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A Geometric Model in 3+1D Space-Time for Electrodynamic Phenomena

Universe 2022, 8(2), 73; https://doi.org/10.3390/universe8020073

by Manfried Faber

Reviewer 1: Anonymous

Reviewer 2: Anonymous

Reviewer 3: Anonymous

Reviewer 4: Anonymous

Universe 2022, 8(2), 73; https://doi.org/10.3390/universe8020073

Submission received: 3 December 2021 / Revised: 2 January 2022 / Accepted: 17 January 2022 / Published: 26 January 2022

Round 1

Reviewer 1 Report

This article provides a concrete realization of a very interesting idea of a possible geometric derivation of classical electrodynamics. To this end, the author introduces, and further develops, a modification of the Skyrme model, which is given by Eq. (17). Similarly to the usual Skyrme model, where every term in the Lagrangian is invariant under the global SU(2)xSU(2) group, isomorphic to the Lorentz group, Lagrangian (17) also possesses this invariance. The stabilization of solitons in model (17) is provided by the potential term, instead of the traditional higher-derivative term of the Skyrme model. While that higher-derivative term is important to fulfill the conditions of Derrick's theorem, which guarantees solitons' stability, the author demonstrates that the potential term in model (17) makes those conditions equally respected. The potential term further yields photon-like solitons, as well as the long-range forces between the solitons. The advantage of this approach is that electron-like solitons, constructed in the same way, automatically have a finite mass and a U(1) charge, which can be expressed in terms of the topological charge of the corresponding soliton. Such photon- and electron-like solitons, stabilized by means of the potential term, appear Lorentz-contracted, similar to heavy ions. Still, these solitons keep possessing the properties of elementary particles, as their collisions can only lead to their annihilation, while their destruction is forbidden by the conservation of topological charge. The author also provides a detailed comparison of the solitons of model (17) with photons and electrons from the point of view of their quantum numbers and the number of their degrees of freedom. He proceeds further to the discussion of macroscopic properties of the model, in comparison with the corresponding properties of classical electrodynamics, arriving in this way at convincing conclusions. In particular, the calculation of the trace of the energy-momentum tensor, Eq. (55), enables the author to evaluate the energy density that can be released in the course of a transition between two vacua of the model, each of which is characterized by specific field configurations. The energy-density release thus obtained agrees reasonably well with the one known from inflationary cosmology, which can certainly be viewed as a remarkable property of the suggested model. Last but not least, the author provides the detailed lists of the main achievements of the model, as well as of the presently open issues. In general, the article is written in a very clear way, all the derivations are self-consistent, and their mathematical logic is transparent and correct. For all the above-given reasons, I fully recommend publication of the article in its present form.

Author Response

I thank this referee for the detailed study of the manuscript and the
exceptionally positive report.

Even if the referee may not agree with the philosophy of the article,
one can clearly see that he appreciated my efforts to write a compact and
nevertheless understandable manuscript which tries also to shed light
on the possible predictions of the model and a comparison with the
phenomena which we observe in our nature.

The report was able to judge conjectures as they are, subjects which
have to be investigated further in hard work. Other referees did not
succeed to do that.

I would like to make the referee aware that I tried to modify the text
according to the wishes of other three referees. I hope that he agrees
to these modifications.

I appreciate very much the open minded report.

With best regards,
Manfried Faber

Reviewer 2 Report

This is an interesting manuscript which intends to describe the electron as a classical field configuration in a Skyrme-like yet local theory subject to the gauge group SU(2). The theory is formulated on connections on the group manifold S_3 (spanning the tangential space and being defined by derivatives of the unit-quaternion group elements). The according action density is the square of the curvature of these connections, and but also requires an ad hoc potential term to obtain stable soliton solutions. The quantum numbers of this configuration (lepton number, charge, spin), which supports an extended energy density, are interpreted as topological invariants, and two Goldstone modes are linked with the propagating electromagnetic field. This paper is interesting and should be published in Universe after the following points are addressed by the author:

1) Personally, I believe that the electron intrinsically is much more quantum than a semiclassical approximation based on the present classical configuration may suggest. As the author himself admits in Sec.6.2 his model cannot be the final say on the nature of the electron. For example, why does the electron appear to be pointlike down to much smaller distances than the classical electron radius that characterizes the spatial extent of the present configuration? What makes the electron a clock in its restframe as de Broglie supposed to derive his wavelength upon Lorentz boost? I urge the author to compare his findings with the works of https://neo-classical-physics.info/uploads/3/0/6/5/3065888/de_broglie_-_hidden_thermodynamics_book.pdf, https://arxiv.org/abs/1709.03873 .

2) I don't understand the following reasoning: The connection Gamma_mu defined in (4) satisfies (10). The area element in the tangential space is defined by (11). Why would one generalize this area element to (12) just to conclude that the connection A_mu 2 Gamma_mu is a pure gauge? To my mind there are some logical gaps from positing that A_mu is a curvature-free gauge field to demanding the saddles and extrema of the action density (17) to describe the dynamics of a curved A_mu, subject to F^2 and Lambda. Maybe I just didn't understand this reasoning well enough. So please make sure this impression of mine about logical gaps can be rectified by adding explanatory text.

3) Below (12): vector(R)_mu nu * vector(sigma) is the element of the algebra, right?

Otherwise, the text is clear and useful, and the subject of research is important: any attempt to question basic assumptions of the Standard Model of Particle Physics (SM) towards the nature of elementary particles is welcome in light of intrinsic contradictions (pointlikeness yet finite mass and spin) and the stark contradiction of predictions about ground-state properties and their cosmological observations, having in mind, of course, that the SM is an excellent framework to describe particle INTERACTIONS.

Therefore, I recommend publication of this ms in Universe after the above points are addressed by the author.

Author Response

I thank the referee for the careful reading, the positive opinion
about the manuscript and the interesting questions raised. It is
interesting to exchange our opinions on these questions.

With pleasure I will try to answer the questions and try to improve the
readability of the article.

----

> I believe that the electron intrinsically is much more quantum than
> a semiclassical approximation based on the present classical
> configuration may suggest. As the author himself admits in Sec.6.2
> his model cannot be the final say on the nature of the electron.

I do not want to give the impression that I believe that an electron
in an hydrogen atom does not behave as a quantum object. This would be
crazy. According to my opinion these quantum properties concern the
trajectories of electrons. As calculations which are not yet published
show, it is sufficient to assume that the electron moves in Compton
time over a distance of a Compton wave length and one can get the
expression for the Zitterbewegung. By the way from this assumption one
can also derive the stationary Schrödinger equation. To get the
correct time dependence and interference one needs further
ingredients. In this respect I tried to formulate a first idea about a
mechanism in Sect.6.2. This part is far away from a complete model.

My present opinion, or conjecture, is that particles are defined by
topology. The electron is extended over a scale of the order of
femtometers. The quantum properties which we know e.g. from the
hydrogen atom and the Schrödinger equation are a result of vacuum
fluctuations of some type unknown up to now.

----

> For example, why does the electron appear to be pointlike down to
> much smaller distances than the classical electron radius that
> characterizes the spatial extent of the present configuration?

In order to learn from experiments about the property of electrons at
distances of the classical electron radius r_cl and smaller we need
much higher energies. As you know, the characteristic relation between
radius and energy is $\hbar c = 200 MeV fm$. 200 MeV electrons are
sensible up to distances of fm, the 100 GeV electrons of LEP up to
distances of 2 attometers.

The solitons of my model are relativistically covariant. A moving
soliton is Lorentz compressed. At LEP electrons reached a gamma factor
of 2*10**5 and therefore could approach another electron in a head-on
collision up to a distance of 0.5*10**(-5)*r_cl = 0.014 am. You can
see the analogous behaviour in the analytical two-particle solutions
of the Sine-Gordon model. For gamma to infinity the distance of
closest approach of two solitons approaches zero.

I have introduced Sect.6.1 to answer this question also for readers.

