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\begin{document}

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\section*{Stückelberg-Petermann renormalization group}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{algebraic_quantum_field_theory}{}\paragraph*{{Algebraic Quantum Field Theory}}\label{algebraic_quantum_field_theory}

[[!include AQFT and operator algebra contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{scaling_transformations_and_running_coupling_constants}{Scaling transformations and ``running coupling constants''}\dotfill \pageref*{scaling_transformations_and_running_coupling_constants} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In [[perturbative quantum field theory]] the \emph{Stückelberg-Petermann renormalization group} (\hyperlink{StiecelbergPetermann53}{Stückelberg-Petermann 53}) is the (original) incarnation of the [[renormalization group]] in the perspective of [[causal perturbation theory]]: It is the group (def. \ref{StueckelbergPetermannRenormalizationGroup} below) of perturbative [[interaction vertex redefinitions]] (def. \ref{InteractionVertexRedefinition}) which is such that any two choices of [[S-matrix]] [[renormalization schemes]] $\mathcal{S}, \mathcal{S}'$ are related by a unique [[vertex redefinition]] $\mathcal{Z}$ via [[precomposition]]

\begin{displaymath}
\mathcal{S}'
  \;=\;
  \mathcal{S} \circ \mathcal{Z}
  \,.
\end{displaymath}
This statement (prop. \ref{AnyTwoSMatrixRenormalizationSchemesDifferByAUniqueVertexRedefinition} below) is also called the \emph{[[main theorem of perturbative renormalization]]}.

If [[scaling transformations]] on [[spacetime]] happen to transform [[renormalization schemes]] into each other, then this [[main theorem of perturbative renormalization]] (prop. \ref{AnyTwoSMatrixRenormalizationSchemesDifferByAUniqueVertexRedefinition}) directly implies that every [[scaling transformation]] uniquely corresponds to a [[interaction vertex redefinition]], or conversely that the [[vertex redefinitions]] are given as funcitons of scale. As such they are also referred to as \emph{[[running coupling constants]]} (def. \ref{GellMannLowTransformations} below). Beware that these do not in general form a [[group]] but a [[group cocycle]], the \emph{[[Gell-Mann-Low renormalization cocycle]]} (prop. \ref{CocyclePropertyOfRunningCouplingConstants} below.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

The elements of the Stückelberg-Petermann renormalization group are \emph{perturbative [[interaction vertex redefinitions]]} (def. \ref{InteractionVertexRedefinition} below). These [[action|act]] on [[S-matrix]] [[renormalization schemes|(``re''-)normalization schemes]] by [[precomposition]] (def. \ref{CausalFactorizationSatisfiedByCompositionOfSMatrixWithVertexRedefinition}) and this action is [[free action|free]] and [[transitive action|transitive]] (prop. \ref{AnyTwoSMatrixRenormalizationSchemesDifferByAUniqueVertexRedefinition} below). In this way these [[vertex redefinitions]] translate between different choices of [[renormalization|(``re''-)normalization]], and as such they form the \emph{Stückelerg-Petermann renormalization group} (def. \ref{StueckelbergPetermannRenormalizationGroup}) 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]] (\href{S-matrix#VacuumFree}{this def.}).

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 (\href{S-matrix#FormalParameters}{this def.}) 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 (\href{S-matrix#eq:CausalAdditivity}{this equation}):

\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]] (\href{S-matrix#VacuumFree}{this def.}) 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 $supp(O_i) \subset \Sigma$ (\href{A+first+idea+of+quantum+field+theory#SpacetimeSupport}{this def.}) 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}
(\hyperlink{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 \href{S-matrix#NotationForTimeOrderedProductsAsGeneralizedFunctions}{this remark}.

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]] (\href{S-matrix#VacuumFree}{this def.}) 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 (\href{S-matrix#LagrangianFieldTheoryPerturbativeScattering}{this def.}), 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'' (\href{S-matrix#BasicConditionsRenormalization}{this prop.}), then so does $\mathcal{S} \circ \mathcal{Z}$.

\end{prop}
(e.g \hyperlink{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]] (\href{S-matrix#DysonCausalFactorization}{this remark}) 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 \href{S-matrix#CausalFactorizationAlreadyImpliesSMatrix}{this prop.} 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 a unique [[vertex redefinition]])}

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

Then for $\mathcal{S}, \mathcal{S}'$ any two [[S-matrix]] schemes (\href{S-matrix#LagrangianFieldTheoryPerturbativeScattering}{this def.}) 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}
(See any of the references at \emph{[[main theorem of perturbative renormalization]]}.)

