Deformed Boson Algebras and W_α,β,ν-Coherent States: A New Quantum Framework

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

18 Jan 2025 - mathematics-3446357-report

Deformed Boson Algebras and W_{α,β,ν}-Coherent States: A New Quantum Framework by R. Droghei

The paper considers the extension of the usual boson Coherent States (CS) to the more general construction that can physically be manifested in the spectral structure of the energy levels (see Fig 1) and in the modifications of the corresponding states (see Fig 2) as well as in deviations from the Poisson distribution, which can be measured using the Mandel parameter (see Figs 5 & 6). The component of the discussed construction is the use of the generalized factorial (1) via the box function that brings in the deformation (2). A few specific examples are presented as  Mittag-Leffler and Wright functions. Such functions are expected to be related to the fractional differential operator (20) that replaces the usual differentiation in the formulation of the creation and annihilation operators using z-D realization of the Heisenberg algebra (3,4,5). In this realization the CS can be defined via (19) as long as the normalization function is well defined (see line 99 and above). This is also related to the positiveness of W in the resolution of the unity (1) as given by the expressions for W below (19). In this respect one arrives at "A New Quantum Framework". From a physics point of view this is a well written paper that seems sufficient for practical explorations and tests of physical phenomena. However, as a paper submitted to journal Mathematics claiming a rigorous Quantum framework, such rigorous proofs seem to be absent in my opinion. First, it is not clear why box function (2) is preferred! It seems to me that one can start with an arbitrary box function [n] to define the sates |n> given after (8) before the line 81 and then to generate all the given expressions. Then of course there is a question of continuity and completeness of the corresponding CS given by (19), and these questions are not addressed satisfactory to me in section 2.2. except for the overview of the known examples.         

 

Thus, the paper is well written, and one of the few papers that come my way for reviewing, there is little to be criticized or object to from physics view point, but to recommend the paper for publication in mathematics journal there should be some major additions and minor changes as follows:

 

Major clarifications needed:

I) The author should address the question of continuity and completeness much rigorously and with more details. The text around (24) is not sufficient in my opinion and has no references to be followed for further details regarding continuity. While the question of completeness does not demonstrate the proof of it for the full range of the parameters α,β,ν.

 

II) The question of the uniqueness for the choice of the box function (2) seems to be open for me. Furthermore, it is not clear why the existence of a unique positive measure corresponding to the moments m_n is essential. Cases where no such measure exists would be of value to see how the problem of normalizability, continuity, and completeness can prevent one from doing calculations within the relevant quantum framework. Furthermore, and example of non-unique but positive measures should also be relevant for understanding some intersting quantum frameworks.

 

III) It seems to me that the range of values for the parameters α,β,ν and their physical meaning are not presented nor discussed. Knowing the physical significants of the deformation parameters is often of a great value for the future developments. Note that there are generalizations of the boson algebras with various other applications in quantum superintegrable systems (see Phys. Rev. A 50 (1994) 3700), nuclear physics initiated q-deformed bosons [J. Phys. A: Math. Gen. 34 (2001) 2999], and algebraic methods for q-deformed many-body systems [Internat. J. Theoret. Phys. 34 (1995), no. 11, p. 2195-2204]. Thus, it will be of value to discuss other explorations of Deformed Boson Algebras, their applications and interpretations of the deformation parameters. 

 

Minor suggestions and comments:

a) Note that the labeling of the equations starts form 1 within each section and does not use section indicators. 

b) Using the name "number operator N" in line 77 and in subsequent text is not quite right since the eigenvalues of this operator are not integer numbers perhaps using quotation marks would alert the reader to this subtle difference. 

c) The paper is mostly concerned with deformation of the standard CS as applied to photons which are massless. What is the meaning of the mass parameter in the expressions such as (13) and (16) and similar places? 

d) In the middle line after line 100 on page 5, the  the ground state is incorrectly defined; 

e) In line 104 and formula (23), there is misplaced factorial sign! 

f) The form of U(x) in (25) and subsequent expressions is not well defined and explained. 

g) In line 124 on page 6 the text " not-trial " is probably 'non-trivial'.

h) Some of the expressions for the variances of q and p in line 169, and in (6) & (7) may have extra power of 2 (note the expressions (4) & (5) seems to be correct).

i) The Carleman’s condition and the question of unique positive measure and range of Convergence/Divergence of the moments series is not clearly explained in the appendix. 

