Let me start with an analogy, namely a mass on a spring. If I am not mistaken,
- first quantization consists of replacing the dynamical variables by their operators and then solving the Schrödinger equation so as to obtain the stationary, energy eigenfunctions of the Hamiltonian, namely the various modes $\psi_k$ the wave function. Any wavefunction can then be expressed as a superposition $\Psi = \sum\limits_k c_k \psi_k$ of those modes. So far, we're only talking about waves, not particles.
- Second quantization consists of replacing the coefficients $c_k$ with the destruction operators [is this a operation a postulate by the way?] to then reveal that $\mid c_k\mid^2$ is an occupation number, or number of particles at the mode $k$.
How does the electromagnetic EM field quantization (i.e., photons) relate to the above? The classical EM field is already a wave, thereby making first quantization seem redundant. What I recall form my quantum optics course, however, is that the Maxwell equations were wrangled until we got the EM energy within a certain volume, then applied first quantization to basically end up with a quadratic potential in the conjugate, dynamical variables. Shuffling around the real and imaginary parts of those variables then produced the ladder operators, and that was the end of the story. That said, I can't relate that "textbook story" to the two quantizations I have outlined above. If I were to draw a (figurative) parallel with a mass on the spring, what would that mass be for the EM field? Would it be
- a fictitious mass at every point along the path of the classical EM wave,
- or a fictitious mass corresponding to the energy stored in the quantization volume?