Products, Scaling Degree, and Extensions of Singular Distributions

Distributions do not form an algebra. A product exists only when an independent construction licenses it—for example, multiplication by a smooth function, an integrable product of regular representatives, or a diagonal pullback satisfying a microlocal condition. If an expression is already a distribution away from a coincidence point and has finite scaling degree, it can be extended across that point without worsening its scaling degree. The extension is unique below the dimensional threshold; at and above the threshold, its finite ambiguity is a sum of delta derivatives.

Required background. Test-Function Spaces, Distributions, Support, and Convergence supplies test-space duality, distributional convergence, and support.

Helpful background. Limits, Completeness, and Modes of Convergence supplies the topology and limit-interchange discipline used to interpret regularized extensions.

The page develops the reusable extension method, not the Epstein–Glaser induction that uses it and not the physical choice of counterterms or renormalization conditions.

Why two distributions cannot simply be multiplied

For $a\in C^\infty(\Omega)$ and $u\in\mathcal D'(\Omega)$, smooth multiplication is defined by

$$\langle au,\varphi\rangle = \langle u,a\varphi\rangle.$$

The test function $a\varphi$ remains smooth and compactly supported. If $f,g\in L^1_{\mathrm{loc}}(\Omega)$ and their ordinary product also satisfies $fg\in L^1_{\mathrm{loc}}(\Omega)$, then

$$\langle u_{fg},\varphi\rangle = \int_\Omega f(x)g(x)\varphi(x)\,\mathrm d^d x$$

defines a regular distribution. The last hypothesis is not automatic. On $\mathbb R$, both

$$f(x)=g(x)=|x|^{-1/2}$$

are locally integrable, but $fg=|x|^{-1}$ is not locally integrable at the origin.

There is also a useful elementary sufficient condition: if the singular supports of $u$ and $v$ are disjoint, a partition of unity can localize every point to a region where at least one factor is smooth. The local products then agree on overlaps and define $uv$. This does not cover coincident singularities.

Tensor product is automatic; diagonal pullback is conditional

The external product

$$u\boxtimes v \in \mathcal D'(\Omega\times\Omega)$$

always exists. A product on the original space would be its pullback along the diagonal embedding

$$\iota:\Omega\longrightarrow\Omega\times\Omega, \qquad \iota(x)=(x,x):$$ $$\boxed{ uv = \iota^*(u\boxtimes v). }$$

The box is a characterization, not a guarantee that the right-hand side exists. Pullback along an embedding is conditional. In the wavefront-set criterion, there must be no covectors

$$(x,\xi)\in\operatorname{WF}(u), \qquad (x,\eta)\in\operatorname{WF}(v)$$

with $\xi+\eta=0$. The notation is only a preview here; Singular Support and Wavefront Sets develops the criterion and its hypotheses. Hörmander 2003, Theorem 8.2.10 is the structural source for this conditional product.

Mollification does not make squaring continuous

Let $\rho\in C_c^\infty(\mathbb R)$ be real with $\int\rho=1$, and set

$$\delta_\varepsilon(x) = \frac{1}{\varepsilon} \rho\!\left(\frac{x}{\varepsilon}\right).$$

Although $\delta_\varepsilon\to\delta_0$ in $\mathcal D'$, the squares obey

$$\begin{aligned} \left\langle \delta_\varepsilon^2,\varphi \right\rangle &= \frac{1}{\varepsilon} \int_{\mathbb R} \rho(y)^2\varphi(\varepsilon y)\,\mathrm dy \\ &= \frac{\varphi(0)}{\varepsilon} \int_{\mathbb R}\rho(y)^2\,\mathrm dy + O(1). \end{aligned}$$

The coefficient diverges and depends on the mollifier. Thus convergence of two regularized factors does not imply convergence of their products, and the symbol $\delta_0^2$ has not been defined. Dyatlov 2022, § 3.2, Remark 3.3, PDF uses this example to show why arbitrary distributional multiplication is unavailable.

