Geometric Justification of the Fundamental Interaction Fields for the Classical Long-Range Forces

Round 1

Reviewer 1 Report

This paper applies the notion of reparametrization invariance to instances of classical fields such as those in electromagnetism. The authors study these behaviors across a wide range of different configurations, ranging from those of monogeneous lagrangians to potential fifth force searches to electromagnetism to the Einstein-Hilbert-Cartan action. It finds that applying these notions allows for the study of potentially new physics, including physics that is potentially relevant to dark matter and the cosmological constant problem.

I find the paper to be well written and well-reasoned. The logical flow of the arguments appear to be correct. Some of the systems studied are a bit far afield from those typically studied (e.g. the potential fifth force stuff), but the analysis seems to have been conducted well and correctly.

As such, I believe the paper should be accepted in its current form, without the need for any further revisions.

Author Response

We are thankful to the reviewer for taking the time to go over our paper. We are very happy to hear that the reviewer finds the paper to be well written, well-reasoned, and with a good logical flow of the arguments. Especially the analysis of our study on the possibility of new forces.

Reviewer 2 Report

The paper deals with the Lagrangian formulation of pointlike as well as extended objects interacting gravitationally and electromagnetically. Using reparametrization invariance as a guiding principle, the authors propose new possible consistent interaction terms. The work is properly motivated and the analysis is thoroughly carried out.
Due to the above, I recommend the paper for publication in its present form.

The only point I had already raised to the authors (but it's just for my own curiosity!)
 concerns the possibility of discussing, within their framework, interactions with p-form gauge potentials of extended objects. The question is motivated by D-brane dynamics in string theory.

The analysis done in general for p-branes is very interesting but perhaps it would be useful to comment on what happens once a p-form gauge potential is included. This would be the typical situation e.g. in string theory D-branes and casting the problem by using the authors' formalism could be extremely valuable. Do the authors have comments regarding this?

 

Author Response

We are thankful to the reviewer for taking the time to go over our paper and to provide constructive feedback.

As we understand, motivated by D-brane dynamics in string theory, the reviewer is particularly interested, for his/her/their own curiosity, in the possibility of discussing interactions with p-form gauge potentials of extended objects within the formalism presented, right?

As a matter of fact, such considerations were part of the initial motivation about the framework and were considered very encouraging. Relevant initial reports were presented at conferences many years ago see the earlier preprint in arxiv.org (math-ph/0210021, math-ph/0210022, math-ph/0311007, math-ph/0512082). Unfortunately, other research projects and activities have taken precedent and the framework was not actively in development. The major concern was the problem of subsequent quantization and how to compare and contrast the Dirac-gamma quantization with other quantization approaches and methods. Recently, however, a suitable alternative approach has produced interesting results arxiv.org/abs/1903.02483. Based on what we know now it is likely that the p-brane potentials would result in a system of constraints in order to provide an integral sub-manifold embedding, the particular operator form of each constraint will have to be carefully analyzed to understand the meaning of the corresponding sub-manifold parametrization parameters. Hopefully, that this is addressing the question asked by the reviewer since it is difficult to say anything more until we do the specific model calculations within string theory or other relevant systems.

Reviewer 3 Report

The paper is very interesting and well written and the authors deal with a topic of actuality. The significance of the content and scientific presentation of the item are high. The conclusions are presented in a clear manner in Conclusions and Discussions. Also, at the end of the paper the authors present eleven interesting Examples and Exercises. I recommend the publication of the paper.

Comments for author File: Comments.pdf

Author Response

We are thankful to the reviewer for taking the time to go over our paper and to provide constructive feedback. We are very happy to hear that the reviewer finds the paper content to be of high significance presented in clear manner.

Regarding the short Section 7 containing relevant theorems framed as problems and exercises, we have chosen this format to maintain short size of the paper overall and to keep the reader focus on the justification rather than overwhelmed by a system of mathematical proof and constructions. We will ask the editor if we should change the section and what formatting would be best suited to the journal style.

This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.

Round 1

Reviewer 1 Report

In this paper, the authors seem to like to discuss about the Lagrangian of ``general'' (extended) objects in the bulk space besed on the reparametrization invariance. They started the arguments by comparing the action (1) with the action (2). The action (1) is the standard proper world length of a single particle and the action (2) is given by the Lagrangian which is the square of the Lagrangian of the action (1). Classically these two actions give equivalent equation of motion. The important difference between the action (1) and the action (2) is that the action (1) has the invariance of the reparametrization of the proper time tau, or the parameter to parametrize the trajectory of the particle but the action (2) does not has the invariance. Philosophically the invariance means that the physics should not depend on the artificial parametrization of the trajectory. As the quantum theory, the invariance becomes much important because the invariance, as well-known, gives a constraint which eliminates the ghost and the constraint is nothing but the Klein-Gordon equation of the particle.
Although the authors claim they are considering about the classical theory but any theory with inconsistency at the quantum level is physically irrelevant and therefore I do not recommend this paper for the publication.

