Distributions and Microlocal Methods

This chapter replaces informal “singular-function” manipulations by continuous functionals on declared test spaces. It begins with distributions, support, and convergence; develops delta sources, weak derivatives, Fourier calculus, and kernels on manifolds; then adds two advanced controls: scaling-degree extension at coincidence and wavefront directions for products and pullbacks. A reader should take only the route required by the operation at hand.

The chapter is a bounded microlocal bridge. It supplies reusable definitions, theorems, examples, and failure tests, but it does not develop propagation of singularities, Hadamard states, the microlocal spectrum condition, Epstein–Glaser induction, or physical renormalization schemes. Those topics are treated in Mathematical QFT, Curved Spacetime, and Renormalization and EFT.

Parent volume: Mathematical Methods

Enter this chapter

The overview has no hard prerequisite. Choose an entry point by identifying what is currently being treated too formally.

Readiness check	Ready	If unsure	Repair and return
Can you state which test space a singular object acts on and what convergence means there?	Enter at the page matching your operation.	Ask whether your probes are compactly supported, rapidly decreasing, or sections on a manifold.	Begin with Test-Function Spaces, Distributions, Support, and Convergence.
Can you derive $H'=\delta_0$ by pairing rather than by pointwise notation?	Enter the localized-source route.	Integrate $-\int_0^\infty\varphi'(x)\,\mathrm dx$ and identify the boundary term.	Read the foundation page, then Delta Distributions, Weak Derivatives, Pullbacks, and Pushforwards.
Can you translate $\mathcal F(\partial_\mu u)$ in the site’s positive-phase convention?	Enter the momentum-space route.	Check whether the multiplier is $+ip_\mu$ or $-ip_\mu$ before using a propagator.	Use Tempered Distributions and Fourier Calculus.
Can you distinguish a tensor-product kernel from its restriction to the diagonal?	Enter the kernel route.	Ask which test density the kernel acts on and whether the diagonal pullback is licensed.	Read Distributional Kernels and Distributions on Manifolds, then the wavefront page if restriction is needed.
Can you explain what data remain after extending a singular distribution across coincidence?	Enter the extension route.	Compare the scaling degree with the relevant dimension or codimension.	Use Products, Scaling Degree, and Extensions of Singular Distributions.
For a product, can two singular covectors sum to zero? For a pullback, is a singular covector annihilated by the transpose differential?	Enter the microlocal route.	Localize, identify the nondecaying Fourier cones, and compute the map’s normal set.	Read the tempered-distribution page, then Singular Support and Wavefront Sets.

Readers who need a connected review of Fourier and Green-kernel reasoning can use Fourier Transforms, Distributions, and Green Kernels and then return to the relevant topic.

Choose a route

The arrows indicate a coherent order for the stated goal. A plus sign means that both branches are useful before the final operation.

Reader goal	Minimum coherent route	Capability at the end
Working distributional calculus	Test-Function Spaces, Distributions, Support, and Convergence → Delta Distributions, Weak Derivatives, Pullbacks, and Pushforwards	Differentiate singular objects, localize support, and use pullback or pushforward only under visible hypotheses
Momentum-space Green kernels	Test-Function Spaces, Distributions, Support, and Convergence → Tempered Distributions and Fourier Calculus	Treat a singular multiplier and its inverse transform as elements of $\mathcal S'$ with phase, normalization, and $i0$ fixed
Coordinate-independent two-point kernels	Test-Function Spaces, Distributions, Support, and Convergence → Distributional Kernels and Distributions on Manifolds; add Tempered Distributions and Fourier Calculus when momentum variables enter	Pass between chart representatives and an intrinsic product-manifold kernel without inventing point values
Local extension at coincidence	Test-Function Spaces, Distributions, Support, and Convergence → Products, Scaling Degree, and Extensions of Singular Distributions	Decide whether a punctured distribution has a unique scaling-preserving extension and list the allowed contact terms
Product or restriction of singular kernels	Test-Function Spaces, Distributions, Support, and Convergence → Tempered Distributions and Fourier Calculus → Singular Support and Wavefront Sets; add Distributional Kernels and Distributions on Manifolds for geometric applications	Test opposite covectors and normal directions before multiplying or restricting
Curved-spacetime preparation	Test-Function Spaces, Distributions, Support, and Convergence → Distributional Kernels and Distributions on Manifolds + Tempered Distributions and Fourier Calculus → Singular Support and Wavefront Sets	Interpret two-point functions as bidistributions and recognize the exact handoff to curved Green functions and Hadamard theory
Local-renormalization preparation	Test-Function Spaces, Distributions, Support, and Convergence → Products, Scaling Degree, and Extensions of Singular Distributions + Tempered Distributions and Fourier Calculus → Singular Support and Wavefront Sets	Separate three questions: whether the off-diagonal product exists, whether it extends, and what additional input fixes its local ambiguity

The hard dependencies are deliberately sparse. Every non-entry leaf uses the test-function foundation. The wavefront page additionally requires tempered Fourier calculus. Fourier analysis, differential geometry, nuclear spaces, and general convergence theory are recommended only where their extra machinery is actually used.