Scaling degree measures the short-distance strength

For $\lambda>0$ and $\varphi\in\mathcal D(\mathbb R^d)$, define

$$\varphi_\lambda(x) = \lambda^{-d}\varphi(x/\lambda), \qquad \langle t_\lambda,\varphi\rangle = \langle t,\varphi_\lambda\rangle.$$

If $t$ is represented by a function, then $t_\lambda$ is represented by $x\mapsto t(\lambda x)$. The scaling degree of $t$ at the origin is

$$\boxed{ \operatorname{sd}_0(t) = \inf \left\{ \omega\in\mathbb R: \lambda^\omega t_\lambda \longrightarrow0 \text{ in }\mathcal D' \text{ as }\lambda\downarrow0 \right\}. }$$

The same definition applies to $t^\circ\in\mathcal D'(\mathbb R^d\setminus\{0\})$: the punctured domain is invariant under positive dilations, so each scaled pairing remains well-defined.

The normalization is fixed by the following examples:

$$\begin{aligned} \operatorname{sd}_0(f)&\leq0 &&\text{for }f\in C^\infty, \\ \operatorname{sd}_0(\delta_0)&=d, \\ \operatorname{sd}_0(\partial^\alpha\delta_0)&=d+|\alpha|, \\ \operatorname{sd}_0(|x|^{-a})&=a &&\text{on }\mathbb R^d\setminus\{0\}. \end{aligned}$$

If $f(0)\neq0$, the first inequality is an equality. Vanishing at the origin can lower the scaling degree. Differentiation and coordinate multiplication satisfy bounds, not universal equalities:

$$\begin{aligned} \operatorname{sd}_0(\partial^\alpha t) &\leq \operatorname{sd}_0(t)+|\alpha|, \\ \operatorname{sd}_0(x^\alpha t) &\leq \operatorname{sd}_0(t)-|\alpha|, \\ \operatorname{sd}_0(ft) &\leq \operatorname{sd}_0(t), \qquad f\in C^\infty. \end{aligned}$$

Scaling degree is unchanged under a local diffeomorphism $\kappa$ satisfying $\kappa(0)=0$ and having invertible derivative at the reference point. It is therefore local short-distance information rather than an artifact of one linear chart; Brunetti and Fredenhagen, §6, especially Propositions 6.5 and 6.8, give the corresponding invariant formulation. Brunetti and Fredenhagen 2000, § 5.1, PDF gives the pointwise definition and proves the displayed calculus properties in Lemma 5.1.

Scaling degree does not decide whether a proposed product exists away from the origin. It becomes relevant only after there is a genuine distribution on the punctured domain.

The point-extension theorem

Let

$$t^\circ \in \mathcal D'(\mathbb R^d\setminus\{0\}), \qquad \omega = \operatorname{sd}_0(t^\circ) <\infty.$$

Then there is an extension $t\in\mathcal D'(\mathbb R^d)$ such that

$$t|_{\mathbb R^d\setminus\{0\}} = t^\circ, \qquad \operatorname{sd}_0(t) = \omega.$$

The uniqueness statement has a dimensional threshold:

$$\boxed{ \omega<d \quad\Longrightarrow\quad \text{the scaling-degree-preserving extension is unique.} }$$

When $\omega\geq d$, define the nonnegative integer

$$N = \left\lfloor\omega-d\right\rfloor.$$

Any two extensions with scaling degree $\omega$ differ by

$$\boxed{ t'-t = \sum_{|\alpha|\leq N} c_\alpha\,\partial^\alpha\delta_0. }$$

There are

$$\binom{d+N}{N}$$

multi-indices in this sum before symmetry or other conditions are imposed. The theorem does not say that every coefficient survives those additional conditions; it says that scaling degree alone cannot determine them.

The ambiguity follows from three structural facts:

1. The difference $t'-t$ vanishes away from the origin, so its support is contained in $\{0\}$.
2. Every distribution supported at one point is a finite sum of delta derivatives.
3. Since $\operatorname{sd}_0(\partial^\alpha\delta_0)=d+|\alpha|$, preserving the original scaling degree allows only $|\alpha|\leq N$.

