The Wrong Equation

The plan from Chapter 0 was simple: rebuild the quantum equation on relativity’s correct relation between energy and motion. Doing it gives a real, named equation – the Klein–Gordon equation – and it is not exactly wrong. It describes genuine particles. But taken as a theory of a single electron, the way the quantum course took the Schrödinger equation, it comes down with two diseases at once. Both are worth catching, because the cure for them is the equation that finally succeeds.

The relativistic energy is a square root: $E = \sqrt{(pc)^2 + (mc^2)^2}$. To build a wave equation you have to feed that expression the quantum recipe, which turns momentum into an instruction to measure how the wave bends in space. But the square root of such an instruction is a monster – loosely, you are being asked for the square root of “wiggliness,” and there is no clean way to take it. So the obvious dodge is to get rid of the root by squaring everything first, and work with $E^2$ instead of $E$.

Squaring to dodge the square root came at a price. A square has two roots – $(-3)^2$ and $3^2$ are both $9$ – so the squared equation quietly describes not just the ordinary positive energies but a whole mirror set of negative ones, a bottomless basement below zero. In classical physics you could shrug and ignore states you never start in. In quantum physics you cannot: anything that can happen eventually does. A particle could tumble down the basement stairs forever, releasing more and more energy with no floor to land on.

That is not a small blemish – it is the same catastrophe, one storey down, that the quantum course showed doomed the classical atom, where an electron should have spiralled into the nucleus radiating endlessly. Use the button in the figure to send a particle cascading down the negative branch and watch the energy it pours out never stop.

The second disease is, if anything, worse. All of quantum mechanics rests on probabilities, and a probability can never be negative – there is no such thing as a $-30\%$ chance of finding the electron in a box. Yet the Klein–Gordon equation’s natural “probability meter” can swing below zero. A quantity that goes negative is not measuring a probability at all, and without a probability the equation has no way to say where the particle is likely to be – which was the entire job.

Two dragons, then: bottomless negative energies, and probabilities that turn negative. Dirac’s insight was to trace the second dragon to its lair – the equation’s two time-derivatives – and to demand an equation with only one, like Schrödinger’s. But relativity will not let him treat time more gently than space. An equation first order in time must be first order in space too, and that innocent-sounding requirement turns out to be impossible with ordinary numbers. What Dirac paid to satisfy it anyway is the next chapter.