Time Evolution in Quantum Mechanics with a Minimal Time Scale

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Journal  Symmetry (ISSN 2073-8994) Manuscript ID symmetry-3303870 Title: Time evolution in quantum mechanics with a minimal time scale

 Dear Author,

The authors developed  a theory of non relativistic quantum mechanics exhibiting a non-zero minimal uncertainty in time. The theory was based on the Page-Wootters  formalism with a modified commutation relations between the time and frequency operator. A modified version of Schrodinger equation is obtained in a continuous and discrete time representation. Both representations are equivalent and describe a time evolution on a lattice of a state vector. The time evolution of couple quantum systems shows  interesting result where the frequency of precession of spins depends on the number of spins composing the system. The existence of a fundamental limit with which can have a measurable time causes entanglement of the spin systems even though the initial system is non-entangled. A valuable description that combined formalism and principle in quantum physics give impressive result in space-time evolution system.

The results are well presented with a good description of the method used. However, some typographical errors are to be corrected for a presentation of the manuscript. My suggestions and recommendations are as follows:

COMMENT 1:

Through the manuscript, some words are poorly written, eg.:

a- Page 1 of line 35, it is written "latticle" instead of lattice.

b- Author can revised the word " behaviour" on page 2 of line 49

c- In page 12 on line 325, this sentence must be completed as follows "The heavier the particle the smaller this speed limit is".

d- Also, Author can revised the word "eventhough" through the text.

e- The following word is poorly written "entaglement" on line 478 of page 21.

COMMENT 2:

The author have used a harmonic potential in the case of the description of a harmonic oscillator. Assuming that the oscillator is in an anharmonic potential in non-linear propagation system, Author can predict its impact on the probability density of states according to the minimum time scale and also on the coherence/decoherence of states? 

Sincerely yours,  

Comments on the Quality of English Language

 

The results are well presented with a good description of the method used. The English language is suited for the description.  

Author Response

Comments 1:
Through the manuscript, some words are poorly written, eg.:
a- Page 1 of line 35, it is written "latticle" instead of lattice.
b- Author can revised the word " behaviour" on page 2 of line 49
c- In page 12 on line 325, this sentence must be completed as follows "The heavier the particle the smaller this speed limit is".
d- Also, Author can revised the word "eventhough" through the text.
e- The following word is poorly written "entaglement" on line 478 of page 21.

Response 1:
Thank you for pointing this out. I have corrected the typos and spelling mistakes which you have noticed and couple others I have found while revising the paper.

Comments 2:
The author have used a harmonic potential in the case of the description of a harmonic oscillator. Assuming that the oscillator is in an anharmonic potential in non-linear propagation system, Author can predict its impact on the probability density of states according to the minimum time scale and also on the coherence/decoherence of states?

Response 2:
It would be possible to predict how the minimal time scale will impact the probability density of states and decoherence of states in the case of an anharmonic potential. I have decided not to investigate such a system in the current work as such system is quite complicated and I might not be able to get any closed formulas for the states of this system undergoing time evolution.

Reviewer 2 Report

Comments and Suggestions for Authors

The author explores the long-standing question of physicists: what is time from a quantum point of view? For this purpose, the commutation relation for the time and frequency operators is considered. An explicit expression for these operators is given. Based on these expressions, the time evolution of some quantum systems is considered.

 

I believe that the article examines an interesting physical question about the nature of time, which has interesting physical consequences, which are considered in this article. The article may be published in the journal "Symmetry".

Author Response

The reviewer had not any comments.

Reviewer 3 Report

Comments and Suggestions for Authors

Report Ms. Symmetry-3303870

Studies of the discretized space-time are  essential for understanding for closing the gap between quantum mechanics and the theory of gravity and cosmology.

