mathematics

Laurent and the Dirac Problem

When observation and theory clash… 

Physicist George Green (1793-1841) needed a way to explain the singularities that arose in electrical potential problems. Oliver Heaviside (1850-1925) found a similar need in his study of electrical impulses in telegraph networks. When that need arose again in the 1930s, this time in the context of quantum mechanics, noted physicist Paul Dirac (1902-1984) decided to formerly define the delta function $\delta(x)$ based in part on the innovations of his predecessors. But this newly rebranded Dirac delta function also fell under the gaze of a skeptical math community, many of whom found its quirky, non-rigorous definition problematic. This post will examine both sides of the issue and the truce that followed.

Dirac’s multipart definition of the delta function is as follows:

$$\begin{eqnarray}
\delta(x) = \begin{cases}
\phantom{-} 0 & \text{if } x \ne 0
\\\phantom{-} +\infty & \text{if } x=0
\end{cases}
\end{eqnarray}$$

which when integrated…

$$\begin{eqnarray}
\int^{\infty}_{-\infty} \delta(x) dx = 1
\end{eqnarray}$$

Hint: both of these things can not be true

It also has this useful sifting (or shifting) property:

$$\begin{eqnarray}
f(a) = \int_{\mathbb{R}} f(z) \delta(a-z) dz
\end{eqnarray}$$

To answer this useful vs troublesome question, we’ll start with a few historical examples of how it arose.

Electrostatics: George Green

Coulomb’s Law describes how the electric field generated by a stationary point charge decreases inversely with the square of the distance. The corresponding electric potential falls off inversely with that distance. But interestingly, the net flux through any sphere that surrounds this charge remains constant regardless of the radius. This phenomenon known as Gauss’s Law can be shown by pairing Coulomb’s Law with the Divergence Theorem [GRF].

Here’s a second curiosity. Despite this result, the divergence of the radial term in this electrical field is zero everywhere except at the point itself.

$$\begin{eqnarray}
\nabla \cdot {\hat r \over r^2}
=
{1 \over r^2}{\partial \over \partial r}(r^2 {1 \over r^2})
\end{eqnarray}$$ $$\begin{eqnarray}
={1 \over r^2}{\partial \over \partial r}(1)
=
0
\end{eqnarray}$$

Since the electrical field lines from a positive point charge radiate outward (or inward if negative), this seems paradoxical. The reason? There’s a singularity in the equation. Or more specifically, since the potential falls with the inverse of distance, when the potential is measured at the charge, it becomes infinite. It wasn’t until much later that this relationship got modeled using the delta function, per se, yet the need for such an expression was clearly evident. In modern terms, this phenomenon is expressed as follows:

$$\begin{eqnarray}
\nabla \cdot {\hat r \over r^2}
=
4\pi \,\delta (0)
\end{eqnarray}$$

Note that above the charge is placed at the origin, 0, but it could be anywhere. Next, compute the divergence of the electric field $E$ and apply the delta sifting property to remove the integral.

$$\begin{eqnarray}
\nabla \cdot  E
=
{ 1 \over 4\pi \epsilon_0} \int_{\Omega} \nabla \cdot {\hat r \over r^2} \rho d\Omega
\end{eqnarray}$$
$$\begin{eqnarray}=
{ 1 \over 4\pi \epsilon_0} \int_{\Omega} 4\pi \,\delta (0) \rho d\Omega
=
{ \rho \over 4\pi \epsilon_0}
\end{eqnarray}$$

Then when you compute the flux by integrating $\nabla \cdot  E$ over the surface you end up with$ {Q_{enc} \over 4\pi \epsilon_0}$ as stated in Gauss’s Law.

George Green in his seminal 1828 paper put forth the idea that one could describe a potential distribution over a region as the linear superposition of individual point charges as described above (though he never uses the word “delta”). He then demonstrated that superimposing (integrating) these terms leads to a solution for Poisson’s Equation. This technique also gave rise to his namesake function, the Green’s function, a huge and interesting subject on its own. Look for more on that here in future posts.