Dyatlov 2022, Theorem 4.19, PDF proves the point-support structure theorem. Brunetti and Fredenhagen 2000, Theorems 5.2–5.3, PDF prove existence, uniqueness below threshold, and the finite ambiguity at and above threshold.

The word “unique” must retain its qualifier. If $\omega<d$, adding a delta derivative still produces an extension in the unrestricted sense, but raises its scaling degree and therefore violates the theorem’s preservation condition.

Taylor subtraction exposes the finite freedom

For $N\geq0$, let

$$\mathcal D_N = \left\{ \varphi\in\mathcal D(\mathbb R^d): \partial^\alpha\varphi(0)=0 \text{ for }|\alpha|\leq N \right\}.$$

Choose functions $w_\alpha\in\mathcal D(\mathbb R^d)$ satisfying

$$\partial^\beta w_\alpha(0) = \delta_{\alpha\beta} \qquad (|\alpha|,|\beta|\leq N),$$

and define the continuous projection

$$W_N\varphi = \varphi - \sum_{|\alpha|\leq N} w_\alpha\,\partial^\alpha\varphi(0).$$

Then $W_N\varphi\in\mathcal D_N$. The scaling bound extends $t^\circ$ uniquely to this vanishing-jet subspace by a cutoff limit. Extending it to all tests requires assigning its values on the removed finite-dimensional Taylor jet:

$$\langle t,\varphi\rangle = \left\langle \overline t_N,W_N\varphi \right\rangle + \sum_{|\alpha|\leq N} b_\alpha\,\partial^\alpha\varphi(0).$$

Here $\overline t_N$ denotes the unique extension on $\mathcal D_N$; it is not shorthand for applying $t^\circ$ directly to a test function whose support meets the origin. Changing $W_N$ or the numbers $b_\alpha$ changes the extension by the allowed delta derivatives. Brunetti and Fredenhagen 2000, § 5.2, Eqs. (38)–(40), PDF give this finite-jet construction.

This procedure outputs a distribution and an explicit finite ambiguity. It does not select a physical subtraction condition.

Threshold examples

Principal value is one of many extensions

On $\mathbb R\setminus\{0\}$, the function $1/x$ has scaling degree $1=d$. Its Cauchy principal value is one scaling-preserving extension, but

$$\operatorname{PV}\frac1x+c\delta_0$$

has the same scaling degree for every $c\in\mathbb C$. The boundary values from the preceding Fourier-calculus page,

$$\frac1{x\pm i0} = \operatorname{PV}\frac1x \mp i\pi\delta_0,$$

are two particular choices. They agree with $1/x$ off the origin and differ only by an allowed contact term.

Power singularities cross the dimension

For $t^\circ(x)=|x|^{-a}$ on $\mathbb R^d\setminus\{0\}$:

• If $a<d$, the function is locally integrable and its scaling-degree-preserving extension is unique.
• If $a=d$, the radial integral is logarithmic and the ambiguity is $c\delta_0$.
• If $a>d$, the possible contact terms include delta derivatives through order $\lfloor a-d\rfloor$.

Exact homogeneity is stronger than finite scaling degree. At certain thresholds, a scaling-degree-preserving extension exists but no extension preserves exact homogeneity; logarithms record this scaling obstruction.

First QFT application: a four-dimensional coincidence singularity

Use a Euclidean relative coordinate so that the multiplication step is ordinary away from coincidence. On $\mathbb R^4$, set

$$\Delta_E = \sum_{j=1}^4\partial_j^2, \qquad G_E(x) = \frac{1}{4\pi^2|x|^2}.$$

This is the massless Euclidean Green distribution normalized by

$$-\Delta_EG_E = \delta^{(4)}.$$

The ordinary square is smooth for $x\neq0$:

$$G_E(x)^2 = \frac{1}{16\pi^4|x|^4}.$$

It is not locally integrable at coincidence, because the radial behavior is

$$\int_0^\varepsilon r^3r^{-4}\,\mathrm dr = \int_0^\varepsilon\frac{\mathrm dr}{r}.$$

Its scaling degree is $4$, equal to the ambient dimension. The point-extension theorem therefore says that a scaling-preserving extension exists and that the entire ambiguity is one multiple of $\delta^{(4)}$.