----

> What makes the electron a clock in its restframe as de Broglie
> supposed to derive his wavelength upon Lorentz boost? I urge the
> author to compare his findings with the works of
> https://neo-classical-physics.info/uploads/3/0/6/5/3065888/de_broglie_-_hidden_thermodynamics_book.pdf,
> https://arxiv.org/abs/1709.03873 .

Thanks for this reference. Just recently I tried to learn about de
Broglie's ideas. But I did not find this book. I inserted the
reference on page 28.

I was thinking and discussing a lot about this question. A friend of
mine urges always that the electron should have an internal clock. In
my model it would be unnatural that the electron has some internal
dynamics acting like a clock. Therefore I try to find, as mentioned
above, a mechanism for the interaction of the electron with the
surrounding quantum fluctuations which results in the Compton
frequency. This does not mean that this is the best explanation. It
only means that this is in the moment the direction which I think
about. I try to proceed in the direction which is mentioned in
Sect. 6.2. It is obvious that I am not able to answer this difficult
question satisfactorily.

In my model the electron has no internal clock. This was a result of
my searches. I had no other choice. I was looking for a three
dimensional generalisation of the Sine-Gordon model with solitons as
hedgehogs. I was trying to find a solution for this intuitive idea.
I did not find another solution than the one which is described in
this article.

----

> I don't understand the following reasoning: The connection Gamma_mu
> defined in (4) satisfies (10). The area element in the tangential
> space is defined by (11). Why would one generalise this area element
> to (12) just to conclude that the connection A_mu 2 Gamma_mu is a
> pure gauge? To my mind there are some logical gaps from positing
> that A_mu is a curvature-free gauge field to demanding the saddles
> and extrema of the action density (17) to describe the dynamics of a
> curved A_mu, subject to F^2 and Lambda. Maybe I just didn't
> understand this reasoning well enough. So please make sure this
> impression of mine about logical gaps can be rectified by adding
> explanatory text.

No. This is not the reasoning.

Short answer: (13) is a side remark. I have modified the text and put
this side remark it in the footnote on page 5. This is a nice example,
where the attempt for a clarification for one group of readers made
the subject unclear for other readers.

Long answer, possibly difficult to read since its details require
longer calculations: When I am defining the connection by
$\partial_\mu Q Q^\dagger$, the usual reaction by colleagues from
field theory is: "This is trivial gauge" and results in field strength
zero. In order to explain to field theoreticians that $\vec\Gamma_\mu$
is not a trivial connection, I am writing Eq.(13).

The difference is just the mentioned factor 2.
If I would modify Eq.(4) by inserting a factor 1/2 by another definition:
$$\partial_\mu Q(x)=:-\mathrm i\vec A_\mu(x)\,\frac{\vec\sigma}{2} Q(x)$$
I would define a trivial connection.

I would like to mention that this problem of the factor 1/2 troubled
me for several holidays and several years in the beginning of the
1990's when I tried to find the present model, first published around
2000. The field Q(x) maps a rectangle in space-time to a parallelogram
on the SU(2)-manifold isomorphic to S^3. It took me so long time until
I understood the difference between transporting Q around a rectangle
and transporting a tangential vector along a rectangle. Parallel
transporting Q defines the connection A_mu and results in a trivial
connection since after closing the path around the rectangle one returns
at the original Q. With this procedure the area of the rectangle in
space-time is related to a quantity (twoform) which is zero. This
vanishing quantity can't define the field strength of a charge.

versus,

parallel transporting a vector along a path defines \Gamma_\mu.
Parallel transporting an algebra valued vector around a rectangle in
space-time results in a rotated algebra valued vector, a calculation
of around one page which I do explicitly in my regular (german) QCD
lecture, see Eq.(3.69) in www.kph.tuwien.ac.at/faber/qcd.pdf. The
difference between initial and final vector is proportional to the
area of the rectangle. The proportionality factor is the field
strength which I was looking for.

Let me continue with the second part of your question:
> To my mind there are some logical gaps from positing that A_mu is a
> curvature-free gauge field to demanding the saddles and extrema of the
> action density (17) to describe the dynamics of a curved A_mu,
> subject to F^2 and Lambda.
There may be a misunderstanding.

I want to use the geometrical quantity (two-form) related to the
AREA=R_{\mu\nu}, of the parallelogram on S^3 as field strength. This
is Eq.(11), left ":=" symbol. Due to the Maurer-Cartan relation (10)
follows the second part of Eq.(11) and further Eq.(12). This
Eq.(12) is the well-known gauge covariant formulation of the field
strength tensor which we are using in QCD. I do not need to use it,
since for practical calculations the original definition (the original
gauge) is the simplest to work with.

----

> 3) Below (12): vector(R)_mu nu * vector(sigma) is the element of the
> algebra, right?

Yes.

The original text was really not very clear. Thanks for this remark.
In this paragraph I want to demonstrate the geometrical realisation of
gauge symmetry. It was a little to short. I hope the text between
Eq.(12) and Eq.(14) is now (hopefully) improved:

"Here we can also observe the geometrical realisation of gauge transformations, $\vec\Gamma_\mu\to\vec\Gamma_\mu^\prime$, as basis changes in the three dimensional euclidean tangential space at $Q$. $\vec R_{\mu\nu}\vec\sigma Q$ is a vector in this three dimensional tangential space at $Q$ on $S^3$, represented by the special basis vectors $\sigma_i Q$. In this special gauge, the algebraic expression for $\vec R_{\mu\nu}$ is defined by Eq.~(\ref{RSU2}). After a basis change the affine connection is modified, $\vec\Gamma_\mu\to\vec\Gamma_\mu^\prime$, and Eq.~(\ref{RSU2}) cannot be used any more. As one can see, performing the basis change explicitely, the more general Eq.~(\ref{RAllg}) is still valid after arbitrary basis rotations in the tangential space at $Q$, after arbitrary local gauge transformations."

I realised, that I did not mention this item before in the list on page 31. Therefore, I added now item 6.

Thanks again for the suggested improvements, they were very helpful.

With best regards,
Manfried Faber

Reviewer 3 Report

This paper proposes an alternative for the theory of Electromagnetism,
QED (the quantized version of Maxwell theory). I specifically mention
QED, as the proposed theoretical framework aims to solve the electron's
"self-energy" problem already at the classical level, instead of at the
quantum level, as is the case in QED.

It is interesting to explore such ideas. But, I do believe that the
author oversells the potential success of his model. There are very
serious questions, some of which are mentioned by the author himself,
and some not.

- First, while mention is made, no serious attempt is made to confront
the model with experiment (even at a classical level). The appearance
of magnetic currents would seem to be a very serious problem. Their
existence is acknowledged, but no discussion of their experimental
impact is given at all. Likewise, the model requires E and B fields
to always be perpendicular -- something that is not required by
Maxwell theory. This is mentioned, but, again, no discussion at all
is given on how to resolve this contradiction (which is what it is).
Mention is made that maybe at a macroscopic level this too-strong
constraint can be avoided, but not how. At this level, this "excuse"
is pure speculation, with nothing to back it up. Moreover, one would
like to see some discussion what "macroscopic" means. Experimental
constraints must exist on what is allowed by what we know about
Maxwell theory -- such constraints must be discussed.

- I was very confused by the speculations about how this model
"interacts" with quantum mechanics, and I suspect most readers will
be. QM, like Maxwell theory (and therefore QED) are huge successes
of theoretical physics. There is some discussion in Sec. 6.2, but
no path to reconciliation is offered. Is the model potential to be
replaced by Eq. (85)?

- This brings me to the potential chosen in Eq. (17), which contains
a "fundamental" scale $r_0$. The origin of this scale is never
discussed in the manuscript. Is this a new constant of nature?
The author clearly sees the model as very fundamental, witness his
assertions of how it puts electrodynamics on a footing with GR.
That means that the nature of $r_0$ needs to be defined--as such,
the potential term introduced seems rather ad hoc, needed in order
to stabilize the solitons the whole model is based on. How does
this term couple to gravity? What does quantizing this term do to
it?

