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\section*{A first idea of quantum field theory -- Renormalization}

\hypertarget{Renormalization}{}\subsection*{{Renormalization}}\label{Renormalization}

In this chapter we discuss the following topics:

\begin{itemize}%
\item \emph{\hyperlink{EpsteinGlaserRenormalization}{Epstein-Glaser normalization}}


\item \emph{\hyperlink{SPRenormalizationGroup}{Stückelberg-Petermann re-normalization}}


\item \emph{\hyperlink{UVRegularizationViaZ}{UV-Regularization via Counterterms}}


\item \emph{\hyperlink{EffectiveQFTFlowWislonian}{Wilson-Polchinski effective QFT flow}}


\item \emph{\hyperlink{RGFlowGeneral}{Renormalization group flow}}


\item \emph{\hyperlink{ScalingTransformatinRGFlow}{Gell-Mann \& Low RG Flow}}



\end{itemize}
$\,$

In the \hyperlink{InteractingQuantumFields}{previous chapter} we have seen that the construction of [[interacting quantum field theory|interacting]] [[perturbative quantum field theories]] is given by perturbative [[S-matrix schemes]] (def. \ref{LagrangianFieldTheoryPerturbativeScattering}), equivalently by [[time-ordered products]] (def. \ref{TimeOrderedProduct}) or equivalently by [[Feynman amplitudes]] (prop. \ref{FeynmanPerturbationSeriesAwayFromCoincidingPoints}). These are uniquely fixed away from coinciding interaction points (prop. \ref{TimeOrderedProductAwayFromDiagonal}) by the given [[local observable|local]] [[interaction]] (prop. \ref{TimeOrderedProductAwayFromDiagonal}), but involve further choices of interactions whenever interaction vertices coincide (prop. \ref{RenormalizationIsInductivelyExtensionToDiagonal}). This choice is called the choice of [[renormalization|(``re''-)normalization]] (def. \ref{ExtensionOfTimeOrderedProoductsRenormalization}) in [[perturbative QFT]].

In this rigorous discussion no ``infinite divergent quantities'' (as in the original informal discussion due to [[Schwinger-Tomonaga-Feynman-Dyson]]) that need to be ``re-normalized'' to finite well-defined quantities are ever considered, instead finite well-defined quantities are considered right away, and the available space of choices is determined. Therefore making such choices is rather a \emph{normalization} of the [[time-ordered products]]/[[Feynman amplitudes]] (as prominently highlighted in \href{causal+perturbation+theoryscatt#Scharf95}{Scharf 95, see title, introduction, and section 4.3}). Actual re-normalization is the the change of such normalizations.

The construction of [[perturbative QFTs]] may be explicitly described by an [[induction|inductive]] [[extension of distributions]] of [[time-ordered products]]/[[Feynman amplitudes]] to coinciding interaction points. This is called

\begin{itemize}%
\item \emph{\hyperlink{EpsteinGlaserRenormalization}{Epstein-Glaser renormalization}}.

\end{itemize}
This inductive construction has the advantage that it gives accurate control over the space of available choices of (``re''-)normalizations (theorem \ref{ExistenceRenormalization} below) but it leaves the nature of the ``new interactions'' that are to be chosen at coinciding interaction points somwewhat implicit.

Alternatively, one may [[vertex redefinition|re-define the interactions]] explicitly (by adding ``[[counterterms]]'', remark \ref{TermCounter} below), depending on a chosen [[UV cutoff]]-scale (def. \ref{CutoffsUVForPerturbativeQFT} below), and construct the [[limit of a sequence|limit]] as the ``cutoff is removed'' (prop. \ref{UVRegularization} below). This is called (``re''-)normalization by

\begin{itemize}%
\item \emph{\hyperlink{UVRegularizationViaZ}{UV-Regularization via Counterterms}}.

\end{itemize}
This still leaves open the question how to choose the [[counterterms]]. For that it serves to understand the \emph{[[relative effective action]]} induced by the choice of [[UV cutoff]] at any given cutoff scale (def. \ref{EffectiveActionRelative} below). This is the perspective of \emph{[[effective quantum field theory]]} (remark \ref{pQFTEffective} below).

The [[infinitesimal]] change of these [[relative effective actions]] follows a universal [[differential equation]], known as \emph{[[Polchinski's flow equation]]} (prop. \ref{FlowEquationPolchinski} below). This makes the problem of (``re''-)normalization be that of solving this [[differential equation]] subject to chosen initial data. This is the perspective on (``re''-)normalization called

\begin{itemize}%
\item \emph{\hyperlink{EffectiveQFTFlowWislonian}{Wilson-Polchinski effective QFT flow}}.

\end{itemize}
The [[main theorem of perturbative renormalization]] (theorem \ref{PerturbativeRenormalizationMainTheorem} below) states that different [[S-matrix schemes]] are precisely related by [[vertex redefinitions]]. This yields the

\begin{itemize}%
\item \emph{\hyperlink{SPRenormalizationGroup}{Stückelberg-Petermann renormalization group}}.

\end{itemize}
If a sub-collection of [[renormalization schemes]] is parameterized by some [[group]] $RG$, then the [[main theorem of perturbative renormalization|main theorem]] implies [[vertex redefinitions]] depending on pairs of elements of $RG$ (prop. \ref{FlowRenormalizationGroup} below). This is known as

\begin{itemize}%
\item \emph{\hyperlink{RGFlowGeneral}{Renormalization group flow}}

\end{itemize}
Specifically [[scaling transformations]] on [[Minkowski spacetime]] yield such a collection of [[renormalization schemes]] (prop. \ref{RGFlowScalingTransformations} below); the corresponding [[renormalization group flow]] is known as

\begin{itemize}%
\item \emph{\hyperlink{ScalingTransformatinRGFlow}{Gell-Mann \& Low RG flow}}.

\end{itemize}
The [[infinitesimal]] behaviour of this flow is known as the \emph{[[beta function]]}, describing the \emph{[[running of the coupling constants]]} with scale (def. \ref{CouplingRunning} below).

$\,$

\textbf{[[Epstein-Glaser renormalization|Epstein-Glaser normalization]]}

The construction of [[perturbative quantum field theories]] around a given [[gauge fixing|gauge fixed]] [[relativistic field theory|relativistic]] [[free field theory|free field]] [[vacuum]] is equivalently, by prop. \ref{InteractingFieldAlgebraOfObservablesIsFormalDeformationQuantization}, the construction of [[S-matrices]] $\mathcal{S}(g S_{int} + j A)$ in the sense of [[causal perturbation theory]] (def. \ref{LagrangianFieldTheoryPerturbativeScattering}) for the given [[local observable|local]] [[interaction]] $g S_{int} + j A$. By prop. \ref{RenormalizationIsInductivelyExtensionToDiagonal} the construction of these [[S-matrices]] is [[induction|inductively]] in $k \in \mathbb{N}$ a choice of [[extension of distributions]] (remark \ref{TimeOrderedProductOfFixedInteraction} and def. \ref{ExtensionOfDistributions} below) of the corresponding $k$-ary [[time-ordered products]] of the [[interaction]] to the locus of coinciding interaction points. An inductive construction of the [[S-matrix]] this way is called \emph{[[Epstein-Glaser renormalization|Epstein-Glaser-(``re''-)normalization]]} (def. \ref{ExtensionOfTimeOrderedProoductsRenormalization}).

By paying attention to the [[scaling degree of distributions|scaling degree]] (def. \ref{ScalingDegree} below) one may precisely characterize the space of choices in the [[extension of distributions]] (prop. \ref{SpaceOfPointExtensions} below): For a given [[local observable|local]] [[interaction]] $g S_{int} + j A$ it is inductively in $k \in \mathbb{N}$ a [[finite dimensional vector space|finite-dimensional]] [[affine space]]. This conclusion is theorem \ref{ExistenceRenormalization} below.

$\,$

\begin{prop}
\label{RenormalizationIsInductivelyExtensionToDiagonal}\hypertarget{RenormalizationIsInductivelyExtensionToDiagonal}{}
\textbf{([[renormalization|(``re''-)normalization]] is [[induction|inductive]] [[extension of distributions|extension]] of [[time-ordered products]] to [[diagonal]])}

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[gauge fixing|gauge-fixed]] [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}).

Assume that for $n \in \mathbb{N}$, [[time-ordered products]] $\{T_{k}\}_{k \leq n}$ of arity $k \leq n$ have been constructed in the sense of def. \ref{TimeOrderedProduct}. Then the time-ordered product $T_{n+1}$ of arity $n+1$ is uniquely fixed on the [[complement]]

\begin{displaymath}
\Sigma^{n+1} \setminus diag(n)
  \;=\;
  \left\{
     (x_i \in \Sigma)_{i = 1}^n
     \;\vert\;
     \underset{i,j}{\exists} (x_i \neq x_j)
  \right\}
\end{displaymath}
of the [[image]] of the [[diagonal]] inclusion $\Sigma \overset{diag}{\longrightarrow} \Sigma^{n}$ (where we regarded $T_{n+1}$ as a [[generalized function]] on $\Sigma^{n+1}$ according to remark \ref{NotationForTimeOrderedProductsAsGeneralizedFunctions}).

\end{prop}
This statement appears in (\href{renormalization#PopineauStora82}{Popineau-Stora 82}), with (unpublished) details in (\href{renormalization#Stora93}{Stora 93}), following personal communication by [[Henri Epstein]] (according to \href{renormalization#Duetsch18}{Dütsch 18, footnote 57}). Following this, statement and detailed proof appeared in (\href{renormalization#BrunettiFredenhagen99}{Brunetti-Fredenhagen 99}).

\begin{proof}
We will construct an [[open cover]] of $\Sigma^{n+1} \setminus \Sigma$ by subsets $\mathcal{C}_I \subset \Sigma^{n+1}$ which are [[disjoint unions]] of [[inhabited set|non-empty]] sets that are in [[causal order]], so that by [[causal factorization]] the time-ordered products $T_{n+1}$ on these subsets are uniquely given by $T_{k}(-) \star_H T_{n-k}(-)$. Then we show that these unique products on these special subsets do coincide on [[intersections]]. This yields the claim by a [[partition of unity]].

We now say this in detail:

For $I \subset \{1, \cdots, n+1\}$ write $\overline{I} \coloneqq \{1, \cdots, n+1\} \setminus I$. For $I, \overline{I} \neq \emptyset$, define the subset

\begin{displaymath}
\mathcal{C}_I
  \;\coloneqq\;
  \left\{
    (x_i)_{i \in \{1, \cdots, n+1\}} \in \Sigma^{n+1}
    \;\vert\;
    \{x_i\}_{i \in I} {\vee\!\!\!\wedge} \{x_j\}_{j \in \{1, \cdots, n+1\} \setminus I}
  \right\}
  \;\subset\;
  \Sigma^{n+1}
  \,.
\end{displaymath}
Since the [[causal order]]-relation involves the [[closed future cones]]/[[closed past cones]], respectively, it is clear that these are [[open subsets]]. Moreover it is immediate that they form an [[open cover]] of the [[complement]] of the [[diagonal]]:

\begin{displaymath}
\underset{ { I \subset \{1, \cdots, n+1\} \atop { I, \overline{I} \neq \emptyset } }   }{\cup}
  \mathcal{C}_I
  \;=\;
  \Sigma^{n+1} \setminus diag(\Sigma)
  \,.
\end{displaymath}
(Because any two distinct points in the [[globally hyperbolic spacetime]] $\Sigma$ may be causally separated by a [[Cauchy surface]], and any such may be deformed a little such as not to intersect any of a given finite set of points. )

Hence the condition of [[causal factorization]] on $T_{n+1}$ implies that [[restriction of distributions|restricted]] to any $\mathcal{C}_{I}$ these have to be given (in the condensed [[generalized function]]-notation from remark \ref{NotationForTimeOrderedProductsAsGeneralizedFunctions}) on any unordered tuple $\mathbf{X} = \{x_1, \cdots, x_{n+1}\} \in \mathcal{C}_I$ with corresponding induced tuples $\mathbf{I} \coloneqq \{x_i\}_{i \in I}$ and $\overline{\mathbf{I}} \coloneqq \{x_i\}_{i \in \overline{I}}$ by

\begin{equation}
T_{n+1}( \mathbf{X} )
  \;=\;
  T(\mathbf{I}) T(\overline{\mathbf{I}})
  \phantom{AA}
  \text{for}
  \phantom{A}
  \mathcal{X} \in \mathcal{C}_I
  \,.
\label{InductiveIdentificationOfTimeOrderedProductAwayFromDiagonal}\end{equation}
This shows that $T_{n+1}$ is unique on $\Sigma^{n+1} \setminus diag(\Sigma)$ if it exists at all, hence if these local identifications glue to a global definition of $T_{n+1}$. To see that this is the case, we have to consider any two such subsets

\begin{displaymath}
I_1, I_2
  \subset
  \{1, \cdots, n+1\}
  \,,
  \phantom{AA}
  I_1, I_2, \overline{I_1}, \overline{I_2} \neq \emptyset
  \,.
\end{displaymath}
By definition this implies that for

\begin{displaymath}
\mathbf{X} \in \mathcal{C}_{I_1} \cap \mathcal{C}_{I_2}
\end{displaymath}
a tuple of spacetime points which decomposes into causal order with respect to both these subsets, the corresponding mixed intersections of tuples are spacelike separated:

\begin{displaymath}
\mathbf{I}_1 \cap \overline{\mathbf{I}_2}
    \;
    {\gt\!\!\!\!\lt}
    \;
  \overline{\mathbf{I}_1} \cap \mathbf{I}_2
  \,.
\end{displaymath}
By the assumption that the $\{T_k\}_{k \neq n}$ satisfy causal factorization, this implies that the corresponding time-ordered products commute:

\begin{equation}
T(\mathbf{I}_1 \cap \overline{\mathbf{I}_2})
  \,
  T(\overline{\mathbf{I}_1} \cap \mathbf{I}_2)
  \;=\;
  T(\overline{\mathbf{I}_1} \cap \mathbf{I}_2)
  \,
  T(\mathbf{I}_1 \cap \overline{\mathbf{I}_2})
  \,.
\label{TimeOrderedProductsOfMixedIntersectionsCommute}\end{equation}
Using this we find that the identifications of $T_{n+1}$ on $\mathcal{C}_{I_1}$ and on $\mathcal{C}_{I_2}$, accrding to \eqref{InductiveIdentificationOfTimeOrderedProductAwayFromDiagonal}, agree on the intersection: in that for $\mathbf{X}  \in \mathcal{C}_{I_1} \cap \mathcal{C}_{I_2}$ we have

\begin{displaymath}
\begin{aligned}
    T( \mathbf{I}_1 ) T( \overline{\mathbf{I}_1} )
    & =
    T( \mathbf{I}_1 \cap \mathbf{I}_2 )
    T( \mathbf{I}_1 \cap \overline{\mathbf{I}_2} )
    \,
    T( \overline{\mathbf{I}_1} \cap \mathbf{I}_2 )
    T( \overline{\mathbf{I}_1} \cap \overline{\mathbf{I}_2} )
    \\
    & =
    T( \mathbf{I}_1 \cap \mathbf{I}_2 )
    \underbrace{
    T( \overline{\mathbf{I}_1} \cap \mathbf{I}_2 )
    T( \mathbf{I}_1 \cap \overline{\mathbf{I}_2} )
    }
    T( \overline{\mathbf{I}_1} \cap \overline{\mathbf{I}_2} )
    \\
    & =
    T( \mathbf{I}_2 )
    T( \overline{\mathbf{I}_2} )
  \end{aligned}
\end{displaymath}
Here in the first step we expanded out the two factors using \eqref{InductiveIdentificationOfTimeOrderedProductAwayFromDiagonal} for $I_2$, then under the brace we used \eqref{TimeOrderedProductsOfMixedIntersectionsCommute} and in the last step we used again \eqref{InductiveIdentificationOfTimeOrderedProductAwayFromDiagonal}, but now for $I_1$.

To conclude, let

\begin{equation}
\left(
    \chi_I
    \in
    C^\infty_{cp}(\Sigma^{n+1}),
    \,
    supp(\chi_I) \subset \mathcal{C}_i
  \right)_{ { I \subset \{1, \cdots, n+1\} } \atop { I, \overline{I} \neq \emptyset } }
\label{PartitionCausalOfUnityForComplementOfDiagonal}\end{equation}
be a [[partition of unity]] subordinate to the [[open cover]] formed by the $\mathcal{C}_I$:

Then the above implies that setting for any $\mathbf{X} \in \Sigma^{n+1} \setminus diag(\Sigma)$

\begin{equation}
T_{n+1}(\mathbf{X})
  \;\coloneqq\;
  \underset{
    { I \in \{1, \cdots, n+1\} }
    \atop
    { I, \overline{I} \neq \emptyset }
  }{\sum}
  \chi_i(\mathbf{X}) T( \mathbf{I} ) T( \overline{\mathbf{I}} )
\label{TimeOrderedProductsAwayFromDiagonalByInduction}\end{equation}
is well defined and satisfies causal factorization.

\end{proof}
\begin{remark}
\label{TimeOrderedProductOfFixedInteraction}\hypertarget{TimeOrderedProductOfFixedInteraction}{}
\textbf{([[time-ordered products]] of fixed [[interaction]] as [[distributions]])}

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[gauge fixing|gauge-fixed]] [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to def. \ref{VacuumFree}, and assume that the [[field bundle]] is a [[trivial vector bundle]] (example \ref{TrivialVectorBundleAsAFieldBundle})

and let

\begin{displaymath}
g S_{int} + j A
  \;\in\;
  LoObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g,j\rangle
\end{displaymath}
be a polynomial [[local observable]] as in def. \ref{FormalParameters}, to be regarded as a [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]]. This means that there is a [[finite set]]

\begin{displaymath}
\left\{
    \mathbf{L}_{int,i}, \mathbf{\alpha}_{i'} \in \Omega^{p+1,0}_\Sigma(E_{\text{BV-BRST}})
  \right\}_{i,i'}
\end{displaymath}
of [[Lagrangian densities]] which are monomials in the field and jet coordinates, and a corresponding finite set

\begin{displaymath}
\left\{
    g_{sw,i} \in C^\infty_{cp}(\Sigma)\langle g \rangle
    \,,\,
    j_{sw,i'} \in C^\infty_{cp}(\Sigma)\langle j \rangle
  \right\}
\end{displaymath}
of [[adiabatic switchings]], such that

\begin{displaymath}
g S_{int} + j A
  \;=\;
  \tau_{\Sigma}
  \left(
    \underset{i}{\sum}
    g_{sw,i} \mathbf{L}_{int,i}
    \;+\;
    \underset{i'}{\sum}
    j_{sw,i'} \mathbf{\alpha}_{i'}
  \right)
\end{displaymath}
is the [[transgression of variational differential forms]] (def. \ref{TransgressionOfVariationalDifferentialFormsToConfigrationSpaces}) of the sum of the products of these [[adiabatic switching]] with these [[Lagrangian densities]].

In order to discuss the [[S-matrix]] $\mathcal{S}(g S_{int} + j A)$ and hence the [[time-ordered products]] of the special form $T_k\left(  \underset{k \, \text{factors}}{\underbrace{g S_{int} + j A, \cdots, g S_{int} + j A }} \right)$ it is sufficient to restrict attention to the [[restriction]] of each $T_k$ to the subspace of [[local observables]] induced by the finite set of [[Lagrangian densities]] $\{\mathbf{L}_{int,i}, \mathbf{\alpha}_{i'}\}_{i,i'}$.

This restriction is a [[continuous linear functional]] on the corresponding space of [[bump functions]] $\{g_{sw,i}, j_{sw,i'}\}$, hence a [[distribution|dstributional]] [[section]] of a corresponding [[trivial vector bundle]].

In terms of this, prop. \ref{RenormalizationIsInductivelyExtensionToDiagonal} says that the choice of [[time-ordered products]] $T_k$ is [[induction|inductively]] in $k$ a choice of [[extension of distributions]] to the [[diagonal]].

If $\Sigma = \mathbb{R}^{p,1}$ is [[Minkowski spacetime]] and we impose the [[renormalization condition]] ``translation invariance'' (def. \ref{RenormalizationConditions}) then each $T_k$ is a distribution on $\Sigma^{k-1} = \mathbb{R}^{(p+1)(k-1)}$ and the [[extension of distributions]] is from the complement of the origina $0 \in \mathbb{R}^{(p+1)(k-1)}$.

\end{remark}
Therefore we now discuss [[extension of distributions]] (def. \ref{ExtensionOfDistributions} below) on [[Cartesian spaces]] from the complement of the origin to the origin. Since the space of choices of such extensions turns out to depend on the \emph{[[scaling degree of distributions]]}, we first discuss that (def. \ref{ScalingDegree} below).

\begin{defn}
\label{RescaledDistribution}\hypertarget{RescaledDistribution}{}
\textbf{([[scaling degree of distributions|rescaled distribution]])}

Let $n \in \mathbb{N}$. For $\lambda \in (0,\infty) \subset \mathbb{R}$ a [[positive number|positive]] [[real number]] write

\begin{displaymath}
\itexarray{
    \mathbb{R}^n
      &\overset{s_\lambda}{\longrightarrow}&
    \mathbb{R}^n
    \\
    x &\mapsto& \lambda x
  }
\end{displaymath}
for the [[diffeomorphism]] given by multiplication with $\lambda$, using the canonical [[real vector space]]-structure of $\mathbb{R}^n$.

