Multiplication of Distributions

Transcription

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

2 quantum field theory Feynman diagram x 2 x 3 x 1 x 4 x 5 x 6 G x 7 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 )

3 quantum field theory Feynman diagram x 2 x 3 x 1 x 4 x 5 x 6 G x 7 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 )

4 quantum field theory Feynman diagram x 2 x 3 x 1 x 4 x 5 x 6 G x 7 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 )

5 quantum field theory Feynman diagram x 2 x 3 x 1 x 4 x 5 x 6 G x 7 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 )

6 quantum field theory Feynman diagram x 2 x 3 x 1 x 4 x 5 x 6 G x 7 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 7d \{x i = x j }) Renormalization: extend the result to D(R 7d )

7 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

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

9 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

10 Multiply Distributions Heaviside step function (x) = 0 for x<0, (x) = 1 for x 0. As a function n = Heaviside distribution h,fi = Z 1 1 (x)f(x)dx = Z f(x)dx If n = then n n 1 = and n = for n > 2

11 regularization Z Mollifier ' such that '(x)dx =1 Distributions are mollified by convolution with (x) = 1 x d ' Mollified Heaviside distribution (x) = Then, Z x 1 (y)dy =lim!0 = 1 2 But 2 =lim diverges!0 2 Very heavy calculations (Colombeau generalized functions)

12 Singular support How detect a singular point in a distribution u? Multiply by a smooth function g 2 D(M) around 0 x 2 M 0 x Look whether gu is smooth or not

13 Singular support Let u be a distribution on M = R d and such that gu is a smooth function. For e (x) =e i x Z d g(x)u(x) =hgu, xi = (2 ) d hgu, e ie i x All the derivatives of gu exist: 8N,9C N,s.t.8, hgu, e i 6 C N (1 + ) N g 2 D(M) The singular support of u is the complement of the set of points x 2 M such that there is a g 2 D(M) with gu a smooth function and g(x) 6= 0

14 Easy products You can multiply a distribution u and a smooth function f hfu,gi = hu, fgi You can multiply two distributions u and v with disjoint singular supports huv, gi = hu, vfgi + hv, u(1 f)gi where f =0 on a neighborhood of the singular support of f =1 on a neighborhood of the singular support of v u

15 Hard products Product of distributions with common singular support Consider 1 u + (x) = x i0 + = i Z 1 0 e ik d More precisely hu +,gi = i Z 1 Its singular support is (u + )={0} 0 ĝ( )d

16 Hard products Product of distributions with common singular support Consider also u (x) = More precisely hu,gi = 1 x + i0 + = Its singular support is i Z 1 0 i Z 1 0 ĝ( )d e ik d (u )={0}

17 Fourier transform Convolution theorem Define the product by Example u + (x) = Square of u + cuv = bu? bv uv = F 1 (bu? bv) 1 x i0 + cu + ( ) =2i ( ) cu 2 +( ) = 2 Z 0 x R ( ) ( )d = 2 ( )

18 Fourier transform Example Product u + (x) = u (x) = u + u 1 x i0 + cu + ( ) =2i ( ) 1 x + i0 + cu ( ) = 2i ( ) x \u + u ( ) =2 Z R ( ) ( )d diverges

19 Fourier transform Interpretation cu + ( ) 0 cu + ( ) ξ bu( ) cu ( ) can be integrable in some direction The non-integrable directions of can be compensated for by the integrable directions of ξ bu( ) bv( )

20 Fourier transform Interpretation cu + ( ) 0 cu + ( ) ξ cu + ( )cu + ( ) 0 ξ u 2 + Integrable : is well-defined

21 Fourier transform Interpretation cu + ( ) 0 cu ( ) ξ cu + ( )cu ( ) ξ u + u Not integrable : is not well-defined

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

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

24 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

25 The wave front set Mikio Sato Lars Valter Hörmander

26 Wave front set A point (x 0, 0 ) 2 T R d does not belong to the wave front set of a distribution u if there is a test function f with f(x 0 ) 6= 0and a conical neighborhood V R d of such that, for every integer N there is a constant for which 0 C N fu( ) c apple C N (1 + ) N for every 2 V 0 ξ 0 V

27 Wave front set The wave front set is a cone: if, then for every (x, ) 2 WF(u) > 0 The wave front set is closed (x, ) 2 WF(u) WF(u + v) WF(u) [ WF(v) The singular support of u is the projection of WF(u) on the first variable

