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?

  • 2
    $\begingroup$ 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. $\endgroup$
    – Conifold
    Apr 4 '17 at 4:18

It's a long story, and students are constantly confused about the point even today. The basic appreciation that a quantized field is an infinite collection of quantum oscillators suitably organized is Pascual Jordan's, of course. But the technical innovations of 2nd quantization methods, the specific "packaging" of them is Dirac's.

In some chronological order, the epochal Dreimännerarbeit, "Zur Quantenmechanic II", Zeitschrift for Physik (1926) (in German) 35 (8) doi:10.1007/BF01379806, by Born, M.; Jordan, P.; Heisenberg, W., gets there on pp 606-615, Ch 4, §3, no "if"s and "but"s!! That section "is Entirely the work of Jordan" reassures one Anthony Duncan in The Conceptual Framework of Quantum Field Theory p 19.

The more familiar language suitably applied to the EM field is Dirac, P. A. M. (1927), "The quantum theory of the emission and absorption of radiation" Proc. R. Soc. Lond. A. Royal Society Publishing 114 (767): 243–65. doi:10.1098/rspa.1927.0039.

But so many strands went into this tapestry that it is hard to parse them out and assess their detailed significance. The most meticulous judgments vindicating Jordan's contributions have been detailed in Silvan S. Schweber's QED and the Men who Made it: Dyson, Feynman, Schwinger, and Tomonaga.

  • I am skipping Fock space itself, as it is basically a technical formalism innovation, and @Conifold covers it in his comment: a well-meaning reader would recognize the logic of Fock space already at the level of the Dreimännerarbeit. The other realizations of field quantization (Wavefunctional quantization, phase-space field quantization, etc.) are mere mathematical curiosities by and large.

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.