nLab extension of distributions

Idea

Given a suitable subspace inclusion $X \hookrightarrow \hat X$ and a distribution $u$ on $X$, then an extension of $u$ to $\hat X$ is a distribution $\hat u$ on $\hat X$ whose restriction of distributions to $X$ coincides with $u$. If $u$ comes from an ordinary smooth function or else if $u$ is properly understood as a generalized function, then this corresponds to an ordinary extension

$$\array{ X &\overset{u}{\longrightarrow}& \mathbb{R} \\ \downarrow & \nearrow_{\mathrlap{\hat u}} \\ \hat X }$$

Regarding that we the distributions are not the maps from the underlying space we need to replace pre-composition $X\to\tilde{X}$ by the appropriate pullback of distributions always defined say if $X\to\tilde{X}$ is a submersion; in the case of an open embedding (see this example) this operation is the restriction of distributions, the operation dual to the operation of extension by zero of test functions.

Definition

For an inclusion of two open sets on a manifold $U\subset V$ there is an operator of extension by zero $E_{V U}: C^\infty_0(U)\to C^\infty_0(V)$ where $E_{V U}(f)(x) = f(x)$ if $x\in U$ and $E_{U V}(f)(x) = 0$ otherwise. The restriction of distributions $\rho_{U V} : \mathcal{D}'(V)\to \mathcal{D}'(U)$ is then defined by

$$\langle \rho_{U V}(\phi), f\rangle = \langle \phi, E_{V U} f\rangle, \,\,\,\,\,\,\,f\in C^\infty_0(U),\,\phi\in \mathcal{D}'(V).$$

(this example).

Now the diagram in the idea section makes sense in the following way: for an open embedding $X\hookrightarrow\hat{X}$, $\hat{u}\in\mathcal{D}'(\hat{X})$ extends $u\in \mathcal{D}'(X)$ if $\rho_{X\hat{X}}(\hat{u}) = u$.

Definition

(extension of distributions)

Let $X \overset{\iota}{\subset} \hat X$ be an inclusion of open subsets of some Cartesian space. This induces the operation of restriction of distributions

$$\mathcal{D}'(\hat X) \overset{\iota^\ast}{\longrightarrow} \mathcal{D}'(X) \,.$$

Given a distribution $u \in \mathcal{D}'(X)$, then an extension of $u$ to $\hat X$ is a distribution $\hat u \in \mathcal{D}'(\hat X)$ such that

$$\iota^\ast \hat u \;=\; u \,.$$

Properties

Point-extensions

Of particular interest is the special case of extension of distributions to a single point $p \in \hat X$, hence where $X = \hat X \setminus \{p\}$ is the complement of that point.

Contrary to ordinary smooth functions, distributions (generalized functions) in general have more than one extension to a point, where the freedom is in choosing a distribution with point support at that point, hence some derivative of the delta distribution supported at that point.

However, if one requires that the scaling degree of the extended distribution at the given point is compatible with that of the original distribution, then there is only a finite set of possible extensions, parameterized by the coefficients of a finite number of derivatives of the delta-distribution (prop. below).

In the construction of perturbative quantum field theories via causal perturbation theory, where the (“operator-valued”) distributions in questions are time-ordered product coefficients in the scattering matrix and where the point being extended to corresponds to the point where an interaction happens, this finite set of choices is identified with the ("re"-)normalization freedom.

We discuss specifically the space of solutions of extending a distribution on the complement $\mathbb{R}^n \setminus \{0\}$ of the origin inside a Cartesian space to the full space $\mathbb{R}^n$.

Proposition

(unique extension of distributions with negative degree of divergence)

For $n \in \mathbb{N}$, let $u \in \mathcal{D}'(\mathbb{R}^n \setminus \{0\})$ be a distribution on the complement of the origin, with negative degree of divergence at the origin

$$deg(u) \lt 0 \,.$$

Then $u$ has a unique extension of distributions $\hat u \in \mathcal{D}'(\mathbb{R}^n)$ to the origin with the same degree of divergence

$$deg(\hat u) = deg(u) \,.$$

(Brunetti-Fredenhagen 00, theorem 5.2, Dütsch 18, theorem 3.35 a))

Proof

Regarding uniqueness:

Suppose $\hat u$ and ${\hat u}^\prime$ are two extensions of $u$ with $deg(\hat u) = deg({\hat u}^\prime)$. Both being extensions of a distribution defined on $\mathbb{R}^n \setminus \{0\}$, this difference has support at the origin $\{0\} \subset \mathbb{R}^n$. By this prop. this implies that it is a linear combination of derivatives of the delta distribution supported at the origin:

$${\hat u}^\prime - \hat u \;=\; \underset{ {\alpha \in \mathbb{N}^n} }{\sum} c^\alpha \partial_\alpha \delta_0$$

for constants $c^\alpha \in \mathbb{C}$. But by this example the degree of divergence of these point-supported distributions is non-negative

$$deg( \partial_\alpha \delta_0) = {\vert \alpha\vert} \geq 0 \,.$$

This implies that $c^\alpha = 0$ for all $\alpha$, hence that the two extensions coincide.

Regarding existence:

Let

$$b \in C^\infty_{cp}(\mathbb{R}^n)$$

be a bump function which is $\leq 1$ and constant on 1 over a neighbourhood of the origin. Write

$$\chi \coloneqq 1 - b \;\in\; C^\infty(\mathbb{R}^n)$$

graphics grabbed from Dütsch 18, p. 108

and for $\lambda \in (0,\infty)$ a positive real number, write

$$\chi_\lambda(x) \coloneqq \chi(\lambda x) \,.$$

Since the product $\chi_\lambda u$ has support of a distribution on a complement of a neighbourhood of the origin, we may extend it by zero to a distribution on all of $\mathbb{R}^n$, which we will denote by the same symbols:

$$\chi_\lambda u \in \mathcal{D}'(\mathbb{R}^n) \,.$$

By construction $\chi_\lambda u$ coincides with $u$ away from a neighbourhood of the origin, which moreover becomes arbitrarily small as $\lambda$ increases. This means that if the following limit exists

$$\hat u \;\coloneqq\; \underset{\lambda \to \infty}{\lim} \chi_\lambda u$$

then it is an extension of $u$.

To see that the limit exists, it is sufficient to observe that we have a Cauchy sequence, hence that for all $b\in C^\infty_{cp}(\mathbb{R}^n)$ the difference

$$(\chi_{n+1} u - \chi_n u)(b) \;=\; u(b)( \chi_{n+1} + \chi_n )$$

becomes arbitrarily small.

It remains to see that the unique extension $\hat u$ thus established has the same scaling degree as $u$. This is shown in (Brunetti-Fredenhagen 00, p. 24).

Proposition

(space of point-extensions of distributions)

For $n \in \mathbb{N}$, let $u \in \mathcal{D}'(\mathbb{R}^n \setminus \{0\})$ be a distribution of degree of divergence $deg(u) \lt \infty$.

Then $u$ does admit at least one extension (def. ) to a distribution $\hat u \in \mathcal{D}'(\mathbb{R}^n)$, and every choice of extension has the same degree of divergence as $u$

$$deg(\hat u) = deg(u) \,.$$

Moreover, any two such extensions $\hat u$ and ${\hat u}^\prime$ differ by a linear combination of partial derivatives of distributions of order $\leq deg(u)$ of the delta distribution $\delta_0$ supported at the origin:

$${\hat u}^\prime - \hat u \;=\; \underset{ { \alpha \in \mathbb{N}^n } \atop { {\vert \alpha\vert} \leq deg(u) } }{\sum} q^\alpha \partial_\alpha \delta_0 \,,$$

for a finite number of constants $q^\alpha \in \mathbb{C}$.

This is essentially (Hörmander 90, thm. 3.2.4). We follow (Brunetti-Fredenhagen 00, theorem 5.3), which was inspired by (Epstein-Glaser 73, section 5). Review of this approach is in (Dütsch 18, theorem 3.35 (b)), see also remark below.

Proof

For $f \in C^\infty(\mathbb{R}^n)$ a smooth function, and $\rho \in \mathbb{N}$, we say that $f$ vanishes to order $\rho$ at the origin if all partial derivatives with multi-index $\alpha \in \mathbb{N}^n$ of total order ${\vert \alpha\vert} \leq \rho$ vanish at the origin:

$$\partial_\alpha f (0) = 0 \phantom{AAA} {\vert \alpha\vert} \leq \rho \,.$$

By Hadamard's lemma, such a function may be written in the form

(1)$$f(x) \;=\; \underset{ {\alpha \in \mathbb{N}^n} \atop { {\vert \alpha \vert} = \rho + 1 } }{\sum} x^\alpha r_\alpha(x)$$

for smooth functions $r_\alpha \in C^\infty_{cp}(\mathbb{R}^n)$.

