Symmetries in Foundation of Quantum Theory and Mathematics

Round 1

Reviewer 1 Report

The work provides valid contributions as well as arguable considerations. For convenience of discussion, we may divide the content in two parts: (1)  discussion of the generalisation of Lie algebra from Poincare symmetry to dS and AdS ones and its application to the cosmological constant problem [sections 2~3]; (2) considerations concerning the so called finite mathematics and its primacy in fundamental physics [sections 4~8]. 

Concerning (1), we draw attention to the following points:

Throughout the study, the concept of a theory being more "fundamental" is in its usage equivalent to the more common one of it being more "general", and including the other as a specific case. This seems to be more or less explicitly acknowledged by the author in his use of words such as "degenerated". Formally the "classical limit"  h → 0 is not free from technical as well as conceptual issues and must be dealt with carefully. Some fundamental confusion seem to underlay the concerns about physical units, as shown in sentences like "We conclude that for quantum theory itself the quantity h is not needed", "because people want to measure [physical quantity] in [physical units]". Although it is rightfully said that geometrical notions are not explicitly introduced by the author, as the names suggest Poincare/dS/AdS Lie algebras are defined consistently with results originating in Riemannian geometry and imply them. While being a minor observation, the author may consider reassessing his claim on this point. On the contrary, the claim that following the author's considerations the "cosmological constant problem and the dark energy problem do not arise" is not supported by the elements provided. Beside the fact that the "cosmological constant problem" is a multi-layered one and what the author refers to is only the strict positivity of Λ, this follows from the different choice of algebra operated by the author at the quantum level, after generalising the Poincare case to include the dS one. The problem is thus only shifted from a seemingly arbitrary choice of Λ to an arbitrary choice of R (which, in principle and in contrast to the author's statement in sec.9, can be chosen in full freedom).

Beside these points and other minor details, we think that the observations made by the author concerning (1) are valuable and may be found useful for the work of other researchers in the field, although it presents strong similarities to ref [1] and maybe the author can provide more original content starting from these elements.

Concerning (2), which is conceptually related to (1) but largely independent of it, we have encountered enough issues and flaws, both in logical and technical in nature, to induce us to suggest reconsidering entirely their submission. To cite only some of them, our attention is drown to

Most eminently, the statement that "regardless of philosophical preferences, finite mathematics is more fundamental than classical one" must be understood and evaluated with context. Being it essentially founded on the simple fact that the infinite set Z is a degenerated case of R_p and more generally on the idea of continuous limit of a discrete set provided with modulo operation, one may notice that various detailed discrete models have been introduced in the past decades in order to model phenomenology, and while the statement is indeed consistent with the definition of "fundamental" adopted by the author, the large literature about the topic may clarify whether such definition of "fundamental" is indeed meaningful in the case considered, in the sense that it is useful to approach the description of physical phenomenology on one side and solving the mathematical technical difficulties arising from the manipulation of continuity. This topic (discrete models for quantum theory and their continuous limit), while very relevant, is not original and the author does not cast further light on it. Besides, the claim that "finite mathematics is more fundamental than classical one if finite mathematics is more pertinent in applications than classical one" seem to introduce a different definition of "being fundamental" with respect to the one introduced before and generates confusion in the following argumentation.  The claim that "at the very fundamental level nature is discrete", which is a overgeneralisation of the discovery and treatment of quantised phenomena. Besides the fact that physical quantities associated with operators with continues spectra are allowed by quantum mechanics, the topic of discreteness versus continuity as being the fundamental qualities of nature is still fairly hotly discussed. The accuracy argument provided, concerning the effective measurement of physical observables, while valid, is still a different one in nature and does not affect subsequent mathematical considerations. In any case, discrete phenomena can be accurately and satisfactorily described by what is referred to as standard/classical mathematics and the author does not provide proof or general arguments about how finite mathematics could avoid "divergences and other inconsistencies" while renouncing to the methods of said standard mathematics. Proof or reference about how finite mathematics (or any other mathematical theory suitable to describe quantum phenomena) can circumvent the fundamental Godel's incompleteness theorems, in contrast to the mathematics employed by quantum theories, are not provided. The argument concerning quantum theory being founded on the principle of verification rather than falsification as any other science is not clear and neither is the similar one concerning standard mathematics. The confusion between the different use of "being more fundamental" (i.e. "being more general", in common use) in the former sections (generalisation of Lie algebra) and the latter ones (functions over ring with elements of Z).

Many others corrections and clarifications would be needed, but it may not contribute to the reviewing process at this stage.

Finally, we notice that in both (1) and (2) there is a significant lack of references to the rich literature of the field, especially in the face of the claims made and topics considered, and more critical contributions with respect to said literature are necessary.

