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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{renormalization} \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{details}{Details}\dotfill \pageref*{details} \linebreak \noindent\hyperlink{InCausalPerturbationTheory}{In causal perturbation theory}\dotfill \pageref*{InCausalPerturbationTheory} \linebreak \noindent\hyperlink{EpsteinGlaserRenormalization}{Epstein-Glaser normalization}\dotfill \pageref*{EpsteinGlaserRenormalization} \linebreak \noindent\hyperlink{SPRenormalizationGroup}{Stückelberg-Petermann renormalization group}\dotfill \pageref*{SPRenormalizationGroup} \linebreak \noindent\hyperlink{UVRegularizationViaZ}{UV-Regularization via Conterterms}\dotfill \pageref*{UVRegularizationViaZ} \linebreak \noindent\hyperlink{EffectiveQFTFlowWislonian}{Wilson-Polchinski effective QFT flow}\dotfill \pageref*{EffectiveQFTFlowWislonian} \linebreak \noindent\hyperlink{RGFlowGeneral}{Renormalization group flow}\dotfill \pageref*{RGFlowGeneral} \linebreak \noindent\hyperlink{ScalingTransformatinRGFlow}{Gell-Mann \& Low RG flow}\dotfill \pageref*{ScalingTransformatinRGFlow} \linebreak \noindent\hyperlink{dimensional_regularization}{Dimensional regularization}\dotfill \pageref*{dimensional_regularization} \linebreak \noindent\hyperlink{conneskreimer_renormalization}{Connes-Kreimer renormalization}\dotfill \pageref*{conneskreimer_renormalization} \linebreak \noindent\hyperlink{BPHZRenormalization}{BPHZ and Hopf-algebraic renormalization}\dotfill \pageref*{BPHZRenormalization} \linebreak \noindent\hyperlink{lattice_renormalization}{Lattice renormalization}\dotfill \pageref*{lattice_renormalization} \linebreak \noindent\hyperlink{OfTheoriesInBVForm}{Of theories in BV-CS form}\dotfill \pageref*{OfTheoriesInBVForm} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{vacuum_energy_and_cosmological_constant}{Vacuum energy and Cosmological constant}\dotfill \pageref*{vacuum_energy_and_cosmological_constant} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak \noindent\hyperlink{General}{Review}\dotfill \pageref*{General} \linebreak \noindent\hyperlink{in_causal_perturbation_theory_2}{In causal perturbation theory}\dotfill \pageref*{in_causal_perturbation_theory_2} \linebreak \noindent\hyperlink{bphz_renormalization}{BPHZ Renormalization}\dotfill \pageref*{bphz_renormalization} \linebreak \noindent\hyperlink{ReferencesInBVFormalism}{In BV formalism}\dotfill \pageref*{ReferencesInBVFormalism} \linebreak \noindent\hyperlink{ReferencesOnCompactifiedConfigurationSpaces}{On compactified configuration spaces}\dotfill \pageref*{ReferencesOnCompactifiedConfigurationSpaces} \linebreak \noindent\hyperlink{operadic_description}{Operadic description}\dotfill \pageref*{operadic_description} \linebreak \noindent\hyperlink{relations_to_motives_polylogarithms_positivity}{Relations to motives, polylogarithms, positivity}\dotfill \pageref*{relations_to_motives_polylogarithms_positivity} \linebreak \hypertarget{Idea}{}\subsection*{{Idea}}\label{Idea} The construction of a [[perturbative quantum field theory]] from a given [[local Lagrangian density]] (rigorously via [[causal perturbation theory]]) involves ambiguities associated with the detailed nature of the quantum processes at point interactions. What is called \emph{renormalization} is making a choice of fixing these ambiguities to produce a [[perturbative quantum field theory]] (\hyperlink{Wightman76}{Wightman et al. 76}). In [[causal perturbation theory]] the renormalization ambiguities are understood as the freedom of extending the [[time-ordered product]] of [[operator-valued distributions]] to the interaction points, in the sense of [[extension of distributions]]. The notorious ``infinities that plague quantum field theory'' arise only if this extension is not handled correctly and has to be fixed. Hence what historically is called ``renormalization'' could from a mathematical point of view just be called ``normalization'' (a point made vividly for instance in [[Finite Quantum Electrodynamics -- The Causal Approach|Scharf 95]], [[Quantum Gauge Theories -- A True Ghost Story|Scharf 01]]). The \emph{[[main theorem of perturbative renormalization theory]]} (theorem \ref{PerturbativeRenormalizationMainTheorem} below) states that any two choices of such (re-)normalizations are uniquely related by a re-definition of the [[interaction]] [[Lagrangian density]] which introduces further point interactions of higher order (``counter terms''), The [[extension of distributions]] of the [[time-ordered product]] may naturally be organized via [[graphs]], the \emph{[[Feynman graphs]]} (\hyperlink{GarciaBondiaLazzarini00}{Garcia-Bondia \& Lazzarini 00}, \hyperlink{Keller10}{Keller 10, chapter IV}), and hence the renormalized perturbative [[S-matrix]] defining the [[perturbative quantum field theory]] is expressed as a [[formal power series]] in renormalized [[Feynman graphs]], the \emph{[[Feynman perturbation series]]} (\hyperlink{Keller10}{Keller 10 (IV.12)}). Historically the [[Feynman perturbation series]] was motivated from intuition about the would-be [[path integral]], and this is still a popular point of view, despite its lack of rigorous formulation. One may understand the axiomatics on the [[S-matrix]] in [[causal perturbation theory]] as defining the result of the path integral without actually doing an integration over field configurations. But while [[path integral quantization]] for [[perturbative quantum field theory]] remains elusive, it has been shown that the (re-)normalized [[perturbative quantum field theory]] thus constructed via [[causal perturbation theory]] is, at least under favorable circumstances, equivalently the ([[Fedosov deformation quantization|Fedosov]]) [[formal deformation quantization]] of the [[covariant phase space]] induced by the given interacting [[Lagrangian density]] (\hyperlink{Collini16}{Collini 16}). This identifies the (re-)normalization freedom with the usual freedom in choosing [[formal deformation quantization]]. This also suggest that the construction of the full [[non-perturbative quantum field theory]] ought to be given by a [[strict deformation quantization]] of the [[covariant phase space]]. But presently no example of such for non-trivial interaction in [[spacetime]] [[dimension]] $\geq 4$ is known. In particular the [[phenomenology|phenomenologically]] interesting case of a complete construction of interacting field theories on 4-dimensional spacetimes is presently unknown. For the case of [[Yang-Mills theory]] this open problem is one of the ``Millenium Problems'' (see at \emph{[[quantization of Yang-Mills theory]]}). The following is taken from \emph{[[geometry of physics -- A first idea of quantum field theory]]}. See there for more backround. \hypertarget{details}{}\subsection*{{Details}}\label{details} There are different formulations of renormalization: \begin{enumerate}% \item \emph{\hyperlink{InCausalPerturbationTheory}{In causal perturbation theory}} \item \emph{\hyperlink{BPHZRenormalization}{BPHZ and Hopf-Algebraic renormalization}} \item \emph{\hyperlink{OfTheoriesInBVForm}{Of theories in BV-CS form}} \end{enumerate} \hypertarget{InCausalPerturbationTheory}{}\subsubsection*{{In causal perturbation theory}}\label{InCausalPerturbationTheory} We discuss (re-)normalization of [[relativistic field theory|relativistic]] [[perturbative QFT]] in the rigorous formulation of [[causal perturbation theory]]/[[perturbative AQFT]]. 