Cavity Quantum Electrodynamics: The Strange Theory of Light in a Box

Appendix F

The Good, the Bad, and the Ugly: Principal Parts, Step and Delta Functions

Is this nonsense? … Not at all. … We have to note that if Q is any function of the time, then (∂/∂t)Q is its rate of increase. If, then, as in the present case, Q is zero before and constant after t = 0, (∂/∂t)Q is zero except when t = 0. It is then infinite. But its total amount is Q. That is to say, (∂/∂t)θ(t) means a function of t which is wholly concentrated at the moment t = 0, of total amount 1. It is impulsive, so to speak. The idea of an impulse is well known in mechanics, and it is essentially the same here. … the function (∂/∂t)θ(t) … involves only the ordinary ideas of differentiation and integration pushed to their limit.

—Oliver Heaviside [281]

Most physics books have either a superficial nonrigorous introduction to this subject, or a very rigorous mathematical treatment. The former often leaves readers pray to many pitfalls. The latter is often difficult to follow, leaving readers when doing practical calculations to work out for themselves how to connect to their real needs the abstract theorems proved. This appendix intends to be different. Here we present a number of nonrigorous physicist-type derivations that connect together many key results regarding delta functions, step functions, principal parts, and their various representations. But rather than presenting only a superficial treatment, our aim is to guide the reader through a number of subtleties and pitfalls that often ...