Application of Extended Dirac Equation to Photon–Electron Interactions and Electron–Positron Collision Processes: A Quantum Theoretical Approach Using a 256 × 256 Matrix Representation

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Applications of extended Dirac equation to photon-electron interactions and electron-positron collision processes: a quantum theoretical approach using a 256x256 matrix representation.

This manuscript revisits the QED calculations, developed in flat Minkowski space long time ago and well reported in the literature over many decades, generalising these calculations to a curved space-time metric by adopting a wider 256x256 matrix representation, therefore introducing more degrees of freedom specifying the space-time curvature.

Even to a non-expert reader a natural question arises: where is the physics?

The author shows the discrepancies derived in the total or angular cross sections of most common QED reactions (Compton scattering, muon pair production, Moeller and Bhabha scattering) when the curved space-time metrics is applied, but without any physical discussion about the real contest, and how the curved metrics can be achieved in a real laboratory set-up. One of the most evident signatures of this approach, missing physics, is the denomination adopted by the author for the specific curved metrics example: “toy metric”. The reader might even feel sort of offended by this “joke”.

 To be more specific: what do the values of the coefficients in the “toy metric” matrix of Eq. 18 represent? Do they correspond to the curved metric at the edge of a black-hole or a star? How can they be reproduced in a laboratory, even in a particle accelerator? Are the written values (1/20, 1/10, 10/9) arbitrary numbers or physical quantities corresponding to a real physical situation?

In appendix H the author writes: “The trial results presented suggest that dependence on the specific toy metric is small, and verification would be difficult without high-precision experiments.” But what kind of experiments should be carried out to reproduce the curved metric specified in Eq.18?

The author must implement a complete discussion about the physical setting and the feasibility of a laboratory experiment in these conditions of curved metric. Otherwise the whole results look more like pure mathematical exercises than a physics study.

Also the discussion about Compton scattering revisited is lacking non only a physics discussion on possible experiments, but also a clear definition of the parameters used in the calculations. Indeed, the angular cross section plotted in Fig.1 depends on the parameter gamma-c, that is nowhere (nowhere) defined in the manuscript. An expert reader can recognize that this parameter is connected to the well known recoil parameter X of the Compton scattering kinematics, but a general reader gets lost in the confusion caused by several missing definitions: like the frequencies of the incident photon, omega, and that of the scattered photon, omega-prime. The author seems not to care about the readability of his manuscript by a general reader not necessarily expert of specific QED nomenclature!

Again, to be more specific (as an expert reader would notice): equation 16 should be joined by the kinematics formula giving omega-prime as a function of omega and the recoil factor, that is: omega-prime = omega / (1+0.5*X(1-Cos(teta))) , where X = 4*hbar*omega / me*c^2 , i.e. the standard Compton formula. Implying that the parameter gamma-c used by the author, and never defined!, is equal to gamma-c = X/2. Therefore, the value of gamma-c=0.173 used by the author for the calculation of the angular cross section plotted in Fig.1, equivalent to X=0.346, implies incident photons of energy 44.2 keV.  It’s been a century since X-rays of this energy have been widely and commonly used in laboratories for research and applications: how is it possible that the discrepancy shown in Fig.1 for large angles has never been measured? At least the author should comment about this issue and, possibly, show some experimental results published in the literature showing at least some indications in favour of this discrepancy. Otherwise it is hard to believe that this effect due to curved metric can be ever measured in a laboratory located on Earth - may be on a star, or a black-hole…

N.B. I checked by myself the Klein-Nishina angular cross section for X=0.346, and found the same result as the blue curve in Figure 1, just to be sure about the correspondence between the undefined parameter gamma-c and the well know Compton recoil parameter X (see for example ref. N. Ranjan et al.  Phys. Rev. AB, 21, 030701 (2018) ).

Last but not least, the author could at least show what is the dependence of the discrepancy in the K-N angular cross section for higher values of the gamma-c (X) parameter: some broader investigation of the dynamic range would help the reader to understand better the underlying physics. Indeed, recent literature on Compton scattering at large recoil (X>>1) points toward approaching maximal acceleration and Unruh radiation at MeV temperature: could this be a way to a curve metric achieved by acceleration instead that with gravity? That would really help the reader with a broader physics consideration (and more general audience, see Ref. Nuclear Instruments and Methods in Physics Research A, 1069 (2024) 169964 ).

In conclusion, I do not consider this manuscript acceptable for publication unless the author thoroughly addresses the aformentioned issues.

