nLab scaling degree of a distribution

Idea

The scaling degree or degree of divergence (Steinmann 71) or more generally the degree (Weinstein 78) of a distribution on Cartesian space $\mathbb{R}^n$ is a measure for how it behaves at the origin $0 \in \mathbb{R}^n$ under rescaling $x \mapsto \lambda x$ of the canonical coordinates.

The concept controls the problem of extension of distributions from the complement $\mathbb{R}^n \setminus \{0\}$ of the origin to all of $\mathbb{R}^n$. Such extensions are important notably in the construction of perturbative quantum field theories via causal perturbation theory, where the freedom in the choice of such extensions models the ("re"-)normalization freedom (“counter-terms”) in the construction.

Definition

Definition

(rescaled distribution)

Let $n \in \mathbb{N}$. For $\lambda \in (0,\infty) \subset \mathbb{R}$ a positive real number write

$$\array{ \mathbb{R}^n &\overset{s_\lambda}{\longrightarrow}& \mathbb{R}^n \\ x &\mapsto& \lambda x }$$

for the diffeomorphism given by multiplication with $\lambda$, using the canonical real vector space-structure of $\mathbb{R}^n$.

Then for $u \in \mathcal{D}'(\mathbb{R}^n)$ a distribution on the Cartesian space $\mathbb{R}^n$ the rescaled distribution is the pullback of $u$ along $m_\lambda$

$$u_\lambda \coloneqq s_\lambda^\ast u \;\in\; \mathcal{D}'(\mathbb{R}^n) \,.$$

Explicitly, this is given by

$$\array{ \mathcal{D}(\mathbb{R}^n) &\overset{ \langle u_\lambda, - \rangle}{\longrightarrow}& \mathbb{R} \\ b &\mapsto& \lambda^{-n} \langle u , b(\lambda^{-1}\cdot (-))\rangle } \,.$$

Similarly for $X \subset \mathbb{R}^n$ an open subset which is invariant under $s_\lambda$, the rescaling of a distribution $u \in \mathcal{D}'(X)$ is is $u_\lambda \coloneqq s_\lambda^\ast u$.

Definition

(scaling degree of a distribution)

Let $n \in \mathbb{N}$ and let $X \subset \mathbb{R}^n$ be an open subset of Cartesian space which is invariant under rescaling $s_\lambda$ (def. ) for all $\lambda \in (0,\infty)$, and let $u \in \mathcal{D}'(X)$ be a distribution on this subset. Then

1. The scaling degree of $u$ is the infimum

   $$sd(u) \;\coloneqq\; inf \left\{ \omega \in \mathbb{R} \;\vert\; \underset{\lambda \to 0}{\lim} \lambda^\omega u_\lambda = 0 \right\}$$

   of the set of real numbers $\omega$ such that the limit of the rescaled distribution $\lambda^\omega u_\lambda$ (def. ) vanishes. If there is no such $\omega$ one sets $sd(u) \coloneqq \infty$.

2. The degree of divergence of $u$ is the difference of the scaling degree by the dimension of the underlying space:

$$deg(u) \coloneqq sd(u) - n \,.$$

Examples

Example

(scaling degree of non-singular distributions)

If $u = u_f$ is a non-singular distribution given by bump function $f \in C^\infty(X) \subset \mathcal{D}'(X)$, then its scaling degree (def. ) is non-positive

$$sd(u_f) \leq 0 \,.$$

Specifically if the first non-vanishing partial derivative $\partial_\alpha f(0)$ of $f$ at 0 occurs at order ${\vert \alpha\vert} \in \mathbb{N}$, then the scaling degree of $u_f$ is $-{\vert \alpha\vert}$.

Proof

By definition we have for $b \in C^\infty_{cp}(\mathbb{R}^n)$ any bump function that

$$\begin{aligned} \left\langle \lambda^{\omega} (u_f)_\lambda, n \right\rangle & = \lambda^{\omega-n} \underset{\mathbb{R}^n}{\int} f(x) g(\lambda^{-1} x) d^n x \\ & = \lambda^{\omega} \underset{\mathbb{R}^n}{\int} f(\lambda x) g(x) d^n x \end{aligned} \,,$$

where in last line we applied change of integration variables.

The limit of this expression is clearly zero for all $\omega \gt 0$, which shows the first claim.

