym(\Gamma)\in\mathbb{N}$
is the symmetry factor of the graph $\Gamma$ to be defined below.
Observe that in the case of polynomial interactions, e.g., $F_{i}(\varphi)=\left\langle \varphi^{k_{i}},f\right\rangle $,
$i\in\left\{ 1,\dots,n\right\} $, and fixed $n\in\mathbb{N}$ only
finitely many of the functional derivatives $\bld{\delta}^{\alpha}$
give non-vanishing contributions to (\ref{eq:TimeOrderedGraphStructure}).
For arbitrary interactions the limiting parameter is the order in
$\hbar$ up to which one wants to compute. One can generate a dependence
on the loop number $\ell(\Gamma)$ for connected graphs $\Gamma$
by absorbing one factor $\hbar$ in each interaction functional,\[
\ell(\Gamma)=\left|E(\Gamma)\right|-\left|V(\Gamma)\right|+1\,,\]
a well known identity from graph theory \citep{GrossYellen2003}.
\begin{example}
As an example regard the threefold time-ordered product of (not necessarily
local) functionals $F,G,H\in\fps{\mathcal{F}(\M)}{\hbar}$. Using
Cauchy's product formula and the Leibniz rule one derives from the
power series expansion (\ref{eq:TimeOrderedProduct}) the following
expression for the threefold time-ordered product \begin{equation}
F\dT G\dT H=\sum_{n=0}^{\infty}\frac{\hbar^{n}}{n!}\sum_{m=0}^{n}\sum_{k=0}^{m}{n \choose m}{m \choose k}\left\langle F^{\left(k+m-k\right)}G_{\left(k\right)}^{\left(n-m\right)}H_{\left(n-m\right)\left(m-k\right)}\right\rangle \,,\label{eq:ExampleThreefoldProduct-1}\end{equation}
where we used the abbreviation $G_{\left(k\right)}:=H_{F}^{\otimes k}*G^{\left(k\right)}$,
cf.~\citep{Keller2009}. The first terms of this expansion are given
by\begin{align*}
F\dT G\dT H & =FGH+\hbar\left\{ \left\langle F^{\left(1\right)}G_{\left(1\right)}H\right\rangle +\left\langle F^{\left(1\right)}GH_{\left(1\right)}\right\rangle +\left\langle FG^{\left(1\right)}H_{\left(1\right)}\right\rangle \right\} \\
 & \quad+\hbar^{2}\left\{ \tfrac{1}{2}\left\langle F^{\left(2\right)}G_{\left(2\right)}H\right\rangle +\left\langle F^{\left(2\right)}G_{\left(1\right)}H_{\left(1\right)}\right\rangle +\tfrac{1}{2}\left\langle F^{\left(2\right)}GH_{\left(2\right)}\right\rangle \right.\\
 & \qquad\qquad\qquad\left.+\left\langle F^{\left(1\right)}G^{\left(1\right)}H_{\left(1\right)\left(1\right)}\right\rangle +\tfrac{1}{2}\left\langle FG^{\left(2\right)}H_{\left(2\right)}\right\rangle +\left\langle F^{\left(1\right)}G_{\left(1\right)}^{\left(1\right)}H_{\left(1\right)}\right\rangle \right\} \\
 & \quad+\cdots\end{align*}
\begin{align*}
\phantom{F\cdot_{T}G\cdot_{T}H} & =\FGH+\hbar\left\{ \FoneGoneH+\FoneGHone+\FGoneHone\right\} \\
 & \quad+\hbar^{2}\left\{ \frac{1}{2}\FtwoGtwoH+\FtwoGoneHone+\frac{1}{2}\FtwoGHtwo+\FoneGoneHtwo+\frac{1}{2}\FGtwoHtwo+\FoneGoneoneHone\right\} \\
 & \quad+\cdots\end{align*}
Observe that the prefactor of each graph is given by its symmetry
factor, \linebreak ${\Sym(\Gamma)^{-1}=\frac{1}{n!}{n \choose m}{m \choose k}=\frac{1}{\left(n-m\right)!k!\left(m-k\right)!}}$,
where for a general graph, $\Gamma\in\mathcal{G}$, $\Sym(\Gamma)$
is the product of the number of possible permutations of edges which
join the same two vertices in $\Gamma$.

The terms in (\ref{eq:ExampleThreefoldProduct-1}) can equivalently
be expressed as a composition of the maps $S_{\Gamma}$ and $\bld{\delta}^{\alpha}$
above,\[
\xymatrix{F\otimes G\otimes H\ar[d]^{\bld{\delta}^{\alpha}}\\
F^{\left(m\right)}\otimes G^{\left(n-m+k\right)}\otimes H^{\left(n-k\right)}\ar[d]^{\frac{1}{\Sym(\Gamma)}S_{\Gamma}}\\
\frac{1}{n!}{n \choose m}{m \choose k}\left\langle F^{\left(k+m-k\right)}G_{\left(k\right)}^{\left(n-m\right)}H_{\left(n-m\right)\left(m-k\right)}\right\rangle \,,}
\]
hence we can write (\ref{eq:ExampleThreefoldProduct-1}) equivalently
as graph expansion,\[
F\dT G\dT H=\sum_{\Gamma\in\mathcal{G}}\frac{\hbar^{\left|E(\Gamma)\right|}}{\Sym(\Gamma)}\left\langle S_{\Gamma},\bld{\delta}^{\alpha}(F\otimes G\otimes H)\right\rangle \,,\quad\alpha\in\mathbb{N}^{3},\quad\left|\alpha\right|=2\left|E(\Gamma)\right|\,.\]
We will properly define these maps, $\bld{\delta}^{\alpha}$ and $S_{\Gamma}$,
in the sequel.
\end{example}
In the case of local functionals $F_{v}\in\fps{\mathcal{F}_{\loc}(\M)}{\hbar}$
we have that the functional derivative can be written in the form,
cf. Eq.~(\ref{eq:LocFuncKernelArbSmallSupp}), \[
F_{v}^{\left(\alpha_{v}\right)}(\varphi)(x_{1},\dots,x_{\alpha_{v}})=\sum_{k}\sum_{i}f_{\varphi}^{v,k,i}(x_{v})\, P_{k}(\partial_{\bld r_{v}})\delta(\bld r_{v})\in\D(\M)\otimes\E_{\mathrm{Dirac}}'(\M^{\alpha_{i}-1})\,,\]
where $x_{v}=\frac{1}{\alpha_{v}}\sum_{k=1}^{\alpha_{v}}x_{k}$ is
the center of mass coordinate and $\bld r_{v}$ are relative coordinates
at vertex $v\in V(\Gamma)$. $P_{k}$ are homogeneous polynomials
of order $k$ in $\alpha_{v}-1$ variables, $\E_{\mathrm{Dirac}}'$
denotes the space of distributions supported at zero, and $\supp(f_{\varphi}^{v,k,i})$
can be chosen arbitrarily small. We want to introduce the short hand
notation\[
\D_{\loc}(\M^{\alpha_{v}}):=\D(\M)\otimes\E_{\mathrm{Dirac}}'(\M^{\alpha_{v}-1})\]
for the space of the $\alpha_{v}$th functional derivative of a local
functional, $F^{\left(\alpha_{v}\right)}(\varphi)\in\D_{\loc}(\M^{\alpha_{v}})$
and call \[
\begin{array}{rccl}
\bld{\delta}^{\alpha}\big|_{\varphi}: & \fps{\mathcal{F}_{\loc}(\M)}{\hbar}^{\otimes\left|V(\Gamma)\right|} & \rightarrow & \bigotimes_{v\in V(\Gamma)}\D_{\loc}(\M^{\alpha_{v}})\\
 & F_{1}\otimes\cdots\otimes F_{n} & \mapsto & F_{1}^{\left(\alpha_{1}\right)}(\varphi)\otimes\cdots\otimes F_{n}^{\left(\alpha_{n}\right)}(\varphi)\quad,\, n=\left|V(\Gamma)\right|\end{array}\]
the \emph{adjacency differential operator}.