- The paper gives a discussion of various rotational properties
of the topological objects identified in this model. But, I miss
a systematic discussion of their properties under Lorentz symmetry,
in order to establish that they conform to the (very well established)
properties of the electron and the photon.

- Another very important property of Maxwell theory is the superposition
principle, which is very well tested in experiment. Does this model
have a superposition principle, and, if so, how does this work? With
the non-linearities present in the basic model, this is not clear to me.

This paper contains many detailed mathematical derivations of the
properties of solitons appearing in this model, their interaction
(to some extent), and their interpretation in terms of the degrees
of freedom of electrodynamics, and it thus constitutes a significant
body of work. But, given the fundamental claims made in this paper,
I think the questions raised above would need to be addressed to
make this paper suitable for publication.

To be fair, some of the questions listed above are raised by the
author himself. But, that does not make them any less urgent!

Author Response

Thanks for reading the manuscript.

Let me start with some philosophical comments:

I am a physicist and not a mathematician. It is now 56 years that I am
curious how nature works. Only after many years of studying physics I
understood that in the most fundamental range of physics we have an
excellent description of nature, but we do not understand the
mechanisms. I do not need to repeat the famous saying of Feynman about
quantum mechanics. I think, the situation of particle physics is not
different. Particle physics does not inform us about the real nature
of particles. We describe particles by harmonic oscillators and this
works very well. But I cannot explain a well-educated physical layman
that particles are harmonic oscillators. Of course this description is
inspired by Fourier transformations. But the real reason could be that
harmonic oscillators have an equidistant spectrum as particle number
operators have. I was reading already several philosophical articles,
written by particle physicists, where they discuss what particles
are. But I think it is very dangerous to base a philosophy on models
which we do not understand deeply.

It would be much easier to get an article accepted if I just presented
a model and did some calculations and approximations and did not
compare the model with nature. Without getting complains by referees I
could write a paper about magnetic monopoles or about physics at the
Planck scale. I am in doubt whether in this way we would proceed in
our understanding of nature and you would not ask me about physical
consequences, more than I tried to explain anyway.

In 1989 a colleague showed me the pendulum experiment to the
Sine-Gordon model. I understood that this is a very intuitive model to
understand the nature of elementary charges and their
interaction. Therefore, I started myself to think about a 3+1D
generalisation of this model with solitons as hedgehogs. It took me
around 10 years to formulate the model, to find a geometrical
formulation of stable solitons with long-range coulombic interaction
and to solve the non-linear equations of motion. This is the first
model of this kind. I was thinking naively that the community would be
interested in a model of charged stable solitons without any
singularity. I was mistaken. In the meantime several books about
monopoles have appeared and no one discusses my model, despite your
claim that I oversell my model.

Meanwhile I learned much more properties of this model. After
giving talks about this model, the colleagues asked me about a
comprehensive formulation of the model. I realised that reading the
original paper and the papers which I wrote afterwards was too time
consuming. I felt the need for a more compact presentation of the
model including the new findings. This paper is the result of this
effort.

As seriously, as I was able to, I have tried a comparison of the
predictions of the model with the properties of nature in Sect. 7.2.
I indicated all pros and cons. Many of these questions are difficult
and probably need years of detailed work. Of course stating clearly
all cons gives an easy possibility to attack the paper. Probably you
would call Achilles stupid, if he showed his vulnerable heel in
public. But a serious discussions needs a discussion about the
cons. Immediate answers are not possible, otherwise I would have
solved it before.

I am retired and do not need papers. My aim is to understand the
mechanism of nature and to have a serious discussion about it. I do
not like to be satisfied by machineries which I do not really
understand. Of course may aim is also to show to the community that
one can understand the phenomena indicated in Sect. 7.2 from a
different perspective.

I know from the referee reports that it is difficult to accept claims
that many phenomena can be explained from such a simple model with
only three degrees of freedom. Such reports are then full of
prejudices, like "ULTIMATELY fails", "would not explain", "Such an
idea fails", "in the LONG RUN they fail", "all alternatives have been
tried and tested with zero success" ,"is not going to hold water
against the tremendous success". All these are real comments and not
my invention. Soon I will put the paper on the ArXiV. If this paper is
rejected, I will enjoy to attach all these comments in the
acknowledgement. It is a pity that the authors of these sayings do not
declare their names, I would like to acknowledge them.

I do not claim that this model is a final answer. I wrote explicitly:
"One should not assume that it can provide a final answer to all
questions, left open by quantum theory and quantum field theory." But
I see no reason to hide the predictions of the model which nicely
agree with nature. From the cons I (or better we) have to learn what
should be improved. I indicated already that this is hard work and
can't be done in one week. But I think that authors writing books
about monopoles should also describe this simple model, closer to the
properties of (electric) monopoles than any other model. This paper
should help grasp the main ideas of this model. After this first
round of reviews there are besides yours three reports with a
classification positive up to very positive.

But to be serious I would prefer to do physics and real calculations
than to write extensive answers to referees with more than 400
lines. But with the present situation with science, this seems to be
the only method which allows one to get some attention. At least the
referees have to give the impression that they have read the manuscript.

I hope that Universe wants to publish a paper about the physical
universe, not only about the universe of our mathematical ideas, which
are far from any physical application.

Now I will concentrate on your detailed questions:

----

"I specifically mention QED, as the proposed theoretical framework
aims to solve the electron's "self-energy" problem already at the
classical level, instead of at the quantum level, as is the case in
QED."

A problem that does not exist must not be solved. In my model there
exists no self-energy problem. I would not call it quantum level, I
would call it perturbative level. For me quantum effects are the
effects produced by quantum fluctuations, as you can see them in the
path integral formulation of quantum mechanics and quantum field
theory. One can see them further in interference effects, as the
double slit experiment. Quantum effects are related to $\hbar$. The
self-energy problem of QED has something to do with the 1/r potential
and the infinite energy density around a point-like charge. It has to
do with infinite momenta of virtual particles. These effects have
nothing to do with actions deviating by $\hbar$ from classical
solutions. They have to do with infinities which appear in a wrong
Lagrangian. Yes, the Lagrangian of Maxwell's electrodynamics for
point-like charges is infinitely wrong. I think that Maxwell himself
knew very well that charges cannot be point-like. To be mathematically
useful the fields and constants in the Lagrangian for point-like
electrons have to be corrected by infinite renormalisation factors. In
this procedure to make it mathematically consistent we learn that the
electric charge is dependent on the momentum transfer, on the strength
of kicks experienced by charges. Since years I myself give lectures on
path integrals, QCD, regularisation and renormalisation, the
Callan-Symanzik equation, Slavnov-Taylor identities, originally
discovered by Gerard 't Hooft .... I know, what I am speaking about.

I get a running of the coupling in my classical model. Recently the
calculations were getting precise enough, that we can compare our
running of the coupling with the Uehling potential, derived in
perturbative QED. I wanted to publish the present paper before
publishing this comparison to QED, in order that I can refer to it
concerning the details of the model. If you don't believe I can send
you our preliminary data where the running of the coupling is in the
ballpark of the predictions of the Uehling potential. We have still to
fight with finite size effects in the calculations on finite
lattices. In order to separate finite size effects from physical
effects we have to do rather extensive calculations. Nevertheless we
see a nice effect.

I have added Sect.6.1 concerning this problem.

----

> It is interesting to explore such ideas.
Fine, I think too.

----

> But, I do believe that the author oversells the potential success of
> his model.

I do not think so. I am indicating all unsolved questions which came
to my mind and I am also explicitly enumerating vulnerable heels. A
confrontation of pros and cons is not an overselling. Sentences like:
"This seems one of the critical questions of the model." and "It is
obvious that the model cannot compete" point rather towards
under-selling. The model could be known to the community since 1999,
hep-th/9910221. I am not overselling the model since obviously almost
nobody knows about it. Obviously I did nothing sell at all.