Author Response

I am grateful to Reviewer for favorable words about my paper.

In the new version of the paper I discuss in more details why, when two fundamental theories are compared, it is posed a problem whether one theory is more fundamental than the other, i.e. the word “fundamental” rather than “general” is used. This is in agreement with the terminology accepted in the literature. A typical example is that in the literature quantum mechanics is called more fundamental than classical mechanics but, at the same time, quantum field theory is called more fundamental than quantum mechanics.

To be honest, it is not clear to me why there may be confusions concerning ħ and physical units. I tried to explain this point clearly.

Probably the main Reviewer’s objection is that, in his/her opinion, the choice of R is arbitrary to the same extent as the choice of Λ. In view of this objection I now discuss in greater details why there is no freedom in the choice of R which is simply the coefficient of proportionality between M^{4μ} and P^{μ} and has nothing to do with the radius of background space. Moreover, background space is only a classical notion and has no physical meaning in quantum theory (as discussed in the paper). The problem why R is as is does not arise because P^{μ} =M^{4μ}/R and  people’s choice is to measure distances in meters. This is one of the key points in my approach. As explained in the paper, the situation with R is fully analogous to that with c and ħ, and for those quantities also there is no freedom of choice.

Of course classical mathematics can describe many discrete phenomena but, as noted in Sec. 10, in my papers I consider problems where it is important whether p is finite or infinite. In particular, I mention my approach to gravity.

I note that Gödel’s incompleteness theorems are based on the fact that the number of natural numbers is infinite. On the other hand, in finite mathematics there are no foundational problems because every statement can be directly verified, at least in principle.

Concerning part (1), I added references [7,8] and [11-16] where experimental data and alternative approaches to the cosmological acceleration problem are discussed. Concerning part (2), I added Sec. 7 and Refs. [24-27] and hope that now the importance of the proof of Statements 1 is clearer.

I give only one Definition when theory A is more fundamental than theory B, and this Definition also applies to the statement that finite mathematics is more fundamental than classical one. As noted in Sec. 9, this fact follows from Statements 1-3. At the beginning of Sec. 9 I say: “As noted in Sec. 5, _finite mathematics is more fundamental than classical one if finite mathematics is more pertinent in applications than classical one.”, and in Sec. 5 I note that finite mathematics is more pertinent in applications if FQT is more fundamental than standard quantum theory.

Now I discuss the words that FQT does not contain divergences and all operators here are well defined.

To be honest, Reviewer’s words about verification and falsification are not clear to me, but in any case, those words refer only to philosophy and the readers can have their own preferences.

In summary, the paper is considerably revised and I hope that now it satisfies the necessary requirements. I would be grateful for any remarks on improving the paper.

Reviewer 2 Report

The article continues the line of research conducted by the author (see publications in the reference list). The author tries to justify the idea that the anti-de Sitter space-time and finite mathematics are more fundamental and thus more appropriate for physics than the standard Minkowski space-time and Newton-Leibnitz calculus. There were a lot of attempts to modify or extend the standard approaches. The present one pretends to justify that the anti-de Sitter space-time and finite mathematics are more "fundamental" than the standard ones. The new paradigm of "fundamentality" is developed. On page 3, it is stated that if “B is a special degenerated case of theory A” then “A is more fundamental than theory B” (see the exact definition on page 3). And this statement is used through out the article to justify further theoretical and phenomenological evaluations. But the very statement is obviously wrong. In fact, it breaks down the formal logic. It is obvious that for any given model B (theory, space-time metric, mathematics, whatever) it is always possible to find an infinite number of extensions which would reproduce B in a certain limit. Moreover those extensions of B can be easily further extended in the same manner. So, for any give theory B, there is an infinite number of theories which are more “fundamental” than B. That is obviously not what we would like to call “fundamental”. Thus, the proposed definition of a fundamental theory is simply wrong. And since the main content of the article is based on application of this definition, I can’t recommend publication.

Author Response

The main Reviewer’s objection is that in his/her opinion my Definition breaks the formal logic because when theory B is fundamental there is an infinite number of theories which are more “fundamental” than B. In Reviewer’s opinion, if theory B is fundamental then there can be no theory more fundamental than B. However, this is not in agreement with the terminology accepted in the literature Moreover, if one accepts Reviewer’s opinion then probably there are no fundamental theories at all because probably for any theory there exists a more fundamental theory.

A typical example is that in the literature quantum mechanics is called more fundamental than classical mechanics but, at the same time, quantum field theory is called more fundamental than quantum mechanics.