 

If the authors are to modify the paper, it will be best to use \color{blue} or any other suitable color to indicate the more substantial changes in the next version no need to show what was the old text, only where are the new substantial changes beyond a few words or major formula modifications.

 

Author Response

We sincerely thank the reviewer for the valuable comments and constructive suggestions. We've incorporated all the requested corrections in the revised manuscript and provided detailed point-by-point responses to each observation in the attached file.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

See attached report

Comments for author File: Comments.pdf

Author Response

COMMENT: The reviewer comments on the definition of the "box function" Eq (2) for the case n=0. The reviewer states that for this particular value of n, [0]_{\alpha,\beta,\nu}=0 so the vacuum state |0> can be annihilated by the A operator.

RESPONSE: I agree with the reviewer and thank him for his helpful comments. I apologize for taking some key concepts of the developed theory for granted and omitting them.  In response to their comments, I will add to the paper some missing definitions related to the box function and the values of the parameters α, β and ν. You can find more details in the attached REPLY REVIEWER 2.pdf.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

This paper is focused on the construction of a new class of generalized coherent states whose basic feature is to replace the factorial n! in the standard formula of boson (or Glauber, or Weyl-Heisnberg) coherent states with a deformed factorial involving Gamma functions depending on three parameters. This approach represents an extension of the work in references [10], [12] and [13] where different generalization of the factiorial formula were considered. The author introduces a deformed boson algebra whose raising/lowering operators are used to define the new coherent states. After discussing the crucial condition for these state to exist relevant to the resolution of the identity, physical properties related to the quantum fluctuations of the new operators and the Mandel parameter are analyzed.

Comments

1) The comment (lines 74-76) "Now we define ...rules" is not clear: Apparently, commutators (3) and (4) are introduced to define the deformed algebra. However, operators A, A^+ (I omit the parameter dependence) were not defined before such formulas. Operators A and A^+ are defined later by means of formulas (16)-(18). This presentation of the algebra is not convincing because it does not clarify the role of commutators and their link with the explicit form of A and A^+. If I understand well, formulas (16)-(18) provide the analytic explicit form of A and A^+ which allows one to satisfy commutators (3) and (4). At most, one can say that (3) and (4) implicitly define A and A^+ which are explicitly given later through formulas (16)-(18). In my opinion, to improve the readibility of the paper, definition 1 and remark 2 should be placed right after formula (5). In any case, this part should be formulated in a more transparent form.

2) The fact that equations (16)-(18) entail the validity of equation (3) is not obvious. This statement should be proven.

3) The definition of D by means of equation (17) is rather ambiguous. Apparently, the generalized derivative D is implicitly defined by the action on z^n where z is a complex-valued parameter. The explicit definition of D is given by equation (20) only for the very specific case z= x^\beta. What about the definition of D for a generic z?

4) the phrase (lines 78-79) is not correct. The "number states |n>" cannot be "an orthonormal basis of the number operator N eigenvectors". Maybe, the author means that the number states |n> form an orthonormal basis of the number operator N (or, otherwise stated, they are eigenstates of N). By the way, it could be useful to remark somewhere that, in the simplest case \alpha= \nu = 0, \beta =1, the symbols N, A and A^+ reduce to the standard operators of the Weyl-Heisenberg algebra.

5) The literature about deformed algebras and coherent states is vast. The reference list should be improved to emphasize the interest for this topic. In J. Phys. A 40 (2007) 9905, for example, coherent states related to f-deformed operators are defined (see also J. Phys. A 37 (2004) 8111). A similar paper on this topic that should be cited is Phys. Scr. 55 (1997) 528. Deformation can be introduced also in algebra su(1,1), see J. Phys. A 42 (2009) 365210 and su(2) see J. Phys. A 26 L871.

