Distributional Kernels and Distributions on Manifolds

A distribution on a manifold is defined without choosing coordinates by letting it act on compactly supported smooth densities. The density supplies the absolute Jacobian that an ordinary Euclidean pairing hides. Likewise, a continuous operator from test inputs to distributions is encoded by one distribution on a product manifold—its Schwartz kernel. These two ideas make two-point functions and Green kernels coordinate independent without pretending that a singular kernel has pointwise values.

Required background. Test-Function Spaces, Distributions, Support, and Convergence supplies the test spaces, duality, support, and continuity used throughout.

Helpful background. Smooth Manifolds, Tangent and Cotangent Bundles, and Tensor Fields supplies charts, tangent and cotangent maps, and tensor bundles; Locally Convex, Nuclear, and Rigged Hilbert Spaces supplies the topological tensor-product machinery behind the kernel theorem. Neither is required to begin, so the needed geometric and continuity data are also stated here.

Test densities absorb coordinate Jacobians

Let $M$ be a smooth $n$-manifold without boundary. Its density bundle is

$$|\Omega_M| = |\Lambda^nT^*M|.$$

Unlike an $n$-form, a density transforms with the absolute value of the Jacobian and can therefore be integrated without orienting $M$. The intrinsic test space is

$$\mathcal D_{\mathrm{dens}}(M) = \Gamma_c^\infty(M;|\Omega_M|).$$

In coordinates $x=(x^1,\ldots,x^n)$, a test density has the form

$$\omega=a(x)|\mathrm d^n x|.$$

If $y$ is another coordinate system, the same density is

$$\omega = b(y)|\mathrm d^n y|, \qquad b(y) = a(x(y)) \left| \det\frac{\partial x}{\partial y} \right|.$$

This transformation law is exactly the change-of-variables factor in an integral. It is geometric data, not an extra convention attached after a calculation.

A sequence $\omega_j$ converges to $\omega$ in $\mathcal D_{\mathrm{dens}}(M)$ when all supports eventually lie in one compact set and, in every chart meeting that set, every derivative of the coordinate coefficients converges uniformly. A finite chart cover suffices on a compact set. The resulting topology is independent of the chosen charts, partition of unity, or auxiliary connection.

Dyatlov 2022, §§ 13.1.7 and 13.2.1, PDF constructs the density bundle, its integral, the test-density topology, and the intrinsic distribution space used below.

Scalar distributions are dual to test densities

Define

$$\mathcal D'(M) = \mathcal D_{\mathrm{dens}}(M)'.$$

Thus a scalar distribution $u$ is a continuous complex-linear functional

$$\omega\longmapsto\langle u,\omega\rangle.$$

There is no hidden complex conjugation in this pairing. A locally integrable scalar function $f$ defines the regular distribution

$$\langle u_f,\omega\rangle = \int_M f\,\omega.$$

This formula is coordinate independent because $\omega$, rather than its coordinate coefficient, is what is integrated.

Let $\kappa:U\to V\subset\mathbb R^n$ be a chart. The coordinate representative of $u|_U$ is characterized by

$$\left\langle \kappa_*u,\varphi \right\rangle = \left\langle u,\, \kappa^*\bigl(\varphi|\mathrm d^n x|\bigr) \right\rangle, \qquad \varphi\in\mathcal D(V).$$

For $u_f$, change of variables gives

$$\kappa_*u_f = u_{f\circ\kappa^{-1}}.$$

There is no additional Jacobian multiplying $f\circ\kappa^{-1}$: the pulled back test density already carries it. Conversely, chart distributions that obey this compatibility law on overlaps glue uniquely by a partition of unity. This local-to-global description also makes restriction, support, and distributional convergence agree with their Euclidean definitions.

Choosing a positive density changes the bookkeeping

A positive smooth density $\mu$ identifies test functions with test densities:

$$\varphi \longmapsto \varphi\mu.$$

Under this choice, a scalar distribution $u$ may be written as the function-test functional

$$u_\mu(\varphi) = \langle u,\varphi\mu\rangle.$$

A Lorentzian metric supplies the canonical positive density

$$\mu_g = \sqrt{|\det g_{\alpha\beta}|}\, |\mathrm d^d x|.$$

In four-dimensional coordinates with the site’s $(+---)$ signature this is usually written $\sqrt{-g}\,\mathrm d^4x$.

The distinction matters already for a point mass. Relative to $\mu$, define the scalar distribution

$$\left\langle\delta_p^\mu,\omega\right\rangle = \frac{\omega(p)}{\mu(p)}, \qquad \left\langle\delta_p^\mu,f\mu\right\rangle=f(p).$$

If $\mu'=a\mu$ with $a>0$, then

$$\delta_p^{\mu'} = \frac{1}{a(p)}\delta_p^\mu.$$

Evaluation $f\mapsto f(p)$ is canonical on test functions, but in the intrinsic type convention it defines a distributional density. Turning it into a scalar distribution acting on test densities requires the choice of $\mu$. Keeping these types visible prevents a missing-Jacobian error.

