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 § Feynman amplitude

x1 x2 x3 x4 x5 x6 x7 ∆ G G(x1, x2)∆(x2, x3)2G(x3, x4)∆(x1, x4)∆(x4, x5)∆(x5, x6)∆(x6, x7)G(x5, x7)

quantum field theory

§ Feynman diagram § Feynman amplitude

x1 x2 x3 x4 x5 x6 x7 ∆ G G(x1, x2)∆(x2, x3)2G(x3, x4)∆(x1, x4)∆(x4, x5)∆(x5, x6)∆(x6, x7)G(x5, x7)

quantum field theory

§ Feynman diagram § Feynman amplitude

x1 x2 x3 x4 x5 x6 x7 ∆ G G(x1, x2)∆(x2, x3)2G(x3, x4)∆(x1, x4)∆(x4, x5)∆(x5, x6)∆(x6, x7)G(x5, x7)

quantum field theory

§ Feynman diagram § Feynman amplitude

x1 x2 x3 x4 x5 x6 x7 ∆ G G(x1, x2)∆(x2, x3)2G(x3, x4)∆(x1, x4)∆(x4, x5)∆(x5, x6)∆(x6, x7)G(x5, x7)

quantum field theory

§ Feynman diagram § Feynman amplitude § Multiply distributions on the largest domain where this is well defined § Renormalization: extend the result to

x1 x2 x3 x4 x5 x6 x7 ∆ G

D(R7d\{xi = xj}) D(R7d)

G(x1, x2)∆(x2, x3)2G(x3, x4)∆(x1, x4)∆(x4, x5)∆(x5, x6)∆(x6, x7)G(x5, x7)

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

• utline

§ 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 § As a function § Heaviside distribution § If then and for θ(x) = 0 for x < 0, θ(x) = 1 for x ≥ 0. θn = θ

hθ, fi = Z ∞

−∞

θ(x)f(x)dx = Z ∞ f(x)dx

θn = θ nθn−1δ = δ nθδ = δ n > 2

regularization

§ Mollifier such that § Distributions are mollified by convolution with § Mollified Heaviside distribution

§ Then, § But diverges § Very heavy calculations (Colombeau generalized functions)

ϕ

Z ϕ(x)dx = 1 ✏(x) = 1 ✏d ' ⇣x ✏ ⌘ θ✏(x) = Z x

−∞

δ✏(y)dy θδ = lim

✏→0 θ✏δ✏ = 1

2δ δ2 = lim

✏→0 δ2 ✏

Singular support

§ How detect a singular point in a distribution ? § Multiply by a smooth function around § Look whether is smooth or not

g ∈ D(M) gu u

x ∈ M

x

§ Let be a distribution on and such that is a smooth function. For § All the derivatives of exist: § The singular support of is the complement of the set

• f points such that there is a with

a smooth function and

u g ∈ D(M) gu M = Rd

eξ(x) = eiξ·x

gu u

x ∈ M

gu

g(x) 6= 0

g ∈ D(M)

g(x)u(x) = hgu, δxi = Z dξ (2π)d hgu, eξie−iξ·x 8N, 9CN, s.t.8ξ, |hgu, eξi| 6 CN(1 + |ξ|)−N

Singular support

Easy products

§ You can multiply a distribution and a smooth function § You can multiply two distributions and with disjoint singular supports where

• on a neighborhood of the singular support of
• on a neighborhood of the singular support of

u

f hfu, gi = hu, fgi

u v

huv, gi = hu, vfgi + hv, u(1 f)gi f = 0 f = 1

v u

Hard products

§ Product of distributions with common singular support § Consider § More precisely § Its singular support is Σ(u+) = {0} u+(x) = 1 x − i0+ = i Z ∞ e−ikξdξ hu+, gi = i Z ∞ ˆ g(ξ)dξ

Hard products

§ Product of distributions with common singular support § Consider also § More precisely § Its singular support is u−(x) = 1 x + i0+ = −i Z ∞ eikξdξ hu−, gi = i Z ∞ ˆ g(ξ)dξ Σ(u−) = {0}