Concluding, this paper provides the definition of a new class of deformed coherent states which should be of interest in the relevant research field and adds information on this topic. In general, I think it deserves to be published. However, the author should clarify various unclear points the most important of which concerns the definition of the generalized derivative D. At present, an explicit formula for D is missing and D is defined only for a very specific case.

Author Response

We sincerely thank the reviewer for the insightful comments and suggestions. We've implemented all the requested corrections in the revised manuscript and provided detailed point-by-point responses in the attached file.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

I am delighted to see that the author has addressed most of the points I have raised and has colored the changes in blue, which makes it easy to read and focus only on the new text. I have only a minor suggestion, and I hope the managing editor can see it applied to the paper. My suggestion is to emphasize the open nature of two key points, which I have asked for more clarification and explanations from the author; however, it seems that these are still open questions: (1) the physical nature and interpretation of the deformation parameters are still unclear; thus, an open question to be clarified in the future;  (2) the uniqueness of the measure needed for the completeness relation is clearly up to a unitary transformation, or it is invariant under such transformations. This begs the question about the meaning of the non-unique measures and their relations to the choice of observables for different observers. These two open topics are important for the future understanding of the quantum mechanical treatment presented. 

Author Response

I sincerely appreciate the reviewer's comments.

In the "Conclusion" section of the manuscript, I added some sentences related to the comments. 

Regards

R.D.

Reviewer 2 Report

Comments and Suggestions for Authors

See attached report

Comments for author File: Comments.pdf

Author Response

Thanks for the comments and  suggestions.

I revised the manuscript following your minor suggestions. I corrected misprints. Thanks.

I made some changes in the text (This time highlighted in red) to address  your final comment.

Regards

R.D.

Reviewer 3 Report

Comments and Suggestions for Authors

The author reacted to most of the referee observations by introducing several clarifying comments which improve the readibility of the paper. Only one point remains, in my opinion, rather unclear which has to do with the explicit definition of operator D and formulas 2.17 and 2.18. If the definition of D can be given only in the very specific case z= x^\beta described by equation 2.18 (where x is real!) then the fundamental condition characterizing coherent states and their properties that z is a complex parameter is completely lost. Note that z=x (real) is a condition one cannot bypass since, in the limiting case \alpha =\nu = 0 and \beta =1, equations 2.19 correctly reproduce the standard definitions of raising and lowering operators where x, in fact, is expected to be real. This is confirmed by a comment of the author about the analogy with the well-known definition of p (see below equation 2.19).

Apparently, the class of new coherent states given by equation 2.12 (involving by construction a complex z) and defined by exploiting the properties of the deformed algebra cannot be associated with a definition of A and A^+ in terms of generalized differential operators (involving fractional derivatives) which is valid for any complex z, and not only for z=x^\beta. At present, equation 2.17 is satisfied only when definition 2.18 is used which involves a real z. Is this a problem related to the present state of art of fractional differential calculus? What about fractional derivatives in the presence of complex variables? The author should remember that this information is certainly not obvious for non expert readers. 

This is the point the author should clarify because it concerns problematic aspects (certainly interesting in the community of coherent-state physics) not sufficiently highlighted in this second version. The problem to find a sufficiently general realization of the deformed-algebra operators in terms of generalized (fractional) differential operators is the central problem of this paper. 

I conclude by observing that, if z is real, the fundamental semiclassical property of Glauber's coherent states to optimize the Heisenberg inequality seems to be lost. So, an obvious question that should be commented (if possible) in this paper concerns definition 2.12. Based on the latter, is it possible to derive the formula for the Heisenberg product of quantum deviations relevant to the operators 2.14?

I hope the author will be able to comment in a more transaprent and convincing way the problematic points I mentioned above and to better explain the open problems related to this research. 

Author Response

I am grateful for the reviewer’s valuable feedback.

In this revised version, I have addressed the confusion between the complex z parameter in coherent states and the real-valued position operator x^beta of the harmonic oscillator. To clarify this distinction, I have redefined the fractional differential operator (which remains real-valued) to introduce a deformed analogy for the harmonic oscillator (Section 2.1). I hope this revised version presents the concepts more clearly and transparently. This time, all modifications to the text are highlighted in red.

Sincerely,

R.D.