Then for $u \in \mathcal{D}'(\mathbb{R}^n)$ a [[distribution]] on the [[Cartesian space]] $\mathbb{R}^n$ the \emph{rescaled distribution} is the [[pullback of a distribution|pullback]] of $u$ along $m_\lambda$

\begin{displaymath}
u_\lambda
  \coloneqq
   s_\lambda^\ast u
  \;\in\;
  \mathcal{D}'(\mathbb{R}^n)
  \,.
\end{displaymath}
Explicitly, this is given by

\begin{displaymath}
\itexarray{
     \mathcal{D}(\mathbb{R}^n)
       &\overset{ \langle  u_\lambda, - \rangle}{\longrightarrow}&
     \mathbb{R}
     \\
     b &\mapsto& \lambda^{-n} \langle u , b(\lambda^{-1}\cdot (-))\rangle
  }
  \,.
\end{displaymath}
Similarly for $X \subset \mathbb{R}^n$ an [[open subset]] which is invariant under $s_\lambda$, the rescaling of a distribution $u \in \mathcal{D}'(X)$ is is $u_\lambda \coloneqq s_\lambda^\ast u$.

\end{defn}
\begin{defn}
\label{ScalingDegree}\hypertarget{ScalingDegree}{}
\textbf{([[scaling degree of a distribution]])}

Let $n \in \mathbb{N}$ and let $X \subset \mathbb{R}^n$ be an [[open subset]] of [[Cartesian space]] which is invariant under [[rescaling]] $s_\lambda$ (def. \ref{RescaledDistribution}) for all $\lambda \in (0,\infty)$, and let $u \in \mathcal{D}'(X)$ be a [[distribution]] on this subset. Then

\begin{enumerate}%
\item The \emph{[[scaling degree of a distribution|scaling degree]]} of $u$ is the [[infimum]]

\begin{displaymath}
sd(u)
  \;\coloneqq\;
  inf
  \left\{
    \omega \in \mathbb{R}
     \;\vert\;
    \underset{\lambda \to 0}{\lim}
     \lambda^\omega u_\lambda
     = 0
  \right\}
\end{displaymath}
of the set of [[real numbers]] $\omega$ such that the [[limit of a sequence|limit]] of the rescaled distribution $\lambda^\omega u_\lambda$ (def. \ref{RescaledDistribution}) vanishes. If there is no such $\omega$ one sets $sd(u) \coloneqq \infty$.


\item The \emph{[[degree of divergence of a distribution|degree of divergence]]} of $u$ is the difference of the scaling degree by the [[dimension]] of the underlying space:



\end{enumerate}
\begin{displaymath}
deg(u) \coloneqq sd(u) - n
     \,.
\end{displaymath}
\end{defn}
\begin{example}
\label{NonSingularDistributionsScalingDegree}\hypertarget{NonSingularDistributionsScalingDegree}{}
\textbf{([[scaling degree of distributions|scaling degree]] of [[non-singular distributions]])}

If $u = u_f$ is a [[non-singular distribution]] given by [[bump function]] $f \in C^\infty(X) \subset \mathcal{D}'(X)$, then its [[scaling degree of a distribution|scaling degree]] (def. \ref{ScalingDegree}) is non-[[positive number|positive]]

\begin{displaymath}
sd(u_f) \leq 0
  \,.
\end{displaymath}
Specifically if the first non-vanishing [[partial derivative]] $\partial_\alpha f(0)$ of $f$ at 0 occurs at order ${\vert \alpha\vert} \in \mathbb{N}$, then the scaling degree of $u_f$ is $-{\vert \alpha\vert}$.

\end{example}
\begin{proof}
By definition we have for $b \in C^\infty_{cp}(\mathbb{R}^n)$ any [[bump function]] that

\begin{displaymath}
\begin{aligned}
    \left\langle
      \lambda^{\omega}
      (u_f)_\lambda,
      n
    \right\rangle
    & =
    \lambda^{\omega-n}
    \underset{\mathbb{R}^n}{\int}
       f(x) g(\lambda^{-1} x)
    d^n x
    \\
    & =
    \lambda^{\omega}
    \underset{\mathbb{R}^n}{\int}
       f(\lambda x) g(x)
    d^n x
  \end{aligned}
  \,,
\end{displaymath}
where in last line we applied [[change of integration variables]].

The limit of this expression is clearly zero for all $\omega \gt 0$, which shows the first claim.

If moreover the first non-vanishing [[partial derivative]] of $f$ occurs at order ${\vert \alpha \vert} = k$, then [[Hadamard's lemma]] says that $f$ is of the form

\begin{displaymath}
f(x)
    \;=\;
  \left( \underset{i}{\prod} \alpha_i ! \right)^{-1}
  (\partial_\alpha f(0))
  \underset{i}{\prod} (x^i)^{\alpha_i}
  +
  \underset{ {\beta \in \mathbb{N}^n} \atop { {\vert \beta\vert} = {\vert \alpha \vert} + 1 }  }{\sum}
  \underset{i}{\prod} (x^i)^{\beta_i}
  h_{\beta}(x)
\end{displaymath}
where the $h_{\beta}$ are [[smooth functions]]. Hence in this case

\begin{displaymath}
\begin{aligned}
    \left\langle
      \lambda^{\omega}
      (u_f)_\lambda,
      n
    \right\rangle
    & =
    \lambda^{\omega + {\vert \alpha\vert }}
    \underset{\mathbb{R}^n}{\int}
      \left( \underset{i}{\prod} \alpha_i ! \right)^{-1}
     (\partial_\alpha f(0))
     \underset{i}{\prod} (x^i)^{\alpha_i}
     b(x)
    d^n x
    \\
    & \phantom{=}
    +
    \lambda^{\omega + {\vert \alpha\vert} + 1}
    \underset{\mathbb{R}^n}{\int}
      \underset{i}{\prod} (x^i)^{\beta_i}
      h_{\beta}(x)
      b(x)
    d^n x
  \end{aligned}
  \,.
\end{displaymath}
This makes manifest that the expression goes to zero with $\lambda \to 0$ precisely for $\omega \gt - {\vert \alpha \vert}$, which means that

\begin{displaymath}
sd(u_f) = -{\vert \alpha \vert}
\end{displaymath}
in this case.

\end{proof}
\begin{example}
\label{DerivativesOfDeltaDistributionScalingDegree}\hypertarget{DerivativesOfDeltaDistributionScalingDegree}{}
\textbf{([[scaling degree of a distribution|scaling degree]] of [[derivative of a distribution|derivatives]] of [[delta-distributions]])}

Let $\alpha \in \mathbb{N}^n$ be a multi-index and $\partial_\alpha \delta \in \mathcal{D}'(X)$ the corresponding [[partial derivative|partial]] [[derivative of distributions|derivatives]] of the [[delta distribution]] $\delta_0 \in \mathcal{D}'(\mathbb{R}^n)$ [[support of a distribution|supported]] at $0$. Then the [[degree of divergence of a distribution|degree of divergence]] (def. \ref{ScalingDegree}) of $\partial_\alpha \delta_0$ is the total order the derivatives

\begin{displaymath}
deg\left(
    {\, \atop \,} \partial_\alpha\delta_0{\, \atop \,}
  \right)
  \;=\;
  {\vert \alpha \vert}
\end{displaymath}
where ${\vert \alpha\vert} \coloneqq \underset{i}{\sum} \alpha_i$.

\end{example}
\begin{proof}
By definition we have for $b \in C^\infty_{cp}(\mathbb{R}^n)$ any [[bump function]] that

\begin{displaymath}
\begin{aligned}
    \left\langle
      \lambda^\omega (\partial_\alpha \delta_0)_\lambda,
      b
    \right\rangle
    & =
    (-1)^{{\vert \alpha \vert}}
    \lambda^{\omega-n}
    \left(
      \frac{
        \partial^{{\vert \alpha \vert}}
      }{
       \partial^{\alpha_1} x^1  \cdots \partial^{\alpha_n}x^n
      }
      b(\lambda^{-1}x)
    \right)_{\vert x = 0}
    \\
    & =
    (-1)^{{\vert \alpha \vert}}
    \lambda^{\omega - n - {\vert \alpha\vert}}
    \frac{
      \partial^{{\vert \alpha \vert}}
    }{
     \partial^{\alpha_1} x^1  \cdots \partial^{\alpha_n}x^n
    }
    b(0)
  \end{aligned}
  \,,
\end{displaymath}
where in the last step we used the [[chain rule]] of [[differentiation]]. It is clear that this goes to zero with $\lambda$ as long as $\omega \gt n + {\vert \alpha\vert}$. Hence $sd(\partial_{\alpha} \delta_0) = n + {\vert \alpha \vert}$.

\end{proof}
\begin{example}
\label{FeynmanPropagatorOnMinkowskiScalingDegree}\hypertarget{FeynmanPropagatorOnMinkowskiScalingDegree}{}
\textbf{([[scaling degree of a distribution|scaling degree]] of [[Feynman propagator]] on [[Minkowski spacetime]])}

Let

\begin{displaymath}
\Delta_F(x)
  \;=\;
  \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
  \frac{+i}{(2\pi)^{p+1}}
  \int
  \int_{-\infty}^\infty
  \frac{
     e^{i k_\mu x^\mu}
  }{
    - k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 + i \epsilon
  }
  \, d k_0 \, d^p \vec k
\end{displaymath}
be the [[Feynman propagator]] for the massive [[free field|free]] [[real scalar field]] on $n = p+1$-dimensional [[Minkowski spacetime]] (prop. \ref{FeynmanPropagatorAsACauchyPrincipalvalue}). Its [[scaling degree of a distribution|scaling degree]] is

\begin{displaymath}
\begin{aligned}
    sd(\Delta_{F})
    & = n - 2
    \\
    & = p -1
  \end{aligned}
  \,.
\end{displaymath}
\end{example}
(\href{renormalization#BrunettiFredenhagen00}{Brunetti-Fredenhagen 00, example 3 on p. 22})

\begin{proof}
Regarding $\Delta_F$ as a [[generalized function]] via the given [[Fourier transform of distributions|Fourier-transform]] expression, we find by [[change of integration variables]] in the Fourier integral that in the scaling limit the Feynman propagator becomes that for vannishing [[mass]], which scales homogeneously:

\begin{displaymath}
\begin{aligned}
    \underset{\lambda \to 0}{\lim}
    \left(
    \lambda^\omega
    \;
    \Delta_F(\lambda x)
    \right)
    & =
    \underset{\lambda \to 0}{\lim}
    \left(
    \lamba^{\omega}
    \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
    \frac{+i}{(2\pi)^{p+1}}
    \int
    \int_{-\infty}^\infty
    \frac{
       e^{i k_\mu \lambda x^\mu}
    }{
      - k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 + i \epsilon
    }
    \, d k_0 \, d^p \vec k
    \right)
    \\
    & =
    \underset{\lambda \to 0}{\lim}
    \left(
    \lambda^{\omega-n}
    \;
    \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
    \frac{+i}{(2\pi)^{p+1}}
    \int
    \int_{-\infty}^\infty
    \frac{
       e^{i k_\mu \lambda x^\mu}
    }{
      - (\lambda^{-2}) k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 + i \epsilon
    }
    \, d k_0 \, d^p \vec k
    \right)
    \\
    & =
    \underset{\lambda \to 0}{\lim}
    \left(
    \lambda^{\omega-n + 2 }
    \;
    \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
    \frac{+i}{(2\pi)^{p+1}}
    \int
    \int_{-\infty}^\infty
    \frac{
       e^{i k_\mu \lambda x^\mu}
    }{
      - k_\mu k^\mu + i \epsilon
    }
    \, d k_0 \, d^p \vec k
    \right)
    \,.
  \end{aligned}
\end{displaymath}
\end{proof}
\begin{prop}
\label{ScalingDegreeOfDistributionsBasicProperties}\hypertarget{ScalingDegreeOfDistributionsBasicProperties}{}
\textbf{(basic properties of [[scaling degree of distributions]])}

Let $X \subset \mathbb{R}^n$ and $u \in \mathcal{D}'(X)$ be a [[distribution]] as in def. \ref{RescaledDistribution}, such that its [[scaling degree of a distribution|scaling degree]] is finite: $sd(u) \lt \infty$ (def. \ref{ScalingDegree}). Then

\begin{enumerate}%
\item For $\alpha \in \mathbb{N}^n$, the [[partial derivative|partial]] [[derivative of distributions]] $\partial_\alpha$ increases scaling degree at most by ${\vert \alpha\vert }$:

\begin{displaymath}
deg(\partial_\alpha u) \;\leq\; deg(u) + {\vert \alpha\vert}
\end{displaymath}

\item For $\alpha \in \mathbb{N}^n$, the [[product of distributions]] with the smooth coordinate functions $x^\alpha$ decreases scaling degree at least by ${\vert \alpha\vert }$:

\begin{displaymath}
deg(x^\alpha u) \;\leq\; deg(u) - {\vert \alpha\vert}
\end{displaymath}

\item Under [[tensor product of distributions]] their scaling degrees add:

\begin{displaymath}
sd(u \otimes v) \leq sd(u) + sd(v)
\end{displaymath}
for $v \in \mathcal{D}'(Y)$ another distribution on $Y \subset \mathbb{R}^{n'}$;


\item $deg(f u) \leq deg(u) - k$ for $f \in C^\infty(X)$ and $f^{(\alpha)}(0) = 0$ for ${\vert \alpha\vert} \leq k-1$;



\end{enumerate}
\end{prop}
(\href{renormalization#BrunettiFredenhagen00}{Brunetti-Fredenhagen 00, lemma 5.1}, \href{renormalization#Duetsch18}{Dütsch 18, exercise 3.34})

\begin{proof}
The first three statements follow with manipulations as in example \ref{NonSingularDistributionsScalingDegree} and example \ref{DerivativesOfDeltaDistributionScalingDegree}.

For the fourth\ldots{}

\end{proof}
\begin{prop}
\label{ScalingDegreeOfProductDistribution}\hypertarget{ScalingDegreeOfProductDistribution}{}
\textbf{([[scaling degree of distributions|scaling degree]] of [[product of distributions|product distribution]])}

Let $u,v \in \mathcal{D}'(\mathbb{R}^n)$ be two [[distributions]] such that

\begin{enumerate}%
\item both have finite [[degree of divergence of a distribution|degree of divergence]] (def. \ref{ScalingDegree})

\begin{displaymath}
deg(u), deg(v) \lt \infty
\end{displaymath}

\item their [[product of distributions]] is well-defined

\begin{displaymath}
u v \in \mathcal{D}'(\mathbb{R}^n)
\end{displaymath}
(in that their [[wave front sets]] satisfy [[Hörmander's criterion]])



\end{enumerate}
then the product distribution has [[degree of divergence of a distribution|degree of divergence]] bounded by the sum of the separate degrees:

\begin{displaymath}
deg(u v)
  \;\leq\;
  deg(u) + deg(v)
  \,.
\end{displaymath}
\end{prop}
With the concept of [[scaling degree of distributions]] in hand, we may now discuss [[extension of distributions]]:

\begin{defn}
\label{ExtensionOfDistributions}\hypertarget{ExtensionOfDistributions}{}
\textbf{([[extension of distributions]])}

Let $X \overset{\iota}{\subset} \hat X$ be an inclusion of [[open subsets]] of some [[Cartesian space]]. This induces the operation of [[restriction of distributions]]

\begin{displaymath}
\mathcal{D}'(\hat X)
    \overset{\iota^\ast}{\longrightarrow}
  \mathcal{D}'(X)
  \,.
\end{displaymath}
Given a [[distribution]] $u \in \mathcal{D}'(X)$, then an \emph{[[extension]]} of $u$ to $\hat X$ is a distribution $\hat u \in \mathcal{D}'(\hat X)$ such that

\begin{displaymath}
\iota^\ast \hat u
  \;=\;
  u
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{ExtensionUniqueNonPositiveDegreeOfDivergence}\hypertarget{ExtensionUniqueNonPositiveDegreeOfDivergence}{}
\textbf{(unique [[extension of distributions]] with negative [[degree of divergence of a distribution|degree of divergence]])}

For $n \in \mathbb{N}$, let $u \in \mathcal{D}'(\mathbb{R}^n \setminus \{0\})$ be a [[distribution]] on the [[complement]] of the origin, with [[negative number|negative]] [[degree of divergence of a distribution|degree of divergence]] at the origin

\begin{displaymath}
deg(u) \lt 0
  \,.
\end{displaymath}
Then $u$ has a \emph{unique} [[extension of distributions]] $\hat u \in \mathcal{D}'(\mathbb{R}^n)$ to the origin with the same degree of divergence

\begin{displaymath}
deg(\hat u) = deg(u)
  \,.
\end{displaymath}
\end{prop}
(\href{renormalization#BrunettiFredenhagen00}{Brunetti-Fredenhagen 00, theorem 5.2}, \href{renormalization#Duetsch18}{Dütsch 18, theorem 3.35 a)})

\begin{proof}
Regarding uniqueness:

Suppose $\hat u$ and ${\hat u}^\prime$ are two extensions of $u$ with $deg(\hat u) = deg({\hat u}^\prime)$. Both being extensions of a distribution defined on $\mathbb{R}^n \setminus \{0\}$, this difference has [[support of a distribution|support]] at the origin $\{0\} \subset \mathbb{R}^n$. By prop. \ref{PointSupportedDistributionsAreSumsOfDerivativesOfDeltaDistibutions} this implies that it is a linear combination of [[derivative of a distribution|derivatives]] of the [[delta distribution]] [[support of a distribution|supported]] at the origin:

\begin{displaymath}
{\hat u}^\prime - \hat u
  \;=\;
  \underset{
    {\alpha \in \mathbb{N}^n}
  }{\sum}
  c^\alpha \partial_\alpha \delta_0
\end{displaymath}
for constants $c^\alpha \in \mathbb{C}$. But by example \ref{DerivativesOfDeltaDistributionScalingDegree} the [[degree of divergence of a distribution|degree of divergence]] of these [[point-supported distributions]] is non-negative

\begin{displaymath}
deg( \partial_\alpha \delta_0)
  =
  {\vert \alpha\vert} \geq 0
  \,.
\end{displaymath}
This implies that $c^\alpha = 0$ for all $\alpha$, hence that the two extensions coincide.

Regarding existence:

Let

\begin{displaymath}
b \in C^\infty_{cp}(\mathbb{R}^n)
\end{displaymath}
be a [[bump function]] which is $\leq 1$ and [[constant function|constant]] on 1 over a [[neighbourhood]] of the origin. Write

\begin{displaymath}
\chi \coloneqq 1 - b \;\in\; C^\infty(\mathbb{R}^n)
\end{displaymath}
\begin{quote}%
graphics grabbed from \href{renormalization#Duetsch18}{Dütsch 18, p. 108}


\end{quote}
and for $\lambda \in (0,\infty)$ a [[positive real number]], write

\begin{displaymath}
\chi_\lambda(x)
  \coloneqq
  \chi(\lambda x)
  \,.
\end{displaymath}
Since the [[product of distributions|product]] $\chi_\lambda u$ has [[support of a distribution]] on a [[complement]] of a [[neighbourhood]] of the origin, we may extend it by zero to a distribution on all of $\mathbb{R}^n$, which we will denote by the same symbols:

\begin{displaymath}
\chi_\lambda u \in \mathcal{D}'(\mathbb{R}^n)
  \,.
\end{displaymath}
By construction $\chi_\lambda u$ coincides with $u$ away from a neighbourhood of the origin, which moreover becomes arbitrarily small as $\lambda$ increases. This means that if the following [[limit of a sequence|limit]] exists

\begin{displaymath}
\hat u
  \;\coloneqq\;
  \underset{\lambda \to \infty}{\lim}
   \chi_\lambda u
\end{displaymath}
then it is an extension of $u$.

To see that the limit exists, it is sufficient to observe that we have a [[Cauchy sequence]], hence that for all $b\in C^\infty_{cp}(\mathbb{R}^n)$ the difference

\begin{displaymath}
(\chi_{n+1} u - \chi_n u)(b)
  \;=\;
  u(b)( \chi_{n+1} + \chi_n )
\end{displaymath}
becomes arbitrarily small.

It remains to see that the unique extension $\hat u$ thus established has the same scaling degree as $u$. This is shown in (\href{extension+of+distributions#BrunettiFredenhagen00}{Brunetti-Fredenhagen 00, p. 24}).

\end{proof}
\begin{prop}
\label{SpaceOfPointExtensions}\hypertarget{SpaceOfPointExtensions}{}
\textbf{(space of [[point-extensions of distributions]])}

For $n \in \mathbb{N}$, let $u \in \mathcal{D}'(\mathbb{R}^n \setminus \{0\})$ be a [[distribution]] of [[scaling degree of a distribution|degree of divergence]] $deg(u) \lt \infty$.