If moreover the first non-vanishing partial derivative of $f$ occurs at order ${\vert \alpha \vert} = k$, then Hadamard's lemma says that $f$ is of the form

$$f(x) \;=\; \left( \underset{i}{\prod} \alpha_i ! \right)^{-1} (\partial_\alpha f(0)) \underset{i}{\prod} (x^i)^{\alpha_i} + \underset{ {\beta \in \mathbb{N}^n} \atop { {\vert \beta\vert} = {\vert \alpha \vert} + 1 } }{\sum} \underset{i}{\prod} (x^i)^{\beta_i} h_{\beta}(x)$$

where the $h_{\beta}$ are smooth functions. Hence in this case

$$\begin{aligned} \left\langle \lambda^{\omega} (u_f)_\lambda, n \right\rangle & = \lambda^{\omega + {\vert \alpha\vert }} \underset{\mathbb{R}^n}{\int} \left( \underset{i}{\prod} \alpha_i ! \right)^{-1} (\partial_\alpha f(0)) \underset{i}{\prod} (x^i)^{\alpha_i} b(x) d^n x \\ & \phantom{=} + \lambda^{\omega + {\vert \alpha\vert} + 1} \underset{\mathbb{R}^n}{\int} \underset{i}{\prod} (x^i)^{\beta_i} h_{\beta}(x) b(x) d^n x \end{aligned} \,.$$

This makes manifest that the expression goes to zero with $\lambda \to 0$ precisely for $\omega \gt - {\vert \alpha \vert}$, which means that

$$sd(u_f) = -{\vert \alpha \vert}$$

in this case.

Example

(scaling degree of derivatives of delta-distributions)

Let $\alpha \in \mathbb{N}^n$ be a multi-index and $\partial_\alpha \delta \in \mathcal{D}'(X)$ the corresponding partial derivatives of the delta distribution $\delta_0 \in \mathcal{D}'(\mathbb{R}^n)$ supported at $0$. Then the degree of divergence (def. ) of $\partial_\alpha \delta_0$ is the total order the derivatives

$$deg\left( {\, \atop \,} \partial_\alpha\delta_0{\, \atop \,} \right) \;=\; {\vert \alpha \vert}$$

where ${\vert \alpha\vert} \coloneqq \underset{i}{\sum} \alpha_i$.

Proof

By definition we have for $b \in C^\infty_{cp}(\mathbb{R}^n)$ any bump function that

$$\begin{aligned} \left\langle \lambda^\omega (\partial_\alpha \delta_0)_\lambda, b \right\rangle & = (-1)^{{\vert \alpha \vert}} \lambda^{\omega-n} \left( \frac{ \partial^{{\vert \alpha \vert}} }{ \partial^{\alpha_1} x^1 \cdots \partial^{\alpha_n}x^n } b(\lambda^{-1}x) \right)_{\vert x = 0} \\ & = (-1)^{{\vert \alpha \vert}} \lambda^{\omega - n - {\vert \alpha\vert}} \frac{ \partial^{{\vert \alpha \vert}} }{ \partial^{\alpha_1} x^1 \cdots \partial^{\alpha_n}x^n } b(0) \end{aligned} \,,$$

where in the last step we used the chain rule of differentiation. It is clear that this goes to zero with $\lambda$ as long as $\omega \gt n + {\vert \alpha\vert}$. Hence $sd(\partial_{\alpha} \delta_0) = n + {\vert \alpha \vert}$.

Example

(scaling degree of Feynman propagator on Minkowski spacetime)

Let

$$\Delta_F(x) \;=\; \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \frac{+i}{(2\pi)^{p+1}} \int \int_{-\infty}^\infty \frac{ e^{i k_\mu x^\mu} }{ - k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 + i \epsilon } \, d k_0 \, d^p \vec k$$

be the Feynman propagator for the massive free real scalar field on $n = p+1$-dimensional Minkowski spacetime (this prop.). Its scaling degree is

$$\begin{aligned} sd(\Delta_{F}) & = n - 2 \\ & = p -1 \end{aligned} \,.$$

(Brunetti-Fredenhagen 00, example 3 on p. 22)

