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}}

\global\long\def\oprod#1{\sideset{}{^{\geq}}\prod_{#1}}

\begin{fmffile}{FFRenHopf02}

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\chapter{\label{cha:ForestFormula}The Epstein-Glaser Forest~Formula}

\begin{center}

\begin{minipage}[t][1.2\totalheight]{0.8\columnwidth}
\begin{flushleft}
\emph{\textquoteleft{}What\textquoteright{}s the sandwich scenario,
Mo?\textquoteright{}}
\par\end{flushleft}

\emph{\textquoteleft{}Ham and cheese; ham and tomato; cheese and tomato.\textquoteright{}}

\emph{\textquoteleft{}And ham, cheese and tomato.\textquoteright{}}

\emph{\textquoteleft{}How did you know?\textquoteright{}}

\emph{\textquoteleft{}You\textquoteright{}ve never noticed how you
group sandwiches into Venn diagrams?\textquoteright{}}

\emph{\textquoteleft{}Do I?\textquoteright{}}

\begin{flushright}
\emph{David Mitchell: Ghostwritten}
\par\end{flushright}
\end{minipage}
\par\end{center}

It was the principle of covariance, understood as the axiom that all
physically relevant concepts must have an analogue in (globally hyperbolic)
curved spacetime, which brought to light the more profound structures
of perturbative renormalization theory in the investigation undertaken
by Brunetti, Dütsch, Fredenhagen, Hollands, and Wald (see references
in the introduction).
\footnote{The covariance principle was made precise in \citep{Brunetti2003}.
And we want to use this footnote to remark that despite its reputation
of being conceptually clear but {}``too far from reality'' to have
predictive power for experiments the algebraic approach and in particular
perturbative Algebraic Quantum Field Theory has lead to falsifiable
predictions in cosmology \citep{Dappiaggi2008}.
} As already said in the introduction of this thesis one of the main
results of their program was the formulation of perturbative Algebraic
Quantum Field Theory (pAQFT), briefly introduced in Chapter~\ref{cha:Setting-pAQFT}.
In this last chapter we will show that the tools of pAQFT and in particular
the precise statement of Stora's main theorem of perturbative renormalization,
augmented by the results on analytic regularization we have gained
in the previous chapters will make it possible to solve the recursive
procedure of Epstein-Glaser renormalization and to prove a forest
formula in the sense of Zimmermann for Epstein-Glaser renormalization.
The result will be independent of the chosen representation and will
in particular be applicable in momentum and in position space. The
main theorem of renormalization, written in termwise form by using
Faà~di~Bruno's formula for the $n$-fold chain rule \citep{FaaDiBruno1855},
implies a recursion relation for the minimally subtracted counterterms
to an analytically regularized $\Sm$-matrix. This recursion relation
will be crucial for the proof of the forest formula.

In 1982 Joni and Rota introduced a bialgebra related to Faà~di~Bruno's
formula \citep{JoniRota1982}. We will use this bialgebra to derive
(a summed up version of) the Connes-Kreimer Hopf algebra of graphs
directly from the main theorem of renormalization. However, in contrast
to the Connes-Kreimer approach the Feynman rules will emerge naturally
and are not assumed to be characters into the commutative ring of
Laurent series with scalar coefficients. The emergent Feynman rules
will rather produce linear maps between spaces of (local) functionals.
On the space of linear maps the construction induces two products,
a symmetrized tensor product, and a non-commutative product, which
is given as the composition of linear maps. Both products, as well
as the coproduct, need to be reflected in the Hopf algebra (of graphs)
in order to encode the algebraic structure of the recursive construction
of counterterms. By giving this derivation we will establish the relation
of the pAQFT formalism to the {}``Hopf algebra school'' which was
not present in the original pAQFT article (cf.~\citep[p.~45]{Brunetti2009}).

After some preliminary remarks on the differential calculus used in
this chapter, we will cite the main theorem of renormalization from
\citep{Brunetti2009} in the second section. The third section will
be devoted to the derivation of a forest formula for regularized Epstein-Glaser
renormalization from Stora's main theorem. The above described Hopf
algebra will be constructed in the fourth section of this chapter.

\section{\label{sec:PreliminariesDifferentialCalculus}Preliminaries on differential
calculus}

We take the elevator in the hierarchy of differential calculi one
floor up and want to regard functional derivatives of the $\Sm$-matrix,
regarded as a map between spaces of (local) functionals,\[
\Sm\equiv\exp_{\dT}:\,\fps{\mathcal{F}_{\loc}(\M)}{\hbar}\rightarrow\fps{\mathcal{F}(\M)}{\hbar}\,,\]
and of the renormalization group transformations $Z\in\R$ to be defined
below as maps,\[
Z:\fps{\mathcal{F}_{\loc}(\M)}{\hbar}\rightarrow\fps{\mathcal{F}_{\loc}(\M)}{\hbar}\,.\]
The $n$-fold derivative of $\Sm$ at the origin \[
\Sm^{\left(n\right)}(0)(F^{\otimes n})=\frac{d^{n}}{d\lambda^{n}}\Sm(\lambda F)\bigg|_{\lambda=0}\]
gives the $n$-fold time-ordered product, i.e. the $n$th coefficient
in the series expansion of $\Sm$, cf.~Equation~(\ref{eq:PerturbationSeries}).
The $n$-fold derivative of $Z\in\R$ gives the counterterm at order
$n$ of causal perturbation theory. We will equivalently use $\Sm^{\left(n\right)}=\Sm^{\left(n\right)}\big|_{0}\equiv\Sm^{\left(n\right)}(0)$,
and likewise for $Z$, wherever there is no risk of confusion.

The mathematically precise definition of such a differential calculus
is quite involved and a focus of research in analysis \citep{Hamilton1982,KrieglMichor1997,Neeb2005}
(taking the stairs here, might be very hard). However, it is enough
for our purposes to assume that a calculus can be defined in such
a way that the corresponding differential, \[
\frac{\delta}{\delta F}:\Sm\mapsto\Sm^{\left(1\right)}(F)\,,\]
fulfills the chain - and the Leibniz rule in the sense below. A calculus
fulfilling the chain rule was defined for locally convex spaces in
\citep{Neeb2005}. And as shown in \citep[Sec.~3.1]{Brunetti2009}
$\mathcal{F}(\M)$ can be endowed with a locally convex topology,
defined as the initial topology of the Hörmander topology on spaces
of distributions with conic wave front set.

Let $Z$ be differentiable at $F$, and $\Sm$ be differentiable at
$Z(F)$, then we want to assume that the derivative of their composition
is given by the chain rule\begin{equation}
\left(\Sm\circ Z\right)^{\left(1\right)}\!(F)=\left(\Sm^{\left(1\right)}\circ Z\right)\!(F)\cdot\left(Z^{\left(1\right)}(F)\right)\equiv\frac{\delta\Sm}{\delta F'}\bigg|_{Z(F)}\cdot\frac{\delta Z}{\delta F'}\bigg|_{F}\,,\label{eq:ChainRule}\end{equation}
where on the right hand side we have a composition of linear maps,
generally denoted by {}``~$\cdot$~'' in this chapter. For the
iteration of the chain rule and the proof of the $n$-fold chain rule
in Lemma~\ref{lem:FaaDiBrunoSetPartitionVersion} we will also need
that the derivative fulfills the Leibniz rule in the following sense,\begin{equation}
\left(Z\otimes Z\right)^{\left(1\right)}=Z^{\left(1\right)}\otimes Z+Z\otimes Z^{\left(1\right)}\,.\label{eq:LeibnizRule}\end{equation}

We call a map \[
\bld{\Psi}:\fps{\mathcal{F}_{\loc}(\M)}{\hbar}\rightarrow\fps{\mathcal{F}(\M)}{\hbar}\]
analytic (at $F$), if the $n$th functional derivative exists for
all $n\in\mathbb{N}$ as a totally symmetric, linear map \[
\bld{\Psi}^{\left(n\right)}(F):\fps{\mathcal{F}_{\loc}(\M)}{\hbar}^{\otimes n}\rightarrow\fps{\mathcal{F}(\M)}{\hbar}\,,\]
and\begin{equation}
\bld{\Psi}^{\left(n\right)}(F):\fps{\mathcal{F}_{\loc}(\M)}{\hbar}^{\otimes n}\rightarrow\fps{\mathcal{F}_{\loc}(\M)}{\hbar}\,,\quad\mbox{ if }\im(\bld{\Psi})\subset\fps{\mathcal{F}_{\loc}(\M)}{\hbar}.\label{eq:-n-foldDerivativeLocalMap}\end{equation}

\section{\label{sec:MainTheoremRenormalization}The Main Theorem of Renormalization}

An important insight in perturbative renormalization theory is the
fact that the freedom in the definition of the $\Sm$-matrix can be
described in terms of the Stückelberg-Petermann renormalization group
\citep{Stueckelberg1953}. Popineau and Stora termed this fact the
{}``main theorem of perturbative renormalization theory'' \citep{StoraPopineau1982}.
One can find it, although not under this name, already in the early
literature of renormalization theory \citep{Gell-MannLow1954,BogoliubovShirkov1959}.
Modern versions are included in \citep{Pinter2001,Grigore2001b}.
The precise statement and proof of this theorem in the algebraic approach
to perturbative QFT \citep{DuetschFredenhagen2004,Duetsch2005} made
it possible to show that the renormalization group of Stückelberg
and Petermann provides a common basis also to other renormalization
groups found in literature \citep{Brunetti2009}. We will give here
a minimalistic review of the basic definitions needed to formulate
the main theorem of renormalization in pAQFT. A more detailed summary,
including a sketch of the proof is contained in Section~4.1 of \citep{Brunetti2009}.

The $\Sm$-matrix\[
\begin{array}{rccl}
\Sm: & \fps{\mathcal{F}_{\loc}(\M)}{\hbar} & \rightarrow & \fps{\mathcal{F}(\M)}{\hbar}\\
 & F & \mapsto & \Sm(F)=\exp_{\dT}(F)\end{array}\]
is analytic at the origin, where its derivatives are given by the
$n$-fold time ordered products. However, $\Sm$ is not unique, but
needs to be defined perturbatively by renormalization. As shown in
\citep{Brunetti2009} the prerequisites needed for a definition of
$\Sm$ within causal perturbation theory can be expressed directly
in terms of properties of the $\Sm$-matrix itself. Let $A,B,F\in\fps{\mathcal{F}_{\loc}(\M)}{\hbar}$,
then $\Sm$ is required to fulfill the following conditions,
\begin{description}[topsep=1mm]
\item [{{[}C1{]}~Causality.}] $\Sm(A+B)=\Sm(A)\star\Sm(B)$, if $\supp(A)\gtrsim\supp(B)$.
\item [{{[}C2{]}~Starting~Element.}] $\Sm(0)=1:\E(\M)\rightarrow1\in\mathbb{C}$,\\
$\phantom{.}\hspace{18.5mm}\Sm^{\left(1\right)}(0)=\id:\fps{\mathcal{F}_{\loc}(\M)}{\hbar}\rightarrow\fps{\mathcal{F}_{\loc}(\M)}{\hbar}$.
\item [{{[}C3{]}~$\varphi$-Locality.}] The value of $\Sm(F)\in\fps{\mathcal{F}(\M)}{\hbar}$
at a given field configuration $\varphi_{0}$ depends only on the
Taylor expansion of $F\in\fps{\mathcal{F}_{\loc}(\M)}{\hbar}$ around
$\varphi_{0}$, \[
\Sm(F)(\varphi_{0})=\Sm(F_{\varphi_{0}}^{\left[N\right]})(\varphi_{0})+\mathcal{O}(\hbar^{N+1})\,,\]
where ${\displaystyle F_{\varphi_{0}}^{\left[N\right]}(\varphi)=\sum_{n=0}^{N}\frac{1}{n!}\left\langle F^{\left(n\right)}(\varphi_{0}),\left(\varphi-\varphi_{0}\right)^{\otimes n}\right\rangle }$
denotes the Taylor expansion of $F$ up to order $N$.
\item [{{[}C4{]}~Field~Independence.}] $\Sm$ depends only implicitly,
i.e. via the interaction $F$, on the field configuration, \[
\forall\psi:{\displaystyle \left\langle \frac{\delta\Sm(F)}{\delta\varphi},\psi\right\rangle =\Sm^{\left(1\right)}\Big|_{F}(\left\langle \frac{\delta F}{\delta\varphi},\psi\right\rangle })\,.\]

\end{description}
While {[}C1{]} and {[}C2{]} are directly related to the inductive
procedure of Epstein-Glaser, condition {[}C3{]} implies that only
finitely many terms will contribute if one cuts the perturbative expansion
of the $\Sm$-matrix at a given order in $\hbar$, see also the discussion
at the end of Chapter~\ref{cha:DimReg-PositionSpace}. This makes
it possible to regard also more general, and in particular non-polynomial
interactions $F\in\fps{\mathcal{F}_{\loc}(\M)}{\hbar}$ in pAQFT.
Furthermore {[}C3{]} implies together with the fourth condition {[}C4{]}
the Wick expansion formula for the time-ordered product of Epstein
and Glaser \citep{Epstein1973}. This is needed to reduce the problem
of renormalizing $\Sm$ to an extension problem for distributions.
See the discussion in \citep[Sec.~4.1]{Brunetti2009} and also \citep[Sec.~4.B]{Keller2009}.