Intrinsic operations use transposes

Smooth multiplication, restriction to open subsets, support, and gluing are defined exactly as duality suggests:

$$\langle au,\omega\rangle = \langle u,a\omega\rangle, \qquad a\in C^\infty(M).$$

For a smooth vector field $X$, distributional differentiation is

$$\boxed{ \langle Xu,\omega\rangle = -\langle u,\mathcal L_X\omega\rangle. }$$

The Lie derivative acts on the whole density. If $\omega=f\mu$, then

$$\mathcal L_X(f\mu) = \bigl(Xf+f\,\operatorname{div}_\mu X\bigr)\mu.$$

The divergence term is therefore compulsory when a coordinate formula uses a nonconstant density.

More generally, if $P$ is a differential operator on scalar functions and $P^t$ is its transpose on compactly supported densities, define

$$\langle Pu,\omega\rangle = \langle u,P^t\omega\rangle.$$

This convention includes all integration-by-parts signs in $P^t$. Bär, Ginoux, and Pfäffle 2007, § 1.1.2, PDF give the corresponding formal-adjoint construction after fixing a smooth volume density.

For a vector bundle $E\to M$, a distributional section of $E$ is

$$\mathcal D'(M;E) = \Gamma_c^\infty \bigl(M;E^*\otimes|\Omega_M|\bigr)'.$$

It acts on compactly supported dual-bundle-valued densities. For complex bundles, $E^*$ here is the complex-linear dual; a Hermitian adjoint or complex conjugation must be introduced explicitly when a physical application requires it.

Tensor products live on product manifolds

If $u\in\mathcal D'(M)$ and $v\in\mathcal D'(N)$, there is a unique tensor product

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

such that

$$\left\langle u\boxtimes v,\omega\boxtimes\eta \right\rangle = \langle u,\omega\rangle \langle v,\eta\rangle.$$

The external product density $\omega\boxtimes\eta$ is naturally a density on $M\times N$. Existence for general, non-decomposable test densities follows from the continuity of iterated pairings; decomposable probes determine the result uniquely.

This operation is always defined, but it is not a pointwise product of distributions on the same manifold. For example, after choosing positive densities on the two factors, $\delta_p^{\mu_M}\boxtimes\delta_q^{\mu_N}$ is supported at $(p,q)$ in $M\times N$; it says nothing about whether two singular objects can be multiplied after their variables are identified.

Dyatlov 2022, § 7.1, Theorem 7.1, PDF proves existence and uniqueness of tensor products and the required iterated-pairing identity.

The Schwartz kernel theorem turns operators into distributions

Let

$$A: \mathcal D_{\mathrm{dens}}(N) \longrightarrow \mathcal D'(M)$$

be a sequentially continuous linear map. The Schwartz kernel theorem gives a unique distribution

$$K_A\in\mathcal D'(M\times N)$$

such that

$$\boxed{ \langle A\eta,\omega\rangle = \left\langle K_A,\omega\boxtimes\eta \right\rangle }$$

for every $\omega\in\mathcal D_{\mathrm{dens}}(M)$ and $\eta\in\mathcal D_{\mathrm{dens}}(N)$. Conversely, every $K\in\mathcal D'(M\times N)$ defines one such operator. Continuity in the declared test topology is essential; an arbitrary algebraic bilinear form does not acquire a distributional kernel.

After choosing positive densities $\mu_M$ and $\mu_N$, set

$$\omega=g\mu_M, \qquad \eta=f\mu_N.$$

The kernel identity becomes

$$\left\langle A(f\mu_N),g\mu_M \right\rangle = \left\langle K_A, (g\mu_M)\boxtimes(f\mu_N) \right\rangle.$$

If $K_A$ is represented by a locally integrable function $k(x,y)$, this pairing may be read as

$$\begin{aligned} \left\langle A(f\mu_N),g\mu_M \right\rangle &= \int_{M\times N} g(x)k(x,y)f(y)\, \mu_M(x)\mu_N(y), \\ (Af)(x) &= \int_N k(x,y)f(y)\,\mu_N(y) \quad\text{for almost every }x. \end{aligned}$$

For a genuinely singular kernel, the first boxed pairing remains the definition and the second line need not exist pointwise.

