Geometry in cotangent space

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The extension of distributions.


In the Stueckelberg ([72]) approach to quantum field theory, renormalization was formulated as a problem of division of distributions. For Epstein–Glaser ([21], [22]) , Stora ([57],[71]), and implicitly in Bogoliubov ([7]), it was for-mulated as a problem of extension of distributions, the latter approach is more general since the ambiguity of the extension is described by the renor-malization group. This procedure was implemented on arbitrary manifolds (hence for curved Lorentzian spacetimes) by Brunetti and Fredenhagen in their groundbreaking paper of 2000 [26]. However, in the mathematical lit-erature, the problem of extension of distributions goes back at least to the work of Hadamard and Riesz on hyperbolic equations ([63],[35]). It became a central argument for the proof of a conjecture of Laurent Schwartz ([65] p. 126,[49]): the problem was to find a fundamental solution E for a linear PDE with constant coefficients in Rn, which means solving the equation P E = δ in the distributional sense. By Fourier transform, this is equiva-lent to the problem of extending P −1 which is a honest smooth function on Rn \ {P = 0} as a distribution on Rn, in such a way that P P −1 = 1 which makes the division a particular case of an extension. This problem set by Schwartz was solved positively by Lojasiewicz and H¨ormander ([40],[68]). Recently, the more general extension problem was revisited in mathematics by Yves Meyer in his wonderful book [53]. In [53], Yves Meyer also explored some deep relations between the extension problem and Harmonic analy-sis (Littlewood–Paley and Wavelet decomposition). The extension problem was solved in [53] on (Rn \ {0}). For the need of quantum field theory, we will extend his method to manifolds. In order to renormalize, one should find some way of measuring the wildness of the singularities of distributions. Indeed, we need to impose some growth condition on distributions because distributions cannot be extended in general! We estimate the wildness of the singularity by first defining an adequate notion of scaling with respect to a closed embedded submanifold I of a given manifold M , as done by Brunetti– Fredenhagen [26]. On Rn+d viewed as the cartesian product Rn × Rd, the scaling is clearly defined by homotheties in the variables corresponding to the second factor Rd. We adapt the definition of Meyer [53] in these vari-ables and define the space of weakly homogeneous distributions of degree s which we call Es.
We are able to represent all elements of Es which are defined on M \ I through a decomposition formula by a family uλ λ∈(0,1] satisfying some specific hypothesis. The distributions uλ λ∈(0,1] are the building blocks of the Es and are the key for the renormalization. We establish the following correspondence uλ 1 dλ → λs(uλ)λ−1 + nice terms, (1.1) λ∈(0,1] 0 λ t ∈ Es → uλ λ ∈ (0,1] where uλ = λ−stλψ, (1.2) the nice terms are distributions supported on the complement of I. However this scaling is only defined in local charts and a scaling around a submanifold I in a manifold M depends on the choice of an Euler vector field. Thus we propose a geometrical definition of a class of Euler vector fields: to any closed embedded submanifold I ⊂ M , we associate the ideal I of smooth functions vanishing on I. A vector field ρ is called Euler vector field if ∀f ∈ I, ρf − f ∈ I2. (1.3)
This definition is clearly intrinsic. We prove that all scalings are equivalent hence all spaces of weakly homogeneous distributions are equivalent and that our definitions are in fact independent of the choice of Euler vector fields. Actually, we prove that all Euler vector fields are locally conjugate by a local diffeomorphism which fixes the submanifold I. So it is enough to study both Es and the extension problem in a local chart. Meyer and Brunetti–Fredenhagen make use of a dyadic decomposition. We use in-stead a continuous partition of unity which is a continuous analog of the Littlewood–Paley decomposition. The continuous partition of unity has many advantages over the discrete approaches: 1) it provides a direct con-nection with the theory of Mellin transform, which allows to easily define meromorphic regularizations; 2) it gives elegant formulas especially for the poles and residues appearing in the meromorphic regularization (see Chapter 7); 3) it is well suited to the study of anomalies (see Chapter 6).
Relationship with other work. In Brunetti–Fredenhagen [26], the scal-ing around manifolds was also defined but they used two different definitions of scalings, then they showed that these actually coincide, whereas we only give one definition which is geometric. In mathematics, we also found some interesting work by Kashiwara–Kawai, where the concept of weak homo-geneity was also defined ([54] Definition (1.1) p. 22).