The freedom in the definition of the $\Sm$-matrix is described by
the Stückelberg-Petermann renormalization group $\R$. In the framework
of perturbative Algebraic Quantum Field Theory $\R$ is the group
of analytic maps\[
Z:\fps{\mathcal{F}_{\loc}(\M)}{\hbar}\rightarrow\fps{\mathcal{F}_{\loc}(\M)}{\hbar}\,,\]
with composition as group operation, and $Z\in\R$ having the following
properties,
\begin{description}[topsep=1mm]
\item [{{[}RG1{]}}] $Z(0)=0$
\item [{{[}RG2{]}~Starting~Element.}] $Z^{\left(1\right)}(0)=\id$
\item [{{[}RG3{]}}] $Z=\id+\mathcal{O}(\hbar)$
\item [{{[}RG4{]}~Locality.}] Let $A,B,C\in\fps{\mathcal{F}_{\loc}(\M)}{\hbar}$
with $\supp(A)\cap\supp(C)=\emptyset$, then\[
Z(A+B+C)=Z(A+B)-Z(B)+Z(B+C)\]

\item [{{[}RG5{]}~$\varphi$-Locality.}] $Z(F)(\varphi_{0})=Z(F_{\varphi_{0}}^{\left[N\right]})(\varphi_{0})+\mathcal{O}(\hbar^{N+1})$
\item [{{[}RG6{]}~Field~Independence.}] $Z$ depends only implicitly
on the field $\varphi$, \[
\forall\varphi\in\E(\M):{\displaystyle \frac{\delta Z}{\delta\varphi}=0}\,.\]

\end{description}
With these definitions at hand, we can now formulate
\begin{thm}
[{Main Theorem of Renormalization, cf.~\citep[Thm.~4.1]{Brunetti2009}}]
\label{thm:MainTheoremRenormalization}Given two $\Sm$-matrices $\Sm$
and $\widehat{\Sm}$ satisfying the conditions Causality, Starting
Element, $\varphi$-Locality, and Field Independence, {[}C1{]}-{[}C4{]},
there exists a unique $Z\in\R$ such that\begin{equation}
\widehat{\Sm}=\Sm\circ Z\,.\label{eq:MainTheoremRenormalization}\end{equation}
Conversely, given an $\Sm$-matrix $\Sm$ satisfying {[}C1{]}-{[}C4{]}
and a $Z\in\R$, then (\ref{eq:MainTheoremRenormalization}) defines
a new $\Sm$-matrix satisfying conditions {[}C1{]}-{[}C4{]}.
\end{thm}
We will be interested in this chapter mainly in a special class of
scattering matrices, which we define now.
\begin{defn}
[{Analytically Regularized $\Sm$-matrix}]\label{def:AnalyticallyRegularizedSmatrix}Any
scattering matrix, $\Sm_{\kappa}$, which fulfills the conditions
{[}C1{]}-{[}C4{]} and depends analytically on an additional parameter
$\kappa\in\Omega\backslash\left\{ 0\right\} \subset\mathbb{C}$, such
that for all $n\in\mathbb{N}$, $n\geq2$, the $n$-fold functional
derivative, \[
\Sm_{\kappa}^{\left(n\right)}(0):\fps{\mathcal{F}_{\loc}(\M)}{\hbar}^{\otimes n}\rightarrow\fps{\mathcal{F}(\M)}{\hbar}\,,\]
is the analytic regularization of a time-ordered product outside the
large diagonal in the sense of Proposition~\ref{pro:RegularizedAmplitude},
we want to call an \emph{analytically regularized $\Sm$-matrix.}
\end{defn}
Observe that the definition implies that the second derivative $\Sm_{\kappa}^{\left(2\right)}(0)$
corresponds to an analytic regularization in the strict sense of Definition~\ref{def:Regularization}.
In the functional framework this implies\[
\pp(\Sm_{\kappa}^{\left(2\right)}(0)):\fps{\mathcal{F}_{\loc}(\M)}{\hbar}^{\otimes2}\rightarrow\fps{\mathcal{F}_{\loc}(\M)}{\hbar}\,,\]
cf.~Section~\ref{sec:MS-For-TProduct}. An example for such an analytically
regularized $\Sm$-matrix is the unique dimensionally regularized
$\Sm$-matrix $\Sm_{\mu,\zeta}$ of Definition~\ref{def:DimRegSMatrix}.
This follows directly from its construction, since it was defined
using the methods of Epstein-Glaser renormalization. However, that
$\Sm_{\mu,\zeta}$ fulfills {[}C1{]}-{[}C4{]} is also readily seen
from its perturbative expansion (\ref{eq:PerturbationSeries}). And
we will take $\Sm_{\mu,\zeta}$ as an example, wherever it is necessary
to introduce a regularization in the discussion below.

By the above theorem, the Stückelberg-Petermann renormalization group
acts transitively on all $\Sm$-matrices fulfilling {[}C1{]}-{[}C4{]}.
Thus, if we want to find a finitely regularized $\Sm$-matrix $\Sm_{\mu,\zeta,\ren}$
which also fulfills {[}C1{]}-{[}C4{]} we will have to construct an
element $Z_{\mu,\zeta}$ of the Stückelberg-Petermann renormalization
group, such that \begin{equation}
\Sm_{\mu,\zeta,\ren}=\Sm_{\mu,\zeta}\circ Z_{\mu,\zeta}\,.\label{eq:FiniteRegSMatrixByRGTransform}\end{equation}
has a limit $\zeta\rightarrow0$ in the set of $\Sm$-matrices. That
is\begin{equation}
\Sm_{\mu,\ren}:=\lim_{\zeta\rightarrow0}\left(\Sm_{\mu,\zeta}\circ Z_{\mu,\zeta}\right)\label{eq:RenSMatrixByRGTransform}\end{equation}
exists in the sense of formal power series in $\hbar$ term by term
in the perturbative expansion; see also \citep[Sec.~5.2]{Brunetti2009}.
In Epstein-Glaser renormalization the construction of these local
counterterms, i.e., the perturbative definition of the map $Z$ has
to be done recursively, i.e., term by term in the perturbative expansion
starting with the counterterm $Z^{\left(2\right)}$ for $\Sm^{\left(2\right)}(0)$.
There will be a choice involved in each step of this recursion and
hence it is impossible to express it in an algorithm which computes,
say, the $n$'th counterterm. However, in the case we dispose of a
regularization, $\Sm_{\mu,\zeta}$, the second term, $\Sm_{\mu,\zeta}^{\left(2\right)}$,
is a regularization in the strict sense, and hence we have a preferred
choice for the local counterterm,\[
Z_{\mu,\zeta}^{\left(2\right)}=-\pp(\Sm_{\mu,\zeta}^{\left(2\right)}):\fps{\mathcal{F}_{\loc}(\M)}{\hbar}^{\otimes2}\rightarrow\fps{\mathcal{F}_{\loc}(\M)}{\hbar}\,.\]
which renormalizes the time ordered product, \[
\Sm_{\mu,\zeta,\ren}^{\left(2\right)}:=\rp(\Sm_{\mu,\zeta}^{\left(2\right)})=\left(1-\pp\right)\Sm_{\mu,\zeta}^{\left(2\right)}\,.\]
It will be show in the following section that this preferred choice
can be done at all orders of perturbation theory in a consistent way,
i.e., with local counterterms at all orders. This, in turn, makes
it possible to solve the recursive renormalization procedure of Epstein-Glaser
in quite the same way as it was done by Zimmermann in 1969 for BPH
in momentum space. We will derive from Equation~(\ref{eq:FiniteRegSMatrixByRGTransform})
a forest formula for Epstein-Glaser renormalization which solves the
recursive construction of counterterms to all orders in causal perturbation
theory. We want to remark that the choices at all orders are unique
in the minimal subtraction scheme, such that this leads to a recursive
procedure, which, in principle, can also be taught to a computer -
in contrast to the original Epstein-Glaser method.

The relation of the presented method to the modern formulation of
renormalization in terms of Hopf algebras will be given in Section~\ref{sec:MoreThanHopfAlgebra}.

\section{\label{sec:EGFF}A Forest Formula for Epstein-Glaser Renormalization}

Since we will stay in the functional framework throughout the derivation
of the forest formula, the result will be valid independent of the
chosen representation, in particular it holds for momentum space as
well as position space, whatever is the best suited representation
for the regularization. Furthermore, it is formulated without regard
to the graph expansion of the time-ordered product. Partitions will
take the place of graphs as the basic combinatorial objects. However,
analogous to the discussion in Section~\ref{sec:MS-For-TProduct},
the forest formula for Epstein-Glaser renormalization also holds in
a graph by graph manner, and then implies Zimmermann's forest formula
of \citep{Zimmermann1969} enhanced by his discussion on spurious
subtractions in \citep{Zimmermann1975}; see also Proposition~\ref{pro:RedundantProjections}
and the discussion thereafter. However, the combinatorial structures
used here will make the role of forests in Zimmermann's formula even
more transparent.

That a version of Zimmermann's forest formula should also exist in
position space was observed before. And the assertion is natural considering
the common origin of BPHZ and Epstein-Glaser renormalization. It has
been shown that Zimmermann's Taylor subtractions with respect to external
momenta of the graphs correspond to the $W$-projections in the Epstein-Glaser
framework \citep{Prange1999,Prange2000}. Gracia-Bondía and Lazzarini
gave a direct translation of this {}``Taylor surgery'' (GB) to position
space by considering a more general test function space for the {}``infrared
regulators'', i.e. the test functions $w_{\alpha}$ of Lemma~\ref{lem:W-Projection-Functions},
in fact they allowed the $w_{\alpha}$ to be distributions of the
Cesàro type \citep{Gracia-Bondia2003,GraciaBondiaLazzarini2003}.
A translation of the complete forest formula to position space was
given by Steinmann in the case of QED \citep{Steinmann2000}. However,
Steinmann's treatment was unsatisfactory in two points. First, Steinmann's
formulation involves the differentiation of (generalized) functions
at singular points. This was recognized by the author himself and
is due to the fact that the implicit regularization
\footnote{I hope this is the only spot in the thesis where I use the word {}``regularization''
only in the sense of {}``making things well-defined''.
} of the momentum space framework, namely the fact that Zimmermann
performs his manipulations on the integral kernel of the convolution
rather than the integral itself, has no counterpart in position space.
Momentum space convolution corresponds to the pointwise product (of
distributions) in position space and it is partly due to this implicit
regularization that momentum space integrals were introduced in perturbative
quantum field theory in the first place \citep{Bogoliubow1957}. Second,
Steinmann regards Quantum Electro Dynamics (QED). The fact that QED
has only one basic vertex of valence three implies that there are
no graphs with less lines but the same set of vertices so that the
spurious subtractions do not occur in QED and other theories {}``of
graphical $\varphi^{3}$-type''. Consequently, Steinmann's version
of the forest formula cannot be considered as a complete translation
of Zimmermann's forest formula (which treats general graphs in $\mathcal{G}$)
to position space. Observe that Zimmermann implements a preferred
choice for the extension at all orders in perturbation theory by performing
his Taylor subtractions always at zero external momentum. In order
to define this rigorously he has to introduce additional maps which
conceal part of the underlying pattern. However, as already remarked
above and as will be clear from the construction below, such a choice
of extension at all orders of perturbation theory is indispensable
for the solution of the recursive procedure of Bogoliubov, Parasiuk
and Hepp, or Epstein and Glaser, respectively. We start by exploring
the termwise structure of the main theorem (Theorem~\ref{thm:MainTheoremRenormalization}),
by applying the $n$-fold derivative to (\ref{eq:MainTheoremRenormalization}).
The Faà~di~Bruno formula arises naturally.

\subsection{Faà~di~Bruno's formula}

In 1855 Francesco Faà~di~Bruno proved a formula for the $n$-fold
chain rule \citep{FaaDiBruno1855}. And it is quite appealing that
this old formula, when applied to Equation~(\ref{eq:MainTheoremRenormalization})
gives a termwise version of the main theorem of perturbative renormalization
(Theorem~\ref{thm:MainTheoremRenormalization}). Considering the
time since its first proof, there are quite a few versions of Faà~di~Bruno's
formula in the literature today. However, in order to keep the relation
to causal perturbation theory and BPHZ renormalization visible at
all steps in our calculation, a set partition version of the form
found in \citep[p.~219]{Johnson2002} seems to be the most appropriate.
We prove here an adjusted version. But let us first give an easy definition,
mainly to fix notation.
\begin{defn}
[Partition, Blocks]\label{def:Partition}By a \emph{partition} $\Ptn$
of a finite set $V$ we mean any set of non-empty, disjoint subsets
$V_{i}\subset V$, $i\in I$, such that\[
V=\dot{\bigcup_{i\in I}}V_{i}\,,\qquad\mbox{that is, }\quad\Ptn=\left\{ V_{i}:i\in I\right\} ,\]
where $\dot{\cup}$ denotes disjoint union. We refer to the non-empty,
disjoint subsets $V_{i}$ as \emph{blocks} of $\Ptn$, and denote
the set of all partitions of $V$ by $\Part V$.
\end{defn}
We generally consider partitions of the set which corresponds to the
set of vertices, $V(\Gamma)$, in the graphical representation, and
in most cases it will be more convenient to regard instead the set
of numbers $\left\{ 1,\dots,n\right\} $. However, this implicit numbering
of vertices, is irrelevant for the derivation due to the symmetry
of the functional derivative briefly introduced in Section~\ref{sec:PreliminariesDifferentialCalculus}.
\begin{lem}
[Main Theorem - termwise]\label{lem:FaaDiBrunoSetPartitionVersion}Let
$\Sm:\fps{\mathcal{F}_{\loc}(\M)}{\hbar}\rightarrow\fps{\mathcal{F}(\M)}{\hbar}$
be an $\Sm$-matrix fulfilling conditions {[}C1{]}-{[}C4{]} and let
$Z:\fps{\mathcal{F}_{\loc}(\M)}{\hbar}\rightarrow\fps{\mathcal{F}_{\loc}(\M)}{\hbar}$
be an element of the Stückelberg-Petermann renormalization group,
$Z\in\R$. Then the $n$th term in the perturbative expansion of the
transformed $\Sm$-matrix, $\widehat{\Sm}=\Sm\circ Z$, is given by
\begin{equation}
\left(\Sm\circ Z\right)^{\left(n\right)}\!(0)=\sum_{\Ptn\in\Part\{1,\dots,n\}}\!\!\Sm^{\left(\left|\Ptn\right|\right)}(0)\cdot\left(\bigotimes_{I\in\Ptn}\left[Z^{\left(\left|I\right|\right)}(0)\right]\right),\label{eq:FaaDiBrunoPartitionVersion}\end{equation}
where the sum is over all partitions $\Ptn$ of the index set $\left\{ 1,\cdots,n\right\} $
into $\left|\Ptn\right|$ blocks, \linebreak[4]and ''~$\cdot$~''
denotes the composition of linear maps,\begin{equation}
\fps{\mathcal{F}_{\loc}(\M)}{\hbar}^{\otimes n}\xrightarrow{\bigotimes_{I\in\Ptn}Z^{\left(\left|I\right|\right)}(0)}\fps{\mathcal{F}_{\loc}(\M)}{\hbar}^{\otimes\left|\Ptn\right|}\xrightarrow[\hphantom{\bigotimes_{I\in\Ptn}Z^{\left(\left|I\right|\right)}(0)}]{\Sm^{\left(\left|\Ptn\right|\right)}(0)}\fps{\mathcal{F}(\M)}{\hbar}.\label{eq:CompositionDiagram}\end{equation}
\end{lem}
\begin{proof}
We prove (\ref{eq:FaaDiBrunoPartitionVersion}) by induction following
\citep{Johnson2002}. For $n=1$ we have,\[
\left(\Sm\circ Z\right)^{\left(1\right)}\!\bigg|_{F}=\Sm^{\left(1\right)}\bigg|_{Z(F)}\cdot Z^{\left(1\right)}\bigg|_{F}\,.\]
However, evaluating at $F=0$ gives an empty assertion ($\id=\id$)
due to the starting element conditions {[}C2{]}, {[}RG1{]}, and {[}RG2{]}.
The first non-trivial contribution is from the second derivative,
which we compute explicitly for illustration,\begin{align*}
\left(\Sm\circ Z\right)^{\left(2\right)}\!\bigg|_{F} & =\left(\Sm^{\left(1\right)}\bigg|_{Z(F)}\cdot Z^{\left(1\right)}\bigg|_{F}\right)^{\left(1\right)}\!\bigg|_{F}\\
 & =\Sm^{\left(2\right)}\bigg|_{Z(F)}\cdot Z^{\left(1\right)}\bigg|_{F}\otimes Z^{\left(1\right)}\bigg|_{F}+\Sm^{\left(1\right)}\bigg|_{Z(F)}\cdot Z^{\left(2\right)}\bigg|_{F}\,,\end{align*}
where the two terms correspond to the two partitions $\left\{ \left\{ 1\right\} ,\left\{ 2\right\} \right\} $
and $\left\{ \left\{ 1,2\right\} \right\} $ of $\left\{ 1,2\right\} $.
Evaluating at $F=0$ gives, again by using {[}C2{]}, {[}RG1{]}, and
{[}RG2{]}, \begin{align*}
\left(\Sm\circ Z\right)^{\left(2\right)}\!(0) & =\Sm^{\left(2\right)}(0)+Z^{\left(2\right)}(0)\,,\end{align*}
and $Z^{\left(2\right)}=\left(\Sm\circ Z\right)^{\left(2\right)}-\Sm^{\left(2\right)}$
is found to be the counterterm at second order.

