Multiplication of Distributions Christian Brouder Institut de - PowerPoint PPT Presentation

Multiplication of Distributions Christian Brouder Institut de Minéralogie, de Physique des Matériaux et de Cosmochimie UPMC, Paris

quantum field theory § Feynman diagram x 2 x 3 ∆ x 4 x 1 x 6 x 5 x 7 G § Feynman amplitude G ( x 1 , x 2 ) ∆ ( x 2 , x 3 ) 2 G ( x 3 , x 4 ) ∆ ( x 1 , x 4 ) ∆ ( x 4 , x 5 ) ∆ ( x 5 , x 6 ) ∆ ( x 6 , x 7 ) G ( x 5 , x 7 )

quantum field theory § Feynman diagram x 2 x 3 ∆ x 4 x 1 x 6 x 5 x 7 G § Feynman amplitude G ( x 1 , x 2 ) ∆ ( x 2 , x 3 ) 2 G ( x 3 , x 4 ) ∆ ( x 1 , x 4 ) ∆ ( x 4 , x 5 ) ∆ ( x 5 , x 6 ) ∆ ( x 6 , x 7 ) G ( x 5 , x 7 )

quantum field theory § Feynman diagram x 2 x 3 ∆ x 4 x 1 x 6 x 5 x 7 G § Feynman amplitude G ( x 1 , x 2 ) ∆ ( x 2 , x 3 ) 2 G ( x 3 , x 4 ) ∆ ( x 1 , x 4 ) ∆ ( x 4 , x 5 ) ∆ ( x 5 , x 6 ) ∆ ( x 6 , x 7 ) G ( x 5 , x 7 )

quantum field theory § Feynman diagram x 2 x 3 ∆ x 4 x 1 x 6 x 5 x 7 G § Feynman amplitude G ( x 1 , x 2 ) ∆ ( x 2 , x 3 ) 2 G ( x 3 , x 4 ) ∆ ( x 1 , x 4 ) ∆ ( x 4 , x 5 ) ∆ ( x 5 , x 6 ) ∆ ( x 6 , x 7 ) G ( x 5 , x 7 )

quantum field theory § Feynman diagram x 2 x 3 ∆ x 4 x 1 x 6 x 5 x 7 G § Feynman amplitude G ( x 1 , x 2 ) ∆ ( x 2 , x 3 ) 2 G ( x 3 , x 4 ) ∆ ( x 1 , x 4 ) ∆ ( x 4 , x 5 ) ∆ ( x 5 , x 6 ) ∆ ( x 6 , x 7 ) G ( x 5 , x 7 ) § Multiply distributions on the largest domain where this is well defined D ( R 7 d \{ x i = x j } ) § Renormalization: extend the result to D ( R 7 d )

Algebraic quantum field theory § Multiplication of distributions • Motivation • The wave front set of a distribution • Application and topology § Extension of distributions (Viet) • Renormalization as the solution of a functional equation • The scaling of a distribution • Extension theorem § Renormalization on curved spacetimes (Kasia) • Epstein-Glaser renormalization • Algebraic structures (Batalin-Vilkovisky, Hopf algebra) • Functional analytic aspects

§ Joint work with Yoann Dabrowski , Nguyen Viet Dang and Frédéric Hélein

outline § Trying to multiply distributions • Singular support • Fourier transfom § The wave front set • Examples • Characteristic functions • Hörmander’s theorem for distribution products § Examples in quantum field theory § Topology

Multiply Distributions § Heaviside step function θ ( x ) = 0 for x < 0 , θ ( x ) = 1 for x ≥ 0 . 0 § As a function θ n = θ § Heaviside distribution Z ∞ Z ∞ h θ , f i = θ ( x ) f ( x ) dx = f ( x ) dx 0 −∞ § If then and for n θ n − 1 δ = δ θ n = θ n θδ = δ n > 2

regularization Z § Mollifier such that ϕ ( x ) dx = 1 ϕ § Distributions are mollified by � ✏ ( x ) = 1 ⇣ x ⌘ convolution with ✏ d ' ✏ § Mollified Heaviside distribution Z x θ ✏ ( x ) = δ ✏ ( y ) dy § Then, −∞ ✏ → 0 θ ✏ δ ✏ = 1 θδ = lim 2 δ δ 2 = lim § But diverges ✏ → 0 δ 2 ✏ § Very heavy calculations (Colombeau generalized functions)

Singular support § How detect a singular point in a distribution ? u 0 § Multiply by a smooth function around g ∈ D ( M ) x ∈ M 0 x § Look whether is smooth or not gu

Singular support M = R d § Let be a distribution on and g ∈ D ( M ) u such that is a smooth function. For e ξ ( x ) = e i ξ · x gu d ξ Z (2 π ) d h gu, e ξ i e − i ξ · x g ( x ) u ( x ) = h gu, δ x i = § All the derivatives of exist: gu | h gu, e ξ i | 6 C N (1 + | ξ | ) − N 8 N, 9 C N , s.t. 8 ξ , § The singular support of is the complement of the set u of points such that there is a with g ∈ D ( M ) x ∈ M gu a smooth function and g ( x ) 6 = 0