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 really 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: \begin{itemize}% \item Normalization \begin{itemize}% \item \emph{\hyperlink{EpsteinGlaserRenormalization}{Epstein-Glaser renormalization}}. \item \emph{\hyperlink{UVRegularizationViaZ}{UV-Regularization via Counterterms}}. \item \emph{\hyperlink{EffectiveQFTFlowWislonian}{Wilson-Polchinski effective QFT flow}}. \end{itemize} \item Re-Normalization \begin{itemize}% \item \emph{\hyperlink{SPRenormalizationGroup}{Stückelberg-Petermann re-normalization}} \item \emph{\hyperlink{RGFlowGeneral}{Renormalization group flow}} \item \emph{\hyperlink{ScalingTransformatinRGFlow}{Gell-Mann \& Low RG Flow}} \end{itemize} \end{itemize} 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 type of construction 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 renormalizations (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]]''), depending on a chosen [[UV cutoff]]-scale, and construct the [[limit of a sequence|limit]] as the ``cutoff is removed''. 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 this it serves to understand the [[relative effective action]] induced by the choice of [[UV cutoff]] at any given cutoff scale. The [[infinitesimal]] change of these [[relative effective actions]] follows a universal [[differential equation]], [[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]] 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 $RG$ this is known as \begin{itemize}% \item \emph{\hyperlink{RGFlowGeneral}{Renormalization group flow}} \end{itemize} Specifically [[scaling transformations]] on [[Minkowski spacetime]] yields such a collection of [[renormalization schemes]]; the corresponding [[renormalization group flow]] is known as \begin{itemize}% \item \emph{\hyperlink{ScalingTransformatinRGFlow}{Gell-Mann \& Low RG flow}}. \end{itemize} $\,$ \hypertarget{EpsteinGlaserRenormalization}{}\paragraph*{{Epstein-Glaser normalization}}\label{EpsteinGlaserRenormalization} 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 \href{S-matrix#InteractingFieldAlgebraOfObservablesIsFormalDeformationQuantization}{this prop.}, the construction of [[S-matrices]] $\mathcal{S}(g S_{int} + j A)$ in the sense of [[causal perturbation theory]] (\href{S-matrix#LagrangianFieldTheoryPerturbativeScattering}{this def.}) 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 (the [[fat diagonal]]). An inductive construction of the [[S-matrix]] this way is called \emph{[[Epstein-Glaser renormalization|Epstein-Glaser-(``re''-)normalization]]} (\href{S-matrix#ExtensionOfTimeOrderedProoductsRenormalization}{this def.}). 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 \href{S-matrix#VacuumFree}{this def.}. 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 \href{S-matrix#TimeOrderedProduct}{this def.}. 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 \href{S-matrix#NotationForTimeOrderedProductsAsGeneralizedFunctions}{this remark}). \end{prop} This statement appears in (\hyperlink{PopineauStora82}{Popineau-Stora 82}), with (unpublished) details in (\hyperlink{Stora93}{Stora 93}), following personal communication by [[Henri Epstein]] (according to \hyperlink{Duetsch18}{Dütsch 18, footnote 57}). Following this, statement and detailed proof appeared in (\hyperlink{BrunettiFredenhagen00}{Brunetti-Fredenhagen 00}). \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 \href{S-matrix#NotationForTimeOrderedProductsAsGeneralizedFunctions}{this remark}) 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 \href{S-matrix#VacuumFree}{this def.}, and assume that the [[field bundle]] is a [[trivial vector bundle]] (\href{A+first+idea+of+quantum+field+theory#TrivialVectorBundleAsAFieldBundle}{this example}) 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]], 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]] (\href{A+first+idea+of+quantum+field+theory#TransgressionOfVariationalDifferentialFormsToConfigrationSpaces}{this def.}) 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'' (\href{S-matrix#RenormalizationConditions}{this def.}) 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]] (\href{Feynman+propagator#FeynmanPropagatorAsACauchyPrincipalvalue}{this prop.}). 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} (\hyperlink{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} (\hyperlink{BrunettiFredenhagen00}{Brunetti-Fredenhagen 00, lemma 5.1}, \hyperlink{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} (\hyperlink{BrunettiFredenhagen00}{Brunetti-Fredenhagen 00, theorem 5.2}, \hyperlink{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 \href{point-supported+distribution#PointSupportedDistributionsAreSumsOfDerivativesOfDeltaDistibutions}{this prop.} 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 \href{scaling+degree+of+a+distribution#DerivativesOfDeltaDistributionScalingDegree}{this example} 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 \hyperlink{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 $\xhi_\lambda u$ 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 (\hyperlink{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 (\hyperlink{Hoermander90}{Hörmander 90, thm. 3.2.4}). We follow (\hyperlink{BrunettiFredenhagen00}{Brunetti-Fredenhagen 00, theorem 5.3}), which was inspired by (\hyperlink{EpsteinGlaser73}{Epstein-Glaser 73, section 5}). Review of this approach is in (\hyperlink{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 \href{scaling+degree+of+a+distribution#ScalingDegreeOfDistributionsBasicProperties}{this prop.}) 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 \href{scaling+degree+of+a+distribution#DerivativesOfDeltaDistributionScalingDegree}{this example} 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 (\hyperlink{BrunettiFredenhagen00}{Brunetti-Fredenhagen 00, p. 25}). \end{proof} \begin{remark} \label{WExtensions}\hypertarget{WExtensions}{} \textbf{(``W-extensions'')} Since in \hyperlink{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 \hyperlink{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 \href{S-matrix#VacuumFree}{this def.}, 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]] $\mathcal{S}$ (\href{S-matrix#LagrangianFieldTheoryPerturbativeScattering}{this def.}) 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]], 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 \href{S-matrix#InteractingFieldAlgebraOfObservablesIsFormalDeformationQuantization}{this prop.}, 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 (the [[fat diagonal]]). \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} $\,$ \hypertarget{SPRenormalizationGroup}{}\paragraph*{{Stückelberg-Petermann renormalization group}}\label{SPRenormalizationGroup} 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]] (\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 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} \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{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 \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} 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]] (\href{S-matrix#VacuumFree}{this def.