An explicit representative, for $\mu>0$ of inverse-length dimension, is

$$R_\mu\!\left(\frac1{|x|^4}\right) = -\frac14\Delta_E \left( \frac{\log(\mu^2|x|^2)}{|x|^2} \right).$$

The function inside the Laplacian is locally integrable. Away from the origin, direct radial differentiation gives

$$\Delta_E \left( \frac{\log(\mu^2|x|^2)}{|x|^2} \right) = -\frac4{|x|^4},$$

so $R_\mu$ really extends $|x|^{-4}$. Dyatlov 2022, Proposition 9.6, PDF also fixes the distributional normalization

$$\Delta_E\frac1{|x|^2} = -4\pi^2\delta^{(4)}.$$

Changing the auxiliary scale changes only the allowed local term:

$$R_{\mu'}\!\left(\frac1{|x|^4}\right) - R_\mu\!\left(\frac1{|x|^4}\right) = \pi^2 \log\!\left(\frac{\mu'^2}{\mu^2}\right) \delta^{(4)}.$$

Consequently, one extension of the squared Green kernel is

$$[G_E^2]_\mu = -\frac{1}{64\pi^4} \Delta_E \left( \frac{\log(\mu^2|x|^2)}{|x|^2} \right),$$

and

$$[G_E^2]_{\mu'} - [G_E^2]_\mu = \frac{1}{16\pi^2} \log\!\left(\frac{\mu'^2}{\mu^2}\right) \delta^{(4)}.$$

This square models the relative-coordinate coincidence singularity in a four-dimensional scalar loop. The mathematical conclusion is exactly that the freedom is local and has the form $C\delta^{(4)}$. Its interpretation as a counterterm, the choice of a subtraction condition, and any claim about schemes or running belong to Local Counterterms and Subdivergence Structure. The theorem-first causal construction continues at Scaling Degree and Extension of Distributions.

Fredenhagen and Rejzner 2012, § 7, PDF give the Lorentzian counterpart: $(\Delta_F)^2$ is defined away from the diagonal by the multiplication theorem, has scaling degree $4$ in four dimensions, and admits a $c\delta$ ambiguity. That off-diagonal product depends on wavefront control; it is not licensed by ordinary pointwise multiplication.

From a point to a diagonal

Near a smooth submanifold $S$, choose local coordinates $(y,z)$ with $S=\{z=0\}$ and scale only the normal variables:

$$(y,z)\longmapsto(y,\lambda z).$$

Let $q=\operatorname{codim}S$. The codimension $q$, rather than the full ambient dimension, becomes the extension threshold. At and above it, the local ambiguity has the form

$$\sum_{|\alpha|\leq\lfloor\omega-q\rfloor} u_\alpha(y)\,\partial_z^\alpha\delta(z),$$

where the coefficients $u_\alpha$ are generally distributions along $S$, not constants or automatically smooth functions. A global theorem also needs uniform transverse scaling control, compatible charts, and microlocal hypotheses.

Brunetti and Fredenhagen, §6, develop this submanifold generalization. The present page uses the point theorem in one flat relative coordinate and stops before Epstein–Glaser induction on nested diagonals.

