Moreover, as explained, for example, in Ref. [17], continuous mathematics itself is a special degenerated case of finite mathematics: the latter becomes the former in the formal limit when the characteristic of the ring or field in finite mathematics goes to infinity. The fact that continuous mathematics describes many data with high accuracy is a consequence of the fact that at the present stage of the Universe the characteristic is very large. There is no doubt that the technique of continuous mathematics is useful in many practical calculations with high accuracy. However, from the above facts it is clear that the problem of substantiation of this mathematics (which was discussed by many famous mathematicians, which has not been solved so far and which probably cannot be solved (e.g. in view of Gödel’s incompleteness theorems)) is not fundamental because continuous mathematics itself, being a special degenerated case of finite mathematics, is not fundamental.

It is also seeming obvious that discrete spectrum is more general than continuous one: the latter can be treated as a formal degenerated special case of the former in a special case when the distances between the levels of the discrete spectrum become (infinitely) small. In physics there are known examples in favor of this point of view. For example, the angular momentum has a pure discrete spectrum which becomes the continuous one in the formal limit ћ→0. Another example is the following. It is known that Poincare symmetry is a special degenerated case of de Sitter symmetry. The procedure when the latter becomes the former is called contraction and is performed as follows. Instead of some four de Sitter angular momenta MdS we introduce standard Poincare four-momentum P such that P= MdS/R where R is a formal parameter which can be called the radius of the world. The spectrum of the operators MdS is discrete, the distances between the spectrum eigenvalues are of the order of ћ and therefore at this stage the Poincare four-momentum P has the discrete spectrum such that the distances between the spectrum eigenvalues are of the order of ћ/R. In the formal limit R→∞ the commutation relations for the de Sitter algebras become the commutation relations for the Poincare algebra and instead of the discrete spectrum for the operators MdS we have the continuous spectrum for the operators P.

I fully agree with Dirac who wrote:

“I learned to distrust all physical concepts as a basis for a theory. Instead one should put one's trust in a mathematical scheme, even if the scheme does not appear at first sight to be connected with physics. One should concentrate on getting an interesting mathematics."

I understand these words such that on quantum level the usual physical intuition does not work, and we can rely only on mathematics. The majority of physicists do not accept this approach and believe that physical meaning (which often is understood simply as common sense) is more important than mathematics. In discussions with me some of them said that the characteristic p in my approach is simply a cutoff parameter. This is an example when finite mathematics is treated in view of continuous mathematics while finite mathematics considerable differs from continuous one. For example, special relativity cannot be treated simply as classical mechanics with the cutoff c for velocities.

As shown in my works, the approach when quantum theory is based on finite mathematics sheds a fully new light on fundamental problems of gravity, particle theory and even mathematics itself. I would be very grateful if Springer accepts my monograph proposal.

15.2. Ответы рецензентов

Reviewer 1

What I do not really see is the fundamentally new aspect. It seems that any finite approximation to the standard continuum theory of gravity, quantum mechanics or quantum field theory more or less gives what the author proposes. But then, any such finite approximation is implemented (though not at a group theoretical level) when making numerical calculations of quantum mechanical (or other) problems on a computer. The criticisms of the mainstream continuum theories are, for my taste, too commonplace and unspecific, or have already been responded to within the usual mainstream theories. Some of the papers cited to support the author's criticism of the mainstream theories are known to present misguided views that have been clarified elsewhere in the literature. It is also not really clear how the author's approach would get around the critisized issues.

In conclusion, I think the book project does not meet the quality expectations of FTPH. I would not like to endorse it, even though FTPH is open to more speculative approaches and non-mainstream philosophical viewpoints.

Reviewer 2

I think that the proposal is kind of esoteric, ignoring 80 years of successful quantum theory. Now, there are problems with QED and QFT in general and they are of various kinds, position operator for photons is one such problem, infinities another one, and the author is only focussing on those. But the first question one would have to address is, when one wants to change the world, how does the world in which we actually live fit into that. The author ignores that or hides the discussion somewhere, where it is hard to find. That’s a second issue, the book is all words, hardly formulas, almost like a book of philosophy. I cannot endorse that proposal.