What you call overselling is obviously the paradigm of my article
which is clearly stated in the aftermath and difficult to digest for
believers in the present paradigms. But physics is not a religion.
At least, I am not in danger to be burned at the stake ;-)

My paradigm is going back to Einstein, who wanted to shift the energy
momentum tensor on the other side. I am giving a simple model which
gives some hints, how to do that.

----

> There are very serious questions, some of which are mentioned by the
> author himself, and some not.
On the first part I agree, on the second part not.
I did not omit open question by purpose.
I may not have treated them due to missing space or time.
You did not indicate which further open question should be mentioned.

----

> First, while mention is made, no serious attempt is made to confront
> the model with experiment (even at a classical level). The
> appearance of magnetic currents would seem to be a very serious
> problem. Their existence is acknowledged, but no discussion of
> their experimental impact is given at all.

This claim is unjustified. It is just the opposite, I try to compare
with experiments. You are right, I did not write all the details which
I know about magnetic currents. I wrote explicitly that abelian
magnetic currents do not contribute to Coulomb and Lorentz forces. I
wrote that they are not quantised, not appearing in quanta. I did not
write that non-abelian magnetic currents appear also inside the
solitons. The electric field of solitons is not originating in a
charge distribution, it originates in non-abelian magnetic currents,
in the curl of \vec B, in dual Amperes law. I did not want to produce
confusion with these interesting details. I did not devote a section
to magnetic currents. There are now comments on abelian magnetic
currents after Eq.(69), (76) and (83). What should be the experimental
impact, if they do not produce forces, are suppressed by minimising
the energy density and seem to be a boundary effect originating from
the topological restrictions for the solutions of the homogeneous
Maxwell equations? In summary, I do not expect experimental
consequences. I expect complications only for the description of
experimental situations, very natural for a non-linear formulation.

----

> Likewise, the model requires E and B fields to always be perpendicular
> -- something that is not required by Maxwell theory. This is
> mentioned, but, again, no discussion at all is given on how to resolve
> this contradiction (which is what it is). Mention is made that maybe
> at a macroscopic level this too-strong constraint can be avoided, but
> not how. At this level, this "excuse" is pure speculation, with
> nothing to back it up. Moreover, one would like to see some
> discussion what "macroscopic" means. Experimental constraints must
> exist on what is allowed by what we know about Maxwell theory -- such
> constraints must be discussed.

To meet your wish, I have added some sentences to Sect. 6.2 and refer
explicitly to the more detailed calculations in Ref.(20). The added
comments show that our treatment of constant electric and magnetic
fields with Maxwell's theory is only an approximation which does not
take into account the atomic structure of capacitors and the hopping
of electrons in a wire. I did not want to blame Maxwell, since the
approximation is ours. In my model this atomistic structure is clearly
reflected in the formulation.

The model is simple in the formulation, but it is non-linear and
therefore not simple for calculations. Such calculations need time. We
wrote already several papers about electromagnetic fields. What I
know, you can read there, see Refs: 20,22,24.

----

> I was very confused by the speculations about how this model
> "interacts" with quantum mechanics, and I suspect most readers will
> be.

Introducing quantum fluctuations and thus quantum effects needs a
proportionality to the Compton wave-length. There is a community which
is very interested in these questions. This is just a first idea how
one could get such a dependence on the mass of quantum particles. You
are right, this needs extensive elaboration - and time. This article
is a good opportunity to state such a first idea.

I would not call an idea a conjecture, if I would have solved it in
detail up to a stage where one can really say whether it is true. As
usual conjectures can be wrong.

----

> QM, like Maxwell theory (and therefore QED) are huge successes
> of theoretical physics.

NO DOUBT! I am not completely stupid to disagree to this statement.
But they are not the end of natural science. We should allow our
fantasy to find further progress. Feynman would still like to
understand.

----

> There is some discussion in Sec. 6.2, but no path to reconciliation is
> offered. Is the model potential to be replaced by Eq. (85)?

There is also a reference given in this paragraph. Maybe you
overlooked it. In this reference the complete derivation is
given. Eq.(85) describes the interaction potential between alpha-waves
and solitons. By the way also alpha-waves are unquantised massive
lumps and could be contributions to dark matter. But possibly you don't
like conjectures.

----

> This brings me to the potential chosen in Eq. (17), which contains
> a "fundamental" scale $r_0$. The origin of this scale is never
> discussed in the manuscript. Is this a new constant of nature?

Sure it is discussed. A potential term has to have dimension of
energy per volume, i.e. length**(-4). Since $\Lambda$ in the
Lagrangian (17) is dimensionless, a dimensionful length has to be
introduced. This is obvious. On page 7 you can find the text: "The
factor $r_0^{-4}$ is necessary by dimensional reasons. $r_0$ is the
length scale responsible for the size and the mass of solitons." r_0
is at the same level a constant of nature as the mass of the electron
and the classical electron radius are constants of nature and at the
same level as the energy released in cosmic inflation, if it really
happened, is a constant of nature. Both are related to r_0, see
Eqs.(32) and (90), if you believe these conjectures.

----

> The author clearly sees the model as very fundamental, witness his
> assertions of how it puts electrodynamics on a footing with GR.

Sorry, I do not really understand this sentence, especially the second
half. Nevertheless, I will try to answer this comment.

The first part of the sentence is polemic. I am investigating a model
and try to compare with the phenomena in nature. As written above, I
think it is important to study models which can be compared to nature.

From my intuition which I gained from the sine-Gordon model, I wanted
to describe charges and electromagnetic phenomena. It turned out that
the three degrees of freedom which are necessary for this model can be
interpreted as rotations of Dreibeins in 3D. Therefore it is natural
to think about a unification of electrodynamics and gravity. By the
way this was an old dream of Einstein. But he did not succeed due to
missing intuition, as Smolin guesses in arXiv:1512.07551. At that time
there was not enough knowledge available about non-abelian theories. I
know, "quod licet Jovi, non licet bovi" and I should stop thinking
about such a unification, as you seem to suggest.

----

> That means that the nature of $r_0$ needs to be defined--as such,
> the potential term introduced seems rather ad hoc, needed in order
> to stabilize the solitons the whole model is based on.

Yes, it is ad hoc as is every term in a Lagrangian. Every model which
we study contains ad hoc assumptions. We were taught in lectures about
these terms and are used to them. Einstein did not explain the origin
of the cosmological constant, he simply introduced it in order
to get a universe which is not expanding. I have devoted a full
section, 6.5, to discuss that the potential term can be interpreted as
a cosmological function. Its average is the cosmological constant. It
is in the right order of magnitude and not 120 orders of magnitude
wrong, as the prediction of QFT. Politely I did not mention in the
paper how many orders the QFT estimate is wrong. One can remove the
contribution of QFT to the cosmological constant by normal
ordering. But 0 is infinitely many orders of magnitude wrong. Normal
ordering does not help. How to judge this? Is this a question,
critical for QFT or do they oversell QFT? I think not.

----

> How does this term couple to gravity?

It is the cosmological constant. Does the cosmological constant have
nothing to do with gravitation? It is the opposite, the cosmological
constant has nothing to do with QFT, see the prediction by QFT,
mentioned above - 120 orders of magnitude wrong.

----

> What does quantizing this term do to it?

I think, you do not share my philosophy which I wrote about
explicitly. No problem with that. But, I would not dare to write a
report about a paper whose philosophy I do not share, at least not a
negative one. I try to find out, whether one can move particle physics
closer to gravitation. There is possibly no need for quantising
gravity. This was not the original idea. This idea is suggested by
the model itself.

----

> The paper gives a discussion of various rotational properties
> of the topological objects identified in this model. But, I miss
> a systematic discussion of their properties under Lorentz symmetry,
> in order to establish that they conform to the (very well established)
> properties of the electron and the photon.

The Lagrangian is Lorentz invariant. From our knowledge of special
relativity we know how such models behave. There is really no further
discussion necessary. The solitons are Lorentz contracted. I discussed
the Lorentz contraction of solitons explicitly in Ref.3.

