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class="existingWikiWord" href="/nlab/show/Lebesgue+space">Lebesgue Spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bornological+vector+space">Bornological Vector Spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/barrelled+topological+vector+space">Barrelled Vector Spaces</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+operator">linear operator</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/bounded+linear+operator">bounded</a>, <a class="existingWikiWord" href="/nlab/show/unbounded+linear+operator">unbounded</a>, <a class="existingWikiWord" href="/nlab/show/self-adjoint+operator">self-adjoint</a>, <a class="existingWikiWord" href="/nlab/show/compact+operator">compact</a>, <a class="existingWikiWord" href="/nlab/show/Fredholm+operator">Fredholm</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectrum+of+an+operator">spectrum of an operator</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operator+algebras">operator algebras</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/functional+calculus">functional calculus</a></li> </ul> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Stone-Weierstrass+theorem">Stone-Weierstrass theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+theory">spectral theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+theorem">spectral theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gelfand+duality">Gelfand duality</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functional+calculus">functional calculus</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Riesz+representation+theorem">Riesz representation theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/measure+theory">measure theory</a></p> </li> </ul> <h2 id="topics_in_functional_analysis">Topics in Functional Analysis</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/basis+in+functional+analysis">Bases</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+theories+in+functional+analysis">Algebraic Theories in Functional Analysis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/an+elementary+treatment+of+Hilbert+spaces">An Elementary Treatment of Hilbert Spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isomorphism+classes+of+Banach+spaces">When are two Banach spaces isomorphic?</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/functional+analysis+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#motivation'>Motivation</a></li> <li><a href='#smoothness'>Smoothness</a></li> <li><a href='#wavefront_set'>Wavefront set</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#properties'>Properties</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#in_quantum_field_theory'>In quantum field theory</a></li> <li><a href='#in_differential_cohomology'>In differential cohomology</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>In <a class="existingWikiWord" href="/nlab/show/microlocal+analysis">microlocal analysis</a>, the <em>wave front set</em> (<a href="#Hoermander70">Hörmander 70</a>) of a <a class="existingWikiWord" href="/nlab/show/generalized+function">generalized function</a> such as a <a class="existingWikiWord" href="/nlab/show/distribution">distribution</a> or a <a class="existingWikiWord" href="/nlab/show/hyperfunction">hyperfunction</a> is a characterization of the singularity structure of the generalized function, hence of how it deviates from being an ordinary smooth function.</p> <p>The wave front set is the sub-bundle of the <a class="existingWikiWord" href="/nlab/show/cotangent+bundle">cotangent bundle</a> that consists of all those <a class="existingWikiWord" href="/nlab/show/direction+of+a+vector">directions</a> (non-zero <a class="existingWikiWord" href="/nlab/show/covectors">covectors</a>) such that the local <a class="existingWikiWord" href="/nlab/show/Fourier+transform">Fourier transform</a> of the distribution is not rapidly decaying in this <a class="existingWikiWord" href="/nlab/show/direction+of+a+vector">direction</a> (<a href="#Hoermander90">Hörmander 90, section 8.1</a>). Such covectors are stable under multiplication by positive scalars, so the wave front set can also be considered as a <a class="existingWikiWord" href="/nlab/show/sub-bundle">sub-bundle</a> of the <a class="existingWikiWord" href="/nlab/show/unit+sphere+bundle">unit sphere bundle</a> of the <a class="existingWikiWord" href="/nlab/show/cotangent+bundle">cotangent bundle</a>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/projection">projection</a> of the wave front set down to the base space is the <a class="existingWikiWord" href="/nlab/show/singular+support+of+a+distribution">singular support</a> of the distribution. The additional information in the “wave front” <a class="existingWikiWord" href="/nlab/show/covectors">covectors</a> over this singular support may be understood as providing the directions of <em>propagation of these singularities</em>. This is made precise by the <em><a class="existingWikiWord" href="/nlab/show/propagation+of+singularities+theorem">propagation of singularities theorem</a></em></p> <p>A notorious issue with <a class="existingWikiWord" href="/nlab/show/distributions">distributions</a> is that, when thought of as generalized functions, generally neither their <span class="newWikiWord">composition of distributions<a href="/nlab/new/composition+of+distributions">?