Nonperturbative Quantization Approach for QED on the Hopf Bundle †
Round 1
Reviewer 1 Report
In this paper the authors consider the Dirac equation and electrodynamics in RxS^3 space time. They derive classical solutions of Dirac and Maxwell equations. It is interesting to find solutions with discrete spectra in a compact space such as S^3. This part deserves publication. It is, however, hard to follow the argument on the non-perturbative quantization of the Dirac and electromagnetic system. The procedure of quantization may not be dependent on whether the spectra are continuous or discrete. A non-perturbative quantization can be performed when we solve equations numerically with discrete spectra on a lattice or a compact space. Anyway we must solve the infinite set of equations for Green's functions by using some method. It may be called the perturbative quantization unless we can solve the equations using non-perturbative methods. We recommend that the authors reconsider perturbative and non-perturbative quantization.
Author Response
Dear Editor,
Thank you very much for the referee reports. We have enclosed an appropriately revised version of our paper.
Sincerely yours,
- Dzhunushaliev, V. Folomeev
First of all, we would like to thank the referees for their careful reading of the manuscript and for the suggestions to improve the paper. We will now address the comments in detail. All our corrections made according to these comments are marked by yellow in the pdf file.
Reviewer #1:
Q1. In this paper the authors consider the Dirac equation and electrodynamics in RxS^3 space time. They derive classical solutions of Dirac and Maxwell equations. It is interesting to find solutions with discrete spectra in a compact space such as S^3. This part deserves publication. It is, however, hard to follow the argument on the non-perturbative quantization of the Dirac and electromagnetic system. The procedure of quantization may not be dependent on whether the spectra are continuous or discrete. A non-perturbative quantization can be performed when we solve equations numerically with discrete spectra on a lattice or a compact space. Anyway, we must solve the infinite set of equations for Green's functions by using some method. It may be called the perturbative quantization unless we can solve the equations using non-perturbative methods. We recommend that the authors reconsider perturbative and non-perturbative quantization.
A1 Perhaps there is some misunderstanding in terms used by us. By the ``perturbative quantization’’, we mean the procedure of calculation of the Green functions using an expansion in a small coupling constant, i.e., using Feynman diagrams. Correspondingly, by the ``nonperturbative quantization’’, we mean the quantization procedure when calculations are performed without an expansion in a coupling constant. In doing so, we write down the Schwinger-Dyson equations that are solved somehow without using an expansion in a coupling constant.
Reviewer 2 Report
The authors extend their reference [6] to include a nonperturbative examination of Maxwell's electrodynamics coupled to a spinor field in a spherical finite space. My comments are listed below.
The authors are not doubt aware of this, but reference to Section numbers should be correct to eliminate the ?s.
Since the space is compact, indices describing the solutions are discrete in k-space. I don't see how this on its own eliminates the Dirac delta functions in real space and hence the singularites. I think this feature is instead due to the self consistent formulation in Section 5. The authors should justify this or pull back on their claims that use of a compact space eliminates the singularities.
It is unclear to me whether real Fermions are coupled to the EM field or only virtual Fermions are present. The physical basis of this key Section should be explained.
Author Response
Dear Editor,
Thank you very much for the referee reports. We have enclosed an appropriately revised version of our paper.
Sincerely yours,
- Dzhunushaliev, V. Folomeev
First of all, we would like to thank the referees for their careful reading of the manuscript and for the suggestions to improve the paper. We will now address the comments in detail. All our corrections made according to these comments are marked by yellow in the pdf file.
Reviewer #2:
Q1 The authors extend their reference [6] to include a nonperturbative examination of Maxwell's electrodynamics coupled to a spinor field in a spherical finite space. My comments are listed below.
The authors are not doubt aware of this, but reference to Section numbers should be correct to eliminate the ?s.
A1 Thank you for showing us this point. This is our fault made when copying the text to the Universe template. We have corrected it. Also, subsections 4.2 and 4.3 of the previous version of the paper are now joined together as one section 5 in the present version of the paper.
Q2 Since the space is compact, indices describing the solutions are discrete in k-space. I don't see how this on its own eliminates the Dirac delta functions in real space and hence the singularites. I think this feature is instead due to the self-consistent formulation in Section 5. The authors should justify this or pull back on their claims that use of a compact space eliminates the singularities.
