A concept in quantum mechanics and in quantum field theory, which is balked at by many, when usually there's no reason for the slightest of double takes. Truly, each of the two quantizations hinted at by the name of this concept are familiar to any physicist, nay to any undergrad. They are very used to working with them in completely classical physics, except that they are used to see each show up in a different situation, and they think of them as entirely intuitive. But when the two show up together when one starts studying quantum mechanics and then quantum field theory, it does feel a little weird and counter-intuitive. I will try to present the two quantization as each comes up in its intuitive sense, and then see how each quantization enter into the description of the other problem.

Consider the classical problem of standing waves on a stretched piece of string clamped at both ends (or any other equivalent situation that satisfies the same equations). The motions allowed can be broken up by frequency, but not all frequencies are represented. Only a discrete set of normal modes turn out to exist. This is first quantization -- the modes of oscillation are discrete. If you say "Why do you call this quantization? This isn't a quantum phenomenon, this is classical!" you are right, but only because that is how we've come to use the adjectives "quantum" and "classical". If semantics were up to me, I would call this a quantum phenomenon.

Indeed, when the waves involved are not string waves but the wavefunctions of a particle in a container ("box" is the technical term), and when we refer to normal modes of the wavefunction for some reason by the crazy name of "quantum states", people come to call the exact same phenomenon as above the quantization of states. But now suppose we want to put more than one particle in our box (assume that the particles do not interact), then one can think of each of the modes, or "states", as being occupied by a certain number of particles -- zero particles, one particle, two particles, and so on. Do you wonder why we can't have a "state" be occupied by six-and-a-half particles? Or any continuous number, for that matter? No, I didn't think you did, because it's intuitive. Again, ridiculous me, I'm going to call this result ("obviously classical") a quantum phenomenon. It is second quantization.

And indeed, if we go back to our waves on the string (or you might like to think of electromagnetic waves now), and I tell you that each normal mode cannot have an arbitrarily continuous amplitude, but must take one of a discrete list of allowed amplitudes, you would surely think of this as non-classical. But it's the same thing, really, as the above restriction of a whole number of particles. Here, we call these things that come in whole numbers "photons" or "phonons" or even "magnons" (depending on context), and we call them the quanta of our waves.

So there, if you can wrap your mind around the equivalence of the two situations (the waves on the string and the particles in the box) and convince yourself that they are just different ways to think about the same thing; that particles are really just quanta of a waving field, that waves are just the wavefunctions associated with "particles" like photons; in fact that particles and waves are just shadows on different walls of Plato's cave of the same higher-dimensional concept; that this thing they are both projections of, and that shall remain unnamed for fear of conjuring either of the two sets of connotations, smells as sweet by any name; then you have grokked quantum field theory. Congratulations!

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