For the induction step regard the derivative of (\ref{eq:FaaDiBrunoPartitionVersion}).
By (\ref{eq:ChainRule})/(\ref{eq:LeibnizRule}) we get, \begin{align}
\left[\Sm^{\left(\left|\Ptn\right|\right)}\!\Big|_{Z(F)}\!\left(\bigotimes_{I\in\Ptn}\left[Z^{\left(\left|I\right|\right)}\Big|_{F}\right]\right)\right]^{\left(1\right)} & =\Sm^{\left(\left|\Ptn\right|+1\right)}\Big|_{Z(F)}\!\left(Z^{\left(1\right)}\Big|_{F}\otimes\bigotimes_{I\in\Ptn}\left[Z^{\left(\left|I\right|\right)}\Big|_{F}\right]\right)\nonumber \\
 & \hspace{-13mm}+\sum_{I'\in\Ptn}\Sm^{\left(\left|\Ptn\right|\right)}\Big|_{Z(F)}\!\left(Z^{\left(\left|I'\right|+1\right)}\Big|_{F}\otimes\bigotimes_{I\in\Ptn\backslash\left\{ I'\right\} }\left[Z^{\left(\left|I\right|\right)}\Big|_{F}\right]\right).\label{eq:FaaDiBrunoPartitionVersion-ProofInductionStep}\end{align}
Any partition $\widetilde{\Ptn}$ of $\left\{ 1,\dots,n+1\right\} $
can be written in terms of a partition $\Ptn$ of\linebreak{}
 $\left\{ 1,\dots,n\right\} $ by either adjoining $\left\{ n+1\right\} $
as a block of its own, or by adding $\left\{ n+1\right\} $ to one
of the blocks in $\Ptn$, i.e., \[
\widetilde{\Ptn}=\Ptn\cup\left\{ \left\{ n+1\right\} \right\} \qquad\mbox{or}\qquad\widetilde{\Ptn}=\left(\Ptn\backslash\left\{ I'\right\} \right)\cup\left\{ I'\cup\left\{ n+1\right\} \right\} \,,\]
for some block $I'\in\Ptn$. Thus equation (\ref{eq:FaaDiBrunoPartitionVersion-ProofInductionStep})
contains all partitions of $\left\{ 1,\dots,n+1\right\} $ which can
be obtained from $\Ptn$. Evaluating at $F=0$ gives the result.
\end{proof}
Equation (\ref{eq:FaaDiBrunoPartitionVersion}) describes the action
of the St{\"u}ckelberg-Petermann group on time-ordered products.
This action followed directly from the main theorem by applying Fa{\`a}~di~Bruno's
formula. Since Equation~(\ref{eq:FaaDiBrunoPartitionVersion}) is
not the most cited version of Faà~di~Bruno's formula, we want to
show that it reduces to the more prevalent versions if we evaluate
$\left(\Sm\circ Z\right)^{\left(n\right)}\!(0)$ at the $n$-fold
tensor power of one and the same interaction functional, $F^{\otimes n}\in\fps{\mathcal{F}_{\loc}(\M)}{\hbar}^{\otimes n}$.
Due to the symmetry of the functional derivative partitions with identical
block sizes will give the same contribution to $\left(\Sm\circ Z\right)^{\left(n\right)}\!(0)(F^{\otimes n})$.
So the question is: How many of them are there? Depending on how one
chooses to sort these partitions, one gets the different versions
of Faà~di~Bruno's formula. As an example we give one of the derivations.

Let $\Ptn\in\Part\{1,\dots,n\}$ be a partition with $\left|\Ptn\right|=k$
blocks. Let $\left(l_{1},\dots,l_{k}\right)\in\mathbb{N}^{k}$ denote
the sizes of these blocks, $l_{1}+\cdots+l_{k}=n$, $l_{i}\geq1$.
There are ${n \choose l_{1},\dots,l_{k}}$ possibilities to distribute
$n$ elements among $k$ different blocks of specified size and order.
\footnote{${n \choose l_{1},\dots,l_{k}}:=\frac{n!}{l_{1}!\cdots l_{k}!}$ denotes
the \emph{multinomial coefficient}, see, e.g., \citep{HarrisHirstMossinghoff2008}.
} However, for a partition the order of the blocks is irrelevant. In
a sum over the multiindex $\left(l_{1},\dots,l_{k}\right)\in\mathbb{N}^{k}$
we thus have to divide by the number of permutations of $\left\{ l_{1},\dots,l_{k}\right\} $
to reduce it to a sum over all partitions. In total we get\begin{align}
 & \left(\Sm\circ Z\right)^{\left(n\right)}\!(F^{\otimes n})\label{eq:FaaDiBrunoCauchyProductFormula}\\
 & \qquad=\sum_{k=1}^{n}\frac{1}{k!}\Sm^{\left(k\right)}\cdot\sum_{\substack{l_{1}+\cdots+l_{k}=n\\
l_{i}\geq1}
}{n \choose l_{1},\dots,l_{k}}Z^{\left(l_{1}\right)}(F^{\otimes l_{1}})\otimes\cdots\otimes Z^{\left(l_{k}\right)}(F^{\otimes l_{k}})\,,\nonumber \end{align}
which was the starting point in \citep{FigueroaGraciaBondiaVarilly2005}
for the derivation of\begin{align}
 & \left(\Sm\circ Z\right)^{\left(n\right)}\!(F^{\otimes n})\label{eq:FaaDiBrunoSortedBlockSize}\\
 & \qquad=\sum_{k=1}^{n}\sum_{\lambda_{1},\dots,\lambda_{n}}\frac{n!}{\lambda_{1}!\cdots\lambda_{n}!}\Sm^{\left(k\right)}\!\cdot\left(\frac{Z^{\left(1\right)}}{1!}\right)^{\otimes\lambda_{1}}\negmedspace\otimes\cdots\otimes\negmedspace\left(\frac{Z^{\left(n\right)}}{n!}\right)^{\otimes\lambda_{n}}\!(F^{\otimes n}).\nonumber \end{align}
Here $\lambda_{l}\in\mathbb{N}_{0}$ denotes the number of blocks
of size $l$. Equation (\ref{eq:FaaDiBrunoSortedBlockSize}) is the
version, which is usually cited as Faà~di~Bruno's formula in the
literature, and often is the starting point for the introduction of
the Faà~di~Bruno bialgebra. We will learn more about this bialgebra
in Section~\ref{sec:MoreThanHopfAlgebra}.

\subsection{Minimal Subtraction}

Regard an analytically regularized $\Sm$-matrix, e.g., $\Sm_{\mu,\zeta}$.
Then Lemma~\ref{lem:FaaDiBrunoSetPartitionVersion} implies a recursion
relation for the counterterms $Z_{\mu,\zeta}^{\left(n\right)}$ in
the minimal subtraction renormalization scheme introduced in the previous
chapter.
\begin{cor}
[Recursion Relation for MS Counterterms]\label{cor:RecursionMSCounterterms}
In the minimal subtraction renormalization scheme (MS), a recursion
relation for the counterterms $Z_{\mu,\zeta}^{\left(n\right)}$ is
given by\begin{equation}
Z_{\mu,\zeta}^{\left(n\right)}=-\pp\sum_{\Ptn\in\Part\{1,\dots,n\}\backslash\left\{ \Ptn_{1}\right\} }\!\!\Sm_{\mu,\zeta}^{\left(\left|\Ptn\right|\right)}\cdot\left(\bigotimes_{I\in\Ptn}Z_{\mu,\zeta}^{\left(\left|I\right|\right)}\right).\label{eq:CountertermsMSRecursionFormula}\end{equation}
The counterterms are local, and all counterterms on the right hand
side are of lower order than $n$, since the only partition in $\Part\{1,\dots,n\}$
with a block containing $n$ elements is removed from the sum, $\Ptn_{1}=\left\{ \left\{ 1,\dots,n\right\} \right\} $.\end{cor}
\begin{proof}
Using the starting element condition {[}C2{]} we get from (\ref{eq:FaaDiBrunoPartitionVersion}),
\[
\left(\Sm_{\mu,\zeta}\circ Z_{\mu,\zeta}\right)^{\left(n\right)}=Z_{\mu,\zeta}^{\left(n\right)}+\sum_{\Ptn\in\Part\{1,\dots,n\}\backslash\left\{ \Ptn_{1}\right\} }\!\!\Sm_{\mu,\zeta}^{\left(\left|\Ptn\right|\right)}\cdot\left(\bigotimes_{I\in\Ptn}Z_{\mu,\zeta}^{\left(\left|I\right|\right)}\right).\]
By assumption $Z_{\mu,\zeta}\in\R$ is a renormalization group transformation
which renders the limit $\zeta\rightarrow0$ of the left hand side
finite, cf.~Equation~(\ref{eq:RenSMatrixByRGTransform}). Thus the
counterterm $Z_{\mu,\zeta}^{\left(n\right)}$ has to subtract at least
the principal part of the sum on the right hand side. In the minimal
subtraction scheme, $Z_{\mu,\zeta}^{\left(n\right)}$ is fixed by
the requirement to remove exactly the principal part, hence formula
(\ref{eq:CountertermsMSRecursionFormula}). The locality of $Z_{\mu,\zeta}^{\left(n\right)}$
is implied directly by the functional calculus, cf.~(\ref{eq:-n-foldDerivativeLocalMap}).
\end{proof}
The expert reader
\footnote{I assume here that $\left\{ \mbox{readers}\right\} \neq\emptyset$.
If you have a proof, please tell me: \href{mailto:kai.johannes.keller@desy.de}{kai.johannes.keller@desy.de}
} readily recognizes the similarity of (\ref{eq:CountertermsMSRecursionFormula})
to the recursive formula for the antipode in the Faà~di~Bruno bialgebra.
However, observe that there are two products involved in the recursion
for the counterterms. The tensor product $\otimes$ and the composition
of linear maps {}``~$\cdot$~''. What might be obvious for the
expert, namely that this is a structure which cannot be described
by a commutative Hopf algebra alone, will be derived {}``by foot''
in Section~\ref{sec:MoreThanHopfAlgebra}. However, let us first
give the derivation of a forest formula for the $n$-fold finitely
regularized time-ordered product $\Sm_{\mu,\zeta,\ren}^{\left(n\right)}$.
The forest formula will solve the inductive construction of the renormalization
group transformation $Z_{\mu,\zeta}$, which renders $\Sm_{\mu,\zeta,\ren}=\Sm_{\mu,\zeta}\circ Z_{\mu,\zeta}$
finite (in the sense of formal power series in $\hbar$) in the limit
$\zeta\rightarrow0$.

\subsection{Derivation of the forest formula}

We begin with the definition of a forest as it was given by Zimmermann.
We will then derive some relations to partitions which will motivate
the definition of an Epstein-Glaser forest and make it possible to
give a transparent proof of the forest formula.

Let us start with Zimmermann's definition, which was contained in
\citep{Zimmermann1969}. However, we incorporate directly his results
from \citep{Zimmermann1975} and will only consider subgraphs, which
are given by a subset of the set of vertices of a graph $\Gamma$
and all lines in $\Gamma$ connecting them, cf. Definition~\ref{def:Subgraph}.
Zimmermann called such subgraphs \emph{full vertex parts}. Observe
that this makes it possible to work directly with the set of vertices
instead of the set of general graphs or full vertex parts. However,
to keep the relation to the original definitions transparent, let
$\Gamma\in\mathcal{G}$ be a graph.

A \emph{$\Gamma$-forest} $U$ is a set of subgraphs $\gamma\subseteq\Gamma$,
such that any two elements $\gamma,\gamma'\in U$ are \emph{non-overlapping},
i.e.\begin{equation}
\mbox{either}\quad\gamma\subset\gamma'\quad\mbox{or}\quad\gamma'\subset\gamma\quad\mbox{or}\quad\gamma\cap\gamma'=\emptyset\,.\label{eq:ForestProperty}\end{equation}
The empty set is referred to as the empty forest. The notation $\gamma\cap\gamma'=\emptyset$
means that $V(\gamma)\cap V(\gamma')=\emptyset$, and it follows from
the definition of a subgraph (Definition~\ref{def:Subgraph}) that
then also the sets of edges are disjoint, $E(\gamma)\cap E(\gamma')=\emptyset$.
A graph $\gamma\in U$ is called \emph{maximal} if there is no other
graph in $U$ containing it. A $\Gamma$-forest $U$ is called \emph{maximal},
if there is no other $\Gamma$-forest containing it. A forest is called
\emph{restricted}, if it contains only {}``divergent graphs'', $\div(\gamma)\geq0$,
cf. (\ref{eq:DegreeOfDivergenceGraph}). Note that a (Feynman-) graph
$\Gamma$ has more than one maximal restricted forest, if and only
if it has \emph{overlapping divergences}, i.e., at least two divergent
subgraphs $\gamma,\gamma'\subset\Gamma$ for which (\ref{eq:ForestProperty})
does not hold. Given a forest $U$ of $\Gamma$ and a subgraph $G\in U$,
we define the set\[
U(G):=\left\{ \gamma\in U:\gamma\subseteq G\right\} ,\]
and note that $U(G)$ is a $G$-forest as well as a $\Gamma$-forest.
\begin{lem}
[Structure of Maximal Forests]\label{lem:StructureMaxForests}Let
$U$ be a maximal $\Gamma$-forest.
\begin{enumerate}
\item \label{enu:ContainsComplement}With any element $G\in U$, $U$ contains
also its vertex complement, $\Gamma\obs G\in U$.
\item \label{enu:MaxSubForest}For any graph $G\in U$ the set $U(G)=\left\{ \gamma\in U:\,\gamma\subseteq G\right\} $
is a maximal $G$-forest.
\item \label{enu:DisjointUnion}The forest $U\backslash\left\{ \Gamma\right\} $
is the disjoint union of two maximal forests. If $G\in U\backslash\left\{ \Gamma\right\} $
is a maximal element, then\[
U\backslash\left\{ \Gamma\right\} =U(G)\dot{\cup}U(\Gamma\obs G)\,.\]

\end{enumerate}
\end{lem}
\begin{proof}
(\ref{enu:ContainsComplement}). Let $G\in U$, then for any element
$\gamma\in U$, we have either $\gamma\subset G$ or $\gamma\subset\Gamma\obs G$.
Hence $U\cup\left\{ \Gamma\obs G\right\} $ is a forest, and by maximality
of $U$: $\Gamma\obs G\in U$.

(\ref{enu:MaxSubForest}). Let $U'$ be a $G$-forest properly containing
$U(G)$. Then there is a subgraph $\gamma'\subset G$ such that $\gamma'\notin U(G)$
does not overlap with any of the elements in $U(G)$. Since $G\in U$
it follows that $\gamma'\notin U$ is non-overlapping with any element
in $U$. Thus $U\dot{\cup}\left\{ \gamma'\right\} $ is a forest,
in contradiction with the maximality of $U$.