While the definition of $\bld{\delta}^{\alpha}$ can be done purely
algebraically, the construction of the distribution $S_{\Gamma}$
on the other hand involves renormalization, i.e., an extension procedure
for distributions. We start from the tensor power\begin{equation}
\widetilde{S_{\Gamma}}=\bigotimes_{e\in E(\Gamma)}H_{F}(e)\,,\label{eq:TensorTProduct}\end{equation}
containing one factor $H_{F}\in\D'(\M\backslash\left\{ 0\right\} )$
for every edge $e$ in $\Gamma$. Hence $\widetilde{S_{\Gamma}}$
is a well-defined distribution in $\D'((\M\backslash\left\{ 0\right\} )^{\left|E(\Gamma)\right|})$
that can be uniquely extended to \linebreak[4]$\D'(\M^{\left|E(\Gamma)\right|})$,
since the Feynman fundamental solution $H_{F}\in\D'(\M\backslash\left\{ 0\right\} )$
has a unique extension $\dot{H}_{F}\in\D'(\M)$ with the same scaling
degree.

The renormalization problem is now to find a restriction $S_{\Gamma}$
of the tensor distribution $\widetilde{S_{\Gamma}}$ to the space
\[
\bigotimes_{v\in V(\Gamma)}\D_{\loc}(\M^{\alpha_{v}})=\D(\M^{\left|V(\Gamma)\right|})\otimes\bigotimes_{v\in V(\Gamma)}\E_{\mathrm{Dirac}}'(\M^{\alpha_{v}-1})\,.\]
The space $\E_{\mathrm{Dirac}}'$ is spanned by the $\delta$-distribution
and its derivatives \linebreak[4] \citep[Thm.~2.3.4]{Hoermander2003},
thus the tensor product \[
\V=\bigotimes_{v\in V(\Gamma)}\E_{\mathrm{Dirac}}'(\M^{\alpha_{v}-1})\]
is graded by the number of derivatives in front of the $\delta$-distributions.\[
\V=\bigoplus_{\left|\vec{k}\right|}\V_{\left|\vec{k}\right|}\,,\quad\left|\vec{k}\right|=\sum_{v\in V(\Gamma)}k_{v}\,.\]
Regard the application of $\widetilde{S_{\Gamma}}$ to the image of
$\bld{\delta}^{\alpha}\big|_{\varphi}$ ,\begin{align*}
\left\langle \widetilde{S_{\Gamma}},\bigotimes_{v\in V(\Gamma)}F_{v}^{\left(\alpha_{v}\right)}\right\rangle  & =\left\langle \widetilde{S_{\Gamma}},\bigotimes_{v\in V(\Gamma)}\sum_{k_{v}}\sum_{I_{v}}f_{\varphi}^{v,k_{v},I_{v}}\, P_{k_{v}}(\partial_{\bld r_{v}})\delta(\bld r_{v})\right\rangle \\
 & =\left\langle \widetilde{S_{\Gamma}},P_{\vec{k}}(\partial_{\vec{\bld r}})\bigotimes_{v\in V(\Gamma)}\sum_{k_{v}}\sum_{I_{v}}f_{\varphi}^{v,k_{v},I_{v}}\,\delta(\bld r_{v})\right\rangle \end{align*}
where $\vec{k}=\left(k_{v}\right)_{v\in V(\Gamma)}$ and $\vec{\bld r}=\left(\bld r_{v}\right)_{v\in V(\Gamma)}$.
We dualize the application of $P_{\vec{k}}(\partial_{\vec{\bld r}})$
and get\[
\left\langle \widetilde{S_{\Gamma}},\bigotimes_{v\in V(\Gamma)}F_{v}^{\left(\alpha_{v}\right)}\right\rangle =\left\langle \widetilde{S_{\left(\Gamma,\vec{k}\right)}}\,,\bigotimes_{v\in V(\Gamma)}\sum_{k_{v}}\sum_{I_{v}}f_{\varphi}^{v,k_{v},I_{v}}\,\delta(\bld r_{v})\right\rangle \]
where according to Lemma~\ref{lem:ScalingDegreeProperties} this
will increase the scaling degree of the distribution by $\left|\vec{k}\right|$,\begin{equation}
\sd(\widetilde{S_{\left(\Gamma,\vec{k}\right)}})=\sd(\widetilde{S_{\Gamma}})+\left|\vec{k}\right|\,.\label{eq:ScalingDegreeWithExternalStructure}\end{equation}
The multiindex $\vec{k}$ thus encodes the derivative couplings (i.e.,
the interaction functionals containing derivatives of the fields)
in the graph $\Gamma$. In the framework of Connes-Kreimer Hopf algebras,
or Feynman graphs in general, $\vec{k}$ sometimes is called the {}``external
structure of the graph'', see \citep{ConnesMarcolli2007} for instance.

The remaining restriction of $\widetilde{S_{\left(\Gamma,\vec{k}\right)}}$
to a distribution in $\D'(\M^{\left|V(\Gamma)\right|})$ can conveniently
be described by the (simplicial) cohomology of the graph $\Gamma$.
For ease of presentation, we will forget about the external structure
$\vec{k}$ for the time being. The algebraic structure to be presented
below can be developed to a large extend without recourse to the external
structure. We will reintroduce $\vec{k}$ by replacing $\Gamma\mapsto\left(\Gamma,\vec{k}\right)$,
where we find it to be relevant for the understanding.

