Hamilton Equations, Commutator, and Energy Conservation^†

Round 1

Reviewer 1 Report

The authors of this paper talks about Hamiltonian, conservation. Heisenberg. etc.

It is well known that the Hamiltonian systems are for energy-conserving world.

There is thus nothing new.

In their abstract, the authors mention Maxwell's equations, but they make no

meaningful statements in the text.

Thus, this paper is not acceptable in the present form.

Author Response

Reply to Reviewer 1:

1.  Yes, it is well known that Hamiltonians are for energy conserving systems.  This paper points out that using energy conserving property of a Hamiltonian, one can derive the classical Hamilton equations succinctly.  The authors have not seen this derivation before.  Instead, most textbooks will start with defining the action, then find the Lagrangian of the system, and derive the Euler-Lagrangian equation.   With the use of Legendre’s transform, then the Hamilton equations are derived.  This derivation is long. But we have presented a more succinct derivation in this work.

2. The new contribution is a succinct way to derive the classical and quantum Hamilton equations.  We have never seen these derivations before. It allows us to quantize a system whose equations of motion can be derived using Hamilton mechanics.

3. We have added an Appendix to illustrate the quantization of a scalar wave equation.  The quantization has been done in Ref. [5] of the present manuscript using commutator.  But here, we have quantized the equation using only quantum Hamilton equations in multi-dimensions making it simpler.  In all the textbooks we have found, the quantization is in the mode space or Fourier space, or the momentum space.

Reviewer 2 Report

In this paper, the similarity between the classical Hamilton equations and their quantum counterpart has been discussed. It proves that the fundamental commutator can be derived from Heisenberg equations of motion and quantum Hamilton equations. Also, the quantum Hamilton equations can be derived from the Heisenberg equation of motion and the fundamental commutator. I have one question for the authors. Is it possible to derive the Heisenberg equations of motion from the fundamental commutator and the quantum Hamilton equations? This paper is well-written with sound theoretical discussion and formulation derivations. I recommend accepting it.

Author Response

Reply to Reviewer 2:

We thank this reviewer for the very constructive review and feedback.  This reviewer wants to know if the Heisenberg equations of motion can be derived from the quantum Hamilton equation.  The answer is yes for the Heisenberg equations of motion for the p and q variables, but no for the general Heisenberg equations of motion (19).  In order to clarify this, we have added the following sentence after (36) of the revised manuscript:

“It is to be noted that since derivations in (20) and (21) can be reversed, then one can start with the 81

quantum Hamilton equations, and derive the Heisenberg equations of motion in (34) and (35).”

Reviewer 3 Report

See attachment.

Comments for author File: Comments.pdf

Author Response

Reply to Reviewer 3:

Again, we thank this reviewer for the very constructive review and feedback.  They ultimately make our paper better and clearer. 

In order to answer Reviewer 3’s questions, we have expanded the revised manuscript, adding the discussions from (9) to (11).  Equation (9) is essentially the equation in Reviewer 3’s report, but with the typos corrected.  We have generalized our derivation to C(t), and it can be shown that our generalization is as general as the equations proposed by Reviewer 3.  We can provide the detail proof offline if reviewer 3 is interested in the proof.  

We also like to emphasize that our new Eq. (10) in the revised manuscript can be transfored to look like the Hamilton equation by using time-stretching transformation similar to the coordinate stretching transformation.  This work is cited as references [9] and [10] of the revised manuscript.  The coordinate stretching transformation has been successfully used in designing perfectly matched layers absorbing boundary condition in many wave physics simulations.

Reviewer 3 also likes to see the extension of this work to multi-dimensions.  We have illustrated this in the new appendix.

Round 2

Reviewer 1 Report

The authors added their appendix to my concerns about what they say about the Maxwell field.  The Maxwell field is not a scalar field.  The quantization of this field has to take care of gauge transformations.  The authors do not seem to understand this.  

Thus, I am not able to change my opinion from the one given in my previous report.  

Author Response

Reply to Reviewer 1:

In both occasions, the Reviewer 1 provided negative feedbacks, but no suggestion as to what we should read, or refer to.  He also did not provide references to back up his reviews.   In his first review, he expressed doubt that our work is connected to the quantization of fields.   We have added an appendix to show this connection, but his review is dismissive, and not very useful for us.  We are willing to learn from Reviewer 1 if he can provide references to stake his claims.   We do not know how to reply to this reviewers.

Reviewer 3 Report

Dear sirs,

In my opinion, the revised manuscript is suitable for publication.

Best regards.

Mario Fusco Girard

Author Response

We thank Reviewers 2 and 3 and the Academic Editor for their constructive feedbacks which eventually motivated us to improve on the revised manuscript.
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