В рецензиях все как обычно: ничего не понимают и понимать не хотят, но раз у меня не QFT, то сразу посылают подальше. Что здесь особенно странно: эта секция в Springer называется FTPH – fundamental theories of physics, и в правилах написано, что надо предлагать что-то фундаментально новое, а не стандартное. Т.е., как бы рецензент должен понимать, что может быть что-то необычное. Но, как обычно, для них правила не писаны и если они не понимают, то сразу отвергают. Например, одна из главных целей моей работы – объяснить, что стандартная непрерывная математика – частный вырожденный случай конечной математики, а не наоборот. Но менталитет этого тупицы такой: он сразу решает, что дискретное это приближение к непрерывному и пишет отрицательный отзыв. Мой ответ на рецензии такой:

Monograph proposal: "Finite Quantum Theory and Applications to Gravity and Particle Theory" by F. M. Lev

Author's Comments on FTPH Reviewer Reports

My first observation is about the attitude of the reviewers from the formal point of view. My experience is that in many cases reviewers do not think that they are bound by the editorial policy of the journal for which they write a report and they believe that they know better what should or should not be published.

The FTPH editorial policy says in particular: «Although the aim of this series is to go beyond established mainstream physics, a high profile and open-minded Editorial Board will evaluate all contributions carefully to ensure a high scientific standard». As follows from this sentence, the reviewers MUST read the author's proposal carefully and at least to have a minimal understanding of what the author proposes. Without this understanding it is not possible to make a conclusion whether «a high scientific standard» is met or not. In addition, the reviewers should be open-minded, i.e. they should accept that in physics different approaches have a right to exist and so they should not reject the proposal only because it is not in the mainstream.

In my proposal I describe the motivation in great details but the reports do not give any indication on whether the reviewers carefully read the proposal, whether they made any efforts to understand it and whether they are qualified to understand.

As I explain, in my approach quantum theory is based on finite mathematics, it is more fundamental than standard continuous mathematics and the latter is a degenerated special case of the former. So for understanding those key statements the reviewers should have at least very basic knowledge in finite mathematics. However, the reports do not show any sign that the reviewers have this knowledge.

Let me quote an extract from my proposal: «… the majority of physicists do not have even a very basic knowledge in finite mathematics. This is not a drawback because everybody knows something and does not know something and it is impossible to know everything. However, many physicists have a mentality that only their vision of physics is correct, they do not accept that different approaches should be published and if they do not understand something or something is not in the spirit of their dogmas then this is pathology or exotics which has nothing to do with physics». This extract fully applies to the reviewer reports.

For example, Reviewer 1 thinks that since my approach is based on discrete mathematics then it is simply an «approximation to the standard continuum theory». First of all, if my approach is only an approximation then it is not FTPH at all. So it should be rejected right away and the remaining part of the report is obsolete. The mentality of the reviewer is that discrete is an approximation to continuous. This mentality is based on standard mathematical education where, for example, integral sums are treated as an approximation to the «true» value obtained by integration. In my proposal I explain why in the given case standard mentality does not work and below will explain this again.

Reviewer 1 writes that «The criticisms of the mainstream continuum theories are, for my taste, too commonplace and unspecific…» First of all, my remarks about problems of those theories are not a criticism but simply a reminder of well-known facts. The reviewer says that this «have already been responded to within the usual mainstream theories» but gives no specifics. For example, does he/she think that the problem of infinities has been already solved? Or in his/her opinion this problem is not important? For example, Weinberg, who is a famous physicist, writes in his textbook on QFT: «Disappointingly this problem appeared with even greater severity in the early days of quantum theory, and although greatly ameliorated by subsequent improvements in the theory, it remains with us to the present day». The title of one Weinberg's paper is «Living with infinities». He also writes that a new theory may be «centuries away». Do those Weinberg statements have been already refuted and if yes then when and where? Do we have quantum gravity where the renormalized perturbation series does not contain infinities?