\subsection{\label{sub:SimplicialCohomology}Simplicial cohomology of a graph
and choice of relative coordinates}

The presentation in this subsection is very much inspired by \citep[Sec.~2.1]{BergbauerBrunettiKreimer2009}.
Let $\mathbb{K}\in\left\{ \mathbb{R},\mathbb{C}\right\} $ be a field.
We define the (simplicial) cohomology $H^{1}(\Gamma,\mathbb{K})$
with coefficients in $\mathbb{K}$ of a connected graph $\Gamma\in\mathcal{G}$
by the exact sequence\begin{equation}
\xymatrix{0\ar[r] & \mathbb{K}\ar@{^{(}->}[r]^{\cc\quad} & \mathbb{K}^{\left|V(\Gamma)\right|}\ar@{->}[r]^{\dc} & \mathbb{K}^{\left|E(\Gamma)\right|}\ar@{->>}[r]^{\oc} & H^{1}(\Gamma,\mathbb{K})\ar[r] & 0}
\,.\label{eq:SimplicialCohomology}\end{equation}
Let $\left\{ a_{v}:v\in V(\Gamma)\right\} $ be a basis of $\mathbb{K}^{\left|V(\Gamma)\right|}$
and $\left\{ b_{e}:e\in E(\Gamma)\right\} $ a basis of $\mathbb{K}^{\left|E(\Gamma)\right|}$.
The maps in (\ref{eq:SimplicialCohomology}) are then defined as the
{}``center of mass'', \[
\cc:x\mapsto x\sum_{v\in V(\Gamma)}a_{v}\,,\]
and \[
\dc:a_{v}\mapsto\sum_{e\in E(\Gamma)}\left(e:v\right)b_{e}\,.\]
One immediately checks that\[
\forall x\in\mathbb{K}:\quad\left(\dc\circ\cc\right)(x)=x\sum_{e\in E(\Gamma)}\sum_{v\in V(\Gamma)}\left(e:v\right)b_{e}=0\,.\]
Furthermore, $\oc\circ\dc=0$ is equivalent to \[
H^{1}(\Gamma,\mathbb{K})=\coker(\dc)=\mathbb{K}^{\left|E(\Gamma)\right|}/\im(\dc)\,,\]
an alternative definition of $H^{1}(\Gamma,\mathbb{K})$. The dimension
of this cohomology is called the first Betti number and gives the
number of independent loops of the graph $\Gamma$, $\dim(H^{1}(\Gamma,\mathbb{K}))=\left|E(\Gamma)\right|-\left|V(\Gamma)\right|+1$.

Let us regard the map $\dc$. The image of a general element $\vec{x}=\sum_{v\in V(\Gamma)}x^{v}a_{v}$
is given by\[
\dc(\vec{x})=\sum_{e\in E(\Gamma)}\left(x^{\target(e)}-x^{\source(e)}\right)b_{e}\,.\]
Thus $\dc$ expresses the coordinates of a given edge $e\in E(\Gamma)$
in terms of the coordinates of the adjacent vertices, $r^{e}=x^{\target(e)}-x^{\source(e)}$.
\begin{example}
Regard the very simple graph with two vertices and one edge, \[
\gamma=\LineX\,.\]
Let $v=\source(e)$ and $w=\target(e)$, and choose a basis $\left\{ a_{v},a_{w}\right\} $
of $\mathbb{K}^{\left|V(\gamma)\right|}$. Then $x\, a_{v}+y\, a_{w}\in\mathbb{K}^{\left|V(\gamma)\right|}$
and we have\[
\dc(x\, a_{v}+y\, a_{w})=\left(y-x\right)b_{e}\,.\]
Thus the pullback of $\dc:\mathbb{K}^{\left|V(\gamma)\right|}\rightarrow\mathbb{K}^{\left|E(\gamma)\right|}\equiv\mathbb{K}$
maps a function $f\in\D(\mathbb{K})$ to\[
\left(\dc^{*}f\right)(x,y)=f(y-x)\,.\]
Consequently, a distribution $u\in\D'(\mathbb{K})$ will be mapped
to $\dc^{*}u\in\D'(\mathbb{K}^{2})$, with\[
\left(\dc^{*}u\right)(f\otimes g)=\int_{\mathbb{K}^{2}}dx\, dy\, u(y-x)\, f(x)\, g(y)=\left\langle f,u*g\right\rangle \,,\]
where $*$ denotes the convolution product and the pullback is defined
in the sense of \citep[Thm.~6.1.2]{Hoermander2003}.
\end{example}
This construction can be lifted to any $\mathbb{K}$-vector space
$V$, by forming the tensor product $\mathbb{K}\otimes V$. We are
interested here in the lift to Minkowski spacetime $\M=\mathbb{R}\otimes\M$.
Thus we have, $\M^{n}=\mathbb{R}^{n}\otimes\M$,\[
\xymatrix{0\ar[r] & \M\ar@{^{(}->}[r]^{\hat{\cc}\quad} & \M^{\left|V(\Gamma)\right|}\ar@{->}[r]^{\hat{\dc}} & \M^{\left|E(\Gamma)\right|}\ar@{->>}[r] & H^{1}(\Gamma,\M)\ar[r] & 0}
\,,\]
i.e., one short exact sequence for each component of $z\in\M$.\addtocounter{thm}{-1}
\begin{example}
[revisited]In terms of this cohomology the Hadamard two point function
is the pullback of the Hadamard solution $H\in\D'(\M)$ by $\dc_{\gamma}$,\[
\left(\hat{\dc}_{\gamma}^{*}H\right)(f,g)=\left\langle f,H*g\right\rangle \,,\]
analogously the Feynman propagator is the pullback of the Feynman
fundamental solution $H_{F}\in\D'(\M\backslash\left\{ 0\right\} )$,
\[
\left(\hat{\dc}_{\gamma}^{*}H_{F}\right)(f,g)=\left\langle f,H_{F}*g\right\rangle \,,\]
$\supp(f)\cap\supp(g)=\emptyset$.
\end{example}
Also translation invariance can be formulated very conveniently in
this cohomological framework. The image of $\cc$ gives all possible
translations of the vertex coordinates by a given vector $a\in\mathbb{K}$.
Hence the orbits of these translations are the elements of the cokernel
$\coker(\cc)=\mathbb{K}^{\left|V(\Gamma)\right|}/\im(\cc$). We can
fix a basis of $\coker(\cc)$ by choosing the coordinates of a vertex
$v_{0}$ and setting $V_{0}:=V(\Gamma)\backslash\left\{ v_{0}\right\} $.
This provides us with a projection\[
\pi_{\Gamma}:\mathbb{K}^{\left|V(\Gamma)\right|}\rightarrow\mathbb{K}^{\left|V_{0}\right|}\equiv\mathbb{K}^{\left|V(\Gamma)\right|-1}\]
and an isomorphism $\phi$ between $\mathbb{K}^{\left|V_{0}\right|}$
and $\coker(\cc)$,\[
\begin{array}{rccl}
\phi: & \mathbb{K}^{\left|V_{0}\right|} & \rightarrow & \coker(\cc)\\
 & a_{v} & \mapsto & a_{v}+\im(\cc)\,.\end{array}\]
All translation invariant functions in $\mathbb{K}^{\left|V(\Gamma)\right|}$
can be seen as generic functions on $\coker(\cc)$, or $\mathbb{K}^{\left|V_{0}\right|}$
respectively. They are related by the pullback via $\pi_{\Gamma}$,
e.g. for smooth functions,\[
\begin{array}{rccl}
\pi_{\Gamma}^{*}: & \E(\mathbb{K}^{\left|V_{0}\right|}) & \rightarrow & \E_{\mathrm{tr.inv.}}(\mathbb{K}^{\left|V(\Gamma)\right|})\\
 & f & \mapsto & \left(\pi_{\Gamma}^{*}f\right)=f\circ\pi_{\Gamma}\,.\end{array}\]
We define the \emph{choice of relative coordinates} in $\mathbb{K}^{\left|E(\Gamma)\right|}$
by \[
\begin{array}{rccl}
\iota_{\Gamma}:=\dc_{\Gamma}\circ\phi: & \mathbb{K}^{\left|V_{0}\right|} & \rightarrow & \mathbb{K}^{\left|E(\Gamma)\right|}\\
 & \sum_{v\in V_{0}}x^{v}a_{v} & \mapsto & \sum_{e\in E(\Gamma)}r^{e}(\vec{x})\, b_{e}\,,\end{array}\]
where $r^{e}(\vec{x})=\sum_{v\in V_{0}}\left(e:v\right)x^{v}$ is
computed to be \[
r^{e}(\vec{x})=\begin{cases}
x^{\target(e)}-x^{\source(e)} & \mbox{if }v_{0}\notin\left\{ \source(e),\target(e)\right\} \\
x^{\target(e)} & \mbox{if }v_{0}=\source(e)\\
-x^{\source(e)} & \mbox{if }v_{0}=\target(e)\,,\end{cases}\]
giving the {}``coordinates of the edges'' relative to $v_{0}$.
In Minkowski spacetime we define correspondingly,\[
\hat{\iota}_{\Gamma}:=\hat{\dc}_{\Gamma}\circ\hat{\phi}:\M^{\left|V_{0}\right|}\rightarrow\M^{\left|E(\Gamma)\right|}\]
as the\emph{ }choice of relative coordinates in $\M^{\left|E(\Gamma)\right|}$.

