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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*{effective quantum field theory} \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{UVRegularization}{(``Re''-)Normalization via UV-Regularization}\dotfill \pageref*{UVRegularization} \linebreak \noindent\hyperlink{RelativeEffectiveAction}{Effective quantum field theory}\dotfill \pageref*{RelativeEffectiveAction} \linebreak \noindent\hyperlink{renormalization_via_wilsonian_rg_flow}{(``Re''-)Normalization via Wilsonian RG flow}\dotfill \pageref*{renormalization_via_wilsonian_rg_flow} \linebreak \noindent\hyperlink{TraditionalInformalDiscussion}{Traditional informal discussion}\dotfill \pageref*{TraditionalInformalDiscussion} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{lightbylight_scattering}{Light-by-light scattering}\dotfill \pageref*{lightbylight_scattering} \linebreak \noindent\hyperlink{rayleigh_scattering}{Rayleigh scattering}\dotfill \pageref*{rayleigh_scattering} \linebreak \noindent\hyperlink{fermi_theory_of_weak_interactions}{Fermi theory of weak interactions}\dotfill \pageref*{fermi_theory_of_weak_interactions} \linebreak \noindent\hyperlink{chiral_perturbation_theory}{Chiral perturbation theory}\dotfill \pageref*{chiral_perturbation_theory} \linebreak \noindent\hyperlink{heavy_quark_effective_field_theory}{Heavy quark effective field theory}\dotfill \pageref*{heavy_quark_effective_field_theory} \linebreak \noindent\hyperlink{neutrino_masses_}{Neutrino masses (?)}\dotfill \pageref*{neutrino_masses_} \linebreak \noindent\hyperlink{StringTheoryAndGravityCoupledToGaugeTheory}{String theory and gravity coupled to gauge theory}\dotfill \pageref*{StringTheoryAndGravityCoupledToGaugeTheory} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak \noindent\hyperlink{ReferencesInCausalPerturbationTheory}{In causal perturbation theory}\dotfill \pageref*{ReferencesInCausalPerturbationTheory} \linebreak \noindent\hyperlink{for_string_theory}{For string theory}\dotfill \pageref*{for_string_theory} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} While fundamental [[physics]] is at some level well described by [[quantum field theory]], a typical [[Lagrangian]] used to define such a QFT can reasonably be expected to define only degrees of freedom and interactions that are relevant up to some given [[energy]] scale. In this perspective one speaks of the theory as being the \emph{effective quantum field theory} of some -- possibly known but possibly unspecified -- more fundamental theory. An example (historically the first to be successfully considered) is the [[Fermi theory of beta decay]] of hadrons: this contains interactions of four [[fermion]]s at a time, for instance a process in which a [[neutron]] decays into a collection consisting of a [[proton]], an [[electron]] and a [[neutrino]]. Later it was discovered that, more fundamentally, this is not a single reaction but is composed out of several other interactions that involve exchanges of [[W-boson]]s between these four particles. Nevertheless, Fermi's original \emph{effective} theory made very precise predictions at energy scales less than 10 [[MeV]]. The reason is that the $W$-boson has mass several orders of magnitude higher than that (about 80 [[GeV]]) and was thus \emph{effectively} invisible at these low energies. The low energy expansion of any unitary, relativistic, [[crossing symmetry|crossing symmetric]] [[S-matrix]] can be described by an effective quantum field theory. In the perspective of effective field theory notably [[renormalizable interaction|non-renormalizable]] [[interaction]] Lagrangians can still make perfect sense as effective theories and give rise to well defined predictions: they can be effective approximations to [[renormalizable interaction|renormalizable]] more fundamental theories. This is sometimes called a [[UV completion]] of the given effective theory. For instance [[quantum gravity]] -- which is notoriously [[non-renormalizable interaction|non-renormalizable]] -- makes perfect sense as an effective field theory (see for instance the introduction in (\hyperlink{DonoghueIntroduction}{Donoghue}). It is in principle possible that there is some more fundamental theory with plenty of excitations at high energies that is however degreewise finite in [[perturbation theory]], whose \emph{effective} description at low energy is given by the [[renormalizable interaction|non-renormalizable]] [[Einstein-Hilbert action]]. (For instance, [[string theory]] is meant to be such a theory.) \hypertarget{details}{}\subsection*{{Details}}\label{details} The concept of effective [[perturbative QFT]] has a precise formulation in the rigoruous context of [[causal perturbation theory]]/[[perturbative AQFT]]: \begin{itemize}% \item \emph{\hyperlink{InCausalPerturbationTheory}{Effective pQFT in Causal perturbation theory}} \end{itemize} Effective quantum field theory has traditioanlly been discussed informally, referring to [[path integral]] intuition: \begin{itemize}% \item \emph{\hyperlink{TraditionalInformalDiscussion}{Traditional informal arguments}-} \end{itemize} \hypertarget{InCausalPerturbationTheory}{}\subsubsection*{{In causal perturbation theory}}\label{InCausalPerturbationTheory} We discuss the rigorous formulation of effective [[perturbative QFT]] in terms of [[causal perturbation theory]]/[[perturbative AQFT]], due to (\hyperlink{BrunettiDuetschFredenhagen09}{Brunetti-Dütsch-Fredenhagen 09, section 5.2}, \hyperlink{Duetsch10}{Dütsch 10}), reviewed in \hyperlink{Duetsch18}{Dütsch 18, section 3.8}). \hypertarget{UVRegularization}{}\paragraph*{{(``Re''-)Normalization via UV-Regularization}}\label{UVRegularization} \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 [[limit of a sequence|limit]] of the $\Delta_{F,\Lambda}$ 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]] of the $\Delta_{F,\Lambda}$ 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{defn} (\hyperlink{Duetsch10}{Dütsch 10, section 4}) example: relativistic momentum cutoff with $\epsilon$-regularization (\hyperlink{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{RelativeEffectiveAction}{}\paragraph*{{Effective quantum field theory}}\label{RelativeEffectiveAction} \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}{\mathcal{S}_\Lambda}{\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.}) being $i \hbar$ times the [[Feynman perturbation series]] of [[Feynman amplitudes]] $\Gamma(g S_{int} + j A)$ 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} \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} \hypertarget{renormalization_via_wilsonian_rg_flow}{}\paragraph*{{(``Re''-)Normalization via Wilsonian RG flow}}\label{renormalization_via_wilsonian_rg_flow} 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]] 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}, following (\hyperlink{Wilson71}{Wilson 71}). \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 $\Lambda \mapsto \Lambda_{F,\Lambda}$ 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{TraditionalInformalDiscussion}{}\subsubsection*{{Traditional informal discussion}}\label{TraditionalInformalDiscussion} Traditional informal discussion of effective field theory proceeds from the following claim \emph{For a given set of asymptotic states, [[perturbation theory]] with the most general [[Lagrangian]] containing all terms allowed by the assumed symmetries will yield the most general [[S-matrix]] elements consistent with analyticity, [[perturbative unitarity]], [[cluster decomposition]] and the assumed symmetries.