Dyatlov 2022, § 7.2, Theorem 7.6, PDF states the Euclidean kernel theorem. The manifold statement follows by localization: choose locally finite, relatively compact chart covers and subordinate partitions of unity $\{\chi_i\}$ on $M$ and $\{\psi_j\}$ on $N$. Apply the Euclidean theorem in product charts to

$$(\omega,\eta) \longmapsto \langle A(\psi_j\eta),\chi_i\omega\rangle,$$

transform each resulting local kernel by the density compatibility law, extend it by zero, and sum the locally finite family. Pairing the sum with $\omega\boxtimes\eta$ recovers $A$ because both partitions sum to one. Decomposable test densities determine a distribution on the product, which proves uniqueness. Dyatlov 2022, § 13.2.2, PDF supplies the chart compatibility and localization rules used in this adaptation. Nuclearity is proof infrastructure, not an extra hypothesis that must be checked anew for each kernel.

The identity already has a singular kernel

Fix a positive density $\mu$ on $M$. The kernel implementing the $\mu$-identification $f\mu\mapsto u_f$ is the diagonal delta $\delta_\Delta^\mu$, characterized by

$$\left\langle \delta_\Delta^\mu, (g\mu)\boxtimes(f\mu) \right\rangle = \int_M g(x)f(x)\,\mu(x).$$

Its support is the diagonal

$$\Delta_M = \{(x,x):x\in M\} \subset M\times M.$$

It is not a function $K(x,y)$, and the notation $\delta_\Delta^\mu(x,y)$ does not license evaluation at $x=y$. This is the global kernel fixed uniquely by the displayed pairing. Its Euclidean local model with Lebesgue density is Dyatlov 2022, Proposition 7.7, PDF.

First QFT application: a two-point bidistribution

Let $(M,g)$ be a globally hyperbolic Lorentzian spacetime with the site’s $(+---)$ signature, and let

$$P = \Box_g+m^2+\xi R$$

for real $m^2$ and $\xi$. With compactly supported tests, $P$ is formally self-adjoint relative to $\mu_g$. Let $\Phi(f)$ denote a real scalar field smeared with $f\in C_c^\infty(M)$, and let $\varpi$ be a state for which

$$W_\varpi(f,h) = \varpi\bigl(\Phi(f)\Phi(h)\bigr)$$

is a continuous bilinear functional in the product test topology. It is complex bilinear after complexifying the real test space. The kernel theorem produces a unique bidistribution

$$W_{\varpi,2} \in \mathcal D'(M\times M)$$

relative to $\mu_g\boxtimes\mu_g$, with

$$\boxed{ \left\langle W_{\varpi,2}, (f\mu_g)\boxtimes(h\mu_g) \right\rangle = W_\varpi(f,h). }$$

This is the precise meaning of the symbolic expression $W_{\varpi,2}(x,y)$. It does not assert that a value exists at any pair of points.

If the field equation is imposed weakly,

$$\Phi(Pf)=0,$$

then the two-point kernel is a bisolution:

$$P_xW_{\varpi,2}=0, \qquad P_yW_{\varpi,2}=0.$$

For example,

$$\begin{aligned} \left\langle P_xW_{\varpi,2}, (f\mu_g)\boxtimes(h\mu_g) \right\rangle &= W_\varpi(Pf,h) \\ &= 0. \end{aligned}$$

State positivity is a separate condition,

$$W_\varpi(\overline f,f)\geq0.$$

Here the conjugation is explicit; neither positivity nor the field equation is a consequence of the kernel theorem alone. The two-point bilinear functional, commutator relation, positivity, and weak-bisolution conditions are compared in Fewster and Rejzner 2019, § 4.3, Eq. (36), PDF. Kay 2023, §§ 1–2, PDF defines metric-volume smearing and $n$-point distributions and derives the weak bisolution property. Kay uses $(-+++)$ and $(\Box_g-m^2)\phi=0$; reversing the metric converts it to the convention used here.

The classification of advanced, retarded, causal, Wightman, and Feynman kernels, and the state-dependent conditions imposed on them, belongs to Green Operators, Causal Propagators, and State-Dependent Two-Point Functions.

Diagonals and kernel compositions are stop signs

A distribution $K\in\mathcal D'(M\times M)$ cannot generally be restricted to the diagonal. The notation

$$K(x,x)$$

would require pulling $K$ back along the embedding $\Delta:M\to M\times M$. Arbitrary embeddings do not admit pullback of every distribution. A diffeomorphism, and more generally a submersion, is safe; diagonal restriction requires information about the singular covectors of $K$.

Likewise, for kernels on $M\times N$ and $N\times L$, the formal composition

$$K_{AB}(x,z) \stackrel{\mathrm{formal}}{=} \int_N K_A(x,y)K_B(y,z)\,\mu_N(y)$$

contains three nontrivial operations: pull both kernels to $M\times N\times L$, multiply them there, and integrate the product along $N$. In the scalar-distribution convention used here, the displayed $\mu_N$ supplies the fiber density needed for that last operation. The product needs a singularity criterion, and the $\mu_N$-weighted fiber pushforward needs a support or properness condition. Kernel notation does not supply either one.

