\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\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*{wavefront set}

\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{motivation}{Motivation}\dotfill \pageref*{motivation} \linebreak
\noindent\hyperlink{smoothness}{Smoothness}\dotfill \pageref*{smoothness} \linebreak
\noindent\hyperlink{wavefront_set}{Wavefront set}\dotfill \pageref*{wavefront_set} \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
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{in_quantum_field_theory}{In quantum field theory}\dotfill \pageref*{in_quantum_field_theory} \linebreak


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

In [[microlocal analysis]], the \emph{wave front set} (\hyperlink{Hoermander70}{H\"o{}rmander 70}) of a [[generalized function]] such as a [[distribution]] or a [[hyperfunction]] is a characterization of the singularity structure of the generalized function, hence of how it deviates from being an ordinary smooth function.

The wave front set is the sub-bundle of the [[cotangent bundle]] that consists of all those [[direction of a vector|directions]] (non-zero [[covectors]]) such that the local [[Fourier transform]] of the distribution is not rapidly decaying in this [[direction of a vector|direction]] (\hyperlink{Hoermander90}{H\"o{}rmander 90, section 8.1}). Such covectors are stable under multiplication by positive scalars, so the wave front set can also be considered as a [[sub-bundle]] of the [[unit sphere bundle]] of the [[cotangent bundle]].

The [[projection]] of the wave front set down to the base space is the [[singular support of a distribution|singular support]] of the distribution. The additional information in the ``wave front'' [[covectors]] over this singular support may be understood as providing the directions of \emph{propagation of these singularities}. This is made precise by the \emph{[[propagation of singularities theorem]]}

A notorious issue with [[distributions]] is that, when thought of as generalized functions, generally neither their [[composition of distributions]] nor their pointwise [[product of distributions]] is defined. However, closer inspection shows that the [[obstruction]] to these operations being defined for any given pair of distributions is exactly characterized by the wave front set:

For instance the [[product of distributions]] is well defined precisely if the sum of their wave front sets does not intersect the zero-section ([[Hörmander's criterion]], \hyperlink{Hoermander90}{H\"o{}rmander 90, theorem 8.2.10}).

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

\hypertarget{motivation}{}\subsubsection*{{Motivation}}\label{motivation}

The definition of wavefront sets is motivated by a version of a [[Paley-Wiener theorem]] that characterizes smooth compactly supported functions ($\mathbb{R}^n \to \mathbb{R}$) by a growth condition on their [[Fourier transform]] $\mathcal{F}$:

\begin{theorem}
\label{}\hypertarget{}{}
\textbf{([[Paley-Wiener-Schwartz theorem]])}

The vector space $C_0^{\infty}(\mathbb{R}^n)$ of [[smooth function|smooth]] [[compact support|compactly supported]] functions ([[bump functions]]) is (algebraically and topologically) [[isomorphism|isomorphic]], via the [[Fourier transform]], to the space of [[entire functions]] $F$ which satisfy the following estimate: there is a positive constant $B$ such that for every [[integer]] $m \gt 0$ there is a constant $C_m$ such that:

\begin{displaymath}
F(z) \le C_m (1 + |z|)^{-m} \exp{ (B \; |\operatorname{Im}(z)|)}
\end{displaymath}
\end{theorem}
\hypertarget{smoothness}{}\subsubsection*{{Smoothness}}\label{smoothness}

We call a smooth compactly supported function that is identically $1$ in a neighbourhood of a point $x_0$ a \textbf{cutoff} function at $x_0$. Let $U \subset \mathbb{R}^n$ be open, we identify the [[cotangent bundle]] of $U$ with $U \times \mathbb{R}^n$. A subset of $U \times \mathbb{R}^n$ is said to be \textbf{conic} if it is stable under the transformation

\begin{displaymath}
(x, \zeta) \mapsto (x, \rho \zeta) \quad \text{with} \; \rho \gt 0
\end{displaymath}
Note that a [[conical set|conic]] subset is uniquely determined by its intersection with the [[unit sphere]] bundle $U\times S^{n-1}$.