Fourier transform

§ Convolution theorem § Define the product by § Example § Square of

c uv = b u ? b v uv = F−1(b u ? b v)

u+(x) = 1 x − i0+ c u+(ξ) = 2iπθ(ξ) c u2

+(ξ) = −2π

Z

R

θ(η)θ(ξ − η)dη = −2πξθ(ξ)

x

u+

Fourier transform

§ Example § Product diverges u+(x) = 1 x − i0+ c u+(ξ) = 2iπθ(ξ) u−(x) = 1 x + i0+ c u−(ξ) = −2iπθ(−ξ) u+u− \ u+u−(ξ) = 2π Z

R

θ(η)θ(η − ξ)dη

x

Fourier transform

§ Interpretation § can be integrable in some direction § The non-integrable directions of can be compensated for by the integrable directions of

ξ ξ

c u+(η) c u+(ξ − η) c u−(ξ − η) b u(η) b v(ξ − η) b u(η)

Fourier transform

§ Interpretation § Integrable : is well-defined

ξ

c u+(η) c u+(ξ − η)

ξ

c u+(η)c u+(ξ − η) u2

+

Fourier transform

§ Interpretation § Not integrable : is not well-defined

c u+(η)

ξ

c u−(ξ − η) c u+(η)c u−(ξ − η)

ξ

u+u−

Fourier transform

§ Define the product by § What if the distributions have no Fourier transform? § The product of distributions is local: near if for in a neighborhood of § How should the integral converge? § Absolute convergence is not enough if we want the Leibniz rule to hold

uv = F−1(b u ? b v) w = uv x

d f 2w = c fu ? c fv

f = 1 x

[ f 2uv(ξ) = 1 (2π)d Z

Rd

c fu(η)c fv(ξ − η)dη

Fourier transform

§ How can the integral converge? § The order of is finite: § If does not decrease along direction , then must decrease faster than any inverse polynomial § Conversely, must compensate for the directions along which does not decrease fast

η c fu(η) fu

|c fu(η)| ≤ C(1 + |η|)m c fv(ξ − η)

c fu(η)

c fv(ξ − η) [ f 2uv(ξ) = 1 (2π)d Z

Rd

c fu(η)c fv(ξ − η)dη

• utline

§ 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 ¡ 1931-­‑2012 ¡ Mikio ¡Sato ¡ 1928-­‑ ¡

Wave front set

§ A point does not belong to the wave front set of a distribution if there is a test function with and a conical neighborhood of such that, for every integer there is a constant for which for every u f (x0, ξ0) ∈ T ∗Rd V ⊂ Rd f(x0) 6= 0 ξ0 N CN ξ ∈ V

ξ0

V

|c fu(ξ)| ≤ CN(1 + |ξ|)−N

Wave front set

§ The wave front set is a cone: if , then for every § The wave front set is closed § § The singular support of is the projection of

• n the first variable

(x, ξ) ∈ WF(u) (x, λξ) ∈ WF(u) λ > 0

WF(u + v) ⊂ WF(u) ∪ WF(v)

WF(u)

u

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 Fourier transform is § Its wave front set is § The distribution is also singular at but its Fourier transform is § Its wave front set is

δ

x = 0 u+(x) = (x − i0+)−1 x = 0 c u+(ξ) = 2iπθ(ξ) WF(u+) = {(0, ξ); ξ > 0} WF(δ) = {(0, ξ); ξ 6= 0} b δ(ξ) = 1

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

1 2 3 4

Distribution product

§ Product of distributions § Hörmander thm: The product of two distributions and is well defined if there is not point such that § The wave front set of the product is [ f 2uv(ξ) = 1 (2π)d Z

Rd

c fu(η)c fv(ξ − η)dη

u v

(x, ξ) ∈ WF(u) (x, −ξ) ∈ WF(v) WF(uv) ⊂ WF(u) ⊕ WF(v) ∪ WF(u) ∪ WF(v)