(\ref{enu:DisjointUnion}) follows from (\ref{enu:ContainsComplement})
and (\ref{enu:MaxSubForest}).
\end{proof}
The combinatorial result, which leads to the forests formula for Epstein-Glaser
renormalization, is that we can write (maximal) forests of the kind
defined above in terms of (complete) sets of partitions of the vertex
set.
\begin{lem}
[Partitions and Forests]\label{lem:MaximalForestsCompleteTotallyOrderedPartitions}The
set of partitions of $n$ elements,\linebreak[4] $\Part\{1,\dots,n\}$,
is a partially ordered set (poset). The partial order is defined by
saying that $\Ptn_{1}$ is finer than $\Ptn_{2}$ (and $\Ptn_{2}$
is coarser than $\Ptn_{1}$),\[
\Ptn_{1}\leq\Ptn_{2}\,,\]
if for any block $I\in\Ptn_{1}$ there is a block $J\in\Ptn_{2}$
containing $I$. $\left(\Part\{1,\dots,n\},\leq\right)$ is a complete
partial order (cpo) with finest element $\left\{ \left\{ 1\right\} ,\dots,\left\{ n\right\} \right\} $
and coarsest element $\left\{ \left\{ 1,\dots,n\right\} \right\} $.

Let $\Pto\subset\Part\{1,\dots,n\}$ be a totally ordered subset.
We call $\Pto$ maximal, if there is no totally ordered subset $\Pto'\subset\Part\{1,\dots,n\}$
containing $\Pto$. The union of any totally ordered subset $\Pto$
is a forest,\[
U(\Pto):=\bigcup_{\Ptn'\in\Pto}\Ptn'\,.\]
$U(\Pto)$ is a maximal forest, if $\Pto$ is maximal.\end{lem}
\begin{proof}
{}``$\leq$'' is reflexive ($\Ptn\leq\Ptn$), antisymmetric ($\Ptn_{1}\leq\Ptn_{2}\,\wedge\,\Ptn_{2}\leq\Ptn_{1}$
$\Rightarrow$ $\Ptn_{1}=\Ptn_{2}$), and transitive ($\Ptn_{1}\leq\Ptn_{2}\,\wedge\,\Ptn_{2}\leq\Ptn_{3}$
$\Rightarrow$ $\Ptn_{1}\leq\Ptn_{3}$), hence a partial order. For
$\left(\Part\{1,\dots,n\},\leq\right)$ to be a cpo, we have to show
that any pair $\Ptn_{1},\Ptn_{2}\in\Part\{1,\dots,n\}$ has a least
upper bound $\Ptn_{1}\sqcup\Ptn_{2}$ and a greatest lower bound $\Ptn_{1}\sqcap\Ptn_{2}$
in $\Part\{1,\dots,n\}$. The assertion is trivial, if $\Ptn_{1}$
and $\Ptn_{2}$ are related by {}``$\leq$'', hence let $\Ptn_{1}$
and $\Ptn_{2}$ not be related by {}``$\leq$''. $\Ptn_{1}\sqcup\Ptn_{2}$
is the partition where all overlapping blocks (cf.~(\ref{eq:ForestProperty}))
are replaced by their union; filled up with the larger blocks of either
$\Ptn_{1}$ or $\Ptn_{2}$. In $\Ptn_{1}\sqcap\Ptn_{2}$ overlapping
blocks are replaced by their intersection and filled up with the smaller
sets of either $\Ptn_{1}$ or $\Ptn_{2}$, see Figure~\ref{fig:PartitionBounds}.
Since $\Ptn_{1}\sqcup\Ptn_{2}$ and $\Ptn_{1}\sqcap\Ptn_{2}$ are
partitions of $\left\{ 1,\dots,n\right\} $, we infer that $\Part\{1,\dots,n\}$
is a cpo.

Regard a totally ordered subset $\Pto\subset\Part\{1,\dots,n\}$ and
let $\Ptn\in\Pto$, then by definition $I\cap J=\emptyset$ for all
$I,J\in\Ptn$. Let $\Ptn,\Ptn'\in\Pto$ be two different partitions,
$\Ptn'\neq\Ptn$, then we have either $I\cap J=\emptyset$, $I\subset J$,
or $J\subset I$ for any pair $\left(I,J\right)\in\Ptn\times\Ptn'$
since $\Pto$ is totally ordered. Thus $I$ and $J$ are non-overlapping,
and $U(\Pto)$ is a forest. Let $\Pto$ be maximal, then it contains
$\left\{ \left\{ 1,\dots,n\right\} \right\} $ and $\left\{ \left\{ 1\right\} ,\dots,\left\{ n\right\} \right\} $,
and with any partition $\Ptn\in\Pto$ it contains all partitions which
can be constructed out of $\Ptn$ by successively dividing any of
its blocks into a pair of disjoint subsets. Let $I=I_{1}\dot{\cup}I_{2}$
be an index set, then there is no partition $\Ptn_{I}$ of $I$ such
that $\left\{ I_{1},I_{2}\right\} \leq\Ptn_{I}\leq\left\{ I\right\} $
and $\left\{ I_{1},I_{2}\right\} \neq\Ptn_{I}\neq\left\{ I\right\} $.
Hence the maximal set $\Pto$ can be constructed out of $\left\{ \left\{ 1,\dots,n\right\} \right\} $
by the successive division procedure described above. Conversely,
let $U$ be a maximal forest, by Lemma~\ref{lem:StructureMaxForests},
$U$ can be constructed in exactly the same way.
\end{proof}
\addtocounter{figure}{1}

\begin{figure}[th]
\includegraphics[width=5cm]{PartitionBounds}

\caption{\label{fig:PartitionBounds}Least upper and greatest lower bound of
a pair of partitions.}

\end{figure}

We will have to regard in the sequel unions of totally ordered sets
of partitions, which is non-trivial, since the set union of two totally
ordered sets of partitions will not be a totally ordered set of partitions,
just as the union of forests is not a forest in general.
\begin{defn}
[Position and Disjoint Union]\label{def:PositionDisjointUnionPto}Let
$\Pto\subset\Part\{1,\dots,n\}$ be a totally ordered set, then we
define the \emph{position} of any element $\Ptn\in\Pto$ by\[
\pos_{\Pto}(\Ptn):=\left|\left\{ \Ptn'\in\Pto:\Ptn\leq\Ptn'\right\} \right|.\]
It is easy to see that $\pos_{\Pto}(\Ptn_{c})=1$, if $\Ptn_{c}$
is the coarsest element of $\Pto$.

Let $\Pto(I)$ and $\Pto(J)$ be totally ordered subsets of $\Part I$
and $\Part J$, respectively. Any subset of \[
\left\{ \Ptn\dot{\cup}\Ptn':\Ptn\in\Pto(I)\mbox{ and }\Ptn'\in\Pto(J)\right\} ,\]
which is a totally ordered set of partitions of $I\dot{\cup}J$, we
call a \emph{disjoint union} of $\Pto(I)$ and $\Pto(J)$.
\end{defn}
Observe that there are forests which do not correspond to a totally
ordered subset of partitions, e.g., the empty forest or any forest
containing just one proper subset of $\left\{ 1,\dots,n\right\} $.
We now come to the definition of an Epstein-Glaser forest; a similar
definition for forests was considered in \citep{Figueroa2005} to
establish the relation to incidence Hopf algebras.
\begin{defn}
[Epstein-Glaser Forest]Let $V=\left\{ 1,\dots,n\right\} $ be a (vertex)
set. Then we call any totally ordered subset $\F$ of the set of partitions,
$\F\subset\Part\{1,\dots,n\}$, containing the finest partition $\Ptn_{n}:=\left\{ \left\{ 1\right\} ,\dots,\left\{ n\right\} \right\} $
of $V$, an \emph{Epstein-Glaser forest (EG forest)}, i.e., $\F$
has the form\[
\F=\left\{ \cdots\geq\Ptn_{n}\right\} ,\quad\Ptn_{n}=\left\{ \left\{ 1\right\} ,\dots,\left\{ n\right\} \right\} .\]
The EG forest containing only $\Ptn_{n}$ we denote by $\F_{n}:=\left\{ \Ptn_{n}\right\} $.
If an EG forest contains the coarsest partition $\Ptn_{1}:=\left\{ \left\{ 1,\dots,n\right\} \right\} $
we call it a \emph{full EG forest} \emph{(full forest)} and write
$\overline{\F}$. If an EG forest does not contain $\Ptn_{1}$, we
call it a \emph{normal EG forest (normal forest)} and write $\underline{\F}$.
For $n=1$ there is just one forest, the one with one vertex, and
we define this forest to be full. For $n>1$ there is a one to one
correspondence between full and normal forests, given by\[
\overline{\F}=\underline{\F}\cup\left\{ \Ptn_{1}\right\} \,.\]
An Epstein-Glaser forest $\F$ is called maximal, if $\F$ is maximal
as a totally ordered set of partitions.\end{defn}
\begin{cor}
\label{cor:NormalForestDisjointUnion}Any normal Epstein-Glaser forest
$\underline{\F}$ is a disjoint union of at least two full Epstein-Glaser
forests. If the coarsest partition in $\underline{\F}$ has $k$ elements,
then $\underline{\F}$ decomposes into $k$ full forests $\overline{\F}_{1},\dots,\overline{\F}_{k}$,
$k\geq2$, and we call $\underline{\F}$ $k$-fold connected. Conversely
one can say that $\underline{\F}$ is a disjoint union of $\overline{\F}_{1},\dots,\overline{\F}_{k}$
in the sense of Definition~\ref{def:PositionDisjointUnionPto},\begin{equation}
\underline{\F}=\dot{\bigcup_{i\in\left\{ 1,\dots,k\right\} }}\overline{\F}_{i}\,.\label{eq:NormalForestDecomposition}\end{equation}
If $I_{i}$ is the block of the coarsest partition in the full forest
$\overline{\F}_{i}$, then $\overline{\F}_{i}=\underline{\F}(I_{i})$.
In this sense the decomposition (\ref{eq:NormalForestDecomposition})
of $\underline{\F}$ is unique. See also Figure~\ref{fig:DecompositionAndDisjointUnion}.
\qed
\end{cor}

\begin{figure}
\includegraphics[height=2cm]{PartitionForests}

\caption{\label{fig:DecompositionAndDisjointUnion}The depicted normal forest
of $A\dot{\cup}B\dot{\cup}C$ decomposes into tree full forests $\overline{\F}_{A}$,
$\overline{\F}_{B}$, and $\overline{\F}_{C}$. Conversely, given
the three full forests, any composition of them, which preserves the
order of partitions in each component separately, gives a different
disjoint union, i.e. a different normal forest of $A\dot{\cup}B\dot{\cup}C$.}

\end{figure}

So far for the combinatorial part, we now have to define the analytic
part, namely, the minimal subtractions in the blocks of a partition.
\begin{defn}
[MS Operator]\label{def:MS-OperatorPartitions}For any partition
$\Ptn\in\Part\{1,\dots,n\}$ define the minimal subtraction operator
(MS operator)\[
-T_{\Ptn}^{\MS}\Sm_{\mu,\zeta}^{\left(n\right)}:=\Sm_{\mu,\zeta}^{\left(\left|\Ptn\right|\right)}\cdot\left(\bigotimes_{I\in\Ptn}R_{\left|I\right|}\left(\Sm_{\mu,\zeta}^{\left(\left|I\right|\right)}\right)\right),\quad\mbox{where }R_{k}=\begin{cases}
\id & \mbox{if }k=1\\
-\pp & \mbox{if }k>1\,,\end{cases}\]
cf.~Definition~\ref{def:MS-OperatorSubsets}.
\end{defn}
This defines the operator $-T_{\Ptn}^{\MS}$ on the whole regularized
time-ordered product $\Sm_{\mu,\zeta}^{\left(n\right)}$. Observe,
however, that the above definition implies that we have chosen another
regularization parameter in each block $I$ of the partition; there
is one operator $R_{\left|I\right|}$ for each block. To consider
all partitions, which possibly contribute to the principal part, we
will regard the $n$-fold regularized time-ordered product as being
regularized in ${n \choose 2}$ different regularization parameters,
one for each pair of vertices. This is certainly possible regarding
the fact that we have {}``regularized the lines'' of each given
diagram, i.e. the propagators. At the stage of only one MS operator
$-T_{\Ptn}^{\MS}$ this consideration is not very important, since
the singularities in different blocks are independent, anyway. However,
the fact that we can choose the regularization parameters freely for
any pair of vertices becomes important as soon as we want to define
products of the MS operators applied to the same regularized time-ordered
product. Such products occur in the forest formula below, and we briefly
discuss one example in order to clarify this point. We choose the
position space representation for convenience. Let $\Ptn_{1}\leq\Ptn_{2}$
be different partitions in $\Part\{1,\dots,n\}$. Let $I_{1}\subset I_{2}$
be a pair of blocks, $I_{1}\in\Ptn_{1}$, $I_{2}\in\Ptn_{2}$, and
regard a special graph contribution, say $S_{\Gamma}^{\mu,\zeta}$,
to $\Sm_{\mu,\zeta}^{\left(n\right)}$ for simplicity, $\left|V(\Gamma)\right|=n$.
Let $\gamma$ be the full vertex part (in $\Gamma$) to the vertex
set $I_{1}$ and let $G$ be the full vertex part of $I_{2}$, then
$\gamma\subset G$. Regard the successive subtraction\[
s_{\Gamma\bbs G}^{\mu,\zeta}\, R_{I_{2}}\left(s_{G\bbs\gamma}^{\mu,\zeta}\, R_{I_{1}}s_{\gamma}^{\mu,\zeta}\right),\hspace{80mm}\]
and insert the definition from the previous chapter, cf.~(\ref{eq:ppSubgraphLaurentSeries}),\begin{align*}
 & =\sum_{n=-\infty}^{-1}\zeta^{n}\frac{1}{2\pi i}\oint_{C}d\xi\,\frac{1}{\xi^{n+1}}\sum_{k=-\infty}^{-1}\xi^{k}\frac{1}{2\pi i}\oint_{C'}d\xi'\,\frac{1}{{\xi'}^{k+1}}s_{\Gamma\bbs G}^{\mu,\zeta}\cdot s_{G\bbs\gamma}^{\mu,\xi}\cdot s_{\gamma}^{\mu,\xi'}\,,\end{align*}
where we had to introduce regularization parameters $\xi$ and $\xi'$
for the subgraphs in order to get independent subtractions in all
subgraphs. With these remarks concerning the regularization, we have
for the composition of MS operators corresponding to related partitions,
$\Ptn\geq\Ptn'$,\[
T_{\Ptn}^{\MS}T_{\Ptn'}^{\MS}\Sm_{\mu,\zeta}^{\left(n\right)}=\Sm_{\mu,\zeta}^{\left(\left|\Ptn\right|\right)}\cdot\left[\bigotimes_{I\in\Ptn}R_{\left|I\right|}\left[\Sm_{\mu,\zeta}^{\left(\left|I'/I\right|\right)}\cdot\left(\bigotimes_{I'\subset I}R_{\left|I'\right|}\left(\Sm_{\mu,\zeta}^{\left(\left|I'\right|\right)}\right)\right)\right]\right],\]
where $\left|I'/I\right|$ denotes the cardinality of $\left\{ I'\in\Ptn':I'\subset I\right\} $.
\begin{thm}
[Forest Formula for Epstein-Glaser Renormalization]\label{thm:EGFF}Let
$T_{\Ptn}^{\MS}$ be the minimal subtraction operator of Definition~\ref{def:MS-OperatorPartitions},
and let the product of two MS operators corresponding to related partitions
be defined as described above. Then\begin{equation}
\Sm_{\mu,\zeta,\ren}^{\left(n\right)}:=\sum_{\F\subset\Part\{1,\dots,n\}}\left(\oprod{\Ptn\in\F}-T_{\Ptn}^{\MS}\right)\Sm_{\mu,\zeta}^{\left(n\right)}\label{eq:EGFF}\end{equation}
gives a finite regularization of the $n$-fold regularized time-ordered
product $\Sm_{\mu,\zeta}^{\left(n\right)}\equiv\Time_{\mu,\zeta}^{n}$.
The sum is taken over all Epstein-Glaser forests. The product of the
operators is taken in the order prescribed by {}``$\geq$'', such
that the coarsest partition in $\F$ stands to the very left.\end{thm}
\begin{proof}
The forest formula implies an expression for the $n$th counterterm
in the renormalization scheme of analytic regularization and minimal
subtraction. We can split (\ref{eq:EGFF}) into a sum over full and
normal forests, \begin{align*}
\Sm_{\mu,\zeta,\ren}^{\left(n\right)} & =\sum_{\underline{\F}}\left(\oprod{\Ptn\in\underline{\F}}-T_{\Ptn}^{\MS}\right)\Sm_{\mu,\zeta}^{\left(n\right)}+\sum_{\overline{\F}}\left(\oprod{\Ptn\in\overline{\F}}-T_{\Ptn}^{\MS}\right)\Sm_{\mu,\zeta}^{\left(n\right)}\,.\end{align*}
Observe that for $n=1$ there are no normal forests, and the first
sum is empty. Since any forest in the second sum on the right hand
side contains the coarsest partition, ${\Ptn_{1}=\left\{ \left\{ 1,\dots,n\right\} \right\} }$,
we can factor out the corresponding $\MS$ operator and get from {[}C2{]},\begin{align*}
\Sm_{\mu,\zeta,\ren}^{\left(n\right)} & =\sum_{\underline{\F}}\left(\oprod{\Ptn\in\underline{\F}}-T_{\Ptn}^{\MS}\right)\Sm_{\mu,\zeta}^{\left(n\right)}+R_{n}\left[\sum_{\underline{\F}}\left(\oprod{\Ptn\in\underline{\F}}-T_{\Ptn}^{\MS}\right)\Sm_{\mu,\zeta}^{\left(n\right)}\right].\end{align*}
We show in the sequel, that\begin{equation}
C_{\mu,\zeta}^{\left(n\right)}:=R_{n}\left[\sum_{\underline{\F}}\left(\oprod{\Ptn\in\underline{\F}}-T_{\Ptn}^{\MS}\right)\Sm_{\mu,\zeta}^{\left(n\right)}\right]\equiv\sum_{\overline{\F}}\left(\oprod{\Ptn\in\overline{\F}}-T_{\Ptn}^{\MS}\right)\Sm_{\mu,\zeta}^{\left(n\right)},\label{eq:ProofLocalCountertermC}\end{equation}
is the local counterterm of regularized causal perturbation theory
in the minimal subtraction scheme. That is $C_{\mu,\zeta}^{\left(n\right)}$
fulfills the recursion relation of Corollary~\ref{cor:RecursionMSCounterterms}.
We proceed by induction.