}). \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, 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]] (\href{S-matrix#RenormalizationConditions}{this def.}) 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} $\,$ \hypertarget{UVRegularizationViaZ}{}\paragraph*{{UV-Regularization via Conterterms}}\label{UVRegularizationViaZ} 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}) 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 \href{S-matrix#VacuumFree}{this def.}), 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} (\hyperlink{Duetsch10}{Dütsch 10, section 4}) example: relativistic momentum cutoff with $\epsilon$-regularization (\href{UV+regularization#KellerKopperSchophaus97}{Keller-Kopper-Schophaus 97, section 6.1}, \hyperlink{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 \href{S-matrix#VacuumFree}{this def.}) 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 $\Delta_{F,\Lambda}$ (\href{star+product#PropagatorStarProduct}{this def.}). 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} (\hyperlink{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 \href{S-matrix#VacuumFree}{this def.}) and let $g S_{int} + j A \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g,j\rangle$ a polynomial [[local observable]], 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]] $\{\mathcal{Z}_\Lambda\}_{\Lambda \in \mathbb{R}_{\geq 0}}$ (\href{Stückelberg-Petermann+renormalization+group#InteractionVertexRedefinition}{this def.}) 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 (\href{S-matrix#LagrangianFieldTheoryPerturbativeScattering}{this def.}); \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]] (\href{S-matrix#ExtensionOfTimeOrderedProoductsRenormalization}{this def.}). \end{prop} This was claimed in (\hyperlink{BrunettiDuetschFredenhagen09}{Brunetti-Dütsch-Fredenhagen 09, (75)}), a proof was indicated in (\hyperlink{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}). 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 \href{S-matrix#VacuumFree}{this def.}) 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 \href{Stückelberg-Petermann+renormalization+group#InteractionVertexRedefinition}{this def.} 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} $\,$ \hypertarget{EffectiveQFTFlowWislonian}{}\paragraph*{{Wilson-Polchinski effective QFT flow}}\label{EffectiveQFTFlowWislonian} 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 \href{S-matrix#VacuumFree}{this def.}) 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 \href{S-matrix#FormalParameters}{this def.}). 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} (\hyperlink{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]] (\href{star+product#SymmetricContribution}{this prop.}) just as for the genuine [[time-ordered product]] on [[regular polynomial observables]] (\href{Wick+algebra#IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise}{this prop.}) that eeach the ``effective time-ordered product'' $\star_{F,\Lambda}$ is [[isomorphism|isomorphic]] to the pointwise product $(-)\cdot (-)$ (\href{A+first+idea+of+quantum+field+theory#Observable}{this def.}) \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 \href{Wick+algebra#eq:OnRegularPolynomialObservablesPointwiseTimeOrderedIsomorphism}{this equation}). 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 \href{S-matrix#OnRegularObservablesPerturbativeSMatrix}{this def.}). 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 \href{S-matrix#VacuumFree}{this def.}) 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} (\hyperlink{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 \href{S-matrix#VacuumFree}{this def.}), 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 \href{S-matrix#VacuumFree}{this def.}) 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]] (\href{S-matrix#InPerturbationTheoryActionEffective}{this def.}) in the sense of the the [[Feynman perturbation series]] of [[Feynman amplitudes]] $\Gamma(g S_{int} + j A)$ (\href{S-matrix#FeynmanPerturbationSeriesAwayFromCoincidingPoints}{this def.}) 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]] (\href{S-matrix#FeynmanPerturbationSeriesAwayFromCoincidingPoints}{this def.}) with [[time-ordered products]] replaced by effective time ordered product at scale $\Lambda$ as in (def. \ref{SMatrixEffective}). \end{prop} (\hyperlink{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 (\href{A+first+idea+of+quantum+field+theory#Observable}{this def.}) \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 \href{S-matrix#LogarithmEffectiveAction}{this prop.}. \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]] (\href{S-matrix#ExtensionOfTimeOrderedProoductsRenormalization}{this def.}). \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 \href{S-matrix#VacuumFree}{this def.}) 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 \href{S-matrix#VacuumFree}{this def.}), 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 $\Delta_{F,\Lambda'}$ (\href{star+product#PropagatorStarProduct}{this def.}). \end{prop} This goes back to (\hyperlink{Polchinski84}{Polchinski 84, (27)}). The rigorous formulation and proof is due to (\hyperlink{BrunettiDuetschFredenhagen09}{Brunetti-Dütsch-Fredenhagen 09, prop. 5.2}, \hyperlink{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 \href{star+product#AssociativeAndUnitalStarProduct}{this prop.}). 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} $\,$ \hypertarget{RGFlowGeneral}{}\paragraph*{{Renormalization group flow}}\label{RGFlowGeneral} 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]] 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]]} (\hyperlink{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 \href{S-matrix#VacuumFree}{this def.}) 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 \href{S-matrix#FormalParameters}{this def.}) \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 (\href{A+first+idea+of+quantum+field+theory#SpacetimeSupport}{this def.}) 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]] (\href{S-matrix#LagrangianFieldTheoryPerturbativeScattering}{this def.}, \href{S-matrix#calSFunctionIsRenormalizationScheme}{this remark}), 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} (\hyperlink{BrunettiDuetschFredenhagen09}{Brunetti-Dütsch-Fredenhagen 09, sections 4.2, 5.1}, \hyperlink{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 \href{S-matrix#LagrangianFieldTheoryPerturbativeScattering}{this def.}). In order to verify the axiom ``[[causal additivity]]'', observe, for convenience, that by \href{S-matrix#CausalFactorizationAlreadyImpliesSMatrix}{this prop.} 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 \href{S-matrix#VacuumFree}{this def.}) 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]] (\href{Stückelberg-Petermann+renormalization+group#AnyTwoSMatrixRenormalizationSchemesDifferByAUniqueVertexRedefinition}{this prop.}) 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} (\hyperlink{BrunettiDuetschFredenhagen09}{Brunetti-Dütsch-Fredenhagen 09, sections 4.2, 5.1}, \hyperlink{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} (\hyperlink{BrunettiDuetschFredenhagen09}{Brunetti-Dütsch-Fredenhagen 09 (69)}, \hyperlink{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} $\,$ \hypertarget{ScalingTransformatinRGFlow}{}\paragraph*{{Gell-Mann \& Low RG flow}}\label{ScalingTransformatinRGFlow} 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 \href{S-matrix#VacuumFree}{this def.