28 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 x =0and its Fourier transform is b ( ) =1 Its wave front set is WF( )={(0, ); 6= 0} The distribution u + (x) =(x i0 + ) 1 is also singular at x =0 but its Fourier transform is cu + ( ) =2i ( ) Its wave front set is WF(u + )={(0, ); > 0}

29 Characteristic function Relation to the Radon transform

30 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

31 Characteristic function Shape and wave front set detection by counting intersections Ω

32 Distribution product Product of distributions [f 2 uv( ) = 1 Z fu( ) c fv( c (2 ) d R d )d Hörmander thm: The product of two distributions u and v is well defined if there is not point (x, ) 2 WF(u) such that (x, ) 2 WF(v) The wave front set of the product is WF(uv) WF(u) WF(v) [ WF(u) [ WF(v) WF(u) WF(v) ={(x, + ); (x, ) 2 WF(u) and (x, ) 2 WF(v)}

33 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

34 QFT: the causal approach Stueckelberg Bogoliubov Radzikowski Klaus Fredenhagen Romeo BruneG Stefan Hollands Robert Wald Kasia Rejzner

35 propagator Product of fields +(x) =h0 '(x)'(0) 0i Singular support {(x, y, t); t 2 x 2 y 2 =0} Wavefront set n + Powers are allowed Wightman propagator Quantization does not need renormalization

36 propagator Time-ordered product of fields F (x) =h0 T '(x)'(0) 0i Feynman propagator Singular support {(x, y, t); t 2 x 2 y 2 =0} Wavefront set n Powers F are allowed away from x =0 Powers n F are forbidden at x =0 Renormalize only at x =0

37 Wave front set Let U R m and V R n be open sets and f : U! V a smooth map. The pull-back of a distribution v 2 D 0 (V ) by f is determined by the wave front set The dual space of a distribution is determined by its wave front set The restriction of a distribution to a submanifold is determined by the wave front set The propagation of singularities is described by the wave front set

38 examples The true propagator is G(x, y) = F (x y) By pull-back by f(x, y) =x y, its wave front set is WF(G) ={ (x, y), (, ) ;(x y, ) 2 WF( F )} In curved space time, the wave front set of the propagator is obtained by pull-back: (x, x), (, ) either for arbitrary 6= 0 (x, y), (, ) or such that there is a null geodesic x y between and, and is obtained by parallel transporting along the geodesic

39 quantum field theory Feynman diagram x 2 x 3 x 1 x 4 x 5 x 6 Feynman amplitude G x 7 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 ) The amplitude is well defined, except on the diagonals It remains to renormalize to define the product on the diagonals The wave front set of the renormalized amplitude can be estimated

40 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

41 Topology For a closed cone T M we define D 0 (U) ={u 2 D 0 (U); WF(u) } We furnish D 0 (U) with a locally convex topology Let E be a vector space over C. A semi-norm on E is a map p : E! R such that p( x) = p(x) for all 2 C and x 2 E p(x + y) apple p(x)+p(y) for all x, y 2 E A locally convex space is a vector space E equipped with a family (p i ) i2i of semi-norms on E The sets V i, = {x 2 E; p i (x) < } form a sub-base of the topology generated by the semi-norms

42 topology The seminorms of D 0 (U) are: p B (u) =sup where is bounded in are the f2b seminorms of the strong topology of u N,V, =sup(1 + k ) N cu (k) for all integers N, closed k2v hu, fi B D(U) D 0 (U) cones V and functions 2 D(U) s.t. supp V \ = ; The second set of seminorms is used to ensure that the Fourier transform of u 2 D 0 (U) around x 2 supp( ) decreases faster than any inverse polynomial: the wave front set of u 2 D 0 (U) is in

43 Topology Thm. (CB, Y. Dabrowski) D 0 (U) is complete D 0 (U) is semi-montel (its closed and bounded subsets are compact) D 0 (U) is semi-reflexive D 0 (U) is nuclear D 0 (U) is a normal space of distributions

44 Topology Thm. (CB, N. V. Dang, F. Hélein) With the topology of D 0 (U) The pull-back is continuous The push-forward is continuous The multiplication of distributions is hypocontinuous The tensor product of distributions is hypocontinuous The duality pairing is hypocontinuous

45 For your attention