\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 \href{S-matrix#TimeOrderedProductsFromSMatrixScheme}{this example}) 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 \href{S-matrix#TimeOrderedProductsFromSMatrixScheme}{this example}) 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{displaymath}
\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}
\end{displaymath}
(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 \href{S-matrix#TimeOrderedProductsFromSMatrixScheme}{this example}).

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 \href{S-matrix#RenormalizationIsInductivelyExtensionToDiagonal}{this prop.} 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}
\begin{defn}
\label{StueckelbergPetermannRenormalizationGroup}\hypertarget{StueckelbergPetermannRenormalizationGroup}{}
\textbf{([[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]] (\href{S-matrix#VacuumFree}{this def.}).

Then prop. \ref{CausalFactorizationSatisfiedByCompositionOfSMatrixWithVertexRedefinition} and prop \ref{AnyTwoSMatrixRenormalizationSchemesDifferByAUniqueVertexRedefinition} together says that (``[[main theorem of perturbative renormalization]]''):

\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]] (\href{S-matrix#LagrangianFieldTheoryPerturbativeScattering}{this def.}, \href{S-matrix#calSFunctionIsRenormalizationScheme}{this remark}) satisfying the [[renormalization condition]] ``field independence'' (\href{S-matrix#BasicConditionsRenormalization}{this prop.}) is a [[torsor]] over this group, meaning that the action is [[regular action|regular]] on this set, in that any two are S-matrix (``re''-)normalization schemes are related by a unique vertex redefinition.



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

Typically one imposes a set of [[renormalization conditions]] (\href{S-matrix#RenormalizationConditions}{this def.}) and then the corresponding [[subgroup]] of [[vertex redefinitions]] preserving these.

\end{defn}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{scaling_transformations_and_running_coupling_constants}{}\subsubsection*{{Scaling transformations and ``running coupling constants''}}\label{scaling_transformations_and_running_coupling_constants}

A priori the Stückelberg-Petermann renormalization group is not about [[scaling transformations]]. But if [[scaling transformations]] happen to produce new S-matrices/renormalization schemes from given ones, then the [[main theorem of perturbative renormalization]] induces for each such scaling transformation a re-definition of interaction Lagrangian densities, this is the [[Gell-Mann-Low renormalization cocycle]] (\hyperlink{GellMannLow54}{Gell-Mann \& Low 54}, \hyperlink{BrunettiDuetschFredenhagen09}{Brunetti-Dütsch-Fredenhagen 09}) for review see (\hyperlink{Duetsch18}{Dütsch 18, section 3.5.3}).

\begin{defn}
\label{GellMannLowTransformations}\hypertarget{GellMannLowTransformations}{}
\textbf{([[running coupling constants]] under [[scale transformations]])}

Let

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

Assume that in fact

\begin{enumerate}%
\item the [[free field]] [[vacuum]] $vac = vac(m)$ depends on a [[mass]] parameter, and with it the choice $\mathcal{S}_{vac(m)}$ of [[renormalization scheme|normalization scheme]],


\item under [[scaling transformations]] on [[local observables]] $\sigma_\rho$ (\hyperlink{Duetsch18}{Dütsch 18, def. 3.19}) we have that with $\mathcal{S}_{vac(m)}$ a [[perturbative S-matrix]] scheme perturbing around $vac(m)$ also

\begin{displaymath}
\sigma_\rho \circ \left(\mathcal{S}_{vac(m/\rho)}\right) \circ \sigma_\rho^{-1}
\end{displaymath}
is a perturbative S-matrix around $L_{kin}(m)$.



\end{enumerate}
In this case the [[main theorem of perturbative renormalization]] (prop. \ref{AnyTwoSMatrixRenormalizationSchemesDifferByAUniqueVertexRedefinition}) says that there exists for each scale $\rho$ a unique [[interaction vertex redefinition]] $\mathcal{Z}^m_\rho$ (def. \ref{InteractionVertexRedefinition}) such that

\begin{displaymath}
\begin{aligned}
    &
    \sigma_\rho
      \circ
    \mathcal{S}_{vac(m/\rho)}
      \circ
    \sigma_\rho^{-1}( g S_{int} + j A )
    \\
    & =
    \mathcal{S}_{vac(m)}(\mathcal{Z}^m_\rho(g S_{int} + j A))
  \end{aligned}
\end{displaymath}
for all [[interaction]] [[action functionals]] $g S_{int} + j A$.

These $\mathcal{Z}^m_\rho$ are the \emph{[[Gell-Mann-Low renormalization cocycle]]} elements.

The [[interaction vertex redefinitions]] $\mathcal{Z}^m_\rho$ as a function of the [[rescaling]] is known as the \emph{[[running coupling constants]]}.