Then $u$ does admit at least one [[extension of distributions|extension]] (def. \ref{ExtensionOfDistributions}) to a distribution $\hat u \in \mathcal{D}'(\mathbb{R}^n)$, and every choice of extension has the same [[degree of divergence of a distribution|degree of divergence]] as $u$

\begin{displaymath}
deg(\hat u) = deg(u)
  \,.
\end{displaymath}
Moreover, any two such extensions $\hat u$ and ${\hat u}^\prime$ differ by a linear combination of [[partial derivatives|partial]] [[derivatives of distributions]] of order $\leq deg(u)$ of the [[delta distribution]] $\delta_0$ [[support of a distribution|supported]] at the origin:

\begin{displaymath}
{\hat u}^\prime
  -
  \hat u
  \;=\;
  \underset{
    { \alpha \in \mathbb{N}^n }
    \atop
    { {\vert \alpha\vert} \leq deg(u) }
  }{\sum}
  q^\alpha
  \partial_\alpha \delta_0
  \,,
\end{displaymath}
for a finite number of constants $q^\alpha \in \mathbb{C}$.

\end{prop}
This is essentially (\href{renormalization#Hoermander90}{Hörmander 90, thm. 3.2.4}). We follow (\href{renormalization#BrunettiFredenhagen00}{Brunetti-Fredenhagen 00, theorem 5.3}), which was inspired by (\hyperlink{EpsteinGlaser73}{Epstein-Glaser 73, section 5}). Review of this approach is in (\href{renormalization#Duetsch18}{Dütsch 18, theorem 3.35 (b)}), see also remark \ref{WExtensions} below.

\begin{proof}
For $f \in C^\infty(\mathbb{R}^n)$ a [[smooth function]], and $\rho \in \mathbb{N}$, we say that \emph{$f$ vanishes to order $\rho$} at the origin if all [[partial derivatives]] with multi-index $\alpha \in \mathbb{N}^n$ of total order ${\vert \alpha\vert} \leq \rho$ vanish at the origin:

\begin{displaymath}
\partial_\alpha f (0) = 0
  \phantom{AAA}
  {\vert \alpha\vert} \leq \rho
  \,.
\end{displaymath}
By [[Hadamard's lemma]], such a function may be written in the form

\begin{equation}
f(x)
  \;=\;
  \underset{
    {\alpha \in \mathbb{N}^n}
    \atop
    { {\vert \alpha \vert} = \rho + 1 }
  }{\sum}
  x^\alpha
  r_\alpha(x)
\label{ForVanishingOrderRhoHadamardExpansion}\end{equation}
for [[smooth functions]] $r_\alpha \in C^\infty_{cp}(\mathbb{R}^n)$.

Write

\begin{displaymath}
\mathcal{D}_\rho(\mathbb{R}^n)
    \hookrightarrow
  \mathcal{D}(\mathbb{R}^n)
    \coloneqq
  C^\infty_{cp}(\mathbb{R}^n)
\end{displaymath}
for the subspace of that of all [[bump functions]] on those that vanish to order $\rho$ at the origin.

By definition this is equivalently the joint [[kernel]] of the [[partial derivative|partial]] [[derivatives of distributions]] of order ${\vert \alpha\vert}$ of the [[delta distribution]] $\delta_0$ [[support of a distribution|supported]] at the origin:

\begin{displaymath}
b \in \mathcal{D}_\rho(\mathbb{R}^n)
  \phantom{AA}
  \Leftrightarrow
  \phantom{AA}
  \underset{
    {\alpha \in \mathbb{N}^n}
      \atop
    { {\vert \alpha\vert} \leq \rho }
  }
  {\forall}
    \left\langle \partial_\alpha \delta_0, b \right\rangle
  = 0
  \,.
\end{displaymath}
Therefore every [[continuous linear map|continuous linear]] [[projection]]

\begin{equation}
p_\rho
   \;\colon\;
  \mathcal{D}(\mathbb{R}^n)
    \longrightarrow
  \mathcal{D}_\rho(\mathbb{R}^n)
\label{ForExtensionOfDistributionsProjectionMaps}\end{equation}
may be obtained from a choice of \emph{dual basis} to the $\{\partial_\alpha \delta_0\}$, hence a choice of smooth functions

\begin{displaymath}
\left\{
    w^\beta \in C^\infty_{cp}(\mathbb{R}^n)
  \right\}_{ { \beta \in \mathbb{N}^n } \atop { {\vert \beta\vert} \leq \rho } }
\end{displaymath}
such that

\begin{displaymath}
\left\langle
    \partial_\alpha \delta_0
    \,,\,
    w^\beta
  \right\rangle
  \;=\;
  \delta_\alpha^\beta
  \phantom{AAA}
  \Leftrightarrow
  \phantom{AAA}
  \partial_\alpha w^\beta(0)
    \;=\;
  \delta_\alpha^\beta
  \phantom{AAAA}
  \text{for}\, {\vert \alpha\vert} \leq \rho
  \,,
\end{displaymath}
by setting

\begin{equation}
p_\rho
  \;\coloneqq\;
  id
  \;-\;
  \left\langle
  \underset{
    {\alpha \in \mathbb{N}^n}
    \atop
    { {\vert \alpha\vert} \leq \rho }
  }{\sum}
  w^\alpha \partial_\alpha \delta_0
  \,,\,
  (-)
  \right\rangle
  \,,
\label{SpaceOfSmoothFunctionsOfGivenVaishingOrderProjector}\end{equation}
hence

\begin{displaymath}
p_\rho
  \;\colon\;
  b
    \mapsto
  b
    -
  \underset{
    {\alpha \in \mathbb{N}^n}
    \atop
    { {\vert \alpha\vert} \leq \rho }
  }{\sum}
  (-1)^{{\vert \alpha\vert}} w^\alpha \partial_\alpha b(0)
  \,.
\end{displaymath}
Together with [[Hadamard's lemma]] in the form \eqref{ForVanishingOrderRhoHadamardExpansion} this means that every $b \in \mathcal{D}(\mathbb{R}^n)$ is decomposed as

\begin{equation}
\begin{aligned}
  b(x)
  & =
  p_\rho(b)(x)
  \;+\;
  (id - p_\rho)(b)(x)
  \\
  & =
  \underset{
    { \alpha \in \mathbb{N}^n }
    \atop
    { {\vert \alpha \vert} = \rho + 1 }
  }{\sum}
  x^\alpha r_\alpha(x)
  \;+\;
  \underset{
    { \alpha \in \mathbb{N}^n }
    \atop
    { {\vert \alpha \vert} \leq \rho }
  }{\sum}
  (-1)^{{\vert \alpha \vert}}
  w^\alpha \partial_\alpha b(0)
  \end{aligned}
\label{ForExtensionOfDistributionsTestFunctionDecomposition}\end{equation}
Now let

\begin{displaymath}
\rho
  \;\coloneqq\;
  deg(u)
  \,.
\end{displaymath}
Observe that (by prop. \ref{ScalingDegreeOfDistributionsBasicProperties}) the [[degree of divergence of a distribution|degree of divergence]] of the [[product of distributions]] $x^\alpha u$ with ${\vert \alpha\vert} = \rho + 1$ is [[negative number|negative]]

\begin{displaymath}
\begin{aligned}
    deg\left(
      x^\alpha u
    \right)
    & =
    \rho - {\vert \alpha \vert}
    \leq
    -1
 \end{aligned}
\end{displaymath}
Therefore prop. \ref{ExtensionUniqueNonPositiveDegreeOfDivergence} says that each $x^\alpha u$ for ${\vert \alpha\vert} = \rho + 1$ has a unique extension $\widehat{ x^\alpha u}$ to the origin. Accordingly the composition $u \circ p_\rho$ has a unique extension, by \eqref{ForExtensionOfDistributionsTestFunctionDecomposition}:

\begin{equation}
\begin{aligned}
  \left\langle
    \hat u
    \,,\,
    b
  \right\rangle
  & =
  \left\langle
    \hat u
    ,
    p_\rho(b)
  \right\rangle
  +
  \left\langle
    \hat u
    ,
    (id - p_\rho)(b)
  \right\rangle
  \\
  & =
  \underset{
    { \alpha \in \mathbb{N}^n }
    \atop
    { {\vert \alpha \vert} = \rho + 1 }
  }{\sum}
  \underset{
    \text{unique}
  }{
  \underbrace{
    \left\langle
     \widehat{x^\alpha u}
      \,,\,
      r_\alpha
    \right\rangle
  }
  }
  \;+\;
  \underset{
    { \alpha \in \mathbb{N}^n }
    \atop
    { {\vert \alpha\vert} \leq \rho }
  }{\sum}
  \underset{
    {  q^\alpha }
    \atop
    { \text{choice} }
  }{
  \underbrace{
  \langle
    \hat u
    \,,\,
    w^\alpha
  \rangle
  }
  }
  \left\langle
    \partial_\alpha \delta_0
    \,,\,
    b
  \right\rangle
  \end{aligned}
\label{ExtensionOfDitstributionsPointFixedAndChoice}\end{equation}
That says that $\hat u$ is of the form

\begin{displaymath}
\hat u
  \;=\;
  \underset{
    \text{unique}
  }{
  \underbrace{
    \widehat{
      u \circ p_\rho
    }
  }
  }
  +
  \underset{
    { \alpha \in \mathbb{N}^n  }
    \atop
    { {\vert \alpha\vert} \leq \rho }
  }{\sum}
  c^\alpha
  \,
  \partial_\alpha \delta_0
\end{displaymath}
for a finite number of constants $c^\alpha \in \mathbb{C}$.

Notice that for any extension $\hat u$ the exact value of the $c^\alpha$ here depends on the arbitrary choice of dual basis $\{w^\alpha\}$ used for this construction. But the uniqueness of the first summand means that for any two choices of extensions $\hat u$ and ${\hat u}^\prime$, their difference is of the form

\begin{displaymath}
{\hat u}^\prime - \hat u
  \;=\;
  \underset{
    { \alpha \in \mathbb{N}^n  }
    \atop
    { {\vert \alpha\vert} \leq \rho }
  }{\sum}
  ( (c')^\alpha - c^\alpha )
  \,
  \partial_\alpha \delta_0
  \,,
\end{displaymath}
where the constants $q^\alpha \coloneqq ( (c')^\alpha - c^\alpha ) \in \mathbb{C}$ are independent of any choices.

It remains to see that all these $\hat u$ in fact have the same degree of divergence as $u$.

By example \ref{DerivativesOfDeltaDistributionScalingDegree} the degree of divergence of the point-supported distributions on the right is $deg(\partial_\alpha \delta_0) = {\vert \alpha\vert} \leq \rho$.

Therefore to conclude it is now sufficient to show that

\begin{displaymath}
deg\left(
    \widehat{
      u \circ p_\rho
    }
  \right)
  \;=\;
  \rho
  \,.
\end{displaymath}
This is shown in (\href{extension+of+distributions#BrunettiFredenhagen00}{Brunetti-Fredenhagen 00, p. 25}).

\end{proof}
\begin{remark}
\label{WExtensions}\hypertarget{WExtensions}{}
\textbf{(``W-extensions'')}

Since in \href{renormalization#BrunettiFredenhagen00}{Brunetti-Fredenhagen 00, (38)} the projectors \eqref{SpaceOfSmoothFunctionsOfGivenVaishingOrderProjector} are denoted ``$W$'', the construction of [[extensions of distributions]] via the proof of prop. \ref{SpaceOfPointExtensions} has come to be called ``W-extensions'' (e.g \href{renormalization#Duetsch18}{Dütsch 18}).

\end{remark}
In conclusion we obtain the central theorem of [[causal perturbation theory]]:

\begin{theorem}
\label{ExistenceRenormalization}\hypertarget{ExistenceRenormalization}{}
\textbf{(existence and choices of [[renormalization|(``re''-)normalization]] of [[S-matrices]]/[[perturbative QFTs]])}

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[gauge fixing|gauge-fixed]] [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]], according to def. \ref{VacuumFree}, such that the underlying [[spacetime]] is [[Minkowski spacetime]] and the [[Wightman propagator]] $\Delta_H$ is translation-invariant.

Then:

\begin{enumerate}%
\item an [[S-matrix scheme]] $\mathcal{S}$ (def. \ref{LagrangianFieldTheoryPerturbativeScattering}) around this vacuum exists;


\item for $g S_{int} + j A \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g,j\rangle$ a [[local observable]] as in def. \ref{FormalParameters}, regarded as an [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]], the space of possible choices of [[S-matrices]]

\begin{displaymath}
\mathcal{S}(g S_{int} + j A)
  \;\in\;
  PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]
\end{displaymath}
hence of the corresponding [[perturbative QFTs]], by prop. \ref{InteractingFieldAlgebraOfObservablesIsFormalDeformationQuantization}, is, [[induction|inductively]] in $k \in \mathbb{N}$, a [[finite dimensional vector space|finite dimensional]] [[affine space]], parameterizing the [[extension of distributions|extension]] of the [[time-ordered product]] $T_k$ to the locus of coinciding interaction points.



\end{enumerate}
\end{theorem}
\begin{proof}
By prop. \ref{FeynmanPropagatorOnMinkowskiScalingDegree} the [[Feynman propagator]] is finite [[scaling degree of a distribution]], so that by prop. \ref{ScalingDegreeOfProductDistribution} the binary [[time-ordered product]] away from the diagonal $T_2(-,-)\vert_{\Sigma^2 \setminus diag(\Sigma)} = (-) \star_{F} (-)$ has finite scaling degree.

By prop. \ref{ScalingDegreeOfProductDistribution} this implies that in the inductive description of the time-ordered products by prop. \ref{RenormalizationIsInductivelyExtensionToDiagonal}, each induction step is the [[extension of distributions]] of finite [[scaling degree of a distribution]] to the point. By prop. \ref{SpaceOfPointExtensions} this always exists.

This proves the first statement.

Now if a polynomial local interaction is fixed, then via remark \ref{TimeOrderedProductOfFixedInteraction} each induction step involved extending a finite number of distributions, each of finite scaling degree. By prop. \ref{SpaceOfPointExtensions} the corresponding space of choices is in each step a finite-dimensional affine space.

\end{proof}
$\,$

\textbf{[[Stückelberg-Petermann renormalization group]]}

A genuine re-normalization is the passage from one [[S-matrix]] [[renormalization scheme|(``re''-)normalization scheme]] $\mathcal{S}$ to another such scheme $\mathcal{S}'$. The [[induction|inductive]] [[Epstein-Glaser renormalization|Epstein-Glaser (``re''-normalization)]] construction (prop. \ref{RenormalizationIsInductivelyExtensionToDiagonal}) shows that the difference between any $\mathcal{S}$ and $\mathcal{S}'$ is inductively in $k \in \mathbb{N}$ a choice of extra term in the [[time-ordered product]] of $k$ factors, equivalently in the [[Feynman amplitudes]] for [[Feynman diagrams]] with $k$ [[vertices]], that contributes when all $k$ of these vertices coincide in [[spacetime]] (prop. \ref{SpaceOfPointExtensions}).

A natural question is whether these additional interactions that appear when several interaction vertices coincide may be absorbed into a re-definition of the original interaction $g S_{int} + j A$. Such an \emph{[[interaction vertex redefinition]]} (def. \ref{InteractionVertexRedefinition} below)

\begin{displaymath}
\mathcal{Z}
  \;\colon\;
  g S_{int} + j A
  \;\mapsto\;
  g S_{int} + j A
  \;+\;
  \text{higher order corrections}
\end{displaymath}
should perturbatively send [[local observables|local]] interactions to local interactions with higher order corrections.

The \emph{[[main theorem of perturbative renormalization]]} (theorem \ref{PerturbativeRenormalizationMainTheorem} below) says that indeed under mild conditions every re-normalization $\mathcal{S} \mapsto \mathcal{S}'$ is induced by such an [[interaction vertex redefinition]] in that there exists a \emph{unique} such redefinition $\mathcal{Z}$ so that for every local interaction $g S_{int} +  j A$ we have that [[scattering amplitudes]] for the interaction $g S_{int} + j A$ computed with the [[renormalization scheme|(``re''-)normalization scheme]] $\mathcal{S}'$ equal those computed with $\mathcal{S}$ but applied to the [[interaction vertex redefinition|re-defined interaction]] $\mathcal{Z}(g S_{int} + j A)$:

\begin{displaymath}
\mathcal{S}'
  \left(
    {\, \atop \,}
    g S_{int} +  j A
    {\, \atop \,}
  \right)
  \;=\;
  \mathcal{S}\left( {\, \atop \,} \mathcal{Z}(g S_{int} + j A) {\, \atop \,} \right)
  \,.
\end{displaymath}
This means that the [[interaction vertex redefinitions]] $\mathcal{Z}$ form a [[group]] under [[composition]] which [[action|acts]] [[transitive action|transitively]] and [[free action|freely]], hence [[regular action|regularly]], on the set of [[S-matrix]] [[renormalization schemes|(``re''-)normalization schemes]]; this is called the \emph{[[Stückelberg-Petermann renormalization group]]} (theorem \ref{PerturbativeRenormalizationMainTheorem} below).

$\,$

\begin{defn}
\label{InteractionVertexRedefinition}\hypertarget{InteractionVertexRedefinition}{}
\textbf{([[perturbative interaction vertex redefinition]])}

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H)$ be a [[gauge fixing|gauge fixed]] [[free field theory|free field]] [[vacuum]] (def. \ref{VacuumFree}).

A \emph{[[perturbative interaction vertex redefinition]]} (or just \emph{[[vertex redefinition]]}, for short) is an [[endofunction]]

\begin{displaymath}
\mathcal{Z}
  \;\colon\;
  LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle
    \longrightarrow
  LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle
\end{displaymath}
on [[local observables]] with formal parameters adjoined (def. \ref{FormalParameters}) such that there exists a sequence $\{Z_k\}_{k \in \mathbb{N}}$ of [[continuous linear functionals]], symmetric in their arguments, of the form

\begin{displaymath}
\left(
    {\, \atop \,}
    LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle
    {\, \atop \,}
  \right)^{\otimes^k_{\mathbb{C}[ [ \hbar, g, j] ]}}
  \longrightarrow
  LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle
\end{displaymath}
such that for all $g S_{int} + j A \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle$ the following conditions hold:

\begin{enumerate}%
\item (perturbation)

\begin{enumerate}%
\item $Z_0(g S_{int + j A}) = 0$


\item $Z_1(g S_{int} + j A) = g S_{int} + j A$


\item and

\begin{displaymath}
\begin{aligned}
 \mathcal{Z}(g S_{int} + j A)
 & =
 Z \exp_\otimes( g S_{int} + j A )
 \\
 &
 \coloneqq
 \underset{k \in \mathbb{N}}{\sum}
 \frac{1}{k!}
 Z_k(
 \underset{
 k \, \text{args}
 }{
 \underbrace{
 g S_{int} + j A , \cdots, g S_{int} + j A
 }
 }
 )
  \end{aligned}
\end{displaymath}


\end{enumerate}

\item (field independence) The [[local observable]] $\mathcal{Z}(g S_{int} + j A)$ depends on the [[field histories]] only through its argument $g S_{int} + j A$, hence by the [[chain rule]]:

\begin{equation}
\frac{\delta}{\delta \mathbf{\Phi}^a(x)}
  \mathcal{Z}(g S_{int} + j A)
  \;=\;
  \mathcal{Z}'_{g S_{int} + j A}
  \left(
      \frac{\delta}{\delta \mathbf{\Phi}^a(x)} (g S_{int} + j A)
  \right)
\label{FieldIndependenceVertexRedefinition}\end{equation}


\end{enumerate}
\end{defn}
The following proposition should be compared to the axiom of \emph{[[causal additivity]]} of the [[S-matrix]] scheme \eqref{CausalAdditivity}:

\begin{prop}
\label{InteractionVertexRedefinitionAdditivity}\hypertarget{InteractionVertexRedefinitionAdditivity}{}
\textbf{(local additivity of [[vertex redefinitions]])}

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H)$ be a [[gauge fixing|gauge fixed]] [[free field theory|free field]] [[vacuum]] (def. \ref{VacuumFree}) and let $\mathcal{Z}$ be a [[vertex redefinition]] (def. \ref{InteractionVertexRedefinition}).

Then for all [[local observables]] $O_0, O_1, O_2 \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]\langle g, j\rangle$ with spacetime support denoted $supp(O_i) \subset \Sigma$ (def. \ref{SpacetimeSupport}) we have

\begin{enumerate}%
\item (local additivity)

\begin{displaymath}
\begin{aligned}
    &
    \left(
      supp(O_1)
      \cap
      supp(O_2)
      =
      \emptyset
    \right)
    \\
    &
      \Rightarrow
    \phantom{AA}
    \mathcal{Z}( O_0 + O_1 + O_2)
    =
    \mathcal{Z}( O_0 + O_1 ) - \mathcal{Z}(O_0) + \mathcal{Z}(O_0 + O_2)
  \end{aligned}
  \,.
\end{displaymath}

\item (preservation of spacetime support)

\begin{displaymath}
supp
  \left(
    {\, \atop \,}
      \mathcal{Z}(O_0 + O_1)
      -
      \mathcal{Z}(O_0)
    {\, \atop \,}
  \right)
  \;\subset\;
  supp(O_1)
\end{displaymath}
hence in particular

\begin{displaymath}
supp
  \left(
    {\, \atop \,}
    \mathcal{Z}(O_1)
    {\, \atop \,}
  \right) = supp(O_1)
\end{displaymath}


\end{enumerate}
\end{prop}
(\href{renormalization#Duetsch18}{Dütsch 18, exercise 3.98})

\begin{proof}
Under the inclusion

\begin{displaymath}
LocObs(E_{\text{BV-BRST}})
    \hookrightarrow
  PolyObs(E_{\text{BV-BRST}})
\end{displaymath}
of [[local observables]] into [[polynomial observables]] we may think of each $Z_k$ as a [[generalized function]], as for [[time-ordered products]] in remark \ref{NotationForTimeOrderedProductsAsGeneralizedFunctions}.

