Bosonic Casimir Effect in an Aether-like Lorentz-Violating Scenario with Higher Order Derivatives

Round 1

Reviewer 1 Report

The manuscript considers the Casimir effect associated with a quantized massive scalar field in the presence of violation of the Lorentz symmetry. As opposed to the previous publications on the subject, it is assumed that violation of the Lorentz symmetry is produced by a term involving the higher order derivatives of the field coupled to a space-like constant vector. The cases when this vector is parallel and orthogonal to the Casimir plates are considered as well as the Dirichlet, Neumann and mixed boundary conditions on the plates. Theadditions to the Casimir result for the energy per unit area arising due to a violation of the Lorentz symmetry are foundin different cases in the form of first expansion terms in powers of a small parameter $l/a$ where $a$ is the separation distance between the plates and $l$ is the characteristic length scale where the Lorentz symmetry is broken [see equations (31), (32), (59), and (60) for the massless case and Dirichlet boundary condition]. In the massive case, the asymptotic expansions in the powers of small and large parameter $ma$ are found. Note that in figures 2 - 5, devoted to the massive case with different boundary conditions, the authors assume that $l/a$=0.01 which is too large to be realistic. It should be remarked in the text that this value of $l/a$ is considered as only an illustrative example.

The manuscript contains new and interesting results. It is recommended for publication after the authors will make the following minor corrections.

1. P.2, the third line. Please replace "M. J. Sparnnaay" with "M. J. Sparnaay".

2. P.3, line 57. It is recommended to add that the vector u^{\mu} is dimensionless [in other case the dimension of the second term on the left-hand side of equation (3) would be incorrect].

3. P.4, line 80. One should add here that the quantity \omega_{{\bf k},n} is defined below.

4. P.4, line 82. Here "for-vector" should be replaced with "four-vector".

5. P.5, line 102. Reference [26] is presented in the list of references incorrectly with missing title. Please replace it with the following:

M. Bordag, G. L. Klimchitskaya, U. Mohideen and V. M. Mostepanenko, Advances in the Casimir Effect (Oxford University Press, Oxford,  2009).       An indication of the volume number is superfluous and should be deleted.

6. P.7, line 122. Please replace "an infinite summ of modified Bessel function" with "an infinite sum of modified Bessel functions".

7. P.20, line 302. Please replace "an direct" with "a direct".

8. P.20, line 308. Please replace "dispersions relations" with "dispersion relations".

 

Please make corrections indicated in the report.

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

Referee's report on the manuscript "Bosonic Casimir effect in an aether-like
Lorentz-violating scenario with higher order derivatives"
by R.A. Dantas, H.F. Santana Mota, E.R. Bezerra de Mello

The authors discuss the Casimir effect for the scalar field theory with the
generalized Klein-Gordon equation (3) where the higher-order term is determined
by a Lorentz-violating vector field. The subject is quite of some interest, and
the results obtained appear to be new, as far as I can see. However, there are
some questions that should be clarified before the paper can be recommended for
publication.

1. I wonder how the structure of the energy-momentum tensor (4) is established.
Is this a canonical current which can be derived from the Noether theorem?
If yes, I am curios how exactly (4) was obtained. Normally, one can derive
the conserved canonical energy-momentum from the invariance of the action (2)
with respect to arbitrary spacetime translations. Then how such invariance can
be demonstrated for the action that depends on a constant vector u? The authors
should prove that (4) is conserved, because if the divergence of this energy
momentum current does not vanish, that would probably mean that in the model
under consideration, the translational symmetry is violated in addition to the
violation of the Lorentz symmetry.

2. The authors analysed three different boundary conditions: Dirichlet, Neumann
and mixed conditions. Although from the mathematical point of view, one can
impose any of these, but physics is an experimental science, as we know. Then
how, in the sense of physics, can one impose the conditions (75), or (78)-(79)
on the scalar field? Please provide explanations.

3. The Casimir effect is manifest by the force per area, acting on the plates,
such as (1). However, rather paradoxically, the authors do not provide any
clear derivation of this crucial physical quantity. If they call the force per
area a "pressure", comments are needed how this pressure is derived. As one
knows since the original work of Casimir, this force is obtained by taking a
derivative of the energy with respect to the distance parameter "a". In this
sense, (42) obviously is not derived from the differentiation of (41), and the
same applies to (45) and (44), and to (71) and (70), and to all other "pressure"
expressions in this paper.

4. Besides the magnitude of the Casimir force, its important feature is the
sign that depends on the geometry and topology of the plates. As far as I can
see, the authors pay no attention to the sign of the Casimir force, and they
do not clearly explain whether the resulting force in their model is attractive
or repulsive, or perhaps its sign depends on the direction of the vector "u"
or/and on the choice of boundary conditions?

5. The last but not least, the authors should provide a discussion of the
physical value of the results obtained. Namely, the scalar field action (2)
obviously describes a toy model, but what can one say about the electromagnetic
field? At least a qualitative comment is needed.

Summarizing, an appropriate revision is required along the lines above.

Minor language editing is required.

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

I have read with great interest the manuscript "Bosonic Casimir effect in an aether-like Lorentz-violating scenario with higher order derivatives". If found this paper interesting, but I have some comments that the authors should address before acceptance in Universe. [2] In Eq. (2), the authors introduce a new model for the scalar field which contains higher derivatives. What is the physical motivation for this choice? Are there other possible ways to introduce higher order derivatives in a Lorentz-violating way? [3] Next the authors solve for the scalar field imposing appropriate boundary conditions upon the plates and then perform an expansion of the energy modes by assuming that the LV parameter is small. Straightforwardly they obtain Eq. (30) for the Casimir energy in the massless case. This result is consistent with previous studies when taking \epsilon =1? This point should be discussed immediately after Eq. (30). [4] Next, the include the effect of mass and obtain Eq. (39) in the limit ma \gg 1. An analysis for the massive Casimir effect in this theory, for \epsilon = 1, where carried out in Refs. [Phys. Rev. D 101, 095011 (2020), Phys. Lett. B 807; 135567 (2020)]. Your result of Eq. (39) reduces to the results of this manuscript in some limit? I think I will vote in favor of publication after the authors address my concerns.

English is ok. Some minor typos detected.

Author Response

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Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Referee's report on the revised manuscript "Bosonic Casimir effect in an
aether-like Lorentz-violating scenario with higher order derivatives"
by R.A. Dantas, H.F. Santana Mota, E.R. Bezerra de Mello

The authors had amended the manuscript in a satisfactory way, responding to
all the questions of my original report. As a result, the revised paper is in
a better shape now and can be recommended for publication in its present form.

Minor language editing may be required.

Reviewer 3 Report

The authors have addressed all my concerns. I recommend the publication of this paper. 

The English is fine.