----

> Another very important property of Maxwell theory is the superposition
> principle, which is very well tested in experiment. Does this model
> have a superposition principle, and, if so, how does this work? With
> the non-linearities present in the basic model, this is not clear to me.

Thanks for this question which I did not answer before. In non-linear
models we do not expect a linear superposition principle. I expect
that linear superposition principle is only valid approximately. We
were discussing such questions in Ref.22. Please read in the
introduction: " This may lead to the conjecture that Maxwell-Dirac
theory is a clever linearisation of a non-linear theory with a smaller
number of fields." After Eq.(69) I have added further remarks
concerning this question.

----

> This paper contains many detailed mathematical derivations of the
> properties of solitons appearing in this model, their interaction
> (to some extent), and their interpretation in terms of the degrees
> of freedom of electrodynamics, and it thus constitutes a significant
> body of work.
This is correct.

----

> But, given the fundamental claims made in this paper, I think the
> questions raised above would need to be addressed to make this paper
> suitable for publication.

I tried to explain in detail all the points you raised. Please
understand that all the conjectures need to be worked out in special
papers, if possible, depending on the number of scientists interested
to work on this model or modifications of it.

Don't mix up, conclusions and conjectures.

> To be fair, some of the questions listed above are raised by the
> author himself. But, that does not make them any less urgent!

I comletely agree. The next question which we are going to solve is
the running of the coupling. Let me not waste further time to
concentrate on this and further interesting questions.

I dare to state finally: Some nice properties of the model give new
insights in possible mechanisms of nature, see the 16 items around
page 31, and the open questions and conjectures give occasions for
further investigations.

I enjoy to have four readers more which possibly think about my
"overselled" model and the mechanisms in nature. Hopefully they do not
stop at that what we were told in lectures ;-)

With best regards,
Manfried Faber

Reviewer 4 Report

I'm sending my report in the attached PDF file. Please, let me know if if there is any problem to open this file.

Comments for author File: Comments.pdf

Author Response

I thank the referee for careful reading of the manuscript and his positive
remarks. I appreciate very much the positive comments.

Thanks for the reference to Vachaspati and Manton. I read Vachaspati
book some years ago: "Kinks and domain walls: An introduction to
classical and quantum solitons" and liked it very much. In Sect. 7.1.
I have added a few sentences with your remarks.

Concerning the further comments:

> Regarding the presentation, there are some aspects that could be im-
> proved. For example, the identification between the linking number
> and the photon number in Sec. 5.4 is not at all clear. I believe
> that this part should not be included because no reason is given to
> support this assumption, besides the fact that both quantities are
> given by integers.

The origin of this model was the remarkable interaction between
solitons and antisolitons in the Sine-Gordon model which give the
impression of an interaction of electrons and positrons. My intention
was to generalise it from 1+1D to 1+3D and to find stable hedgehog
solutions which react like electrons and positrons. From this
intuition it took me around 10 years to formulate the model. I had no
idea, that I would get three topological quantum numbers. Of course I
expected something like the charge. I did not expect to find a
geometrical realisation of spin. Further I found two Goldstone
bosons, see Sect.5.4 and a corresponding topological quantum number,
corresponding to the Hopf number. Fibers of constant $\vec$ n-field
which are circulating around themselves, as described in "Charges and
Electromagnetic radiation as topological excitations". We have a
system of charges and their electromagnetic fields. Looking for
phenomena in nature for massless excitations of these electromagnetic
fields propagating with the speed of light, excitations which have two
types of chiralities, left-handed and right-handed? Interacting with
solitons they can modify the spin of solitons. Isn't it reasonable to
compare these excitations to photons. I agree that I could have
written more on this subject. But somewhere there has to be and
end. Photons is a very reach field and needs extensive detailed
investigations. We have devoted to this field already several papers,
Ref. 20, 22 and 24.

----

> However, some expressions used in this manuscript from the very beginning
> may confuse the reader. For example, referring to the “electrodynamic limit”
> and then pointing to differences with electrodynamics is confusing.
> A similar issue applies to the title “A geometric model in 3+1D space-time
> for electrodynamic phenomena”. Also, before Eq, (82), it is stated that
> configurations “describe electromagnetic waves”. Maybe there is a more
> appropriate term for this limit, to better describe the current preliminary
> status of this program.

Please, observe, that I make a careful difference between the terms
Maxwell's electrodynamics (MEdyn) and electrodynamics. With MEdyn I
refer to the formulation of electrodynamic phenomena with the usual
gauge field $A_\mu$. I do not use MEdyn, since I am using a (Hodge)
dual formulation. With electrodynamics I mean electrodynamic
phenomena. With those phenomena I want to compare the quantities of my
model. The title "Electrodynamic limit" was used already long ago,
2004, in a paper, Ref.~20 of this paper, which Sasha Kobushkin wrote
together with me in Physical Review entitled "Electrodynamic limit in
a model for charged solitons", Phys.Rev.D69(2004)116002. Sasha died in
2020, I miss him very much.

I tried to clarify my usage of the terms "electrodynamics" and
"Maxwell's electrodynamics" with a clarification of the terms in the last
sentence of Sect. 3.1.:
"It is an interesting subject to investigate which predictions of this
dual electrodynamics differ from Maxwell's electrodynamics (MEdyn) and
which predictions disagree with the experiments to electrodynamics (Edyn)."

With best regards,
Manfried Faber

Round 2

Reviewer 3 Report

The nature of the author's response to my referee report is rather polemic,
which is unfortunate. Let me try to summarize my basic issues with the
manuscript.

QED (as embedded in the Standard Model) has been a spectacular success,
experimentally. This, of course, does not mean that it leaves no questions -- we do not understand what nature looks like at smaller distances than can currently be probed experimentally. The author proposes an alternative theory to QED, and clearly claims, or suggests to claim that it solves a number of fundamental questions not answered by QED -- notably that of the self-energy of the electron. This is a very strong claim, and given the success of QED, the "burden of proof" is on the author, and a large one indeed. I do not think the present paper succeeds in making the case.

As I wrote in my previous report, QED has been experimentally tested to an astonishing degree, and I believe that an alternative theory of electrodynamics should be scrutinized for its ability to meet these experimental tests. While I do appreciate that not every problem can be solved in one paper, I would expect that in the many years the author has spent on this model, more progress would have been made. Not only is such progress not reported here, but also no clear delineation is provided of what precise, quantitative tests should be considered to potentially falsify the author's model. For any proposed theory beyond what we currently know and understand, this is a first, and most important, requirement. While several paragraphs have been added to the revised version, no new information in this regard has been added.

Let me also point out something that the author knows, but that might have gotten lost in the work on this model. According to QED, the issue of the self-energy of the electron has everything to do with quantum fluctuations! Indeed, it is the quantization of electrodynamics that solves the problem, and the modern way of understanding how that works is with the help of the Wilsonian renormalization group. Of course, one might say that this "pushes the problem" toward shorter distances, where QED is not a complete theory (even though its embedding in the full Standard Model
extends its range of validity to much shorter distances). The statement by the author, that "These effects have nothing to do with actions deviating by $\hbar$ from classical solutions." is clearly not correct in the context of QED as a quantum field theory -- it is directly contradicted by how the renormalization group works. In fact, suppose the author's model would be
correct, and thus the "self energy problem" would be solved at a classical level, how would quantization of the model modify these conclusions? In order to address the running of the QED coupling, not only should the potential between charged particles be considered classically, but quantum corrections should be considered as well. Apart from pointing out the issue of quantization, very little is said about the quantization of this model. The way I understand Sec. 6.3 of the manuscript is that quantization presents a rather big problem for the model!

The author relates that he was inspired by the interesting properties of the Sine-Gordon model. This is an interesting toy model indeed, and has helped us understand many aspects of QFT. If the author's model can be viewed as a generalization (in a broad sense) of such models to 3+1 dimenions, it might be fun to work out its consequences, and consider it as another model in our "tool box" of toy models, aspects of which can possibly teach us something interesting in the mathematical physics of field theory.