A2 This is not the case. The ansatz (5) for the spinor field is written not in a k-space but in Minkowski space. We calculated numerically the propagator (69) in the particular case of t=t’, \chi=\chi’, \theta=\theta’, \varphi=\varphi’; it turns out that for the corresponding choice of the parameter f_{nml}, in calculating the propagator, the series converge ensuring the finite value of the fermion propagator, which means that the Dirac delta function is absent in the fermion propagator on the given compact space. This is pointed out in the text after eq. (69): lines 183-187.
Q3 It is unclear to me whether real Fermions are coupled to the EM field or only virtual Fermions are present. The physical basis of this key Section should be explained.
A3 This can be seen from eq. (75): if the nonzero spinors $<\hat \psi>$ are present, then the fermions are real. If the spinors $<\hat \psi>=0$ but eq. (77) gives the nonzero dispersion $<\hat \psi> <\hat \psi> \neq 0$, this means that the fermions are virtual.
Author Response File: 
Author Response.docx
Reviewer 3 Report
The authors have found solutions for Dirac Equation and Maxwell's equations separately for R XS^3 manifold with Hopf fibration of S^3. The interacting case and Green's functions have been framed but not yet solved.
Some of the results have to be compared with existing work in this field e.g.
(i) M. Carmeli, S. Malin Foundations of Physics Vol 15 (1985) 1019; where they have solved the Dirac equation in this topology.
(ii) D. Alves et al Physics Letters B 773 (2017) 412 where EM fields have been addressed in the Hopf Fibers.
(iii) K. Busse Ph. D. Dissertation http://sundoc.bibliothek.uni-halle.de/diss-online/98/98H152/prom.pdf, where the Maxwell fields have been solved in S^1 X S^3, and the S^3 solutions should be compared with the ones in this paper.
I have not checked the calculations explicitly, but they seem to be correct. Except for some section numbering in the text (line 56, line 171 etc) most of the text is well written.
I would suggest that some physical insights be provided for the solutions graphed in Figs 1 & 2. Why do the fields have maxima etc.
Author Response
Dear Editor,
Thank you very much for the referee reports. We have enclosed an appropriately revised version of our paper.
Sincerely yours,
- Dzhunushaliev, V. Folomeev
First of all, we would like to thank the referees for their careful reading of the manuscript and for the suggestions to improve the paper. We will now address the comments in detail. All our corrections made according to these comments are marked by yellow in the pdf file.
Reviewer #3
Q1 The authors have found solutions for Dirac Equation and Maxwell's equations separately for R XS^3 manifold with Hopf fibration of S^3. The interacting case and Green's functions have been framed but not yet solved.
Some of the results have to be compared with existing work in this field e.g.
(i) M. Carmeli, S. Malin Foundations of Physics Vol 15 (1985) 1019; where they have solved the Dirac equation in this topology.
A1 We are grateful to the referee for showing us the papers related to the subject of the studies performed in our paper. After eq. (35) (lines 86-89), we have added this reference and compared the results obtained by us with those given in that paper.
(ii) D. Alves et al Physics Letters B 773 (2017) 412 where EM fields have been addressed in the Hopf Fibers.
At the end of Sec. 4 (lines 149-156), we have added this reference and compared the results obtained by us with those given in that paper.
(iii) K. Busse Ph. D. Dissertation http://sundoc.bibliothek.uni-halle.de/diss-online/98/98H152/prom.pdf, where the Maxwell fields have been solved in S^1 X S^3, and the S^3 solutions should be compared with the ones in this paper.
At the end of Sec. 4(lines 149-156), we have added this reference and discussed the results obtained by us and in that dissertation.
Q2 I have not checked the calculations explicitly, but they seem to be correct. Except for some section numbering in the text (line 56, line 171 etc) most of the text is well written.
A2 We have corrected these misprints.
Q3 I would suggest that some physical insights be provided for the solutions graphed in Figs 1 & 2. Why do the fields have maxima etc.
A3 In Sec. 4.1.1 (lines 114-118), we discuss the physical meaning of the solutions shown in Figs. 1 and 2.
Author Response File: Author Response.docx
Round 2
Reviewer 1 Report
I think that the paper can be published after the revision of the manuscript.