How the six pages fit together

The chapter follows the lifecycle of a singular expression.

1. Declare its test space. A distribution is known through pairings, support, order, and convergence—not through values at points.
2. Define safe operations by duality. Derivatives move to the test function; smooth multipliers act on the probe; pullbacks and pushforwards expose rank and properness requirements.
3. Choose the correct global growth class. Schwartz probes make $\mathcal S'$ stable under Fourier transformation and turn constant-coefficient equations into multiplier equations.
4. Make coordinate type explicit. Test densities absorb absolute Jacobians, and the Schwartz kernel theorem represents continuous operators by distributions on product manifolds.
5. Separate existence from extension. A product must first be licensed away from coincidence. Finite scaling degree then controls extension across the missing set and bounds its contact-term freedom.
6. Resolve directional singularity. Localized Fourier cones form the wavefront set. Opposite covectors diagnose products, while normal covectors diagnose pullbacks and restrictions.

These are distinct questions. Temperedness controls behavior at infinity; singular support controls where smoothness fails; scaling degree controls short-distance strength; and the wavefront set controls cotangent direction. None can replace the others.

Shared conventions

The following choices apply across the chapter.

Issue	Convention	Check
Distribution pairing	$\langle u,\varphi\rangle$ is complex-linear in the test object	No implicit conjugation appears in a distributional duality formula
Test spaces	$\mathcal D=C_c^\infty$ and $\mathcal S$ is the Schwartz space	An operation must preserve the declared test space
Fourier transform	$\widetilde f(p)=\int e^{+ip\cdot x}f(x)\,\mathrm d^d x$ and $f(x)=\int(2\pi)^{-d}e^{-ip\cdot x}\widetilde f(p)\,\mathrm d^d p$	$\mathcal F(\partial_\mu u)=-ip_\mu\mathcal Fu$
Metric	QFT-facing spacetime formulas use $(+---)$	$p^2=(p^0)^2-\lvert\mathbf p\rvert^2$
Boundary values	$1/(s\pm i0)=\operatorname{PV}(1/s)\mp i\pi\delta(s)$	Reversing $i0$ reverses the on-shell delta term
Manifolds	Scalar distributions act on compactly supported smooth densities	Coordinate Jacobians belong to the density transformation law
Delta constraints	Absolute Jacobians and regular-value hypotheses are explicit	$\delta(g)$ is not licensed at a critical zero by the simple-root rule
Wavefront set	Fiber cones use the site’s positive Fourier phase	$(x+i0)^{-1}$ has the negative one-dimensional frequency cone
Microlocal direction	Wavefront elements are nonzero covectors	The zero section is excluded and no metric is needed for the definition

Sources using $e^{-i\xi\cdot x}$ reflect every wavefront covector relative to the last two rows. The product and pullback criteria survive that consistent reflection, but future/past labels must not be copied without translating the Fourier phase.

Exact page guide

Test-Function Spaces, Distributions, Support, and Convergence

The foundation page has no hard prerequisite. It defines $\mathcal D(\Omega)$, its convergence topology, continuous linear functionals, regular distributions, support, singular support, order, restriction, convergence, and smooth multiplication. It explains why a quantum field is meaningfully smeared rather than evaluated at a point. Every later leaf depends on this page.

Reader role: entry foundation. Most useful continuation: Delta Distributions, Weak Derivatives, Pullbacks, and Pushforwards.

Delta Distributions, Weak Derivatives, Pullbacks, and Pushforwards

The delta page requires the foundation. It derives delta derivatives, jump terms, weak-versus-distributional derivatives, interface sources, regular-level-set constraints, coarea factors, submersion pullbacks, and proper-on-support pushforwards. Its free-scalar contact term and mass-shell measure are controlled examples; coincident operator products and full phase-space physics remain elsewhere.

Reader role: core operational page. Most useful continuation: Tempered Distributions and Fourier Calculus.

Tempered Distributions and Fourier Calculus

The tempered page requires the foundation; Fourier Series, Fourier Transforms, and Plancherel Theory is recommended preparation. It defines $\mathcal S$ and $\mathcal S'$, extends the site’s Fourier transform by duality, derives differentiation, multiplication, convolution, and normalization rules, and constructs $i0$ boundary values. Its Feynman-kernel check yields

$$(\Box+m^2)\Delta_F = -i\delta^{(d)}$$

without treating the inverse transform as a pointwise improper integral.