 

Author Response

We sincerely thank Reviewer 1 for the thorough and insightful review. All comments have been carefully addressed and substantial revisions have been made. Please see the attached detailed response document for point-by-point responses. Summary of main revisions: 1. "Toy metric" replaced with "illustrative test metric" throughout; its role as a diagnostic tool clarified in new Section 6.3. 2. New subsection "Scope of This Paper" (Section 1.7) added to delineate contributions and limitations. 3. New subsection "Comparison of Conventional and Extended Dirac Equations" (Section 1.8) added with visual matrix comparison. 4. Missing definitions (γc, ω, ω', θ, X) added in Section 5.2.1 with kinematic relation (Equation (17)). 5. Figure 1 caption expanded with quantitative distinction between test metric deviation (δg ~ 10⁻¹) and realistic values (δg ~ GM/(rc²) ~ 10⁻⁴⁰). 6. References [18] (Ranjan et al.) and [20] (Curatolo et al.) added per Reviewer's recommendation. All revisions are shown in blue text in the revised manuscript.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The paper shows a matrix-based extension/reformulation of the Dirac operator and QED scattering theory using a 256×256 representation derived from the full gamma-matrix basis, with emphasis on unification and computational transparency and reproducibility. The unification of calculation rules through matrix products and traces, direct comparison with standard flat-space results via fixed trace normalization, and the ease of parameter scanning over metric components makes a strong case that the whole workflow is suitable for automation and easy to reproduce. A brief estimation on the computational cost of the enlarged matrix traces relative to conventional methods might be helpful (although not compulsory) to reinforce these points even more firmly. In addition, while the outlined future directions are appropriate, a short clarification of the intended scope of the present results would somewhat improve clarity and prevent over-interpretation. The points listed in this review appear addressable without major restructuring, and the manuscript would benefit from minor revisions with contextualization in mind.


Minor issues:

No superscript 1 needed in affiliation. This is the only institution
Why is the first figure in the appendix cited as A2, should it be A1? Same for A3

 

additional assessments:

1) The article could benefit more from explaining the reasoning for high-dimensional representation and cross comparison to established methods
2) Author could include the discussion of similar physical phenomena where metric corrections can be applied, to avoid the proposed solution appearing too abstract.
3) Author could somewhat improve physical insight, the balance leans over mathematical formalism. 

Author Response

We sincerely thank Reviewer 2 for the careful and encouraging evaluation. All comments have been carefully addressed. Please see the attached detailed response document for point-by-point responses. Summary of main revisions: 1. Computation time comparison table (Table A1) added in Appendix D. 2. New subsection "Scope of This Paper" (Section 1.7) added to clarify the intended scope of results. 3. Superscript "1" removed from affiliation. 4. Appendix figure numbering corrected (A2→A1, A3→A2, A4→A3). 5. New subsection (Section 1.8) with visual matrix comparison and Table 2 added to explain the reasoning for the 256×256 representation. 6. Comparison table (Table 1) added for vierbein formalism vs. present method. 7. New subsection "Positioning of the Illustrative Test Metric" (Section 6.3) added. All revisions are shown in blue text in the revised manuscript.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

This paper describes a new formulation of the Dirac equation in spacetime by defining position dependent gamma matrices using the spacetime metric. This method avoids the traditional use of Vierbeins and spin connections used in curved space times, which makes the calculations involved more familiar (matrix products and traces) and suitable for computation. The method is applied to several practical situations including Compton, Møller and Bhabha scattering as well as muon pair production.  The results of the present calculations compare favorably with the more traditional methods in a flat spacetime. In particular the Klein-Nishina formula is explicitly derived from the method.  Toy position independent metrics are used to test the sensitivity of the calculations to geometric information.  

The new method presented in the paper is novel and conceptually straightforward, however it represents a very large increase in the size of the matrices involved, 4x4 matrices to 256x256. With such a large matrix 256x256 compared to 4x4 using other methods, it seems there might be some computational limitations or difficulties.  For example, it seems that maintaining normalization may be difficult with such a large matrix.  In section 6.4 (Limitations and Future work) the author states: “ Whether transparency comparable to conventional methods can be maintained while keeping background treatment consistent.”  I would be interested in further discussion of this, is it related to the size of the matrices involved?  Does the author see complications in keeping the method transparent and consistent? I think a further discussion of these issues could be useful here especially since it is claimed that the method is a rapid, practical alternative to other methods.