Hence if

\begin{displaymath}
O_j = \underset{\Sigma}{\int} j^\infty_\Sigma( \mathbf{L}_j )
\end{displaymath}
is the [[transgression of variational differential forms|transgression]] of a [[Lagrangian density]] $\mathbf{L}$ we get

\begin{displaymath}
Z_k(
    (O_1 + O_2 + O_3)
    ,
    \cdots
    ,
    (O_1 + O_2 + O_3)
  )
  =
  \underset{
    j_1, \cdots, j_k \in \{0,1,2\}
  }{\sum}
  \underset{\Sigma^{k}}{\int}
    Z( \mathbf{L}_{j_1}(x_1) , \cdots , \mathbf{L}_{j_k}(x_k) )
  \,.
\end{displaymath}
Now by definition $Z_k(\cdots)$ is in the subspace of [[local observables]], i.e. those [[polynomial observables]] whose [[coefficient]] [[distributions]] are [[support of a distribution|supported]] on the [[diagonal]], which means that

\begin{displaymath}
\frac{\delta}{\delta \mathbf{\Phi}^a(x)}
  \frac{\delta}{\delta \mathbf{\Phi}^b(y)}
  Z_{k}(\cdots)
  =
  0
  \phantom{AA}
  \text{for}
  \phantom{AA}
  x \neq y
\end{displaymath}
Together with the axiom ``field independence'' \eqref{FieldIndependenceVertexRedefinition} this means that the support of these generalized functions in the [[integrand]] here must be on the [[diagonal]], where $x_1 = \cdots = x_k$.

By the assumption that the spacetime supports of $O_1$ and $O_2$ are disjoint, this means that only the summands with $j_1, \cdots, j_k \in \{0,1\}$ and those with $j_1, \cdots, j_k \in \{0,2\}$ contribute to the above sum. Removing the overcounting of those summands where all $j_1, \cdots, j_k \in \{0\}$ we get

\begin{displaymath}
\begin{aligned}
    & Z_k\left(
      {\, \atop \,}
      (O_1 + O_2 + O_3)
      ,
      \cdots
      ,
      (O_1 + O_2 + O_3)
      {\, \atop \,}
    \right)
    \\
    & =
    \underset{
      j_1, \cdots, j_k \in \{0,1\}
    }{\sum}
    \underset{\Sigma^{k}}{\int}
      Z( \mathbf{L}_{j_1}(x_1) , \cdots , \mathbf{L}_{j_k}(x_k) )
    \\
    & \phantom{=}
    -
    \underset{
      j_1, \cdots, j_k \in \{0\}
    }{\sum}
    \underset{\Sigma^{k}}{\int}
      Z( \mathbf{L}_{j_1}(x_1) , \cdots , \mathbf{L}_{j_k}(x_k) )
    \\
    & \phantom{=}
    -
    \underset{
      j_1, \cdots, j_k \in \{0,2\}
    }{\sum}
    \underset{\Sigma^{k}}{\int}
      Z( \mathbf{L}_{j_1}(x_1) , \cdots , \mathbf{L}_{j_k}(x_k) )
    \\
    & =
    Z_k\left( {\, \atop \,} (O_0 + O_1), \cdots, (O_0 + O_1) {\, \atop \,}\right)
    -
    Z_k\left( {\, \atop \,} O_0, \cdots,  O_0 {\, \atop \,} \right)
    +
    Z_k\left( {\, \atop \,} (O_0 + O_2), \cdots, (O_0 + O_2) {\, \atop \,} \right)
  \end{aligned}
  \,.
\end{displaymath}
This directly implies the claim.

\end{proof}
As a corollary we obtain:

\begin{prop}
\label{CausalFactorizationSatisfiedByCompositionOfSMatrixWithVertexRedefinition}\hypertarget{CausalFactorizationSatisfiedByCompositionOfSMatrixWithVertexRedefinition}{}
\textbf{([[composition]] of [[S-matrix]] scheme with [[vertex redefinition]] is again [[S-matrix]] scheme)}

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H)$ be a [[gauge fixing|gauge fixed]] [[free field theory|free field]] [[vacuum]] (def. \ref{VacuumFree}) and let $\mathcal{Z}$ be a [[vertex redefinition]] (def. \ref{InteractionVertexRedefinition}).

Then for

\begin{displaymath}
\mathcal{S}
  \;\colon\;
  LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle
    \longrightarrow
  PolyObs(E_{\text{BV-BRST}})_{mc}((\hbar))[ [ g,j ] ]
\end{displaymath}
and [[S-matrix]] scheme (def. \ref{LagrangianFieldTheoryPerturbativeScattering}), the [[composition|composite]]

\begin{displaymath}
\mathcal{S} \circ \mathcal{Z}
  \;\colon\;
  LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle
    \overset{\mathcal{Z}}{\longrightarrow}
  LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle
    \overset{\mathcal{S}}{\longrightarrow}
  PolyObs(E_{\text{BV-BRST}})_{mc}((\hbar))[ [ g,j ] ]
\end{displaymath}
is again an [[S-matrix]] scheme.

Moreover, if $\mathcal{S}$ satisfies the [[renormalization condition]] ``field independence'' (prop. \ref{BasicConditionsRenormalization}), then so does $\mathcal{S} \circ \mathcal{Z}$.

\end{prop}
(e.g \href{renormalization#Duetsch18}{Dütsch 18, theorem 3.99 (b)})

\begin{proof}
It is clear that [[causal order]] of the spacetime supports implies that they are in particular [[disjoint subset|disjoint]]

\begin{displaymath}
\left(
    {\, \atop \,}
    supp(O_1)
    {\vee\!\!\!\wedge}
    supp(O_2)
    {\, \atop \,}
  \right)
  \phantom{AA}
  \Rightarrow
  \phantom{AA}
  \left(
    {\, \atop \,}
    supp(O_1)
      \cap
    supp(O_)
    \;=\;
    \emptyset
    {\, \atop \,}
  \right)
\end{displaymath}
Therefore the local additivity of $\mathcal{Z}$ (prop. \ref{InteractionVertexRedefinitionAdditivity}) and the [[causal factorization]] of the [[S-matrix]] (remark \ref{DysonCausalFactorization}) imply the causal factorization of the composite:

\begin{displaymath}
\begin{aligned}
    \mathcal{S}
    \left(
      {\, \atop \,}
      \mathcal{Z}(O_1 + O_2)
      {\, \atop \,}
    \right)
    & =
    \mathcal{S}
    \left(
      {\, \atop \,}
      \mathcal{Z}(O_1)
      +
      \mathcal{Z}(O_2)
      {\, \atop \,}
    \right)
    \\
    & =
    \mathcal{S}
    \left(
      {\, \atop \,}
      \mathcal{Z}(O_1)
      {\, \atop \,}
    \right)
    \,
    \mathcal{S}
    \left(
      {\, \atop \,}
      \mathcal{Z}(O_2)
      {\, \atop \,}
    \right)
    \,.
  \end{aligned}
\end{displaymath}
But by prop. \ref{CausalFactorizationAlreadyImpliesSMatrix} this implies in turn [[causal additivity]] and hence that $\mathcal{S} \circ \mathcal{Z}$ is itself an S-matrix scheme.

Finally that $\mathcal{S} \circ \mathcal{Z}$ satisfies ``field indepndence'' if $\mathcal{S}$ does is immediate by the [[chain rule]], given that $\mathcal{Z}$ satisfies this condition by definition.

\end{proof}
\begin{prop}
\label{AnyTwoSMatrixRenormalizationSchemesDifferByAUniqueVertexRedefinition}\hypertarget{AnyTwoSMatrixRenormalizationSchemesDifferByAUniqueVertexRedefinition}{}
\textbf{(any two [[S-matrix]] [[renormalization schemes]] differ by unique [[vertex redefinition]])}

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H)$ be a [[gauge fixing|gauge fixed]] [[free field theory|free field]] [[vacuum]] (def. \ref{VacuumFree}).

Then for $\mathcal{S}, \mathcal{S}'$ any two [[S-matrix]] schemes (def. \ref{LagrangianFieldTheoryPerturbativeScattering}) which both satisfy the [[renormalization condition]] ``field independence'', the there exists a unique [[vertex redefinition]] $\mathcal{Z}$ (def. \ref{InteractionVertexRedefinition}) relating them by [[composition]], i. e. such that

\begin{displaymath}
\mathcal{S}'
  \;=\;
  \mathcal{S} \circ \mathcal{Z}
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
By applying both sides of the equation to linear combinations of local observables of the form $\kappa_1 O_1 + \cdots + \kappa_k O_k$ and then taking [[derivatives]] with respect to $\kappa$ at $\kappa_j = 0$ (as in example \ref{TimeOrderedProductsFromSMatrixScheme}) we get that the equation in question implies

\begin{displaymath}
(i \hbar)^k
  \frac{
    \partial^k
  }{
    \partial \kappa_1 \cdots \partial \kappa_k
  }
  \mathcal{S}'( \kappa_1 O_1 + \cdots + \kappa_k O_k )
  \vert_{\kappa_1, \cdots, \kappa_k = 0}
  \;=\;
  (i \hbar)^k
  \frac{
    \partial^k
  }{
    \partial \kappa_1 \cdots \partial \kappa_k
  }
  \mathcal{S} \circ \mathcal{Z}( \kappa_1 O_1 + \cdots + \kappa_k O_k )
  \vert_{\kappa_1, \cdots, \kappa_k = 0}
\end{displaymath}
which in components means that

\begin{displaymath}
\begin{aligned}
    T'_k( O_1, \cdots, O_k )
    & =
    \underset{
      2 \leq n \leq k
    }{\sum}
      \frac{1}{n!}
      (i \hbar)^{k-n}
      \underset{
        { { I_1 \sqcup \cdots \sqcup I_n } \atop { =  \{1, \cdots, k\}, } }
        \atop
        { I_1, \cdots, I_n \neq \emptyset }
      }{\sum}
      T_n
      \left(
        {\, \atop \,}
        Z_{{\vert I_1\vert}}\left( (O_{i_1})_{i_1 \in I_1} \right),
        \cdots,
        Z_{{\vert I_n\vert}}\left( (O_{i_n})_{i_n \in I_n} \right),
        {\, \atop \,}
      \right)
  \\
  & \phantom{=}
    +
    Z_k( O_1,\cdots, O_k )
  \end{aligned}
\end{displaymath}
where $\{T'_k\}_{k \in \mathbb{N}}$ are the [[time-ordered products]] corresponding to $\mathcal{S}'$ (by example \ref{TimeOrderedProductsFromSMatrixScheme}) and $\{T_k\}_{k \in \mathcal{N}}$ those correspondong to $\mathcal{S}$.

Here the sum on the right runs over all ways that in the composite $\mathcal{S} \circ \mathcal{Z}$ a $k$-ary operation arises as the composite of an $n$-ary time-ordered product applied to the ${\vert I_i\vert}$-ary components of $\mathcal{Z}$, for $i$ running from 1 to $n$; except for the case $k = n$, which is displayed separately in the second line

This shows that if $\mathcal{Z}$ exists, then it is unique, because its coefficients $Z_k$ are [[induction|inductively]] in $k$ given by the expressions

\begin{equation}
\begin{aligned}
    & Z_k( O_1,\cdots, O_k )
    \\
    & =
    T'_k( O_1, \cdots, O_k )
    \;-\;
    \underset{
      (T \circ \mathcal{Z}_{\lt k})_k
    }{
    \underbrace{
    \underset{
      2 \leq n \leq k
    }{\sum}
      \frac{1}{n!}
      (i \hbar)^{k-n}
      \underset{
        { { I_1 \sqcup \cdots \sqcup I_n } \atop { =  \{1, \cdots, k\}, } }
        \atop
        { I_1, \cdots, I_n \neq \emptyset }
      }{\sum}
      T_n
      \left(
        Z_{{\vert I_1\vert}}( (O_{i_1})_{i_1 \in I_1} ),
        \cdots,
        Z_{{\vert I_n\vert}}( (O_{i_n})_{i_n \in I_n} ),
      \right)
    }
    }
  \end{aligned}
\label{MainTheoremPerturbativeRenormalizationInductionStep}\end{equation}
(The symbol under the brace is introduced as a convenient shorthand for the term above the brace.)

Hence it remains to see that the $Z_k$ defined this way satisfy the conditions in def. \ref{InteractionVertexRedefinition}.

The condition ``perturbation'' is immediate from the corresponding condition on $\mathcal{S}$ and $\mathcal{S}'$.

Similarly the condition ``field independence'' follows immediately from the assumoption that $\mathcal{S}$ and $\mathcal{S}'$ satisfy this condition.

It only remains to see that $Z_k$ indeed takes values in [[local observables]]. Given that the [[time-ordered products]] a priori take values in the larrger space of [[microcausal polynomial observables]] this means to show that the spacetime support of $Z_k$ is on the [[diagonal]].

But observe that, as indicated in the above formula, the term over the brace may be understood as the coefficient at order $k$ of the [[exponential series]]-expansion of the [[composition|composite]] $\mathcal{S} \circ \mathcal{Z}_{\lt k}$, where

\begin{displaymath}
\mathcal{Z}_{\lt k}
  \;\coloneqq\;
  \underset{
    n \in \{1, \cdots, k-1\}
  }{\sum}
  \frac{1}{n!} Z_n
\end{displaymath}
is the truncation of the [[vertex redefinition]] to degree $\lt k$. This truncation is clearly itself still a vertex redefinition (according to def. \ref{InteractionVertexRedefinition}) so that the composite $\mathcal{S} \circ \mathcal{Z}_{\lt k}$ is still an [[S-matrix]] scheme (by prop. \ref{CausalFactorizationSatisfiedByCompositionOfSMatrixWithVertexRedefinition}) so that the $(T \circ \mathcal{Z}_{\lt k})_k$ are [[time-ordered products]] (by example \ref{TimeOrderedProductsFromSMatrixScheme}).

So as we solve $\mathcal{S}' = \mathcal{S} \circ \mathcal{Z}$ inductively in degree $k$, then for the induction step in degree $k$ the expressions $T'_{\lt k}$ and $(T \circ \mathcal{Z})_{\lt k}$ agree and are both time-ordered products. By prop. \ref{RenormalizationIsInductivelyExtensionToDiagonal} this implies that $T'_{k}$ and $(T \circ \mathcal{Z}_{\lt k})_{k}$ agree away from the diagonal. This means that their difference $Z_k$ is supported on the diagonal, and hence is indeed local.

\end{proof}
In conclusion this establishes the following pivotal statement of [[perturbative quantum field theory]]:

\begin{theorem}
\label{PerturbativeRenormalizationMainTheorem}\hypertarget{PerturbativeRenormalizationMainTheorem}{}
\textbf{([[main theorem of perturbative renormalization]] -- [[Stückelberg-Petermann renormalization group]] of [[vertex redefinitions]])}

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H)$ be a [[gauge fixing|gauge fixed]] [[free field theory|free field]] [[vacuum]] (def. \ref{VacuumFree}).

\begin{enumerate}%
\item the [[vertex redefinitions]] $\mathcal{Z}$ (def. \ref{InteractionVertexRedefinition}) form a [[group]] under [[composition]];


\item the set of [[S-matrix]] [[renormalization schemes|(``re''-)normalization schemes]] (def. \ref{LagrangianFieldTheoryPerturbativeScattering}), remark \ref{calSFunctionIsRenormalizationScheme}) satisfying the [[renormalization condition]] ``field independence'' (prop. \ref{BasicConditionsRenormalization}) is a [[torsor]] over this group, hence equipped with a [[regular action]] in that

\begin{enumerate}%
\item the set of [[S-matrix schemes]] is [[inhabited set|non-empty]];


\item any two [[S-matrix]] [[renormalization scheme|(``re''-)normalization schemes]] $\mathcal{S}$, $\mathcal{S}'$ are related by a \emph{unique} [[vertex redefinition]] $\mathcal{Z}$ via [[composition]]:

\begin{displaymath}
\mathcal{S}' \;=\; \mathcal{S} \circ \mathcal{Z}
  \,.
\end{displaymath}


\end{enumerate}


\end{enumerate}
This group is called the \emph{[[Stückelberg-Petermann renormalization group]]}.

Typically one imposes a set of [[renormalization conditions]] (def. \ref{RenormalizationConditions}) and considers the corresponding [[subgroup]] of [[vertex redefinitions]] preserving these conditions.

\end{theorem}
\begin{proof}
The [[group]]-[[structure]] and [[regular action]] is given by prop. \ref{CausalFactorizationSatisfiedByCompositionOfSMatrixWithVertexRedefinition} and prop. \ref{AnyTwoSMatrixRenormalizationSchemesDifferByAUniqueVertexRedefinition}. The existence of S-matrices follows is the statement of [[Epstein-Glaser renormalization|Epstein-Glaser (``re''-)normalization]] in theorem \ref{ExistenceRenormalization}.

\end{proof}
$\,$

\textbf{[[UV-regularization|UV-Regularization]] via [[counterterms]]}

While [[Epstein-Glaser renormalization]] (prop. \ref{RenormalizationIsInductivelyExtensionToDiagonal}) gives a transparent picture on the space of choices in [[renormalization|(``re''-)normalization]] (theorem \ref{ExistenceRenormalization}) the physical nature of the higher interactions that it introduces at coincident interaction points (via the [[extensions of distributions]] in prop. \ref{SpaceOfPointExtensions}) remains more implicit. But the [[main theorem of perturbative renormalization]] (theorem \ref{PerturbativeRenormalizationMainTheorem}), which re-expresses the \emph{difference} between any two such choices as an [[interaction vertex redefinition]], suggests that already the choice of [[renormalization|(``re''-)normalization]] itself should have an incarnation in terms of [[interaction vertex redefinitions]].

This may be realized via a construction of [[renormalization|(``re''-)normalization]] in terms of \emph{[[UV-regularization]]} (prop. \ref{UVRegularization} below): For any choice of ``[[UV-cutoff]]'', given by an approximation of the [[Feynman propagator]] $\Delta_F$ by [[non-singular distributions]] $\Delta_{F,\Lambda}$ (def. \ref{CutoffsUVForPerturbativeQFT} below) there is a unique ``[[effective S-matrix]]'' $\mathcal{S}_\Lambda$ induced at each cutoff scale (def. \ref{SMatrixEffective} below). While the ``UV-limit'' $\underset{\Lambda \to \infty}{\lim} \mathcal{S}_\Lambda$ does not in general exist, it may be ``regularized'' by applying suitable [[interaction vertex redefinitions]] $\mathcal{Z}_\Lambda$; if the higher-order corrections that these introduce serve to ``[[counterterms|counter]]'' (remark \ref{TermCounter} below) the coresponding UV-divergences.

This perspective of [[renormalization|(``re''-)normalization via]] via \emph{[[counterterms]]} is often regarded as the primary one. Its elegant proof in prop. \ref{UVRegularization} below, however relies on the [[Epstein-Glaser renormalization]] via inductive [[extensions of distributions]] and uses the same kind of argument as in the proof of the [[main theorem of perturbative renormalization]] (theorem \ref{PerturbativeRenormalizationMainTheorem} via prop. \ref{AnyTwoSMatrixRenormalizationSchemesDifferByAUniqueVertexRedefinition}) that establishes the [[Stückelberg-Petermann renormalization group]].

$\,$

\begin{defn}
\label{CutoffsUVForPerturbativeQFT}\hypertarget{CutoffsUVForPerturbativeQFT}{}
\textbf{([[UV cutoffs]])}

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[gauge fixing|gauge fixed]] [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] over [[Minkowski spacetime]] $\Sigma$ (according to def. \ref{VacuumFree}), where $\Delta_H = \tfrac{i}{2}(\Delta_+ - \Delta_-) + H$ is the corresponding [[Wightman propagator]] inducing the [[Feynman propagator]]

\begin{displaymath}
\Delta_F \in \Gamma'_{\Sigma \times \Sigma}(E_{\text{BV-BRST}} \boxtimes E_{\text{BV-BRST}})
\end{displaymath}
by $\Delta_F = \tfrac{i}{2}(\Delta_+ + \Delta_-) + H$.