Write

$$\mathcal{D}_\rho(\mathbb{R}^n) \hookrightarrow \mathcal{D}(\mathbb{R}^n) \coloneqq C^\infty_{cp}(\mathbb{R}^n)$$

for the subspace of that of all bump functions on those that vanish to order $\rho$ at the origin.

By definition this is equivalently the joint kernel of the partial derivatives of distributions of order ${\vert \alpha\vert}$ of the delta distribution $\delta_0$ supported at the origin:

$$b \in \mathcal{D}_\rho(\mathbb{R}^n) \phantom{AA} \Leftrightarrow \phantom{AA} \underset{ {\alpha \in \mathbb{N}^n} \atop { {\vert \alpha\vert} \leq \rho } } {\forall} \left\langle \partial_\alpha \delta_0, b \right\rangle = 0 \,.$$

Therefore every continuous linear projection

$$p_\rho \;\colon\; \mathcal{D}(\mathbb{R}^n) \longrightarrow \mathcal{D}_\rho(\mathbb{R}^n)$$

may be obtained from a choice of dual basis to the $\{\partial_\alpha \delta_0\}$, hence a choice of smooth functions

$$\left\{ w^\beta \in C^\infty_{cp}(\mathbb{R}^n) \right\}_{ { \beta \in \mathbb{N}^n } \atop { {\vert \beta\vert} \leq \rho } }$$

such that

$$\left\langle \partial_\alpha \delta_0 \,,\, w^\beta \right\rangle \;=\; \delta_\alpha^\beta \phantom{AAA} \Leftrightarrow \phantom{AAA} \partial_\alpha w^\beta(0) \;=\; \delta_\alpha^\beta \phantom{AAAA} \text{for}\, {\vert \alpha\vert} \leq \rho \,,$$

by setting

(2)$$p_\rho \;\coloneqq\; id \;-\; \left\langle \underset{ {\alpha \in \mathbb{N}^n} \atop { {\vert \alpha\vert} \leq \rho } }{\sum} w^\alpha \partial_\alpha \delta_0 \,,\, (-) \right\rangle \,,$$

hence

$$p_\rho \;\colon\; b \mapsto b - \underset{ {\alpha \in \mathbb{N}^n} \atop { {\vert \alpha\vert} \leq \rho } }{\sum} (-1)^{{\vert \alpha\vert}} w^\alpha \partial_\alpha b(0) \,.$$

Together with Hadamard's lemma in the form (1) this means that every $b \in \mathcal{D}(\mathbb{R}^n)$ is decomposed as

(3)$$\begin{aligned} b(x) & = p_\rho(b)(x) \;+\; (id - p_\rho)(b)(x) \\ & = \underset{ { \alpha \in \mathbb{N}^n } \atop { {\vert \alpha \vert} = \rho + 1 } }{\sum} x^\alpha r_\alpha(x) \;+\; \underset{ { \alpha \in \mathbb{N}^n } \atop { {\vert \alpha \vert} \leq \rho } }{\sum} (-1)^{{\vert \alpha \vert}} w^\alpha \partial_\alpha b(0) \end{aligned}$$

Now let

$$\rho \;\coloneqq\; deg(u) \,.$$

Observe that (by this prop.) the degree of divergence of the product of distributions $x^\alpha u$ with ${\vert \alpha\vert} = \rho + 1$ is negative

$$\begin{aligned} deg\left( x^\alpha u \right) & = \rho - {\vert \alpha \vert} \leq -1 \end{aligned}$$