}, and consider a decomposition into a set $Spec$ of field species (\href{S-matrix#VerticesAndFieldSpecies}{this def.}) 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} (\hyperlink{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 (\href{Feynman+propagator#FeynmanPropagatorAsACauchyPrincipalvalue}{this prop.}) 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 \href{S-matrix#VacuumFree}{this def.} equipped with a decomposition into a set $Spec$ of field species (\href{S-matrix#VerticesAndFieldSpecies}{this def.}) 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 \href{renormalization+group+flow#FlowRenormalizationGroup}{this prop.}. \end{prop} (\hyperlink{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} $\,$ \hypertarget{dimensional_regularization}{}\paragraph*{{Dimensional regularization}}\label{dimensional_regularization} Discussion of renormalization via [[dimensional regularization]] in the rigorous context of [[causal perturbation theory]] is due to \hyperlink{DuetschFredenhagenKellerRejzner14}{Dütsch-Fredenhagen-Keller-Rejzner 14, section 4}. \hypertarget{conneskreimer_renormalization}{}\paragraph*{{Connes-Kreimer renormalization}}\label{conneskreimer_renormalization} Discussion of the Connes-Kreimer Hopf algebraic renormalization in [[causal perturbation theory]] is in \hyperlink{DuetschFredenhagenKellerRejzner14}{Dütsch-Fredenhagen-Keller-Rejzner 14, section 5}. (\ldots{}) $\,$ \hypertarget{BPHZRenormalization}{}\subsubsection*{{BPHZ and Hopf-algebraic renormalization}}\label{BPHZRenormalization} \textbf{The phenomenon} In the study of [[perturbative quantum field theory]] one is concerned with functions -- called \emph{amplitudes} -- that take a collection of graphs -- called [[Feynman graph]]s -- to [[Laurent polynomial]]s in a complex variable $z$ -- called the \textbf{(dimensional) [[regularization (physics)|regularization]] parameter} -- \begin{displaymath} Amplitude : CertainGraphs \to LaurentPolynomials \end{displaymath} and wishes to extract a ``meaningful'' finite component when evaluated at vanishing regularization parameter $z = 0$. A prescription -- called \textbf{renormalization scheme} -- for adding to a given amplitude in a certain recursive fashion further terms -- called \textbf{counterterms} -- such that the resulting modified amplitude -- called the \textbf{renormalized amplitude} -- is finite at $z=0$ was once given by physicists and is called the \textbf{BPHZ-procedure} . This procedure justifies itself mainly through the remarkable fact that the numbers obtained from it match certain numbers measured in particle accelerators to fantastic accuracy. \textbf{Its combinatorial Hopf-algebraic interpretation} The combinatorial Hopf algebraic approach to perturbative quantum field theory, see for instance \begin{itemize}% \item Hector Figueroa, [[Jose Gracia-Bondia]], \emph{Combinatorial Hopf algebras in quantum field theory I} (\href{http://arxiv.org/abs/hep-th/0408145}{arXiv}), \end{itemize} starts with the observation that the BPHZ-procedure can be understood \begin{itemize}% \item by noticing that there is secretly a natural [[group]] structure on the collection of amplitudes; \item which is induced from the fact that there is secretly a natural [[Hopf algebra]] structure on the vector space whose basis consists of graphs; \item and with respect to which the BPHZ-procedure is simply the [[Birkhoff decomposition]] of group valued functions on the circle into a divergent and a finite part. \end{itemize} The Hopf algebra structure on the vector space whose basis consists of graphs can be understood most conceptually in terms of [[pre-Lie algebra|pre-Lie algebras]]. \textbf{The Connes-Kreimer theorem} A [[Birkhoff decomposition]] of a loop $\phi : S^1 \to G$ in a complex group $G$ is a continuation of the loop to \begin{itemize}% \item a [[holomorphic function]] $\phi_+$ on the standard disk inside the circle; \item a holomorphic function $\phi_-$ on the complement of this disk in the projective [[complex plane]] \item such that on the unit circle the original loop is reproduced as \begin{displaymath} \phi = \phi_+ \cdot (\phi_-)^{-1} \,, \end{displaymath} with the product and the inverse on the right taken in the group $G$. Notice that by the assumption of holomorphicity $\phi_+(0)$ is a well defined element of $G$. \end{itemize} \begin{theorem} \label{}\hypertarget{}{} \textbf{(Connes-Kreimer)} \begin{enumerate}% \item If $G$ is the group of [[character]]s on any graded connected commutative [[Hopf algebra]] $H$ \begin{displaymath} G = Hom(H,\mathbb{C}) \end{displaymath} then the Birkhoff decomposition always exists and is given by the formula \begin{displaymath} \phi_- : (X \in H) \mapsto Counit(X) - PolePartOf( Product(\phi_- \otimes \phi) \circ (1 \otimes (1 - Counit)) \circ Coproduct (X) ) \,. \end{displaymath} \item There is naturally the structure of a [[Hopf algebra]], $H = Graphs$, on the graphs considered in [[quantum field theory]]. As an algebra this is the free commutative algebra on the ``1-particle irreducible graphs''. Hence QFT amplitudes can be regarded as characters on this Hopf algebra. \item The BPHZ renormalization-procedure for amplitudes is nothing but the first item applied to the special case of the second item. \end{enumerate} \end{theorem} \begin{proof} The proof is given in \begin{itemize}% \item [[Alain Connes]], [[Dirk Kreimer]], \emph{Renormalization in quantum field theory I} (\href{http://arxiv.org/abs/hep-th/9912092}{arXiv}) \end{itemize} \end{proof} \textbf{The Hopf-algebra perspective on QFT} This result first of all makes [[Hopf algebra]] an organizational principle for (re-)expressing familiar operations in [[quantum field theory]]. Computing the renormalization $\phi_+$ of an amplitude $\phi$ amounts to using the above formula to compute the counterterm $\phi_-$ and then evaluating the right hand side of \begin{displaymath} \underbrace{\phi_+}_{renormalized amplitude} = \underbrace{\phi}_{amplitude} \underbrace{\cdot}_{convolution product} \underbrace{\phi_-}_{counterterm} \,, \end{displaymath} where the product is the group product on characters, hence the [[convolution product]] of characters. Every elegant reformulation has in it the potential of going beyond mere reformulation by allowing to see structures invisible in a less natural formulation. For instance [[Dirk Kreimer]] \href{http://golem.ph.utexas.edu/category/2007/03/recent_developments_in_quantum.html#c010903}{claims} that the Hopf algebra language allows him to see patterns in perturbative quantum gravity previously missed. \textbf{Gauge theory and BV-BRST with Hopf algebra} Walter von Suijlekom is thinking about the Hopf-algebraic formulation of [[BV theory|BRST-BV methods]] in nonabelian [[gauge theory]] In his nicely readable \begin{itemize}% \item [[Walter von Suijlekom]], \emph{Renormalization of gauge fields using Hopf algebra}, (\href{http://arxiv.org/abs/0801.3170}{arXiv}) \end{itemize} he reviews the central idea: the [[BRST]] formulation