Then a choice of \emph{[[UV cutoffs]] for [[perturbative QFT]]} around this vacuum is a collection of [[non-singular distributions]] $\Delta_{F,\Lambda}$ parameterized by [[positive real numbers]]

\begin{displaymath}
\itexarray{
    (0, \infty)
      &\overset{}{\longrightarrow}&
    \Gamma_{\Sigma \times \Sigma,cp}(E_{\text{BV-BRST}} \boxtimes E_{\text{BV-BRST}})
    \\
    \Lambda &\mapsto& \Delta_{F,\Lambda}
  }
\end{displaymath}
such that:

\begin{enumerate}%
\item each $\Delta_{F,\Lambda}$ satisfies the following basic properties

\begin{enumerate}%
\item (translation invariance)

\begin{displaymath}
\Delta_{F,\Lambda}(x,y) = \Delta_{F,\Lambda}(x-y)
\end{displaymath}

\item (symmetry)

\begin{displaymath}
\Delta^{b a}_{F,\Lambda}(y, x)
  \;=\;
  \Delta^{a b}_{F,\Lambda}(x, y)
\end{displaymath}
i.e.

\begin{displaymath}
\Delta_{F,\Lambda}^{b a}(-x)
  \;=\;
  \Delta_{F,\Lambda}^{a b}(x)
\end{displaymath}


\end{enumerate}

\item the $\Delta_{F,\Lambda}$ interpolate between zero and the Feynman propagator, in that, in the [[Hörmander topology]]:

\begin{enumerate}%
\item the [[limit of a sequence|limit]] as $\Lambda \to 0$ exists and is zero

\begin{displaymath}
\underset{\Lambda \to \infty}{\lim} \Delta_{F,\Lambda}
  \;=\;
  0
  \,.
\end{displaymath}

\item the [[limit of a sequence|limit]] as $\Lambda \to \infty$ exists and is the [[Feynman propagator]]:

\begin{displaymath}
\underset{\Lambda \to \infty}{\lim} \Delta_{F,\Lambda}
  \;=\;
  \Delta_F
  \,.
\end{displaymath}


\end{enumerate}


\end{enumerate}
\end{defn}
(\href{renormalization#Duetsch10}{Dütsch 10, section 4})

\begin{example}
\label{}\hypertarget{}{}
\textbf{(relativistic momentum cutoff)}

Recall from \href{Feynman+propagator#FeynmanPropagatorAsACauchyPrincipalvalue}{this prop.} that the [[Fourier transform of distributions]] of the [[Feynman propagator]] for the [[real scalar field]] on [[Minkowski spacetime]] $\mathbb{R}^{p,1}$ is,

\begin{displaymath}
\begin{aligned}
  \widehat{\Delta}_F(k)
  & =  
  \frac{+i}{(2\pi)^{p+1}}
  \frac{
     1
  }{
    - \eta(k,k) - \left( \tfrac{m c}{\hbar} \right)^2 + i 0
  }
  \end{aligned}
\end{displaymath}
To produce a [[UV cutoff]] in the sense of def. \ref{CutoffsUVForPerturbativeQFT} we would like to set this function to zero for [[wave numbers]] $\vert \vec k\vert$ (hence [[momenta]] $\hbar\vert \vec k\vert$) larger than a given $\Lambda$.

This needs to be done with due care: First, the [[Paley-Wiener-Schwartz theorem]] (prop. \ref{PaleyWienerSchwartzTheorem}) says that $\Delta_{F,\Lambda}$ to be a test function and hence compactly supported, its [[Fourier transform of distributions|Fourier transform]] $\widehat{\Delta}_{F,\Lambda}$ needs to be smooth and of bounded growth. So instead of multiplying $\widehat{\Delta}_F$ by a [[step function]] in $k$, we may multiply it with an exponential damping.

\end{example}
(\href{UV+regularization#KellerKopperSchophaus97}{Keller-Kopper-Schophaus 97, section 6.1}, \href{renormalization#Duetsch18}{Dütsch 18, example 3.126})

\begin{defn}
\label{SMatrixEffective}\hypertarget{SMatrixEffective}{}
\textbf{([[effective S-matrix scheme]])}

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[gauge fixing|gauge fixed]] [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] (according to def. \ref{VacuumFree}) and let $\left\{ \Delta_{F,\Lambda}\right\}_{\Lambda \in [0,\infty)}$ be a choice of [[UV cutoffs]] for [[perturbative QFT]] around this vacuum (def. \ref{CutoffsUVForPerturbativeQFT}).

We say that the \emph{[[effective S-matrix scheme]]} $\mathcal{S}_\Lambda$ at cutoff scale $\Lambda \in [0,\infty)$

\begin{displaymath}
\itexarray{
    PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]
     &\overset{\mathcal{S}_{\Lambda}}{\longrightarrow}&
    PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]
    \\
    O &\mapsto& \mathcal{S}_\Lambda(O)
  }
\end{displaymath}
is the [[exponential series]]

\begin{equation}
\begin{aligned}
    \mathcal{S}_\Lambda(O)
    & \coloneqq
    \exp_{F,\Lambda}\left( \frac{1}{i \hbar} O \right)
    \\
    & =
    1
      +
    \frac{1}{i \hbar} O
      +
    \frac{1}{2} \frac{1}{(i \hbar)^2} O \star_{F,\Lambda} O
      +
    \frac{1}{3!} \frac{1}{(i \hbar)^3} O \star_{F,\Lambda} O \star_{F,\Lambda} 0
      +
    \cdots
  \end{aligned}
  \,.
\label{EffectiveSMatrixScheme}\end{equation}
with respect to the [[star product]] $\star_{F,\Lambda}$ induced by the $\Delta_{F,\Lambda}$ (def. \ref{PropagatorStarProduct}).

This is evidently defined on all [[polynomial observables]] as shown, and restricts to an endomorphism on [[microcausal polynomial observables]] as shown, since the contraction coefficients $\Delta_{F,\Lambda}$ are [[non-singular distributions]], by definition of [[UV cutoff]].

\end{defn}
(\href{renormalization#Duetsch10}{Dütsch 10, (4.2)})

\begin{prop}
\label{UVRegularization}\hypertarget{UVRegularization}{}
\textbf{([[renormalization|(``re''-)normalization]] via [[UV regularization]])}

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[gauge fixing|gauge fixed]] [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] (according to def. \ref{VacuumFree}) and let $g S_{int} + j A \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g,j\rangle$ a polynomial [[local observable]] as in def. \ref{FormalParameters}, regarded as an [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]].

Let moreover $\{\Delta_{F,\Lambda}\}_{\Lambda \in [0,\infty)}$ be a [[UV cutoff]] (def. \ref{CutoffsUVForPerturbativeQFT}); with $\mathcal{S}_\Lambda$ the induced [[effective S-matrix schemes]] \eqref{EffectiveSMatrixScheme}.

Then

\begin{enumerate}%
\item there exists a $[0,\infty)$-parameterized [[interaction vertex redefinition]] $\{\mathcal{Z}_\Lambda\}_{\Lambda \in \mathbb{R}_{\geq 0}}$ (def. \ref{InteractionVertexRedefinition}) such that the [[limit of a sequence|limit]] of [[effective S-matrix schemes]] $\mathcal{S}_{\Lambda}$ \eqref{EffectiveSMatrixScheme} applied to the $\mathcal{Z}_\Lambda$-[[vertex redefinition|redefined interactions]]

\begin{displaymath}
\mathcal{S}_\infty
  \;\coloneqq\;
  \underset{\Lambda \to \infty}{\lim}
  \left(
    \mathcal{S}_\Lambda \circ \mathcal{Z}_\Lambda
  \right)
\end{displaymath}
exists and is a genuine [[S-matrix scheme]] around the given vacuum (def. \ref{LagrangianFieldTheoryPerturbativeScattering});


\item every [[S-matrix scheme]] around the given vacuum arises this way.



\end{enumerate}
These $\mathcal{Z}_\Lambda$ are called \emph{[[counterterms]]} (remark \ref{TermCounter} below) and the composite $\mathcal{S}_\Lambda \circ \mathcal{Z}_\Lambda$ is called a \emph{[[UV regularization]]} of the [[effective S-matrices]] $\mathcal{S}_\Lambda$.

Hence [[UV-regularization]] via [[counterterms]] is a method of [[renormalization|(``re''-)normalization]] of [[perturbative QFT]] (def. \ref{ExtensionOfTimeOrderedProoductsRenormalization}).

\end{prop}
This was claimed in (\href{renormalization#BrunettiDuetschFredenhagen09}{Brunetti-Dütsch-Fredenhagen 09, (75)}), a proof was indicated in (\href{renormalization#DuetschFredenhagenKellerRejzner14}{Dütsch-Fredenhagen-Keller-Rejzner 14, theorem A.1}).

\begin{proof}
Let $\{p_{\rho_{k}}\}_{k \in \mathbb{N}}$ be a sequence of projection maps as in \eqref{ForExtensionOfDistributionsProjectionMaps} defining an [[Epstein-Glaser renormalization|Epstein-Glaser (``re''-)normalization]] (prop. \ref{RenormalizationIsInductivelyExtensionToDiagonal}) of [[time-ordered products]] $\{T_k\}_{k \in \mathbb{N}}$ as [[extensions of distributions]] of the $T_k$, regarded as distributions via remark \ref{TimeOrderedProductOfFixedInteraction}, by the choice $q_k^\alpha = 0$ in \eqref{ExtensionOfDitstributionsPointFixedAndChoice}.

We will construct that $\mathcal{Z}_\Lambda$ in terms of these projections $p_\rho$.

First consider some convenient shorthand:

For $n \in \mathbb{N}$, write $\mathcal{Z}_{\leq n} \coloneqq \underset{1 \in \{1, \cdots, n\}}{\sum} \frac{1}{n!} Z_n$. Moreover, for $k \in \mathbb{N}$ write $(T_\Lambda \circ \mathcal{Z}_{\leq n})_k$ for the $k$-ary coefficient in the expansion of the composite $\mathcal{S}_\Lambda \circ \mathcal{Z}_{\leq n}$, as in equation \eqref{MainTheoremPerturbativeRenormalizationInductionStep} in the proof of the [[main theorem of perturbative renormalization]] (theorem \ref{PerturbativeRenormalizationMainTheorem}, via prop. \ref{AnyTwoSMatrixRenormalizationSchemesDifferByAUniqueVertexRedefinition}).

In this notation we need to find $\mathcal{Z}_\Lambda$ such that for each $n \in \mathbb{N}$ we have

\begin{equation}
\underset{\Lambda \to \infty}{\lim}
  \left(
    T_\Lambda \circ \mathcal{Z}_{\leq n, \Lambda}
  \right)_n
  \;=\;
  T_n
  \,.
\label{CountertermsInductionAssumption}\end{equation}
We proceed by [[induction]] over $n \in \mathbb{N}$.

Since by definition $T_0 = const_1$, $T_1 = id$ and $Z_0 = const_0$, $Z_1 = id$ the statement is trivially true for $n = 0$ and $n = 1$.

So assume now $n \in \mathbb{N}$ and $\{Z_{k}\}_{k \leq n}$ has been found such that \eqref{CountertermsInductionAssumption} holds.

Observe that with the chosen renormalizing projection $p_{\rho_{n+1}}$ the time-ordered product $T_{n+1}$ may be expressed as follows:

\begin{equation}
\begin{aligned}
  T_{n+1}(O, \cdots, O)
  & =
  \left\langle
    \underset{  {\mathbf{I} \in \{1, \cdots, n+1\} } \atop { \mathbf{I}, \overline{\mathbf{I}} \neq \emptyset }  }{\sum}
    \!\!\chi_i(\mathbf{X})\,
    \left(
      {\, \atop \,}
      T_{{\vert \mathbf{I}\vert}} (\mathbf{I})
      {\, \atop \,}
    \right)
      \star_{F}
    \left(
      {\, \atop \,}
      T_{ {\vert \overline{\mathbf{I}} \vert} } ( \overline{\mathbf{I}} )
      {\, \atop \,}
    \right)
    \,,\,
    p_{\rho_k}(O \otimes \cdots \otimes O)
  \right\rangle
  \\
  & =
  \left\langle
  \underset{\Lambda \to \infty}{\lim}
    \underset{  {\mathbf{I} \in \{1, \cdots, n+1\} } \atop { \mathbf{I}, \overline{\mathbf{I}} \neq \emptyset }  }{\sum}
    \!\!\chi_i(\mathbf{X})\,
    \left(
      {\, \atop \,}
      T_{{\vert \mathbf{I}\vert}} (\mathbf{I})
      {\, \atop \,}
    \right)
      \star_{F,\Lambda}
    \left(
      {\, \atop \,}
      T_{ {\vert \overline{\mathbf{I}} \vert} } ( \overline{\mathbf{I}} )
      {\, \atop \,}
    \right)
    \,,\,
    p_{\rho_k}(O \otimes \cdots \otimes O)
  \right\rangle
  \end{aligned}
  \,.
\label{RenormalizedSMatrixAsLimitOfEffectiveSMatricesEvaluatedOnProjection}\end{equation}
Here in the first step we inserted the causal decomposition \eqref{TimeOrderedProductsAwayFromDiagonalByInduction} of $T_{n+1}$ in terms of the $\{T_k\}_{k \leq n}$ away from the diagonal, as in the proof of prop. \ref{RenormalizationIsInductivelyExtensionToDiagonal}, which is admissible because the image of $p_{\rho_{n+1}}$ vanishes on the diagonal. In the second step we replaced the star-product of the Feynman propagator $\Delta_F$ with the limit over the star-products of the regularized propagators $\Delta_{F,\Lambda}$, which converges by the nature of the [[Hörmander topology]] (which is assumed by def. \ref{CutoffsUVForPerturbativeQFT}).

Hence it is sufficient to find $Z_{n+1,\Lambda}$ and $K_{n+1,\Lambda}$ such that

\begin{equation}
\begin{aligned}
  \left\langle
    \left(
      T_{\Lambda} \circ \mathcal{Z}_{\Lambda}
    \right)_{n+1}
    \,,\,
    (-, \cdots, -)
  \right\rangle
  & =
  \left\langle
    \underset{  {\mathbf{I} \in \{1, \cdots, n+1\} } \atop { \mathbf{I}, \overline{\mathbf{I}} \neq \emptyset }  }{\sum}
    \!\!\chi_i(\mathbf{X})\,
    \left(
      {\, \atop \,}
      T_{{\vert \mathbf{I}\vert}} (\mathbf{I})
      {\, \atop \,}
    \right)
      \star_{F,\Lambda}
    \left(
      {\, \atop \,}
      T_{ {\vert \overline{\mathbf{I}} \vert} } ( \overline{\mathbf{I}} )
      {\, \atop \,}
    \right)
    \,,\,
    p_{\rho_{k}}\left( -, \cdots, - \right)
  \right\rangle
  \\
  & \phantom{=}
  +
  K_{n+1,\Lambda}(-, \cdots, -)
  \end{aligned}
\label{CountertermsAndCorrectionTerm}\end{equation}
subject to these two conditions:

\begin{enumerate}%
\item $\mathcal{Z}_{n+1,\Lambda}$ is local;


\item $\underset{\Lambda \to \infty}{\lim} K_{n+1,\Lambda} = 0$.



\end{enumerate}
Now by expanding out the left hand side of \eqref{CountertermsAndCorrectionTerm} as

\begin{displaymath}
(T_\Lambda \circ \mathcal{Z}_\Lambda)_{n+1}
  \;=\;
  Z_{n+1,\Lambda}
    \;+\;
  (T_\Lambda \circ Z_{\leq n, \Lambda})_{n+1}
\end{displaymath}
(which uses the condition $T_1 = id$) we find the unique solution of \eqref{CountertermsAndCorrectionTerm} for $Z_{n+1,\Lambda}$, in terms of the $\{Z_{\leq n,\Lambda}\}$ and $K_{n+1,\Lambda}$ (the latter still to be chosen) to be:

\begin{equation}
\begin{aligned}
  \left\langle
    Z_{n+1,\Lambda}
    ,
    (-,\cdots, -)
  \right\rangle
  & =
  \left\langle
    \underset{  {\mathbf{I} \in \{1, \cdots, n+1\} } \atop { \mathbf{I}, \overline{\mathbf{I}} \neq \emptyset }  }{\sum}
    \!\!\chi_i(\mathbf{X})\,
    \left(
      {\, \atop \,}
      T_{{\vert \mathbf{I}\vert}} (\mathbf{I})
      {\, \atop \,}
    \right)
      \star_{F,\Lambda}
    \left(
      {\, \atop \,}
      T_{ {\vert \overline{\mathbf{I}} \vert} } ( \overline{\mathbf{I}} )
      {\, \atop \,}
    \right)
    \,,\,
    p_{\rho_{n+1}}(-, \cdots, -)
  \right\rangle
  \\
  & \phantom{=}
  -
  \left\langle
    \left(
      T_{\Lambda} \circ \mathcal{Z}_{\leq n,\Lambda}
    \right)_{n+1}
    \,,\,
    (-, \cdots, -)
  \right\rangle
  \\
  & \phantom{=}
  +
  \left\langle
    K_{n+1, \Lambda},
    (-, \cdots, -)
  \right\rangle
  \end{aligned}
  \,.
\label{CountertermOrderByOrderInTermsOfCorrectionTerm}\end{equation}
We claim that the following choice works:

\begin{equation}
\begin{aligned}
  K_{n+1, \Lambda}(-, \cdots, -)
  & \coloneqq
  \left\langle
    \left(
      T_{\Lambda} \circ \mathcal{Z}_{\leq n, \Lambda}
    \right)_{n+1}
    \,,\,
    p_{\rho_{n+1}}(-, \cdots, -)
  \right\rangle
  \\
  & \phantom{=}
  -
  \left\langle
    \underset{  {\mathbf{I} \in \{1, \cdots, n+1\} } \atop { \mathbf{I}, \overline{\mathbf{I}} \neq \emptyset }  }{\sum}
    \!\!\chi_i(\mathbf{X})\,
    \left(
      {\, \atop \,}
      T_{{\vert \mathbf{I}\vert}} (\mathbf{I})
      {\, \atop \,}
    \right)
      \star_{F,\Lambda}
    \left(
      {\, \atop \,}
      T_{ {\vert \overline{\mathbf{I}} \vert} } ( \overline{\mathbf{I}} )
      {\, \atop \,}
    \right)
    \,,\,
    p_{\rho_{n+1}}(-, \cdots, -)
  \right\rangle
  \end{aligned}
  \,.
\label{LocalityCorrection}\end{equation}
To prove this, we need to show that 1) the resulting $Z_{n+1,\Lambda}$ is local and 2) the limit of $K_{n+1,\Lambda}$ vanishes as $\Lambda \to \infty$.

First regarding the locality of $Z_{n+1,\Lambda}$: By inserting \eqref{LocalityCorrection} into \eqref{CountertermOrderByOrderInTermsOfCorrectionTerm} we obtain

\begin{displaymath}
\begin{aligned}
    \left\langle
      Z_{n+1,\Lambda}
      \,,\,
      (-,\cdots,-)
    \right\rangle
    & =
    \left\langle
      \left(
        T_{\Lambda} \circ \mathcal{Z}_{\leq n}
      \right)_{n+1}
      \,,\,
      p(-, \cdots, -)
    \right\rangle
    -
    \left\langle
      \left(
        T_{\Lambda} \circ \mathcal{Z}_{\leq n}
      \right)_{n+1}
      \,,\,
      (-, \cdots, -)
    \right\rangle
    \\
    & =
    \left\langle
      \left(
        T_{\Lambda} \circ \mathcal{Z}_{\leq n}
      \right)_{n+1}
      \,,\,
      ( p_{\rho_{n+1}} - id)(-, \cdots, -)
    \right\rangle
  \end{aligned}
\end{displaymath}
By definition $p_{\rho_{n+1}} - id$ is the identity on test functions (adiabatic switchings) that vanish at the diagonal. This means that $Z_{n+1,\Lambda}$ is [[support of a distribution|supported]] on the diagonal, and is hence local.