Reader role: core momentum-space page. Most useful continuation: Singular Support and Wavefront Sets.

Distributional Kernels and Distributions on Manifolds

The manifold-kernel page requires the foundation; Smooth Manifolds, Tangent and Cotangent Bundles, and Tensor Fields and Locally Convex, Nuclear, and Rigged Hilbert Spaces are recommended preparation. It uses test densities to make scalar distributions coordinate independent, defines distributional bundle sections, and states the manifold Schwartz kernel theorem. The diagonal delta shows that even the identity operation can have a singular kernel. A curved-spacetime two-point bidistribution is the bounded QFT bridge.

Reader role: core depth page. Most useful continuation: Singular Support and Wavefront Sets.

Products, Scaling Degree, and Extensions of Singular Distributions

The extension page requires the foundation; Limits, Completeness, and Modes of Convergence is recommended preparation. It first explains why distributions do not form an algebra, then defines scaling degree and proves the point-extension threshold. If $t^\circ\in\mathcal D'(\mathbb R^d\setminus\{0\})$ has finite scaling degree $\omega$, its scaling-preserving extension is unique for $\omega<d$. For $\omega\geq d$, two such extensions differ by

$$\sum_{|\alpha|\leq\lfloor\omega-d\rfloor} c_\alpha\partial^\alpha\delta_0.$$

The four-dimensional Euclidean Green-kernel square makes that local freedom explicit without choosing a physical renormalization scheme.

Reader role: advanced bridge. Most useful continuation: Singular Support and Wavefront Sets.

Singular Support and Wavefront Sets

The wavefront page requires the foundation and tempered Fourier calculus. It defines rapid decay in a localized open cone, proves that the base projection of $\operatorname{WF}(u)$ is $\operatorname{sing\,supp}u$, and contrasts the opposite one-sided wavefront sets of $(x\pm i0)^{-1}$. It states the no-opposite-covector product criterion and the normal-set pullback criterion. A relative-time kernel is restrictable to a fixed-time slice but not canonically to its diagonal, illustrating why position alone is insufficient.

Reader role: advanced microlocal bridge. Most useful continuation: Wavefront-Set Products, Pullbacks, and Pushforwards.

One Green kernel across the chapter

The free scalar Feynman kernel is a compact coherence test. In the site’s conventions,

$$\widetilde\Delta_F(p) = \frac{i}{p^2-m^2+i0} \in\mathcal S'(\mathbb R^d), \qquad \Delta_F = \mathcal F^{-1}\widetilde\Delta_F.$$

Each chapter layer answers a different question:

1. Distribution: pairing with a test function defines the singular multiplier and inverse transform.
2. Delta calculus: applying $\Box+m^2$ gives the localized source $-i\delta^{(d)}$.
3. Fourier calculus: the $+i0$ boundary value separates a principal-value part from an on-shell delta and fixes the inverse.
4. Kernel theorem: translation invariance may be written as the bidistribution $K(x,y)=\Delta_F(x-y)$ rather than as pointwise data.
5. Wavefront test: avoiding the kernel’s normal or opposite singular covectors licenses the canonical pullback or product theorem; failure of that sufficient condition does not prove universal nonexistence.
6. Extension: if a legitimate off-diagonal distribution has finite scaling degree and needs extension across coincidence, scaling degree determines the finite local ambiguity.

The order matters. Scaling degree cannot manufacture a product that was undefined off the diagonal, and a wavefront set cannot choose the contact terms of an extension. Likewise, the Green equation alone does not determine the physical interpretation of the boundary condition.

Chapter synthesis

The minimum chapter-scale conclusions are:

• a singular object is defined by its action on a named test space;
• distributional differentiation always exists, while a weak derivative in a specified function space is an additional representation claim;
• pullback transports against a map and needs rank or wavefront transversality, while pushforward transports with the map and needs support/properness plus the correct density type;
• $\mathcal S'$ is Fourier stable, whereas an arbitrary element of $\mathcal D'$ need not have a Fourier transform;
• a kernel is a distribution on a product manifold, not automatically a function of two points;
• external tensor products are always defined, but pointwise products and diagonal restrictions are conditional;
• finite scaling degree guarantees an extension and bounds its delta-derivative ambiguity, but does not select the coefficients; and
• the wavefront set adds nonzero cotangent directions to singular support, making product and pullback obstructions visible.

Together these statements answer the organizing question: enter through working distributions when the issue is meaning, differentiation, support, Fourier inversion, or kernels; enter through the microlocal bridge when the issue is a coincident product, extension, or restriction of singular data.

For the distribution-theory spine, compare Dyatlov 2022, Chapters 2–7 and 10–13, PDF and Hörmander 2003, Chapters III, VII, and VIII. The extension and product criteria used later are developed in Brunetti and Fredenhagen 2000, §§ 5–6, PDF, Brouder, Dang, and Hélein 2014, wavefront-set examples and products, PDF, and Fredenhagen and Rejzner 2012, § 7 and Appendix A, PDF.