For $n=1$ we have by {[}C2{]} and the definition of $R_{1}$,\[
C_{\mu,\zeta}^{\left(1\right)}=R_{1}\Sm_{\mu,\zeta}^{\left(1\right)}=\id=Z_{\mu,\zeta}^{\left(1\right)}\,.\]
Thus $C_{\mu,\zeta}^{\left(1\right)}=\id:\fps{\mathcal{F}_{\loc}(\M)}{\hbar}\rightarrow\fps{\mathcal{F}_{\loc}(\M)}{\hbar}$
is local, and we can assume that $C_{\mu,\zeta}^{\left(k\right)}=Z_{\mu,\zeta}^{\left(k\right)}$
for all $k<n$. For the induction step we have to show that\[
R_{n}\left[\sum_{\underline{\F}}\left(\oprod{\Ptn\in\underline{\F}}-T_{\Ptn}^{\MS}\right)\Sm_{\mu,\zeta}^{\left(n\right)}\right]=-\pp\sum_{\Ptn\in\Part\{1,\dots,n\}\backslash\left\{ \Ptn_{1}\right\} }\!\!\Sm_{\mu,\zeta}^{\left(\left|\Ptn\right|\right)}\cdot\left(\bigotimes_{I\in\Ptn}C_{\mu,\zeta}^{\left(\left|I\right|\right)}\right).\]
for $n\geq2$. By the definition of $R_{n}$ this is the case, if\begin{equation}
\sum_{\underline{\F}}\left(\oprod{\Ptn\in\underline{\F}}-T_{\Ptn}^{\MS}\right)\Sm_{\mu,\zeta}^{\left(n\right)}=\sum_{\Ptn\in\Part\{1,\dots,n\}\backslash\left\{ \Ptn_{1}\right\} }\!\!\Sm_{\mu,\zeta}^{\left(\left|\Ptn\right|\right)}\cdot\left(\bigotimes_{I\in\Ptn}C_{\mu,\zeta}^{\left(\left|I\right|\right)}\right).\label{eq:EGFF-Proof-A}\end{equation}
Regard the left hand side of this equation. Any normal forest is a
disjoint union of at least two full forests (Corollary~\ref{cor:NormalForestDisjointUnion}).
Hence we can write the sum over all normal forests as \begin{align}
\mbox{lhs} & =\sum_{k=2}^{n}\,\sum_{\substack{\Ptn'\in\Part\{1,\dots,n\}\\
\left|\Ptn'\right|=k}
}\,\sum_{\underline{\F}=\dot{\bigcup}_{I\in\Ptn'}\underline{\F}(I)}\left(\,\oprod{\Ptn\in\underline{\F}}-T_{\Ptn}^{\MS}\right)\Sm_{\mu,\zeta}^{\left(n\right)}\,.\nonumber \\
\intertext{The\, product\, splits\, and\, we\, get} & =\sum_{k=2}^{n}\sum_{\substack{\Ptn'\in\Part\{1,\dots,n\}\\
\left|\Ptn'\right|=k}
}\sum_{\underline{\F}=\dot{\bigcup}_{I\in\Ptn'}\underline{\F}(I)}\bigotimes_{I\in\Ptn'}\left(\,\,\,\,\oprod{\Ptn\in\underline{\F}(I)}-T_{\Ptn}^{\MS}\right)\Sm_{\mu,\zeta}^{\left(n\right)}\,.\label{eq:EGFF-Proof-B}\end{align}
The sum over all normal, i.e., multiply connected forests $\underline{\F}$
can be performed by summing over all full forests $\underline{\F}(I)$
in its connected components, cf.~Corollary~\ref{cor:NormalForestDisjointUnion}.
And we have to perform the sum in these components, in order to get
a well-defined expression for (\ref{eq:EGFF-Proof-B}),\[
\mbox{lhs}=\sum_{k=2}^{n}\sum_{\substack{\Ptn'\in\Part\{1,\dots,n\}\\
\left|\Ptn'\right|=k}
}\bigotimes_{I\in\Ptn'}\sum_{\underline{\F}(I)}\left(\,\,\,\,\oprod{\Ptn\in\underline{\F}(I)}-T_{\Ptn}^{\MS}\right)\Sm_{\mu,\zeta}^{\left(n\right)}\,.\]
Inserting the inductive assumption, $C_{\mu,\zeta}^{\left(k\right)}=Z_{\mu,\zeta}^{\left(k\right)}$
$\forall k<n$, gives the desired result ($\mbox{lhs}=\mbox{rhs}$).
\end{proof}

\begin{cor}
Let $\F_{1},\dots,\F_{c}$ be the maximal forests of the vertex set
$\left\{ 1,\dots,n\right\} $. Then we can write (\ref{eq:EGFF})
equivalently as\begin{equation}
\Sm_{\mu,\zeta,\ren}^{\left(n\right)}=\sum_{\substack{\emptyset\neq\left\{ i_{1},\dots,i_{\nu}\right\} \\
\subset\left\{ 1,\dots,c\right\} }
}\left(-1\right)^{\nu+1}\left(\,\,\,\,\oprod{\Ptn\in\F_{i_{1}}\cap\cdots\cap\F_{i_{\nu}}}\left(1-T_{\Ptn}^{\MS}\right)\right)\Sm_{\mu,\zeta}^{\left(n\right)}\,.\label{eq:EGFF-1-T}\end{equation}
\end{cor}
\begin{proof}
This is merely another way of summing up the contributions to (\ref{eq:EGFF}).
Multiplying out the factors $\left(1-T_{\Ptn}^{\MS}\right)$ gives
the result, cf. \citep[Thm.~3.3]{Zimmermann1969}.
\end{proof}
{}
\begin{cor}
[EG Forest Formula in terms of graphs]Let $\Gamma$ be a graph with
$n$ vertices, $\left|V(\Gamma)\right|=n$. For any partition $\Ptn$
of $V(\Gamma)$ the action of the MS operator on the level of graphs
is given by \[
-T_{\Ptn}^{\MS}\left\langle s_{\Gamma}^{\mu,\zeta}\right\rangle :=\left\langle s_{\Gamma/\Ptn}^{\mu,\zeta}\right\rangle \left(\bigotimes_{I\in\Ptn}R_{\left|I\right|}\left\langle s_{\gamma_{I}}^{\mu,\zeta}\right\rangle \right),\quad\mbox{where }R_{k}=\begin{cases}
\id & \mbox{if }k=1\\
-\pp & \mbox{if }k>1\,,\end{cases}\]
where $\Gamma/\Ptn$ is the graph with the blocks $I\in\Ptn$ as vertices
and as lines all lines in $\Gamma$ which connect different blocks
of $\Ptn$. For each block $I\in\Ptn$, the graph $\gamma_{I}$ is
the full vertex part of $I$. $\left\langle s_{\Gamma}^{\mu,\zeta}\right\rangle $
denotes the analytically regularized amplitude in any representation
(momentum or position space). Then the limit\[
\left\langle s_{\Gamma,\ren}^{\mu}\right\rangle =\lim_{\zeta\rightarrow0}\sum_{\F\subset\Part\{1,\dots,n\}}\left(\oprod{\Ptn\in\F}-T_{\Ptn}^{\MS}\right)\left\langle s_{\Gamma}^{\mu,\zeta}\right\rangle \]
is well-defined and gives a UV finite, i.e., renormalized amplitude.\end{cor}
\begin{proof}
The MS operators are tensor products of the corresponding operators
on sets (Definition~\ref{def:MS-OperatorSubsets}) and thus linear.
Hence the corollary is a direct consequence of the discussion given
in Section~\ref{sec:MS-For-TProduct} and the above Theorem~\ref{thm:EGFF}.
\end{proof}
{}
\begin{cor}
[Prepared Amplitude]\label{cor:PreparedAmplitude}Let $\Gamma$ be
a graph with $n$ vertices, $n>1$, i.e., $\left\langle S_{\Gamma}^{\mu,\zeta},\bld{\delta}^{\alpha}\right\rangle $
with $\alpha\in\mathbb{N}^{n}$ and $\left|\alpha\right|=2\left|E(\Gamma)\right|$
is a contribution to the $n$-fold, regularized time ordered product
\[
\Sm_{\mu,\zeta}^{\left(n\right)}:\fps{\mathcal{F}_{\loc}(\M)}{\hbar}^{\otimes n}\rightarrow\fps{\mathcal{F}(\M)}{\hbar}.\]
Then the prepared amplitude to $\Gamma$ is given by,\begin{equation}
S_{\Gamma,\prep}^{\mu,\zeta}:=\sum_{\substack{\emptyset\neq\left\{ i_{1},\dots,i_{\nu}\right\} \\
\subset\left\{ 1,\dots,c\right\} }
}\left(-1\right)^{\nu+1}\left(\,\,\,\,\oprod{\Ptn\in\F_{i_{1}}\cap\cdots\cap\F_{i_{\nu}}\backslash\left\{ \Ptn_{1}\right\} }\left(1-T_{\Ptn}^{\MS}\right)\right)S_{\Gamma}^{\mu,\zeta}\,.\qed\label{eq:TransInvPreparedAmplitude}\end{equation}

\end{cor}

\section{\label{sec:MoreThanHopfAlgebra}More than Hopf Algebra}

The investigation of the combinatorial structure of perturbative renormalization
theory is a vivid field of research in mathematical physics and for
the characterization of the underlying pattern Hopf algebras, and
in particular the \linebreak[4]Faà~di~Bruno bialgebra introduced
by Joni and Rota in 1982 \citep{JoniRota1982}, became more and more
important in recent years \citep{Figueroa2005}. We will show in this
last section, that there is a more intricate pattern underlying the
combinatorial structure of renormalization than is described by a
Hopf algebra. However, the relation to the Hopf algebra of graphs
originally encountered by Connes and Kreimer in BPHZ \citep{Kreimer1998,Connes2000,Connes2001}
and later by Gracia-Bondía, Lazzarini, and Pinter in Epstein-Glaser
renormalization \citep{Gracia-Bondia2000,Pinter2000b} will become
transparent. The attractive feature of our derivation is that we can
understand the emerging Hopf algebraic structure as a direct consequence
of the main theorem of renormalization (Theorem~\ref{thm:MainTheoremRenormalization}).
The elements of this Hopf algebra will be differential operators which
give the time-ordered products, when acting on a $\Sm$-matrix, and
local counterterms when acting on an element of the Stückelberg-Petermann
renormalization group. Hence, in a graphical representation they correspond
to sums of graphs with the same set of vertices. The Hopf algebra
structure for individual graphs is regained by linearity. The regularized
Feynman rules and the renormalization map will appear naturally as
soon as one specializes to an analytically regularized $\Sm$-matrix.
This is a major difference to the Connes-Kreimer approach, where the
Feynman rules had to be put by hand as characters into the commutative
ring of Laurent series. We will show, however, that the recursion
formula for minimally subtracted counterterms (\ref{eq:CountertermsMSRecursionFormula}),
which was seen to be a direct consequence of the main theorem, cannot
be described within the commutative Hopf algebra of Connes and Kreimer
\citep{Connes2000,Connes2001}.

As a first step we will use the Faà~di~Bruno Hopf algebra introduced
by Joni and Rota to derive the commutative, non-cocommutative Hopf
algebra of graphs described briefly above. It will be clear from the
given presentation that one needs an additional, non-commutative,
composition product, to get back the recursion formula for the counterterms
(Corollary~\ref{cor:RecursionMSCounterterms}). We will sketch in
the last section how this composition product can be implemented into
the Hopf algebra to describe algebraically the construction of counterterms
in pAQFT. An interpretation of the maps in terms of graph operations
will be given in the end.