However, the author clearly does not present his model as such. Clearly, the author believes that his model is a theory of electrodynamics. I am sorry to say, but given the lack of quantitative comparisons with a very extensive amount of experimental knowledge about the properties of charged particles and electromagnetic fields, he has not made his case. As it is, the only "success" I see is the claimed resolution of the problem of a point-particle self energy in Maxwell theory -- a problem for which an alternate, and experimentally very successful solution is provided by QED. Moreover, it is not clear that this solution survives quantization of the model.

If the author would be interested in reformulating his paper as a paper about an interesting mathematical model, without the claims that it describes electromagnetic phenomena, it might be suitable for publication as a paper in mathematical physics. But, the author clearly makes the claim that his
model provides an alternative theory of electrodynamics, without providing any proof that it fares better, experimentally, than QED, nor how it resolves many of the theoretical problems the model raises.

Author Response

Dear Editor,

Please would you like to forward the attached letter to the scientific editor.

With best regards,
Manfried Faber

------------------------------------------------------
Dear scientific editor,

Thanks for the report of referee#3.

I want to emphasise that with referee#3 there is a conflict of schools.

I suggested a model which suggests to geometrise particle physics and
this discuss a first step in this direction.
Referee#3 wants to follow the common paradigm to quantise gravity.

Since we do not know the final answer, both ways are equally justified.
It is unjustified to reject my view.

In the first report Referee#3 asked questions which are extensively
discussed in the manuscript.
I am rather in doubt that he really studied the article carefully.

In this second report he did not even mention the modifications of the
manuscript which I did according to his wishes after his first report.
There is no reaction to these efforts to satisfy his former wishes.
This new report contains even more general comments without real substance.

It is not easy to react to such statements in a constructive way.
He requests a general reformulation. Obviously, he dislikes even the
title.

In the reviewing procedure three of reports were clearly positive.
One report was negative. I answered every detail of all four reports
and inserted modifications, where the were requested.

In the new report referee#3 did not give any detailed suggestion
how to improve the manuscript.
He wrote very general statements of discontent only, which give me
no indications how to react.
In principle he asks me to redraw the article.

I don't want to do fulfil this request based on the statements of the
other 3 referees:

1st referee:
I fully recommend publication of the article in its present form.

2nd referee:
" ...the text is clear and useful, and the subject of research is important..."

4th referee:
"...the developments in this work are very interesting and should be accepted for publication in Universe..."

I hope that Universe keeps its promises and ignores the statements of
prejudiced referees.

With best regards,
Manfried Faber

Despite it does not make much sense to answer this second report of referee#3
I will comment them one by one:

QED (as embedded in the Standard Model) has been a spectacular success,
experimentally.

No doubt. Please read in my manuscript: "This model tries to give an
idea about a possible direction of future investigations. One should
not assume that it can provide a final answer to all questions which
are left open by quantum theory and quantum field theory. Both
theories have a tremendous success. It is obvious that the model
cannot compete with their excellent and precise predictions"

----

This, of course, does not mean that it leaves no questions -- we do not
understand what nature looks like at smaller distances than can currently
be probed experimentally.

Yes.
----

The author proposes an alternative theory to QED

I see no scientif reason, why this should not be allowed?

and clearly claims, or suggests to claim that it solves a number of
fundamental questions not answered by QED -- notably that of the
self-energy of the electron. This is a very strong claim, and
given the success of QED, the "burden of proof" is on the author,
and a large one indeed. I do not think the present paper succeeds
in making the case.

This is not true.
I am clearly enumerating, what prediction agrees with experiment
and also with QED and what disagrees.
There is no other claim.

----

As I wrote in my previous report, QED has been experimentally tested
to an astonishing degree

No doubt.
But there are also some questions which are not answered.

----

and I believe that an alternative theory of electrodynamics should
be scrutinized for its ability to meet these experimental tests.

Sure.
----

While I do appreciate that not every problem can be solved in one
paper, I would expect that in the many years the author has spent
on this model, more progress would have been made. Not only is
such progress not reported here, but also no clear delineation is
provided of what precise, quantitative tests should be considered
to potentially falsify the author's model.

This comment is not related to the manuscript.
It is unnecessary to deteriorate the work of colleagues.
----
For any proposed
theory beyond what we currently know and understand, this is a
first, and most important, requirement. While several paragraphs
have been added to the revised version, no new information in this
regard has been added.

In 10 days one can not get new results.
The new paragraphs were answers to the questions of all 4 referees.
Such a global critics to the answers does not help to improve a paper.
----

Let me also point out something that the author knows, but that
might have gotten lost in the work on this model. According to QED,
the issue of the self-energy of the electron has everything to do
with quantum fluctuations! Indeed, it is the quantization of
electrodynamics that solves the problem, and the modern way of
understanding how that works is with the help of the Wilsonian
renormalization group. Of course, one might say that this "pushes
the problem" toward shorter distances, where QED is not a complete
theory (even though its embedding in the full Standard Model extends
its range of validity to much shorter distances). The statement by
the author, that "These effects have nothing to do with actions
deviating by $\hbar$ from classical solutions." is clearly not
correct in the context of QED as a quantum field theory -- it is
directly contradicted by how the renormalization group works. In
fact, suppose the author's model would be correct, and thus the
"self energy problem" would be solved at a classical level, how
would quantization of the model modify these conclusions? In order
to address the running of the QED coupling, not only should the
potential between charged particles be considered classically, but
quantum corrections should be considered as well.

I did not forget what I teach every year in my lectures on QCD and the
renormalisation group.
This is a general statement which is not related to the manuscript.
----

Apart from pointing out the issue of quantization, very little is said about
the quantization of this model. The way I understand Sec. 6.3 of the manuscript is that

I wrote clearly in the introduction:
"it may be worthwhile to think about models which give finite results already at the classical level."

How quantum effects could enter is discussed in Section 6, with the title
"Open questions and Conjectures"
----

quantization presents a rather big problem for the model!

Sure it is a difficult question. But to imply that it is unsolavble, would
be a prejudice.

The author relates that he was inspired by the interesting
properties of the Sine-Gordon model. This is an interesting toy
model indeed, and has helped us understand many aspects of QFT. If
the author's model can be viewed as a generalization (in a broad
sense) of such models to 3+1 dimenions, it might be fun to work out
its consequences, and consider it as another model in our "tool box"
of toy models, aspects of which can possibly teach us something
interesting in the mathematical physics of field theory.

This concerns the answer which tried to explain some surroundings to the
referee. Since this does not concern the manuscript, no further anwer is
necessary.
----
However, the author clearly does not present his model as such.

This is a general statement.
It is a pity, that there is no single example given.
If the referee does not indicate what he really means, its impossible
to react.
----

Clearly, the author believes that his model is a theory of
electrodynamics.

The title of the paper is "A geometric model ...
Nowhere concerning my work appears the word theory.
----

I am sorry to say, but given the lack of
quantitative comparisons with a very extensive amount of
experimental knowledge about the properties of charged particles and
electromagnetic fields, he has not made his case. As it is, the
only "success" I see is the claimed resolution of the problem of a
point-particle self energy in Maxwell theory -- a problem for which
an alternate, and experimentally very successful solution is
provided by QED. Moreover, it is not clear that this solution
survives quantization of the model.

There are 5+17 items which compare the model with Maxwell's electrodynamics
and with phenomena in nature. This means 22 comparisons.
Therefore, this statement is wrong.
----
If the author would be interested in reformulating his paper as a
paper about an interesting mathematical model,

This is a very general statement.
The referee does not explain which statements should be reformulated.
----

without the claims that
it describes electromagnetic phenomena, it might be suitable for
publication as a paper in mathematical physics.

It seems that the referee does not like to compare a model with electromagnetic phenomena. The title of the paper is:
"A geometric model in 3+1D space-time for electrodynamic phenomena".
----

But, the author
clearly makes the claim that his model provides an alternative theory
of electrodynamics, without providing any proof that it fares better,
experimentally, than QED, nor how it resolves many of the theoretical
problems the model raises.

