A Model for the Maxwell Equations Coupled with Matter Based on Solitons^†

Round 1

Reviewer 1 Report

The comments are included in the attached .pdf file.

Comments for author File: Comments.pdf

Author Response

please see Attachment. 

Author Response File: Author Response.pdf

Reviewer 2 Report

In the submitted manuscript, the purpose of the author was to present a model for nonlinear complex scalar field interacting with electromagnetic fields and to show the existence of solitary waves in this system which behaviour at low energies resembles a behaviour of pointwise charged particles in an electromagnetic field.

Concerning the material of the manuscript, it seems necessary to note the following.

1) The manuscript is not written well and not enough clearly. In particular,

   (a) The author presents the equation of motion of a charged particle in the electromagnetic field in a not relativistic invariant form (1) and discusses then its inconsistency with the Maxwell equations. However, this equation can be written in a relativistic invariant form (see, e.g. L. D. Landau, E. M. Lifshitz (1975). The Classical Theory of Fields. Vol. 2 (4th ed.). Butterworth-Heinemann. ISBN 978-0-7506-2768-9.).

   (b) It is not clear, what means the following author's phrase in the first paragraph of the Introduction:
   ``... the Maxwell equations are relativistic invariant and hence the inertial mass/energy of a charged material point is infinite.''

   (c) In the last phrase of the Introduction one can find one more puzzle:
   ``The original point of this paper is the introduction of a very simple interaction between the "matter field" and the e.m. field which produces identical particles ...'' --
   -- It is not clear here, why this interaction PRODUCES the particles and why these ``particles'' are IDENTICAL.

2) Unfortunately, the author does not use the four-dimensional tensor and vector notations which would simplify many expressions in the manuscript.

3) There are some ``strange'' points in the text. In particular,

   (a) In the last three lines on the page 3, we read: ``''There are two possible candidates which are ... invariant for the transformation (8):''.
   --- However, one can note that (8) is not a transformation.

   (b) Just above the equation (9), we read: ``Notice that these vectors are controvariant.'' --- It is an error because the vectrs at the last line of the page 3 and at the first line of the page 4 are covariant vectors.  And, just after the equation (9) we read that these vectors ``are covariant with respect to the time-coordinate;'' --- It seems to be incorrect to say about a covariance of some object with respect to some coordinate.

4) As it was mentioned by the author, the basic model determined by the Lagrangian (9) (see the text above (22)) is not gauge invariant with respect to the general gauge transformations of electromagnetic field. It is easy to see that the corresponding equations (11)-(14) are also not gauge invariant, and therefore, the suggested model of interaction of the charged scalar field with electromagnetic fields can be considered only as some toy model which, speaking generally, is not physically acceptable.

5) The ``matrix'' notation, used in (55) and later, was not explained.

6) On the page 22, in Definition 12, a division of one vector by another one is not a correct operation.

7) I think that it is a difficult task to read this manuscript in all details, many of which seem to be not necessary for understanding of the main points.

Besides that, it can be noted also that reading this manuscript, it is very difficult to summarize all definitions and assumptions which lead to such result as, e.g., the equation (81). On the other hand, these assumptions seem to be very restrictive and therefore, the reults are not so important as it may seem from the title and abstract of the manuscript.

All the points mentioned above does not allow me to be inclined to recommend this manuscript for publication.

Author Response

Please see the attachment. 

Author Response File: Author Response.pdf

Reviewer 3 Report

A classical model of the Maxwell equations coupled with matter field by the Klein-Gordon equation is studied. The author proved the existence of hylomorphic solitons and, under a couple of suitable assumptions, the low energy behavior of those solitons with some similarity with charged particles in the classical electromagnetic field.
This result is new and interesting. This paper is well-written and accessible at least to readers of the book [7].
The reviewer recommends its publication provided that the following two points are fixed:

1. p1. "Lorentz" should be "Lorenz." They are different.
2. p10. The author should explain $U_{loc}$.

Author Response

Please see the attachment. 

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

The author addressed all of my comments and I recommend this paper for publication.

Author Response

No comment!

Reviewer 2 Report

In the revised version of the manuscript, the author made many obviously necessary corrections mentioned in the previous report. However, the authors explanations concerning the other (and even more important) points mentioned in that report, can not be considered as enough satisfactory. In the explanations of this, given below, the numeration corresponds to numeration of the comments in the previous report:

1a) In the equation (1) the author used incorrect form of the equation of motion of charged particle in the external electromagnetic field. The correct form of these equations of motion is well known in Special and General Relativity theories and it is consistent with Maxwell equations. Moreover, the relativistically invariant form of these equations of motion was confirmed by many experiments and used, e.g. in the calculations of parameters of particle accelerators.  There is an author's error to use the incorrect form of equations of motion (1) as motivation for his farther considerations.

1b) It is incorrect to relate (see the text after eq. (1)) the divergency of electromagnetic energy of a point charge with relativistic invariance of Maxwell equations. The problem of divergency of this energy is well known and for its solution the notion of classical electron radius was introduced long ago.

1c) Here the author's answer also is not satisfactory and consists of a set of author's specualtions, which show, however, that the actual answer is absent. However, this point has not been changed in the new text of the manuscript. 

4) The author has agreed that the model of interaction suggested in the manuscript is a ``toy model'' and that it is not gauge invariant. However, this means (in a complete disagreement with the manuscript title) that this model can not have any relation to Maxwell equations and to coupling of matter with electromagnetic fields. On the other hand, a classical model for interaction of matter with Maxwell fields is well known and its description can be found in many classical field theory and continuous mechanics textbooks.

Thus, I think that the above comments explain in enough measure why my point of view has not been changed by the author's explanations and changes in the text of the manuscript and why, as before, I can not recommend the revised version of this manuscript for publication.

Author Response

see attachment.

Author Response File: Author Response.pdf