</a></span> nor their pointwise <a class="existingWikiWord" href="/nlab/show/product+of+distributions">product of distributions</a> is defined. However, closer inspection shows that the <a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a> to these operations being defined for any given pair of distributions is exactly characterized by the wave front set:</p> <p>For instance the <a class="existingWikiWord" href="/nlab/show/product+of+distributions">product of distributions</a> is well defined precisely if the sum of their wave front sets does not intersect the zero-section (<a class="existingWikiWord" href="/nlab/show/H%C3%B6rmander%27s+criterion">Hörmander's criterion</a>, <a href="#Hoermander90">Hörmander 90, theorem 8.2.10</a>).</p> <h2 id="definition">Definition</h2> <h3 id="motivation">Motivation</h3> <p>The definition of wavefront sets is motivated by a version of a <a class="existingWikiWord" href="/nlab/show/Paley-Wiener+theorem">Paley-Wiener theorem</a> that characterizes smooth compactly supported functions (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}^n \to \mathbb{R}</annotation></semantics></math>) by a growth condition on their <a class="existingWikiWord" href="/nlab/show/Fourier+transform">Fourier transform</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℱ</mi></mrow><annotation encoding="application/x-tex">\mathcal{F}</annotation></semantics></math>:</p> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Paley-Wiener-Schwartz+theorem">Paley-Wiener-Schwartz theorem</a>)</strong></p> <p>The vector space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>C</mi> <mn>0</mn> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_0^{\infty}(\mathbb{R}^n)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth</a> <a class="existingWikiWord" href="/nlab/show/compact+support">compactly supported</a> functions (<a class="existingWikiWord" href="/nlab/show/bump+functions">bump functions</a>) is (algebraically and topologically) <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a>, via the <a class="existingWikiWord" href="/nlab/show/Fourier+transform">Fourier transform</a>, to the space of <a class="existingWikiWord" href="/nlab/show/entire+functions">entire functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> which satisfy the following estimate: there is a positive constant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> such that for every <a class="existingWikiWord" href="/nlab/show/integer">integer</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">m \gt 0</annotation></semantics></math> there is a constant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>m</mi></msub></mrow><annotation encoding="application/x-tex">C_m</annotation></semantics></math> such that:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>≤</mo><msub><mi>C</mi> <mi>m</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mo stretchy="false">|</mo><mi>z</mi><mo stretchy="false">|</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>m</mi></mrow></msup><mi>exp</mi><mrow><mo stretchy="false">(</mo><mi>B</mi><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mo lspace="0em" rspace="thinmathspace">Im</mo><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mo stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex"> F(z) \le C_m (1 + |z|)^{-m} \exp{ (B \; |\operatorname{Im}(z)|)} </annotation></semantics></math></div></div> <h3 id="smoothness">Smoothness</h3> <p>We call a smooth compactly supported function that is identically <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math> in a neighbourhood of a point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x_0</annotation></semantics></math> a <strong>cutoff</strong> function at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x_0</annotation></semantics></math>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊂</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">U \subset \mathbb{R}^n</annotation></semantics></math> be open, we identify the <a class="existingWikiWord" href="/nlab/show/cotangent+bundle">cotangent bundle</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>×</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">U \times \mathbb{R}^n</annotation></semantics></math>. A subset of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>×</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">U \times \mathbb{R}^n</annotation></semantics></math> is said to be <strong>conic</strong> if it is stable under the