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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*{scaling degree of a distribution} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{functional_analysis}{}\paragraph*{{Functional analysis}}\label{functional_analysis} [[!include functional analysis - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} The \emph{scaling degree} or \emph{degree of divergence} (\hyperlink{Steinmann71}{Steinmann 71}) or more generally the \emph{degree} (\hyperlink{Weinstein78}{Weinstein 78}) of a [[distribution]] on [[Cartesian space]] $\mathbb{R}^n$ is a measure for how it behaves at the origin $0 \in \mathbb{R}^n$ under [[rescaling]] $x \mapsto \lambda x$ of the canonical [[coordinates]]. The concept controls the problem of [[extension of distributions]] from the [[complement]] $\mathbb{R}^n \setminus \{0\}$ of the origin to all of $\mathbb{R}^n$. Such extensions are important notably in the construction of [[perturbative quantum field theories]] via [[causal perturbation theory]], where the freedom in the choice of such extensions models the [[renormalization|(``re''-)normalization]] freedom (``counter-terms'') in the construction. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \begin{defn} \label{RescaledDistribution}\hypertarget{RescaledDistribution}{} \textbf{(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} \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \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( \lambda^{\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} \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \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} (\hyperlink{BrunettiFredenhagen00}{Brunetti-Fredenhagen 00, special case of lemma 6.6}) \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[support of a distribution]] \item [[order of a distribution]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} The concept of scaling degree is due to \begin{itemize}% \item O. Steinmann, \emph{Perturbation Expansions in Axiomatic Field Theory}, volume 11 of Lecture Notes in Physics, Springer, Berlin Springer Verlag, 1971. \end{itemize} and the more general concept of \emph{degree} due to \begin{itemize}% \item [[Alan Weinstein]], \emph{The order and symbol of a distribution}, Trans. Amer. Math. Soc. 241, 1--54 (1978). \end{itemize} Review and further developments in the context of [[renormalization|(``re''-)normalization]] in [[causal perturbation theory]]/[[pAQFT]] is in \begin{itemize}% \item [[Romeo Brunetti]], [[Klaus Fredenhagen]], section 5.1 of \emph{Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds}, Commun. Math. Phys. 208 : 623-661, 2000 (\href{https://arxiv.org/abs/math-ph/9903028}{math-ph/9903028}) \item Dorothea Bahns, Micha Wrochna, \emph{On-shell extension of distributions} (\href{https://arxiv.org/abs/1210.5448}{arXiv:1210.5448}) \item [[Michael Dütsch]], def. 3.32 of \emph{[[From classical field theory to perturbative quantum field theory]]}, 2018 \end{itemize} [[!redirects scaling degree of distributions]] [[!redirects degree of divergence of a distribution]] [[!redirects degree of divergence of distributions]] [[!redirects degree of a distribution]] [[!redirects degree of distributions]] \end{document}