scaling degree of a distribution



The scaling degree or degree of divergence (Steinmann 71) or more generally the degree (Weinstein 78) of a distribution on Cartesian space n\mathbb{R}^n is a measure for how it behaves at the origin 0 n0 \in \mathbb{R}^n under rescaling xλxx \mapsto \lambda x of the canonical coordinates.

The concept controls the problem of extension of distributions from the complement n{0}\mathbb{R}^n \setminus \{0\} of the origin to all of n\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.



(rescaled distribution)

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

n s λ n x λx \array{ \mathbb{R}^n &\overset{s_\lambda}{\longrightarrow}& \mathbb{R}^n \\ x &\mapsto& \lambda x }

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

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

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

Explicitly, this is given by

𝒟( n) u λ, b λ nu,b(λ 1()). \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 nX \subset \mathbb{R}^n an open subset which is invariant under s λs_\lambda, the rescaling of a distribution u𝒟(X)u \in \mathcal{D}'(X) is is u λs λ *uu_\lambda \coloneqq s_\lambda^\ast u.


(scaling degree of a distribution)

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

  1. The scaling degree of uu is the infimum

    sd(u)inf{ω|limλ0λ ωu λ=0} 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 ω\omega such that the limit of the rescaled distribution λ ωu λ\lambda^\omega u_\lambda (def. ) vanishes. If there is no such ω\omega one sets sd(u)sd(u) \coloneqq \infty.

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

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



(scaling degree of non-singular distributions)

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

sd(u f)0. sd(u_f) \leq 0 \,.

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


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

λ ω(u f) λ,n =λ ωn nf(x)g(λ 1x)d nx =λ ω nf(λx)g(x)d nx, \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 ω>0\omega \gt 0, which shows the first claim.

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

f(x)=(iα i!) 1( αf(0))i(x i) α i+β n|β|=|α|+1i(x i) β ih β(x) 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 βh_{\beta} are smooth functions. Hence in this case

λ ω(u f) λ,n =λ ω+|α| n(iα i!) 1( αf(0))i(x i) α ib(x)d nx =+λ ω+|α|+1 ni(x i) β ih β(x)b(x)d nx. \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 λ0\lambda \to 0 precisely for ω>|α|\omega \gt - {\vert \alpha \vert}, which means that

sd(u f)=|α| sd(u_f) = -{\vert \alpha \vert}

in this case.


(scaling degree of derivatives of delta-distributions)

Let α n\alpha \in \mathbb{N}^n be a multi-index and αδ𝒟(X)\partial_\alpha \delta \in \mathcal{D}'(X) the corresponding partial derivatives of the delta distribution δ 0𝒟( n)\delta_0 \in \mathcal{D}'(\mathbb{R}^n) supported at 00. Then the degree of divergence (def. ) of αδ 0\partial_\alpha \delta_0 is the total order the derivatives

deg( αδ 0)=|α| deg\left( {\, \atop \,} \partial_\alpha\delta_0{\, \atop \,} \right) \;=\; {\vert \alpha \vert}

where |α|iα i{\vert \alpha\vert} \coloneqq \underset{i}{\sum} \alpha_i.


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

λ ω( αδ 0) λ,b =(1) |α|λ ωn( |α| α 1x 1 α nx nb(λ 1x)) |x=0 =(1) |α|λ ωn|α| |α| α 1x 1 α nx nb(0), \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 λ\lambda as long as ω>n+|α|\omega \gt n + {\vert \alpha\vert}. Hence sd( αδ 0)=n+|α|sd(\partial_{\alpha} \delta_0) = n + {\vert \alpha \vert}.


(scaling degree of Feynman propagator on Minkowski spacetime)


Δ F(x)=limϵ(0,)ϵ0+i(2π) p+1 e ik μx μk μk μ(mc) 2+iϵdk 0d pk \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+1n = p+1-dimensional Minkowski spacetime (this prop.). Its scaling degree is

sd(Δ F) =n2 =p1. \begin{aligned} sd(\Delta_{F}) & = n - 2 \\ & = p -1 \end{aligned} \,.

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


Regarding Δ F\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:

limλ0(λ ωΔ F(λx)) =limλ0(λ ωlimϵ(0,)ϵ0+i(2π) p+1 e ik μλx μk μk μ(mc) 2+iϵdk 0d pk) =limλ0(λ ωnlimϵ(0,)ϵ0+i(2π) p+1 e ik μλx μ(λ 2)k μk μ(mc) 2+iϵdk 0d pk) =limλ0(λ ωn+2limϵ(0,)ϵ0+i(2π) p+1 e ik μλx μk μk μ+iϵdk 0d pk). \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}



(basic properties of scaling degree of distributions)

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

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

    deg( αu)deg(u)+|α| deg(\partial_\alpha u) \;\leq\; deg(u) + {\vert \alpha\vert}
  2. For α n\alpha \in \mathbb{N}^n, the product of distributions with the smooth coordinate functions x αx^\alpha decreases scaling degree at least by |α|{\vert \alpha\vert }:

    deg(x αu)deg(u)|α| deg(x^\alpha u) \;\leq\; deg(u) - {\vert \alpha\vert}
  3. Under tensor product of distributions their scaling degrees add:

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

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

  4. deg(fu)deg(u)kdeg(f u) \leq deg(u) - k for fC (X)f \in C^\infty(X) and f (α)(0)=0f^{(\alpha)}(0) = 0 for |α|k1{\vert \alpha\vert} \leq k-1;

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


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

For the fourth…


(scaling degree of product distribution)

Let u,v𝒟( n)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)< deg(u), deg(v) \lt \infty
  2. their product of distributions is well-defined

    uv𝒟( n) 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(uv)deg(u)+deg(v). deg(u v) \;\leq\; deg(u) + deg(v) \,.

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


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

Last revised on March 28, 2018 at 11:20:37. See the history of this page for a list of all contributions to it.