Delta Distributions, Weak Derivatives, Pullbacks, and Pushforwards

The Dirac delta is not an infinitely high, infinitely narrow function. It is the continuous linear functional that evaluates a test function at a point. Once that definition is taken seriously, differentiation, localized sources, constraint surfaces, and changes of variables all follow from duality. The main lesson is equally important: every pullback or pushforward of a distribution comes with a geometric support or rank hypothesis.

Required background. Test-Function Spaces, Distributions, Support, and Convergence supplies test-function pairings, continuity, support, and distributional convergence.

We work first on open subsets of Euclidean space, pair distributions linearly with complex-valued test functions, and use absolute Jacobians because those pairings use Lebesgue measure. The first QFT application is the delta contact term in the free-scalar Green equation; the positive-energy mass shell is a second bounded application in the site’s $(+---)$ metric convention. The QFT interpretation of coincident operator products and contact terms belongs to Coincident Products and Contact Terms. Mathematical criteria for products and extensions remain in Products, Scaling Degree, and Distribution Extensions.

A delta distribution is an evaluation map

Let $\Omega\subseteq\mathbb R^d$ be open and let $a\in\Omega$. The Dirac distribution at $a$ is

$$\delta_a\in\mathcal D'(\Omega), \qquad \langle\delta_a,\varphi\rangle=\varphi(a), \qquad \varphi\in\mathcal D(\Omega).$$

Evaluation is continuous in the test-function topology: if $\operatorname{supp}\varphi\subseteq K\Subset\Omega$, then

$$|\langle\delta_a,\varphi\rangle| \leq \sup_K|\varphi|$$

when $a\in K$, while the pairing vanishes when $a\notin K$. The notation

$$\int_{\mathbb R^d} \delta^{(d)}(x-a)\varphi(x)\,\mathrm d^d x = \varphi(a)$$

is therefore a pairing identity, not an instruction to assign a pointwise value to $\delta^{(d)}$.

Translations and invertible linear changes of variables give the first useful calculus rule. If $A\in GL(d,\mathbb R)$ and $b\in\mathbb R^d$, then

$$\delta^{(d)}(Ax-b) = \frac{1}{|\det A|} \delta^{(d)}(x-A^{-1}b)$$

as distributions in $x$. Indeed, for every test function $\varphi$,

$$\begin{aligned} \int_{\mathbb R^d} \delta^{(d)}(Ax-b)\varphi(x)\,\mathrm d^d x &= \frac{1}{|\det A|} \int_{\mathbb R^d} \delta^{(d)}(y-b)\varphi(A^{-1}y)\,\mathrm d^d y \\ &= \frac{\varphi(A^{-1}b)}{|\det A|}. \end{aligned}$$

The absolute value is compulsory. Reversing orientation does not reverse a Lebesgue density. Differential forms carry orientation signs; the scalar distributions used here do not.

Distributional differentiation is integration by parts

For a multi-index $\alpha$, define the derivative of $u\in\mathcal D'(\Omega)$ by

$$\boxed{ \langle\partial^\alpha u,\varphi\rangle = (-1)^{|\alpha|} \langle u,\partial^\alpha\varphi\rangle. }$$

The right-hand side is again a continuous linear functional on $\mathcal D(\Omega)$. Thus every distribution has derivatives of every order, the derivatives commute, and differentiation is sequentially continuous:

$$u_n\longrightarrow u\text{ in }\mathcal D' \quad\Longrightarrow\quad \partial^\alpha u_n\longrightarrow\partial^\alpha u.$$

For a regular distribution $u_f$ with $f\in C^1$, this definition agrees with the classical derivative after integration by parts. It also implies

$$\operatorname{supp}(\partial^\alpha u) \subseteq \operatorname{supp}u,$$

because a test function supported where $u$ vanishes has all its derivatives supported there as well. Dyatlov 2022, § 3.1, PDF proves these facts and works out the Heaviside and delta examples below.

Delta derivatives record jets

In one dimension,

$$\langle\delta_a^{(n)},\varphi\rangle = (-1)^n\varphi^{(n)}(a).$$

More generally,

$$\langle\partial^\alpha\delta_a,\varphi\rangle = (-1)^{|\alpha|}\partial^\alpha\varphi(a).$$

The delta probes the value of a test function; its derivatives probe the successive Taylor coefficients at the same point. They remain supported at $\{a\}$.