Proof

Regarding $\Delta_F$ as a generalized function via the given Fourier-transform expression, we find by change of integration variables in the Fourier integral that in the scaling limit the Feynman propagator becomes that for vannishing mass, which scales homogeneously:

$$\begin{aligned} \underset{\lambda \to 0}{\lim} \left( \lambda^\omega \; \Delta_F(\lambda x) \right) & = \underset{\lambda \to 0}{\lim} \left( \lambda^{\omega} \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \frac{+i}{(2\pi)^{p+1}} \int \int_{-\infty}^\infty \frac{ e^{i k_\mu \lambda x^\mu} }{ - k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 + i \epsilon } \, d k_0 \, d^p \vec k \right) \\ & = \underset{\lambda \to 0}{\lim} \left( \lambda^{\omega-n} \; \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \frac{+i}{(2\pi)^{p+1}} \int \int_{-\infty}^\infty \frac{ e^{i k_\mu \lambda x^\mu} }{ - (\lambda^{-2}) k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 + i \epsilon } \, d k_0 \, d^p \vec k \right) \\ & = \underset{\lambda \to 0}{\lim} \left( \lambda^{\omega-n + 2 } \; \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \frac{+i}{(2\pi)^{p+1}} \int \int_{-\infty}^\infty \frac{ e^{i k_\mu \lambda x^\mu} }{ - k_\mu k^\mu + i \epsilon } \, d k_0 \, d^p \vec k \right) \,. \end{aligned}$$

Properties

Proposition

(basic properties of scaling degree of distributions)

Let $X \subset \mathbb{R}^n$ and $u \in \mathcal{D}'(X)$ be a distribution as in def. , such that its scaling degree is finite: $sd(u) \lt \infty$ (def. ). Then

1. For $\alpha \in \mathbb{N}^n$, the partial derivative of distributions $\partial_\alpha$ increases scaling degree at most by ${\vert \alpha\vert }$:

   $$deg(\partial_\alpha u) \;\leq\; deg(u) + {\vert \alpha\vert}$$

2. For $\alpha \in \mathbb{N}^n$, the product of distributions with the smooth coordinate functions $x^\alpha$ decreases scaling degree at least by ${\vert \alpha\vert }$:

   $$deg(x^\alpha u) \;\leq\; deg(u) - {\vert \alpha\vert}$$

3. Under tensor product of distributions their scaling degrees add:

   $$sd(u \otimes v) \leq sd(u) + sd(v)$$

   for $v \in \mathcal{D}'(Y)$ another distribution on $Y \subset \mathbb{R}^{n'}$;

4. $deg(f u) \leq deg(u) - k$ for $f \in C^\infty(X)$ and $f^{(\alpha)}(0) = 0$ for ${\vert \alpha\vert} \leq k-1$;

(Brunetti-Fredenhagen 00, lemma 5.1, Dütsch 18, exercise 3.34)

Proof

The first three statements follow with manipulations as in example and example .

For the fourth…

Proposition

(scaling degree of product distribution)

Let $u,v \in \mathcal{D}'(\mathbb{R}^n)$ be two distributions such that

1. both have finite degree of divergence (def. )

   $$deg(u), deg(v) \lt \infty$$

2. their product of distributions is well-defined

   $$u v \in \mathcal{D}'(\mathbb{R}^n)$$

   (in that their wave front sets satisfy Hörmander's criterion)

then the product distribution has degree of divergence bounded by the sum of the separate degrees:

$$deg(u v) \;\leq\; deg(u) + deg(v) \,.$$

(Brunetti-Fredenhagen 00, special case of lemma 6.6)

Related concepts

• support of a distribution

• order of a distribution

References

The concept of scaling degree is due to

• O. Steinmann, Perturbation Expansions in Axiomatic Field Theory, volume 11 of Lecture Notes in Physics, Springer, Berlin Springer Verlag, 1971.

and the more general concept of degree due to

• Alan Weinstein, The order and symbol of a distribution, Trans. Amer. Math. Soc. 241, 1–54 (1978).

Review and further developments in the context of ("re"-)normalization in causal perturbation theory/pAQFT is in

• Romeo Brunetti, Klaus Fredenhagen, section 5.1 of Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds, Commun. Math. Phys. 208 : 623-661, 2000 (math-ph/9903028)

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

• Michael Dütsch, def. 3.32 of From classical field theory to perturbative quantum field theory, 2018