of [[Yang-Mills theory]] manifests itself at the level of the resulting \emph{bare} i.e. unnormalized amplitudes in certain relations satisfied by these, the \textbf{Slavnov-Taylor identities} . Renormalization of gauge theories is consistent only if these relations are still respected by renormalized amplitudes, too. We can reformulate this in terms of Hopf algebra now: the relations between amplitudes to be preserved under renormalization must define a [[Hopf ideal]] in the Hopf algebra of graphs. Walter von Suijlekom proves this to be the case for Slavnov-Taylor in his \href{http://arxiv.org/PS_cache/arxiv/pdf/0801/0801.3170v1.pdf#page=12}{theorem 9 on p. 12} As a payoff, he obtains a very transparent way to prove the generalization of \textbf{Dyson's formula} to nonabelian gauge theory, which expresses renormalized Green's functions in terms of unrenormalized Green's functions ``at bare coupling''. This is his \href{http://arxiv.org/PS_cache/arxiv/pdf/0801/0801.3170v1.pdf#page=12}{corollary 12 on p. 13}. In the context of [[BV theory|BRST-BV quantization]] these statements are subsumed, he says, by the structure encoded in the Hopf ideal which corresponds to imposing the BV-master equation. See also (\hyperlink{Suijlekom}{Suijlekom}). \hypertarget{lattice_renormalization}{}\subsubsection*{{Lattice renormalization}}\label{lattice_renormalization} \begin{itemize}% \item [[lattice renormalization]] \end{itemize} \hypertarget{OfTheoriesInBVForm}{}\subsubsection*{{Of theories in BV-CS form}}\label{OfTheoriesInBVForm} In (\hyperlink{Costello}{Costello 07}) a renormalization procedure is discussed that applies to [[theory (physics)|theories]] that are given by [[action functionals]] which can be given in the form \begin{displaymath} S(\phi) = \langle \phi , Q \phi \rangle + I(\phi) \end{displaymath} where \begin{enumerate}% \item the [[field (physics)|fields]] $\phi$ are sections of a graded [[field bundle]] $E$ on which $Q$ is a [[differential]], $\langle -,-\rangle$ a compatible [[antibracket]] pairing such that $(E,Q, \langle \rangle)$ is a [[free field theory]] (as discussed there) in [[BV-BRST formalism]]; \item $I$ is an [[interaction]] that is at least cubic. \end{enumerate} These are action functionals that are well adapted to [[BV-BRST formalism]] and for which there is a [[quantization]] to a [[factorization algebra of observables]]. Most of the fundamental [[theory (physics)|theories]] in [[physics]] are of this form, notably [[Yang-Mills theory]]. In particular also all theories of [[schreiber:infinity-Chern-Simons theory]]-type coming from binary [[invariant polynomials]] are perturbatively of this form, notably ordinary 3d [[Chern-Simons theory]]. For a discussion of just the simple special case of 3d CS see (\hyperlink{Costello11}{Costello 11, chapter 5.4 and 5.14}). For comparison of the following with other renormalization schemes, see at (\hyperlink{Costello}{Costello 07, section 1.7}). \begin{enumerate}% \item \hyperlink{TheSetup}{The setup} \item \hyperlink{OperatorKernelsAndPropagators}{Operator (heat) kernels and propagators} \item \hyperlink{TheRenormalizationGroupOperator}{The renormalization group operator} \item \hyperlink{ThePathIntegral}{The path integral} \item \hyperlink{RenormalizedAction}{Renormalized action} \item \hyperlink{Renormalization}{Renormalization} \end{enumerate} \textbf{The setup} \begin{defn} \label{}\hypertarget{}{} A \textbf{[[free field theory]]} $(E, \langle, -,-\rangle. Q)$: \textbf{[[kinematics]]}: \begin{itemize}% \item A [[smooth manifold]] $X$ (``[[spacetime]]''/``[[worldvolume]]''); \item a $\mathbb{Z}$-[[graded object|graded]] [[complex numbers|complex]] [[vector bundle]] $E \to X$ (the ``[[field bundle]]'' containing also in general [[antifields]] and [[ghosts]]); \item equipped with a [[bundle]] [[homomorphism]] (the ``[[antibracket]] [[density]]'') \begin{displaymath} \langle -,-\rangle_{loc} \;\colon\; E \times E \to Dens_X \end{displaymath} from the fiberwise [[tensor product]] of $E$ with itself to the compex [[density bundle]] which is fiberwise \begin{itemize}% \item non-degenerate \item anti-symmetric \item of degree -1 \end{itemize} \end{itemize} Write $\mathcal{E}_c \coloneqq \Gamma_{cp}(E)$ for the space of [[sections]] of the [[field bundle]] of [[compact support]]. Write \begin{displaymath} \langle -,-\rangle \;\colon\; \mathcal{E}_c \otimes \mathcal{E}_c \to \mathbb{C} \end{displaymath} for the induced pairing on sections \begin{displaymath} \langle \phi, \psi\rangle = \int_{x \in X} \langle \phi(x), \psi(x)\rangle_{loc} \,. \end{displaymath} The paring being non-degenerate means that we have an [[isomorphism]] $E \stackrel{\simeq}{\to} E^* \otimes Dens_X$ and we write \begin{displaymath} E^! \coloneqq E^* \otimes Dens_X \,. \end{displaymath} \textbf{[[dynamics]]} \begin{itemize}% \item A [[differential operator]] on sections of the [[field bundle]] \begin{displaymath} Q \;\colon\; \mathcal{E} \to \mathcal{C} \end{displaymath} of degree 1 such that \begin{enumerate}% \item $(\mathcal{E}, Q)$ is an [[elliptic complex]]; \item $Q$ is [[self-adjoint operator|self-adjoint]] with respect to $\langle -,-\rangle$ in that for all [[field (physics)|fields]] $\phi,\psi \in \mathcal{E}_c$ of homogeneous degree we have $\langle \phi , Q \psi\rangle = (-1)^{{\vert \phi\vert}} \langle Q \phi, \psi\rangle$. \end{enumerate} \end{itemize} From this data we obtain: \begin{itemize}% \item The [[action functional]] $S \colon \mathcal{E}_c \to \mathbb{C}$ of this corresponding free field theory is \begin{displaymath} S \;\colon\; \phi \mapsto \int_X \langle \phi, Q \phi\rangle \,. \end{displaymath} \item The [[classical BV-complex]] is the [[symmetric algebra]] $Sym \mathcal{E}^!$ of sections of $E^!$ equipped with the induced action of the differential $Q$ and the pairing \begin{displaymath} \{\alpha,\beta\} \coloneqq \int_{x \in X} \langle \alpha(x), \beta(x)\rangle \,. \end{displaymath} \end{itemize} \end{defn} \begin{defn} \label{}\hypertarget{}{} An \textbf{[[interaction]]} term $I \in \cdots$ (\ldots{}) \end{defn} \begin{defn} \label{LaplaceOperatorFromGaugeFixing}\hypertarget{LaplaceOperatorFromGaugeFixing}{} A \textbf{[[gauge fixing operator]]} $Q^{GF}$ such that \begin{displaymath} H \coloneqq [Q, Q^{GF}] \end{displaymath} is a [[generalized Laplace operator]]. \end{defn} \textbf{Operator (heat) kernels and propagators} \begin{defn} \label{ConvolutionOfOperatorKernel}\hypertarget{ConvolutionOfOperatorKernel}{} For $K = \sum K' \otimes K'' \in \mathcal{E} \otimes \mathcal{E}$ the corresponding \textbf{convolution operator} $K \star \colon \mathcal{E} \to \mathcal{E}$ is \begin{displaymath} K \star e = (-1)^{\vert e\vert} \sum K' \otimes \langle K'', e\rangle \,. \end{displaymath} \end{defn} \begin{defn} \label{HeatKernel}\hypertarget{HeatKernel}{} For $H \colon \mathcal{E} \to \mathcal{E}$ a [[linear operator]], a \textbf{[[heat kernel]]} for it is a [[function]] $K_{(-)} \colon \mathbb{R}_{\gt 0} \to \mathcal{E}\otimes \mathcal{E}$ such that for each $t \in \mathbb{R}_{\gt 0}$ the convolution with $K_t$, def. \ref{ConvolutionOfOperatorKernel}, reproduces the exponential of $t\cdotH$: \begin{displaymath} K_t \star = \exp(-t H) \;\;\;\;\; \forall t \in \mathbb{R}_{\gt 