Extension and renormalization.

Notation, definitions.

We work in Rn+d with coordinates (x, h), I = Rn ×{0} is the linear subspace {h = 0}. For any open set U ⊂ R n+d, we denote by D(U ) the space of test functions supported on U and for all compact K ⊂ U , we denote by DK (U ) the subset of all test functions in D(U ) supported on K. We also use the seminorms:
∀ϕ ∈ D(Rn+d), πk(ϕ) := sup ∂αϕ L∞(Rn+d), |α| k
∀ϕ ∈ C∞(Rn+d), ∀K ⊂ Rd, πk,K (ϕ) := sup sup |∂αϕ(x)|. |α| k x∈K
We denote by D ′(U ) the space of distributions defined on U . The duality pairing between a distribution t and a test function ϕ is denoted by t, ϕ .
For a function, we define ϕλ(x, h) = ϕ(x, λh). For the vector field ρ = hj ∂h∂j , the following formula
ϕλ = e(log λ)ρ⋆ϕ, shows the relation between ρ and the scaling. Once we have defined the scaling for test functions, for any distribution f , we define the scaled distri-bution fλ: ∀ϕ ∈ D(Rn+d), fλ, ϕ = λ−d f, ϕλ−1 .
If f were a function, this definition would coincides with the naive scaling fλ(x, h) = f (x, λh).
We give a definition of weakly homogeneous distributions in flat space following [53]. We call a subset U ⊂ Rn +d ρ-convex if (x, h) ∈ U =⇒ ∀λ ∈ (0, 1], (x, λh) ∈ U . We insist on the fact that since we pick λ > 0, a ρ-convex domain may have empty intersection with I.
Definition 1.2.1 Let U be an arbitrary ρ-convex open subset of Rn+d. Es(U ) is defined as the space of distributions t such that t ∈ D′(U ) and ∀ϕ ∈ D(U ), ∃C(ϕ), sup |λ−s tλ, ϕ | C(ϕ). λ∈(0,1]
In the quantum field theory litterature, the wildness of distributions is mea-sured by the Steinman scaling degree. We prefer the definition of Meyer, which exploits the properties of bounded sets in the space of distributions (this is related to bornological properties of D′(U )).
We denote by dλλ the multiplicative measure on [0, 1]. We shall now give a definition of a class of maps λ uλ with value in the space of distributions.
Definition 1.2.2 For all 1 p , we define Lpdλ ([0, 1],
space of families (uλ)λ∈(0,1] of distributions such that ∀ ϕ ∈ D (U), λ uλ, ϕ ∈ Lpdλ ([0, 1], C). (1.4)
The H¨ormander trick. We recall here the basic idea of Littlewood–Paley analysis ([53] p. 14). Pick a function χ which depends only on h such that χ = 1 when |h| 2 and χ = 0 for |h| 3. The Littlewood–Paley function ψ(•) = χ(•) − χ(2•) is supported on the annulus 1 |h| 3. Then the idea is to rewrite the plateau function χ using the trick of the telescopic series = χ(•) − χ(2•) + • • • + χ(2j •) − χ(2j+1•) + • • • and deduce a dyadic partition of unity ∞ 1 = (1 − χ) + ψ(2j .) j=0
Our goal in this paragraph is to derive a continuous analog of the dyadic partition of unity. Let χ ∈ C ∞(Rn+d) such that χ = 1 in a neighborhood N1 of I and χ vanishes outside a neighborhood N2 of N1. This implies χ satisfies the following constraint: for all compact set K ⊂ Rn, ∃(a, b) ∈ R 2 such that b > a > 0 and χ|(K×Rd)∩{|h| a} = 1, χ|(K×Rd)∩{|h| b} = 0. We find a convenient formula (inspired by [41] equation (8.5.1) p. 200 and [53] Formula (5.6) p. 28) for χ as an integral over a scale space indexed by λ ∈ (0, 1].