\subsection{The Hopf Algebra}

As shown by Joni and Rota,\linebreak[1] Faà~di~Bruno's formula
for the chain rule gives rise to a natural bialgebra structure, which
the authors called the Faà~di~Bruno bialgebra \citep{JoniRota1982}.
In modern presentations it is often introduced as a bialgebra of the
coefficients in (\ref{eq:FaaDiBrunoSortedBlockSize}), cf.~\citep{Figueroa2005}.
However, to keep the correspondence to causal perturbation theory
transparent also in this last section of the present thesis we regard
instead directly the partition version given in Lemma~\ref{lem:FaaDiBrunoSetPartitionVersion}.
Apart from that we follow essentially the steps of \citep{Figueroa2005}
as far as the Hopf algebra structure is concerned.

Regard Faà~di~Bruno's formula (\ref{eq:FaaDiBrunoPartitionVersion})
in the termwise form of the main theorem of renormalization (Lemma~\ref{lem:FaaDiBrunoSetPartitionVersion}).
We denote the coefficients by \[
a_{n}(\Sm):=\Sm^{\left(n\right)}(0)\qquad\mbox{and}\qquad a_{n}(Z):=Z^{\left(n\right)}(0)\]
and get\begin{equation}
a_{n}(\Sm\circ Z)=\sum_{\Ptn\in\Part\{1,\dots,n\}}a_{\left|\Ptn\right|}(\Sm)\cdot\bigodot_{I\in\Ptn}a_{\left|I\right|}(Z)\,.\label{eq:FaaDiBrunoCoefficients}\end{equation}
We want to make the symmetry of the functional derivative explicit
here and replaced the tensor product in (\ref{eq:FaaDiBrunoPartitionVersion})
by the symmetrized tensor product\[
\bigodot_{i=1}^{k}A_{i}:=\frac{1}{k!}\sum_{\sigma\in\Perm(k)}\bigotimes_{i=1}^{k}A_{\sigma(i)}\,,\]
where $\Perm(k)$ denotes the group of permutations of $k$ elements.
Observe that besides this commutative product ($\odot$), there is
a second, non-commutative product in formula (\ref{eq:FaaDiBrunoCoefficients}).
Namely, the composition of linear maps\begin{equation}
\linc:a_{\left|\Ptn\right|}(\Sm)\otimes\bigodot_{I\in\Ptn}a_{\left|I\right|}(Z)\mapsto a_{\left|\Ptn\right|}(\Sm)\cdot\bigodot_{I\in\Ptn}a_{\left|I\right|}(Z)\,,\label{eq:CompositionLinearMaps}\end{equation}
cf.~(\ref{eq:CompositionDiagram}). This second product is absent
if we regard the coefficients in (\ref{eq:FaaDiBrunoCoefficients})
as scalars. And we will sketch how to implement this additional non-commutative
product into the Hopf algebra in the next section. Let us first regard
the commutative part.

Regard the coefficients $a_{k}$ in (\ref{eq:FaaDiBrunoCoefficients})
as differential operators\begin{equation}
a_{k}:\Sm\mapsto a_{k}(\Sm)\equiv\Sm^{\left(k\right)}(0)\quad\mbox{and}\quad a_{k}:Z\mapsto a_{k}(Z)\equiv Z^{\left(k\right)}(0)\,.\label{eq:FdBCoefficientsDifferentialOperators}\end{equation}
The operators will produce multi-linear maps on local functionals;
with local image in the case they act on $Z\in\R$,\[
a_{k}(Z):\fps{\mathcal{F}_{\loc}(\M)}{\hbar}^{\otimes k}\rightarrow\fps{\mathcal{F}_{\loc}(\M)}{\hbar}\,,\]
and with possibly non-local image if they act of $\Sm$ fulfilling
{[}C1{]}-{[}C4{]},\[
a_{k}(\Sm):\fps{\mathcal{F}_{\loc}(\M)}{\hbar}^{\otimes k}\rightarrow\fps{\mathcal{F}(\M)}{\hbar}\,.\]
Denote by $\Halg$ the space of these coefficients $a_{k}$. Since
the $Z^{\left(k\right)}$ and $\Sm^{\left(k\right)}$ (evaluation
at zero understood) are linear maps on tensor products of local functionals,
$\Halg$ carries a natural $\mathbb{C}$-vector space structure induced
by the $\mathbb{C}$-vector space structure on local functionals.
As already remarked above, the symmetry of the functional derivative
induces a commutative product on $\Halg$,\[
\begin{array}{rccl}
M_{\odot}: & \Halg\otimes\Halg & \rightarrow & \Halg\\
 & (a_{k}\otimes a_{l}) & \mapsto & a_{k}\odot a_{l}:=\tfrac{1}{2}\left[a_{k}\otimes a_{l}+a_{l}\otimes a_{k}\right],\end{array}\]
where we set\begin{equation}
\left(a_{k}\odot a_{l}\right)(Z):=a_{k}(Z)\odot a_{l}(Z)\,,\label{eq:CummutativeHproduct}\end{equation}
and likewise for $\Sm$. We regard $\Halg$ as the free, commutative
algebra generated by the $a_{k}$, and commit the usual abuse of notation
by using the same symbol for the (symmetrized) tensor product of linear
maps on the right hand side of (\ref{eq:CummutativeHproduct}) and
the commutative product of the algebra $\left(\Halg,\odot\right)$
on the left. It will be clear from the context, where we mean which.
$\left(\Halg,\odot\right)$ is a unital algebra with unit \[
\openone:Z\mapsto\openone(Z):=\id\,,\qquad\openone:\Sm\mapsto\openone(\Sm):=\id\,,\]
where $\id:\fps{\mathcal{F}_{\loc}(\M)}{\hbar}\rightarrow\fps{\mathcal{F}_{\loc}(\M)}{\hbar}$
denotes the identity map on the space of local functionals. We denote
the corresponding unit map by\[
\begin{array}{rccl}
e: & \mathbb{C} & \rightarrow & \Halg\\
 & \alpha & \mapsto & \alpha\,\openone\,.\end{array}\]
We set $\Halg\odot\openone=\Halg$. Joni and Rota interpreted Faà~di~Bruno's
formula as a coproduct rule for the coefficients $a_{k}$ , and we
can do exactly the same thing here, by defining the coproduct, $\Delta:\Halg\rightarrow\Halg\otimes\Halg$,
as\begin{equation}
\left(\Delta a_{n}\right):=\sum_{\Ptn\in\Part\{1,\dots,n\}}a_{\left|\Ptn\right|}\otimes\left(\bigodot_{I\in\Ptn}a_{\left|I\right|}\right).\label{eq:FaaDiBrunoCoproduct}\end{equation}
It is obvious from (\ref{eq:FaaDiBrunoCoefficients}) that this coproduct
is induced by\[
\left(\Delta a_{n}\right)(\Sm\otimes Z)=a_{n}(\Sm\circ Z)\,,\]
and we break with the tradition of flipping the arguments of $\Delta a_{n}$
here,
\footnote{see, e.g. page 2 of \citep{FigueroaGraciaBondiaVarilly2005}.
} since, in contrast to the chain rule for functions or formal power
series with scalar coefficients, we have a composition of linear maps
with a prescribed order rather than a commutative product on the right
hand side of Faà~di~Bruno's formula, cf.~(\ref{eq:CompositionLinearMaps}).
Consequently we have the linear part of the coproduct on the left
hand side of $\otimes$. Equipping $\Halg$ with the counit, defined
on generators \[
\begin{array}{rccl}
\overline{e}: & \Halg & \rightarrow & \mathbb{C}\\
 & a_{n} & \mapsto & \overline{e}(a_{n}):=\begin{cases}
1 & \mbox{if }a_{n}=\openone\\
0 & \mbox{else}\,,\end{cases}\end{array}\]
gives the usual commutative, non-cocommutative Faà~di~Bruno bialgebra,
however, now interpreted in terms of the functional differential operators
(\ref{eq:FdBCoefficientsDifferentialOperators}).

It will be helpful for the construction of the antipode to first discuss
the natural gradings on $\Halg$.
\footnote{Two articles by Kastler were very helpful in learning about the relevance
of grading and other Hopf related topics \citep{Kastler2000,Kastler2004}.
They are probably not the standard references to be cited at this
point, however, they contain explicit proofs of the results from Hopf
algebra theory needed here. See also more standard literature like
\citep{Sweedler1969,Abe1977}. Since we will not make any connection
to more advanced structures in algebraic geometry, the given references
will fully suffice for the discussion in this section.
} As any tensor algebra, $\Halg$ is graded by the number of factors,\[
\begin{array}{rccl}
\deg^{\odot}: & \Halg & \rightarrow & \mathbb{N}\\
 & \bigodot_{i=1}^{k}a_{l_{i}} & \mapsto & \deg^{\odot}(a_{l_{1}}\odot\cdots\odot a_{l_{k}}):=k\,.\end{array}\]
That is, $\Halg$ can be written as the direct sum \[
\Halg=\bigoplus_{k=0}^{\infty}\Halg^{\odot k}\quad,\quad\Halg^{\odot k}=\left\{ a\in\Halg:\deg^{\odot}(a)=k\right\} ,\quad\Halg^{\odot0}=\mathbb{C}\,.\]
Subordinate to this tensor algebra grading is a naturally induced
grading of the individual $a_{l}\in\Halg^{\odot1}$ given by the order
of the derivative (minus one), \[
\deg^{v}(a_{l}):=l-1\,,\]
and we have\[
\Halg^{\odot1}=\bigoplus_{n=0}^{\infty}\Halg_{n}^{\odot1}\quad,\quad\Halg_{n}^{\odot1}=\left\{ a\in\Halg^{\odot1}:\deg^{v}(a)=n\right\} .\]

Observe that $\deg^{v}+1$ corresponds to the order of derivative
of $\Sm$ (or $Z$) at zero. This determines the number of interaction
functionals in the argument of the corresponding linear map\[
a_{n}(\Sm)=\Sm^{\left(n\right)}(0):\fps{\mathcal{F}_{\loc}(\M)}{\hbar}^{\otimes n}\rightarrow\fps{\mathcal{F}(\M)}{\hbar}\,,\quad n=\deg^{v}+1\,,\]
and hence $\deg^{v}+1$ is the number of vertices of the graphs contributing
to the graph expansion of $\Sm^{\left(n\right)}(0)\equiv\Time_{n}$,
cf.~Equation~(\ref{eq:TimeOrderedGraphStructure}).

This grading by vertex number can be extended to the tensor product
$\Halg^{\odot k}$ by setting\[
\deg^{v}(a_{k}\odot a_{l}):=\deg^{v}(a_{k})+\deg^{v}(a_{l})\,,\quad\mbox{i.e.,}\quad\deg^{v}(a_{l_{1}}\odot\cdots\odot a_{l_{k}})=\sum_{i=1}^{k}l_{i}-k\,.\]
With this definition also $\deg^{v}$ is an algebra grading of the
algebra $\Halg$, and we have\[
\Halg=\bigoplus_{n=0}^{\infty}\Halg_{n}\quad,\quad\Halg_{n}=\left\{ a\in\Halg:\deg^{v}(a)=n\right\} \,.\]
Furthermore the vertex grading $\deg^{v}$ is compatible with the
coproduct (\ref{eq:FaaDiBrunoCoproduct}),\[
\deg^{v}(\Delta a_{n})=\left|\Ptn\right|-1+\sum_{I\in\Ptn}\left(\left|I\right|-1\right)=n-1\quad\forall\Ptn\in\Part\{1,\dots,n\}\,,\]
and hence $\left(\Halg,\odot,e,\Delta,\overline{e}\right)$ is graded
as a bialgebra. The starting element conditions {[}C2{]} and {[}RG2{]},
\[
a_{1}(\Sm)=\Sm^{\left(1\right)}(0)=\id\quad\mbox{and}\quad a_{1}(Z)=Z^{\left(1\right)}(0)=\id\]
imply that $a_{1}=\openone$ and hence $\Halg$ is $\mathbb{N}_{0}$-graded
connected as a bialgebra, i.e., $\Halg_{0}=\mathbb{C}$ (we implicitly
identify $\openone=\openone\odot\openone$ here). It is a well-known
fact of Hopf algebra theory that any $\mathbb{N}_{0}$-graded connected
bialgebra possesses an antipode \citep[Prop.~2.7]{Kastler2000},\[
\ap:\Halg\rightarrow\Halg\,,\]
and thus $\Halg$ is a Hopf algebra. We will now derive a recursion
formula for this antipode. By definition, the antipode $\ap$ of a
Hopf algebra is the inverse of the identity with respect to the induced
convolution product on the Hopf algebra automorphisms $\Aut(\Halg)$.
The convolution product on $\Aut(\Halg)$ is induced by the product
and coproduct on $\Halg$, we denote it by\[
\phi\oast\psi:=M_{\odot}\circ\left(\phi\otimes\psi\right)\circ\Delta\,,\quad\phi,\psi\in\Aut(\Halg)\,.\]
It is a standard computation to prove that \[
e\circ\overline{e}:\Halg\rightarrow\Halg\]
defines a unit in the algebra $\left(\Aut(\Halg),\oast\right)$. A
similar computation will be done below for the second product on the
Hopf algebra, so we leave it out here. The antipode of an $\mathbb{N}_{0}$-graded
connected bialgebra can then be constructed directly from its defining
condition,\[
\id_{\Halg}\oast\ap=e\circ\overline{e}\,.\]
We have\begin{align*}
M_{\odot}\circ\left(\id_{\Halg}\otimes\ap\right)\circ\Delta(a_{n}) & =e(\overline{e}(a_{n}))\\
\sum_{\Ptn\in\Part\{1,\dots,n\}}a_{\left|\Ptn\right|}\odot\left(\bigodot_{I\in\Ptn}\ap(a_{\left|I\right|})\right) & =\begin{cases}
\openone & \mbox{if }n=1\\
0 & \mbox{else}\,,\end{cases}\end{align*}
since, in particular, $\ap$ is an algebra homomorphism. From the
case $n=1$ we get $\ap(a_{1})=\openone$, and since there is only
one partition of $\left\{ 1,\dots,n\right\} $ with one block, $\Ptn_{1}=\left\{ \left\{ 1,\dots,n\right\} \right\} $,
we infer by connectedness of $\Halg$, i.e., by $a_{1}=\openone$
that\begin{equation}
\ap(a_{n})=-\sum_{\Ptn\in\Part\{1,\dots,n\}\backslash\left\{ P_{1}\right\} }a_{\left|\Ptn\right|}\odot\left(\bigodot_{I\in\Ptn}\ap(a_{\left|I\right|})\right).\label{eq:CKantipode}\end{equation}
Observe the similarity to the recursion for the counterterms in Corollary~\ref{cor:RecursionMSCounterterms}.
However, observe also that the composition structure of (\ref{eq:CountertermsMSRecursionFormula})
is completely absent in (\ref{eq:CKantipode}). This is no problem,
if we regard the Feynman rules as characters of $\left(\Halg,\odot\right)$
into a commutative ring of Laurent series with scalar coefficients,
as it was done in \citep{Connes2000,Connes2001}. We want to emphasize
the relation of the Hopf algebra $\left(\Halg,\odot,e,\Delta,\overline{e},\ap\right)$
to the Connes-Kreimer Hopf algebra of graphs. The elements of $\Halg$
are differential operators whose order is determined by the vertex
grading $\deg^{v}$. By implementing the graph expansion (\ref{eq:TimeOrderedGraphStructure}),
the elements $a_{k}\in\Halg$ can be interpreted as sums over graphs
with the same set of vertices. By linearity of the maps and the fact
that only finitely many graphs contribute to the perturbative expansion
at a given order of $\hbar$, {[}C3{]}, we can break the Hopf algebra
structure down to the level of graphs. However, since the structure
for the algebraic construction of counterterms is not complete yet,
we will give a more detailed account of this interpretation only at
the end of the next section.