0} \,. \end{displaymath} \end{defn} \begin{prop} \label{UniquenessOfHeatKernel}\hypertarget{UniquenessOfHeatKernel}{} For $H$ a [[generalized Laplace operator]] such as the $[Q,Q^{GF}]$ of def. \ref{LaplaceOperatorFromGaugeFixing} there is a unique [[heat kernel]] which is moreover a [[smooth function]] of $t$. \end{prop} \begin{defn} \label{DifferentialOperatorByKernel}\hypertarget{DifferentialOperatorByKernel}{} For $\phi = \sum \phi' \otimes \phi'' \in Sym^2 \mathcal{E}$ write \begin{displaymath} \partial_{\phi} \coloneqq \frac{1}{2} \sum \partial_{\phi''} \partial_{\phi'} \,. \end{displaymath} \end{defn} This is (\hyperlink{Costello}{Costello 07, p. 32}). \begin{prop} \label{}\hypertarget{}{} \begin{displaymath} [\partial_\phi, Q] = \partial_{Q \phi} \,. \end{displaymath} \end{prop} \textbf{The renormalization group operator} \begin{defn} \label{}\hypertarget{}{} For $Q^{GF}$ the [[gauge fixing operator]] of def. \ref{LaplaceOperatorFromGaugeFixing}, and $K_t$ the [[heat kernel]] of the corresponding [[generalized Laplace operator]] $H = [Q, Q^{GF}]$ by prop. \ref{UniquenessOfHeatKernel}, write for $\epsilon, T \in \mathbb{R}_{\gt 0}$ \begin{displaymath} P(\epsilon, T) \coloneqq \int_{\epsilon}^T (Q^{GF} \otimes 1) K_t \; d t \;\;\;\;\; \in \mathcal{E} \otimes \mathcal{E}\,. \end{displaymath} \end{defn} (\hyperlink{Costello}{Costello 07, p. 33}) \begin{defn} \label{RenormalizationOperator}\hypertarget{RenormalizationOperator}{} Write \begin{displaymath} \Gamma(P(\epsilon,T), S) \coloneqq \hbar log \left( \exp\left(\hbar \partial_{P(\epsilon,T)}\right) \exp\left(I/\hbar\right) \right) \;\; \in \mathcal{O}(\mathcal{E}, \mathbb{C}[ [ \hbar ] ]) \,, \end{displaymath} where $\partial_{\cdots}$ is given by def. \ref{DifferentialOperatorByKernel}. \end{defn} (\hyperlink{Costello}{Costello 07, def. 6.6.1}) \textbf{The path integral} \begin{prop} \label{PathIntegralByLimitInDimension0}\hypertarget{PathIntegralByLimitInDimension0}{} If $X = *$ is the [[point]], then the [[path integral]] over the [[action functional]] exists as an ordinary [[integral]] and is equal to \begin{displaymath} \hbar log \int_{x \in (Im Q^{GF})_{\mathbb{R}}} \exp\left( \tfrac{1}{2} \langle x, Q x\rangle + I(x + a) d \mu \right) = \Gamma(P(0,\infty), I)(a) \end{displaymath} \end{prop} This is (\hyperlink{Costello}{Costello 07, lemma 6.6.2}). \begin{remark} \label{}\hypertarget{}{} For $X$ of positive [[dimension]], the [[limit]] \begin{displaymath} \underset{\epsilon \to 0}{\lim} \Gamma(P(\epsilon,\infty), I)(a) \end{displaymath} does not in general exist. Renormalization is the process of adding $\hbar$-corrections to the action -- the [[counterterms]] -- such as to make it exist after all. In this case we may regard the limit, by prop. \ref{PathIntegralByLimitInDimension0}, as the \emph{definition} of the [[path integral]]. \end{remark} \textbf{Renormalized action} \begin{defn} \label{RenormalizedActionAndCounterterms}\hypertarget{RenormalizedActionAndCounterterms}{} Given the [[action functional]] $S \colon \phi \mapsto \frac{1}{2}\langle \phi, Q phi\rangle + I(\phi)$, a \textbf{renormalization} is a [[power series]] \begin{displaymath} I^R(\hbar, \epsilon) = I(\hbar) - \underset{{i \gt 0} \atop { k \geq k}}{\sum} \hbar^i I^{CT}_{i,k}(\epsilon) \end{displaymath} such that the [[limit]] \begin{displaymath} \underset{\epsilon \to 0}{\lim} \Gamma(P(\epsilon,T), I^R(\epsilon)) \,, \end{displaymath} by def. \ref{RenormalizationOperator}, exists. \end{defn} The $I^{CT}_{i,k}$ are called the \emph{[[counterterms]]}. \textbf{Renormalization} \begin{defn} \label{RenormalizationScheme}\hypertarget{RenormalizationScheme}{} A [[renormalization scheme]] is a decomposition of functions $\mathcal{A}$ on $(0,1)$, as a [[vector space]], into a [[direct sum]] \begin{displaymath} \mathcal{A} \simeq \mathcal{A}_{\geq 0} \oplus \mathcal{A}_{\gt 0} \end{displaymath} such that the functions $f \in \mathcal{A}_{\geq 0}$ are non-singular in that $\underset{\epsilon \to 0}{\lim} f(\epsilon)$ exists. \end{defn} Hence this is a choice of picking the [[singularities]] in functions that are not necessarily defined at $\epsilon = 0$. \begin{theorem} \label{}\hypertarget{}{} Given any choice of [[renormalization scheme]], def. \ref{RenormalizationScheme}, there exists a unique choice of [[counterterms]] $\{I^{CT}_{k,i}\}$, def. \ref{RenormalizedActionAndCounterterms} such that \begin{itemize}% \item each counterterm is in the chosen $\mathcal{A}_{\gt 0}$ as a function of $\epsilon$; \item for $k \gt 0$ the [[germ]] of a counterterm at $x \in X$ depends only on the [[germ]] of the given field theory data $(E, Q, Q^{GF})$ at that point. \end{itemize} \end{theorem} This is (\hyperlink{Costello}{Costello 07, theorem B, p. 38}). \begin{defn} \label{}\hypertarget{}{} Given a renormalization $I^{R}$, write for all $T \in \mathbb{R}_{\ht 0}$ \begin{displaymath} \Gamma^R(P(0,T), I) \coloneqq \underset{\epsilon \to 0}{\to} \Gamma(P(\epsilon, R), I - I^{CT}(\epsilon)) \,. \end{displaymath} \end{defn} \begin{remark} \label{}\hypertarget{}{} We think of $\Gamma^R(P(0,T), I)$ as the renormalized [[effective action]] of the original action at [[scale]] $T$. \end{remark} \begin{prop} \label{}\hypertarget{}{} The [[renormalization group flow]] \begin{displaymath} \Gamma^R(P(0,T'), I) = \Gamma(P(T,T'), \; \Gamma^R(P(0,T), I )) \end{displaymath} holds. \end{prop} (\hyperlink{Costellon}{Costello 07, lemma 9.0.6}). \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{vacuum_energy_and_cosmological_constant}{}\subsubsection*{{Vacuum energy and Cosmological constant}}\label{vacuum_energy_and_cosmological_constant} The renormalization freedom in [[perturbative QFT|perturbative]] [[quantization]] of [[gravity]] ([[perturbative quantum gravity]]) induces freedom in the choice of [[vacuum expectation value]] of the [[stress-energy tensor]] and hence in the [[cosmological constant]]. For details see \href{cosmological+constant#InPerturbativeQuantumGravity}{there}. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[renormalization group]], [[renormalization group flow]] \item [[renormalization scheme]] \item [[renormalon]] \item [[regularization (physics)]] \item [[perturbation theory]] \item [[BV quantization]] \item [[gauge fixing]], [[Gribov ambiguity]] \item [[cosmic Galois group]] \item [[differential renormalization]] \end{itemize} \hypertarget{References}{}\subsection*{{References}}\label{References} After the original informal suggestions by [[Schwinger-Tomonaga-Feynman-Dyson]] \begin{itemize}% \item [[Freeman Dyson]], \emph{The raditation theories of Tomonaga, Schwinger and Feynman}, Phys. Rev. 75, 486, 1949 (\href{http://web.ihep.su/dbserv/compas/src/dyson49b/eng.pdf}{pdf}) \end{itemize} the mathematics of renormalization was finally understood and summarized in the 1975 Erice Majorana School: \begin{itemize}% \item G. Velo and [[Arthur Wightman]] (eds.) \emph{Renormalization Theory} Proceedings of the 1975 Erice summer school, NATO ASI Series C 23, D. Reidel, Dordrecht, 1976 \end{itemize} which included [[causal perturbation theory]], [[BPHZ renormalization]], proof of the [[forest formula]] and the [[BRST complex]] method for [[gauge theory]]. Little advancement happened until the identification of [[Hopf algebra]] structure in the [[forest formula]] due to \begin{itemize}% \item [[Dirk Kreimer]], \emph{On the Hopf algebra structure of perturbative quantum field theories}, Adv. Theor. Math. Phys. 2 , 303 (1998) (\href{https://arxiv.org/abs/q-alg/9707029}{q-alg/9707029}) \end{itemize} This finally triggered the formulation of [[causal perturbation theory]] in terms of [[Feynman diagrams]] in \begin{itemize}% \item [[Jose Gracia-Bondia]], S. Lazzarini, \emph{Connes-Kreimer-Epstein-Glaser Renormalization} (\href{https://arxiv.org/abs/hep-th/0006106}{arXiv:hep-th/0006106}) \item [[Kai Keller]], chapter IV of \emph{Dimensional Regularization in Position Space and a Forest Formula for Regularized Epstein-Glaser Renormalization}, PhD thesis (\href{https://arxiv.org/abs/1006.2148}{arXxiv:1006.2148}) \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} Thus the original ``dark art'' of the construction of [[perturbative quantum field theory]] via renormalization finds a complete rigorous formulation. In \begin{itemize}% \item [[Giovanni Collini]], \emph{Fedosov Quantization and Perturbative Quantum Field Theory} (\href{https://arxiv.org/abs/1603.09626}{arXiv:1603.09626}) \end{itemize} it is shown that (at least under favorable circumstances) the construction of [[perturbative quantum field theory]] via [[causal perturbation theory]] is equivalently [[Fedosov deformation quantization]] of the given interacting [[Lagrangian density]], thus identifying the renormalization freedom with the Freedom in choosing a [[deformation quantization]]. \hypertarget{General}{}\subsubsection*{{Review}}\label{General} A textbook account of [[renormalization|(``re''-)normalization]] in [[causal perturbation theory]] is in \begin{itemize}% \item [[Michael Dütsch]], chapters 3 and 4 of \emph{[[From classical field theory to perturbative quantum field theory]]}, 2018 \end{itemize} Informal introductions include \begin{itemize}% \item G. Peter Lepage, \emph{What is Renormalization?}, talk 1989 (\href{http://arxiv.org/abs/hep-ph/0506330}{arXiv:hep-ph/0506330}) \item [[Steven Weinberg]], \emph{Effective Field Theory, Past and Future} (\href{http://arxiv.org/abs/0908.1964}{arXiv:0908.1964}) \item [[R. E. Borcherds]], \emph{Renormalization and quantum field theory}, (\href{http://arxiv.org/abs/1008.0129}{arxiv/1008.0129}) \item [[Arnold Neumaier]], \emph{Renormalizatin without infinities -- an elementary tutorial} (\href{http://www.mat.univie.ac.at/~neum/ms/ren.pdf}{pdf}) \end{itemize} \hypertarget{in_causal_perturbation_theory_2}{}\subsubsection*{{In causal perturbation theory}}\label{in_causal_perturbation_theory_2} The idea of Epstein-Glaser renormalization is due to \begin{itemize}% \item [[Henri Epstein]], [[Vladimir Glaser]], \emph{[[The Role of locality in perturbation theory]]}, Annales Poincar\'e{} Phys. Theor. A 19 (1973) 211. \end{itemize} following precursors in \begin{itemize}% \item [[Ernst Stückelberg]], [[André Petermann]], , \emph{La normalisation des constants dans la theorie des quanta}, Helv. Phys. Acta 26, 499 (1953); and earlier references therein (see also \emph{[[Stückelberg-Petermann renormalization group]]}) \item [[Nikolay Bogoliubov]], [[Dmitry Shirkov]],, \emph{Introduction to the Theory of Quantized Fiels}, New York: John Wiley and Sons, 1976, 3rd edition \end{itemize} This was formulated in terms of splittings of distributions. The equivalent formulation in terms of [[extensions of distributions]] is due to \begin{itemize}% \item G. Popineau and [[Raymond Stora]], \emph{A pedagogical remark on the main theorem of perturbative renormalization theory}, Nucl. Phys. B 912 (2016), 70–78, preprint: LAPP–TH, Lyon (1982). \item [[Raymond Stora]], \emph{Differential algebras in Lagrangean field theory}, Lectures at ETH, Zürich, 1993, unpublished \item [[Romeo Brunetti]], [[Klaus Fredenhagen]], \emph{Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds}, Commun. Math. Phys. 208:623-661 (2000) (\href{http://arxiv.org/abs/math-ph/9903028}{arXiv:math-ph/9903028}) \end{itemize} Exposition includes \begin{itemize}% \item [[Christian Brouder]], \emph{Multiplication of distributions}, 2010 ([[BrouderProductOfDistributions.pdf:file]]) \end{itemize} The resulting [[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 Epstein-Glaser/Stückelberg-Petermann to the [[renormalization group flow]] of \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} and the [[effective quantum field theory]] of \begin{itemize}% \item [[Kenneth Wilson]], \emph{Renormalization group and critical phenomena} , I., Physical review B 4(9) (1971). \item [[Joseph Polchinski]], \emph{Renormalization and effective Lagrangians} , Nuclear Phys. B B231, 1984 (\href{http://max2.physics.sunysb.edu/~rastelli/2016/Polchinski.pdf}{pdf}) \end{itemize} is due to \begin{itemize}% \item [[Romeo Brunetti]], [[Michael Dütsch]], [[Klaus Fredenhagen]], section 5.2 of \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]], \emph{Connection between the renormalization groups of Stückelberg-Petermann and Wilson}, Confluentes Mathematici, Vol. 4, No. 1 (2012) 12400014 (\href{https://arxiv.org/abs/1012.5604}{arXiv:1012.5604}) \item [[Michael Dütsch]], [[Klaus Fredenhagen]], [[Kai Keller]], [[Katarzyna Rejzner]], appendix A of \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} Review is in \begin{itemize}% \item [[Michael Dütsch]], section 3.8 of \emph{[[From classical field theory to perturbative quantum field theory]]}, 2018 \end{itemize} For more see at \emph{\href{perturbation+theory#ReferencesInAQFT}{perturbation theory -- In AQFT}}. Applications to the renormalization of [[Yang-Mills theory]] on curved [[background field|background]] [[spacetimes]] is accomplished in \begin{itemize}% \item [[Stefan Hollands]], \emph{Renormalized Quantum Yang-Mills Fields in Curved Spacetime}, Rev.Math.Phys.20:1033-1172,2008 (\href{https://arxiv.org/abs/0705.3340}{arXiv:0705.3340}) \end{itemize} \hypertarget{bphz_renormalization}{}\subsubsection*{{BPHZ Renormalization}}\label{bphz_renormalization} BPHZ renormalization was introduced in particular in \begin{itemize}% \item [[Klaus Hepp]], \emph{Th\'e{}orie de la Renormalisation} Lect. Notes in Phys. Springer (1969) \end{itemize} Review includes \begin{itemize}% \item Klaus Sibold, \emph{Bogoliubov-Parasiuk-Hepp-Zimmermann renormalization scheme}, Scholarpedia, 5(5):7306. \href{http://dx.doi.org/10.4249/scholarpedia.7306}{doi:10.4249/scholarpedia.7306} \item [[Alain Connes]], [[Matilde Marcolli]] \emph{[[Noncommutative Geometry, Quantum Fields and Motives]]} \end{itemize} The original articles on this are \begin{itemize}% \item [[Alain Connes]], [[Dirk Kreimer]], \emph{Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem}, Comm. Math. Phys. \textbf{210} (2000), no. 1, 249--273, \href{http://arxiv.org/abs/hep-th/9912092}{hep-th/9912092}, \href{http://www.ams.org/mathscinet-getitem?mr=1748177}{MR2002f:81070}, \href{http://dx.doi.org/10.1007/s002200050779}{doi}, \emph{II. The $\beta$-function, diffeomorphisms and the renormalization group}, Comm. Math. Phys. \textbf{216} (2001), no. 1, 215--241; \href{http://arxiv.org/abs/hep-th/0003188}{hep-th/0003188}, \href{http://www.ams.org/mathscinet-getitem?mr=1748177}{MR2002f:81071}, \href{http://dx.doi.org/10.1007/PL00005547}{doi} \end{itemize} An introduction and review to the Hopf-algebraic description of renormalization is in \begin{itemize}% \item Christian Brouder, \emph{Quantum field theory meets Hopf algebra} (\href{http://de.arxiv.org/abs/hep-th/0611153}{arxiv:hep-th/0611153}) \end{itemize} A textbook treatment is \begin{itemize}% \item [[Dirk Kreimer]], \emph{Knots and Fenyman diagrams} , Cambridge Lecture Notes in Physics. 