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Table of contents :

1 The extension of distributions
1.1 Introduction
1.2 Extension and renormalization
1.2.1 Notation, definitions
1.2.2 From bounded families to weakly homogeneous distributions.
1.3 Extension of distributions
1.3.1 Removable singularity theorems
1.4 Euler vector fields
1.4.1 Invariances
1.5 Appendix
2 A prelude to the microlocal extension
2.0.1 Introduction
2.1 Geometry in cotangent space
2.2 Geometric and metric topological properties
2.3 The counterterms are conormal distributions
2.4 Counterexample
2.5 Appendix
3 The microlocal extension. 
3.1 Dynamics in cotangent space
3.1.1 Definitions
3.2 Main theorem
3.2.1 Proof of the main theorem
3.2.2 The renormalized version of the main theorem
3.3 Appendix
3.3.1 Estimates for the product of a distribution and a smooth function
4 Stability of the microlocal extension
4.1 Notation, definitions
4.2 The product of distributions
4.2.1 Approximation and coverings
4.2.2 The product is bounded
4.2.3 The soft landing condition is stable by sum
4.3 The pull-back by diffeomorphisms
4.3.1 The symplectic geometry of the vector fields tangent to I and of the diffeomorphisms leaving I invariant
4.3.2 The pull-back is bounded
4.3.3 The action of Fourier integral operators
4.4 Appendix
5 The two point function h0|φ(x)φ(y)|0i
5.1 The flat case
5.1.1 The Poisson kernel, the Wick rotation and the subordination identity
5.1.2 Oscillatory integral
5.2 The holomorphic family
5.3 Pull-backs and the exponential map
5.3.1 The wave front set of the pull-back
5.3.2 The pull back of the phase function
5.4 The construction of the parametrix
5.4.1 The meaning of the asymptotic expansions
5.4.2 The invariance properties of the Beltrami operator g and of gradient vector fields
5.4.3 The function 􀀀 and the vectors ρ1, ρ2
5.4.4 The main theorem
6 The recursive construction of the renormalization
6.0.5 Introduction
6.1 Hopf algebra, T product and ⋆ product
6.1.1 The polynomial algebra of fields
6.1.2 Comparison of our formalism and the classical formalism from physics textbooks
6.1.3 Hopf algebra bundle over Mn
6.1.4 Deformation of the polynomial algebra of fields
6.1.5 The construction of ⋆
6.1.6 The associativity of ⋆
6.1.7 Wick’s property
6.1.8 Recovering Feynman graphs
6.2 The causality equation
6.2.1 Definition of the time-ordering operator
6.2.2 The Causality theorem
6.2.3 Consistency condition
6.3 The geometrical lemma for curved space time
6.4 The recursion
6.4.1 Polarized conic sets
6.4.2 Localization and enlarging the polarization
6.4.4 The scaling properties of translation invariant conicsets
6.4.5 Thickening sets
6.4.6 The μlocal properties of the two point function
6.4.7 Pull-back of good cones
6.4.8 The wave front set of the product tn is contained in a good cone 􀀀n
6.4.9 We define the extension tn and control WF(tn)
7 A conjecture by Bennequin. 
7.1 Parametrizing the wave front set of the extended distributions.
7.2 Morse families and Lagrangians
7.3 A conjectural formula
8 Anomalies and residues. 
8.1 Introduction
8.2 Currents and renormalisation
8.2.1 Notation and definitions
8.2.2 From Taylor polynomials to local counterterms via the notion of moments of a compactly supported distribution
8.2.3 The results of Chapter 1
8.3 Renormalization, local counterterms and residues
8.3.1 The ambiguities of the operator Rε and the moments of a distribution T
8.3.2 The geometric residues
8.3.3 Stability of geometric residues
9 The meromorphic regularization. 
9.1 Introduction
9.2 Fuchsian symbols
9.2.1 Constant coefficients Fuchsian operators
9.2.2 Fuchsian symbols currents
9.2.3 The solution of a variable coefficients Fuchsian equation is a Fuchsian symbol
9.2.4 Stability of the concept of approximate Fuchsians
9.3 Meromorphic regularization as a Mellin transform
9.3.1 The meromorphic extension
9.4 The Riesz regularization
9.5 The log and the 1-parameter RG


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