Also in section 6.4 it is stated: “Extension to general covariant form including connection is needed.”  Does this mean that the method may not be fully covariant.  I know that this is future work, but I think it could be good to discuss this a little more.  Does the author foresee difficulties with this and is there a plan to implement this?

In section 1.2 the method’s relevance to atomic physics processes.  While relativistic formulations are needed for the processes mentioned, it isn’t quite clear to me why a curved spacetime approach is useful. It seems that this might be relevant near a black hole for example, but not necessarily in a typical laboratory setting. Are there processes where the effect of the gravitational field or maybe gravitational waves might be important. Is there a specific physical situation the author plans to study in future work?  I think it would be useful if the author could identify specifically a situation or otherwise address this point since it could strengthen the relevance here.

The paper is well written and clear.  It contains a novel approach to studying scattering processes in curved space-time.  Currently only toy metrics are used, but it seems that it could be extended to real situations. I recommend it for publication. I would be interested in seeing the above points addressed if possible.

 

Assessments:

1) The clarity and necessity of the proposed formalism relative to existing approaches.

The method is clear and does use familiar mathematical objects, which I do see as an advantage of the approach.  My concern is that it leads to a very large matrix 256x256 and it isn’t clear to me that this is a big advantage over other methods.  The method does reproduce results for flat spacetime which shows the method is valid, but it is hard to say how it will do in real position dependent situations.  The author plans to consider this in the future so I see the paper as an introduction to the method and a proof of concept, which is fine with me.

2) The strength of its relevance to atomic and photon–matter physics.

Since the method is meant for curved spacetime it seems the approach is more relevant to extreme situations such as atomic or photon-matter processes occurring near a black hole for example.  I don’t know if it is really relevant to any typical processes in a normal environment.  I do think it is an interesting approach that could be useful for studying these processes in a strong gravitational field or gravitational wave. 

3) Whether the presentation appropriately balances mathematical development with physical insight.

I do think the presentation appropriately makes this balance at this point of the methods development.  Toy metrics are used at the current level, but they do indicate that the processes can look slightly different in a curved space time and the author indicates this.  I think it will be interesting if the author can extend the method to a real position dependent metric and apply it to a real physical situation.  

Author Response

We sincerely thank Reviewer 3 for the thoughtful and encouraging evaluation, and for the recommendation for publication. All comments have been carefully addressed. Please see the attached detailed response document for point-by-point responses. Summary of main revisions: 1. Normalization and transparency concerns addressed; computation time comparison (Table A1) added. 2. Covariance discussion expanded in Conclusions (Section 7, lines 422–428), clarifying that the method is fully covariant and that extension to position-dependent metrics with connection coefficients is planned. 3. Physical motivation based on general covariance explicitly stated in new Section 1.7; visual comparison added in new Section 1.8. 4. Connection to strong gravitational field environments (neutron stars, black hole accretion disks) and gravitational waves discussed in Section 6.3 and Conclusions. 5. "Toy metric" replaced with "illustrative test metric" throughout; positioning clarified in Section 6.3. All revisions are shown in blue text in the revised manuscript.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

I acknowledge the author took properly under consideration my comments and recommendations and I believe the manuscript is now acceptable for publication. Its readability by a general reader is now very much improved and quite good.

There is only a very minor detail that the author may consider to adjust: reference n.20 reports a wrong title of the published paper, though with the correct link to the journal. The correct reference is:  Full inverse Compton Scattering: Total transfer of energy and momentum from electrons to photons, L. Serafini, V. Petrillo, S.Samsam, Nucl. Instrum. Methods Phys. Res. A 2024, 1069, 169964. https://doi.org/10.1016/j.nima.2024.169964.

I recommend the author to fix this typo, to avoid confusion in the reader. I do not need to revise the manuscript after this correction.

 

Author Response

We sincerely thank Reviewer 1 for the positive evaluation and for recommending our manuscript for publication. We are very grateful for the detailed and constructive comments throughout the review process, which have greatly improved the manuscript.

As pointed out, Reference [20] contained an incorrect title and author list. This has been corrected to:

Serafini, L.; Petrillo, V.; Samsam, S. Full inverse Compton Scattering: Total transfer of energy and momentum from electrons to photons. Nucl. Instrum. Methods Phys. Res. A 2024, 1069, 169964. https://doi.org/10.1016/j.nima.2024.169964

The correction has been made in the revised manuscript.