Review prompts

Test-space diagnosis. Given a formula containing a delta, a polynomially growing function, and a compactly supported source, state whether each pairing belongs naturally to $\mathcal D'$ or $\mathcal S'$. Success names the probe space and explains why the proposed operation preserves it. Use Test-Function Spaces, Distributions, Support, and Convergence to repair a missing test-space or topology declaration.

Derivative and contact term. Starting from a piecewise $C^1$ function with one jump, derive its distributional derivative and identify when it fails to be an $L^p_{\mathrm{loc}}$ weak derivative. Repair with the delta page if the jump coefficient or function-space qualifier is missing: Delta Distributions, Weak Derivatives, Pullbacks, and Pushforwards.

Fourier convention. Derive $\mathcal F(\partial_\mu u)=-ip_\mu\mathcal Fu$ and recover $\mathcal F1=(2\pi)^d\delta_0$. Success uses the positive forward phase and the linear distribution pairing rather than copying a sign from another source. Check the distributional rule in Tempered Distributions and Fourier Calculus; repair missing function-level Fourier preparation with Fourier Series, Fourier Transforms, and Plancherel Theory.

Kernel type. Explain why a two-point bidistribution is represented by a kernel on $M\times M$ but need not have a value $K(x,x)$. Success identifies diagonal evaluation as a pullback and names the missing transversality test. Use Distributional Kernels and Distributions on Manifolds for the kernel type and Singular Support and Wavefront Sets to repair a missing diagonal pullback test.

Extension threshold. For $t^\circ(x)=|x|^{-d-2}$ on the punctured space, compute the scaling degree and list the delta derivatives allowed in a scaling-preserving extension. Success uses $N=\lfloor\omega-d\rfloor=2$ and then states which further symmetries might reduce the coefficients. The relevant result is in Products, Scaling Degree, and Extensions of Singular Distributions.

Directional comparison. Compare $(x+i0)^{-1}$ and $(x-i0)^{-1}$. Success gives the same singular support, opposite one-sided wavefront cones in the site’s positive-phase convention, and explains why their mixed product fails the no-opposite-covector test. Use Singular Support and Wavefront Sets for the criterion and Tempered Distributions and Fourier Calculus to repair an untranslated Fourier phase.

Full operation check. A proposed kernel composition contains pullbacks, a product, and a fiber integration. Success checks each operation separately and stops if wavefront transversality or proper support is absent. Combine Distributional Kernels and Distributions on Manifolds with Singular Support and Wavefront Sets; the latter is the repair route when a characteristic covector or normal-set condition has not been checked.

QFT handoffs

The chapter stops when physical or theorem-first specialist input becomes essential.

• Quantum Fields as Operator-Valued Distributions develops the physical interpretation of smeared fields.
• Scalar Propagators, Ordered Correlators, and Sources treats time ordering and the physical propagator taxonomy.
• Green Operators, Causal Propagators, and State-Dependent Two-Point Functions develops curved-spacetime Green and two-point kernels.
• Wavefront-Set Products, Pullbacks, and Pushforwards develops detailed covector bookkeeping, proper pushforwards, kernel composition, and developed QFT restrictions.
• Scaling Degree and Extension of Distributions gives the theorem-first causal-renormalization construction.
• Local Counterterms and Subdivergence Structure treats counterterm interpretation, subdivergences, and locality conditions.

For operator domains, locally convex topology, and spectral measures, continue to Functional and Spectral Analysis. To choose another mathematical route, return to Mathematical Methods.

References

• Christian Brouder, Nguyen Viet Dang, and Frédéric Hélein, A Smooth Introduction to the Wavefront Set, PDF, Journal of Physics A 47 (2014), 443001. This supplies a positive-phase treatment of localized Fourier cones, examples, products, and pullbacks.
• 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. This is the specialist source for scaling degree and extensions at points and submanifolds.
• Semyon Dyatlov, Lecture Notes for 18.155: Differential Analysis, PDF, Chapters 2–7 and 10–13, MIT, 2022. These chapters develop distributions, support, differentiation, tensor-product kernels, pullbacks, tempered distributions, Fourier calculus, and distributions on manifolds.
• Klaus Fredenhagen and Katarzyna Rejzner, Perturbative Algebraic Quantum Field Theory, PDF, § 7 and Appendix A, 2012. This provides the QFT-facing bridge from wavefront-controlled products to scaling-degree extension.
• Lars Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, 2nd ed., Springer, 2003. Chapters III, VII, and VIII are the structural sources for distribution theory, Fourier analysis, wavefront sets, products, and pullbacks.