We now want to define $S_{\Gamma}\in\D'(\M^{\left|V(\Gamma)\right|})$
as the pullback of $\widetilde{S_{\Gamma}}$ via $\hat{\dc}_{\Gamma}$,
\[
S_{\Gamma}=\hat{\dc}_{\Gamma}^{*}\,\widetilde{S_{\Gamma}}\,.\]
Let us regard the case of the unextended amplitude $\widetilde{S_{\Gamma}}\in\D'(\left(\M\backslash\left\{ 0\right\} \right)^{\left|E(\Gamma)\right|})$.
Each edge corresponds to a Feynman propagator $H_{F}$ and any set
of edges joining the same two vertices will have the same coordinate
$r^{e}(\vec{x})$. This introduces powers of $H_{F}$, which are well-defined
distributions only outside the origin, $\left(H_{F}\right)^{k}\in\D'(\M\backslash\left\{ 0\right\} )$.
As a consequence the pullback $S_{\Gamma}$ is a well-defined distribution
only outside the large diagonal \[
\DIAG=\left\{ \vec{x}\in\M^{\left|V(\Gamma)\right|}|\,\exists v,w\in V(\Gamma),v\neq w:\, x_{v}=x_{w}\right\} \,,\]
$S_{\Gamma}\in\D'(\M^{\left|V(\Gamma)\right|}\backslash\DIAG)$. As
remarked before a restriction of $\widetilde{S_{\Gamma}}$ by means
of a wave front set argument, i.e. by applying \citep[Thm.~8.2.4]{Hoermander2003},
is not possible due to the wave front set of $H_{F}$. A restriction
of $\widetilde{S_{\Gamma}}$, or equivalently an extension of $\hat{\dc}_{\Gamma}^{*}\widetilde{S_{\Gamma}}$
to $\D'(\M^{\left|V(\Gamma)\right|})$, will involve renormalization.
In the case of even dimensions, $d\in2\mathbb{N}$, the amplitude
is a tensor power of the Feynman propagator (\ref{eq:FeynmanEvenDimensions}),\[
\widetilde{S_{\Gamma}^{\mu}}:=\bigotimes_{e\in E(\Gamma)}H_{F}^{\mu}(e)\in\D'(\left(\M\backslash\left\{ 0\right\} \right)^{\left|E(\Gamma)\right|})\,,\]
and hence depends on an additional parameter $\mu$ of mass dimension
one.

We review briefly the Epstein-Glaser induction for constructing the
extension $S_{\Gamma}$ in order to discuss the renormalization freedom
in the cohomological framework advertised here.

\subsection{\label{sub:EpsteinGlaserInduction}The Epstein-Glaser induction}

Having defined what we mean by a graph $\Gamma\in\mathcal{G}$, we
define an \emph{Epstein-Glaser subgraph (EG subgraph)} $\gamma\subseteq\Gamma$
to be a subset of the set of vertices $V(\gamma)\subseteq V(\Gamma)$
together with all lines in $\Gamma$ connecting them,\[
E(\gamma)=\left\{ e\in E(\Gamma):\left\{ \source(e),\target(e)\right\} \subset V(\gamma)\right\} .\]
The orientation of $\gamma$ is inherited from $\Gamma$. The first
step of the Epstein-Glaser induction is to choose extensions for all
EG subgraphs with two vertices, $\left|V(\gamma)\right|=2$. In this
case we have translation invariant distributions $\hat{\dc}_{\gamma}^{*}\widetilde{S_{\gamma}}\in\D'(\M^{2}\backslash\Diag)$
($\Diag=\left\{ \vec{x}\in\M^{\left|V(\gamma)\right|}|\,\forall v,w\in V(\Gamma):\, x_{v}=x_{w}\right\} $
denotes the thin diagonal), which correspond to generic distributions
$\hat{\iota}_{\gamma}^{*}\widetilde{S_{\gamma}}\in\D'(\M\backslash\left\{ 0\right\} )$.
The scaling degree of these distributions is given by their number
of lines $\sd(\hat{\iota}_{\gamma}^{*}\widetilde{S_{\gamma}})=\left|E(\gamma)\right|\left(d-2\right)$,
and we can choose a (possibly unique) extension according to Theorem~\ref{thm:ExtensionScalingDegree}.
By translation invariance this gives extensions $S_{\gamma}\in\D'(\M^{2})$
of the distributions $\hat{\dc}_{\gamma}^{*}S_{\gamma}\in\D'(\M^{2}\backslash\Diag)$.
By causality, i.e. relation (\ref{eq:CausalityProducts}) of time-ordered
and algebra product, these extensions define the (translation-invariant)
restrictions of all EG subgraphs with three vertices up to the thin
diagonal.