13. Cambridge: Cambridge University Press. \item Joseph C. Varilly, \emph{The interface of noncommutative geometry and physics}, \href{http://arxiv.org/abs/hep-th/0206007}{hep-th/0206007} \end{itemize} Some heavywheight automated computations using this formalism are discussed in \begin{itemize}% \item D. J. Broadhurst, [[Dirk Kreimer]], \emph{Renormalization automated by Hopf algebra} (\href{http://arxiv.org/abs/hep-th/9810087}{arXiv:hep-th/9810087}) \end{itemize} See also \begin{itemize}% \item W. van Suijlekom, \emph{Representing Feynman graphs on BV-algebras} , (\href{http://arxiv.org/abs/0807.0999}{arXiv}) \end{itemize} \begin{itemize}% \item [[Dirk Kreimer]], Karen Yeats, \emph{Diffeomorphisms of quantum fields}, \href{https://arxiv.org/abs/1610.01837}{arxiv/1610.01837} \end{itemize} \hypertarget{ReferencesInBVFormalism}{}\subsubsection*{{In BV formalism}}\label{ReferencesInBVFormalism} Discussion of renormalization in the context of [[BV-BRST formalism]] is for [[relativistic field theory]] in the rigorous framework of [[causal perturbation theory]]/[[perturbative AQFT]] due to \begin{itemize}% \item [[Klaus Fredenhagen]], [[Kasia Rejzner]], \emph{Batalin-Vilkovisky formalism in the functional approach to classical field theory}, Commun. Math. Phys. 314(1), 93--127 (2012) (\href{https://arxiv.org/abs/1101.5112}{arXiv:1101.5112}) \item [[Klaus Fredenhagen]], [[Kasia Rejzner]], \emph{Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory}, Commun. Math. Phys. 317(3), 697--725 (2012) (\href{https://arxiv.org/abs/1110.5232}{arXiv:1110.5232}) \item [[Katarzyna Rejzner]], \emph{Batalin-Vilkovisky formalism in locally covariant field theory} (\href{https://arxiv.org/abs/1111.5130}{arXiv:1111.5130}) \item [[Katarzyna Rejzner]], \emph{Remarks on local symmetry invariance in perturbative algebraic quantum field theory} (\href{https://arxiv.org/abs/1301.7037}{arXiv:1301.7037}) \end{itemize} and surveyed in \begin{itemize}% \item [[Kasia Rejzner]], section 7 of \emph{Perturbative algebraic quantum field theory} Springer 2016 (\href{https://link.springer.com/book/10.1007%2F978-3-319-25901-7}{web}) \end{itemize} Discussion for [[Euclidean field theory]] is in \begin{itemize}% \item [[Kevin Costello]], \emph{[[Renormalization and Effective Field Theory]]} AMS (2011) (\href{http://www.ams.org/publications/authors/books/postpub/surv-170}{publisher webpage}) \end{itemize} building on \begin{itemize}% \item [[Kevin Costello]], \emph{Renormalisation and the Batalin-Vilkovisky formalism} (\href{http://arxiv.org/abs/0706.1533}{arXiv:0706.1533}) \end{itemize} See also at \emph{[[factorization algebra of observables]]}. A vaguely related approach earlier appeared in \begin{itemize}% \item [[Dmitry Tamarkin]], \emph{A formalism for the renormalization procedure} (\href{http://arxiv.org/abs/math/0312219}{arXiv:math/0312219}) \end{itemize} \hypertarget{ReferencesOnCompactifiedConfigurationSpaces}{}\subsubsection*{{On compactified configuration spaces}}\label{ReferencesOnCompactifiedConfigurationSpaces} For [[Euclidean field theory]], an alternative to regarding [[propagators]]/[[time-ordered products]]/[[Feynman amplitudes]] as [[distributions of several variables]] with [[singularities]] at (in particular) coincident points, one may [[pullback of differential forms|pullback]] these distributions to [[smooth functions]] on [[Fulton-MacPherson compactifications]] of [[configuration spaces]] of points and study renormalization in that perspective. This approach was originally considered specifically for [[Chern-Simons theory]] in \begin{itemize}% \item [[Scott Axelrod]], [[Isadore Singer]], \emph{Chern--Simons Perturbation Theory II}, J. Diff. Geom. 39 (1994) 173-213 (\href{http://arxiv.org/abs/hep-th/9304087}{arXiv:hep-th/9304087}) \end{itemize} which was re-amplified in \begin{itemize}% \item [[Raoul Bott]], [[Alberto Cattaneo]], Remark 3.6 in \emph{Integral invariants of 3-manifolds}, J. Diff. Geom., 48 (1998) 91-133 (\href{https://arxiv.org/abs/dg-ga/9710001}{arXiv:dg-ga/9710001}) \item [[Alberto Cattaneo]], [[Pavel Mnev]], Remark 11 in \emph{Remarks on Chern-Simons invariants}, Commun.Math.Phys.293:803-836,2010 (\href{https://arxiv.org/abs/0811.2045}{arXiv:0811.2045}) \end{itemize} A systematic development of [[perturbative quantum field theory]] from this perspective is discussed in \begin{itemize}% \item [[Christoph Bergbauer]], [[Romeo Brunetti]], [[Dirk Kreimer]], \emph{Renormalization and resolution of singularities} (\href{https://arxiv.org/abs/0908.0633}{arXiv:0908.0633}) \item [[Christoph Bergbauer]], \emph{Renormalization and resolution of singularities}, talks as IHES and Boston, 2009 (\href{http://www.emg.uni-mainz.de/Dateien/bergbauer.pdf}{pdf}) \item [[Marko Berghoff]], \emph{Wonderful renormalization}, 2014 (\href{http://www2.mathematik.hu-berlin.de/%7Ekreimer/wp-content/uploads/Berghoff-Marko.pdf}{pdf}, \href{https://doi.org/10.18452/17160}{doi:10.18452/17160}) \item [[Marko Berghoff]], \emph{Wonderful compactifications in quantum field theory}, Communications in Number Theory and Physics Volume 9 (2015) Number 3 (\href{https://arxiv.org/abs/1411.5583}{arXiv:1411.5583}) \end{itemize} For more see at \emph{[[Feynman amplitudes on compactified configuration spaces of points]]}. \hypertarget{operadic_description}{}\subsubsection*{{Operadic description}}\label{operadic_description} \begin{itemize}% \item [[Jean-Louis Loday]], Nikolay M. Nikolov, \emph{Operadic construction of the renormalization group} (\href{http://arxiv.org/abs/1202.1206}{arXiv:1202.1206}) \end{itemize} \hypertarget{relations_to_motives_polylogarithms_positivity}{}\subsubsection*{{Relations to motives, polylogarithms, positivity}}\label{relations_to_motives_polylogarithms_positivity} \begin{itemize}% \item [[Alexander Goncharov]], Marcus Spradlin, C. Vergu, Anastasia Volovich, \emph{Classical polylogarithms for amplitudes and Wilson loops}, Phys.Rev.Lett.105:151605,2010 \href{http://arxiv.org/abs/1006.5703}{arxiv/1006.5703} \item [[Nima Arkani-Hamed]], Jacob L. Bourjaily, Freddy Cachazo, [[Alexander Goncharov]], Alexander Postnikov, Jaroslav Trnka, \emph{Scattering amplitudes and the positive Grassmannian}, \href{http://arxiv.org/abs/1212.5605}{arxiv/1212.5605} \item [[Spencer Bloch]], [[Hélène Esnault]], [[Dirk Kreimer]], \emph{On motives associated to graph polynomials}, Commun.Math.Phys. \textbf{267} (2006) 181-225 \href{http://arxiv.org/abs/math/0510011}{math.AG/0510011} \href{http:&&dx.doi.org/10.1007/s00220-006-0040-2}{doi} \end{itemize} [[!redirects renormalizable]] [[!redirects renormalizable theory]] [[!redirects Epstein-Glaser renormalization]] [[!redirects Epstein-Glaser renormalizations]] [[!redirects EG renormalization]] [[!redirects EG renormalizations]] [[!redirects EG-renormalization]] [[!redirects EG-renormalizations]] [[!redirects BPHZ renormalization]] [[!redirects Hopf-algebraic renormalization]] [[!redirects Hopf algebra renormalization]] \end{document}