If $g\in C^\infty(\Omega)$, multiplication and differentiation obey the usual Leibniz rule:

$$\partial_j(gu) = (\partial_j g)u+g\,\partial_j u.$$

This is not a new pointwise product. It follows from the already defined smooth multiplication $\langle gu,\varphi\rangle=\langle u,g\varphi\rangle$ and the product rule for the smooth test function $g\varphi$.

Heaviside functions turn jumps into localized terms

Let $H$ be the locally integrable Heaviside function,

$$H(x)= \begin{cases} 0,&x<0,\\ 1,&x>0. \end{cases}$$

Its value at the single point $0$ is irrelevant to the associated regular distribution. For every $\varphi\in\mathcal D(\mathbb R)$,

$$\begin{aligned} \langle H',\varphi\rangle &= -\int_0^\infty\varphi'(x)\,\mathrm dx \\ &= \varphi(0) = \langle\delta_0,\varphi\rangle. \end{aligned}$$

Therefore

$$H'=\delta_0.$$

Suppose the two one-sided pieces extend to $C^1$ functions $f_-$ and $f_+$ on a neighborhood of $a$, and write

$$f(x) = H(a-x)f_-(x)+H(x-a)f_+(x).$$

With $[f]_a=f_+(a)-f_-(a)$, its distributional derivative is

$$\partial_x f = H(a-x)f_-'(x) + H(x-a)f_+'(x) + [f]_a\delta_a.$$

The ordinary derivatives describe the two open regions, while the delta remembers the jump. A second derivative would also contain a $[f]_a\delta_a'$ term, so higher derivatives retain progressively more interface data.

“Distributional derivative” and “weak derivative” are not synonyms

Every $f\in L^1_{\mathrm{loc}}(\Omega)$ has a distributional derivative. One says that $g$ is the weak derivative of $f$ in a specified function space, for example $g\in L^p_{\mathrm{loc}}$, only when

$$\int_\Omega g\varphi\,\mathrm d^d x = -\int_\Omega f\,\partial_j\varphi\,\mathrm d^d x$$

for every test function and $g$ has the stated regularity. A function with a nonzero jump has a perfectly good distributional derivative, but its delta term has no $L^p_{\mathrm{loc}}$ representative for $1\leq p\leq\infty$. The extra function-space claim therefore fails. This is the Sobolev-space usage in Evans 2010, Chapter 5, § 5.2.1.

Interfaces convert conservation laws into matching conditions

Distributional differentiation makes surface sources explicit. Let $g:\Omega\to\mathbb R$ be smooth with $\mathrm dg\neq0$ on $\Sigma=g^{-1}(0)$, and let $J^\mu_+$ and $J^\mu_-$ be smooth currents on the two sides. Set

$$J^\mu = H(g)J^\mu_+ + H(-g)J^\mu_-.$$

The chain rule for the regular-value pullback of $H$ gives

$$\partial_\mu J^\mu = H(g)\partial_\mu J^\mu_+ + H(-g)\partial_\mu J^\mu_- + \delta(g)\,\partial_\mu g \bigl(J^\mu_+-J^\mu_-\bigr).$$

If both bulk currents are conserved, the global current is conserved without an added surface source precisely when the normal flux is continuous:

$$\left. \partial_\mu g \bigl(J^\mu_+-J^\mu_-\bigr) \right|_\Sigma =0.$$

This calculation is independent of how $g$ is rescaled by a smooth positive factor: the transformation of $\delta(g)$ cancels the transformation of its normal covector on $\Sigma$. It is the basic mechanism behind localized sources, junction conditions, and delta contact terms. The operator-product and time-ordering consequences are developed at the Coincident Products and Contact Terms page.