For a generic EG subgraph $\gamma\subseteq\Gamma$ we make the assumption
that the restrictions of all EG subgraphs of $\gamma$ with less vertices
have already been chosen (induction hypothesis). The causality condition
then gives a translation invariant distribution $\hat{\dc}_{\gamma}^{*}\widetilde{S}_{\gamma}\in\D'(\M^{\left|V(\gamma)\right|}\backslash\Diag)$
which corresponds to a generic distribution $\hat{\iota}_{\gamma}^{*}\widetilde{S_{\gamma}}\in\D'(\M^{\left|V(\gamma)\right|-1}\backslash\left\{ 0\right\} )$.
The scaling degree and hence the degree of divergence of this distribution
is completely fixed by the structure of the graph, cf. (\ref{eq:FeynmanScalingDegree}),\begin{equation}
\div(\gamma)=\div(\hat{\iota}_{\gamma}^{*}\widetilde{S_{\gamma}})=\left|E(\gamma)\right|\left(d-2\right)-\left(\left|V(\gamma)\right|-1\right)d\,,\qquad d=\dim(\M)\,.\label{eq:DegreeOfDivergenceGraph}\end{equation}
We call $\gamma$ superficially convergent if $\div(\gamma)<0$, logarithmically
divergent if \linebreak[4]$\div(\gamma)=0$ and divergent of degree
$\div(\gamma)$ otherwise. Again by Theorem~\ref{thm:ExtensionScalingDegree}
there is a choice to be made in the extension of $\hat{\iota}_{\gamma}^{*}\widetilde{S_{\gamma}}$
in the case $\div(\gamma)\geq0$. The inductive procedure of Epstein-Glaser
will thus lead to an extension $S_{\Gamma}\in\D'(\M^{\left|V(\Gamma)\right|})$
of $\hat{\dc}_{\Gamma}^{*}\widetilde{S_{\Gamma}}\in\D'(\M^{\left|V(\Gamma)\right|}\backslash\DIAG)$.
As suggested above, we will refer to any such extension of $\hat{\dc}_{\Gamma}^{*}\widetilde{S_{\Gamma}}$
as \emph{restriction} of $\widetilde{S_{\Gamma}}\in\D'((\M\backslash\left\{ 0\right\} )^{\left|E(\Gamma)\right|})$.

In the case of couplings which involve derivatives of the fields,
also the external structure of $\gamma$ has to be taken into account,
cf.~(\ref{eq:ScalingDegreeWithExternalStructure}), \[
\div\!\left(\gamma,\vec{k}\right)=\div(\gamma)+\left|\vec{k}\right|\,.\]
This introduces an additional freedom in the choice of the extension
in each step, but does otherwise not change the inductive procedure.

The combination of all choices involved in the inductive construction
of a restriction $S_{\Gamma}\in\D'(\M^{\left|V(\Gamma)\right|})$
of $\widetilde{S_{\Gamma}}$ make up the Stückelberg-Petermann renormalization
group acting on local functionals, cf. \citep{Brunetti2009}. We will
see in the next section that the freedom in this construction is considerably
restricted, if we replace in (\ref{eq:TensorTProduct}) the Feynman
propagator by its dimensionally regularized counterpart.

\section{\label{sec:TheRegularizedAmplitude}The Regularized Amplitude}

The aim of this section is to construct a regularization of the above
defined amplitude $\hat{\dc}_{\Gamma}^{*}\widetilde{S_{\Gamma}^{\mu}}\in\D'(\M^{\left|V(\Gamma)\right|}\backslash\DIAG)$
by applying the Epstein-Glaser reduction procedure to\begin{equation}
\widetilde{S_{\Gamma}^{\mu,\zeta}}:=\bigotimes_{e\in E(\Gamma)}H_{F}^{m,\mu,\zeta}(e)\,.\label{eq:RegularizedTensorDistribution}\end{equation}
We will see in the sequel that $\widetilde{S_{\Gamma}^{\mu,\zeta}}$
has a unique restriction $S_{\Gamma}^{\mu,\zeta}\in\D'(\M^{\left|V(\Gamma)\right|})$
by means of Corollary~\ref{cor:ExtensionHeterogeneousDistribution}.
This will provide a regularization of the original amplitude $\hat{\dc}_{\Gamma}^{*}\widetilde{S_{\Gamma}^{\mu}}$
outside the large diagonal.

Regard the dimensionally regularized Feynman fundamental solution
$H_{F}^{m,\mu,\zeta}\in\D'(\M\backslash\left\{ 0\right\} )$ constructed
in Chapter~\ref{cha:DimRegHadamard}. The expansion of $H_{F}^{m,\mu,\zeta}$
in powers of the mass parameter $m^{2}$ follows directly from the
expansion of the analytic Hadamard function (\ref{eq:DimRegAnalyticHadamardSeriesM}),\begin{align}
H_{F}^{m,\mu,\zeta}(x) & =\left(-1\right)^{\frac{d-2}{2}}\left(2\pi\right)^{-\frac{d}{2}}2^{\frac{2-\left(d+\zeta\right)}{2}}\frac{\pi}{2\sin(\zeta\tfrac{\pi}{2})}\cdot\nonumber \\
 & \hspace{-5mm}\cdot\left[\sum_{s=0}^{\infty}\left\{ \left(\frac{2}{\mu\sqrt{-x^{2}+i0}}\right)^{\zeta}\frac{1}{s!\,\Gamma(-\frac{\zeta}{2}-\frac{d-4}{2}+s)}-\frac{\theta(s-\frac{d-3}{2})}{\left(s-\frac{d-2}{2}\right)!\,\Gamma(\frac{\zeta}{2}+s+1)}\right\} \right.\cdot\label{eq:DimRegFeynmanSeriesM}\\
 & \hspace{60mm}\cdot\left.\left(\frac{\sqrt{-x^{2}+i0}}{2}\right)^{2s-\left(d-2\right)}\left(m^{2}\right)^{s}\right]\!,\nonumber \\
 & =:\sum_{s=0}^{\infty}H_{F}^{s,\mu,\zeta}(x)\,\left(m^{2}\right)^{s}\,,\nonumber \end{align}
where we used the common shorthand $f(x^{2}-i0)=\lim_{\varepsilon\rightarrow0^{+}}f(x^{2}-i\varepsilon)$.
The coefficients of this series, $H_{F}^{s,\mu,\zeta}\in\D'(\M\backslash\left\{ 0\right\} )$,
are sums of a distributional and a smooth part, both of which are
homogeneous, but of different degree. The distributional part is homogeneous
of degree $2s-\left(d+\zeta-2\right)$, whereas the smooth part is
identically zero for $s<\frac{d-3}{2}$ and homogeneous of degree
$2s-\left(d-2\right)$ otherwise.