\begin{defn}
\label{}\hypertarget{}{}
Let $f$ be a distribution and $(x_0, \zeta_0)$ with $\zeta_0 \neq 0$ be a point of the cotangent bundle of $U$. $f$ is \textbf{smooth} in $(x_0, \zeta_0)$ if there is a cutoff function $\chi$ in $x_0$ and an open cone $\Gamma_0$ in $\mathbb{R}^n$ containing $\zeta_0$ such that for every $m \gt 0$ there is a nonnegative constant $C_m$ such that for all $\zeta \in \Gamma_0$:

\begin{displaymath}
\| \mathcal{F}(\chi f) (\zeta)   \| \le C_m (1 + \| \zeta \|)^{-m}
\end{displaymath}
where $\mathcal{F}(\chi f)$ is the [[Fourier transform]] (of the variable $\zeta$) of the function $\chi f$ (of the variable $x$).

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
A distribution $f$ is smooth in a [[conical set|conic]] subset $\Gamma$ of the cotangent bundle of $U$ if $f$ is smooth in a neighbourhood of every point in $\Gamma$.

\end{defn}
\hypertarget{wavefront_set}{}\subsubsection*{{Wavefront set}}\label{wavefront_set}

Let $U \subseteq \mathbb{R}^n$ be an open subset, $T^* U$ its cotangent bundle and $f$ be a distribution on $U$. The complement of the union of all [[conical set|conic]] subsets of $T^* U$ where $f$ is smooth is the \textbf{wavefront set $WF(f)$}. Since the wavefront set is therefore itself [[conical set|conic]], it is equivalently determined by a subset of the unit sphere bundle of $T^* U$.

(\hyperlink{Hoermander70}{H\"o{}rmander 70 (2.4.1)}, \hyperlink{Hoermander90}{H\"o{}rmander 90, section 8.1})

This definition turns out to make invariant sense (\hyperlink{Hoermander90}{H\"o{}rmander 90, p. 256}).

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{example}
\label{WaveFrontOfDeltaDistribution}\hypertarget{WaveFrontOfDeltaDistribution}{}
\textbf{(wave front set of [[delta distribution]])}

For $n \in \mathbb{N}$, consider the [[delta distribution]]

\begin{displaymath}
\delta(0) \in \mathcal{D}'(\mathbb{R}^n)
\end{displaymath}
on $n$-dimensional [[Cartesian space]], given by [[evaluation]] at the origin. Its wave front set is

\begin{displaymath}
WF(\delta(0))
  = 
  \left\{
    (0,k)
    \;\vert\;
    k \in \mathbb{R}^n \setminus \{0\}
  \right\}
  \subset
  \mathbb{R}^n \times \mathbb{R}^n
  \simeq
  T^\ast \mathbb{R}^n 
  \,.
\end{displaymath}
\end{example}
\begin{proof}
First of all the [[singular support of a distribution|singular support]] of $\delta(0)$ is clearly $singsupp(\delta(0)) = \{0\}$, hence the wave front set vanishes over $\mathbb{R}^n \setminus \{0\}$.

At the origin, any bump function $b$ supported around the origin with $b(0) = 1$ satisfies $b \cdot \delta(0) = \delta(0)$ and hence the wave front set over the origin is the set of covectors along which the [[Fourier transform of distributions|Fourier transform]] $\hat \delta(0)$ does not suitably decay. But this Fourier transform is in fact a [[constant function]] and hence does not decay in any direction.