First QFT application: a free-scalar contact term

Let $\phi$ be a free real scalar field in $d$-dimensional Minkowski space, and define its time-ordered two-point distribution by

$$D_F(x-y) = \langle0|T\{\phi(x)\phi(y)\}|0\rangle.$$

For bosonic operators, differentiating the step functions in the time-ordering symbol gives

$$\begin{aligned} \partial_{x^0}T\{A(x)B(y)\} &= T\{\partial_0 A(x)B(y)\} \\ &\quad + \delta(x^0-y^0)[A(x),B(y)]. \end{aligned}$$

Use the equal-time canonical relations

$$\begin{aligned} [\phi(t,\mathbf x),\phi(t,\mathbf y)]&=0, \\ [\pi(t,\mathbf x),\phi(t,\mathbf y)] &= -i\delta^{(d-1)}(\mathbf x-\mathbf y), \qquad \pi=\partial_0\phi. \end{aligned}$$

The first time derivative of $D_F$ has no contact term because the equal-time field commutator vanishes. Differentiating once more produces

$$\begin{aligned} \partial_{x^0}^2D_F(x-y) &= \langle0| T\{\partial_0^2\phi(x)\phi(y)\} |0\rangle \\ &\quad - i\delta(x^0-y^0) \delta^{(d-1)}(\mathbf x-\mathbf y). \end{aligned}$$

Spatial derivatives do not differentiate the time-ordering step functions. Combining this identity with the free equation $(\Box+m^2)\phi=0$ gives

$$(\Box_x+m^2)D_F(x-y) = -i\delta^{(d)}(x-y).$$

The field obeys the homogeneous equation away from coincidence, but its time-ordered product is an inhomogeneous Green distribution. The localized term is created by differentiating the ordering prescription and using the canonical commutator; it is not an ordinary pointwise source. Schwartz 2014, § 6.2, pp. 75–77, and § 7.1, Eqs. (7.9)–(7.10) gives this normalization and sign. The interpretation of such coincident-point terms continues at Coincident Products and Contact Terms.

Pullback requires a rank or singularity condition

Let $\Phi:U\to V$ be a smooth map between open subsets of Euclidean spaces. For a smooth function $f$ there is always a classical pullback

$$\Phi^*f=f\circ\Phi.$$

The same assertion is false for an arbitrary distribution. A map can send an entire region into the singular support of the object being pulled back, and a distribution has no pointwise values there to compose.

A diffeomorphism is safe and, more generally, so is any submersion for which $\mathrm d\Phi_x$ is surjective at every $x\in U$.

For a diffeomorphism $\Phi:U\to V$, duality with change of variables fixes the formula

$$\langle\Phi^*u,\varphi\rangle = \left\langle u,\, |\det D\Phi^{-1}|\, \varphi\circ\Phi^{-1} \right\rangle.$$

Here the determinant is evaluated at the argument of the test function on $V$. If $u=u_f$ is regular, this formula reduces to $\Phi^*u_f=u_{f\circ\Phi}$, exactly as required. Applied to a delta, it gives

$$\Phi^*\delta_b = \frac{\delta_{\Phi^{-1}(b)}} {|\det D\Phi(\Phi^{-1}(b))|}.$$

For a projection

$$\pi:\mathbb R^{n+k}\longrightarrow\mathbb R^n, \qquad \pi(x,y)=x,$$

the submersion pullback is characterized by

$$\langle\pi^*u,\varphi\rangle = \left\langle u,\, x\longmapsto \int_{\mathbb R^k}\varphi(x,y)\,\mathrm d^k y \right\rangle.$$

Every submersion is locally a projection after a diffeomorphism. This factorization produces a unique sequentially continuous map

$$\Phi^*:\mathcal D'(V)\longrightarrow\mathcal D'(U)$$

that agrees with composition on locally integrable functions. Dyatlov 2022, Theorem 10.2, PDF gives the local submersion construction; Hörmander 2003, Theorem 6.1.1 supplies the structural treatment.

Submersion is a sufficient condition for pulling back every distribution, not a necessary condition for pulling back one particular distribution. The sharper later criterion compares the covectors normal to $\Phi$ with the distribution’s wavefront set. Singular Support and Wavefront Sets develops the mathematical criterion; the theorem-level criteria for products, pullbacks, and pushforwards in QFT belong to Wavefront-Set Products, Pullbacks, and Pushforwards.

