Motivating quantum field theory: the boosted particle in a box

It is a maxim often stated, yet rarely illustrated, that the combination of special relativity and quantum mechanics necessarily leads to quantum field theory. An elementary illustration is provided, using the familiar particle in a box, boosted to relativistic speeds. It is shown that quantum fluctuations of momentum lead to energy fluctuations, that are inexplicable without a framework that endows the vacuum with dynamical degrees of freedom and allows particle creation/annihilation.