Author Response

We are glad to read that the reviewer recognizes that “Philosophically the invariance means that the physics should not depend on the artificial parametrization of the trajectory.” It seems that this important principle is on par with the coordinate independence but has not received the appropriate recognition and attention.

Recognizing this principle within the framework of extended objects has been a key motivation for the current research; however, at its current form, the paper dedicates only a sub-section as an additional illustration of that principle. So, we have modified the paper accordingly by changing the abstract and the text as well (see lines 6&7 and lines 114-119).

Furthermore, the reviewer is raising important questions about the consequences of the principle of such invariance and its role within quantum theory. We have to say that the consequences seem to be far-reaching, and we have begun research in this direction. However, we had to contain our excitement to only a few additional statements (see the text page 6 lines 219-223 and the footnote1 there) because the detailed discussion of the relevant results pertained to quantum physics is outside the scope of the paper. Nevertheless, we have elaborated on our results by adding the text in lines 314-322 on page 8, along with two new references relevant to the treatment of constraints within the description of quantum systems see [38,39].

Some additional changes are present on page 19, line 701, where footnote #3 has been added, and on page 21, lines 776-777, along with a new reference [25].

Reviewer 2 Report

Referee report for the manuscript

Geometric Justification of the Fundamental Interaction Fields for the Classical Long-Range Forces

by

Vesselin Gueorguiev and Andre Maeder

In this manuscript the authors introduce the notion of Reparametrization Invariance and discuss the consequences of enforcing this principle
in the basic formulation for fundamental interactions. Is is shown that Long-Range forces, namely electromagnetism and gravitation are derived from
geometrical arguments based in the aforementioned principle. Moreover, the text addresses the possibility of additional long range forces which have not been
detected in nature but may explain cosmological phenomena like inflation or dark matter/dark energy effects.

I consider the text is well written and logically structured, in a pedagogical way. Included are discussions on the different aspects of
reparametrization invariance through the canonical-form of the first-order homogeneous Lagrangians in the velocity or the generalized velocity. The manuscript
ends with a section of problems and questions that I find very instructive for the reader.

Interestingly, the results strongly rely on the mathematical formulation of the theory, with the physical law seen as a consequence.

My complain on the manuscript is the lack of concrete predictions or applications of the theory in physical phenomena. I think this aspect can be improved or devoted
to a follow up. All in all, I find the work of good quality, worth for publication.

Author Response

We are glad to read that the reviewer has received the impression that “the results strongly rely on the mathematical formulation of the theory, with the physical law seen as a consequence,” since this has been one of our goals, which is usually hard to convey without drowning the reader in too much mathematical formalism.

Furthermore, the reviewer is raising important questions about the lack of concrete predictions or applications of the theory in physical phenomena. We have to say that the consequences seem to be far-reaching, and we have begun research in this direction. However, we had to contain our excitement to only a few additional statements (see page 6 lines 219-223 and the footnote1 there, lines 314-322 on page 8, and also page 19, line 701) because this is still a work in progress and preparation that is devoted to follow-up papers on the implications of the reparametrization invariance.

Some additional changes are present in the abstract line 6&7 and page 3, lines 114-119, and on page 21, lines 776-777, along with new references [25,38,39].

Reviewer 3 Report

The present manuscript offers a very well discussion about the potency of the principle of reparametrization invariance realised via the canonical-form of the first-order.

I agree with the publication of this manuscript, but I have some minor comments:

• Along the text there are phrases in "BOLD". They should be homogenaised according to the rest of the text.
• I suggest to add a couple of references in the Conclusions related to the following quotes: 
• After: " as well as interactions that are not yet experimentally discovered"
• After: "Thus, perhaps relevant to the dark matter and dark energy phenomena.
• After "may be relevant to the inflation processes in the early universe"
• I suggest the following references:

• Cosmology Intertwined I: Perspectives for the Next Decade

  Eleonora Di Valentino (Manchester U.), Luis A. Anchordoqui (New York U.), Ozgur Akarsu (Istanbul, Tech. U.), Yacine Ali-Haimoud (New York U.), Luca Amendola (Heidelberg U.) et al. e-Print: 2008.11283 [astro-ph.CO]

Author Response

We are glad to read that the reviewer recognizes the potency of the principle of reparametrization invariance and has provided us with a comments that improve our manuscript. We are particularly  grateful for bringing our attention to the publication “Cosmology Intertwined I: Perspectives for the Next Decade,” we have included it as reference [25] on page 21 & 22 lines 776-777, 787, 789 as suggested.

Furthermore, following the reviewer suggestion the phrases in "BOLD" were instead emphasized using \emph or normal text within the section 6. Conclusions and Discussions.

Some additional changes are present in the abstract line 6&7 and page 3, page 6 lines 219-223 and the footnote1 there, lines 314-322 on page 8, and also page 19, line 701, along with the new references [25,38,39].

Round 2

Reviewer 1 Report

In spite of the reply by the authors, the problem of the ghost is well-studied in the last century especially for the one-dimensional (particle) and the two-dimensional (string) objects and well-known. Therefore the problem is not future nor forthcoming problem but past already-solved problem. Therefore the claim in this paper is physically irrelevant and I do not recommend this paper for the publication.