The author explores the consequences of a minimum time scale in the context of non-relativistic quantum mechanics for several representative systems. The paper fits well into the current research activities in this evolving field. As such, the work is worth publishing. However, before publication the author may consider the following issues:

    Constraining Eq.(4.17) from above by the speed of light as upper limit results in kappa(m), more precisely in fact  depending on the Compton wave length. Implicitly, that is considered to some extent in the following discussion by estimating v_{max} for a few  selected  values of m. The author should spend a few comments if kappa is indeed depending on the system under consideration and what that would that mean for the quantum mechanics of the specific system and in general.
    Under relativistic covariance one needs in fact a concept for integrating into quantum mechanics a minimal invariant eigenzeit element |d\tau| > L which amounts to correlating minimal time and  space. Moreover, the uncertainty relation needs to be updated. Those questions are surely outside of the present work but a few remarks on that broader horizon would be highly welcome, eventually as an addendum to the summary.
    Optionally, the author should consider to simplify the text by defining for the systems studied in the paper once and for all energies, wave numbers, and other recurring variables instead of endlessly repeating the same type of expressions. A bit of „streamlining“ the text would help the reader immensely to concentrate on the scientific content of the (valuable) work instead of being distracted by disentangling notional overkill and research  issues.

Author Response

Comments 1:
Constraining Eq.(4.17) from above by the speed of light as upper limit results in kappa(m), more precisely in fact  depending on the Compton wave length. Implicitly, that is considered to some extent in the following discussion by estimating v_{max} for a few  selected  values of m. The author should spend a few comments if kappa is indeed depending on the system under consideration and what that would that mean for the quantum mechanics of the specific system and in general.

Response 1:
Thank you for your comment. I would not say that constraining v_{max} from above by the speed of light is meaningful. Since I do not consider relativistic quantum mechanics, the speed of light is not an upper speed limit. It is possible to consider relativistic quantum mechanics with minimal time and length scales and investigate how the propagation of wave packets would be affected by the minimal uncertainties in time and position. In Section 5 at lines 487-490 I have added a new paragraph commenting on the dependence of the parameter κ on the system undergoing time evolution. I have not explicitly assumed that the parameter κ is dependent on the system, although it might be possible that κ will depend on the system. However, the investigation of such possibility was outside of the scope of the current paper.

Comments 2:
Under relativistic covariance one needs in fact a concept for integrating into quantum mechanics a minimal invariant eigenzeit element |d\tau| > L which amounts to correlating minimal time and  space. Moreover, the uncertainty relation needs to be updated. Those questions are surely outside of the present work but a few remarks on that broader horizon would be highly welcome, eventually as an addendum to the summary.

Response 2:
In Section 5 I have modified the last paragraph giving a more detailed comment on a relativistic quantum mechanics exhibiting a non-zero uncertainty in time and space.

Comments 3:
Optionally, the author should consider to simplify the text by defining for the systems studied in the paper once and for all energies, wave numbers, and other recurring variables instead of endlessly repeating the same type of expressions. A bit of „streamlining“ the text would help the reader immensely to concentrate on the scientific content of the (valuable) work instead of being distracted by disentangling notional overkill and research  issues.

Response 3:
Thank you for your advice. I have decided not change the notation used in the paper as I believe it is sufficiently clear and I do not know how I could improve it.

Reviewer 4 Report

Comments and Suggestions for Authors

Good work, well developped on mathematical ground.  About the physics I have a suggestion not mandatory since in my opinion the paper deserves publication as it is.

I suggest to compare your results with the very nice theory of the Caldirola Chronon performed in the second half of XX century by the late Piero Caldirola,one of the last pupil of the Enrico Fermi theoretical school.

Maybe you can add more insights to your conclusions.

Anyway my compliments for your work.

Sincerely Yours.                       

Author Response

Comments 1:
I suggest to compare your results with the very nice theory of the Caldirola Chronon performed in the second half of XX century by the late Piero Caldirola,one of the last pupil of the Enrico Fermi theoretical school.

Response 1:
Thank you for pointing this out. In the Introduction at the end of the second paragraph of page 2 I have added three sentences commenting on the theory of P. Caldirola. I have also added a reference to the paper of this author.