Method and stop conditions

For a proposed singular product or extension:

1. License the product. Identify a smooth factor, an integrable product, disjoint singular supports, or a valid diagonal-pullback theorem. Scaling degree does not create the initial product.
2. Name the missing set. Begin with a distribution on the complement of a point or submanifold. An expression undefined even there cannot be repaired by the extension theorem.
3. Compute the scaling degree. Fix the dilation convention and compare the result with the dimension or codimension of the missing set.
4. Preserve the qualifier. Below threshold, uniqueness means uniqueness among extensions with the original scaling degree.
5. List the finite ambiguity. Use $N=\lfloor\omega-q\rfloor$, where $q=d$ for a point in $\mathbb R^d$ and $q=\operatorname{codim}S$ for a transverse submanifold theorem. Retain only normal delta derivatives through order $N$ before applying further conditions; along $S$, their coefficients can themselves be distributions.
6. Separate mathematics from physical input. Symmetry, locality, covariance, subtraction conditions, and experiment may reduce or fix the coefficients; scaling degree alone does not.
7. Stop at the stated scope. Do not infer a beta function, running coupling, or renormalization scheme from the appearance of the auxiliary scale $\mu$ alone.

Exercises

1. Compute the scaling degrees of $1$, $x^k$, $\delta_0$, and $\partial^\alpha\delta_0$ on $\mathbb R^d$.

   Check

   Under $x\mapsto\lambda x$, the first two representatives become $1$ and $\lambda^kx^k$, so their scaling degrees are $0$ and $-k$. The delta scales as $\lambda^{-d}\delta_0$, while each derivative adds one inverse power:

   $$\operatorname{sd}_0(\delta_0)=d, \qquad \operatorname{sd}_0(\partial^\alpha\delta_0) = d+|\alpha|.$$

2. Let $t^\circ(x)=|x|^{-d-3/2}$ on the punctured space. Which contact terms can occur in an extension preserving its scaling degree?

   Check

   Here $\omega=d+3/2$, so

   $$N = \left\lfloor\omega-d\right\rfloor = 1.$$

   Two extensions may differ by

   $$c\,\delta_0 + \sum_{j=1}^d c_j\,\partial_j\delta_0.$$

   Rotational or parity invariance could remove the vector coefficients, but that conclusion uses an additional symmetry requirement.

3. Explain why $\operatorname{PV}(1/x)+c\delta_0$ does not contradict the uniqueness theorem.

   Check

   The punctured distribution $1/x$ has scaling degree $\omega=1$ in dimension $d=1$. It is exactly at, not below, the uniqueness threshold. Since $N=\lfloor1-1\rfloor=0$, one delta term is allowed.

4. Verify the scale dependence of $R_\mu(|x|^{-4})$.

   Check

   The logarithms differ by the constant $\log(\mu'^2/\mu^2)$. Therefore

   $$\begin{aligned} R_{\mu'}-R_\mu &= -\frac14 \log\!\left(\frac{\mu'^2}{\mu^2}\right) \Delta_E\frac1{|x|^2} \\ &= \pi^2 \log\!\left(\frac{\mu'^2}{\mu^2}\right) \delta^{(4)}. \end{aligned}$$

   The difference has support only at the removed point and is exactly the ambiguity allowed at scaling degree $4$.

References

• Romeo Brunetti and Klaus Fredenhagen, Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds, PDF, §§ 5–6, Communications in Mathematical Physics 208 (2000), 623–661. Journal record. This is the specialist source for scaling degree, point-extension theorems, finite-jet subtraction, and the extension to submanifolds.
• Semyon Dyatlov, Lecture Notes for 18.155: Differential Analysis, PDF, §§ 3.2 and 4.4 and Proposition 9.6, MIT, 2022. These notes supply smooth multiplication, the mollified-product counterexample, the structure theorem for point-supported distributions, and the Euclidean fundamental-solution normalization.
• Klaus Fredenhagen and Katarzyna Rejzner, Perturbative Algebraic Quantum Field Theory, PDF, § 7 and Appendix A, 2012. This is the QFT source for the off-diagonal square of the Feynman propagator, its four-dimensional scaling degree and local ambiguity, and an independent statement of the distribution product and extension theorems.
• Lars Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, 2nd ed., § 8.2, Springer, 2003. Theorems 8.2.4 and 8.2.10 are the structural sources for pullback and multiplication under wavefront-set hypotheses.