Vacuum Polarization in a Zero-Width Potential: Self-Adjoint Extension

Round 1

Reviewer 1 Report

The paper studies the vacuum fluctuations of a massless neutral scalar quantum field
in presence of a delta-type interaction concentrated at a point of 3-dimensional Euclidean
space. The renormalized vacuum polarization and energy-momentum tensor are computed
making reference to two different descriptions of the singular potential: firstly, the authors
consider a heuristic realization, derived by formal perturbation theory; next, a mathematically
rigorous formulation in terms of self-adjoint extensions of symmetric operators is examined. While the calculation of the vacuum polarization is original, the computation of the
renormalized energy-momentum tensor for the setting under analysis is not new.
Despite this fact, it is worth remarking that the authors employ a different regularization
scheme deriving results in agreement with those reported in the preceding literature.
All things considered, I believe that the content of the paper can be of interest for the
community working on Casimir physics and I therefore recommend its publication.
However, I would suggest some minor modifications, which could help to improve
the quality of the paper. 1) A careful and thorough revision of the English language is necessary. 2) The regularization method used by the authors is clearly a variant of the standard
''point-splitting'' approach. This should be mentioned explicitly somewhere in the text. 3) For the reader's convenience, I would recommend adding some details about
the derivation of Eq. (10), (12) and (16). In alternative, some precise bibliographical
reference should be indicated. 4) When considering the perturbation theory description of the delta potential,
is there any reason why the renormalized energy-momentum tensor should vanish
identically in the conformal theory with  ξ = 1/6? 5) As acknowledged by the authors themselves, the renormalized energy-momentum
tensor for the model under analysis was previously studied in Ref. [10].
This fact should be mentioned also in the Introduction, highlighting the similarities
and the differences with the cited work. 6) I would suggest to discuss in a more systematic way the relations between the results
derived with the two descriptions of the delta potential, pointing out the connections
between the related results. 7) Similar configurations were previously analyzed in the following references,
which should be added to the Bibliography and mentioned contextually in the paper:
- S. Albeverio, G. Cognola, M. Spreafico and S. Zerbini, J. Math. Phys. 51, 063502 (2010).
- D. Fermi, Mod. Phys. Lett. A, 35, 2040008 (2020).
- A. Scardicchio, Phys. Rev. D 72, 065004 (2005).
- M. Spreafico and S. Zerbini, Rep. Math. Phys. 63, 163 (2009). 8) The reminder term in the asymptotic expansion (55) is missing.

Author Response

We thank Referee for the careful reading and reviewing of our manuscript. Concerning the remarks made:   1) We have fixed some typos and tried to improve the language  2) We have pointed out this fact on lines 66-69 of the resubmitted version 3) References on the popular textbooks were added just before the mentioned eqns. 4) This fact is also interesting for us. We can not answer about the conceptual nature of it now (and propose the argument which does not somehow reproduce our derivation here). But in order to don’t confuse the reader, we have added few words (lines 131-136) to emphasize that we compute the EMT just in the 1st perturbational order, not completely 5) We put a sensence in the Introduction (lines 69-71); combined with the previous lines, it is intended to clarify the difference in approaches 6) Few sentences were added into the conclusion section (lines 299-306) 7) We thank Referee for bringing our attention to these works. They were added as Refs. 26-29. The corresponding sentense was added into the Introduction (lines 40-44) 8) In order to don’t add an extra (almost empty) line for the remainder into the equation, we have added a note in the text just after the Eqn.(55) (line 252)

Reviewer 2 Report

This manuscript considers the effects of vacuum polarization
near the pointlike source of a zero-range potential. The model case
of a massless real scalar field is considered. The arising infinities
are removed by means of the dimensional regularization. As a result,
the renormalized vacuum expectation values of the energy-momentum 
tensor are found and compared with the results of perturbative 
computations. The authors take due care to fine mathematical details. 
Specifically, they consider self-adjoint extensions of the Hermitian 
operators.

This manuscript can be recommended for publication after taking into
account the following remarks.

1. In line 2 above Equation (1) the words "of subtracted points."
should be replaced with "of subtracted points \bf y." In other case
Equation (1) remains unclear. In line 1 above Equation (1) the term 
"hamiltonian" should start with a capital letter "Hamiltonian". 

2. Some of the formulas look nonelegant. For instance, the external 
brackets in Equations (2) and (3) should be replaced with the square
\brackets, as well as in many equations below. The first factor in 
the denominator of (10) should be not in square brackets, as in the 
manuscript, but in the round brackets. The same about the factor 
[2\zeta -1/2] and the first factor in denominator of (15), and in 
many other equations. These should be corrected throughout the 
manuscript.

Author Response

We thank Referee for careful reading, review of our manuscript and the remarks made   Concerning the remarks:   1) We agree. The changes were made.  All textlike Hamiltonians (as well as Laplacians) throughout the text begin with capital letter now. 2) We agree. We try to change all primary parentheses to round ones, the external parentheses to square ones   Some typos were fixed throughout the text.