The fact that we found the Hopf algebra structure in the sums of graphs
is in accordance with the results of Brouder and Frabetti, who found
in different examples (including gauge theories) that Connes and Kreimer's
Hopf algebra structure is preserved when one sums up the graph contributions
at certain orders or perturbation theory \citep{BrouderFrabetti2000,BrouderFrabetti2001,Frabetti2007},
see also \citep{Suijlekom2007c}. Brouder and Frabetti, in collaboration
with Krattenthaler and Menous, respectively, also observed the relation
to the Faà~di~Bruno Hopf algebra \citep{BrouderFrabettiKra2006,BrouderFrabettiMenous2009},
however, the relation to the main theorem of perturbative renormalization
as proven in \citep{DuetschFredenhagen2004,Duetsch2005,Brunetti2009}
was, to the best of my knowledge, unobserved before.

We will now incorporate the non-commutative composition structure
into the commutative Hopf algebra $\left(\Halg,\odot,e,\Delta,\overline{e},\ap\right)$
constructed above.

\subsection{Algebraic Construction of Counterterms}

In contrast to the Connes-Kreimer approach to renormalization, in
our approach the Feynman rules are naturally induced as evaluation
maps of the differential operators $a_{n}\in\Halg$. The basic evaluation
operator, which gives the $n$-fold time-ordered product, and thus
corresponds to the Feynman rules is given by \[
\feyn:a_{n}\mapsto a_{n}(\Sm)=\Sm^{\left(n\right)}\equiv\Time^{n}\,.\]
The image of $\feyn$ is a multi-linear map between spaces of (local)
functionals. On linear maps there are two natural products, one is
the (symmetrized) tensor product discussed above, and the other is
the composition. Composition of linear maps is a non-commutative operation
in the generic case, and as a consequence it is impossible to derive
the action of the counterterms $Z^{\left(n\right)}$ on the time-ordered
products $\Sm^{\left(n\right)}$ described by Lemma~\ref{lem:FaaDiBrunoSetPartitionVersion}
from the commutative Hopf algebra $\left(\Halg,\odot,e,\Delta,\overline{e},\ap\right)$
alone. Regard once again the expression given in the lemma,

\[
\left(\Sm\circ Z\right)^{\left(n\right)}\!(0)=\sum_{\Ptn\in\Part\{1,\dots,n\}}\!\!\Sm^{\left(\left|\Ptn\right|\right)}(0)\cdot\left(\bigodot_{I\in\Ptn}\left[Z^{\left(\left|I\right|\right)}(0)\right]\right),\]
and observe that we need both, the commutative product $\odot$ and
the \linebreak[4]non-commutative composition {}``~$\cdot$~'',
as well as the coproduct $\Delta$ for defining the action of $\R$
on itself and on the set of $\Sm$-matrices term by term in an algebraic
fashion.

We want to incorporate the composition as an additional product in
the commutative, non-cocommutative Hopf algebra constructed above,
\[
\left(\Halg,\odot,e,\Delta,\overline{e},\ap\right)\]
with generators $a_{k}\in\Halg$, $k\in\mathbb{N}$. We define a map\[
\linc:\Halg\otimes\Halg\rightarrow\Halg\,,\]
induced by the composition of linear maps (\ref{eq:CompositionLinearMaps}),
\[
\linc(a_{k}\otimes\bigodot_{i=1}^{k}a_{l_{i}})=a_{k}\ocomp\bigodot_{i=1}^{k}a_{l_{i}}\,,\qquad\mbox{with}\qquad\left(a_{k}\ocomp\bigodot_{i=1}^{k}a_{l_{i}}\right)(Z):=a_{k}(Z)\cdot\bigodot_{i=1}^{k}a_{l_{i}}(Z)\,.\]
Observe that the application on $\Sm$ is ill-defined in the generic
case, since the derivatives $\Sm^{\left(n\right)}$ do not have local
images for $n>1$. The composition product $\ocomp$ can be seen as
(the dual of) the termwise group action of the Stückelberg-Petermann
group on itself. We implement compatibility with the vertex grading
by defining \begin{align*}
\deg^{v}(a_{k}\ocomp\bigodot_{i=1}^{k}a_{l_{i}}) & :=\deg^{v}(a_{k})+\deg^{v}(\bigodot_{i=1}^{k}a_{l_{i}})\\
 & =k-1+\sum_{i=1}^{k}\left(l_{i}-1\right)=\sum_{i=1}^{k}l_{i}-1,\end{align*}
and with the Hopf algebra unit \[
\openone\ocomp a_{k}=a_{k}\ocomp\openone=a_{k}\,,\]
in accordance with {[}RG2{]}. The product is also compatible with
the coproduct, in the sense that we can define another convolution
on $\Aut(\Halg)$,\[
\phi\bullet_{\linc}\psi:=\linc\circ\left(\phi\otimes\psi\right)\circ\Delta\,.\]
We check that $e\circ\overline{e}$ is a (both sided) identity for
this convolution,\begin{align*}
\left(e\circ\overline{e}\bullet_{\linc}\psi\right)(a_{k}) & =\sum_{\Ptn\in\Part\left\{ 1,\dots,k\right\} }e\circ\overline{e}\left(a_{\left|\Ptn\right|}\right)\ocomp\psi(\bigodot_{I\in\Ptn}a_{\left|I\right|})\\
 & =\sum_{\Ptn\in\Part\left\{ 1,\dots,k\right\} }\delta_{\left|\Ptn\right|,1}\openone\ocomp\psi(\bigodot_{I\in\Ptn}a_{\left|I\right|})\\
 & =\psi(a_{k})\,,\end{align*}
where $\delta_{\left|\Ptn\right|,1}$ is the Kronecker-$\delta$.
Observe that $\left|\Ptn\right|=1$ implies $\Ptn=\left\{ \left\{ 1,\dots,k\right\} \right\} $.
Conversely we get\begin{align*}
\left(\phi\bullet_{\linc}e\circ\overline{e}\right)(a_{k}) & =\sum_{\Ptn\in\Part\left\{ 1,\dots,k\right\} }\phi\left(a_{\left|\Ptn\right|}\right)\ocomp\bigodot_{I\in\Ptn}e\circ\overline{e}(a_{\left|I\right|})\\
 & =\sum_{\Ptn\in\Part\left\{ 1,\dots,k\right\} }\phi\left(a_{\left|\Ptn\right|}\right)\ocomp\bigodot_{I\in\Ptn}\delta_{\left|I\right|,1}\\
 & =\phi(a_{k})\,.\end{align*}
Here the result is obtained since $\left|I\right|=1$ $\forall I\in\Ptn$
implies $\Ptn=\left\{ \left\{ 1\right\} ,\dots,\left\{ k\right\} \right\} $.
We define the $Z$-Feynman rules,\[
\feyn_{Z}(a_{k}):=a_{k}(Z)\equiv Z^{\left(k\right)}(0):\fps{\mathcal{F}_{\loc}(\M)}{\hbar}^{\otimes k}\rightarrow\fps{\mathcal{F}_{\loc}(\M)}{\hbar}\,.\]
These are algebra homomorphisms with respect to both algebra products,\[
\feyn_{Z}:\left(\Halg,\odot\right)\rightarrow\left(\Lin,\odot\right)\qquad\mbox{and}\qquad\feyn_{Z}:\left(\Halg,\ocomp\right)\rightarrow\left(\Lin,\cdot\right)\,,\]
where we denoted by $\Lin$ the space of multi-linear maps between
spaces of local functionals. We get at order $n$ of causal perturbation
theory the finite renormalizations, the changes of renormalization
scheme, or the action of the Stückelberg-Petermann renormalization
group on itself,\[
\left(\feyn_{Z_{1}}\bullet_{\linc}\feyn_{Z_{2}}\right)(a_{n})=\sum_{\Ptn\in\Part\{1,\dots,n\}}Z_{1}^{\left(\left|\Ptn\right|\right)}\cdot\left(\bigodot_{I\in\Ptn}Z_{2}^{\left(\left|I\right|\right)}\right).\]

Let us compute the right sided antipode of the convolution $\bullet_{\linc}$,\begin{align*}
\id_{\Halg}\bullet_{\linc}\ap_{\linc}(a_{n}) & =e\circ\overline{e}(a_{n})\\
\sum_{\Ptn\in\Part\left\{ 1,\dots,n\right\} }a_{\left|\Ptn\right|}\ocomp\ap_{\linc}(\bigodot_{I\in\Ptn}a_{\left|I\right|}) & =\delta_{n,1}\end{align*}
From $n=1$ we get $\ap_{\linc}(a_{1})=\openone$. Furthermore we
get from the connectedness of the Hopf algebra, i.e., from $a_{1}=\openone$,\[
\ap_{\linc}(a_{n})=-\sum_{\Ptn\in\Part\left\{ 1,\dots,n\right\} \backslash\left\{ \Ptn_{1}\right\} }a_{\left|\Ptn\right|}\ocomp\ap_{\linc}(\bigodot_{I\in\Ptn}a_{\left|I\right|})\,,\]
where, as before, $\Ptn_{1}=\left\{ \left\{ 1,\dots,n\right\} \right\} $.
Observe the similarity to the result of Connes-Kreimer in the commutative
case, i.e., the recursion formula (\ref{eq:CKantipode}) for the antipode
$\ap$. And observe the difference, the composition product $\ocomp$.
The augmented Hopf algebra\[
\left(\Halg,\odot,e,\Delta,\overline{e},\ocomp,\ap_{\linc}\right)\]
constructed above can be interpreted as the algebraic dual of the
St{\"u}ckelberg-Petermann renormalization group. We want to discuss
now the action of this algebraic dual on the (regularized) time-ordered
products, and its relation to the original formulation of the Connes-Kreimer
theory of renormalization.

Given the nice circumstance that we have a preferred renormalization
prescription at all orders of perturbation theory as it is provided
by any analytic regularization of the $\Sm$-matrix combined with
minimal subtraction. Then we define the regularized Feynman rules
as\[
\feyn_{\mu,\zeta}(a_{k}):=a_{k}(\Sm_{\mu,\zeta})\equiv\Sm_{\mu,\zeta}^{\left(k\right)}:\fps{\mathcal{F}_{\loc}(\M)}{\hbar}^{\otimes k}\rightarrow\fps{\mathcal{F}(\M)}{\hbar}\,.\]
Since the derivatives of $\Sm_{\mu,\zeta}$ have non-local images
in the generic case, we have that, in contrast to the $Z$-Feynman
rules above, the regularized Feynman rules cannot be iterated. This
is reflected in the algebraic setting by the fact that the regularized
Feynman rules are algebra homomorphisms with respect to $\odot$,
but not with respect to $\ocomp$, $ $\[
\feyn_{\mu,\zeta}(a_{k}\odot a_{l})=\feyn_{\mu,\zeta}(a_{k})\odot\feyn_{\mu,\zeta}(a_{l})\,.\]
Following the idea of Kreimer \citep{Kreimer1999}, we define\[
\ap_{\linc,\MS}^{\feyn_{\mu,\zeta}}:=R\circ\feyn_{\mu,\zeta}\circ\ap_{\linc}\,,\]
where $R$ denotes minimal subtraction, i.e., the {}``renormalization
map'',\[
R(\Sm_{\mu,\zeta}^{\left(k\right)})=\begin{cases}
\id & \mbox{for }k=1\\
\pp(\Sm_{\mu,\zeta}^{\left(k\right)}) & \mbox{for }k>1\,.\end{cases}\]
Assuming that $R$ is only applied to prepared time-ordered products,
we have that $R(\Sm_{\mu,\zeta}^{\left(k\right)})$ is a multi-linear
map from local functionals to local functionals, and then $R\circ\feyn_{\mu,\zeta}$
defines an algebra homomorphism of $\left(\Halg,\odot,\ocomp\right)$
with respect to both products\[
R\circ\feyn_{\mu,\zeta}\left(a_{k}\ocomp\bigodot_{i=1}^{k}a_{l_{i}}\right)=R(\Sm_{\mu,\zeta}^{\left(k\right)})\cdot\left(\bigodot_{i=1}^{k}R(\Sm_{\mu,\zeta}^{\left(l_{i}\right)})\right).\]
That $R$ is applied only to prepared time-ordered products is guaranteed
by the recursive definition of the antipode. In particular $R$ itself
is a homomorphism of the symmetrized tensor product $\odot$, which
makes the Rota-Baxter argument of \citep{Kreimer1999} redundant in
the presented framework (see Remark~\ref{rem:RotaBaxter} below).
We infer that $\ap_{\linc,\MS}^{\feyn,\mu,\zeta}$ is an algebra homomorphism\[
\ap_{\linc,\MS}^{\feyn,\mu,\zeta}:\left(\Halg,\odot\right)\rightarrow\left(\Lin,\odot\right)\,.\]
We have that $\ap_{\linc,\MS}^{\feyn_{\mu,\zeta}}(a_{1})=\id$, the
identity on local functionals, and for $n>1$, \[
\ap_{\linc,\MS}^{\feyn_{\mu,\zeta}}(a_{n})=-R\sum_{\Ptn\in\Part\left\{ 1,\dots,n\right\} \backslash\left\{ \Ptn_{1}\right\} }a_{\left|\Ptn\right|}(\Sm_{\mu,\zeta})\cdot\ap_{\linc,\MS}^{\feyn_{\mu,\zeta}}(\bigodot_{I\in\Ptn}a_{\left|I\right|})\,.\]
This is just the recursion formula we got for the counterterms (\ref{eq:CountertermsMSRecursionFormula})
by applying the minimal subtraction condition to the main theorem
of renormalization in the form of Lemma~\ref{lem:FaaDiBrunoSetPartitionVersion}.
Observe that one loses the information on the product $\ocomp$ if
one regards the amplitudes, or regularized time-ordered products as
elements in a commutative ring of Laurent series, only. However, similar
as in Connes-Kreimer theory of renormalization, we can define the
algebra homomorphism \begin{equation}
\ap_{\linc,\ren}^{\feyn_{\mu,\zeta}}=\feyn_{\mu,\zeta}\:\bullet_{\linc}\:\ap_{\linc,\MS}^{\feyn_{\mu,\zeta}}\,,\label{eq:FiniteAntipode}\end{equation}
which gives the finitely regularized $n$-fold time ordered product,
when applied to a generator $a_{n}\in\Halg$,\begin{equation}
\ap_{\linc,\ren}^{\feyn_{\mu,\zeta}}(a_{n})=\Sm_{\mu,\zeta,\ren}^{\left(n\right)}.\label{eq:FiniteAntipodeTimeOrdProd}\end{equation}
However, although (\ref{eq:FiniteAntipode}) gives the correct result
in the general case, i.e., for arbitrary local interactions, this
is merely a compact notation for the successive subtraction of counterterms
in the sense of BPH rather than a forest formula in the sense of Zimmermann
\citep{Zimmermann1969,Zimmermann1975} or Theorem~\ref{thm:EGFF}.
Solving the recursion would be equivalent to giving a closed formula
for the character $\ap_{\linc,\ren}^{\feyn_{\mu,\zeta}}$ in (\ref{eq:FiniteAntipodeTimeOrdProd}).
This was done in Section~\ref{sec:EGFF}, although not in this abstract
algebraic setting. After the remark on Rota-Baxter maps, we give a
graphical interpretation of the maps constructed above.