In the revised version of the paper I discuss this question in greater details. First I note that a theory is treated as fundamental if it deals with foundation of physics and/or mathematics, and this is an accepted terminology. Also it is accepted that when two fundamental theories are compared one can pose a problem whether one of them is more fundamental than the other (i.e. the word “fundamental” rather than “general” is used). Therefore Reviewer’s phrase “That is obviously not what we would like to call “fundamental”” reflects his/her treatment of the word “fundamental” but does not mean that my Definition breaks the formal logic.

In any case, my understanding is that the question raised in Reviewer’s objection is only a matter of terminology but not physics or mathematics. For example, if  “fundamental” is replaced by “general” then all my results remain.

Reviewer 3 Report

Dear Author, In my opinion, the introduction has not been very well organized - its recent form is not suitable for the review paper. Therefore, first of all, it should be rewritten and modified. Then, in my opinion, the reference list is not complete. Moreover, I observed, that you use old observational data in your work. In your opinion is it not possible/interesting/natural to take into account new observational data? Moreover, is it not interesting/natural to have some discussion taking into account several observational data-sets? Even you can consider data that could be obtained from future missions. In other words, in my opinion, the resent manuscript, first of all, should be revised and resubmitted back and only after this it would be possible to make the final evolution of the work to make a decision. Regards, Referee

Author Response

Dear Referee,

I tried to revise the paper following your recommendations.

My first problem was that nothing specific is said on “the introduction has not been very well organized”. The main idea of the new version of introduction is the following. The editorial policy of Symmetry says that “Felix Klein's Erlangen Program, and continuous symmetry” are in the scope of the journal, and the key point in the paper is that I propose the approach to symmetry opposite to that in standard usage of the Program. This new approach is used throughout the whole paper and all results of the paper are based on this approach.

I added Refs. [7,8] and [11-16] with the words “and references cited therein”. Those references contain extensive literature and consider different approaches to the problem of cosmological acceleration and discussions of the observational data. I note that my approach to cosmological acceleration can be treated as a solution of the pure mathematical problem on what happens if one accepts my treatment of de Sitter symmetry. As I note, my solution gives a possible scenario while there are different approaches to the problem. The problem is very complicated and I believe that different approaches have a right to exist if those approaches do not contradict the existing knowledge. In my understanding, at present it is not clear what approach is the most promising.

As far as the discussion of the observational data is concerned, I am not an expert in this field, and the paper is not a review on the cosmological acceleration problem. It is rather a review on my treatment of the Erlangen Program, and cosmological acceleration is only one of examples. I would be grateful if somebody agrees to write a review on this problem and to include my results. This might be his/her own paper or our joint paper.

I also added Sec. 7 and Refs. [24-27] and hope that now the importance of the proof of Statements 1 is clearer.

The other referee reports raise some objections and the strongest of them are that in their opinion: a) The choice of R is arbitrary to the same extent as the choice of Λ; b) My Definition breaks the formal logic. In view of those objections I explain in greater detail that

There is no freedom in the choice of R, and this is one of the key points in my approach. The situation with R is fully analogous to that with c and ħ, and for those quantities also there is no freedom of choice. My Definition does not break the formal logic and, in any case, the question raised by Reviewer is only a question of terminology but not physics or mathematics.

In summary, the paper is considerably revised and I hope that now it satisfies the necessary requirements. I would be grateful for any remarks on improving the paper.

Round 2

Reviewer 1 Report

While the new version of the manuscript provides a better exposition of the thesis and claims of the author, unfortunately our criticisms concerning these very claims have largely been ignored or left unanswered.

In particular, there is still confusion concerning the meaning of a theory being more fundamental than another and a theory being more general than another (these two terms are even used interchangeably towards the of the manuscript). Just to provide one helpful example to clarify this point, the author states that “Usually RT is treated as more fundamental than NT because RT describes experimental data better”, which is not true nor conforms to the terminology adopted in literature. RT is considered more fundamental than NT because it is founded on more fundamental principles (namely, the invariance of c for inertial reference systems and the principle of relativity), as per the brief and general definition of “fundamental” provided by the author. Secondly, to sustain this claim on theoretical grounds, the author cites an argument concerning the existence of a finite parameter in RT which is absent in NT. While the author implicitly adopts this view, the same does not provide a clarification why this would relate to the more fundamental nature of RT. More fundamental principles do not necessarily require introducing extra parameters in mathematical theories. The argument is offered as an underlaying reason for the proposal of the definition of “A being more fundamental than B”, a concept which reduces to that of “B being a limit of A”, which is in fact closely associated to that of “A being more general than B” (reason why we discuss “de-generate” cases). This does not reduce simply to a matter of choice of words, because it leads to different theoretical choices by drawing the argument’s strength from theories which are indeed more fundamental than others, such as quantum mechanics against classical one (again, not because of the presence of Planck’s constant).