I suggest a model, compare it with nature, indicate clearly pros and cons.
What is wrong about this aim?
I am a physicist and want to understand how nature works.

With best regards,
Manfried Faber

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 author has provided a geometrical model for the electromagnetic phenomena. The idea is not new: the author himself presented it in ref. [2], and in-fact there are numerous variations of this from the early days of Maxwell electrodynamics. Unfortunately however, viewing electrodynamics completely as a geometrical effect, while fascinating, ultimately fails to explain most of the subtle effects of electrodynamics. For example no geometrical model would be able to explain Lamb shift, magnetic moment of the electron etc. The work presented here touches on the "classical" aspects of the subject and tries to explain them using some geometrical properties. As an example, the author describes the properties of the photon using two Goldstone bosons. Such an idea fails to explain the short-distance behavior of the photon. Additionally, the stable solitonic excitations are mapped to the electrons. Such a picture will fail to explain the electic-magnetic duality (for example the Montonen-Olive type duality).

The author also views gravity completely as classical and remarks on why he differs from the rest of the community. The point is that, viewing gravity completely as classical fails to mesh even with Einstein's unified model. For example, if we take five-dimensional classical gravitational background and dimensionally reduce to our four-dimensional space-time, then we get both gravity and Maxwell electrodynamics in four-dimensions. However if gravity is purely classical, then the dimensional reduction will make the Maxwell electrodynamics completely classical. This, as discussed above, will fail to explain many if the subtle electromagnetic effects. Thus gravity by itself cannot be completely captured by classical geometry.

Thus, in my opinion the author has tried to bring back old discredited models of gravity and electromagnetism. While these viewpoints may explain some of the phenomena, but in the long run they fail to satisfactorily explain many subtle effects which can only be explained using the quantum picture. Therefore I cannot recommend the paper.

Author Response

I thank the referees for careful reading of the manuscript, for their positive and also for their critical remarks. I tried to consider them as good as possible. Hopefully this helped to improve the paper.

I think the three referees agree that one should not overestimate the predictive power of the model.

I have therefore added after enumeration of many of the properties (also the questionable ones) the paragraph:

"This model tries to give an idea about a possible direction of future investigations. One should not assume that it can provide a final answer to all questions which are left open by quantum theory and quantum field theory. It is obvious that the model cannot compete with their excellent and precise predictions and the century-long efforts of numerous scientists to describe the properties of nature in the most fundamental domain. But this does not mean that one should not think about scenarios which finally could lead to deeper understanding of the mechanisms in nature and which could allow us to answer some open questions, to answer how the nature works to produce the interesting riddles which we could not solve yet. Some of these questions concerning electrodynamics and quantum theory are enumerated in the above mentioned critical remarks. Many may still be missing. Essential open questions are also those related to the three other fundamental forces, gravitational, strong and weak interactions."

I hope that these comments are clear enough to underline that one (and I) should not overestimate the model, despite the fun I have with it.

I thank the referees for many detailed remarks, which I tried to satisfy, if possible.

--------------------

- In the title itself, it seems to be more in line with the Minkowski treatment adopted in the paper to write 3+1D instead of 4D.
Answer:
Thanks, the misleading 4D is substituted by 3+1D everywhere.

--------------------

- On page 2, beginning of the last third `.. .. by two topological quantum numbers corresponding to charge and spin quantum numbers' (plural).
Answer:
This is done.

--------------------

- Same page 3, Eq.(3) a sign of : is missing in the second equality.
Answer:
Thanks for this careful observation.

--------------------

- Same page 3, the footnote1. Though standard enough, it might be worth defining the symbol of x in the products \vec{a}x\vec{b}, as it is used throughout the paper.
Answer:
I have inserted a corresponding comment in this footnote
:"We are using the vector symbol for the three coefficients of the imaginary quaternionic units and of the su(2) algebra, $\vec q\vec\sigma$ is a short notation for $q_i\sigma_i$. For simplicity we omit the dot for such three dimensional scalar products, if it does not lead to ambiguities."

--------------------

- On page 9, just before Eq.(28), though made clear by the resulting Eq.(28), it would be more pedagogic to indicate with respect to what the variations of Eq.(27) are considered.
Answer:
Yes. I inserted "By the variation of the energy functional~(\ref{...}) with respect to $\alpha(\rho)$ we get the differential equation for the minimum of the energy"

--------------------

Same page footnote 3, `With bold characters' could be better than `With bold phases' .
Answer:
Sure, phases -> characters, is modified.

--------------------

- On page 10, an example of a possible amelioration, here as in other places, could be to write, just after Eq.(32), `..is specific to the chosen..' instead of `..is specific for the chosen..' .
Answer:
I have modified the text according to your suggestion.

--------------------

- On page 10, something confusing. After Eq.(30), shouldn't it be written `total' instead of `radial' ? Wouldn't radial refer to $\vec{E_r}\vec{E_r}$ as in Eq.(27)first line?.. whereas in Eq.(31) as in the last line of Eq.(27) the 3 energy density contributions are listed, in the same order, radial, tangential and potential, and not just the radial only ??
Answer:
I tried to clarify this term which means the $r$ dependent energy density
"we get the three contributions to the radial energy density, the $r$ dependent spherical integrals in Eq.~(27),"

--------------------

- On page 12, after Eq.(39), write ` These relations may etc..'.
Answer:
This is done.

--------------------

- On page 14, 1st line.. this is not quite true indeed. See K. Huang and D. F. Stump,
Phys. Rev. Lett. 37, 545 (1976); Phys. Rev. D15, 3660 (1977).
Answer:
Thanks for indicating this imprecise statement. I was thinking of QCD. The indicated references treat other theories, where my statement is incorrect. I have modified it.
The new sentence is:
"In distinction to the internal color property in QCD and the Standard Model, spin contributes to the total angular momentum."

--------------------

- On page 14 the end.. we can combine the 4 single-particle congs. Would be nice to indicate on the basis of which principle such a combination is made possible (don't think that it is a non-linear superposition principle!?)..
Answer:
There is no simple superposition principle yet, despite the addition rule of topological quantum numbers. The real solution has to be found by a minimisation procedure. More details are now inserted in the first lines on page 15. In the sine-Gordon model they have found analytical solutions for such two-particle configurations. Of course one can dream that due to the simple solution for single-particle configurations the same can be found in this model.

--------------------

- On page 15, last sentence: very interesting; and challenging !
Answer:
Thanks for this pleasing remark.

--------------------

- On page 24, 5th line, not easy to make sense of $2^*4$..
Answer:
The new comment reads:
"Up to the different sign, this Lagrangian is formally identical to the Lagrangian of Maxwell's electrodynamics with 2*4 physical dofs of the wave-field in radiation gauge when the time component and the longitudinal component of the gauge field are removed."

--------------------

- On page 25 last paragraph write `hundreds' instead of `hunderts'. Same page, by the end.. Note that the Couder experiment has been refuted lastly : `An experimental Boost for QuantumWeirdness', Quanta Magazine, Oct.11th, 2018. https://www.quantamagazine.org/famous-experiment-dooms-pilot-wave-alternative-toquantum-weirdnes
s-20181011/.
Answer:
I have softened the statement concerning Couder.
"lead to the famous interference pattern" -> "lead to an interference pattern"
Now it does not say that the pattern is that expected from quantum mechanics.
It just says that the experiment leads to some interference pattern.
Even if it the experiment is criticised, it is inspiring.

--------------------

- On page 28, last paragraph isn't it Georgi (Howard) instead of Georgia ?
Answer:
Sure! It is corrected.

--------------------

- On page 29 last point 2. The mass of solitons is completely due to field energy see Eq.(27).. and potential too, no?
Answer:
Yes, but this is also energy which is expressed in degrees of freedom of the soliton field. I think, this sentence is correct.

--------------------

- On page 31, Aftermath. Wheeler `Matter etc.. tells matter how to move'.
Answer:
Wheeler may have said this sentence in different versions. Somewhere I was reading that the version which I wrote before was the correct one. Now I checked carefully and found a written version in Wheeler's biography, which I cite now.