\end{proof}
\begin{example}
\label{WaveFrontSetOfHeavisideDistribution}\hypertarget{WaveFrontSetOfHeavisideDistribution}{}
\textbf{(wave front set of [[Heaviside distribution]])}

Let $H \in \mathcal{D}'(\mathbb{R}^1)$ be the [[Heaviside distribution]] given by

\begin{displaymath}
\langle H, b\rangle \coloneqq \int_0^\infty b(x)\, d x
  \,.
\end{displaymath}
Its wave front set is

\begin{displaymath}
WF(H) = \{(0,k) \vert k \neq 0\}
  \,.
\end{displaymath}
\end{example}
\begin{example}
\label{}\hypertarget{}{}
For $(X,e)$ a [[globally hyperbolic spacetime]] and $P$ a [[hyperbolic differential operator]] such as the [[wave operator]]/[[Klein-Gordon operator]], then the [[propagation of singularities theorem]] says that the wave front set of any solution $f$ to $P f = 0$ is a union of [[lightlike]] [[geodesics]] and their [[cotangent vectors]].

Specifically for the [[Klein-Gordon operator]] such ditributional solutions include the [[causal propagator]] and the [[Feynman propagator]].

\end{example}
\begin{example}
\label{WaveFrontOfTensorProductDistribution}\hypertarget{WaveFrontOfTensorProductDistribution}{}
\textbf{(wave front set of [[tensor product distribution]])}

Let $u \in \mathcal{D}'(X)$ and $v \in \mathcal{D}'(Y)$ be two distributions. then the wave front set of their [[tensor product distribution]] $u \otimes v \in \mathcal{D}'(X \times Y)$ satisfies

\begin{displaymath}
WF(u \otimes v)
  \;\subset\;
  \left(
     WF(u) \times WF(v)
  \right)
    \cup
  \left(
    \left( supp(u) \times \{0\} \right)
    \times WF(v)
  \right)
    \cup
   \left(
     WF(u)
     \times
     \left(
       supp(v) \times \{0\}
     \right)
   \right)
  \,,
\end{displaymath}
where $supp(-)$ denotes the [[support of a distribution]].

\end{example}
(\hyperlink{Hoermander90}{H\"o{}rmander 90, theorem 8.2.9})

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

\begin{prop}
\label{EmptyWaveFrontSetCorrespondsToOrdinaryFunction}\hypertarget{EmptyWaveFrontSetCorrespondsToOrdinaryFunction}{}
\textbf{(empty wave front set corresponds to ordinary functions)}

The wave front set of a [[compactly supported distribution]] is empty precisely if the distribution comes from an ordinary [[smooth function]] (hence a [[bump function]]).

\end{prop}
e.g. (\hyperlink{Hoermander90}{H\"o{}rmander 90, below (8.1.1)})

\begin{prop}
\label{DerivativeOfDistributionRetainsOrShrinksWaveFrontSet}\hypertarget{DerivativeOfDistributionRetainsOrShrinksWaveFrontSet}{}
\textbf{([[derivative of distributions]] retains or shrinks wave front set)}

Taking [[derivatives of distributions]] retains or shrinks the wave front set:

For $u \in \mathcal{D}'(\mathbb{R}^n)$ a distribution and $\alpha \in \mathbb{N}^n$ a multi-index with $D^\alpha$ denoting the corresponding [[partial derivative|partial]] [[derivative of distributions]], then

\begin{displaymath}
WF(D^\alpha u) \subset WF(u)
  \,.
\end{displaymath}
Hence if $P$ is any [[differential operator]] with [[smooth function]] [[coefficients]], then

\begin{displaymath}
WF(P u) \subset WF(u)
  \,.
\end{displaymath}
\end{prop}
(\hyperlink{Hoermander90}{H\"o{}rmander 90, (8.1.10) (8.1.11), p. 256})

\begin{prop}
\label{WaveFrontSetOfCompactlySupportedDistributions}\hypertarget{WaveFrontSetOfCompactlySupportedDistributions}{}
\textbf{([[wave front set]] of [[convolution of distributions|convolution of]] [[compactly supported distributions]])}