Another issue, related to the author’s claim that R is not as arbitrary as Λ, is that of background space “not existing in quantum theory”, which does not correspond to the reality of present knowledge and research, at least in the sense and context in which it is stated. The explicit “absence of space” from the author’s treatment and other treatments referred to is due to the simple fact that no choice of representation is made at that stage, as it is not needed. In the case of the S-matrix formulation of QFT, the “absence of space” is due to a different choice of representation than the "position/field configuration" one. Furthermore, as already noticed in the first review, the construction of AdS/dS algebras originally rely on the corresponding AdS/dS metric choices. Even if these algebras are taken then as fundamental and geometrical effects derived from them, this is not an argument in favour of the "absence of space". These two aspects lead to grave misunderstandings that undermine the validity of the claims made throughout the work.

Finally, some fundamental confusion underlies the discussion concerning the relevance of the choice of physical units for the cosmological constant problem(s). At the origin of such confusion there might be the fact that setting physical constants equal to “1” makes them seemingly disappear from the equations (and the problem of their value with them), while this choice is simply comfortable in order to making calculations on papers and does not affect the physics in any way. The question concerning the value of the cosmological constant is independent on such choice, as the matter is not the numerical value itself, but its physical significance in terms of cosmological observations. The author’s argument may have value if we were to ask why the distance from the tip of one’s nose to the tip of the fingers of one’s arm are about one yard, but this is not the case. Furthermore, Λ does not appear in physics in a way analogous to that in which physical constants such as c and h appear, which is one of the reasons why the former may be expected to have a phenomenological origin while the latter do not.

These concerns add to the ones we already exposed in the preview review and others that have emerged in the new version or that were already present in the previous one and that we have no opportunity to discuss in this occasion.

As a final remark, rather than comment that “Reviewer’s words about verification and falsification are not clear to me, but in any case, those words refer only to philosophy and the readers can have their own preferences”, the author might have found more fruitful in the reviewing process to ask for a better elucidation of the reviewer’s comments rather than dismiss them as unclear and “referring only to philosophy”, especially on such a crucial matter as the principle of falsifiability in science (a principle which, in any case, the methodology of quantum physics satisfies). If this were so, the author might as well have avoided to include his own considerations, equally referring only to philosophy, in the submitted manuscript.

Author Response

Please find attached my response to your report. 

Author Response File: Author Response.pdf

Reviewer 2 Report

The author tries to develop his own model of physics. He pretends to make it more "fundamental" in comparison with the standard Quantum Field Theory (QFT). It is good and important that the new model contains QFT as a limiting case. The present paper looks as a preliminary formulation of the basic features of the new model. Many serious questions about compatibility with QFT, general relativity and observations remain open. In the revised version, the author took into account the main remarks of my report.

Still, I can't agree with his definition of more and less fundamental theories (on page 3) because it formally allows to call an extremely complicated useless theory being more "fundamental" than an elegant simple one just under the condition that the simple one is a limiting case of the extremely complicated one, while both might be equally good in description of nature. Thus, the suggested definition of fundamental theories is not formally correct. It doesn't agree also with the common treatment of the word "fundamental", see e.g. the Merriam-Webster dictionary. Nevertheless, the author is allowed to invent his own definitions.

But I would agree to allow publication of this article, since it opens new directions for further work and discussions.

Author Response

An essential part of the criticism in your report and in the report of the other reviewer was related to my definition when theory A is more fundamental than theory B. After the first round of reports I have revised the presentation of this point. As I noted, in my understanding the choice between "fundamental" and "general" is only a matter of terminology, and my physical and mathematical results do not depend on the choice. However, in the second round of reports both reviewers still insist that my treatment of "fundamental" is not correct. In view of those remarks I have replaced "fundamental" by "general" throughout the paper. Now I am not saying that a theory is or is not fundamental. I consider only comparisons of two theories and my Definition specifies when theory A is more general than theory B. I believe that now all requirements concerning this matter are met and there can be no problems with logic. Such a choice is not fully in the spirit of usual terminology because in the literature, when two theories are compared, the word "fundamental" is often used. However, I hope that both reviewers agree that the author can make a choice on what terminology to use if the choice is logically correct.

In view of the other reviewer’s remarks, I have also added the discussion of locality in quantum theory and Refs. [1-7].

I am very grateful for your positive words on my paper and your agreement to allow publication.

Reviewer 3 Report

Dear Author,

in my opinion, now it can be accepted to be published. In my opinion, you did enough changes and clarifications.

 

Regards, Referee

Author Response

I am very grateful for your advice to accept the paper. One of the reviewers also advises acceptance but the other reviewer still has objections. I have revised the paper taking into account his/her objections, added Refs. [1-7] and sent a detailed response.