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I am wondering the difference between the canonical quantization and the second quantization in quantum field theory.

For example, a harmonic chain, one can write down its lagrangian density $\mathcal{L}(\phi,\partial\phi/\partial t, \nabla\phi)$, where $\phi(\mathbf{r},t)$ is the classical field. By replacing classical field $\phi(\mathbf{r},t)$ with quantum field $\hat{\phi}(\mathbf{r},t)$, one do the canonical quantization, which finally leads to the Hamiltonian \begin{equation} H = \sum_k \omega_k\left( \hat{a}_k^\dagger \hat{a}_k + \frac{1}{2} \right). \end{equation}

However, from my understanding, the above result is obvious in second quantization where a one-particle operator can be written as the a creation-annihilation operator pair times the matrix element between these two states.

So my question is what's the difference of the two concepts?

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Second quantization is a term used to describe the quantization of the fields in order to describe situations with variable number of particles. You can do this in more than one way and one of the ways is to quantize the fields by introducing the field operators which correspond to classical field and conjugate momentum operator. Of course to know how to use these operators you have to define the states and commutation relations. This is canonical quantization. But you can also use path integral formalism or some other methods.

So second quantization is actually just quantum theory used on systems with many degrees of freedom like many particle systems. But these methods can also be used in systems like crystals where you describe vibrations of the crystal using these field theory methods. In relativistic situations where you have variable number of particles this is very useful.

In describing these quantum fields you can use the ideas of Fourier analysis and describe the field in Fourier or momentum space. In this way you can actually use quantum harmonic oscillator formalism describing the field as a bunch of simple quantum harmonic oscillators with definite frequencies.

So if we are talking about electromagnetic field we can see it as a bunch of oscillators of certain frequency. Energy levels of quantum oscillators are quantized ad so become the EM field. So for a certain frequency you can be at level 1,2 or 1679 or something. This corresponds to a number of photons of that energy. Number of photons is related to a intensity of EM field (square of its amplitude). But QHO is solved and so if our fields have the same general mathematical description we can use similar formalism. That is the algebra of ladder operators which are called creation and annihilation operators in QFT.

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  • $\begingroup$ Sorry for the late reply. Do you mean that second quantization is a general concept which makes the classic field to be quantized. However, canonical quantization is just one kind of way to do this? $\endgroup$ – user8036269 Aug 9 at 12:32
  • $\begingroup$ Yes!! You said it better😁 $\endgroup$ – Žarko Tomičić Aug 9 at 12:34
  • $\begingroup$ But for example some people say that operator method of description of harmonic oscilator is the so called second quantisation and doing it via schroding equation is like first but that term second quantization is actually Just non sense. $\endgroup$ – Žarko Tomičić Aug 9 at 12:36

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