Regard now the finite tensor powers of the dimensionally regularized
Feynman distribution. The expansion of $\left(H_{F}^{m,\mu,\zeta}\right)^{\otimes k}$,
$k=\left|E(\Gamma)\right|$, in $m^{2}$ follows directly from (\ref{eq:DimRegFeynmanSeriesM})
and Cauchy's product formula,\begin{align}
\left(H_{F}^{m,\mu,\zeta}\right)^{\otimes k} & =\sum_{s_{k}=0}^{\infty}\left(m^{2}\right)^{s_{k}}\sum_{s_{k-1}=0}^{s_{k}}\cdots\sum_{s_{1}=0}^{s_{2}}H_{F}^{\left(s_{k}-s_{k-1}\right),\mu,\zeta}\otimes\cdots\otimes H_{F}^{s_{1},\mu,\zeta}\label{eq:DimRegFeynmanTensorPowerSeriesM}\\
 & =:\sum_{s_{k}=0}^{\infty}\left(m^{2}\right)^{s_{k}}\left(H_{F}^{\mu,\zeta}\right)^{\otimes\vec{s}},\nonumber \end{align}
where $\vec{s}=\left(s_{k}-s_{k-1},\dots,s_{1}\right)\in\mathbb{N}^{k}$.
The scaling degree of the coefficient of $\left(m^{2}\right)^{s_{k}}$
is given by the sum of the scaling degrees of the individual factors,
cf. Lemma~\ref{lem:ScalingDegreeProperties} and Remark~\ref{rem:ScalingDegreeHomogeneity},\begin{equation}
\sd(\left(H_{F}^{\mu,\zeta}\right)^{\otimes\vec{s}})=k\left(d+\Re(\zeta)-2\right)-2s_{k}\,.\label{eq:ScalingDegreeCoefficientM}\end{equation}
Hence the scaling degree of the coefficients become arbitrarily small
as one regards higher powers of $m^{2}$. Thus according to Theorem~\ref{thm:ExtensionScalingDegree}
these coefficients will have unique restrictions to arbitrary subdiagonals.
\footnote{Keep in mind that the product of distributions can be defined as the
restriction of their tensor product to subdiagonals, cf.~\citep[Thm.~8.2.10]{Hoermander2003}.
} In other words, only a finite number of coefficients in (\ref{eq:DimRegFeynmanTensorPowerSeriesM})
need renormalization. This is one of the advantages of the concept
of a {}``scaling expansion'' introduced in \citep{HollandsWald2002};
see also \citep{Hollands2004}, \citep[p.~1310ff]{DuetschFredenhagen2004}.
The coefficients in (\ref{eq:DimRegFeynmanTensorPowerSeriesM}) which
need renormalization when restricted to subdiagonals are in general
not homogeneous, since the coefficients $H_{F}^{s,\mu,\zeta}$ in
(\ref{eq:DimRegFeynmanSeriesM}) are not homogeneous. However, since
we regard graphs with a finite number of edges, they are certainly
heterogeneous of finite order. If we assume $\zeta\notin\mathbb{Q}_{+}$
their multidegree contains no integer number. We can construct a restriction
of $\widetilde{S_{\Gamma}^{\mu,\zeta}}$ by the same procedure described
in Section~\ref{sub:EpsteinGlaserInduction}, with the only difference
that we have a preferred choice of the extension at each order according
to Corollary~\ref{cor:ExtensionHeterogeneousDistribution}, namely
the extensions which are heterogeneous of the same multidegree. Thus
we are lead to a unique restriction $S_{\Gamma}^{\mu,\zeta}\in\D'(\M^{\left|V(\Gamma)\right|})$.
\begin{prop}
[Regularization outside $\DIAG$]\label{pro:RegularizedAmplitude}The
restriction of $\widetilde{S_{\Gamma}^{\mu,\zeta}}$ is \linebreak[4]uniquely
defined by the above homogeneity condition and gives a distribution
$S_{\Gamma}^{\mu,\zeta}\in\D'(\M^{\left|V(\Gamma)\right|})$. \textup{$S_{\Gamma}^{\mu,\zeta}$}
is a regularization of $\hat{\dc}_{\Gamma}^{*}\widetilde{S_{\Gamma}^{\mu}}\in\D'(\M^{\left|V(\Gamma)\right|}\backslash\DIAG)$,
in the sense that\[
\forall f\in\D(\M^{\left|V(\Gamma)\right|}\backslash\DIAG):\quad\lim_{\zeta\rightarrow0}\left\langle S_{\Gamma}^{\mu,\zeta},f\right\rangle =\left\langle \hat{\dc}_{\Gamma}^{*}\widetilde{S_{\Gamma}^{\mu}},f\right\rangle \,.\]
We will refer to $S_{\Gamma}^{\mu,\zeta}\in\D'(\M^{\left|V(\Gamma)\right|})$
as the dimensionally regularized amplitude of $\Gamma$. By translation
invariance it naturally corresponds to a unique dimensionally regularized
amplitude in relative coordinates, $s_{\Gamma}^{\mu,\zeta}\in\D'(\M^{\left|V(\Gamma)\right|-1})$,
which is the unique extension of $\hat{\iota}_{\Gamma}^{*}\widetilde{S_{\Gamma}^{\mu,\zeta}}\in\D'(\left(\M\backslash\left\{ 0\right\} \right)^{\left|V(\Gamma)\right|-1})$.\end{prop}
\begin{proof}
The first part follows from the construction above. By the discussion
at the end of the previous chapter we have that the unique extensions
$\dot{H}_{F}^{m,\mu,\zeta}\in\D'(\M)$ is a regularization of $H_{F}^{m,\mu}\in\D'(\M\backslash\left\{ 0\right\} )$,
cf. (\ref{eq:RegularizedFeynman}). And by continuity of the maps
involved we get the assertion,\begin{align*}
\hat{\dc}_{\Gamma}^{*}\left(\bigotimes_{e\in E(\Gamma)}H_{F}^{m,\mu}(e)\right) & =\hat{\dc}_{\Gamma}^{*}\left(\bigotimes_{e\in E(\Gamma)}\lim_{\zeta\rightarrow0}H_{F}^{m,\mu,\zeta}(e)\right)\\
 & =\lim_{\zeta\rightarrow0}\hat{\dc}_{\Gamma}^{*}\left(\bigotimes_{e\in E(\Gamma)}H_{F}^{m,\mu,\zeta}(e)\right)\end{align*}
Observe that the extension map $\hat{\dc}_{\Gamma}^{*}\widetilde{S_{\Gamma}^{\mu,\zeta}}\mapsto S_{\Gamma}^{\mu,\zeta}$
is also continuous by Theorem~\ref{thm:HomogeneousExtension}.
\end{proof}
We have that $S_{\Gamma}^{\mu,\zeta}$ is a regularization of $\hat{\dc}_{\Gamma}^{*}\widetilde{S_{\Gamma}^{\mu}}$
in a broader sense of the word, since $\hat{\dc}_{\Gamma}^{*}\,\widetilde{S_{\Gamma}^{\mu}}$
is defined only in the complement of the large diagonal. However,
the regularization $S_{\Gamma}^{\mu,\zeta}\in\D'(\M^{\left|V(\Gamma)\right|})$
comes with a natural renormalization prescription, defined at any
order of (causal) perturbation theory: minimal subtraction (MS). This
has already been introduced on the conceptual level in Section~\ref{sec:RegularizationOfDistributions},
and we will see in the next chapter, how minimal subtraction is to
be applied to the regularized graph amplitudes $s_{\Gamma}^{\mu,\zeta}$
and $S_{\Gamma}^{\mu,\zeta}$, respectively. The complete renormalization
of the graph amplitudes will then be discussed in Chapter~\ref{cha:ForestFormula}
and it will be useful, for the derivation of the underlying combinatorial
structure to collect all the different contributions to the perturbative
expansion in the definition of a unique dimensionally regularized
$\Sm$-matrix, defined as a map on local functionals, cf.~Section~\ref{sec:TimeOrderedProduct}.
\begin{defn}
[{Dimensionally Regularized $\Sm$-matrix}]\label{def:DimRegSMatrix}Let
\[
\Gamma_{H_{F}^{m,\mu,\zeta}}':=\frac{1}{2}\int dx\, dy\, H_{F}^{m,\mu,\zeta}(x,y)\frac{\delta^{2}}{\delta\varphi(x)\,\delta\varphi(y)}\,,\]
be the dimensionally regularized Feynman bidifferential operator.
Define the regularized time-ordering operator\[
\Time_{\mu,\zeta}:=\exp(\hbar\Gamma_{H_{F}^{m,\mu,\zeta}}')\,,\]
and the dimensionally regularized time-ordered product on local functionals
$F,G\in\fps{\mathcal{F}_{\loc}(\M)}{\hbar}$\[
F\cdot_{\Time_{\mu,\zeta}}G:=\Time_{\mu,\zeta}\left(\Time_{\mu,\zeta}^{-1}F\cdot\Time_{\mu,\zeta}^{-1}G\right)\,.\]
Then we define the dimensionally regularized $\Sm$-matrix as\[
\Sm_{\mu,\zeta}(F):=\exp_{\cdot_{\Time_{\mu,\zeta}}}(F)=\sum_{n=0}^{\infty}\frac{1}{n!}\Time_{\mu,\zeta}^{n}(F^{\otimes n})\,,\qquad F\in\fps{\mathcal{F}_{\loc}(\M)}{\hbar}\,,\]
where $\Time_{\mu,\zeta}^{n}$ denotes the uniquely extended regularized
$n$-fold time-ordered product constructed (graph by graph) by Epstein-Glaser
induction.
\end{defn}
Inserting (\ref{eq:TimeOrderedGraphStructure}) we can write the regularized
$\Sm$-matrix also in terms of a graph expansion\begin{equation}
\Sm_{\mu,\zeta}(F)=\sum_{n=0}^{\infty}\frac{1}{n!}\sum_{\alpha\in\mathbb{N}^{n}}\sum_{\Gamma\in\mathcal{G}_{\alpha}}\frac{\hbar^{\left|E(\Gamma)\right|}}{\Sym(\Gamma)}\left\langle S_{\Gamma}^{\mu,\zeta},\bld{\delta}^{\alpha}\left(F^{\otimes n}\right)\right\rangle \,,\qquad F\in\fps{\mathcal{F}_{\loc}(\M)}{\hbar}.\label{eq:PerturbationSeries}\end{equation}
This expansion is often referred to as the perturbative expansion
of the $\Sm$-matrix. And we want to remark that the sum over all
graphs at a fixed order $n$ of causal perturbation theory is finite,
if we assume that $F\in\fps{\mathcal{F}_{\loc}(\M)}{\hbar}$ is a
polynomial interaction functional. This remains valid, if $F$ contains
derivatives of the field. The order of causal perturbation theory
is given by the number of vertices of the graphs contributing to (\ref{eq:PerturbationSeries}),
irrespective of the fact if they are connected or not. Conversely,
the sum is finite at each order $\mathcal{O}($$\hbar^{\left|E(\Gamma)\right|}$),
and we repeat the remark that this is in essence the {}``loop order'',
if we regard only graphs with a fixed number of connected components
$c(\Gamma)$. The order is given by the Betti number of the graph
\citep{GrossYellen2003}, \[
\ell(\Gamma)=\left|E(\Gamma)\right|-\left|V(\Gamma)\right|+c(\Gamma)\,,\]
if we {}``hide'' one power of $\hbar$ in the interaction functional
$F$.

