Two of the reviews are quite positive but the third not. Naturally the editor decided against publication.

Reviewer 1 report:

The paper is very interesting indeed. Continuing the classical analysis of spin-orbit coupling in hydrogenlike atoms, the author suggests including the ‘hidden energy’ of an orbiting electron into the total energy balance. He notices that ‘hidden energy’ has been introduced by Hnizdo to obtain a relativistically covariant description of classical electrodynamics with ‘hidden momentum’. Although the ‘hidden energy’ is not a well recognized quantity to the moment and, in particular, is not accounted in the standard expression for the electromagnetic energy-momentum tensor, simple examples of a motion of a magnetic dipole in an external electric field (to be omitted in this report) show a reality of ‘hidden energy’ in the same extend, like ‘hidden momentum’.

In this respect I completely support the idea of the author to involve ‘hidden energy’ in the analysis of non-Coulomb interactions of the classical hydrogenlike atoms. It is shown that the inclusion of ‘hidden energy’ leaves the spin-orbit interval unmodified, when ‘hidden angular momentum’ along with the kinetic angular momentum is accounted in the modified second Bohr postulate. This result implies, in particular, that ‘hidden momentum’ should be added to kinetic momentum in the de Broglie relationship, too. If we believe that the classical approach to hydrogenlike atoms is able to give the same value of spin-orbit splitting, like the relativistic quantum mechanics, we must recognize the validity of the results of this paper, which, in general, look non-trivial. In particular, for a particle with spin resting in an external electric field and possessing the ‘hidden momentum’ P_hidden, its de Broglie wavelength λ is no longer infinite, but rather is defined by the relationship

λ= h/P_hidden

Correspondingly, it seems that the Heisenberg uncertainty principle should also include the ‘hidden momentum’.

Thus the present work, like any other interesting paper, induces new interesting questions and certainly deserves to be published.

As the whole, the paper has been written in a clear enough way, but I recommend some minor amendments:

1. Abstract, after the words “In response to a comment,…” the reference to Lush comment should be indicated; otherwise, the sense of this phrase is unclear.

2. First paragraph of Introduction, the sentence “Hidden momentum refers to mechanical momentum of a current carrying body…” should be modified as “Hidden momentum refers to mechanical momentum of a current carrying non-conducting body…”. This specification is essential, because for conducting body we get its polarization in an external electric field instead of ‘hidden momentum’.

3. Section 4, second like: “Reference 3” should be replaced by “Reference 7”.

If the author agrees with these remarks, I recommend the paper for publication in Am. J. Phys.

Reviewer 2 did not provide a report, according to the AJP editor. (However, Reviewer 4 refers to a comment by Reviewer 2.)

Reviewer 3 report:

This paper is a careful study of the implications of the hidden momentum for the spin-orbit coupling in a semiclassical (Bohr) model of the hydrogen atom.

The author shows that the correct empirical value of the spin-orbit coupling is obtained when not only the forces due to the moving electron's electric dipole moment and hidden momentum but also its hidden energy are included, if the orbit-quantizing postulate of Bohr is reformulated in terms of the total orbital angular momentum, i.e., kinetic plus hidden, of the electron. This analysis demonstrates nicely that the assumption that the existence of intrinsic magnetic moment of the electron implies that it has the same hidden linear momentum in an external electric field as a classical macroscopic current-carying body is consistent with the correct value of the spin-orbit coupling in the semi-classical hydrogen atom. I recommend publication.

There are some very minor things that can be taken care of in the proofs. I think that the -/+ sign of the term delta_{+/-} in Eq. (30) should be replaced by just + since the term itself is defined with the -/+ sign in equation (13);"Reference 3" in the 1st sentence of Section IV should presumably read "Reference 7"; some references do not have page numbers.

Reviewer 4 report:

I don’t think I have ever found myself in such drastic disagreement with other reviewers (referees 1 and 3) as in the case of this paper. However, I do agree with referee #2 that the paper is not self-contained and that it should not be published, although my reasons – explained below - are of a different nature. I don’t mind too much having to pick up a previous issue of the AJP to acquaint myself with the paper by Kholmetskii, Missevitch, and Yarman (KMY), but being forced to making guesses about the contents of references 2 and 7 in Lush’s manuscript is almost asking for a rejection.

Now, most of these objections to MS 23826 are fairly subjective. I am recommending against publication based solely on the physics contained in the paper. Both Lush’s paper as well as the original KMY article that initiated this exchange are plagued with misunderstandings about spin-orbit coupling, reference frames, Thomas precession, and hidden momentum/energy. In my opinion, KMY should not have been accepted for publication. But since it is already in print and this is not a review of KMY, I will constrain myself to Lush’s manuscript. Lush is correct when he states that (in the laboratory frame) the dynamical equation should be “Eq. (28) of Reference 1, with appropriate modification to account for the halving of the non-Coulomb force due to the presence of hidden momentum.” Unfortunately, in Section V he adds a hidden energy contribution H± to the spin-orbit V± without realizing that the latter already contains the hidden energy contribution. The source of the confusion here, as in KMY, seems to be the mixing of energy contributions calculated in different frames, which leads to a double-counting and the necessity for a modification of the Bohr postulate in order to bring the prediction back into agreement with experiment. In anticipation of rebuttals to my comments, I would suggest, following reviewer #2, that KMY and Lush “duke it out in private”. More specifically, they should answer the following questions.

1.Have they checked their claims against the Bargmann-Michel-Telegdi equation? This is a relativistic equation, so all hidden energy/momentum terms are included ab initio. The Bohr postulates are irrelevant here, since the basic problem is one of classical energy contributions.

2.Instead of modifying Bohr’s postulates, why not go directly to the Schrödinger equation with the energy terms they claim should be included? Will they have to modify the postulates of quantum mechanics too?

In essence, then, my recommendation against publication stems from my conviction that these papers represent a step backward in our understanding of the spin-orbit coupling and the Thomas precession.

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