The Dirac-delta

The Dirac-delta functional δ(x–a) is an extremely useful mathematical object despite its abstract nature. It was proposed by Paul Dirac – an electrical engineer who decided to become a physicist and shared the 1933 Nobel Prize in Physics with Erwin Schrodinger. The story went that mathematicians did not like the way the Dirac-delta was derived and defined by its inventor, which was probably considered not rigorous enough, but in the end mathematicians recognized its beauty and importance. Paul Dirac also wrote a quantum mechanics textbook, The Principles of Quantum Mechanics, which was written concisely and beautifully at the same time.

The Dirac-delta exemplifies a proof that when an idea anchors several things at once, it needs not be explicit. It is a subtle object since it is defined by an integral representation:

∫ dx f(x)δ(x–a) = f(a),

where the integral is performed over an interval that includes x = a. δ(x–a) = 0 in almost everywhere except at x = a. Despite this extremely localized effect, it is so strong that it never gives a chance to f(x), as it is integrated over the interval, to yield anything other than its value at x = a. It is difficult to explicitly write what δ(x–a) is. There are several representations of the Dirac delta, e.g., a Gaussian peak and a Lorentzian peak, but they are still approximations to what the integral representation sufficiently captures.

Using the integral representation, we can derive three basic properties of the Dirac-delta:

1. ∫ dx f(x) δ⁽ⁿ⁾(x–a) = (-1)ⁿ f ⁽ⁿ⁾(a), where the (n) superscript indicates an nth derivative;

2. ∫ f(x) δ(ax–b) = a^-1 f(b/a);

3. ∫ dx f(x) δ(g(x)) = ∑ f(x_i)/g^'(x_i), where x_i are roots of g(x) and the ' superscript indicates a first derivative.

The marvel of the Dirac-delta is its ability to concisely sum up a complete set of continuous functions for the space of nonperiodic functions. It is used to represent a Fourier transform:

δ(x–y) = (2π)^-1 ∫ dk exp[ik(x–y)],

where the integral is over –∞ < k < ∞. This result is obtained from the Fourier series representation of the Dirac-delta and from extending the domain of the Fourier series representation to cover the entire space (–∞ < x < ∞).

The integral representation would become

∫ dx f(x) δ(x–y) = (2π)^-1 ∫ dx f(x) ∫ dk exp[ik(x–y)]

                          = (2π)^-1 ∫ dk exp[–iky] f(k) = f(y),

which defines both Fourier transform and its inverse of f(x). This is the most concise way to define a Fourier transform I know - as clearly explained by Mark Swanson's Path Integrals and Quantum Processes. A complete derivation of Fourier transform and its math details can found in Ian Sneddon's Fourier Transforms, which is a terrific source book on Fourier transform.

The Dirac-delta is used to model effects of periodic diffraction slits, to obtain Green's functions of various differential equations, and many more. It has wide-ranging applications, from specific mathematical models to fundamental equations.