Second we need to show that $\underset{\Lambda \to \infty}{\lim} K_{n+1,\Lambda} = 0$:

By applying the analogous causal decomposition \eqref{TimeOrderedProductsAwayFromDiagonalByInduction} to the regularized products, we find

\begin{equation}
\begin{aligned}
  &
  \left\langle
    (T_\Lambda \circ \mathcal{Z}_{\leq n, \Lambda})_{n+1}
    \,,\,
    p_{\rho_{n+1}}(-,\cdots,-)
  \right\rangle
  \\
  & =
  \left\langle
    \underset{  {\mathbf{I} \in \{1, \cdots, n+1\} } \atop { \mathbf{I}, \overline{\mathbf{I}} \neq \emptyset }  }{\sum}
    \!\!\chi_i(\mathbf{X})\,
    \left(
      {\, \atop \,}
      (T_{\Lambda} \circ \mathcal{Z}_{\leq n, \Lambda})_{{\vert \mathbf{I}\vert}} (\mathbf{I})
      {\, \atop \,}
    \right)
      \star_{F,\Lambda}
    \left(
      {\, \atop \,}
      (T_{\Lambda} \circ \mathcal{Z}_{\leq n, \Lambda})_{ {\vert \overline{\mathbf{I}} \vert} } ( \overline{\mathbf{I}} )
      {\, \atop \,}
    \right)
    \,,\,
    p_{\rho_{n+1}}(-,\cdots,-)
  \right\rangle
  \,.
  \end{aligned}
\label{InductionStepForCounterterms}\end{equation}
Using this we compute as follows:

\begin{equation}
\begin{aligned}
  &
  \left\langle
    \underset{\Lambda \to \infty}{\lim}
    (T_\Lambda \circ \mathcal{Z}_{\leq n, \Lambda})_{n+1}
    \,,\,
    p_{\rho_{n+1}}(-,\cdots,-)
  \right\rangle
  \\
  & =
  \left\langle
    \underset{\Lambda \to \infty}{\lim}
    \underset{  {\mathbf{I} \in \{1, \cdots, n+1\} } \atop { \mathbf{I}, \overline{\mathbf{I}} \neq \emptyset }  }{\sum}
    \!\!\chi_i(\mathbf{X})\,
    \left(
      {\, \atop \,}
      (T_{\Lambda} \circ \mathcal{Z}_{\leq n, \Lambda})_{{\vert \mathbf{I}\vert}} (\mathbf{I})
      {\, \atop \,}
    \right)
      \star_{F,\Lambda}
    \left(
      {\, \atop \,}
      (T_{\Lambda} \circ \mathcal{Z}_{\leq n, \Lambda})_{ {\vert \overline{\mathbf{I}} \vert} } ( \overline{\mathbf{I}} )
      {\, \atop \,}
    \right)
    \,,\,
    p_{\rho_{n+1}}(-,\cdots,-)
  \right\rangle
  \\
  & =
  \left\langle
    \underset{  {\mathbf{I} \in \{1, \cdots, n+1\} } \atop { I, \overline{\mathbf{I}} \neq \emptyset }  }{\sum}
    \chi_i(\mathbf{X})\,
    \underset{
      T_{{\vert \mathbf{I}\vert}}(\mathbf{I})
    }{
    \underbrace{
    \left(
      \underset{\Lambda \to \infty}{\lim}
      (T_{\Lambda} \circ \mathcal{Z}_{\leq n, \Lambda})_{{\vert \mathbf{I}\vert}} (\mathbf{I})
    \right)
    }}
    \left(
      \underset{\Lambda \to \infty}{\lim}
      \star_{F,\Lambda}
    \right)
    \underset{
      T_{{\vert \overline{\mathbf{I}}\vert}}(\overline{\mathbf{I}})
    }{
    \underbrace{
    \left(
      \underset{\Lambda \to \infty}{\lim}
      (T_{\Lambda} \circ \mathcal{Z}_{\leq n, \Lambda})_{ {\vert \overline{\mathbf{I}} \vert} } ( \overline{\mathbf{I}} )
    \right)
    }}
    \,,\,
    p_{\rho_{n+1}}(-,\cdots,-)
  \right\rangle
  \\
  & =
  \left\langle
    \underset{\Lambda \to \infty}{\lim}
    \underset{  {\mathbf{I} \in \{1, \cdots, n+1\} } \atop { I, \overline{\mathbf{I}} \neq \emptyset }  }{\sum}
    \chi_i(\mathbf{X})\,
    T_{ { \vert \mathbf{I} \vert } }( \mathbf{I} )
      \star_{F,\Lambda}
    T_{ {\vert  \overline{\mathbf{I}} \vert} }( \overline{\mathbf{I}} )
    \,,\,
    p_{\rho_{n+1}}(-,\cdots,-)
  \right\rangle
  \end{aligned}
  \,.
\label{CorrectionTermForCountertermsVanishesAsCutoffIsRemoved}\end{equation}
Here in the first step we inserted \eqref{InductionStepForCounterterms}; in the second step we used that in the [[Hörmander topology]] the [[product of distributions]] preserves limits in each variable and in the third step we used the induction assumption \eqref{CountertermsInductionAssumption} and the definition of [[UV cutoff]] (def. \ref{CutoffsUVForPerturbativeQFT}).

Inserting this for the first summand in \eqref{LocalityCorrection} shows that $\underset{\Lambda \to \infty}{\lim} K_{n+1, \Lambda} = 0$.

In conclusion this shows that a consistent choice of [[counterterms]] $\mathcal{Z}_\Lambda$ exists to produce \emph{some} S-matrix $\mathcal{S} = \underset{\Lambda \to \infty }{\lim} (\mathcal{S}_\Lambda \circ \mathcal{Z}_\Lambda)$. It just remains to see that for \emph{every} other S-matrix $\widetilde{\mathcal{S}}$ there exist counterterms $\widetilde{\mathcal{Z}}_\lambda$ such that $\widetilde{\mathcal{S}} = \underset{\Lambda \to \infty }{\lim} (\mathcal{S}_\Lambda \circ \widetilde{\mathcal{Z}}_\Lambda)$.

But by the [[main theorem of perturbative renormalization]] (theorem \ref{PerturbativeRenormalizationMainTheorem}) we know that there exists a [[vertex redefinition]] $\mathcal{Z}$ such that

\begin{displaymath}
\begin{aligned}
    \widetilde{\mathcal{S}}
    & = \mathcal{S} \circ \mathcal{Z}
    \\
    & =
    \underset{\Lambda \to \infty}{\lim}
    \left(
      \mathcal{S}_\Lambda \circ \mathcal{Z}_\Lambda
    \right)
    \circ
    \mathcal{Z}
    \\
    & =
    \underset{\Lambda \to \infty}{\lim}
    (
      \mathcal{S}_\Lambda
        \circ
      (
        \underset{
          \widetilde{\mathcal{Z}}_\Lambda
        }{
        \underbrace{
        \mathcal{Z}_\Lambda
          \circ
        \mathcal{Z}
        }
        }
      )
    )
  \end{aligned}
\end{displaymath}
and hence with counterterms $\mathcal{Z}_\Lambda$ for $\mathcal{S}$ given, then counterterms for any $\widetilde{\mathcal{S}}$ are given by the composite $\widetilde{\mathcal{Z}}_\Lambda \coloneqq \mathcal{Z}_\Lambda \circ \mathcal{Z}$.

\end{proof}
\begin{remark}
\label{TermCounter}\hypertarget{TermCounter}{}
\textbf{([[counterterms]])}

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[gauge fixing|gauge fixed]] [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] (according to def. \ref{VacuumFree}) and let $\left\{ \Delta_{F,\Lambda}\right\}_{\Lambda \in [0,\infty)}$ be a choice of [[UV cutoffs]] for [[perturbative QFT]] around this vacuum (def. \ref{CutoffsUVForPerturbativeQFT}).

Consider

\begin{displaymath}
g S_{int} + j A
  \;\in\;
  LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g,j\rangle
\end{displaymath}
a [[local observable]], regarded as an [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]].

Then prop. \ref{UVRegularization} says that there exist [[vertex redefinitions]] of this [[interaction]]

\begin{displaymath}
\mathcal{Z}_\Lambda(g S_{int} + j A)
  \;\in\;
  LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g,j\rangle
\end{displaymath}
parameterized by $\Lambda \in [0,\infty)$, such that the [[limit of a sequence|limit]]

\begin{displaymath}
\mathcal{S}_\infty(g S_{int} + j A)
  \;\coloneqq\;
  \underset{\Lambda \to \infty}{\lim}
  \mathcal{S}_\Lambda\left( \mathcal{Z}_\Lambda( g S_{int} + j A )\right)
\end{displaymath}
exists and is an [[S-matrix]] for [[perturbative QFT]] with the given [[interaction]] $g S_{int} + j A$.

In this case the difference

\begin{displaymath}
\begin{aligned}
    S_{counter, \Lambda}
    & \coloneqq
    \left(
      g S_{int} + j A
    \right)
    \;-\;
    \mathcal{Z}_{\Lambda}(g S_{int} + j A)
    \;\;\;\;\;\in\;
    Loc(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g^2, j^2, g j\rangle
  \end{aligned}
\end{displaymath}
(which by the axiom ``perturbation'' in def. \ref{InteractionVertexRedefinition} is at least of second order in the [[coupling constant]]/[[source field]], as shown) is called a choice of \emph{[[counterterms]]} at cutoff scale $\Lambda$. These are new interactions which are added to the given interaction at cutoff scale $\Lambda$

\begin{displaymath}
\mathcal{Z}_{\Lambda}(g S_{int} + j A)
  \;=\;
  g S_{int} + j A
  \;+\;
  S_{counter,\Lambda}
  \,.
\end{displaymath}
In this language prop. \ref{UVRegularization} says that for every free field vacuum and every choice of local interaction, there is a choice of counterterms to the interaction that defines a corresponding [[renormalization|(``re''-)normalized]] [[perturbative QFT]], and every [[renormalization|(re``-)normalized]] [[perturbative QFT]] arises from some choice of counterterms.

\end{remark}
$\,$

\textbf{[[effective quantum field theory|Wilson-Polchinski effective QFT flow]]}

We have seen \hyperlink{UVRegularizationViaZ}{above} that a choice of [[UV cutoff]] induces [[effective S-matrix schemes]] $\mathcal{S}_\Lambda$ at cutoff scale $\Lambda$ (def. \ref{SMatrixEffective}). To these one may associated non-local [[relative effective actions]] $S_{eff,\Lambda}$ (def. \ref{EffectiveActionRelative} below) which are such that their effective [[scattering amplitudes]] at scale $\Lambda$ coincide with the true scattering amplitudes of a genuine [[local observable|local]] interaction as the cutoff is removed. This is the Wilsonian picture of \emph{[[effective quantum field theory]]} at a given cutoff scale (remark \ref{pQFTEffective} below). Crucially the ``flow'' of the [[relative effective actions]] with the cutoff scale satisfies a [[differential equation]] that in itself is independent of the full UV-theory; this is \emph{[[Polchinski's flow equation]]} (prop. \ref{FlowEquationPolchinski} below). Solving this equation for given choice of initial value data is hence another way of choosing [[renormalization|(``re''-)normalization]] constants.

$\,$

\begin{prop}
\label{EffectiveSmatrixSchemeInvertible}\hypertarget{EffectiveSmatrixSchemeInvertible}{}
\textbf{([[effective S-matrix schemes]] are [[inverse|invertible functions]])}

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[gauge fixing|gauge fixed]] [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] (according to def. \ref{VacuumFree}) and let $\left\{ \Delta_{F,\Lambda}\right\}_{\Lambda \in [0,\infty)}$ be a choice of [[UV cutoffs]] for [[perturbative QFT]] around this vacuum (def. \ref{CutoffsUVForPerturbativeQFT}).

Write

\begin{displaymath}
PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]\langle g,j\rangle
    \hookrightarrow
  PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]
\end{displaymath}
for the subspace of the space of [[formal power series]] in $\hbar, g, j$ with [[coefficients]] [[polynomial observables]] on those which are at least of first order in $g,j$, i.e. those that vanish for $g, j = 0$ (as in def. \ref{FormalParameters}).

Write moreover

\begin{displaymath}
1 + PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]\langle g,j\rangle
    \hookrightarrow
  PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]
\end{displaymath}
for the subspace of polynomial observables which are the sum of 1 (the multiplicative unit) with an observable at least linear n $g,j$.

Then the [[effective S-matrix schemes]] $\mathcal{S}_\Lambda$ (def. \ref{SMatrixEffective}) [[restriction|restrict]] to [[linear isomorphisms]] of the form

\begin{displaymath}
PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]\langle g,j\rangle
   \underoverset{\simeq}{\phantom{AA}\mathcal{S}_\Lambda \phantom{AA} }{\longrightarrow}
  1
    +
  PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]\langle g,j\rangle
  \,.
\end{displaymath}
\end{prop}
(\href{renormalization#Duetsch10}{Dütsch 10, (4.7)})

\begin{proof}
Since each $\Delta_{F,\Lambda}$ is symmetric (def. \ref{CutoffsUVForPerturbativeQFT}) if follows by general properties of [[star products]] (prop. \ref{SymmetricContribution}) just as for the genuine [[time-ordered product]] on [[regular polynomial observables]] (prop. \ref{IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise}) that eeach the ``effective time-ordered product'' $\star_{F,\Lambda}$ is [[isomorphism|isomorphic]] to the pointwise product $(-)\cdot (-)$ (def. \ref{Observable})

\begin{displaymath}
A_1 \star_{F,\Lambda} A_2
  \;=\;
  \mathcal{T}_\Lambda
  \left(
    \mathcal{T}_\Lambda^{-1}(A_1)
    \cdot
    \mathcal{T}_\Lambda^{-1}(A_2)
  \right)
\end{displaymath}
for

\begin{displaymath}
\mathcal{T}_\Lambda
  \;\coloneqq\;
  \exp
  \left(
    \tfrac{1}{2}\hbar
    \underset{\Sigma}{\int}
    \Delta_{F,\Lambda}^{a b}(x,y)
    \frac{\delta^2}{\delta \mathbf{\Phi}^a(x) \delta \mathbf{\Phi}^b(y)}
  \right)
\end{displaymath}
as in \eqref{OnRegularPolynomialObservablesPointwiseTimeOrderedIsomorphism}.

In particular this means that the [[effective S-matrix]] $\mathcal{S}_\Lambda$ arises from the [[exponential series]] for the pointwise product by [[conjugation]] with $\mathcal{T}_\Lambda$:

\begin{displaymath}
\mathcal{S}_\Lambda
  \;=\;
  \mathcal{T}_\Lambda \circ \exp_\cdot\left( \frac{1}{i \hbar}(-) \right) \circ \mathcal{T}_\Lambda^{-1}
\end{displaymath}
(just as for the genuine S-matrix on [[regular polynomial observables]] in def. \ref{OnRegularObservablesPerturbativeSMatrix}).

Now the exponential of the pointwise product on $1 + PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g,j\rangle$ has as [[inverse function]] the [[natural logarithm]] [[power series]], and since $\mathcal{T}$ evidently preserves powers of $g,j$ this [[conjugation|conjugates]] to an inverse at each UV cutoff scale $\Lambda$:

\begin{equation}
\mathcal{S}_\Lambda^{-1}
  \;=\;
  \mathcal{T}_\Lambda
    \circ
  \ln\left( i \hbar (-) \right)
    \circ
  \mathcal{T}_\Lambda^{-1}
  \,.
\label{InverseOfEffectiveSMatrixByLogarithm}\end{equation}
\end{proof}
\begin{defn}
\label{EffectiveActionRelative}\hypertarget{EffectiveActionRelative}{}
\textbf{([[relative effective action]])}

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[gauge fixing|gauge fixed]] [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] (according to def. \ref{VacuumFree}) and let $\left\{ \Delta_{F,\Lambda}\right\}_{\Lambda \in [0,\infty)}$ be a choice of [[UV cutoffs]] for [[perturbative QFT]] around this vacuum (def. \ref{CutoffsUVForPerturbativeQFT}).

Consider

\begin{displaymath}
g S_{int} + j A
  \;\in\;
  LocObs(E_{\text{BV-BrST}})[ [ \hbar, g, j] ]\langle g, j\rangle
\end{displaymath}
a [[local observable]] regarded as an [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]].

Then for

\begin{displaymath}
\Lambda,\, \Lambda_{vac} \;\in\; (0, \infty)
\end{displaymath}
two [[UV cutoff]]-scale parameters, we say the \emph{[[relative effective action]]} $S_{eff, \Lambda, \Lambda_0}$ is the image of this interaction under the [[composition|composite]] of the [[effective S-matrix scheme]] $\mathcal{S}_{\Lambda_0}$ at scale $\Lambda_0$ \eqref{EffectiveSMatrixScheme} and the [[inverse function]] $\mathcal{S}_\Lambda^{-1}$ of the [[effective S-matrix scheme]] at scale $\Lambda$ (via prop. \ref{EffectiveSmatrixSchemeInvertible}):

\begin{equation}
S_{eff,\Lambda, \Lambda_0}
  \;\coloneqq\;
  \mathcal{S}_{\Lambda}^{-1} \circ \mathcal{S}_{\Lambda_0}(g S_{int} + j A)
  \phantom{AAA}
  \Lambda, \Lambda_0 \in [0,\infty)
  \,.
\label{RelativeEffectiveActionComposite}\end{equation}
For chosen [[counterterms]] (remark \ref{TermCounter}) hence for chosen [[UV regularization]] $\mathcal{S}_\infty$ (prop. \ref{UVRegularization}) this makes sense also for $\Lambda_0 = \infty$ and we write:

\begin{equation}
S_{eff,\Lambda}
  \;\coloneqq\;
  S_{eff,\Lambda, \infty}
  \;\coloneqq\;
  \mathcal{S}_{\Lambda}^{-1} \circ \mathcal{S}_{\infty}(g S_{int} + j A)
  \phantom{AAA}
  \Lambda  \in [0,\infty)
\label{RelativeEffectiveActionRelativeToInfinity}\end{equation}
\end{defn}
(\href{renormalization#Duetsch10}{Dütsch 10, (5.4)})

\begin{remark}
\label{pQFTEffective}\hypertarget{pQFTEffective}{}
\textbf{([[effective quantum field theory]])}

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[gauge fixing|gauge fixed]] [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] (according to def. \ref{VacuumFree}), let $\left\{ \Delta_{F,\Lambda}\right\}_{\Lambda \in [0,\infty)}$ be a choice of [[UV cutoffs]] for [[perturbative QFT]] around this vacuum (def. \ref{CutoffsUVForPerturbativeQFT}), and let $\mathcal{S}_\infty = \underset{\Lambda \to \infty}{\lim} \mathcal{S}_\Lambda \circ \mathcal{Z}_\Lambda$ be a corresponding [[UV regularization]] (prop. \ref{UVRegularization}).

Consider a [[local observable]]

\begin{displaymath}
g S_{int} + j A
  \;\in\;
  LocObs(E_{\text{BV-BrST}})[ [ \hbar, g, j] ]\langle g, j\rangle
\end{displaymath}
regarded as an [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]].

Then def. \ref{CutoffsUVForPerturbativeQFT} and def. \ref{EffectiveActionRelative} say that for any $\Lambda \in (0,\infty)$ the [[effective S-matrix]] \eqref{EffectiveSMatrixScheme} of the [[relative effective action]] \eqref{RelativeEffectiveActionComposite} equals the genuine [[S-matrix]] $\mathcal{S}_\infty$ of the genuine [[interaction]] $g S_{int} + j A$:

\begin{displaymath}
\mathcal{S}_\Lambda( S_{eff,\Lambda} )
  \;=\;
  \mathcal{S}_\infty\left( g S_{int} + j A  \right)
  \,.
\end{displaymath}
In other words the [[relative effective action]] $S_{eff,\Lambda}$ encodes what the actual [[perturbative QFT]] defined by $\mathcal{S}_\infty\left( g S_{int} + j A  \right)$ \emph{effectively} looks like at [[UV cutoff]] $\Lambda$.

Therefore one says that $S_{eff,\Lambda}$ defines \emph{[[effective quantum field theory]]} at [[UV cutoff]] $\Lambda$.

Notice that in general $S_{eff,\Lambda}$ is \emph{not a [[local observable|local]] [[interaction]]} anymore: By prop. \ref{EffectiveSmatrixSchemeInvertible} the [[image]] of the [[inverse]] $\mathcal{S}^{-1}_\Lambda$ of the [[effective S-matrix]] is [[microcausal polynomial observables]] in $1 + PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]\langle g,j\rangle$ and there is no guarantee that this lands in the subspace of [[local observables]].

Therefore [[effective quantum field theories]] at finite [[UV cutoff]]-scale $\Lambda \in [0,\infty)$ are in general \emph{not} [[local field theories]], even if their [[limit of a sequence|limit]] as $\Lambda \to \infty$ is, via prop. \ref{UVRegularization}.

\end{remark}
\begin{prop}
\label{EffectiveActionAsRelativeEffectiveAction}\hypertarget{EffectiveActionAsRelativeEffectiveAction}{}
\textbf{([[effective action]] is [[relative effective action]] at $\Lambda = 0$)}

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[gauge fixing|gauge fixed]] [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] (according to def. \ref{VacuumFree}) and let $\left\{ \Delta_{F,\Lambda}\right\}_{\Lambda \in [0,\infty)}$ be a choice of [[UV cutoffs]] for [[perturbative QFT]] around this vacuum (def. \ref{CutoffsUVForPerturbativeQFT}).

Then the [[relative effective action]] (def. \ref{EffectiveActionRelative}) at $\Lambda = 0$ is the actual [[effective action]] (def. \ref{InPerturbationTheoryActionEffective}) in the sense of the the [[Feynman perturbation series]] of [[Feynman amplitudes]] $\Gamma(g S_{int} + j A)$ (def. \ref{FeynmanPerturbationSeriesAwayFromCoincidingPoints}) for [[connected graph|connected]] [[Feynman diagrams]] $\Gamma$:

\begin{displaymath}
\begin{aligned}
    S_{eff,0}
    & \coloneqq\;
    S_{eff,0,\infty}
    \\
    & = S_{eff} \;\coloneqq\;
    \underset{\Gamma \in \Gamma_{conn}}{\sum}
    \Gamma(g S_{int} + j A)
    \,.
  \end{aligned}
\end{displaymath}
More generally this holds true for any $\Lambda \in [0, \infty) \sqcup \{\infty\}$

\begin{displaymath}
\begin{aligned}
    S_{eff,0,\Lambda}
    & =
    \underset{\Gamma \in \Gamma_{conn}}{\sum}
    \Gamma_\Lambda(g S_{int} + j A)
    \,,
  \end{aligned}
\end{displaymath}
where $\Gamma_\Lambda( g S_{int} + j A)$ denotes the evident version of the [[Feynman amplitude]] (def. \ref{FeynmanPerturbationSeriesAwayFromCoincidingPoints}) with [[time-ordered products]] replaced by effective time ordered product at scale $\Lambda$ as in (def. \ref{SMatrixEffective}).