Delta constraints are pullbacks to regular level sets

Let $g:\Omega\to\mathbb R$ be smooth and suppose $0$ is a regular value:

$$\nabla g(x)\neq0 \qquad \text{whenever }g(x)=0.$$

Then $\Sigma=g^{-1}(0)$ is a smooth hypersurface and the pullback $g^*\delta_0$, conventionally written $\delta(g)$, is well defined. To see why the regular-value hypothesis is enough, choose an open neighborhood $W$ of $\Sigma$ on which $\mathrm dg$ never vanishes. The submersion theorem defines $(g|_W)^*\delta_0$; it is supported on the closed set $\Sigma$, so localization extends it uniquely by zero to $\Omega$. The coarea formula gives

$$\boxed{ \langle\delta(g),\varphi\rangle = \int_\Sigma \frac{\varphi(x)}{|\nabla g(x)|}\, \mathrm dS(x). }$$

The norm and surface measure in this formula are Euclidean because the ambient pairing uses $\mathrm d^d x$. The distribution depends on the normalization of the defining function: $\delta(a g)=|a|^{-1}\delta(g)$ on the zero set for a smooth nowhere-zero factor $a$. The geometric combination $\delta(g)\,\mathrm dg$ is unchanged when $a>0$, while $|\nabla g|\delta(g)$ is independent of the defining function altogether.

In one dimension, if the zeros $x_i$ of $g$ meeting $\operatorname{supp}\varphi$ are simple, the formula becomes

$$\delta(g(x)) = \sum_{g(x_i)=0} \frac{\delta(x-x_i)}{|g'(x_i)|}.$$

For a smooth map $F:\Omega\to\mathbb R^k$ of full rank $k$ along $F^{-1}(0)$, the codimension-$k$ version is

$$\left\langle\delta^{(k)}(F),\varphi\right\rangle = \int_{F^{-1}(0)} \frac{\varphi(x)}{J_F(x)}\, \mathrm d\mathcal H^{d-k}(x),$$

where

$$J_F(x) = \sqrt{\det\!\bigl(DF_x\,DF_x^{\mathsf T}\bigr)}.$$

This is the precise meaning of imposing $k$ independent smooth constraints inside an ordinary integral. The same localization argument reduces the construction to a submersion near $F^{-1}(0)$. Federer 1969, § 3.2.22 supplies the codimension-$k$ coarea formula. Dyatlov 2022, Proposition 10.12, PDF proves the hypersurface formula.

The regular-value hypothesis cannot be deleted. For $g(x)=x^2$, the only zero is critical and the simple-zero rule would demand division by zero. The expression $\delta(x^2)$ is not a canonical pullback of $\delta_0$. Assigning one requires additional extension data; it is not justified by the delta change-of-variables rule.

Pushforward requires properness on support

Pullback transports a generalized function against the direction of a map. Pushforward transports a localized distribution in the direction of the map. Let $\Phi:U\to V$ be smooth and $u\in\mathcal D'(U)$. Assume that $\Phi$ is proper on $\operatorname{supp}u$:

$$\Phi^{-1}(K)\cap\operatorname{supp}u \quad\text{is compact for every compact }K\subset V.$$

Then the pushforward is defined by

$$\langle\Phi_*u,\psi\rangle = \langle u,\psi\circ\Phi\rangle.$$

Although $\psi\circ\Phi$ need not have compact support in all of $U$, its intersection with $\operatorname{supp}u$ is compact. Multiplying it by a cutoff equal to $1$ near that intersection makes the right-hand side a legitimate distributional pairing, independent of the cutoff. A compactly supported $u$ therefore has a pushforward under every smooth $\Phi$.

Three examples separate this operation from pullback:

$$\Phi_*\delta_a=\delta_{\Phi(a)}.$$

For the projection $\pi(x,y)=x$ and a compactly supported regular distribution $u_f$,

$$\pi_*u_f = u_{f_{\mathrm{fib}}}, \qquad f_{\mathrm{fib}}(x) = \int_{\mathbb R^k}f(x,y)\,\mathrm d^k y.$$

For a diffeomorphism and a regular distribution,

$$\Phi_*u_f = u_{\widetilde f}, \qquad \widetilde f(y) = f(\Phi^{-1}(y)) |\det D\Phi^{-1}(y)|.$$

Thus a pushforward integrates a density along fibers and includes the Jacobian appropriate to that integration. A pullback of the regular function $f$ is instead just $f\circ\Phi$. The proper-on-support definition is given explicitly in Dinh and Sibony 2005, § 2.2, PDF.