transformation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>ζ</mi><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>ρ</mi><mi>ζ</mi><mo stretchy="false">)</mo><mspace width="1em"></mspace><mtext>with</mtext><mspace width="thickmathspace"></mspace><mi>ρ</mi><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> (x, \zeta) \mapsto (x, \rho \zeta) \quad \text{with} \; \rho \gt 0 </annotation></semantics></math></div> <p>Note that a <a class="existingWikiWord" href="/nlab/show/conical+set">conic</a> subset is uniquely determined by its intersection with the <a class="existingWikiWord" href="/nlab/show/unit+sphere">unit sphere</a> bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>×</mo><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">U\times S^{n-1}</annotation></semantics></math>.</p> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> be a distribution and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>ζ</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x_0, \zeta_0)</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ζ</mi> <mn>0</mn></msub><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\zeta_0 \neq 0</annotation></semantics></math> be a point of the cotangent bundle of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is <strong>smooth</strong> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>ζ</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x_0, \zeta_0)</annotation></semantics></math> if there is a cutoff function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi></mrow><annotation encoding="application/x-tex">\chi</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x_0</annotation></semantics></math> and an open cone <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\Gamma_0</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> containing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ζ</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\zeta_0</annotation></semantics></math> such that for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">m \gt 0</annotation></semantics></math> there is a nonnegative constant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>m</mi></msub></mrow><annotation encoding="application/x-tex">C_m</annotation></semantics></math> such that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ζ</mi><mo>∈</mo><msub><mi>Γ</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\zeta \in \Gamma_0</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">‖</mo><mi>ℱ</mi><mo stretchy="false">(</mo><mi>χ</mi><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo><mo stretchy="false">‖</mo><mo>≤</mo><msub><mi>C</mi> <mi>m</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mo stretchy="false">‖</mo><mi>ζ</mi><mo stretchy="false">‖</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>m</mi></mrow></msup></mrow><annotation encoding="application/x-tex"> \| \mathcal{F}(\chi f) (\zeta) \| \le C_m (1 + \| \zeta \|)^{-m} </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℱ</mi><mo stretchy="false">(</mo><mi>χ</mi><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{F}(\chi f)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Fourier+transform">Fourier transform</a> (of the variable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math>) of the function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">\chi f</annotation></semantics></math> (of the variable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>).</p> </div> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>A distribution <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is smooth in a <a class="existingWikiWord" href="/nlab/show/conical+set">conic</a> subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math> of the cotangent bundle of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is smooth in a neighbourhood of every point in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math>.</p> </div> <h3 id="wavefront_set">Wavefront set</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊆</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">U \subseteq \mathbb{R}^n</annotation></semantics></math> be an open subset, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>T</mi> <mo>*</mo></msup><mi>U</mi></mrow><annotation encoding="application/x-tex">T^* U</annotation></semantics></math> its cotangent bundle and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> be a distribution on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>. The complement of the union of all <a class="existingWikiWord" href="/nlab/show/conical+set">conic</a> subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>T</mi> <mo>*</mo></msup><mi>U</mi></mrow><annotation encoding="application/x-tex">T^* U</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is smooth is the <strong>wavefront set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>WF</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">WF(f)</annotation></semantics></math></strong>. Since the wavefront set is therefore itself <a class="existingWikiWord" href="/nlab/show/conical+set">conic</a>, it is equivalently determined by a subset of the unit sphere bundle of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>T</mi> <mo>*</mo></msup><mi>U</mi></mrow><annotation encoding="application/x-tex">T^* U</annotation></semantics></math>.