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有篇近作，7月2日投出，8月17日接受！

European Physical Journal C，算来也是物理学顶级一区刊物！

论文的基本框架都在这里：

General covariant geometric momentum, gauge potential and a Dirac fermion on a two-dimensional sphere

https://arxiv.org/abs/1905.07735

这篇论文不仅我们基本不懂，而且审稿人编辑肯定也不懂。这不妨碍我们一起莫名的激动并赞赏这件工作！

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Cover Letter

Dear Editors of The European Physical Journal C,

It is well-known that curved manifold can induce somegauge structure, but how this gauge structure appears, e.g., for a Dirac fermion on the curved graphene, and how it affects the motion of the particle,is not clear. Our approach starts with that** the symmetry expressed by the Poisson or Dirac brackets in classical mechanics preserves in quantum mechanics**; and so the Hamiltonian is determined by the symmetry, and we establish a general framework to explore these problems. As two examples, we show that for the Dirac fermion on a two-dimensional sphere, the gauge field enters the momentum, but does not exhibit in the Hamiltonian. The details arein the following.

In2011, we proposed a *geometric momentum *for non-relativistic particles constrained on the curved surface (PRA 84, 042101(2011)) as the proper form of momentum. This concept together with *geometric potential* is now widely accepted for it has stood both theoretical examinations and experimental testifications. Present paper develops such a geometric momentum to be general covariant, otherwise it is inapplicable to spin particles on curved surface. First of all, we need to establish a quite general formalism to accommodate the *general covariant geometric momentum*(GCGM). This is in fact the framework of Dirac canonical quantization, in which we make some close examinations, emphasizing the fundamental role of symmetry indicated by both the fundamental quantum conditions and dynamical ones.

It has been recognized, through construction of quantum field on curved manifolds,that some gauge structures emerge, our GCGM makes the gauge structure transparent. Two important and interesting demonstrations of the GCGM are: 1, the generalized total angular momentum for a Dirac fermionon two-dimensional sphere, **which reproduce the same result as that given by 2003 Nobel prize laureate Abrikosov via the purely consideration of dynamics of the particle**; 2, **the GCGM together with the dynamical quantum conditions clears up a controversial issue as whether there is geometric potential for the Dirac Fermion on the two-dimensional sphere, and our answer to this question is that no geometric potential is acceptable.**

The problem we discuss in present paper is of broad interests, and **present work is of fundamental contributions into physics. It relates to the topologic state on surface of the low-dimensional materials, and gauge theory of gravitation, and fundamental problem in quantum mechanics.** So, it is appropriate for publicationin your journal.

Sincerelyyours,

Q.H. Liu

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**一审意见**

Referee: 1

Comments to the Author

The paper by Liu et al. deals with the definition of the general covariant geometric momentum, which is the first part of the paper, and the study of Dirac equation on a sphere, which cover the second part..

The paper is interesting, and deserves publication. I have just some minor points:

1) May be the authors wish to enlarge the presentation of Eq. (1).

2) From Eq. (1) it appears that the canonical commutation relation are modified. They have on the right handed side a quantity different from 1. This should have non trivial consequences, I suppose. Is it?

3) To better understand the model, the author should also comment the meaning of (1) in the case in which the index i is different from index j

4) The construction studied in the paper has any connection with the generalized uncertainty principle?

Referee: 2

Comments to the Author

The paper deals with the quantization of generally covariant systems when constrained on submanifolds of flat spaces. After a general introduction to this problem, and offering quite an extensive list of cogent references, it faces a specific problem, namely whether Dirac fermions on a sphere do carry the so-called 'geometric potential'. In other words, whether the 'standard' quantization in curved space techniques (leading there to their eq (28), or the Hamiltonian (29)), need be corrected when the two dimensional manifold is thought of as coming from constraints in a larger three-dimensional flat space.

I found the paper sound, the calculations seem correct, the logic is convincing. I have two possibly minor concerns:

i) could they improve a little the English, by checking some expressions (e.g., right in the Abstract 'Once the momentum is made general covariance'...), and by making their sentences shorter, sharper and clearer?

ii) since they cite work where Dirac fermions are studied on 2d surfaces of negative curvature, could they address this problem here too? or at least, comment on that?

Besides this, I believe that the paper deserves publication in EPJC.

——————**接下来，论文被接受！**

Dear Prof. Liu:

It is a pleasure to accept your manuscript entitled "Generally covariant geometric momentum, gauge potential and a Dirac fermion on a two-dimensional sphere" in its current form for publication in the European Physical Journal C. The paper will also be published electronically under the journal's title.

Thank you for your fine contribution. On behalf of the Editors of the European Physical Journal C, we look forward to your continued contributions to the Journal.

Sincerely,

Prof. Kostas Skenderis

Editor in Chief, European Physical Journal C

Associate Editor

Comments to the Author:

(There are no comments.)

Referee: 1

Comments to the Author

(There are no comments.)

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