Author Response File: Author Response.txt

Reviewer 2 Report

This paper proposes an entirely new description of QED. The new description is dual (in the sense of Hodge duality) to the usual one. It is based on the scalar field in eq.(2) and the Lagrangian (17) within an underlying SU(2) geometry. The author studies the soliton solutions of this model and is able to show that they are stable and have topological numbers, which can be interpreted as quantum numbers to be possibly associated with quantization of charge, spin and angular momentum. The author suggests that one of these solitons could be interpreted as the electron. In the limit where the potential $q_0$ vanishes the model becomes similar in some respects to QED, with Coulomb and Lorentz forces and U(1) symmetry. The rest of the paper consists of speculations about quantum effects, possible applications to describe inflation and cosmological constant.

As I understand this paper is a review of mostly previous results obtained by the authors and his collaborators. As such it is well written and clear and, although the author's point of view is rather exotic with respect to the generally accepted Maxwell electrodynamics, it is fairly enough presented with many pro's and a few con's. My impressions is that the author's attention is focused mostly on the quantum mechanical aspects of its model and skips over the quantum field theory ones. For instance, I don't see where quantum effects are in his model and on what basis the author claims that the model should be finite when considering them. And I don't see any hint of how one can describe, for example, scattering problems, that is asymptotic states and S-matrix.

I think the paper can be published on Universe, provided the author presents in a more sober and detached way the difficulties of interpreting his model as candidate for a realistic QED.

Author Response

I think the three referees agree that one should not overestimate the predictive power of the model.

I have therefore added after enumeration of many of the properties (also the questionable ones) the paragraph:

I hope that these comments are clear enough to underline that one (and I) should not overestimate the model, despite the fun I have with it.

I thank the referees for many detailed remarks, which I tried to satisfy, if possible.

--------------------

- In the title itself, it seems to be more in line with the Minkowski treatment adopted in the paper to write 3+1D instead of 4D.
Answer:
Thanks, the misleading 4D is substituted by 3+1D everywhere.

--------------------

- On page 2, beginning of the last third `.. .. by two topological quantum numbers corresponding to charge and spin quantum numbers' (plural).
Answer:
This is done.

--------------------

- Same page 3, Eq.(3) a sign of : is missing in the second equality.
Answer:
Thanks for this careful observation.

--------------------

Same page footnote 3, `With bold characters' could be better than `With bold phases' .
Answer:
Sure, phases -> characters, is modified.

--------------------

- On page 12, after Eq.(39), write ` These relations may etc..'.
Answer:
This is done.

--------------------

- On page 15, last sentence: very interesting; and challenging !
Answer:
Thanks for this pleasing remark.

--------------------

- On page 28, last paragraph isn't it Georgi (Howard) instead of Georgia ?
Answer:
Sure! It is corrected.

--------------------

Reviewer 3 Report

To the author

The paper offers a review of a research trend extending over more than 2 decades of efforts, and supported by a number of papers published in serious scientific reviews.

This research is rooted in the seminal ideas of H.T.R. Skyrme which, since then, have long been exploited to study Baryonic structures with a real success at the qualitative level. Numbers only are problematic in these approaches.

The more fundamental objects of electromagnetism (electrons, photons) are concerned within the current paper and more contacts with numbers are available; and accordingly more comparisons with experiments are feasible, some of them already encouraging.

Besides the possibility of understanding the 2 long range forces of the Universe (gravitation and electromagnetism) in a very simple and unified way, this approach is challenging for physics at a qualitative and fundamental level. This is because elementary particles show up in a totally different disguise, as compared to the definitions coming from quantum field theories.

The paper deserves to be published in the review Universe where it should find interested readers once the minor corrections and suggestions proposed below are considered by the author.

- In the title itself, it seems to be more in line with the Minkowski treatment adopted in the paper to write 3+1D instead of 4D.

- On page 2, beginning of the last third `.. .. by two topoligical quantum numbers corresponding to charge and spin quantum numbers' (plural).

- On page 3 line 4th, Einstein's would be better than `Einsteins'. This same typo appears at several other places in relation to `Sommerfelds' for example and could be corrected (an example is after Eq.(17)); or with Maxwell later on..

- Same page 3, Eq.(3) a sign of : is missing in the second equality.

- Same page 3, the footnote1. Though standard enough, it might be worth defining the symbol of x in the products \vec{a}x\vec{b}, as it is used throughout the paper.

- On page 4, just before Eq.(7), why is the $C_\mu$ non-abelian vector field a dual one? Or do we have to get the explanation on page 11, mid page?

- On page 9, just before Eq.(28), though made clear by the resulting Eq.(28), it would be more pedagogic to indicate with respect to what the variations of Eq.(27) are considered.

Same page footnote 3, `With bold characters' could be better than `With bold phases' .

- On page 10, an example of a possible amelioration, here as in other places, could be to write, just after Eq.(32), `..is specic to the chosen..' instead of `..is specic for the chosen..' .

- On page 12, after Eq.(39), write ` These relations may etc..'.

- On page 14, 1st line.. this is not quite true indeed. See K. Huang and D. F. Stump,

Phys. Rev. Lett. 37, 545 (1976); Phys. Rev. D15, 3660 (1977).

- On page 15, last sentence: very interesting; and challenging !

- On page 19, the sentence after Eq.(55).. if the force density vanishes this implies that the overall force on the closed classical system vanishes also, as it should. A sufficient condition for sure, but not necessary. So isn't it still a bit astonishing..?

- On page 24, 5th line, not easy to make sense of $2^*4$..

- On page 28, last paragraph isn't it Georgi (Howard) instead of Giorgi ?

- On page 29 last point 2. The mass of solitons is completely due to field energy see Eq.(27).. and potential too, no?

- On page 31, Aftermath. Wheeler `Matter etc.. tells matter how to move'.

The author may want to take these minor suggestions into account so as to improve somewhat the overall presentation of a very interesting paper.

Author Response

I think the three referees agree that one should not overestimate the predictive power of the model.

I have therefore added after enumeration of many of the properties (also the questionable ones) the paragraph:

I hope that these comments are clear enough to underline that one (and I) should not overestimate the model, despite the fun I have with it.

I thank the referees for many detailed remarks, which I tried to satisfy, if possible.

--------------------

- In the title itself, it seems to be more in line with the Minkowski treatment adopted in the paper to write 3+1D instead of 4D.
Answer:
Thanks, the misleading 4D is substituted by 3+1D everywhere.

--------------------

- On page 2, beginning of the last third `.. .. by two topological quantum numbers corresponding to charge and spin quantum numbers' (plural).
Answer:
This is done.

--------------------

- Same page 3, Eq.(3) a sign of : is missing in the second equality.
Answer:
Thanks for this careful observation.

--------------------

Same page footnote 3, `With bold characters' could be better than `With bold phases' .
Answer:
Sure, phases -> characters, is modified.

--------------------

- On page 12, after Eq.(39), write ` These relations may etc..'.
Answer:
This is done.

--------------------

- On page 15, last sentence: very interesting; and challenging !
Answer:
Thanks for this pleasing remark.

--------------------

- On page 28, last paragraph isn't it Georgi (Howard) instead of Georgia ?
Answer:
Sure! It is corrected.

--------------------

Round 2

Reviewer 1 Report

I looked through the replies from the author. Unfortunately the replies do not answer the main concerns that I have. Let me repeat my previous concern: we do not have options to propose an alternative model for electrodynamics because all alternatives have been tried and tested with zero success. Even string theory reproduces the quantum field theory view of the universe. Thus, although I think that author has tried hard to push his alternative theory, this is not going to hold water against the tremendous success of the QFT viewpoint of our universe. Therefore I cannot recommend the paper.

Reviewer 2 Report

The revised version is satisfactory. I recommend to publish the paper on Universe

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A Geometric Model in 3+1D Space-Time for Electrodynamic Phenomena

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