\end{fmffile}

config: arxiv


TITLE: What causes the specific pattern of sparkly light from a laser pointer
QUESTION [8 upvotes]: I have a fairly cheap green laser pointer that produces a fairly big spot of light (actually two spots most of the time!). I'd guess the size of the spot is about 1/100th of the distance from the end of the pointer to the surface it hits (e.g. after 1 metre the size is approximately 1 cm).
If the pointer is steady and I sit still I see a pattern of sparkly light (on the wall, table top, etc.). Although it sparkles, each tiny point stays in the same spot within the circle of laser light. If I move my head left or right the pattern moves with me but disappears at the edge of the spot, while a "new" part of the pattern appears on the other edge. What causes this effect?
I wear contact lenses and I've noticed exactly the same thing whether I'm wearing them or not, no matter the distance. Because each point emitted by the laser should be a true point source it should not be possible for it to appear blurry. My personal observations agree with this.

REPLY [0 votes]: I just did a test, on the suggestion of the wikipedia article: shining the laser on the surface of an opaque liquid - a glass of milk - and the speckle pattern is almost invisible, while on the surface of the table next to the glass, it's very distinct.


TITLE: How many pairs $(p, m)$ exist, where $p$ is a prime and $m$ is an integer, such that $\frac{m^3-pm+1}{m^2+pm+2}$ is an integer.
QUESTION [1 upvotes]: How many pairs $(p, m)$ exist, where $p$ is a prime and $m$ is an integer, such that $$\frac{m^3-pm+1}{m^2+pm+2}$$ is an integer? I've found three pairs by myself, but I'm not sure if that's all. What's the good algorithm for solving this type of problems?

