The alternative free tasting: First quantization of light and the photon's wavefunction

We shall begin by developing an extreme light quantum theory, and forget for the time all connection with Maxwell equations. …Our present problem is to find the wave equation for the de Broglie waves of the quantum; the theory of the field we shall then obtain by a suitable (second) quantization of the de Broglie amplitudes.

—Julius Robert Oppenheimer [472]

Welcome to our free tasting of alternative wines. So far we have presented the traditional route to QED. From the classical Maxwell equations, canonical quantization takes us straight to many-body (i.e., second-quantized) quantum electrodynamics. This is quite a shortcut on the usual route taken in elementary quantum mechanics of nonrelativistic point particles, where one first has to construct a single-particle first-quantized theory (e.g., Schrödinger equation) from which to build the many-body second-quantized theory. Why not follow the ordinary longer route from first to second quantization for electromagnetic radiation as well? The main problem is that first quantization is really good only in nonrelativistic quantum mechanics. Pair creation out of the vacuum implies that we always have a many-body problem in the relativistic case. Now, unlike matter, there is no nonrelativistic limit for radiation. The photon has no rest mass and it always travels at the speed of light. There is no inertial frame where a photon is at rest. As a ...

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