The Spectral Condition, Plane Waves, and Harmonic Analysis in de Sitter and Anti-de Sitter Quantum Field Theories

Round 1

Reviewer 1 Report

It is well known in mathematics that de Sitter spacetime is a maximally symmetric Lorentzian manifold with constant positive scalar curvature. Instead, anti-de Sitter space has a negative scalar curvature. From a physical point of view, initially, de Sitter's spacetime seemed an unrealistic solution of Einstein equations describing an empty universe filled only with dark energy. The inflationary model and the discovery of the accelerating expansion of the universe, have, instead,  now led to a revival of interest in de Sitter simmetry. There are some researchers who even hypothesize that the laws of physics are not invariant under the Poincaré group of special relativity, but under the symmetry group of de Sitter space. Furthermore, an ever-increasing number of research groups are dealing with quantum field theory in de Sitter and anti-de Sitter spacetime. For this reason I consider this paper a very interesting review on this topic. It is very well written. The paper is clear to read and seems mathematically correct. Finally it is well referenced. Given what has been said, I believe the paper deserves publication in this form.

Author Response

Thank you for your review.

Reviewer 2 Report

The article provides a brief historical and technical review on the spectral condition on maximally symmetric spacetimes. It is well written and it gives and nice historical background on the field. On the other hand, I find the article a bit too technical, serving only as a review for specialists and not for the ones who desire to enter the field. To broaden the audience, and hence increase the paper's impact, I suggest the following.

1) Include more discussions around important concepts (be it equations, definitions or theorems) whenever possible. In particular, there should be a final section with general discussions and conclusions. It is not appropriate to end the paper with an equation.

2) When for whatever reason 1) is not possible, increase the number of references, quoting them appropriately along text in those parts where the explanation is too concise.

I am happy to recommend this review article for publication after the above improvements.

Author Response

I have added a short concluding section

Reviewer 3 Report

In this paper, the author presents a review on spectral conditions in Minkowski, de Sitter (dS) and Anti-de Sitter (AdS) space-time.

In particular, the author focused on plane wave solutions compatible with analyticity as a standpoint for discussions of harmonic analysis. 

After an interesting historical review in the introduction about dS solution of Einstein's field equation, the author discussed spectral conditions in the  axiomatic QFT approach comparing the cases of the three space-time backgrounds aforementioned. 

I think the paper is well written and clear even for not expert in the field. A review on these topics can be useful for having possible insights on quantum field theories on curved space-time. 

Thus, I accept this work in the current form. 

Author Response

Thank you for your review.