What is the difference between first quantization and second quantization and where does the name second quantization come from?


It is very complicated to construct a consistent quantum theory from scratch. One of the most general methods to do so is to take a classical theory, and imitate some of its ingredients - the most important being the symplectic structure and the generators of symmetries. This is called quantisation, because the quantum theory is built using a classical one. One should point out that the classical theory is used only for consistency, and there is no need for it to have a physical meaning: we are only interested in the quantum theory; the classical one is just a formal tool. For example, the classical theory need not have anything to do with a possible macroscopic limit of the quantum one. The two theories are conceptually unrelated, and the quantum one is not subject to the existence of the classical one.

If the phase-space variables of the classical theories are trajectories, we call the process of quantisation "first". If the phase-space variables are fields, we call it "second" quantisation. This is just a historical name, without any deep meaning. The process of quantisation itself is identical in 1st and 2nd quantisation, so the two classes are not really fundamentally different. The only real difference is that in the first case we have a fixed and finite number of degrees of freedom, while in the second case we have an arbitrarily large (but also finite) number of degrees of freedom, and we are interested in the limit of infinite degrees of freedom, if it exists.

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    $\begingroup$ There is a historical explanation for the names first and second quantisations. It comes from the fact that the wave-function (in first quantisation thus) was thought as a (complex valued) field at the beginning of the 20-th century. When people tried to quantise the classical field (QED was the only existing one at that time), they thought they need to quantise the wave-function... Hence they thought they need to quantise a second time what was already quantised, hence the second quantisation process. It turns out to be incorrect, but the two names survive. $\endgroup$ – FraSchelle May 3 '17 at 8:18
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    $\begingroup$ Also, an important difference between first and second quantisation is that the first one applies in a Hilbert space, whereas the second one applies in a Fock space. You clearly said it in your good answer, without mentioning the names in use in physics literature. So I just put this in a comment: a wave-function has value in a Hilbert space, the creation and annihilation operators have values in a Fock space. $\endgroup$ – FraSchelle May 3 '17 at 8:20
  • $\begingroup$ @FraSchelle wrong $\endgroup$ – Anthonny Aug 19 '17 at 1:49
  • $\begingroup$ @Anthonny excellent $\endgroup$ – FraSchelle Aug 21 '17 at 7:09
  • $\begingroup$ @AccidentalFourierTransform What does it mean for fields to be phase space variables? I understand how trajectories (i.e., position and momentum) live in phase space, but just can't picture that for fields. $\endgroup$ – Tfovid Jun 13 at 8:27

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