Let $u,v \in \mathcal{E}'(\mathbb{R}^n)$ be two [[compactly supported distributions]]. Then the [[wave front set]] of their [[convolution of distributions]] is

\begin{displaymath}
WF(u \star v)
  \;=\;
  \left\{
    (x + y, k)
    \;\vert\;
    (x,k) \in WF(u) \,\text{and}\, (y,k) \in WF(u)
  \right\}
  \,.
\end{displaymath}
\end{prop}
(\href{convolution+of+distributions#Bengel77}{Bengel 77, prop. 3.1})

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

\begin{itemize}%
\item [[ultraviolet divergence]]

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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

The concept of wave front set is due to

\begin{itemize}%
\item [[Lars Hörmander]], \emph{Linear differential operators}, Actes Congr. Int. Math. Nice 1970, 1, 121-133 (\href{http://www.mathunion.org/ICM/ICM1970.1/Main/icm1970.1.0121.0134.ocr.pdf}{pdf})

\end{itemize}
A textbook account for distributions on open subsets of [[Euclidean space]] is in

\begin{itemize}%
\item [[Lars Hörmander]], section 8.1 of \emph{The analysis of linear partial differential operators}, vol. I, Springer 1983, 1990

\end{itemize}
and for distributions more generally on smooth manifolds is in

\begin{itemize}%
\item [[Lars Hörmander]], \emph{The analysis of linear partial differential operators}, vol. III, Springer 1994

\end{itemize}
A history of the concept of wave front sets with extensive pointers to the literature is given in \hyperlink{Hoermander90}{H\"o{}rmander 90, p. 322-324}.

See also

\begin{itemize}%
\item Wikipedia: \href{http://en.wikipedia.org/wiki/Wavefront_set}{wavefront set}

\end{itemize}
\hypertarget{in_quantum_field_theory}{}\subsubsection*{{In quantum field theory}}\label{in_quantum_field_theory}

The application of [[microlocal analysis]] via wave front sets to the discussion of [[n-point functions]] in [[quantum field theory]] and especially [[quantum field theory on curved spacetimes]] originates with the results of

\begin{itemize}%
\item [[Johann Duistermaat]], [[Lars Hörmander]], sections 6.5, 6.6 of \emph{Fourier integral operators II}, Acta Mathematica 128, 183-269, 1972 (\href{https://projecteuclid.org/euclid.acta/1485889724}{Euclid})

\end{itemize}
which were first picked up in

\begin{itemize}%
\item C. Moreno, \emph{Spaces of positive and negative frequency solutions of field equations in curved space- times. I. The Klein-Gordon equation in stationary space-times, II. The massive vector field equations in static space-times}, J. Math. Phys. 18, 2153-61 (1977), J. Math. Phys. 19, 92-99 (1978)


\item [[Jonathan Dimock]], \emph{Scalar quantum field in an external gravitational background}, J. Math. Phys. 20, 2549-2555 (1979)



\end{itemize}
and brought into context with the [[Hadamard distributions]] needed for the [[construction]] of [[Wick algebras]] in

\begin{itemize}%
\item [[Marek Radzikowski]], \emph{Micro-local approach to the Hadamard condition in quantum field theory on curved space-time}, Commun. Math. Phys. 179 (1996), 529--553 (\href{http://projecteuclid.org/euclid.cmp/1104287114}{Euclid})

\end{itemize}
A textbook account amplifying this usage (on [[Minkowski spacetime]]) in the mathematically rigorous construction of [[perturbative quantum field theory]] via [[causal perturbation theory]] is in

\begin{itemize}%
\item [[Günter Scharf]], \emph{[[Finite Quantum Electrodynamics -- The Causal Approach]]}, Berlin: Springer-Verlag, 1995, 2nd edition


\item [[Günter Scharf]], \emph{[[Quantum Gauge Theories -- A True Ghost Story]]}, Wiley 2001



\end{itemize}
For more see the references at \emph{[[locally covariant perturbative quantum field theory]]}.

[[!redirects wavefront sets]]

[[!redirects wave front set]] [[!redirects wave front sets]]

[[!redirects wave-front set]] [[!redirects wave-front sets]]



\end{document}