# History of the fock space based quantization of fields

In almost all Quantum Field Theories textbooks the same approach to quantization is presented as the first example: one considers the scalar real Klein-Gordon field $\phi$ and just write it as

$$\phi(x) =\int_{} \dfrac{d^3 p}{(2\pi^3)\sqrt{2\omega_p}} (a(p) e^{-i p_\mu x^\mu}+a^\dagger(p) e^{i p_\mu x^\mu})$$

being $a(p),a^\dagger(p)$ operators satisfying $[a(p),a^\dagger(q)]=(2\pi)^3\delta(p-q)$ and $[a(p),a(q)]=[a^\dagger(p),a^\dagger(q)]=0$.

Then textbooks continue by stating that $a(p),a^\dagger(q)$ are annihilation and creation operators in a Fock Space.

Putting this all into a more organized way, what one has done here is: considering one wants to make classical fields turn to quantum fields obeying the canonical commutation relations, one possible approach is this one where the Hilbert space turns out to be a Fock Space.

It turns out that this is not the only way. Unlike QM, there are infinitely many non equivalent representations of the CCR.

My question here is: what is the history of this specific approach based on the Fock Space? How Physicists first derived it? How it was discovered that one approach to achieve a representation of the CCR for a system of uncountably many degrees of freedom, i.e., a field, was this one based on the Fock Space? What was the first derivations of the Fock Space itself that appears in this aproach?

• Did you look at Fock's original paper? It is cited at the beginning of Wikipedia's article on Fock space:"It is named after V. A. Fock who first introduced it in his 1932 paper Konfigurationsraum und zweite Quantelung"[Configuration space and second quantization]. Here's direct link. – Conifold Apr 4 '17 at 4:18