\end{prop}
(\href{renormalization#Duetsch18}{Dütsch 18, (3.473)})

\begin{proof}
Observe that the [[effective S-matrix scheme]] at scale $\Lambda = 0$ \eqref{EffectiveSMatrixScheme} is the [[exponential series]] with respect to the pointwise product (def. \ref{Observable})

\begin{displaymath}
\mathcal{S}_0(O) = \exp_\cdot( O )
  \,.
\end{displaymath}
Therefore the statement to be proven says equivalently that the [[exponential series]] of the [[effective action]] with respect to the pointwise product is the [[S-matrix]]:

\begin{displaymath}
\exp_\cdot\left( \frac{1}{i \hbar} S_{eff} \right)
  \;=\;
  \mathcal{S}_\infty\left( g S_{int} + j A \right)
  \,.
\end{displaymath}
That this is the case is the statement of prop. \ref{LogarithmEffectiveAction}.

\end{proof}
The definition of the [[relative effective action]] $\mathcal{S}_{eff,\Lambda} \coloneqq \mathcal{S}_{eff,\Lambda, \infty}$ in def. \ref{EffectiveActionRelative} invokes a choice of [[UV regularization]] $\mathcal{S}_\infty$ (prop. \ref{UVRegularization}). While (by that proposition and the [[main theorem of perturbative renormalization]], theorem \ref{PerturbativeRenormalizationMainTheorem} )this is guaranteed to exist, in practice one is after methods for constructing this without specifying it a priori.

But the collection [[relative effective actions]] $\mathcal{S}_{eff,\Lambda, \Lambda_0}$ for $\Lambda_0 \lt \infty$ ``flows'' with the cutoff-parameters $\Lambda$ and in particular also with $\Lambda_0$ (remark \ref{GroupoidOfEFTs} below) which suggests that examination of this flow yields information about full theory at $\mathcal{S}_\infty$.

This is made precise by \emph{[[Polchinski's flow equation]]} (prop. \ref{FlowEquationPolchinski} below), which is the [[infinitesimal]] version of the ``Wilsonian RG flow'' (remark \ref{GroupoidOfEFTs}). As a [[differential equation]] it is \emph{independent} of the choice of $\mathcal{S}_{\infty}$ and hence may be used to solve for the Wilsonian RG flow without knowing $\mathcal{S}_\infty$ in advance.

The freedom in choosing the initial values of this differential equation corresponds to the [[renormalization|(``re''-)normalization freedom]] in choosing the [[UV regularization]] $\mathcal{S}_\infty$. In this sense ``Wilsonian RG flow'' is a method of [[renormalization|(``re''-)normalization]] of [[perturbative QFT]] (def. \ref{ExtensionOfTimeOrderedProoductsRenormalization}).

\begin{remark}
\label{GroupoidOfEFTs}\hypertarget{GroupoidOfEFTs}{}
\textbf{(Wilsonian [[groupoid]] of [[effective quantum field theories]])}

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[gauge fixing|gauge fixed]] [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] (according to def. \ref{VacuumFree}) and let $\left\{ \Delta_{F,\Lambda}\right\}_{\Lambda \in [0,\infty)}$ be a choice of [[UV cutoffs]] for [[perturbative QFT]] around this vacuum (def. \ref{CutoffsUVForPerturbativeQFT}).

Then the [[relative effective actions]] $\mathcal{S}_{eff,\Lambda, \Lambda_0}$ (def. \ref{EffectiveActionRelative}) satisfy

\begin{displaymath}
S_{eff, \Lambda', \Lambda_0}
  \;=\;
  \left(
    \mathcal{S}_{\Lambda'}^{-1}
    \circ
    \mathcal{S}_\Lambda
  \right)
  \left(
    S_{eff, \Lambda, \Lambda_0}
  \right)
  \phantom{AAA}
  \text{for}
  \,
  \Lambda,\Lambda' \in [0,\infty)
  \,,\,
  \Lambda_0 \in [0,\infty) \sqcup \{\infty\}
  \,.
\end{displaymath}
This is similar to a [[group]] of UV-cutoff scale-transformations. But since the [[composition]] operations are only sensible when the UV-cutoff labels match, as shown, it is really a [[groupoid]] [[groupoid action|action]].

This is often called the \emph{Wilsonian RG}.

\end{remark}
We now consider the [[infinitesimal]] version of this ``flow'':

\begin{prop}
\label{FlowEquationPolchinski}\hypertarget{FlowEquationPolchinski}{}
\textbf{([[Polchinski's flow equation]])}

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[gauge fixing|gauge fixed]] [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] (according to def. \ref{VacuumFree}), let $\left\{ \Delta_{F,\Lambda}\right\}_{\Lambda \in [0,\infty)}$ be a choice of [[UV cutoffs]] for [[perturbative QFT]] around this vacuum (def. \ref{CutoffsUVForPerturbativeQFT}), such that $\Lambda \mapsto \Delta_{F,\Lambda}$ is [[differentiable function|differentiable]].

Then for \emph{every} choice of [[UV regularization]] $\mathcal{S}_\infty$ (prop. \ref{UVRegularization}) the corresponding [[relative effective actions]] $S_{eff,\Lambda}$ (def. \ref{EffectiveActionRelative}) satisfy the following [[differential equation]]:

\begin{displaymath}
\frac{d}{d \Lambda}
  S_{eff,\Lambda}
  \;=\;
  -
  \frac{1}{2} \frac{1}{i \hbar}
  \frac{d}{d \Lambda'}
  \left(
    S_{eff,\Lambda}
      \star_{F,\Lambda'}
    S_{eff,\Lambda}
  \right)\vert_{\Lambda' = \Lambda}
  \,,
\end{displaymath}
where on the right we have the [[star product]] induced by $\Delta_{F,\Lambda'}$ (def. \ref{PropagatorStarProduct}).

\end{prop}
This goes back to (\hyperlink{Polchinski84}{Polchinski 84, (27)}). The rigorous formulation and proof is due to (\href{renormalization#BrunettiDuetschFredenhagen09}{Brunetti-Dütsch-Fredenhagen 09, prop. 5.2}, \href{renormalization#Duetsch10}{Dütsch 10, theorem 2}).

\begin{proof}
First observe that for any [[polynomial observable]] $O \in PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]$ we have

\begin{displaymath}
\begin{aligned}
  &
  \frac{1}{(k+2)!}
  \frac{d}{d \Lambda}
  (
    \underset{
      k+2 \, \text{factors}
    }{
    \underbrace{
      O
        \star_{F,\Lambda}
        \cdots
        \star_{F,\Lambda}
      O
    }
    }
  )
  \\
  & =
  \frac{1}{(k+2)!}
  \frac{d}{d \Lambda}
  \left(
    prod
    \circ
    \exp\left(
      \hbar
      \underset{1 \leq i \lt j \leq k}{\sum}
      \left\langle
         \Delta_{F,\Lambda}
         ,
         \frac{\delta}{\delta \mathbf{\Phi}_i}
         \frac{\delta}{\delta \mathbf{\Phi}_j}
      \right\rangle
    \right)
    (
      \underset{
        k + 2 \, \text{factors}
      }{
      \underbrace{
        O \otimes \cdots \otimes O
      }
      }
    )
  \right)
  \\
  & =
  \underset{
    = \frac{1}{2} \frac{1}{k!}
  }{
  \underbrace{
    \frac{1}{(k+2)!}
    \left(
      k + 2
      \atop
      2
    \right)
  }}
  \left(
    \frac{d}{d \Lambda}
    O
     \star_{F,\Lambda}
    O
  \right)
    \star_{F,\Lambda}
  \underset{
    k \, \text{factors}
  }{
  \underbrace{
  O
    \star_{F,\Lambda}
    \cdots
    \star_{F,\Lambda}
  O
  }
  }
  \end{aligned}
\end{displaymath}
Here $\frac{\delta}{\delta \mathbf{\Phi}_i}$ denotes the functional derivative of the $i$th tensor factor of $O$, and the binomial coefficient counts the number of ways that an unordered pair of distinct labels of tensor factors may be chosen from a total of $k+2$ tensor factors, where we use that the [[star product]] $\star_{F,\Lambda}$ is commutative (by symmetry of $\Delta_{F,\Lambda}$) and associative (by prop. \ref{AssociativeAndUnitalStarProduct}).

With this and the defining equality $\mathcal{S}_\Lambda(S_{eff,\Lambda}) =  \mathcal{S}(g S_{int} + j A)$ \eqref{RelativeEffectiveActionRelativeToInfinity} we compute as follows:

\begin{displaymath}
\begin{aligned}
    0
    & =
    \frac{d}{d \Lambda} \mathcal{S}(g S_{int} + j A)
    \\
    & =
    \frac{d}{d \Lambda}
    \mathcal{S}_\Lambda(S_{eff,\Lambda})
    \\
    & =
    \left(
      \frac{1}{i \hbar}
      \frac{d}{d \Lambda} S_{eff,\Lambda}
    \right)
      \star_{F,\Lambda}
    \mathcal{S}_\Lambda(S_{eff,\Lambda})
    +
    \left(
      \frac{d}{d \Lambda}
      \mathcal{S}_{\Lambda}
   \right)
   \left( S_{eff, \Lambda} \right)
    \\
    & =
    \left(
      \frac{1}{i \hbar}
      \frac{d}{d \Lambda} S_{eff,\Lambda}
    \right)
      \star_{F,\Lambda}
    \mathcal{S}_\Lambda(S_{eff,\Lambda})
    \;+\;
    \frac{1}{2}
    \frac{d}{d \Lambda'}
    \left(
      \frac{1}{i \hbar} S_{eff,\Lambda}
        \star_{F,\Lambda'}
      \frac{1}{i \hbar} S_{eff, \Lambda}
    \right)
    \vert_{\Lambda' = \Lambda}
      \star_{F,\Lambda}
    \mathcal{S}_\Lambda
    \left(
      S_{eff, \Lambda}
    \right)
  \end{aligned}
\end{displaymath}
Acting on this equation with the multiplicative inverse $(-) \star_{F,\Lambda} \mathcal{S}_\Lambda( - S_{eff,\Lambda} )$ (using that $\star_{F,\Lambda}$ is a commutative product, so that exponentials behave as usual) this yields the claimed equation.

\end{proof}
$\,$

\textbf{[[renormalization group flow]]}

In [[perturbative quantum field theory]] the construction of the [[scattering matrix]] $\mathcal{S}$, hence of the [[interacting field algebra of observables]] for a given [[interaction]] $g S_{int}$ [[perturbation theory|perturbing]] around a given [[free field theory|free field]] [[vacuum]], involves choices of \emph{normalization} of [[time-ordered products]]/[[Feynman diagrams]] (traditionally called \emph{[[renormalization|``re''-normalizations]]}) encoding new [[interactions]] that appear where several of the original interaction vertices defined by $g S_{int}$ coincide.

Whenever a [[group]] $RG$ [[action|acts]] on the space of [[observables]] of the theory such that [[conjugation]] by this action takes [[renormalization scheme|(``re''-)normalization schemes]] into each other, then these choices of [[renormalization|(``re''-)normalization]] are parameterized by -- or ``flow with'' -- the elements of $RG$. This is called \emph{renormalization group flow} (prop. \ref{FlowRenormalizationGroup} below); often called \emph{RG flow}, for short.

The archetypical example here is the [[group]] $RG$ of [[scaling transformations]] on [[Minkowski spacetime]] (def. \ref{ScalingTransformations} below), which induces a [[renormalization group flow]] (prop. \ref{RGFlowScalingTransformations} below) due to the particular nature of the [[Wightman propagator]] resp. [[Feynman propagator]] on [[Minkowski spacetime]] (example \ref{ScalarFieldMassDimensionOnMinkowskiSpacetime} below). In this case the choice of [[renormalization|(``re''-)normalization]] hence ``flows with scale''.

Now the \emph{[[main theorem of perturbative renormalization]] (theorem \ref{PerturbativeRenormalizationMainTheorem}) states that (if only the basic [[renormalization condition]] called ``field independence'' is satisfied) any two choices of [[renormalization scheme|(``re''-)normalization schemes]] $\mathcal{S}$ and $\mathcal{S}'$ are related by a unique [[interaction vertex redefinition]] $\mathcal{Z}$, as}

\begin{displaymath}
\mathcal{S}' = \mathcal{S} \circ \mathcal{Z}
  \,.
\end{displaymath}
Applied to a parameterization/flow of renormalization choices by a group $RG$ this hence induces an [[interaction vertex redefinition]] as a function of $RG$. One may think of the shape of the interaction vertices as fixed and only their ([[adiabatic switching|adiabatically switched]]) [[coupling constants]] as changing under such an [[interaction vertex redefinition]], and hence then one has [[coupling constants]] $g_j$ that are parameterized by elements $\rho$ of $RG$:

\begin{displaymath}
\mathcal{Z}_{\rho_{vac}}^\rho
  \;\colon\;
  \{g_j\}
    \mapsto
  \{g_j(\rho)\}
\end{displaymath}
This dependendence is called \emph{running of the coupling constants} under the renormalization group flow (def. \ref{CouplingRunning} below).

One example of [[renormalization group flow]] is that induced by [[scaling transformations]] (prop. \ref{RGFlowScalingTransformations} below). This is the original and main example of the concept (\hyperlink{GellMannLow54}{Gell-Mann \& Low 54})

In this case the [[running of the coupling constants]] may be understood as expressing how ``more'' [[interactions]] (at higher energy/shorter [[wavelength]]) become visible (say to [[experiment]]) as the scale resolution is increased. In this case the dependence of the coupling $g_j(\rho)$ on the parameter $\rho$ happens to be [[differentiable function|differentiable]]; its [[logarithm|logarithmic]] [[derivative]] (denoted ``$\psi$'' in \hyperlink{GellMannLow54}{Gell-Mann \& Low 54}) is known as the \emph{[[beta function]]} (\hyperlink{Callan70}{Callan 70}, \hyperlink{Symanzik70}{Symanzik 70}):

\begin{displaymath}
\beta(g) \coloneqq \rho \frac{\partial g_j}{\partial \rho}
  \,.
\end{displaymath}
The [[running of the coupling constants]] is not quite a [[representation]] of the [[renormalization group flow]], but it is a ``twisted'' representation, namely a [[group cocycle|group 1-cocycle]] (prop. \ref{CocycleRunningCoupling} below). For the case of [[scaling transformations]] this may be called the \emph{[[Gell-Mann-Low renormalization cocycle]]} (\href{renormalization#BrunettiDuetschFredenhagen09}{Brunetti-Dütsch-Fredenhagen 09}).

$\,$

\begin{prop}
\label{FlowRenormalizationGroup}\hypertarget{FlowRenormalizationGroup}{}
\textbf{([[renormalization group flow]])}

Let

\begin{displaymath}
vac
    \;\coloneqq\;
  (E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )
\end{displaymath}
be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] (according to def. \ref{VacuumFree}) around which we consider [[interacting field theory|interacting]] [[perturbative QFT]].

Consider a [[group]] $RG$ equipped with an [[action]] on the [[Wick algebra]] of [[off-shell]] [[microcausal polynomial observables]] with formal parameters adjoined (as in def. \ref{FormalParameters})

\begin{displaymath}
rg_{(-)}
  \;\colon\;
  RG \times PolyObs(E_{\text{BV-BRST}})_{mc}((\hbar))[ [ g, j ] ]
  \longrightarrow
  PolyObs(E_{\text{BV-BRST}})_{mc}((\hbar))[ [ \hbar, g, j ] ]
  \,,
\end{displaymath}
hence for each $\rho \in RG$ a [[continuous linear map]] $rg_\rho$ which has an [[inverse]] $rg_\rho^{-1} \in RG$ and is a [[homomorphism]] of the [[Wick algebra]]-product (the [[star product]] $\star_H$ induced by the [[Wightman propagator]] of the given vauum $vac$)

\begin{displaymath}
rg_\rho( A_1 \star_H A_2 )
  \;=\;
  rg_\rho(A_1) \star_H rg_\rho(A_2)
\end{displaymath}
such that the following conditions hold:

\begin{enumerate}%
\item the action preserves the subspace of [[off-shell]] polynomial [[local observables]], hence it [[restriction|restricts]] as

\begin{displaymath}
rg_{(-)}
  \;\colon\;
  RG \times LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]\langle g,j\rangle
  \longrightarrow
  LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]\langle g,j\rangle
\end{displaymath}

\item the action respects the [[causal order]] of the spacetime support (def. \ref{SpacetimeSupport}) of local observables, in that for $O_1, O_2 \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]$ we have

\begin{displaymath}
\left(
    supp(O_1)
    \,{\vee\!\!\!\wedge}\,
    supp(O_2)
  \right)
  \phantom{A}
    \Rightarrow
  \phantom{A}
  \left(
    supp(rg_\rho(O_1))
    \,{\vee\!\!\!\wedge}\,
    supp(rg_\rho(O_2))
  \right)
\end{displaymath}
for all $\rho \in RG$.



\end{enumerate}
Then:

The operation of [[conjugation]] by this action on [[observables]] induces an [[action]] on the [[set]] of [[S-matrix]] [[renormalization schemes]] (def. \ref{LagrangianFieldTheoryPerturbativeScattering}, remark \ref{calSFunctionIsRenormalizationScheme}), in that for

\begin{displaymath}
\mathcal{S}
  \;\colon\;
  LocObs(E_{\text{BV-BRST}})[ [ \hbar , g, j] ]
    \longrightarrow
  PolyObs(E_{\text{BV-BRST}})( (\hbar) )[ [ g, j] ]
\end{displaymath}
a perturbative [[S-matrix scheme]] around the given [[free field theory|free field]] [[vacuum]] $vac$, also the [[composition|composite]]

\begin{displaymath}
\mathcal{S}^\rho
  \;\coloneqq\;
  rg_\rho \circ \mathcal{S} \circ rg_{\rho}^{-1}
  \;\colon\;
  LocObs(E_{\text{BV-BRST}})[ [ \hbar , g, j] ]
    \longrightarrow
  PolyObs(E_{\text{BV-BRST}})( (\hbar) )[ [ g, j] ]
\end{displaymath}
is an [[S-matrix]] scheme, for all $\rho \in RG$.

More generally, let

\begin{displaymath}
vac_\rho
    \;\coloneqq\;
  (E_{\text{BV-BRST}}, \mathbf{L}'_\rho, \Delta_{H,\rho} )
\end{displaymath}
be a collection of [[gauge fixing|gauge fixed]] [[free field theory|free field]] [[vacua]] parameterized by elements $\rho \in RG$, all with the same underlying [[field bundle]]; and consider $rg_\rho$ as above, except that it is not an [[automorphism]] of any [[Wick algebra]], but an [[isomorphism]] between the [[Wick algebra]]-structures on various vacua, in that

\begin{equation}
rg_{\rho}( A_1 \star_{H, \rho^{-1} \rho_{vac}} A_2 )
  \;=\;
  rg_{\rho}(A_1) \star_{H, \rho_{vac}} rg_{\rho}(A_2)
\label{IntertwiningWickProductsActionRG}\end{equation}
for all $\rho, \rho_{vac} \in RG$

Then if

\begin{displaymath}
\{ \mathcal{S}_{\rho} \}_{\rho \in RG}
\end{displaymath}
is a collection of [[S-matrix schemes]], one around each of the [[gauge fixing|gauge fixed]] [[free field theory|free field]] [[vacua]] $vac_\rho$, it follows that for all pairs of group elements $\rho_{vac}, \rho \in RG$ the [[composition|composite]]

\begin{equation}
\mathcal{S}_{\rho_{vac}}^\rho
  \;\coloneqq\;
  rg_\rho \circ \mathcal{S}_{\rho^{-1}\rho_{vac}} \circ rg_\rho^{-1}
\label{RGConjugateSmatrix}\end{equation}
is an [[S-matrix scheme]] around the vacuum labeled by $\rho_{vac}$.

Since therefore each element $\rho \in RG$ in the [[group]] $RG$ picks a different choice of [[renormalization|normalization]] of the [[S-matrix]] scheme around a given vacuum at $\rho_{vac}$, we call the assignment $\rho \mapsto \mathcal{S}_{\rho_{vac}}^{\rho}$ a \emph{[[renormalization group flow|re-normalization group flow]]}.

\end{prop}
(\href{renormalization#BrunettiDuetschFredenhagen09}{Brunetti-Dütsch-Fredenhagen 09, sections 4.2, 5.1}, \href{renormalization#Duetsch18}{Dütsch 18, section 3.5.3})

\begin{proof}
It is clear from the definition that each $\mathcal{S}^{\rho}_{\rho_{vac}}$ satisfies the axiom ``perturbation'' (in def. \ref{LagrangianFieldTheoryPerturbativeScattering}).

In order to verify the axiom ``[[causal additivity]]'', observe, for convenience, that by prop. \ref{CausalFactorizationAlreadyImpliesSMatrix} it is sufficient to check [[causal factorization]].

