Distributions

Prerequisites: Calculus, Fourier Transform. Full path (6 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?
• FAQ
• History

Intuitive

Physicists often want to talk about objects that ordinary functions cannot describe: a point charge, a force that acts in a single instant, a wave with exactly one frequency spread over all of space. The charge density of a point charge would have to be zero everywhere except at one point and still integrate to the total charge. No function does that.

The way out is to stop asking for the value of the object at each point and only ask what it does when we average it against smooth, well-behaved functions. A measuring device never reads off a field at a mathematical point anyway. It always averages over a small region. A distribution is a rule that assigns a number to each such smooth averaging function. Every ordinary function defines a distribution this way, but there are many more, the Dirac delta being the most famous one: it simply returns the value of the averaging function at one point.

With this point of view one can differentiate things that have jumps or spikes, Fourier transform things that do not decay, and give a meaning to many formulas that physicists used long before they were made rigorous. A distribution is a generalized function.

Concrete

Test functions and distributions

Let $\mathcal{D}(\mathbb{R}^n)=C_c^\infty(\mathbb{R}^n)$ be the space of smooth functions with compact support, called test functions. A distribution $T$ is a linear map $\varphi\mapsto T[\varphi]$ from test functions to complex numbers that is continuous in the sense that $T[\varphi_k]\to 0$ whenever the supports of the $\varphi_k$ stay in one fixed compact set and all derivatives of $\varphi_k$ go to zero uniformly.

Every locally integrable function $f$ defines a distribution $$T_f[\varphi]=\int f(x)\varphi(x)\,d^nx .$$ The Dirac delta at the point $a$ is $$\delta_a[\varphi]=\varphi(a),$$ and physicists write this as $\int \delta(x-a)\varphi(x)\,dx=\varphi(a)$. There is no function $\delta(x)$ for which this integral holds; the integral sign is notation.

Derivatives

The derivative of a distribution is defined by moving the derivative onto the test function, which is what integration by parts would give for a differentiable $f$ (the boundary terms vanish because $\varphi$ has compact support): $$T'[\varphi]=-T[\varphi'].$$ Every distribution can therefore be differentiated arbitrarily often. As an example take the Heaviside step function $\Theta(x)$, equal to $1$ for $x>0$ and $0$ for $x<0$: $$\Theta'[\varphi]=-\int_0^\infty \varphi'(x)\,dx=\varphi(0)-\varphi(\infty)=\varphi(0),$$ so $\Theta'=\delta$. In the same way $\delta'[\varphi]=-\varphi'(0)$.

Useful rules

• Multiplication by a smooth function $g$: $(gT)[\varphi]=T[g\varphi]$. Hence $g(x)\delta(x)=g(0)\delta(x)$ and $x\,\delta(x)=0$.
• Composition: if $g$ is smooth and has only simple zeros $x_i$ with $g'(x_i)\neq 0$, then $\delta(g(x))=\sum_i \delta(x-x_i)/|g'(x_i)|$.
• Products of two distributions are in general not defined. For example $\delta(x)^2$ has no meaning. This is the mathematical origin of many divergences in quantum field theory, where propagators are distributions and products of them at the same point appear.

Fourier transform

With the convention $\tilde f(k)=\int f(x)e^{-ikx}\,dx$ the Fourier transform is defined on tempered distributions (see below) by $\tilde T[\varphi]=T[\tilde\varphi]$. This gives $$\int e^{-ikx}\,dx = 2\pi\,\delta(k),\qquad \tilde\delta(k)=1 .$$ A plane wave, which is not square integrable, thus has a delta function as its Fourier transform.

• No-Nonsense Electrodynamics by Jakob Schwichtenberg, appendix "Delta Distribution"

Abstract

The space $\mathcal{D}(\Omega)=C_c^\infty(\Omega)$ for an open set $\Omega\subset\mathbb{R}^n$ carries the inductive limit topology of the Fréchet spaces $\mathcal{D}_K(\Omega)$ of test functions supported in compact sets $K\subset\Omega$. The space of distributions $\mathcal{D}'(\Omega)$ is its topological dual. A linear functional $T$ is a distribution exactly when for every compact $K$ there are $C$ and $m$ with $$|T[\varphi]|\le C\sum_{|\alpha|\le m}\sup_{x\in K}|\partial^\alpha\varphi(x)|\quad\text{for all }\varphi\in\mathcal{D}_K(\Omega).$$ If one $m$ works for all $K$, the smallest such $m$ is the order of $T$; $\delta$ has order $0$.

For Fourier analysis one uses the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ of smooth functions which, together with all derivatives, decay faster than any power. It is a Fréchet space with the seminorms $\sup_x|x^\beta\partial^\alpha\varphi(x)|$, and $\mathcal{D}\subset\mathcal{S}$ is dense. Its dual $\mathcal{S}'(\mathbb{R}^n)$ is the space of tempered distributions. The Fourier transform is a topological isomorphism of $\mathcal{S}$, and by duality also of $\mathcal{S}'$. The chain $$\mathcal{S}(\mathbb{R}^n)\subset L^2(\mathbb{R}^n)\subset\mathcal{S}'(\mathbb{R}^n)$$ is the standard example of a rigged Hilbert space and gives a precise meaning to the improper eigenvectors $|x\rangle$ and $|p\rangle$ of Dirac notation.

Some structural results:

• Support and structure: a distribution with support in a single point $a$ is a finite linear combination of derivatives $\partial^\alpha\delta_a$.
• Schwartz kernel theorem: every continuous linear map $\mathcal{D}(\mathbb{R}^m)\to\mathcal{D}'(\mathbb{R}^n)$ is given by a unique kernel $K\in\mathcal{D}'(\mathbb{R}^n\times\mathbb{R}^m)$. This justifies writing operators as integral kernels such as $\langle x|A|y\rangle$.
• Products: there is no associative product on $\mathcal{D}'$ extending the pointwise product of continuous functions and compatible with differentiation (Schwartz impossibility result). Products can be defined when the singularities are compatible, which is made precise by the wave front set $\mathrm{WF}(T)\subset T^*\Omega\setminus 0$. Hörmander's criterion: $T_1T_2$ is well defined if $\mathrm{WF}(T_1)\oplus\mathrm{WF}(T_2)$ contains no point of the form $(x,0)$.

In quantum field theory a field is an operator valued distribution: $\phi[f]=\int\phi(x)f(x)\,d^4x$ is a (densely defined) operator for each test function $f$, while $\phi(x)$ at a point is not. This is built into the Wightman axioms, where the vacuum expectation values $\langle 0|\phi(x_1)\cdots\phi(x_n)|0\rangle$ are tempered distributions. Renormalization in position space can be phrased as the problem of extending products of distributions such as $\Delta_F(x)^2$ from $\mathbb{R}^4\setminus\{0\}$ to all of $\mathbb{R}^4$ (Epstein and Glaser).

Why is it interesting?

Distributions are the correct mathematical tools to describe fields in quantum field theory.

FAQ

History

Most people associate the origin of the delta function with the great physicist Paul Dirac, but in fact, it was introduced into mathematics 100 years earlier by Fourier and Cauchy. See WHO INVENTED DIRAC’S DELTA FUNCTION? MIKHAIL G. KATZ AND DAVID TALL

Path to this page

1. Calculus
2. Imaginary Numbers
3. Trigonometric Functions
4. Exponential Function
5. Euler's Formula
6. Fourier Transform

Leads to: Axiomatic Quantum Field Theory.