Without properness, $\psi\circ\Phi$ may probe a noncompact part of $\operatorname{supp}u$, so an ordinary distribution need not be able to act on it. For example, pushing the constant regular distribution on $\mathbb R^2$ through the projection $(x,y)\mapsto x$ would require the divergent fiber integral $\int_{\mathbb R}\mathrm dy$. A regulator or a different distribution space would be extra structure, not part of the definition above.

Secondary QFT application: the positive-energy mass shell

Use the site’s $(+---)$ convention in $d$ spacetime dimensions:

$$p^2=(p^0)^2-|\mathbf p|^2, \qquad E_{\mathbf p} = \sqrt{|\mathbf p|^2+m^2}, \qquad m>0.$$

At fixed $\mathbf p$, the mass-shell constraint has two simple roots:

$$(p^0)^2-E_{\mathbf p}^2 = (p^0-E_{\mathbf p})(p^0+E_{\mathbf p}).$$

The one-dimensional regular-zero formula therefore gives

$$\delta(p^2-m^2) = \frac{ \delta(p^0-E_{\mathbf p}) + \delta(p^0+E_{\mathbf p}) } {2E_{\mathbf p}}.$$

The shell has two components separated by the open strip $|p^0|<m$. Choose $\chi\in C^\infty(\mathbb R)$ with

$$\chi(p^0)=0\quad\text{for }p^0\leq-\frac m2, \qquad \chi(p^0)=1\quad\text{for }p^0\geq\frac m2,$$

and define

$$\delta_+(p^2-m^2) = \chi(p^0)\delta(p^2-m^2), \qquad \delta_-(p^2-m^2) = \bigl(1-\chi(p^0)\bigr)\delta(p^2-m^2).$$

These distributions do not depend on the chosen interpolation of $\chi$, because the shell does not meet that interpolation region. The conventional notation

$$\delta_\pm(p^2-m^2) = \theta(\pm p^0)\delta(p^2-m^2)$$

is shorthand for this smooth localization, not an unrestricted product with a discontinuous function.

Selecting the future sheet now yields, for every test function $\Psi$,

$$\begin{aligned} \int_{\mathbb R^d} \mathrm d^d p\, \delta_+(p^2-m^2)\Psi(p) &= \int_{\mathbb R^{d-1}} \frac{\mathrm d^{d-1}\mathbf p}{2E_{\mathbf p}}\, \Psi(E_{\mathbf p},\mathbf p). \end{aligned}$$

Equivalently, let

$$\iota_+:\mathbb R^{d-1}\longrightarrow\mathbb R^d, \qquad \iota_+(\mathbf p)=(E_{\mathbf p},\mathbf p).$$

Then the equality of measures on $\mathbb R^d$ is

$$\delta_+(p^2-m^2)\,\mathrm d^d p = (\iota_+)_* \left( \frac{\mathrm d^{d-1}\mathbf p}{2E_{\mathbf p}} \right).$$

This measure is invariant under proper orthochronous Lorentz transformations. Factors of $(2\pi)^{-(d-1)}$ are conventional and have deliberately not been included. With the common scattering convention, one writes

$$\mathrm d\Pi_p = \frac{\mathrm d^{d-1}\mathbf p} {(2\pi)^{d-1}2E_{\mathbf p}}.$$

This is the localized shell constraint underlying relativistic phase-space integrals. The complete multiparticle construction belongs to Lorentz-Invariant Phase Space. For $d=4$, Schwartz 2014, § 5.1, p. 61, Eq. (5.21) gives $\mathrm d^3\mathbf p/[(2\pi)^3 2E_{\mathbf p}]$. The displayed $d$-dimensional version follows from the same one-variable delta calculation.