So consider $O_1, O_2 \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g,j\rangle$ two local observables whose spacetime support is in [[causal order]].

\begin{displaymath}
supp(O_1)
  \;{\vee\!\!\!\wedge}\;
  supp(O_2)
  \,.
\end{displaymath}
We need to show that the

\begin{displaymath}
\mathcal{S}_{\rho_{vac}}^{\rho}(O_1 + O_2)
  =
  \mathcal{S}_{\rho_{vac}}^\rho(O_1)
    \star_{H,\rho_{vac}}
  \mathcal{S}_{vac_e}^\rho(O_2)
\end{displaymath}
for all $\rho, \rho_{vac} \in RG$.

Using the defining properties of $rg_{(-)}$ and the [[causal factorization]] of $\mathcal{S}_{\rho^{-1}\rho_{vac}}$ we directly compute as follows:

\begin{displaymath}
\begin{aligned}
    \mathcal{S}_{\rho_{vac}}^\rho(O_1 + O_2)
    & =
    rg_\rho
      \circ
    \mathcal{S}_{\rho^{-1} \rho_{vac}}
      \circ
    rg_\rho^{-1}( O_1 + O_2 )
    \\
    & =
    rg_\rho
    \left(
      {\, \atop \,}
      \mathcal{S}_{\rho^{-1}\rho_{vac}}
      \left(
        rg_\rho^{-1}(O_1) + rg_\rho^{-1}(O_2)
      \right)
      {\, \atop \,}
    \right)
    \\
    & =
    rg_\rho
    \left(
      {\, \atop \,}
      \left(
        \mathcal{S}_{\rho^{-1}\rho_{vac}}\left(rg_\rho^{-1}(O_1)\right)
      \right)
        \star_{H, \rho^{-1} \rho_{vac}}
      \left(
        \mathcal{S}_{ \rho^{-1} \rho_{vac} }\left(rg_\rho^{-1}(O_2)\right)
      \right)
      {\, \atop \,}
    \right)
    \\
    & =
    rg_\rho
    \left(
      {\, \atop \,}
      \mathcal{S}_{\rho^{-1} \rho_{vac}}\left(rg_{\rho^{-1}}(O_1)\right)
      {\, \atop \,}
    \right)
    \star_{H, \rho_{vac}}
    rg_\rho
    \left(
      {\, \atop \,}
      \mathcal{S}_{\rho^{-1} \rho_{vac}}\left( rg_\rho^{-1}(O_2)\right)
      {\, \atop \,}
    \right)
    \\
    & =
    \mathcal{S}^\rho_{\rho_{vac}}( O_1 )
      \,
      \star_{H, \rho_{vac}}
      \,
     \mathcal{S}_{\rho_{vac}}^\rho(O_2)
    \,.
  \end{aligned}
\end{displaymath}
\end{proof}
\begin{defn}
\label{CouplingRunning}\hypertarget{CouplingRunning}{}
\textbf{([[running coupling constants]])}

Let

\begin{displaymath}
vac
  \coloneqq
  vac_e
    \;\coloneqq\;
  (E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )
\end{displaymath}
be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] (according to def. \ref{VacuumFree}) around which we consider [[interacting field theory|interacting]] [[perturbative QFT]], let $\mathcal{S}$ be an [[S-matrix]] scheme around this vacuum and let $rg_{(-)}$ be a [[renormalization group flow]] according to prop. \ref{FlowRenormalizationGroup}, such that each re-normalized [[S-matrix scheme]] $\mathcal{S}_{vac}^\rho$ satisfies the [[renormalization condition]] ``field independence''.

Then by the [[main theorem of perturbative renormalization]] (theorem \ref{PerturbativeRenormalizationMainTheorem}, via prop. \ref{AnyTwoSMatrixRenormalizationSchemesDifferByAUniqueVertexRedefinition}) there is for every [[pair]] $\rho_1, \rho_2 \in RG$ a unique [[interaction vertex redefinition]]

\begin{displaymath}
\mathcal{Z}_{\rho_{vac}}^{\rho}
  \;\colon\;
  LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]
    \longrightarrow
  LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]
\end{displaymath}
which relates the corresponding two [[S-matrix]] schemes via

\begin{equation}
\mathcal{S}_{\rho_{vac}}^{\rho}
  \;=\;
  \mathcal{S}_{\rho_{vac}}
    \circ
  \mathcal{Z}_{\rho_{vac}}^\rho
  \,.
\label{SMatrixScemesRelatedByRunningFunction}\end{equation}
If one thinks of an [[interaction]] vertex, hence a [[local observable]] $g S_{int}+ j A$, as specified by the ([[adiabatic switching|adiabatically switched]]) [[coupling constants]] $g_j \in C^\infty_{cp}(\Sigma)\langle g \rangle$ multiplying the corresponding [[interaction]] [[Lagrangian densities]] $\mathbf{L}_{int,j} \in \Omega^{p+1,0}_\Sigma(E_{\text{BV-BRST}})$ as

\begin{displaymath}
g S_{int}
  \;=\;
  \underset{j}{\sum} \tau_\Sigma \left( g_j \mathbf{L}_{int,j} \right)
\end{displaymath}
(where $\tau_\Sigma$ denotes [[transgression of variational differential forms]]) then $\mathcal{Z}_{\rho_1}^{\rho_2}$ exhibits a dependency of the ([[adiabatic switching|adiabatically switched]]) [[coupling constants]] $g_j$ of the [[renormalization group flow]] parameterized by $\rho$. The corresponding functions

\begin{displaymath}
\mathcal{Z}_{\rho_{vac}}^{\rho}(g S_{int})
    \;\colon\;
   (g_j)
     \mapsto
   (g_j(\rho))
\end{displaymath}
are then called \emph{[[running coupling constants]]}.

\end{defn}
(\href{renormalization#BrunettiDuetschFredenhagen09}{Brunetti-Dütsch-Fredenhagen 09, sections 4.2, 5.1}, \href{renormalization#Duetsch18}{Dütsch 18, section 3.5.3})

\begin{prop}
\label{CocycleRunningCoupling}\hypertarget{CocycleRunningCoupling}{}
\textbf{([[running coupling constants]] are [[group cocycle]] over [[renormalization group flow]])}

Consider [[running coupling constants]]

\begin{displaymath}
\mathcal{Z}_{\rho_{vac}}^{\rho}
   \;\colon\;
   (g_j)
     \mapsto
   (g_j(\rho))
\end{displaymath}
as in def. \ref{CouplingRunning}. Then for all $\rho_{vac}, \rho_1, \rho_2 \in RG$ the following equality is satisfied by the ``running functions'' \eqref{SMatrixScemesRelatedByRunningFunction}:

\begin{displaymath}
\mathcal{Z}_{\rho_{vac}}^{\rho_1 \rho_2}
  \;=\;
  \mathcal{Z}_{\rho_{vac}}^{\rho_1}
    \circ
  \left(
    \sigma_{\rho_1}
      \circ
    \mathcal{Z}_{\rho^{-1} \rho_{vac}}^{\rho_2}
      \circ
    \sigma_{\rho_1}^{-1}
  \right)
  \,.
\end{displaymath}
\end{prop}
(\href{renormalization#BrunettiDuetschFredenhagen09}{Brunetti-Dütsch-Fredenhagen 09 (69)}, \href{renormalization#Duetsch18}{Dütsch 18, (3.325)})

\begin{proof}
Directly using the definitions, we compute as follows:

\begin{displaymath}
\begin{aligned}
    \mathcal{S}_{\rho_{vac}}
      \circ
    \mathcal{Z}_{\rho_{vac}}^{\rho_1 \rho_2}
    & =
    \mathcal{S}_{\rho_{vac}}^{\rho_1 \rho_2 }
    \\
    & =
    \sigma_{\rho_1}
      \circ
    \underset{
      =
      \mathcal{S}_{\rho_1^{-1} \rho_{vac}}^{\rho_2}
        =
      \mathcal{S}_{\rho_1^{-1} \rho_{vac}}
        \circ
      \mathcal{Z}_{\rho_1^{-1} \rho_vac}^{\rho_2}
    }{
    \underbrace{
      \sigma_{\rho_2}
        \circ
      \mathcal{S}_{\rho_2^{-1}\rho_1^{-1}\rho_{vac}}
        \circ
      \sigma_{\rho_2}^{-1}
    }}
      \circ
    \sigma_{\rho_1}^{-1}
    \\
    & =
    \underset{
      =
      \mathcal{S}_{\rho_{vac}}
        \circ
      \mathcal{Z}_{\rho_{vac}}^{\rho_1}
        \circ
      \sigma_{\rho_1}
    }{
    \underbrace{
      \sigma_{\rho_1}
        \circ
      \mathcal{S}_{\rho_1^{-1} \rho_{vac}}
        \circ
      \overset{ = id }{
        \overbrace{
          \sigma_{\rho_1}^{-1}
            \circ
          \sigma_{\rho_1}
        }
    }
    }}
      \circ
      \mathcal{Z}_{\rho_1^{-1} \rho_{vac}}^{\rho_2}
      \circ
      \sigma_{\rho_1}^{-1}
    \\
    & =
    \mathcal{S}_{\rho_{vac}}
      \circ
    \mathcal{Z}_{\rho_{vac}}^{\rho_1}
      \circ
    \underbrace{
      \sigma_{\rho_1}
        \circ
      \mathcal{Z}_{\rho_1^{-1} \rho_{vac}}^{\rho_2}
        \circ
      \sigma_{\rho_1}^{-1}
    }
  \end{aligned}
\end{displaymath}
This demonstrates the equation between vertex redefinitions to be shown after [[composition]] with an S-matrix scheme. But by the uniqueness-clause in the [[main theorem of perturbative renormalization]] (theorem \ref{PerturbativeRenormalizationMainTheorem}) the composition operation $\mathcal{S}_{\rho_{vac}} \circ (-)$ as a function from [[vertex redefinitions]] to S-matrix schemes is [[injective function|injective]]. This implies the equation itself.

\end{proof}
$\,$

\textbf{[[Gell-Mann-Low renormalization cocycles|Gell-Mann \& Low RG flow]]}

We discuss (prop. \ref{RGFlowScalingTransformations} below) that, if the field species involved have well-defined [[mass dimension]] (example \ref{ScalarFieldMassDimensionOnMinkowskiSpacetime} below) then [[scaling transformations]] on [[Minkowski spacetime]] (example \ref{ScalingTransformations} below) induce a [[renormalization group flow]] (def. \ref{FlowRenormalizationGroup}). This is the original and main example of [[renormalization group flows]] (\hyperlink{GellMannLow54}{Gell-Mann\& Low 54}).

\begin{example}
\label{ScalingTransformations}\hypertarget{ScalingTransformations}{}
\textbf{([[scaling transformations]] and [[mass dimension]])}

Let

\begin{displaymath}
E \overset{fb}{\longrightarrow} \Sigma
\end{displaymath}
be a [[field bundle]] which is a [[trivial vector bundle]] over [[Minkowski spacetime]] $\Sigma = \mathbb{R}^{p,1} \simeq_{\mathbb{R}} \mathbb{R}^{p+1}$.

For $\rho \in (0,\infty) \subset \mathbb{R}$ a [[positive real number]], write

\begin{displaymath}
\itexarray{
    \Sigma
      &\overset{\rho}{\longrightarrow}&
    \Sigma
    \\
    x &\mapsto& \rho x
  }
\end{displaymath}
for the operation of multiplication by $\rho$ using the [[real vector space]]-[[structure]] of the [[Cartesian space]] $\mathbb{R}^{p+1}$ underlying [[Minkowski spacetime]].

By [[pullback of differential forms|pullback]] this acts on [[field histories]] ([[sections]] of the [[field bundle]]) via

\begin{displaymath}
\itexarray{
    \Gamma_\Sigma(E)
      &\overset{\rho^\ast}{\longrightarrow}&
    \Gamma_\Sigma(E)
    \\
    \Phi &\mapsto& \Phi(\rho(-))
  }
  \,.
\end{displaymath}
Let then

\begin{displaymath}
\rho
    \mapsto
  vac_\rho
    \;\coloneqq\;
  (E_{\text{BV-BRST}}, \mathbf{L}'_{\rho}, \Delta_{H,\rho} )
\end{displaymath}
be a 1-parameter collection of [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacua]] on that field bundle, according to def. \ref{VacuumFree}, and consider a decomposition into a set $Spec$ of field species (def. \ref{VerticesAndFieldSpecies}) such that for each $sp \in Spec$ the collection of [[Feynman propagators]] $\Delta_{F,\rho,sp}$ for that species \emph{scales homogeneously} in that there exists

\begin{displaymath}
dim(sp) \in \mathbb{R}
\end{displaymath}
such that for all $\rho$ we have (using [[generalized functions]]-notation)

\begin{equation}
\rho^{ 2 dim(sp) }  \Delta_{F, 1/\rho, sp}( \rho x )
  \;=\;
  \Delta_{F,sp, \rho = 1}(x)
  \,.
\label{FeynmanPropagatorScalingBehaviour}\end{equation}
Typically $\rho$ rescales a [[mass]] parameter, in which case $dim(sp)$ is also called the \emph{[[mass dimension]]} of the field species $sp$.

Let finally

\begin{displaymath}
\itexarray{
    PolyObs(E)
    &
      \overset{
        \sigma_\rho
      }{\longrightarrow}
    &
    PolyObs(E)
    \\
    \mathbf{\Phi}_{sp}^a(x)
      &\mapsto&
    \rho^{- dim(sp)} \mathbf{\Phi}^a( \rho^{-1} x )
  }
\end{displaymath}
be the [[function]] on [[off-shell]] [[polynomial observables]] given on [[field observables]] $\mathbf{Phi}^a(x)$ by [[pullback of differential forms|pullback]] along $\rho^{-1}$ followed by multiplication by $\rho$ taken to the negative power of the [[mass dimension]], and extended from there to all [[polynomial observables]] as an [[associative algebra|algebra]] [[homomorphism]].

This constitutes an [[action]] of the [[group]]

\begin{displaymath}
RG
    \coloneqq
  \left(
    \mathbb{R}_+, \cdot
  \right)
\end{displaymath}
of [[positive real numbers]] (under [[multiplication]]) on [[polynomial observables]], called the group of \emph{[[scaling transformations]]} for the given choice of field species and [[mass]] parameters.

\end{example}
(\href{renormalization#Duetsch18}{Dütsch 18, def. 3.19})

\begin{example}
\label{ScalarFieldMassDimensionOnMinkowskiSpacetime}\hypertarget{ScalarFieldMassDimensionOnMinkowskiSpacetime}{}
\textbf{([[mass dimension]] of [[scalar field]])}

Consider the [[Feynman propagator]] $\Delta_{F,m}$ of the [[free field theory|free]] [[real scalar field]] on [[Minkowski spacetime]] $\Sigma = \mathbb{R}^{p,1}$ for [[mass]] parameter $m \in (0,\infty)$; a [[Green function]] for the [[Klein-Gordon equation]].

Let the group $RG \coloneqq (\mathbb{R}_+, \cdots)$ of [[scaling transformations]] $\rho \in \mathbb{R}_+$ on [[Minkowski spacetime]] (def. \ref{ScalingTransformations}) act on the mass parameter by inverse multiplication

\begin{displaymath}
(\rho , \Delta_{F,m})
    \mapsto
  \Delta_{F,\rho^{-1}m}(\rho (-))
  \,.
\end{displaymath}
Then we have

\begin{displaymath}
\Delta_{F,\rho^{-1}m}(\rho (-))
  \;=\;
  \rho^{-(p+1) + 2}
  \Delta_{F,1}(x)
\end{displaymath}
and hence the corresponding [[mass dimension]] (def. \ref{ScalingTransformations}) of the [[real scalar field]] on $\mathbb{R}^{p,1}$ is

\begin{displaymath}
dim(\text{scalar field})
  =
  (p+1)/2 - 1
  \,.
\end{displaymath}
\end{example}
\begin{proof}
By prop. \ref{FeynmanPropagatorAsACauchyPrincipalvalue} the [[Feynman propagator]] in question is given by the [[Cauchy principal value]]-formula (in [[generalized function]]-notation)

\begin{displaymath}
\begin{aligned}
  \Delta_{F,m}(x)
  & =
  \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
  \frac{+i}{(2\pi)^{p+1}}
  \int
  \int_{-\infty}^\infty
  \frac{
     e^{i k_\mu x^\mu}
  }{
    - k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon
  }
  \, d k_0 \, d^p \vec k
  \,.
  \end{aligned}
\end{displaymath}
By applying [[change of integration variables]] $k \mapsto \rho^{-1} k$ in the [[Fourier transform of distributions|Fourier transform]] this becomes

\begin{displaymath}
\begin{aligned}
  \Delta_{F,\rho^{-1}m}(\rho x)
  & =
  \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
  \frac{+i}{(2\pi)^{p+1}}
  \int
  \int_{-\infty}^\infty
  \frac{
     e^{i k_\mu \rho x^\mu}
  }{
    - k_\mu k^\mu - \left( \rho^{-1} \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon
  }
  \, d k_0 \, d^p \vec k
  \\
  & =
  \rho^{-(p+1)}
  \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
  \frac{+i}{(2\pi)^{p+1}}
  \int
  \int_{-\infty}^\infty
  \frac{
     e^{i k_\mu x^\mu}
  }{
    - \rho^{-2} k_\mu k^\mu - \rho^{-2} \left( \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon
  }
  \, d k_0 \, d^p \vec k
  \\
  & =
  \rho^{-(p+1)+2}
  \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
  \frac{+i}{(2\pi)^{p+1}}
  \int
  \int_{-\infty}^\infty
  \frac{
     e^{i k_\mu x^\mu}
  }{
    - k_\mu k^\mu -  \left( \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon
  }
  \, d k_0 \, d^p \vec k
  \\
  & =
  \rho^{-(p+1) + 2}
  \Delta_{F,m}(x)
  \end{aligned}
\end{displaymath}
\end{proof}
\begin{prop}
\label{RGFlowScalingTransformations}\hypertarget{RGFlowScalingTransformations}{}
\textbf{([[scaling transformations]] are [[renormalization group flow]])}

Let

\begin{displaymath}
vac \coloneqq vac_m
  \coloneqq (E_{\text{BV-BRST}}, \mathbf{L}', \Delta_{H,m})
\end{displaymath}
be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacua]] on that field bundle, according to def. \ref{VacuumFree} equipped with a decomposition into a set $Spec$ of field species (def. \ref{VerticesAndFieldSpecies}) such that for each $sp \in Spec$ the collection of [[Feynman propagators]] the corresponding field species has a well-defined [[mass dimension]] $dim(sp)$ (def. \ref{ScalingTransformations})

Then the [[action]] of the [[group]] $RG \coloneqq (\mathbb{R}_+, \cdot)$ of [[scaling transformations]] (def. \ref{ScalingTransformations}) is a [[renormalization group flow]] in the sense of prop. \ref{FlowRenormalizationGroup}.

\end{prop}
(\href{renormalization#Duetsch18}{Dütsch 18, exercise 3.20})

\begin{proof}
It is clear that rescaling preserves [[causal order]] and the [[renormalization condition]] of ``field indepencen''.

The condition we need to check is that for $A_1, A_2 \in PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]$ two [[microcausal polynomial observables]] we have for any $\rho, \rho_{vac} \in \mathbb{R}_+$ that

\begin{displaymath}
\sigma_\rho
  \left(
    A_1 \star_{H, \rho^{-1} \rho_{vac} c} A_2
  \right)
  \;=\;
  \sigma_\rho(A_1)
    \star_{H,\rho_{vac}}
  \sigma_\rho(A_2)
  \,.
\end{displaymath}
By the assumption of decomposition into free field species $sp \in Spec$, it is sufficient to check this for each species $\Delta_{H,sp}$. Moreover, by the nature of the [[star product]] on [[polynomial observables]], which is given by iterated contractions with the [[Wightman propagator]], it is sufficient to check this for one such contraction.

Observe that the scaling behaviour of the [[Wightman propagator]] $\Delta_{H,m}$ is the same as the behaviour \eqref{FeynmanPropagatorScalingBehaviour} of the correspponding [[Feynman propagator]]. With this we directly compute as follows:

\begin{displaymath}
\begin{aligned}
    \sigma_\rho (\mathbf{\Phi}(x))
     \star_{F, \rho_{vac} m}
    \sigma_\rho (\mathbf{\Phi}(y)
    & =
    \rho^{-2 dim }
    \mathbf{\Phi}(\rho^{-1} x)
      \star_{F, \rho_{vac} m}
    \mathbf{\Phi}(\rho^{-1} y)
    \\
    & =
    \rho^{-2 dim }
    \Delta_{F, \rho_{vac} m}(\rho^{-1}(x-y))
    \\
    & =
    \Delta_{F, \rho^{-1}\rho_{vac}m }(x,y) \mathbf{1}
    \\
    & =
    rg_{\rho}\left(
     \Delta_{F, \rho^{-1}\rho_{vac}m }(x,y) \mathbf{1}
    \right)
    \\
    & =
    rg_{\rho}
    \left(
      \mathbf{\Phi}(x)
        \star_{F, \rho^{-1} \rho_{vac} m}
      \mathbf{\Phi}(y)
    \right)
  \end{aligned}
  \,.
\end{displaymath}
\end{proof}
$\,$

This concludes our discussion of [[renormalization]].



\end{document}