Easy products § You can multiply a distribution and a smooth u function f h fu, g i = h u, fg i § You can multiply two distributions and with disjoint u v singular supports h uv, g i = h u, vfg i + h v, u (1 � f ) g i where • on a neighborhood of the singular support of f = 0 v • on a neighborhood of the singular support of f = 1 u

Hard products § Product of distributions with common singular support § Consider Z ∞ 1 e − ik ξ d ξ u + ( x ) = x − i 0 + = i 0 § More precisely Z ∞ h u + , g i = i g ( � ξ ) d ξ ˆ 0 § Its singular support is Σ ( u + ) = { 0 }

Hard products § Product of distributions with common singular support § Consider also Z ∞ 1 e ik ξ d ξ u − ( x ) = x + i 0 + = − i 0 § More precisely Z ∞ h u − , g i = � i g ( ξ ) d ξ ˆ 0 § Its singular support is Σ ( u − ) = { 0 }

Fourier transform § Convolution theorem uv = b c u ? b v uv = F − 1 ( b § Define the product by u ? b v ) § Example 1 u + ( x ) = u + ( ξ ) = 2 i πθ ( ξ ) c x − i 0 + § Square of u + x 0 Z c u 2 + ( ξ ) = − 2 π θ ( η ) θ ( ξ − η ) d η = − 2 πξθ ( ξ ) R

Fourier transform § Example 1 u + ( x ) = u + ( ξ ) = 2 i πθ ( ξ ) c x − i 0 + 1 u − ( x ) = c u − ( ξ ) = − 2 i πθ ( − ξ ) x + i 0 + § Product u + u − x Z diverges u + u − ( ξ ) = 2 π \ θ ( η ) θ ( η − ξ ) d η R

Fourier transform Interpretation § c u + ( η ) 0 c u + ( ξ − η ) ξ c u − ( ξ − η ) ξ § can be integrable in some direction u ( η ) b § The non-integrable directions of can be compensated u ( η ) b for by the integrable directions of v ( ξ − η ) b

Fourier transform Interpretation § c u + ( η ) 0 c u + ( ξ − η ) ξ u + ( η ) c c u + ( ξ − η ) 0 ξ § Integrable : is well-defined u 2 +

Fourier transform Interpretation § c u + ( η ) 0 c u − ( ξ − η ) ξ u + ( η ) c c u − ( ξ − η ) ξ § Not integrable : is not well-defined u + u −

Fourier transform uv = F − 1 ( b § Define the product by u ? b v ) § What if the distributions have no Fourier transform? § The product of distributions is local: near if w = uv x d f 2 w = c fu ? c for in a neighborhood of f = 1 fv x § How should the integral converge? Z 1 [ fu ( η ) c c f 2 uv ( ξ ) = fv ( ξ − η ) d η (2 π ) d R d § Absolute convergence is not enough if we want the Leibniz rule to hold

Fourier transform § How can the integral converge? Z 1 [ fu ( η ) c c f 2 uv ( ξ ) = fv ( ξ − η ) d η (2 π ) d R d | c § The order of is finite: fu ( η ) | ≤ C (1 + | η | ) m fu c § If does not decrease along direction , then fu ( η ) η c must decrease faster than any inverse fv ( ξ − η ) polynomial c § Conversely, must compensate for the directions fu ( η ) c along which does not decrease fast fv ( ξ − η )

outline § Trying to multiply distributions • Singular support • Fourier transfom § The wave front set • Examples • Characteristic functions • Hörmander’s theorem for distribution products § Examples in quantum field theory § Topology

The wave front set Lars ¡Valter ¡Hörmander ¡ Mikio ¡Sato ¡ 1931-­‑2012 ¡ 1928-­‑ ¡

Wave front set § A point does not belong to the ( x 0 , ξ 0 ) ∈ T ∗ R d wave front set of a distribution if there is a test u function with and a conical f f ( x 0 ) 6 = 0 neighborhood of such that, for every V ⊂ R d ξ 0 integer there is a constant for which C N N | c fu ( ξ ) | ≤ C N (1 + | ξ | ) − N V for every ξ ∈ V ξ 0 0

Wave front set § The wave front set is a cone: if , ( x, ξ ) ∈ WF( u ) then for every ( x, λξ ) ∈ WF( u ) λ > 0 § The wave front set is closed § WF( u + v ) ⊂ WF( u ) ∪ WF( v ) § The singular support of is the projection of WF( u ) u on the first variable

examples § The wavefront set describes in which direction the distribution is singular above each point of the singular support § The Dirac function is singular at and its x = 0 δ b Fourier transform is δ ( ξ ) = 1 § Its wave front set is WF( δ ) = { (0 , ξ ); ξ 6 = 0 } u + ( x ) = ( x − i 0 + ) − 1 § The distribution is also singular at but its Fourier transform is u + ( ξ ) = 2 i πθ ( ξ ) c x = 0 § Its wave front set is WF( u + ) = { (0 , ξ ); ξ > 0 }

Characteristic function Relation to the Radon transform •

Characteristic function Characteristic function of a disk: the wave front set is • perpendicular to the edge The wave front set is used in edge detection for • machine vision and image processing

Characteristic function Shape and wave front set detection by counting • intersections Ω 4 3 2 1 0