A Conformally Invariant Derivation of Average Electromagnetic Helicity
Round 1
Reviewer 1 Report
This paper offers a novel and interesting derivation of the (conserved) electromagnetic helicity. The main ingredient is the use of an inner product linked to the conformal invariance of the Maxwell theory. The paper is direct, well-written, and self-contained. The final result agrees with the local expression for the average helicity given in Eq. (1), as first obtained by M.G. Calkin in 1965.
I think that, for completeness, the list of references should also include the derivation obtained by Deser and Teitelboim in the seventies [Phys. Rev. D13, 1592 (1975)] as well as the more recent ones [J. Bernabeu and J. Navarro-Salas, Symmetry 11, 1191 (2019).
Author Response
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Author Response.pdf
Reviewer 2 Report
Paper tracking is hampered by the lack of a detailed description of the vector C
There should be an Appendix for a relation between the definition for average helicity (pg 1) in theory without C and with C
Author Response
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Reviewer 3 Report
This manuscript presents a number of different expressions for the integrated magnetic helicity of a free photon system. A catalog of equivalent expressions might be of modest interest, but the current version of the paper is very basic. Moreover, the presentation is confusingly incomplete on a number of points.
The paper begins, with its first (unnumbered) equation introducing a nonstandard electric vector potential C. One can safely infer that the curl of this new vector potential is the electric field strength, but this still leaves many questions about the structure of this new function unanswered. Certainly, such a electric vector potential cannot exist in general, but does it exist only in the complete absence of charge? What does the paper mean by gauge transformations when there are two vector potentials? These questions (particularly the second one, since gauge invariance is brought up several more times in the course of the manuscript) are important, but they are never answered.
The notation is unnecessarily clumsy in other ways as well. It uses both the microscopic fundamental fields E and B, and the auxiliary fields D and H. These are introduce entirely unnecessary notational complexity, since the paper is really only concerned with the behavior of the fields in vacuum, in which there is no physically meaningful distinction between the fundamental and auxiliary fields. The discussion of the use of real versus complex fields is also potentially confusing, since again there is no physical content involved. Whether or not certain quantities are real is dictated entirely by the choice of conventions and how the Fourier coefficients are defined.
Mention of the Fourier coefficients brings me to the most fundamental issue that I have with the manuscript. The title promises a conformally invariant treatment of total helicity. However, the actual action of conformal transformations is never even discussed in the paper. There is a brief discussion of what it means for quantities to be conformally invariant. However, the invariance of the helicity is never verified, and it is not even explained how that invariance could be concretely verified. The manuscript simply takes as given the fact that the integrated total helicity of an ensemble of free photons is conformally invariant.
The quantity is indeed invariant, but this is fairly obvious for the noninteracting fields. What makes the integrated magnetic helicity interesting in the general case is that it is a gauge invariant global observable, even in the presence of charged sources and currents, and in spite of the fact that its definition involves the gauge-dependent vector potential A. Moreover, the total magnetic helicity is related to topological invaraints of higher gauge groups and other potentially interesting quantities.
Most of the manipulations in this manuscript are just elementary manipulations of the Fourier transforms of the fields and potentials. The dictionary these provide is modestly interesting (although, as noted above, the inclusion of expressions involving the auxiliary fields instead of the fundamental E and B is superfluous in a vacuum background); however, there is nothing at all surprising or unexpected about any of the paper's expressions.
As a result of these inadequate explanations and the limited value of the existing content, I would not recommend publication of this paper in its current form.
Author Response
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Author Response File: Author Response.pdf
Round 2
Reviewer 3 Report
This paper is improved, and it could, in principle, be published without too much further emendation. However, the work is still suffers from being fundamentally made up of a sequence of essentially elementary manipulations of Fourier transforms. The results are not subtle or unexpected in any way, and thus they are not going to be of better than marginal interest. I therefore leave it up to the editor to decide whether this paper is sufficiently valuable to be published in Symmetry.
The authors' response also suggests an unfortunately poor understanding of the nature of helicity in electromagnetism, belaboring a supposed distinction between "electromagnetic helicity" and "magnetic helicity." For a theory containing only freely propagating photons, there is no physical distinction between the two. Apart from unphysical boundary terms, the only possible difference between a magnetic helicity defined in terms of A and B and the one in the paper is the normalization. This is apparent from the the fact that, in the free theory, either potential A or C is sufficient to determine both the electric and magnetic fields, as shown, e.g., in eq. (4) of ref. [7]. The fact that an expression like first one in this manuscript (which is, mystifyingly, still not endowed with an equation number) is rather superfluous is further pointed out in ref. [3], where the equation in questions only appears in a footnoted observation that there actually exists a whole family of equivalent expressions for the conserved quantity when only free fields are involved.
My previous mentions of the "magnetic helicity" specifically were thus motivated by the physical equivalence of the helicity concepts when only free propagating waves are present, along with the fact that the integrated magnetic expression remains gauge invariant when sources are present, whereas the electric helicity term is not even defined in the presence of charges. The paper should give a more accurate impression as to the nature of these quantities.
In the author's reply, he has also argued for retaining the use of the auxiliary fields D and H, which have been used in some other papers, such as ref. [21]. If this is to be done, then the paper must emphasize that its results are apparently only usable when there is an infinite, linear (and isotropic) medium. The way the auxiliary fields appear in the manuscript suggests that the results should carry over to situations in which electromagnetic waves are able to traverse across and reflect from material interfaces (where bound sources will develop). The mention of evanescent solutions further makes it seem that this interpretation is meant. The whole discussion surrounding the macroscopic fields needs to be laid out more carefully to avoid giving readers incorrect impressions.
Author Response
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Author Response File: Author Response.pdf