Singular Support and Wavefront Sets develops the directional criterion for pullback and products. Products, Scaling Degree, and Extensions of Singular Distributions develops the extension ambiguity that appears when singular products fail at a diagonal.

Operation checklist and stop conditions

Before using a distribution or kernel on a manifold, identify:

1. Geometric type. Does the object act on functions, densities, or dual-bundle-valued densities? A coordinate coefficient without its transformation law is not a global distribution.
2. Test topology. Are support and all-derivative convergence controlled? Kernel uniqueness requires continuity in that topology.
3. Density choice. If formulas use $\sqrt{-g}\,\mathrm d^4x$ or a point delta, record the positive density that identifies scalar tests with test densities.
4. Kernel meaning. A kernel is a distribution on a product manifold. Do not infer pointwise values from the notation $K(x,y)$.
5. Map and singular set. Diffeomorphism and submersion pullbacks are safe. Stop before an embedding, diagonal restriction, or coincident-point limit unless a sharper theorem applies.
6. Composition. Verify the pullback, product, and proper pushforward separately. Formal integration over an intermediate variable is not a definition for arbitrary singular kernels.

Exercises

1. Let $\mu'=a\mu$ for a positive smooth function $a$. Express the same test density $\omega=f\mu$ as $f'\mu'$ and verify the transformation of $\delta_p^\mu$.

   Check

   Equality $\omega=f\mu=f'\mu'$ gives $f'=f/a$. Hence

   $$\langle\delta_p^{\mu'},\omega\rangle = f'(p) = \frac{f(p)}{a(p)} = \left\langle \frac{\delta_p^\mu}{a(p)},\omega \right\rangle.$$

   The scalar delta changes because its test object is a density; the function-evaluation rule remains unchanged after the new identification.

2. Let $f\in C^1(M)$, let $u_f$ be its regular distribution, and let $X$ be a vector field. Show that the intrinsic derivative $Xu_f$ is the regular distribution represented by $Xf$.

   Check

   Write $\omega=g\mu$. Compact support and the divergence theorem give

   $$\begin{aligned} \langle Xu_f,g\mu\rangle &= -\int_M f\, \bigl(Xg+g\,\operatorname{div}_\mu X\bigr)\mu \\ &= \int_M (Xf)g\,\mu = \langle u_{Xf},g\mu\rangle. \end{aligned}$$

   For nonsmooth $f$, the first line remains the definition even when the last line has no classical representative.

3. Use the diagonal kernel to evaluate the operator associated with $\delta_\Delta^\mu$ on $f\mu$.

   Check

   For every test $g$,

   $$\left\langle A(f\mu),g\mu \right\rangle = \left\langle \delta_\Delta^\mu, (g\mu)\boxtimes(f\mu) \right\rangle = \int_M gf\,\mu.$$

   Therefore $A(f\mu)=u_f$. The kernel is singular even though the induced operation is the familiar identity after the $\mu$-identification.

4. Which assumption, besides the kernel theorem, is used to prove $P_xW_{\varpi,2}=0$?

   Check

   One needs the field relation $\Phi(Pf)=0$ and formal self-adjointness of $P$ relative to $\mu_g$. The kernel theorem only represents the continuous bilinear functional; it does not impose a differential equation or state positivity.

References

• Christian Bär, Nicolas Ginoux, and Frank Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization, PDF, §§ 1.1.1–1.1.2, European Mathematical Society, 2007. This is an independent structural source for test-section topology, distributional sections, formal adjoints, support, and singular support. It fixes a smooth volume density; the page translates its function-test convention to intrinsic test densities.
• Semyon Dyatlov, Lecture Notes for 18.155: Differential Analysis, §§ 7.1–7.2 and 13.1–13.2, PDF, MIT, 2022. This is the teaching and theorem-level source for tensor products, distributional kernels, the Schwartz kernel theorem, densities, distributions and distributional sections on manifolds, and safe submersion pullback.
• Christopher J. Fewster and Kasia Rejzner, Algebraic Quantum Field Theory—an Introduction, PDF, § 4.3, 2019. Equation (36) and the discussion following it provide an independent check of the two-point bilinear functional, positivity, commutator relation, and weak-bisolution structure.
• Bernard S. Kay, Quantum Field Theory in Curved Spacetime, §§ 1–2, PDF, 2nd ed., 2023. This is the QFT source for metric-volume smearing, $n$-point distributions, positivity, and the weak bisolution property. Kay uses $(-+++)$; the page explicitly translates the field equation to $(+---)$.