Therefore prop. says that each $x^\alpha u$ for ${\vert \alpha\vert} = \rho + 1$ has a unique extension $\widehat{ x^\alpha u}$ to the origin. Accordingly the composition $u \circ p_\rho$ has a unique extension, by (3):

$$\begin{aligned} \left\langle \hat u \,,\, b \right\rangle & = \left\langle \hat u , p_\rho(b) \right\rangle + \left\langle \hat u , (id - p_\rho)(b) \right\rangle \\ & = \underset{ { \alpha \in \mathbb{N}^n } \atop { {\vert \alpha \vert} = \rho + 1 } }{\sum} \underset{ \text{unique} }{ \underbrace{ \left\langle \widehat{x^\alpha u} \,,\, r_\alpha \right\rangle } } \;+\; \underset{ { \alpha \in \mathbb{N}^n } \atop { {\vert \alpha\vert} \leq \rho } }{\sum} \underset{ \text{choice} }{ \underbrace{ \langle \hat u \,,\, w^\alpha \rangle } } \left\langle \partial_\alpha \delta_0 \,,\, b \right\rangle \end{aligned}$$

That says that $\hat u$ is of the form

$$\hat u \;=\; \underset{ \text{unique} }{ \underbrace{ \widehat{ u \circ p_\rho } } } + \underset{ { \alpha \in \mathbb{N}^n } \atop { {\vert \alpha\vert} \leq \rho } }{\sum} c^\alpha \, \partial_\alpha \delta_0$$

for a finite number of constants $c^\alpha \in \mathbb{C}$.

Notice that for any extension $\hat u$ the exact value of the $c^\alpha$ here depends on the arbitrary choice of dual basis $\{w^\alpha\}$ used for this construction. But the uniqueness of the first summand means that for any two choices of extensions $\hat u$ and ${\hat u}^\prime$, their difference is of the form

$${\hat u}^\prime - \hat u \;=\; \underset{ { \alpha \in \mathbb{N}^n } \atop { {\vert \alpha\vert} \leq \rho } }{\sum} ( (c')^\alpha - c^\alpha ) \, \partial_\alpha \delta_0 \,,$$

where the constants $q^\alpha \coloneqq ( (c')^\alpha - c^\alpha ) \in \mathbb{C}$ are independent of any choices.

It remains to see that all these $\hat u$ in fact have the same degree of divergence as $u$.

By this example the degree of divergence of the point-supported distributions on the right is $deg(\partial_\alpha \delta_0) = {\vert \alpha\vert} \leq \rho$.

Therefore to conclude it is now sufficient to show that

$$deg\left( \widehat{ u \circ p_\rho } \right) \;=\; \rho \,.$$

This is shown in (Brunetti-Fredenhagen 00, p. 25).

Remark

(“W-extensions”)

Since in Brunetti-Fredenhagen 00, (38) the projectors (2) are denoted “$W$”, the construction of extensions of distributions via the proof of prop. has come to be called “W-extensions” (e.g Dütsch 18).

Examples

Powers of Feynman propagators

Extension of powers of Feynman propagators on globally hyperbolic spacetimes to the diagonal are worked out explicitly in (Hollands 07, section 3.4).

References

The argument for the characterization of the point extension of distributions goes back to

• Henri Epstein and Vladimir Glaser, section 5 of The role of locality in perturbation theory, Annales Poincaré Phys. Theor. A 19 (1973) 211 (Numdam)

thereby laying the foundation for causal perturbation theory. A textbook account in functional analysis is in

• Lars Hörmander, theorem 3.2.4 of The Analysis of Linear Partial Differential Operators I (Springer, 1990, 2nd ed.)

A more concise formulation and proof is due to

• Romeo Brunetti, Klaus Fredenhagen, section 5.2 of Microlocal analysis and interacting quantum field theories: Renormalization on Physical Backgrounds, Commun. Math. Phys. 208 : 623-661, 2000 (math-ph/9903028)

reviewed in

• Michael Dütsch, theorem 3.35 of From classical field theory to perturbative quantum field theory, 2018

Exposition of the application to renormalization of Feynman diagrams is in

• Christian Brouder, Multiplication of distributions, 2010 (pdf)

The refinement to the point-extension problem for distributions in the solution space of a given system of differential equations is discussed in

• Dorothea Bahns, Michał Wrochna, On-shell extension of distributions, arXiv:1210.5448

Examples and applications to renormalization in perturbative quantum field theory are discussed for instance in

• Stefan Hollands, sections 3.3 and 3.4 of Renormalized Quantum Yang-Mills Fields in Curved Spacetime, Rev. Math. Phys. 20:1033-1172, 2008 (arXiv:0705.3340)

For more on extension of distributions in renormalization see the references at causal perturbation theory and locally covariant perturbative quantum field theory.