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\begin{document}

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\section*{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{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

In [[perturbative quantum field theories]] various concepts of ``renormalization groups'' describe the choices of [[renormalization|(``re''-)normalization]] and their behaviour under [[scaling transformations]] or choices of cutoffs.

There are at least three different concepts referred to as ``the renormalization group'', only the first is in general really a [[group]]:

\begin{enumerate}%
\item the [[Stückelberg-Petermann renormalization group]] (\hyperlink{StueckelbergPetermann53}{Stückelberg-Petermann 53}, historically the origin of the concept)

this is literally the [[group]] of \emph{re-normalizations}, whose elements relate any two given [[renormalization scheme|normalization schemes]] $\mathcal{S}$ and $\mathcal{S}'$ by [[precomposition]] with a transformation $\mathcal{Z}$ of the space of [[local observable|local]] [[interaction]] [[action functionals]];


\item [[renormalization group flow]], say along [[scaling transformations]] yielding the [[Gell-Mann-Low renormalization cocycle]]


\item the [[Wilsonian RG]] of [[effective quantum field theories]] defined with a [[UV cutoff]].



\end{enumerate}
(e.g. \hyperlink{BrunettiDuetschFredenhagen09}{Brunetti-Dütsch-Fredenhagen 09, p. 10})

In more detail:

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]].

Then a [[perturbative S-matrix]] scheme/[[renormalization scheme|(``re''-)normalization scheme]] around this vacuum (\href{S-matrix#LagrangianFieldTheoryPerturbativeScattering}{this def.}) is a function

\begin{displaymath}
\itexarray{
    LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]\langle g, j \rangle
      & \overset{\mathcal{S}_{vac}}{\longrightarrow} &
    PolyObs(E_{\text{BV-BRST}})_{mc}( ( \hbar ) )[ [ g, j ] ]
    \\
    g S_{int} + j A
     &\mapsto&
    \mathcal{S}_{vac}(g S_{int} + j A)
  }
\end{displaymath}
from [[local observables]], regared as [[adiabatic switching|adiatically switched]] [[interaction]] [[action functionals]] to [[Wick algebra]]-elements $\mathcal{S}( g S_{int} + j A)$, encoding [[scattering amplitudes]] in the given vacuum $\mathbf{L}'$ for the given interaction $g S_{int} + j A$, with formal parameters adjoined as indicated.

The \emph{[[Stückelberg-Petermann renormalization group]]} is a group of transformations

\begin{displaymath}
\itexarray{
    LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j \rangle
      &\overset{Z}{\longrightarrow}& 
    LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]\langle g, j \rangle
    \\
    g S_{int} + J A
      &\mapsto& 
    \mathcal{Z}(g S_{int} + j A)
  }
\end{displaymath}
such that for $\mathcal{S}$ and $\mathcal{S}'$ two [[renormalization schemes|normalization schemes]]/[[S-matrix]] schemes, there is a unique $\mathcal{Z}$ relating them by [[precomposition]], in that

\begin{equation}
\mathcal{S}(g S_{int} + j A)
    \;=\; 
  \mathcal{S}'\left( \mathcal{Z}(g S_{int} + j A) \right)
\label{SMatricesRelatedBySPRenormalizationGroupElement}\end{equation}
for all $g S_{int} + j A$. This is the \emph{[[main theorem of perturbative renormalization]]}. Hence this says that any two ways of choosing [[interactions]] at coincident interaction points are related by a re-definition of the original [[interaction]] $g S_{int} + j A$.

Now it may happen that

\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 above statement of the [[main theorem of perturbative renormalization]] implies with \eqref{SMatricesRelatedBySPRenormalizationGroupElement} that there exists a unique transformation $\mathcal{Z}^m_\rho$ of the space of [[local observable|local]] [[interaction]] [[action functionals]] 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 $g S_{int} + j A$.

These $\mathcal{Z}^m_\rho$ are the \emph{[[Gell-Mann-Low cocycle]]} elements. They do not actually form a [[group]], unless $m = 0$, but 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}
(\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}
To conclude, it is now sufficient to see that the perturbative S-matrix $S_{vac(m)}$, as a function form [[interaction]] [[Lagrangian densities]] to [[Wick algebra]]-elements, is an [[injective function]]. (\ldots{})

\end{proof}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[renormalization]]


\item [[renormalization group flow]]


\item [[asymptotic safety]]


\item [[cosmic Galois group]]



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

The [[Stückelberg-Petermann renormalization group]] 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}
The relation of the [[Stückelberg-Petermann renormalization group]] to [[renormalization group flow]] ([[Gell-Mann-Low renormalization cocycles]])

\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})

\end{itemize}
as well as to [[Wilsonian RG]] of [[effective quantum field theories]] is due to

\begin{itemize}%
\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})


\item [[Michael Dütsch]], [[Klaus Fredenhagen]], [[Kai Keller]], [[Katarzyna Rejzner]], \emph{Dimensional Regularization in Position Space, and a Forest Formula for Epstein-Glaser Renormalization}, J. Math. Phy. 55(12), 122303 (2014) (\href{https://arxiv.org/abs/1311.5424}{arXiv:1311.5424})



\end{itemize}
reviewed in

\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 Kiyoshi Higashijima, Kazuhiko Nishijima, \emph{Renormalization Groups of Gell-Mann and Low and of Callan and Symanzik}, Progress in theoretical physics, vol. 64, no. 6, December 1980 (\href{https://academic.oup.com/ptp/article-pdf/64/6/2179/5275368/64-6-2179.pdf}{pdf})


\item Assaf Shomer, \emph{A pedagogical explanation for the non-renormalizability of gravity} (\href{https://arxiv.org/abs/0709.3555}{arXiv:0709.3555})


\item Wikipedia \emph{\href{http://en.wikipedia.org/wiki/Renormalization_group}{Renormalization group}}



\end{itemize}
[[!redirects renormalization groups]]



\end{document}