</p> <p>(<a href="#Hoermander70">Hörmander 70 (2.4.1)</a>, <a href="#Hoermander90">Hörmander 90, section 8.1</a>)</p> <p>This definition turns out to make invariant sense (<a href="#Hoermander90">Hörmander 90, p. 256</a>).</p> <h2 id="examples">Examples</h2> <div class="num_example" id="WaveFrontOfDeltaDistribution"> <h6 id="example">Example</h6> <p><strong>(wave front set of <a class="existingWikiWord" href="/nlab/show/delta+distribution">delta distribution</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, consider the <a class="existingWikiWord" href="/nlab/show/delta+distribution">delta distribution</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>δ</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>∈</mo><mi>𝒟</mi><mo>′</mo><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \delta(0) \in \mathcal{D}'(\mathbb{R}^n) </annotation></semantics></math></div> <p>on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a>, given by <a class="existingWikiWord" href="/nlab/show/evaluation">evaluation</a> at the origin. Its wave front set is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>WF</mi><mo stretchy="false">(</mo><mi>δ</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><mi>k</mi><mo>∈</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>∖</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo><mo>}</mo></mrow><mo>⊂</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>×</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>≃</mo><msup><mi>T</mi> <mo>*</mo></msup><msup><mi>ℝ</mi> <mi>n</mi></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> 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 \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof">Proof</h6> <p>First of all the <a class="existingWikiWord" href="/nlab/show/singular+support+of+a+distribution">singular support</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>δ</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\delta(0)</annotation></semantics></math> is clearly <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>singsupp</mi><mo stretchy="false">(</mo><mi>δ</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">singsupp(\delta(0)) = \{0\}</annotation></semantics></math>, hence the wave front set vanishes over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>∖</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}^n \setminus \{0\}</annotation></semantics></math>.</p> <p>At the origin, any bump function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math> supported around the origin with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">b(0) = 1</annotation></semantics></math> satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>⋅</mo><mi>δ</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mi>δ</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">b \cdot \delta(0) = \delta(0)</annotation></semantics></math> and hence the wave front set over the origin is the set of covectors along which the <a class="existingWikiWord" href="/nlab/show/Fourier+transform+of+distributions">Fourier transform</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>δ</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\hat \delta(0)</annotation></semantics></math> does not suitably decay. But this Fourier transform is in fact a <a class="existingWikiWord" href="/nlab/show/constant+function">constant function</a> and hence does not decay in any direction.</p> </div> <div class="num_example" id="WaveFrontSetOfHeavisideDistribution"> <h6 id="example_2">Example</h6> <p><strong>(wave front set of <a class="existingWikiWord" href="/nlab/show/Heaviside+distribution">Heaviside distribution</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>∈</mo><mi>𝒟</mi><mo>′</mo><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H \in \mathcal{D}'(\mathbb{R}^1)</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/Heaviside+distribution">Heaviside distribution</a> given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mi>H</mi><mo>,</mo><mi>b</mi><mo stretchy="false">⟩</mo><mo>≔</mo><msubsup><mo>∫</mo> <mn>0</mn> <mn>∞</mn></msubsup><mi>b</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mi>d</mi><mi>x</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \langle H, b\rangle \coloneqq \int_0^\infty b(x)\, d x \,. </annotation></semantics></math></div> <p>Its wave front set is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>WF</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mi>k</mi><mo>≠</mo><mn>0</mn><mo stretchy="false">}</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> WF(H) = \{(0,k) \vert