\end{defn}
\begin{prop}
\label{CocyclePropertyOfRunningCouplingConstants}\hypertarget{CocyclePropertyOfRunningCouplingConstants}{}
\textbf{(cocycle property of [[running coupling constants]])}

In the situation of def. \ref{GellMannLowTransformations}, the [[Gell-Mann-Low renormalization cocycles]] ([[running coupling constants]]) $\mathcal{Z}^m_\rho$ satisfy the relation

\begin{displaymath}
\mathcal{Z}^m_{\rho_1 \rho_2}
  \;=\;
  \mathcal{Z}^m_{\rho_1}
    \circ
  \left(
    \sigma_{\rho_1}
      \circ
    \mathcal{Z}^{m/\rho_1}_{\rho_2}
      \circ
    \sigma_{\rho_2}
  \right)
\end{displaymath}
Hence only for vanishing [[mass]] do these ``renormalization cocycles'' themselves form an actual [[renormalization group]].

\end{prop}
(\hyperlink{BrunettiDuetschFredenhagen09}{Brunetti-Dütsch-Fredenhagen 09 (69)}, \hyperlink{Duetsch18}{Dütsch 18 (3.325)})

\begin{proof}
From the definition we have

\begin{displaymath}
\begin{aligned}
    \mathcal{S}_{vac(m)}
      \circ
    \mathcal{Z}^m_{\rho_1 \rho_2}
    & =
    \sigma_{\rho_1}
      \circ
    \underset{
      \mathcal{S}_{vac(m/\rho_1)}
        \circ
      \mathcal{Z}^{m/\rho_1}_{\rho_2}
    }{
    \underbrace{
      \sigma_{\rho_2}
        \circ
      \mathcal{S}_{vac(m/\rho_1\rho_2)}
        \circ
      \sigma_{\rho_2}^{-1}
    }}
      \circ
    \sigma_{\rho_1}^{-1}
    \\
    & =
    \underset{
      =
      \mathcal{S}_{vac(m)}
        \circ
      \mathcal{Z}^m_{\rho_1}
        \circ
      \sigma_{\rho_1}
    }{
    \underbrace{
      \sigma_{\rho_1}
        \circ
      \mathcal{S}_{vac(m/\rho_1)}
        \circ
      \overset{ = id }{
        \overbrace{
          \sigma_{\rho_1}^{-1}
            \circ
          \sigma_{\rho_1}
        }
    }
    }}
      \circ
    \mathcal{Z}^{m/\rho_1}_{\rho_2}
      \circ
    \sigma_{\rho_1}^{-1}
    \\
    & =
    \mathcal{S}_{vac(m)}
      \circ
    \mathcal{Z}^m_{\rho_1}
      \circ
    \sigma_{\rho_1}
      \circ
    \mathcal{Z}^{m/\rho_1}_{\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]] 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}
\hypertarget{references}{}\subsection*{{References}}\label{references}

(See also the references at \emph{[[main theorem of perturbative renormalization]]}.)

The original article on the Stückelberg-Petermann renormalization group is

\begin{itemize}%
\item [[Ernst Stückelberg]], [[André Petermann]], \emph{La normalisation des constantes dans la theorie des quanta}, Helv. Phys. Acta 26 (1953), 499–520

\end{itemize}
The relation of the Stückelberg-Petermann renormalization group to [[scale transformations]] and the [[Gell-Mann-Low renormalization cocycle]] is due to

\begin{itemize}%
\item [[Murray Gell-Mann]] and F. E. Low, \emph{Quantum Electrodynamics at Small Distances}, Phys. Rev. 95 (5) (1954), 1300–1312 (\href{http://www.fafnir.phyast.pitt.edu/py3765/GellManLow.pdf}{pdf})


\item [[Romeo Brunetti]], [[Michael Dütsch]], [[Klaus Fredenhagen]], \emph{Perturbative Algebraic Quantum Field Theory and the Renormalization Groups}, Adv. Theor. Math. Physics 13 (2009), 1541-1599 (\href{https://arxiv.org/abs/0901.2038}{arXiv:0901.2038})



\end{itemize}
Review includes

\begin{itemize}%
\item [[Michael Dütsch]], section 3.5.1 of \emph{[[From classical field theory to perturbative quantum field theory]]}, 2018

\end{itemize}
See also

\begin{itemize}%
\item [[Michael Dütsch]], [[Klaus Fredenhagen]], \emph{Action Ward Identity and the Stückelberg-Petermann renormalization group}, Prog. Math. 251:113-124,2007

\end{itemize}
[[!redirects Stückelberg-Petermann renormalization groups]]

[[!redirects Stueckelberg-Petermann renormalization group]] [[!redirects Stueckelberg-Petermann renormalization groups]]



\end{document}