\vspace{-1mm}
\begin{rem}
[Rota-Baxter Maps]\label{rem:RotaBaxter}For the reader, less familiar
with the {}``Hopf algebra school'' in renormalization theory, we
probably have to remark here that a linear map fulfilling the algebraic
relation\begin{equation}
R(a)R(b)=R\left[R(a)b\right]+R\left[aR(b)\right]-R(ab)\,,\label{eq:RotaBaxter}\end{equation}
is called a \emph{Rota-Baxter map of weight one}. Examples of such
maps are the projections in a \emph{Birkhoff sum}, i.e., an algebra
which splits into a direct sum of algebras, both closed under multiplication,\[
A=A^{+}\oplus A^{-}.\]
The Laurent series with scalar coefficients are elements of a Birkhoff
sum. It is straight forward to show the above claim that any linear
projection in a Birkhoff sum, which projects to one of its components
is Rota-Baxter of weight one. Denote by $a^{+}+a^{-}\in A^{+}\oplus A^{-}$
the elements of a Birkhoff sum, then by linearity and the fact that
the components are closed under multiplication we have\begin{align*}
\left[\left(a^{+}+a^{-}\right)\left(b^{+}+b^{-}\right)\right]^{-} & =\left[a^{-}b^{+}+a^{+}b^{-}\right]^{-}+a^{-}b^{-}\\
 & =\left[a^{-}b^{+}+a^{+}b^{-}+2a^{-}b^{-}\right]^{-}-a^{-}b^{-}\\
 & =\left[a^{-}b\right]^{-}+\left[ab^{-}\right]^{-}-a^{-}b^{-}.\end{align*}
Following a suggestion by Brouder, Kreimer used this property for
the projection to the principal part of a Laurent series in order
to solve the {}``multiplicativity constraints'', see~\citep[Sec.~3]{Kreimer1999}.
Kreimer's {}``multiplicativity constraints'' were originally formulated
to have the map \[
\pp\circ\phi\circ\ap:\left(\Halg,\odot\right)\rightarrow\mathcal{L}^{-}\,,\]
defined as a character of the commutative Hopf algebra to the ring
of scalar Laurent series $\mathcal{L}$. $\phi:\Halg\rightarrow\Lrt$
denotes the character which induces Kreimer's (regularized) Feynman
rules, $\ap$ is the antipode of the commutative Hopf algebra, and
$\pp:\Lrt^{+}\oplus\Lrt^{-}\rightarrow\Lrt^{-}$ is the projection
to the principal part. Kreimer was able to show that the Rota-Baxter
condition implies the multiplicativity of $R\circ\phi\circ\ap$ (Prop.~2,
loc. cit.). More advanced topics related to Rota-Baxter algebras in
Physics and Mathematics partially induced by Kreimer's observation
can be found, e.g., in \citep{Ebrahimi-FardGuo2007}.

Observe, however, that in the framework advocated here all maps, and
in particular $R$, are homomorphisms of the commutative algebra $\left(\Halg,\odot\right)$,
so that a Rota-Baxter argument is not necessary, since any algebra
homomorphism, trivially, is a Rota-Baxter map of weight one.
\end{rem}

\subsection{Graphs}

The role of the additional composition product $\ocomp$ and antipode
$\ap_{\linc}$ in \[
\left(\Halg,\odot,e,\Delta,\overline{e},\ap,\ocomp,\ap_{\linc}\right)\,.\]
may become clearer if we break them down to the graph level. This
graphwise interpretation is regained, if we insert for $\Sm$ its
perturbative expansion (\ref{eq:PerturbationSeries}), and regard
the corresponding operations on the level of the graph contributions.
This can be done since all maps involved are linear and since by condition
{[}C3{]} all sums are finite at each fixed order of $\hbar$. Remember
that we only regard full vertex parts as subgraphs.

In accordance with the structure of $\left(\Halg,\odot,e\right)$,
we regard the abstract algebra of graphs with disjoint union $\dot{\cup}$
as product and the empty set $\emptyset$ as unit. Let $\Gamma\in\mathcal{G}$
be a graph with $n$ vertices, hence a contribution to $a_{n}(\Sm$).
Let $\Part^{c}V(\Gamma)$ be the set of all connected partitions of
the vertex set of $\Gamma$. By connected partition we mean a partition
$\Ptn$ whose blocks $I\in\Ptn$ give rise to connected full vertex
parts $\gamma_{I}\subset\Gamma$. We can restrict to connected partitions,
since the principal parts of the regularized amplitudes corresponding
to disconnected graphs vanish. Denote by $\Gamma/\Ptn$ the graph,
which has the blocks $I\in\Ptn$ as vertices and as lines the lines
in $\Gamma$, which connect different blocks of $\Ptn$. Let $\gamma_{I}$,
$I\in\Ptn$, be the full vertex part of the block $I\in\Ptn$. Then
we can write the coproduct on the level of graphs as\[
\Delta\Gamma=\sum_{\Ptn\in\Part^{c}V(\Gamma)}\Gamma/\Ptn\otimes\dot{\bigcup_{I\in\Ptn}}\gamma_{I}\,.\]
Applying the map $\id_{\Halg}\otimes\ap_{\linc}$ gives\[
\left(\id_{\Halg}\otimes\ap_{\linc}\right)\circ\Delta\Gamma=\sum_{\Ptn\in\Part^{c}V(\Gamma)}\Gamma/\Ptn\otimes\dot{\bigcup_{I\in\Ptn}}\ap_{\linc}(\gamma_{I})\,,\]
which corresponds to (recursively) computing the counterterms for
the connected subgraphs $\gamma_{I}$, e.g., in DimReg+MS. Applying
the composition product $\linc$, inserts the counterterms $\ap_{\linc}(\gamma_{I})$
at the vertices of $\Gamma/\Ptn$ to give one contribution to the
{}``renormalized graph'',\[
\id_{\Halg}\bullet_{\linc}\ap_{\linc}=\sum_{\Ptn\in\Part^{c}V(\Gamma)}\Gamma/\Ptn\ocomp\dot{\bigcup_{I\in\Ptn}}\ap_{\linc}(\gamma_{I})\,.\]
It is one of the results of the present thesis that this procedure,
performed on the level of analytically regularized amplitudes in arbitrary
representation (momentum or position space) leads to finitely regularized,
i.e., UV convergent integrals, and local counterterms in all orders
of perturbation theory.

\end{fmffile}


text:
{\bf Problem.} What is the positive integer $N$ for which $$ 22^2 \times 55^2 = 10^2 \times N^2 \ ? $$
{\bf Level.} Level 3
{\bf Type.} Prealgebra
{\bf Solution.} \begin{align*}
10^2\times N^2&=22^2\times55^2\\
&=\left(2^2\cdot11^2\right)\times\left(5^2\cdot11^2\right)\\
&=\left(2^2\cdot5^2\right)\times\left(11^2\cdot11^2\right)\\
&=10^2\times \left(11^2\right)^2\\
&=10^2\times 121^2
\end{align*} So $N=\boxed{121}$.

set_name: MATH


TITLE: If $(x - \frac {1}{x})^2 = 2$ and $x > 0$ what is $x^4 - \frac{1}{x^4}$?
QUESTION [0 upvotes]: If $(x - \frac {1}{x})^2 = 2$ and $x > 0$ what is $x^4 - \frac{1}{x^4}$?
I tried factorising the $x^4 - \frac{1}{x^4}$ expression and I got $(x^2 - \frac {1}{x^2})(x - \frac {1}{x})(x + \frac {1}{x})$ but I'm not sure what to do from here.
Thanks!

REPLY [0 votes]: $(x - \frac {1}{x})^2 = 2\implies x^2+\frac{1}{x^2}-2=2\implies x^2+\frac{1}{x^2}=4$
Again $(x + \frac {1}{x})^2=(x - \frac {1}{x})^2+4x\cdot\frac{1}{x}=2+4=6\implies x + \frac {1}{x}=\sqrt6\qquad \text{as}\quad x > 0$
$(x - \frac {1}{x})^2=2\implies x - \frac {1}{x}=\sqrt2 \qquad \text{as}\quad x > 0$
Now $x^4 - \frac{1}{x^4}=(x^2 - \frac{1}{x^2})(x^2 + \frac{1}{x^2})=(x^2 + \frac{1}{x^2})(x - \frac{1}{x})(x + \frac{1}{x})=4\cdot(\sqrt2) \cdot(\sqrt6)=8\sqrt3 $.


TITLE: Number of check bits for single-error-correcting binary linear code
QUESTION [3 upvotes]: This is Exercise 23 of the textbook "Abstract Algebra: Theory and Applications" by Thomas W. Judson, 2016; Page 111. Chapter 8 "Algebraic Coding Theory" mainly deals with binary linear code.

Exercise 23: How many check positions are needed for a single error-correcting code with 20 information positions? With 32 information positions?

My attempt: Suppose that the number of check bits is $r$. Then we have
$$2^r - 1 \ge r + 20$$ and $$2^r - 1 \ge r + 32$$ respectively. Therefore, $r$ are $5$ and $6$, respectively.
However, the (partial) answer given in the textbook (Page 331) is $r = 6$ for the case of 20 information bits:

For 20 information positions, at least 6 check bits are needed to ensure an error-correcting code.

What is wrong with my argument?
And how to solve this problem (for both cases of 20 and 32 information bits)?

REPLY [3 votes]: The answer in the book doesn't seem correct. $5$ parity bits can cover a block of $2^5-1=31$ bits, of which $31-5=26$ are information bits, so it should be more than enough to cover $20$ information bits. This Wiki article explains a popular linear code that achieves maximal efficiency in terms off the number of parity bits needed.
Let's explicitly build Hamming code that covers $20$ information bits.
We have a block of $25$ bits, numbered $b_1 \ldots b_{25}$. $5$ of those bits are parity bits, specifically $b_1, b_2, b_4, b_8, b_{16}$. The rest are information bits.
Expressions for the parity bits are:
\begin{array}{llcl} (1)& b_1 &=& b_3 \oplus b_5 \oplus \ldots \oplus b_{25} \\ (2)& b_2 &=& b_3 \oplus b_6 \oplus b_7 \oplus b_{10} \oplus b_{11} \oplus \ldots \oplus b_{22} \oplus b_{23} \\ (3)& b_4 &=& b_5 \oplus b_6 \oplus b_7 \oplus b_{12} \oplus b_{13} \oplus b_{14} \oplus b_{15} \oplus b_{20} \oplus b_{21} \oplus b_{22} \oplus b_{23} \\ (4)& b_8 &=& b_9 \oplus b_{10} \oplus \ldots \oplus b_{15} \oplus b_{24} \oplus b_{25} \\ (5)& b_{16} &=& b_{17} \oplus b_{18} \oplus \ldots \oplus b_{25} \end{array}
It is easy to see that each bit participates in a unique subset of these expressions. Specifically, bit $b_k$ participates in line $i$ iff the $i$th digit in the binary representation of $k$ is $1$.
Now if we have a possibly corrupt block $b'_1 \ldots b'_{25}$ where exactly one $b'_i$ is different from $b_i$, some of the equalities above will not hold. If we write down $0$ for each equality that holds and $1$ for each one that doesn't, we get exactly the number of the flipped bit, in binary. If all hold, there is no error.


TITLE: Linear operator with invariant subspace - Representation wrt to a basis [Kolman, 6.3.12]
QUESTION [1 upvotes]: A nonempty subspace $S$ of $V$ is called invariant under $L$ means : $L(S) \subseteq S$.
  Let $L: V \rightarrow V$ be a linear operator with invariant subspace $S.$
  Show that if $\dim S = m$ and $\dim V = n$, then $L$ has a representation with respect to a basis $\beta$ for $V$ of the form
  $\begin{bmatrix}
        \color{#009900}{A_{m \times m}} & B_{m \times (n - m)} \\
     \color{#009900}{O_{(n - m) \times m}} & C_{(n - m) \times (n - m)} \\
       \end{bmatrix}$.
Answer: Since $\dim S = m$ is given, let $\beta = \{\mathbf{v_i}\}_{1 \le i \le m}$ be an ordered basis for $S$.
  Since any lin-ind subset of a vector space can be extended to a basis for this same vector space proper (Basis Extension Theorem),
  thus $T = \{\mathbf{v_i, v_{m + 1}, \cdots, v_n}\}_{1 \le i \le m}$ is an ordered basis for $V$.
  Since  $L(S) \subseteq S$, thus $\forall \; 1 \le j \le m, \; L(\mathbf{v_j}) \in S$.
  $\Longrightarrow L(\mathbf{v_j}) = \sum\limits_{1 \le j \le m}a_j\mathbf{v_j} + \sum\limits_{(m+1) \le k \le n}0\mathbf{v_k} $.
  $\Longrightarrow [L(\mathbf{v_j})]_T = \begin{bmatrix}
    a_1 & a_2 & \ldots & a_m & 0 & \ldots & 0 \\
    \end{bmatrix}^T$.

The above $[L(\mathbf{v_j})]_T$ holds for $\forall \; 1 \le j \le m$, so it describes only $\color{#009900}{\text{the first $m$ columns of the matrix representation}}$? What about submatrices $B, C$?
Source: P398, 6.3.12, Elementary Linear Algebra by Kolman. WARNING: Book's odiously vile.

Supplementary dated Sep 12 2013 :
Doesn't the solution prove that the matrix representation of $L$ is only  $\begin{bmatrix}
        \color{#009900}{A_{m \times m}} \\ \color{#009900}{O_{(On - m) \times m}}  \\
       \end{bmatrix}$? Despite our apathy towards $B,C$, mustn't we conclude with $\begin{bmatrix}
        \color{#009900}{A_{m \times m}} & B_{m \times (n - m)} \\ \color{#009900}{O_{(n - m) \times m}} & C_{(n - m) \times (n - m)} \\
       \end{bmatrix}$ ?

REPLY [1 votes]: You are correct: the proof describes only the first $m$ columns. This is all that is necessary. The goal is to find a matrix representation where the lower $(n-m)\times m$ corner vanishes. To verify this property, we only need to consider the first $m$ columns. We don't care what $A,B,C$ are.
Regarding your edit, we know from the problem statement $L$ is an operator from an $n$-dimensional vector space to itself, so it must be an $n\times n$ matrix (after we fix a basis). We can partition any $n\times n$ matrix into the following form (I have violated alphabetical order to conform with the problem statement)
$$\begin{bmatrix}
        {A_{m \times m}} & B_{m \times (n - m)} \\ {D_{(n - m) \times m}} & C_{(n - m) \times (n - m)} \\
       \end{bmatrix}.$$
Think of taking a $n\times n$ matrix and drawing a vertical line after the $m$th column and a horizontal line after the $m$th row. This gives you $4$ block matrices (here $A,B,C,D$).
Your problem asks us to show that if we fix a basis for $V$ as stipulated in the problem statement, then $D$ is the zero matrix $O$. To show this, we must only examine the first $m$ columns, since $D$ is contained in the first $m$ columns.