k \neq 0\} \,. </annotation></semantics></math></div></div> <div class="num_example"> <h6 id="example_3">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,e)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/globally+hyperbolic+spacetime">globally hyperbolic spacetime</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/hyperbolic+differential+operator">hyperbolic differential operator</a> such as the <a class="existingWikiWord" href="/nlab/show/wave+operator">wave operator</a>/<a class="existingWikiWord" href="/nlab/show/Klein-Gordon+operator">Klein-Gordon operator</a>, then the <a class="existingWikiWord" href="/nlab/show/propagation+of+singularities+theorem">propagation of singularities theorem</a> says that the wave front set of any solution <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mi>f</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">P f = 0</annotation></semantics></math> is a union of <a class="existingWikiWord" href="/nlab/show/lightlike">lightlike</a> <a class="existingWikiWord" href="/nlab/show/geodesics">geodesics</a> and their <a class="existingWikiWord" href="/nlab/show/cotangent+vectors">cotangent vectors</a>.</p> <p>Specifically for the <a class="existingWikiWord" href="/nlab/show/Klein-Gordon+operator">Klein-Gordon operator</a> such ditributional solutions include the <a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a> and the <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a>.</p> </div> <div class="num_example" id="WaveFrontOfTensorProductDistribution"> <h6 id="example_4">Example</h6> <p><strong>(wave front set of <a class="existingWikiWord" href="/nlab/show/tensor+product+distribution">tensor product distribution</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>∈</mo><mi>𝒟</mi><mo>′</mo><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">u \in \mathcal{D}'(X)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>∈</mo><mi>𝒟</mi><mo>′</mo><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">v \in \mathcal{D}'(Y)</annotation></semantics></math> be two distributions. then the wave front set of their <a class="existingWikiWord" href="/nlab/show/tensor+product+distribution">tensor product distribution</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>⊗</mo><mi>v</mi><mo>∈</mo><mi>𝒟</mi><mo>′</mo><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">u \otimes v \in \mathcal{D}'(X \times Y)</annotation></semantics></math> satisfies</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>WF</mi><mo stretchy="false">(</mo><mi>u</mi><mo>⊗</mo><mi>v</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>⊂</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mi>WF</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>×</mo><mi>WF</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>∪</mo><mrow><mo>(</mo><mrow><mo>(</mo><mi>supp</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo><mo>)</mo></mrow><mo>×</mo><mi>WF</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>∪</mo><mrow><mo>(</mo><mi>WF</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>×</mo><mrow><mo>(</mo><mi>supp</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> 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) \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>supp</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">supp(-)</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/support+of+a+distribution">support of a distribution</a>.</p> </div> <p>(<a href="#Hoermander90">Hörmander 90, theorem 8.2.9</a>)</p> <h2 id="properties">Properties</h2> <div class="num_prop" id="EmptyWaveFrontSetCorrespondsToOrdinaryFunction"> <h6 id="proposition">Proposition</h6> <p><strong>(empty wave front set corresponds to ordinary functions)</strong></p> <p>The wave front set of a <a class="existingWikiWord" href="/nlab/show/compactly+supported+distribution">compactly supported distribution</a> is empty precisely if the distribution comes from an ordinary <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> (hence a <a class="existingWikiWord" href="/nlab/show/bump+function">bump function</a>).</p> </div> <p>e.g. (<a href="#Hoermander90">Hörmander 90, below (8.1.1)</a>)</p> <div class="num_prop" id="DerivativeOfDistributionRetainsOrShrinksWaveFrontSet"> <h6 id="proposition_2">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/derivative+of+distributions">derivative of distributions</a> retains or shrinks wave front set)</strong></p> <p>Taking <a class="existingWikiWord" href="/nlab/show/derivatives+of+distributions">derivatives of distributions</a> retains or shrinks the wave front set:</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>∈</mo><mi>𝒟</mi><mo>′</mo><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">u \in \mathcal{D}'(\mathbb{R}^n)</annotation></semantics></math> a distribution and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>∈</mo><msup><mi>ℕ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\alpha \in \mathbb{N}^n</annotation></semantics></math> a multi-index with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>α</mi></msup></mrow><annotation encoding="application/x-tex">D^\alpha</annotation></semantics></math> denoting the corresponding <a class="existingWikiWord" href="/nlab/show/partial+derivative">partial</a> <a class="existingWikiWord" href="/nlab/show/derivative+of+distributions">derivative of distributions</a>, then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>WF</mi><mo stretchy="false">(</mo><msup><mi>D</mi> <mi>α</mi></msup><mi>u</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>WF</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> WF(D^\alpha u) \subset WF(u) \,. </annotation></semantics></math></div> <p>Hence if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> is any <a class="existingWikiWord" href="/nlab/show/differential+operator">differential operator</a> with <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a>, then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>WF</mi><mo stretchy="false">(</mo><mi>P</mi><mi>u</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>WF</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> WF(P u) \subset WF(u) \,. </annotation></semantics></math></div></div> <p>(<a href="#Hoermander90">Hörmander 90, (8.1.10) (8.1.11), p. 256</a>)</p> <div class="num_prop" id="WaveFrontSetOfCompactlySupportedDistributions"> <h6 id="proposition_3">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/wave+front+set">wave front set</a> of <a class="existingWikiWord" href="/nlab/show/convolution+of+distributions">convolution of</a> <a class="existingWikiWord" href="/nlab/show/compactly+supported+distributions">compactly supported distributions</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>,</mo><mi>v</mi><mo>∈</mo><mi>ℰ</mi><mo>′</mo><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">u,v \in \mathcal{E}'(\mathbb{R}^n)</annotation></semantics></math> be two <a class="existingWikiWord" href="/nlab/show/compactly+supported+distributions">compactly supported distributions</a>. Then the <a class="existingWikiWord" href="/nlab/show/wave+front+set">wave front set</a> of their <a class="existingWikiWord" href="/nlab/show/convolution+of+distributions">convolution of distributions</a> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>WF</mi><mo stretchy="false">(</mo><mi>u</mi><mo>⋆</mo><mi>v</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mrow><mo>{</mo><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>WF</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mtext>and</mtext><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>WF</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> WF(u \star v) \;=\; \left\{ (x + y, k) \;\vert\; (x,k) \in WF(u) \,\text{and}\, (y,k) \in WF(u) \right\} \,. </annotation></semantics></math></div></div> <p>(<a href="convolution+of+distributions#Bengel77">Bengel 77, prop. 3.1</a>)</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/ultraviolet+divergence">ultraviolet divergence</a></li> </ul> <h2 id="references">References</h2> <h3 id="general">General</h3> <p>The concept of wave front set is due to</p> <ul> <li id="Hoermander70"><a class="existingWikiWord" href="/nlab/show/Lars+H%C3%B6rmander">Lars Hörmander</a>, <em>Linear differential operators</em>, Actes Congr. Int. Math. Nice 1970, 1, 121-133 (<a href="http://www.mathunion.org/ICM/ICM1970.1/Main/icm1970.1.0121.0134.ocr.pdf">pdf</a>)</li> </ul> <p>A textbook account for distributions on open subsets of <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a> is in</p> <ul> <li id="Hoermander90"><a class="existingWikiWord" href="/nlab/show/Lars+H%C3%B6rmander">Lars Hörmander</a>, section 8.1 of <em>The analysis of linear partial differential operators</em>, vol. I, Springer 1983, 1990</li> </ul> <p>and for distributions more generally on smooth manifolds is in</p> <ul> <li id="Hoermander94"><a class="existingWikiWord" href="/nlab/show/Lars+H%C3%B6rmander">Lars Hörmander</a>, <em>The analysis of linear partial differential operators</em>, vol. III, Springer 1994</li> </ul> <p>A history of the concept of wave front sets with extensive pointers to the literature is given in <a href="#Hoermander90">Hörmander 90, p. 322-324</a>.