REPLY [5 votes]: Following up on lulu's comment: once you know that $p=2$, you can simplify the expression.
$$\frac{m^3-2m+1}{m^2+2m+2} = m+ \frac{-2m^2-4m+1}{m^2+2m+2} = m-2+ \frac{5}{m^2+2m+2}$$
So $m^2+2m+2$ is $1,5, -1$ or $-5$.
You can solve these four quadratic equations to obtain all solutions. (Or alternatively, you can observe that $m^2+2m+2=(m+1)^2+1$, so $m$ is between $-3$ and $1$, and check all five possibilities.)
It is also unclear to me if $p=-2$ should be allowed or not. In that case, a similar calculation works, too.


TITLE: Automorphisms of projective spaces, and the Axiom of Choice
QUESTION [3 upvotes]: It is known that upon not accepting the Axiom of Choice (AC), there exist models of ZF in which there are projective spaces (over a division ring) with a trivial automorphism group. (This is a truly remarkable result, if I may say.)
Now let $G$, more generally, be any given group. Do there exist models of ZF (without AC) in which there are projective spaces $\mathbb{P}$ over some division ring, for which the automorphism group of $\mathbb{P}$ is isomorphic to $G$ ?

REPLY [6 votes]: On abstract metamathematical grounds (having nothing to do with projective spaces), I claim that it is relatively consistent with ZFC and indeed with ZFC+Con(ZF) and much more that the answer to your question is negative. This doesn't mean the answer is negative, since it could also be consistent that the answer is positive, making the property independent of ZFC+Con(ZF). It could also be that a positive answer to your question is a consequence of a stronger theory, such as ZFC plus large cardinals.
The role of Con(ZF) is important for your question. In particular, your initial remark is not quite stated correctly, since what we need for the initial result is not to reject AC, but (presumably) to assume Con(ZF), in order to get the model of ZF as you mentioned.
But I shall argue that neither Con(ZF) nor indeed any iteration of consistency statements Con${}^n$(ZF) is sufficient in general to get your more general statement. The reason is that it is consistent with these theories that some groups $G$ simply cannot appear in a model of ZF at all.
To see this, observe first that Con(ZF) and the stronger iterated statements are strictly weaker than the assertion that there is an $\omega$-standard model of ZF. One can see that it is weaker because all $\omega$-standard models fulfill the same consistency statements as the ambient model, and so if Con${}^n$(ZF) implied the existence of an $\omega$-standard model of ZF, then it would imply its own consistency, contrary to the incompleteness theorem.
So if ZF+Con${}^n$(ZF) is consistent, then there is a model of this theory that thinks there are no $\omega$-standard models of ZF. In such a set-theoretic world, there are many models of ZF, but no $\omega$-standard models of ZF. But in this case, no model of ZF could have a group that is isomorphic to the standard integers $\mathbb{Z}$, because this can happen only in an $\omega$-standard model.
In summary, if ZF+Con(ZF) is consistent, then there are models of this theory in which the answer to your question is no.


TITLE: Generators increasing the word metric
QUESTION [7 upvotes]: Let $G$ be an infinite group with a finite generating set $T$ which is symmetric ($T=T^{-1}$) and let $|\cdot|$ be the corresponding word metric, i.e. $|g|$ is the minimal number of (not necessarily distinct) elements from $T$ one needs to multiply to get $g$.
Is it true that for every $g\in G$ there is a $t\in T$ such that $|gt|>|g|$?

REPLY [9 votes]: Consider $G=\mathbb{Z}$ with $T=\{\pm2,\pm3\}$. Then $|1|=2$, and $|1+t|\leq 2$ for all $t\in T$.


text:
\subsection{Notation}
For a vector $X=(x_{1},\dots,x_{d})^{T}\in \R^{d}$, $\|X\|$ denotes the Euclidean norm, $\E|X^{\otimes k}|<\infty$ denotes that $\E |x_{i_{1}} \cdots x_{i_{k}}|<\infty$ for all integer $i_{1},\dots, i_{k}\in \{1,\dots, d\}$.
For tensors $A, B\in {\R^{d}}^{\otimes k}$, their inner product equals $\inner{A}{B}\define \sum_{1\leq i_{j}\leq d}a_{i_{1},\dots,i_{k}}b_{i_{1},\dots,i_{k}}$, where $a_{i_{1},\dots,i_{k}}$ and $b_{i_{1},\dots,i_{k}}$ are elements of $A$ and $B$.
The operator norm of $A$ (for $k\geq 2$) induced by the Euclidean norm is denoted by $\|A\|\define \sup\{ \inner{A}{\gamma_{1}\otimes \dots \otimes \gamma_{k}}: \| \gamma_{j}\|=1, \gamma_{j}\in \R^{d}, j=1,\dots,k\}.$ The Frobenius norm is $\|A\|_{\mathrm{F}}=\sqrt{\inner{A}{A}}$. The maximum norm is $\|A\|_{\max}\define \max \{|a_{i_{1},\dots,i_{k}}|: i_{1},\dots, i_{k}\in \{1,\dots, d\}\}$.
 For a function $f:\R^{d}\mapsto \R$ and $h\in \R^{d}$,  $f^{(s)}(x)h^{s}$ denotes the higher-order directional derivative $(h^{T}\nabla)^{s}f(x)$. $\phid(x)$ denotes  the p.d.f. of the standard normal distribution in $\R^{d}$.
 $C,c$ denote positive generic constants. The abbreviations p.d. and p.s.d. denote positive definite and positive semi-definite matrices correspondingly.


TITLE: $3$ mutually tangent circles
QUESTION [1 upvotes]: $\textbf{Problem}$: In $\triangle{ABC}$, $AB=3$, $BC=4$, and $CA=5$. Additionally, we have mutually tangent circles $X$, $Y$, and $Z$ inside the triangle that are tangent to $\{AB, BC\}$, $\{BC, CA\}$, and $\{CA, AB\}$ respectively. Determine the sum of the radii of the circles $X$, $Y$, and $Z$.
$\textbf{Thoughts}$:
I drew a diagram. However, I do not know asymptote. I'm aware that drawing a diagram is best in these types of geometry problems, but my diagram (which was grotesque) did not help me.

An addendum by Jack: the above diagram depicts Steiner's construction of the Malfatti circles of a $3-4-5$ triangle; the blue lines are the bitangents mentioned by the linked Wikipedia article.

REPLY [1 votes]: The hint.
Let $a$, $b$ and $c$ be radius of circles.
Thus, we need to solve the following system.
$$a\sqrt5+2\sqrt{ab}+b=3,$$ $$3c+2\sqrt{bc}+b=4,$$ $$a\sqrt5+2\sqrt{ac}+3c=5.$$