} This is due to (\hyperlink{Weinberg79}{Weinberg 1979}) and (\hyperlink{Leutwyler94}{Leutwyler94}); reviewed in \hyperlink{Pich}{Pich, p. 6}. Based on this, one argues to obtains an effective approximation to a given more fundamental theory (which may or may not be actually known) by \begin{enumerate}% \item choosing the (sub)set of fields to be considered; \item writing down a [[Lagrangian]] \begin{displaymath} L_{eff} = \sum_i c_i O_i \end{displaymath} that contains \emph{all} the possible polynomial interaction terms $O_i$ of these fields scaled by their expected/known energy scale $[O_i] = d_i$, up to a maximal energy scale (this will in general contain lots of direct interaction that in the fundamental theory are really compound interactions) with $c_i \propto \frac{1}{\Lambda^{d_i - dim X}}$; \item finally one fixes all the coupling constants of all these interactions by \begin{itemize}% \item either deriving them from a known fundamental theory by \emph{integrating out} higher energy effects in that theory; \item or, otherwise, measuring them in the laboratory. The point being that due to the energy cutoff, this is guaranteed to be a finite number of parameters. After these have been determined, all remaining quantities given by the Lagrangian are then predictions of the effective theory. \end{itemize} \end{enumerate} \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{lightbylight_scattering}{}\subsubsection*{{Light-by-light scattering}}\label{lightbylight_scattering} (\hyperlink{Pich}{Pich, section 2.1}) \hypertarget{rayleigh_scattering}{}\subsubsection*{{Rayleigh scattering}}\label{rayleigh_scattering} (\hyperlink{Pich}{Pich, section 2.2}) \hypertarget{fermi_theory_of_weak_interactions}{}\subsubsection*{{Fermi theory of weak interactions}}\label{fermi_theory_of_weak_interactions} \begin{itemize}% \item [[Fermi theory of beta decay]] \end{itemize} (\hyperlink{Pich}{Pich, section 2.3}) \hypertarget{chiral_perturbation_theory}{}\subsubsection*{{Chiral perturbation theory}}\label{chiral_perturbation_theory} [[chiral perturbation theory]] is an effective approximation of [[QCD]] in the light [[quark]] sector. \hypertarget{heavy_quark_effective_field_theory}{}\subsubsection*{{Heavy quark effective field theory}}\label{heavy_quark_effective_field_theory} (\ldots{}) \hypertarget{neutrino_masses_}{}\subsubsection*{{Neutrino masses (?)}}\label{neutrino_masses_} On [[neutrino]] [[masses]] and the [[standard model of particle physics]] as an effective field theory: \begin{quote}% I also noted at the same time that interactions between a pair of lepton doublets and a pair of scalar doublets can generate a neutrino mass, which is suppressed only by a factor $M^{-1}$, and that therefore with a reasonable estimate of $M$ could produce observable neutrino oscillations. The subsequent confirmation of neutrino oscillations lends support to the view of the Standard Model as an effective field theory, with M somewhere in the neighborhood of $10^{16} GeV$. (\hyperlink{Weinberg09}{Weinberg 09, p. 15}) \end{quote} \hypertarget{StringTheoryAndGravityCoupledToGaugeTheory}{}\subsubsection*{{String theory and gravity coupled to gauge theory}}\label{StringTheoryAndGravityCoupledToGaugeTheory} The [[string scattering amplitudes]] for [[superstrings]] are finite (fully proven so for low loop order and with various plausibility arguments for higher loop order, see at \emph{[[string scattering amplitudes]]} for more), hence define a UV-complete [[S-matrix]]. The corresponding low energy effective field theories are theories of [[supergravity]] coupled to [[gauge theory]]. ([[type II supergravity]], [[heterotic supergravity]]). See also at \emph{\href{string+theory+FAQ#WhatIsStringTheory}{string theory FAQ -- What is string theory?}}. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[effective action]] \item [[higher order curvature corrections]] \item [[Stückelberg-Petermann renormalization group]] \item [[threshold correction]] \item [[swampland]] \item \href{string+theory+FAQ#RelationshipBetweenQuantumFieldTheoryAndStringTheory}{string theory FAQ -- What is the relationship between quantum field theory and string theory?