WF(u) ⊕ WF(v) = {(x, ξ + η); (x, ξ) ∈ WF(u) and (x, η) ∈ WF(v)}

• utline

§ 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

QFT: the causal approach

Stueckelberg ¡ Bogoliubov ¡

Klaus ¡Fredenhagen ¡ Kasia ¡Rejzner ¡ Romeo ¡BruneG ¡ Stefan ¡Hollands ¡ Robert ¡Wald ¡

Radzikowski ¡

propagator

Wightman propagator

§ Product of fields § Singular support § Wavefront set § Powers are allowed § Quantization does not need renormalization ∆+(x) = h0|ϕ(x)ϕ(0)|0i {(x, y, t); t2 − x2 − y2 = 0} ∆n

+

propagator

Feynman propagator

§ Time-ordered product of fields § Singular support § Wavefront set § Powers are allowed away from § Powers are forbidden at § Renormalize only at {(x, y, t); t2 − x2 − y2 = 0} ∆F (x) = h0|T

• ϕ(x)ϕ(0)
• |0i

∆n

F

x = 0

∆n

F

x = 0 x = 0

Wave front set

§ Let and be open sets and a smooth map. § The pull-back of a distribution by 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

U ⊂ Rm V ⊂ Rn f : U → V f

v ∈ D0(V )

examples

§ The true propagator is § By pull-back by , its wave front set is § In curved space time, the wave front set of the propagator is obtained by pull-back:

• either for arbitrary
• or such that there is a null geodesic

between and , and is obtained by parallel transporting along the geodesic

G(x, y) = ∆F (x − y)

WF(G) = {

• (x, y), (ξ, −ξ)
• ; (x − y, ξ) ∈ WF(∆F )}

ξ 6= 0

• (x, y), (ξ, −η)
• x

y η ξ

• (x, x), (ξ, −ξ)
• f(x, y) = x − y

quantum field theory

§ Feynman diagram § Feynman amplitude § 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

x1 x2 x3 x4 x5 x6 x7 ∆ G G(x1, x2)∆(x2, x3)2G(x3, x4)∆(x1, x4)∆(x4, x5)∆(x5, x6)∆(x6, x7)G(x5, x7)

• utline

§ 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

Topology

§ For a closed cone we define § We furnish with a locally convex topology § Let be a vector space over . A semi-norm on is a map such that

• for all and
• for all

§ A locally convex space is a vector space equipped with a family of semi-norms on § The sets form a sub-base of the topology generated by the semi-norms

Γ ⊂ T ∗M E p : E → R

p(λx) = |λ|p(x)

E

x ∈ E p(x + y) ≤ p(x) + p(y) x, y ∈ E

C

λ ∈ C

E

(pi)i∈I

E

Vi,✏ = {x ∈ E; pi(x) < ✏}

D0

Γ(U)

D0

Γ(U) = {u ∈ D0(U); WF(u) ⊂ Γ}

topology

§ The seminorms of are:

• where is bounded in are the

seminorms of the strong topology of

• for all integers , closed

cones and functions s.t.

§ The second set of seminorms is used to ensure that the Fourier transform of around decreases faster than any inverse polynomial: the wave front set of is in

||u||N,V,χ = sup

k∈V

(1 + |k|)N|c uχ(k)| pB(u) = sup

f∈B

|hu, fi|

B D(U)

D0(U)

N V χ ∈ D(U) suppχ × V ∩ Γ = ∅

Γ x ∈ supp(χ) D0

Γ(U)

u ∈ D0

Γ(U)

u ∈ D0

Γ(U)

Topology

• Thm. (CB, Y. Dabrowski)
• is complete
• is semi-Montel (its closed and bounded subsets

are compact)

• is semi-reflexive
• is nuclear
• is a normal space of distributions

D0

Γ(U)

D0

Γ(U)

D0

Γ(U)

D0

Γ(U)

D0

Γ(U)

Topology

• Thm. (CB, N. V. Dang, F. Hélein)

With the topology of

• 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

D0

Γ(U)

For your attention