Signal Theory: Oliver Heaviside

Heaviside proposed the idea of a step function as a way to model certain electric signals (Heaviside worked to improve telegraph networks). In doing so, he described the delta as the derivative of the step function and used it as such in his symbolic calculus. This argument makes sense heuristically in that the rate of change on both sides of $a$ is zero but at $a$ the function has a discontinuity, which could be considered a derivative of $\infty$.

$$\begin{eqnarray} H(x):=\begin{cases}
0,\ \text{for}\ x < a \\ \\ 1,\ \text{for}\ x\ge a \end{cases} \end{eqnarray}$$

Fourier/Cauchy Observations

Joseph Fourier (1768-1830) stumbled upon the delta sifting property while working on heat diffusion problems in unbounded regions. For the unbounded scenario he took what is now known as the Fourier Series and transformed it from a sum to an integral, now known as the Fourier Transform. He then noted that the Fourier Transform applied to its inverse (in what today would be called a convolution) produces an expression that is precisely the delta sifting property. Cauchy (1789-1857) generalized this a bit further resulting in the following representation:

$$\begin{eqnarray}
f(x)=\frac{1}{2\pi} \int_{\mathbb{R}} e^{iyx}\left(\int_{\mathbb{R}} e^{-iy\xi }f(\xi)\ d\xi \right)dy
\end{eqnarray}$$

This then gives the following representation of the delta function:

$$\begin{eqnarray}
\delta(x-w)=\frac{1}{2\pi} \int_{\mathbb{R}} e^{i(x-w)\xi}\ d\xi
\end{eqnarray}$$

Okay so now we have several compelling examples in support of this weird function. What’s the problem then?

The problem with the delta function as defined is that the integral expression $\int^{\infty}_{-\infty} \delta(x) dx = 1$ makes no mathematical sense given the piecewise definition of $\delta(x)$. After all, a single point has measure zero, therefore its effect on the integral is also zero per the rules of Lebesque integration. More specifically, the integrand is said to be zero almost everywhere, thus so is the value of the integral. Yet miraculously it’s not.

I have trouble with Dirac. This balancing on the dizzying path between genius and madness is awful. — Albert Einstein

Despite these objections, Dirac proceeded to make heavy use of the delta function, a nod to the Kronecker delta function, in his seminal text The Principles of Quantum Mechanics and in subsequent research. And it was Dirac who formalized the equations in the form presented at the beginning of this article. He at least conceded that as functions they were “improper.” He further suggested that they be thought of as the limit of a sequence of functions whose value converges to the desired delta. But this had problems too. To that end, John von Neumann bluntly proclaimed Dirac’s delta to be “a mathematical fiction.”

John Von Neumann


Laurent Schwartz

This conundrum finally got sorted out in the late 1940s by a French mathematician named Laurent Schwartz using a technique he called Distribution Theory. Schwartz’s theory offered much more than a simple fix for the Dirac delta though; it was an enhancement to the language of mathematical analysis itself. And while his work expanded on ideas that had been floating around already, he was the first to rigorously articulate it as a single theory.

Here’s the snarky preface from Schwartz’s original textbook, Théorie Des Distributions:

Translated:

It is more than 50 years since the engineer Heaviside introduced his rules of symbolic calculus in a daring mémoire where unjustified mathematical calculations were used for the solution of physics problems. This symbolic calculus, or operational calculus, has not stopped developing since and is used as a basis for the theoretical studies of electricians. The engineers systematically use it each with its own design, with a more or less calm conscience. It became a technique ‘which is not rigorous but which succeeds well.’ Since Dirac’s introduction of the famous function …the formulas of the symbolic calculus have become even more unacceptable for the rigor of mathematicians…

Ouch. He goes on to add…

How to explain the success of these methods? When such a contradictory situation arises, it is very rare that a new mathematical theory does not result from it which justifies, in a modified form, the language of physicists; this is even an important source of progress in mathematics and physics.

In the work that followed, Dirac’s officium non grata got rebranded a generalized function and all was well again. Schwartz in turn was awarded the Fields Medal for his efforts.

The details of this theory are the subject of our next post.

Hyperfunctions

But wait, there’s more!!

A Japanese mathematician named Mikio Sato invented a variant of distribution theory in 1958 called Hyperfunctions. In the preface of the Urs Graf textbook on Hyperfunctions, the author argues that Sato’s theory is more intuitive than Schwartz’s because it doesn’t rely on an extensive background in functional analysis. He also concedes that while this clarity holds for problems in the complex plane, for higher dimensional problems the mathematics becomes orders of magnitude more difficult (which is probably why it never caught on).

One fun side effect of this theory is that it offers a simple explanation of the relation between the Dirac sifting property and the Cauchy integral theorem, something I noticed early on without finding much discussion in the literature—until I discovered Hyperfunctions.

$ \frac{1}{2 \pi i}\oint_C \frac{f(z)}{z-a} dz = f(a) $

Turns out with hyperfunctions, one can define the delta in terms of a function that is exactly that integrand thus preserving both the sifting formula and Cauchy’s theorem. If interested in any of this, check out that textbook link.

For more on the evolution of delta functions, please see this nice summary by Nicholas Wheeler at Reed College. Here’s another comprehensive article by Katz and Tall on this subject .
More on George Green
More on Paul Dirac
More on Mikio Sato
More on Electrostatics: GRF

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