# Talk:Second quantization

## Reviewer B

Trying to make this article conform to my taste would require a major revision and rewriting. But since the author(s) of an item (and the editor in chief of the Scholarpedia) should bear primarily the responsibility for its style and content, I will restrict myself to a few comments and minor corrections.

1) General comments and remarks on the content

Mathematically, second quantization is a functor from the Hilbert category to itself which sends a 1-particle Hilbert space H to its bosonic (or fermionic) Fock space F that is the completion of the symmetric (or antisymmetric) tensor algebra over H (see, e.g., the text of John Baez, available at http://math.ucr.edu/home/baez/categories.html). Any unitary operator in H gives rise to a unitary operator in F in an obvious manner. Working in such an algebraic setting one would, of course, never have ill defined infinite products of creation operators as in the first formula after the relations (A, B, C) in the section entitled "Second Quantization of the Bose-Einstein Assemblies". (By the way, “Ensemble” appears to be more appropriate than “Assembly” in this context.) Then one would not have to appeal to finiteness of the energy in order to exclude infinite product states (as it is done in the section on "Quantization of the Free Bose-Einstein Fields"). Since such an algebraic definition only applies to non-interacting systems, if the author chooses to cite it, he should explain that physicists actually go beyond it when introducing particle interactions. Even then I would not relate “compositeness to the tensor product", since a bound state, typically, does not belong to the tensor product of its constituents.

In the concluding section on "Other Statistics" it would be clearer to mention non-abelian permutation group statistics (called parastatistics) rather than just refer to Green's paper, and to contrast them further with the abelian braid group statistics corresponding to the anyons. The story of the discovery of anyons is more complicated than alluded to in the text and can be found in the article: The ancestry of the "anyon," by L. Biedenharn, E. Lieb, B. Simon, F. Wilczek, Physics Today (L) AUG 90.

2) Terminological and other minor remarks

- I would describe second quantization as a method rather than an algorithm; “non-relativistic and field theoretic framework” would sound better than “ ... field theory ...” (in the introductory paragraph).

- The first sentence, “Second quantization is a basic ...” should not be repeated in the beginning of the section "History".

- Speaking there of superselection sectors one should cite G.C. Wick, A.S. Wightman, E.P. Wigner, The intrinsic parity of elementary particles, Phys. Rev. 88 (1952) 101-105.

- Rather than talking of “sum rules” (in the section "The Bose-Einstein Statistics, the State Space") one should say that states are labelled by partitions of the integer N.

- The accepted order “Fermi-Dirac” (rather than “Dirac-Fermi”) statistics is also historically justified.

- “Vacuum property” is more common than “Fock property”; the same is true for “normal ordering” instead of “Wick ordering”.

- One should say “an arbitrary annihilation operator” (rather than a “generic ...”) since “generic” is used as opposed to exceptional (and one has no exceptional operators here).

3) Concluding, I would like to reiterate that it is the author who is responsible for the article. I am trying to be helpful but the above remarks are by no means exhaustive. The author may or may not take them into account (in particular those in point 1), and I do not need to see any subsequent version.

## Author's answer to reviewer B

I thank the Referee for his kind report. These are my answers/comments:

The aim of scholarpedia is to present basic ideas of modern physics in a way accessible to under-graduated physics students. Categories are not a common tool for working physicists and hence the functor language is excluded a priori.

What I do know for sure is that the identification of the state space with a Fock space in QFT requires an energy condition as it was shown by Haag in his seminal paper on relativistic scattering theory (see Haag’s book in the references). Of course this does not refer to a free theory. The lack of triviality of the condition also appears in the difficulties in the definition of an asymptotic state space in QED.

The sentence about the compositeness-tensor product connection appears in the study of the non-relativistic many particle problem where the connection extends also to bound states. The hydrogen atom wave function, belongs, forgetting spin, to L^2(R^6) which coincides with L^2(R^3)x L^2(R^3).

The section “other statistic”s was meant to give an indication from the point of view of second quantization of most recent developments. The reference to Green’s paper was chosen since Green refers explicitly to second quantization. A secondary purpose of the section is to give an idea of the role of a space with low dimensionality.

A general discussion about quantum statistics is out of the foreseen subject.

The word “assembly” was used by Dirac in his paper on the quantum theory of radiation.

The term Fock property is commonly used, see e.g. Streater’s cited papers, and in my opinion gives the idea that CCR are not sufficient to identify a Fock representation. I have added "vacuum property" as an alternative wording.

I have modified the sentence: “a generic annihilation” canceling the word “generic”, I have re-ordered Dirac-Fermi into Fermi- Dirac, and I have introduced “normal ordering” together with Wick ordering. I have introduced the WWW reference for superselection rules. Sum rule has become: constraint. I have inserted Biedenharn et al in the Further Readings

I have avoided using the same sentence in the abstract and in the introduction.