Operation checklist and stop conditions

Before manipulating a delta or another distribution, identify five pieces of data:

1. Test space and meaning. Is the object in $\mathcal D'$, $\mathcal S'$, or a more specialized dual? A statement such as $\delta(0)=\infty$ gives no pairing and therefore defines nothing.
2. Derivative claim. Distributional derivatives always exist. A weak derivative in $L^p$ is an additional representation claim and may fail when a jump produces a delta.
3. Map and direction. A pullback needs a submersion or a valid singularity criterion; a pushforward needs properness on support. Keep the absolute Jacobian in pullback formulas.
4. Product. Smooth functions may multiply distributions, but the rules on this page do not define $\delta^2$ or an arbitrary product of singular distributions. The extension problem belongs to Products, Scaling Degree, and Distribution Extensions.
5. Pairing test. After moving the proposed operation to the test function, is the resulting probe smooth and compactly supported where required?

Stop when one of these checks fails. Writing one line of duality usually exposes the missing hypothesis before formal delta notation can hide it.

Exercises

1. Let $f(x)=H(x-a)(x-a+1)$. Compute its first distributional derivative.

   Check

   The smooth factor has value $1$ at $a$. The Leibniz rule gives

   $$\partial_x f = (x-a+1)\delta_a+H(x-a) = \delta_a+H(x-a).$$

   Pairing directly with a test function gives the same boundary term.

2. Let $a>0$. Determine $\delta(x^2-a^2)$ and explain why the corresponding formula does not apply at $a=0$.

   Check

   The zeros are $x=\pm a$, and $|(x^2-a^2)'|=2a$ at each one. Hence

   $$\delta(x^2-a^2) = \frac{\delta(x-a)+\delta(x+a)}{2a}.$$

   At $a=0$, the sole zero is critical because the derivative of $x^2$ vanishes there. The regular-zero pullback theorem gives no distribution $\delta(x^2)$.

3. For $\Phi(x)=\alpha x+\beta$ with $\alpha\neq0$, compare $\Phi^*\delta_b$ and $\Phi_*\delta_a$.

   Check

   Pullback solves the constraint $\Phi(x)=b$ and includes its Jacobian:

   $$\Phi^*\delta_b = \frac{1}{|\alpha|} \delta_{\frac{b-\beta}{\alpha}}.$$

   Pushforward simply transports the point mass:

   $$\Phi_*\delta_a = \delta_{\alpha a+\beta}.$$

   The two operations point in opposite directions and answer different questions.

4. Integrate $\Psi(p^0,\mathbf p)$ over the negative-energy mass shell.

   Check

   The negative-sheet localization $\delta_-$ retains only the root $p^0=-E_{\mathbf p}$:

   $$\int\mathrm d^d p\, \delta_-(p^2-m^2)\Psi(p) = \int \frac{\mathrm d^{d-1}\mathbf p}{2E_{\mathbf p}}\, \Psi(-E_{\mathbf p},\mathbf p).$$

   The Jacobian is positive on both sheets because it is $|2p^0|=2E_{\mathbf p}$.

References

• Tien-Cuong Dinh and Nessim Sibony, Introduction to the Theory of Currents, § 2.2, PDF, dated September 21, 2005. This supplies the explicit pushforward definition under properness on support.
• Semyon Dyatlov, Lecture Notes for 18.155: Differential Analysis, PDF, §§ 3.1–3.2 and 10.1, MIT, 2022. These are the teaching sources for distributional derivatives, smooth multiplication, submersion pullbacks, the chain rule, and hypersurface deltas.
• Lawrence C. Evans, Partial Differential Equations, 2nd ed., Chapter 5, § 5.2.1, American Mathematical Society, 2010. Book record. This fixes the function-space meaning of weak derivatives used here.
• Herbert Federer, Geometric Measure Theory, § 3.2.22, Springer, 1969. Book record. This is the source for the coarea formula and its codimension-$k$ Jacobian.
• Lars Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, 2nd ed., Chapters 3 and 6, Springer, 2003. Book record. This is the structural source for differentiation, composition with smooth maps, and the limits of unrestricted pullback.
• Matthew D. Schwartz, Quantum Field Theory and the Standard Model, § 5.1, p. 61, Eq. (5.21); § 6.2, pp. 75–77; and § 7.1, Eqs. (7.9)–(7.10), Cambridge University Press, 2014. Book record. This is the QFT source for the free-scalar contact term and for the four-dimensional on-shell factor $\mathrm d^3\mathbf p/[(2\pi)^3 2E_{\mathbf p}]$; the page derives the latter’s $d$-dimensional extension.