Distributions and the Schwartz Space

A distribution is a "generalized function" — an object like the Dirac delta that acts on test functions but need not have pointwise values. Distributions make rigorous the delta functions, Green's functions, and — crucially — the operator-valued distributions that quantum fields literally are. This page builds on normed-banach.md; the Hilbert-space refinement is rigged-hilbert-space.md, and the Fourier side connects to complex analysis.

Test functions and distributions

Fix a space of well-behaved test functions:

• $\mathcal{D}=C_c^{\infty }$: smooth, compactly supported functions;
• $\mathcal{S}$: the Schwartz space of smooth functions decaying (with all derivatives) faster than any polynomial.

A distribution is a continuous linear functional on test functions — an element of the dual space ${\mathcal{D}}^{\prime }$ (or ${\mathcal{S}}^{\prime }$). Rather than evaluate at points, a distribution $T$ is known by its pairings $⟨T,\phi ⟩$ against all test functions $\phi$. Every locally integrable function $f$ is a distribution via $⟨f,\phi ⟩=\int f\phi$, so distributions genuinely generalize functions.

The Dirac delta

The archetype is the Dirac delta $$⟨\delta ,\phi ⟩=\phi (0),$$ the functional "evaluate at $0$". It is not a function (no locally integrable $f$ reproduces it), but it is a perfectly good distribution — the rigorous meaning of the physicist's $\delta (x)$ with $\int \delta \phi =\phi (0)$. Its translates, derivatives, and the identity $\delta (x)=\frac{1}{2\pi }\int e^{ikx}dk$ (a Fourier statement, below) are all distributional. The Sokhotski–Plemelj formula $\frac{1}{x\mp i\epsilon }=\mathrm{P}\frac{1}{x}\pm i\pi \delta (x)$ is an identity in ${\mathcal{D}}^{\prime }$.

Distributional derivatives

Every distribution is infinitely differentiable, by moving the derivative onto the test function (motivated by integration by parts): $$⟨T^{\prime },\phi ⟩=-⟨T,{\phi }^{\prime }⟩.$$ So non-differentiable and even discontinuous functions have derivatives as distributions: the derivative of the step function $\Theta$ is ${\Theta }^{\prime }=\delta$, and the second derivative of $|x|$ is $2\delta$. This is what lets differential equations be solved in a generalized sense and is the backbone of the weak formulations used throughout PDE and physics.

Tempered distributions and the Fourier transform

The tempered distributions ${\mathcal{S}}^{\prime }$ (duals of the Schwartz space) are the right class for Fourier analysis: because the Fourier transform maps $\mathcal{S}\to \mathcal{S}$ bijectively, it extends by duality to ${\mathcal{S}}^{\prime }\to {\mathcal{S}}^{\prime }$, $$⟨\hat{T},\phi ⟩=⟨T,\hat{\phi }⟩.$$ Then $\hat{\delta }=1$ and $\hat{1}=2\pi \delta$ — the plane-wave completeness relation. Tempered distributions are exactly the objects quantum fields are valued in: QFT/postulates.md defines a field as an operator-valued tempered distribution, $\varphi (f)=\int \varphi (x)f(x)d^{4}x$ for $f\in \mathcal{S}({\mathbb{M}}^4)$ — smearing against a test function is what makes the field a well-defined operator, since $\varphi (x)$ at a sharp point is too singular.

Green's functions and fundamental solutions

A fundamental solution of a linear differential operator $L$ is a distribution $G$ with $$LG=\delta .$$ Then $Lu=f$ is solved by convolution $u=G\ast f$. These Green's functions are distributions (they typically have $1/r$ or $\log$ singularities): the Coulomb potential $-\frac{1}{4\pi r}$ is the fundamental solution of the Laplacian ($-{\nabla }^{2}G=\delta$), and the Feynman propagator is the fundamental solution of the Klein–Gordon operator with the $i\epsilon$ prescription selecting the causal one. Propagators throughout QFT are distributional Green's functions.

References

• Reed & Simon, Methods of Modern Mathematical Physics I, Ch. V; II, Ch. IX.
• Hörmander, The Analysis of Linear Partial Differential Operators I.
• Strichartz, A Guide to Distribution Theory and Fourier Transforms.