</p> <p>See also</p> <ul> <li>Wikipedia: <a href="http://en.wikipedia.org/wiki/Wavefront_set">wavefront set</a></li> </ul> <h3 id="in_quantum_field_theory">In quantum field theory</h3> <p>The application of <a class="existingWikiWord" href="/nlab/show/microlocal+analysis">microlocal analysis</a> via wave front sets to the discussion of <a class="existingWikiWord" href="/nlab/show/n-point+functions">n-point functions</a> in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> and especially <a class="existingWikiWord" href="/nlab/show/quantum+field+theory+on+curved+spacetimes">quantum field theory on curved spacetimes</a> originates with the results of</p> <ul> <li id="DuistermaatHoermander72"><a class="existingWikiWord" href="/nlab/show/Johann+Duistermaat">Johann Duistermaat</a>, <a class="existingWikiWord" href="/nlab/show/Lars+H%C3%B6rmander">Lars Hörmander</a>, sections 6.5, 6.6 of <em>Fourier integral operators II</em>, Acta Mathematica 128, 183-269, 1972 (<a href="https://projecteuclid.org/euclid.acta/1485889724">Euclid</a>)</li> </ul> <p>which were first picked up in</p> <ul> <li> <p>C. Moreno, <em>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</em>, J. Math. Phys. 18, 2153-61 (1977), J. Math. Phys. 19, 92-99 (1978)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jonathan+Dimock">Jonathan Dimock</a>, <em>Scalar quantum field in an external gravitational background</em>, J. Math. Phys. 20, 2549-2555 (1979)</p> </li> </ul> <p>and brought into context with the <a class="existingWikiWord" href="/nlab/show/Hadamard+distributions">Hadamard distributions</a> needed for the <a class="existingWikiWord" href="/nlab/show/construction">construction</a> of <a class="existingWikiWord" href="/nlab/show/Wick+algebras">Wick algebras</a> in</p> <ul> <li id="Radzikowski96"> <p><a class="existingWikiWord" href="/nlab/show/Marek+Radzikowski">Marek Radzikowski</a>, <em>Micro-local approach to the Hadamard condition in quantum field theory on curved space-time</em>, Commun. Math. Phys. <strong>179</strong> (1996) 529-553 [<a href="https://doi.org/10.1007/BF02100096">doi:10.1007/BF02100096</a>, <a href="http://projecteuclid.org/euclid.cmp/1104287114">euclid:cmp/1104287114</a>]</p> </li> <li id="BrunettiFredenhagen00"> <p><a class="existingWikiWord" href="/nlab/show/Romeo+Brunetti">Romeo Brunetti</a>, <a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, <em>Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds</em>, Commun. Math. Phys. <strong>208</strong> (2000) 623-661 [<a href="https://arxiv.org/abs/math-ph/9903028">math-ph/9903028</a>, <a href="https://doi.org/10.1007/s002200050004">doi:10.1007/s002200050004</a>]</p> </li> </ul> <p>A textbook account amplifying this usage (on <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a>) in the mathematically rigorous construction of <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a> via <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a> is in</p> <ul> <li id="Scharf95"> <p><a class="existingWikiWord" href="/nlab/show/G%C3%BCnter+Scharf">Günter Scharf</a>, <em><a class="existingWikiWord" href="/nlab/show/Finite+Quantum+Electrodynamics+--+The+Causal+Approach">Finite Quantum Electrodynamics – The Causal Approach</a></em>, Berlin: Springer-Verlag, 1995, 2nd edition</p> </li> <li id="Scharf01"> <p><a class="existingWikiWord" href="/nlab/show/G%C3%BCnter+Scharf">Günter Scharf</a>, <em><a class="existingWikiWord" href="/nlab/show/Quantum+Gauge+Theories+--+A+True+Ghost+Story">Quantum Gauge Theories – A True Ghost Story</a></em>, Wiley 2001</p> </li> </ul> <p>For more see the references at <em><a class="existingWikiWord" href="/nlab/show/locally+covariant+perturbative+quantum+field+theory">locally covariant perturbative quantum field theory</a></em>.</p> <h3 id="in_differential_cohomology">In differential cohomology</h3> <p>Wave-front sets of <a class="existingWikiWord" href="/nlab/show/currents+%28distribution+theory%29">currents</a> play a role in the construction of “geometric cycles” for <a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a> by actual <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a>-classes equipped with <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometric</a> data:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ulrich+Bunke">Ulrich Bunke</a>, <a class="existingWikiWord" href="/nlab/show/Thomas+Schick">Thomas Schick</a>, Ingo Schroeder, Moritz Wiethaup, §4.2.6 in: <em>Landweber exact formal group laws and smooth cohomology theories</em>, Algebr. Geom. Topol. <strong>9</strong> (2009) 1751-1790 [<a href="https://arxiv.org/abs/0711.1134">arXiv:0711.1134</a>, <a href="https://doi.org/10.2140/agt.2009.9.1751">doi:10.2140/agt.2009.9.1751</a>]</li> </ul> <p>and in the <a class="existingWikiWord" href="/nlab/show/Hodge-filtered+differential+cohomology">Hodge-filtered</a> version:</p> <ul> <li id="Haus22"> <p><a class="existingWikiWord" href="/nlab/show/Knut+Bjarte+Haus">Knut Bjarte Haus</a>, §2.6 in: <em>Geometric Hodge filtered complex cobordism</em>, PhD thesis (2022) [<a href="https://ntnuopen.ntnu.no/ntnu-xmlui/handle/11250/3017489">ntnuopen:3017489</a>]</p> </li> <li id="HausQuick22"> <p><a class="existingWikiWord" href="/nlab/show/Knut+Bjarte+Haus">Knut Bjarte Haus</a>, <a class="existingWikiWord" href="/nlab/show/Gereon+Quick">Gereon Quick</a>, §2.10 of: <em>Geometric Hodge filtered complex cobordism</em> [<a href="https://arxiv.org/abs/2210.13259">arXiv:2210.13259</a>]</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on July 3, 2023 at 08:41:52. 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