Position in Minimal Length Quantum Mechanics

Round 1

Reviewer 1 Report

Report on the manuscript Universe-1491798

“Position in minimal length quantum mechanics” by P. Bosso

This article is a nice study of some subtle point in the GUP program i.e. generalizing the Heisenberg Uncertainty Principle and the position-momentum commutator to take into account a postulated minimal length scale arising from quantum gravity. I have only a few minor comments suggestions:

• Equation (2) implies equation (1) only if one does not modify the momentum operator. This should be mentioned. For example, if the ansatz function in (1) and (2) is f(p) =1 + b p² then one could equally well get the commutator in (1) by taking q=iℏ d/dp (the standard position) and p₀ = p(1+b/3 p² ) where p₀ is the modified momentum.
• Equations (10) and (11) deal with the quasi-position representation of the position and momentum operators. This is nice since most work on GUP uses the momentum representation for positon and momentum. How does this quasi-positon representation compare with the one of KMM of reference 19 from the present work? Equation (50) of KMM (reference 19) gives the quasi-positon representation for position. If I set f(p)=1 + b p² (the KMM choice) in equation (10) of the present work do I recover (50) from reference 19? If not, why? Also equation (49) of reference 19 appears different from the quasi-position representation of the momentum operator given in equation (11) of the present work. If this is true is there a reason? Finally, if I take (10) and (11) and use these to calculate the commutator do I get the same result as in (1)? I think you do, but wanted to make sure.
• Equation (19) appears to give the modified momentum p₀ in terms of the standard momentum p. (p₀ is the modified momentum correct?). Now if I do a first order expansion of equation (19) I get p₀ ~(-δ+ (δ+ε)p)/ε + δ/ε --> (δ+ε)p/ε (arctan (x) ~ x). However shouldn’t the first, order small limit of (19) reduce to the standard momentum i.e p₀ ~ p ? Also the arctan modification of momentum is interesting implying an upper limit to momentum. Such an approach to GUP with an arctan modified momentum was recently proposed in Michael Bishop, et al.,   Lett. B 816, 136265 (2021).

Overall this is a good, well written and interesting piece of work. If the author can answer the questions/suggestions above, I would be happy to recommend publication.

 

Author Response

I thank the Referee for her/his comments. Please find my reply in the attached file.

Author Response File: Author Response.pdf

Reviewer 2 Report

The paper is a short, less technical review of earlier work on the quasiposition representation in quantum theories with minimal length. The author displays all relevant relations between this and the momentum representation, in particular the Fourier transform, choosing two different paths on the way there. He thus derives a modified de Broglie relation. In that vein, he applies the LQGUP to the particle-in-a-box problem and the potential barrier thereby making apparent, how the minimal length manifests itself in the respective cases. Over all, this can be seen as a pedagogical introduction into quantum mechanics with a minimum length.

Language and style of presentation are satisfactory. The same goes for the depth of technicalities. I suggest a couple of minor explanatory additions, however:

1. It should be made clear in the introduction, that generalized uncertainty relations can be derived from different perspectives (as e. g. in Lake et al - Class.Quant.Grav. 36 (2019) 15, 155012) too, not only from a deformed Heisenberg algebra, while the paper applies solely to the latter.
2. It should be mentioned, that the function p_0, introduced in eq. (4), may be understood as conjugate momentum coordinate. The Fourier transform (eq. (6a)), for example, is just an integral over p_0. 
3. The function \chi in eq. (7) is undefined. That should be done somewhere.

Once these changes are implemented, the paper will, in my view, be suitable for publication.

Author Response

I thank the Referee for her/his comments. Please find my reply in the attached file.

Author Response File: Author Response.pdf

Reviewer 3 Report

The paper can not be accepted since it almost completely duplicates the author's paper

Bosso, P. Position in Models of Quantum Mechanics with a Minimal Length. Phys. Sci. Forum 2021, 2, 35. https://doi.org/10.3390/ECU2021-09275

and does not contain any new results.

Author Response

I thank the Referee for her/his comments. Please find my reply in the attached file.

Author Response File: Author Response.pdf

Round 2

Reviewer 3 Report

The refined version of the paper looks much more appropriate for publishing.  While reading, I encountered at least two typos - in eq. (15) and inside Fig. 2 caption. Also I can’t find the change 3.(b) in the text. It is about the 4th paragraph of Section 4. Currently this section consists of three paragraph only. Finally, I strongly recommend to include the reference "Bosso, P. Position in Models of Quantum Mechanics with a Minimal Length. Phys. Sci. Forum 2021, 2, 35. https://doi.org/10.3390/ECU2021-09275" into the final version. 
To summarize, the paper can not be published in the present form. However, it would be recommended to publish after taking into account the aforementioned points and making the appropriate improvings.

Author Response

I thank the Referee for her/his comments. Please find my response in the attached file.

Author Response File: Author Response.pdf