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\begin{document}

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\section*{interaction vertex redefinition}

[[!redirects vertex redefinitions]]

\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{references}{References}\dotfill \pageref*{references} \linebreak


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

In [[perturbative quantum field theory]] an ([[adiabatic switching|adiabatically switch]]) [[interaction]] [[action functional]] is a [[local observable]] of the form

\begin{displaymath}
g S_{int} + j A
  \;\in\;
  LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle
  \,.
\end{displaymath}
In the [[Feynman perturbation series]] this is interpreted as the label of [[vertices]] in [[multigraphs]] underlying [[Feynman diagrams]] which encode [[field (physics)|field]] [[interactions]] at that point.

In the process of [[renormalization|(re-)normalization]] of [[perturbative QFT]] one considers redefinitions of such ``interaction vertices'' by terms of higher order in the [[coupling constant]] $g$ ad [[source field]] $j$:

\begin{displaymath}
\mathcal{Z}
  \;\colon\;
  g S_{int} + j A
  \;\mapsto\;
  g S_{int} + j A
  +
  \underset{
    \mathcal{O}(g^2, j^2, g j)
  }{
  \underbrace{   
    S_{counter}
  }}
  \,.
\end{displaymath}
In the context of [[effective QFT]] via [[UV cutoffs]], the correction term $S_{counter}$ is called a \emph{[[counterterm]]}.

Under mild assumptions, such \emph{interaction vertex redefinitions} form a [[group]], called the \emph{[[Stückelberg-Petermann renormalization group]]}. See there for more.

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

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

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}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The original discussion is due to

\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}
in the context of the [[Stückelberg-Petermann renormalization group]].

For more see the references there and those at \emph{[[main theorem of perturbative renormalization]]}.

[[!redirects perturbative interaction vertex redefinition]] [[!redirects perturbative interaction vertex redefinitions]]

[[!redirects interaction vertex redefinition]] [[!redirects interaction vertex redefinitions]]

[[!redirects vertex redefinition]] [[!redirects vertex redefinitions]]



\end{document}