} \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \hypertarget{general}{}\subsubsection*{{General}}\label{general} The modern picture of effective low-energy QFT goes back to \begin{itemize}% \item L. P. Kadanoff, \emph{Scaling laws for Ising models near $T_c$} , Physica 2 (1966); \item [[Kenneth Wilson]], \emph{Renormalization group and critical phenomena 1. Renormalization group and the Kadanoff scaling picture} , , Physical review B 4(9) (1971) (\href{https://doi.org/10.1103/PhysRevB.4.3174}{doi:10.1103/PhysRevB.4.3174}) \item [[Kenneth Wilson]], \emph{Renormalization group and critical phenomena. 2. Phase space cell analysis of critical behavior}, Phys. Rev. B4 , 3184 (1971). \href{https://doi.org/10.1103/PhysRevB.4.3184}{doi:10.1103/PhysRevB.4.3184} \item [[Joseph Polchinski]], \emph{Renormalization and effective Lagrangians} , Nuclear Phys. B B231, 1984 (\href{http://max2.physics.sunysb.edu/~rastelli/2016/Polchinski.pdf}{pdf}) \item [[Steven Weinberg]], Physica 96 A (1979) 327 \item H. Leutwyler, Ann. Phys., NY 235 (1994) 165. \end{itemize} Review includes \begin{itemize}% \item [[Steven Weinberg]], \emph{Effective Field Theory, Past and Future} (\href{http://arxiv.org/abs/0908.1964}{arXiv:0908.1964}) \item A. Pich, \emph{Effective Field Theory} (\href{http://arxiv.org/abs/hep-ph/9806303}{arXiv:hep-ph/9806303}) \item [[Abdelmalek Abdesselam]], \emph{QFT, RG, and all that, for mathematicians, in eleven pages} (\href{https://arxiv.org/abs/1311.4897}{arXiv:1311.4897}) \item [[Daniel Freed]], \emph{Lecture 5 of [[Five lectures on supersymmetry]]} \end{itemize} A standard textbook adopting this perspective is \begin{itemize}% \item [[Steven Weinberg]], \emph{The Quantum Theory of Fields} (Cambridge University Press,Cambridge,1995). \end{itemize} whose author describes his goal as: \begin{quote}% This is intended to be a book on quantum field theory for the era of effective field theory. \end{quote} Another book which takes the effective-field-theory approach to QFT is \begin{itemize}% \item Anthony Zee, \emph{Quantum Field Theory in a Nutshell} (Princeton University Press, second edition, 2010). \end{itemize} Discussion with an eye towards [[condensed matter physics]] is in \begin{itemize}% \item [[Ramamurti Shankar]], \emph{Effective Field Theory in Condensed Matter Physics} in \emph{Conceptual Foundations of Quantum Field Theory}, 1999 (\href{http://arxiv.org/abs/cond-mat/9703210}{arXiv:cond-mat/9703210}) \end{itemize} and with an eye towards [[particle physics]] and the [[standard model of particle physics]]: \begin{itemize}% \item [[Alexey Petrov]], Andrew E Blechman, \emph{Effective Field Theories}, World Scientific 2016 (\href{https://doi.org/10.1142/8619}{doi:10.1142/8619}) \end{itemize} The point that perturbatively [[renormalization|non-renormalizable]] theories may be regarded as effective field theories at each energy scale was highligted in \begin{itemize}% \item J. Gomis and [[Steven Weinberg]], \emph{Are nonrenormalizable gauge theories renormalizable?}, Nucl. Phys. B 469 (1996) 473 (\href{https://arxiv.org/abs/hep-th/9510087}{arXiv:hep-th/9510087}) \end{itemize} Notably the theory of [[gravity]] based on the standard [[Einstein-Hilbert action]] may be regarded as just an effective QFT, which makes some of its notorious problems be non-problems: \begin{itemize}% \item [[John Donoghue]], \emph{Introduction to the Effective Field Theory Description of Gravity} (\href{http://arxiv.org/abs/gr-qc/9512024}{arXiv:gr-qc/9512024}) \item Mario Atance, Jose Luis Cortes, \emph{Effective Field Theory of pure Gravity and the Renormalization Group} (\href{http://arxiv.org/abs/hep-th/9604076}{arXiv:hep-th/9604076}) \end{itemize} and in the context of [[perturbation theory]] in [[AQFT]]: \begin{itemize}% \item [[Romeo Brunetti]], [[Klaus Fredenhagen]], [[Katarzyna Rejzner]], \emph{Quantum gravity from the point of view of locally covariant quantum field theory} (\href{http://arxiv.org/abs/1306.1058}{arXiv:1306.1058}) \end{itemize} Comments on this point are also in \begin{itemize}% \item [[Jacques Distler]], blog posts \begin{itemize}% \item \emph{\href{http://golem.ph.utexas.edu/~distler/blog/archives/000639.html}{Motivation}} \item \emph{\href{http://golem.ph.utexas.edu/~distler/blog/archives/001255.html}{Effective field theory and gravity}} \end{itemize} \item [[Kevin Costello]], \emph{[[Renormalization and Effective Field Theory]]} \end{itemize} See also \begin{itemize}% \item [[Alastair Hamilton]], \emph{On two constructions of an effective field theory} (\href{http://arxiv.org/abs/1502.05790}{arXiv:1502.05790}) \end{itemize} \hypertarget{ReferencesInCausalPerturbationTheory}{}\subsubsection*{{In causal perturbation theory}}\label{ReferencesInCausalPerturbationTheory} Discussion of [[perturbative QFT|perturbative]] effective QFT in the rigorous context of [[causal perturbation theory]]/[[perturbative AQFT]] and its relation to the [[Stückelberg-Petermann renormalization group]] 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} reviewed in \begin{itemize}% \item [[Michael Dütsch]], section 3.8 of \emph{[[From classical field theory to perturbative quantum field theory]]}, 2018 \end{itemize} See also \begin{itemize}% \item Georg Keller, Christoph Kopper, Clemens Schophaus, \emph{Perturbative Renormalization with Flow Equations in Minkowski Space}, Helv.Phys.Acta 70 (1997) 247-274 (\href{https://arxiv.org/abs/hep-th/9605137}{arXiv.hep-th/9605137}) \end{itemize} \hypertarget{for_string_theory}{}\subsubsection*{{For string theory}}\label{for_string_theory} Discussion of the effective field theories induced by [[string theory]] includes the following: Via [[string scattering amplitudes]]: \begin{itemize}% \item [[Ralph Blumenhagen]], [[Dieter Lüst]], [[Stefan Theisen]], \emph{String Scattering Amplitudes and Low Energy Effective Field Theory}, chapter 16 in \emph{Basic Concepts of String Theory} Part of the series Theoretical and Mathematical Physics pp 585-639 Springer 2013 (\href{https://link.springer.com/content/pdf/bfm%3A978-3-642-29497-6%2F1.pdf}{TOC pdf}, \href{http://www.springer.com/gp/book/9783642294969}{publisher page}) \end{itemize} Via [[string field theory]]: \begin{itemize}% \item R. Brustein, S.P.De Alwis, \emph{Renormalization group equation and non-perturbative effects in string-field theory}, Nuclear Physics B Volume 352, Issue 2, 25 March 1991, Pages 451-468 () \item Brustein and K. Roland, “Space-time versus world sheet renormalization group equation in string theory,” Nucl. Phys. B372, 201 (1992) () \item [[Ashoke Sen]], \emph{Wilsonian Effective Action of Superstring Theory}, J. High Energ. Phys. (2017) 2017: 108 (\href{https://arxiv.org/abs/1609.00459}{arXiv:1609.00459}) \end{itemize} Discussion of possible criteria for which effective field theory do \emph{not} arise as effective field theories of a string theory: \begin{itemize}% \item Eran Palti, \emph{The Swampland: Introduction and Review}, lecture notes (\href{https://arxiv.org/abs/1903.06239}{arXiv:1903.06239}) \end{itemize} For more see at \emph{[[landscape of string theory vacua]]}. [[!redirects effective quantum field theories]] [[!redirects effective QFT]] [[!redirects effective QFTs]] [[!redirects UV cutoff]] [[!redirects UV cutoffs]] [[!redirects UV cut-off]] [[!redirects UV cut-offs]] [[!redirects UV-cutoff]] [[!redirects UV-cutoffs]] [[!redirects UV cutoff scale]] [[!redirects UV cutoff scales]] [[!redirects UV cut-off scale]] [[!redirects UV cut-off scales]] [[!redirects UV-cutoff scale]] [[!redirects UV-cutoff scales]] [[!redirects UV regularization]] [[!redirects UV regularizations]] [[!redirects UV-regularization]] [[!redirects UV-regularizations]] [[!redirects effective S-matrix]] [[!redirects effective S-matrices]] [[!redirects effective S-matrix scheme]] [[!redirects effective S-matrix schemes]] [[!redirects effective field theory]] [[!redirects effective field theories]] [[!redirects effective low-energy quantum field theory]] [[!redirects effective low-energy field theory]] [[!redirects UV-completion]] [[!redirects UV completion]] [[!redirects UV-completions]] [[!redirects UV completions]] \end{document}