A Valid Quantization of a Half-Harmonic Oscillator Field Theory

Round 1

Reviewer 1 Report

Please see attached file

Comments for author File: Comments.pdf

Author Response

I would have to write this report twice  because it didn’t accept my first report. Then  it failed to offer me the comments again. I remember that I fixed the main problems and  answered the several questions  BIG words have been replaced. Sorry for that.

 

   

Reviewer 2 Report

This paper presents a discussion of affine quantization applied to the half harmonic oscillator, mainly in a field theory context. Several problems of standard treatments may be avoided in this way, which implies current interest in the results. Before I recommend publication, I suggest that the following questions be addressed to improve clarity.

page 1: In footnote 1, "eliminates the ground state eigenvalue" is confusing because the new Hamiltonian certainly has a ground state. I suppose the intended meaning is that the non-zero ground state eigenvalue is eliminated. In the same context, the ordering (P+iQ)(P-iQ) is not unique. If the opposite ordering of the two factors is chosen, the system retains a non-zero ground state eigenvalue.

page 1, last line: "CQ quantization" doubles quantization.

page 2, top: The half harmonic oscillator can also be quantized using Schroedinger quantization with a boundary condition at q=0, which results in a different spectrum than proposed here. While there may alwayss be quantization ambiguities, it would be good to comment further on the discrepancy already in the introduction.

page 2, equation (1): It seems that commutation relations of Q, P and P^+ have been used to arrive at the final result in this equation. Since P does not equal P^+, do both P and P^+ retain the standard commutation relations with Q?

page 3, section 1.2: The "next state" \psi_g of (3) puts each of the N oscillators in their first excited state. Shouldn't the first excited state of the full system have only one of the N oscillators excited while the rest remain in their ground state?

page 6: The scaling in the top line changes the commutation relations of pi and phi or of kappa and phi. There is an ambiguity in whether the commutation relations given in the first paragraph of section 3 are to be applied before or after the scaling transformation and the limit where A goes to infinity.

page 6, bottom: In "the results using AQ for the same models, have shown strong presence of the interaction term when it is present" is it clear whether the interactions are from the new 1/phi^2 term as in (15) or at least partially also from the polynomial potential that has been studied in CQ? These are results of other papers, but it would be good to comment more on them also in the present context.

Author Response

p. 1 The zero eigenvalue was only to show that even \int [pi(x)^2+ phi(x)^2]/2 dx can eliminate an infinite “eigenvalue”  and leave regular behavior after that. & p. 1 quantization fixed & p. 2 if you use CQ then the first derivative at q=0 is discontinues & p. 2 no. Try  int_1^1  [f(x) g(x)]’ dx & p. 3 next is used differently  & p. 6; problem not seen & p.  6 the  (3/4)\hbar^2 is independent of The potential, which could be Q^16 or Q^160, etc.

Reviewer 3 Report

Starting from full harmonic and half harmonic oscillators, the author constructs the canonical and affine quantization of scalar field theories that satisfy \phi>0 or the more realistic condition \phi\not 0. The procedure followed by the author suggests a new (alternative) hbar-term with a coefficient of 2 instead of the standard factor of 3/4. Although the results are not dazzling they have some interest and I think that the paper deserves publication [although I feel that the introduction (and background material) could be improved].

Author Response

A resumable review, but  I find  find it OK

 

Reviewer 4 Report

In the manuscript "A Valid Quantization of a Half-Harmonic Oscillator Field Theory" the author attempts to use a field theory approach for the half and full harmonic oscillator problem and use canonical and affine quantization to examine the approach. Apart from the comments that I have listed in what follows, the manuscript is worth consideration for publication only after some major revisions are applied to it. The text is plain, though in some parts English language Grammar must be improved.  Here are a list of my comments:

1- The introduction is a quite brief, I suggest more information to be added to the introduction section with the use of proper references. The introduction is simply too short and lacks previous research that can guide the reader into what the problem is that he or she is looking at.

2- The notation used should be consistent, for example the usual Hamiltonian is represented as a function of generalized momentum and generalized position coordinates which are indicated by lower case letters, whereas the cursive Hamiltonian is used for functions of transformed momenta and position; this notation should be respected throughout the text, including for example lines 5 and 8 of the second paragraph of section 1.1  

3- In equation 4, the "pi" and "phi" functions have not been defined.

4- The mathematical analysis can be credited; however what is clearly absent from this manuscript is the physical interpretation of the results introduced by the author. In many parts of the text, the author simply states mathematical expressions without a discussion on the physics. 

5- In equation (12) the method is a well known scheme to produce quantum Hamiltonians out of Classical ones; but using expressions such as "YES!" in a scientific manuscript is not professional. The same follows for the first sentence of the second paragraph of section three; GIGANTIC is awkward.

6- A three dimensional analysis of this work is not seen, and the author only presents a one-dimensional model and the related discussion. It would be of great improvement if a three dimensional generalization is also introduced (perhaps in an appendix section where it can be summarized). 

Author Response

Comments 1-4 or points of vi

Comments is granted with nicer terms

 Comment 6 Is easy. Since there no derivatives  you take points ………  or  …….. as a two dimensional  space                          
                 ……..

the referee  can imagine a 3+dimensional story.

 

 

Round 2

Reviewer 4 Report

The author has not completely responded to my comments, and has only attempted to make minor corrections. I would suggest publication only after the author has made the necessary revision.

Author Response

\begin{document}
1. add  to the first point\v
2. clarified by using $H'(pq,q)=H(p,q)$, etc.\v
3. $\pi$ and $\vp$ defined\v
4. the physics is clearly implied when certain  operator Hamiltonians are either valid or invalid.
You cannot talk about physics with invalid operators! \v
5. original BAD terms removed already, and replaced with proper items \v
6. since there are no derivatives, as many dimensions can be built from points beside
 points like  .   . etc.
 
         \hskip9.29em .   .

\end{document}