As I note in the proposal, several famous physicists discussed a possibility that fundamental quantum theory will be based on finite mathematics and one of the arguments is that in this case infinities cannot exist in principle. Reviewer 1 says that «Some of the papers cited to support the author's criticism of the mainstream theories are known to present misguided views that have been clarified elsewhere in the literature» but does not give any explanation on what is misguided, what has been clarified and no references are given.

Reviewer 1 says: «It is also not really clear how the author's approach would get around the criticized issues». I do not see any meaning in this statement because the reviewer does not say specifically what is not clear to him/her and, as noted above, there is no indication that he/she has at least a basic understanding of my approach. Scientific ethics imply that any negative statement should be substantiated, i.e. the words «too commonplace», «unspecific», «not really clear», «speculative» and others should be explained.

In summary, the report of Reviewer 1 contains nothing specific, contradicts scientific ethics and fully contradicts the FTPH policy because he/she recommends rejection without any understanding of my approach and results.

The report of Reviewer 2 also does not follow standards of scientific ethics. He/she says that I ignore «80 years of successful quantum theory». This is a very serious accusation but no explanation is given. Does he/she think that any attempt to improve the theory means ignoring it? In particular, does he/she think that relativistic mechanics ignores nonrelativistic one? Or does quantum theory ignore classical one? He/she also thinks «that the proposal is kind of esoteric» but again does not explain why he/she thinks so.

In contrast to Reviewer 1, Reviewer 2 acknowledges that there are problems with the photon position operator and with infinities but says that «the author is only focusing on those». This immediately shows that, in full contradiction to the FTPH policy, Reviewer 2 even did not carefully read my abstract where it is indicated what problems are discussed. Reviewer 2 says: «But the first question one would have to address is, when one wants to change the world, how does the world in which we actually live fit into that. This sentence is fully puzzled. Why does he/she think that I want to change the world? If I show that standard photon position operator is inconsistent then does it mean that I want to change the world? Does it mean that any improvement of standard theory means changing the world?

Reviewer 2 says The author ignores that or hides the discussion somewhere, where it is hard to find». Why was it hard for the reviewer to find? Was it hard to read the title of paper [15]?

Then he/she writes: «…the book is all words, hardly formulas, almost like a book of philosophy». Probably Reviewer 1 read only the introductory chapter because the other chapters contain extensive mathematical derivations of new results which have never been published. The existing version of the manuscript contains 259 pages. Again, in contradiction to scientific ethics, Reviewer 2 does not explain how many pages he/she treats as «all words» and how many as «hardly formulas».

In summary, my conclusion on the report of Reviewer 2 is absolutely the same as the conclusion on the report of Reviewer 1.

In view of the FTPH policy, the author should submit to FTPH a fundamentally new approach, not just a variation of mainstream one. So the reviewers should be ready that standard mentality is not sufficient for understanding the proposal. In particular, standard mentality that discrete is only an approximation to continuous, does not imply in the given case. In my proposal I tried to explain this point and below will try to explain again.

The notions of infinitely small, continuity etc. were proposed by Newton and Leibniz approximately 370 years ago. At that time people did not know about atoms and elementary particles and believed that any object can be divided by arbitrarily large numbers of arbitrarily small parts. But now it is obvious that when we reach the level of atoms and elementary particles then standard division loses its meaning and one cannot obtain arbitrarily small parts. It is immediately clear from this observation that the notions of infinitely small and continuity are not fundamental on quantum level. Moreover, it is rather strange to think that fundamental quantum theory should be based on mathematics involving infinitely small and continuity. The founders of quantum theory were highly educated physicists but they used only standard continuous mathematics, and even now discrete and finite mathematics is not a part of standard mathematical education at physics departments. For understanding my statement that finite mathematics is more fundamental than standard continuous one and that the latter is a degenerated special case of the former (see e.g. paper[16]), at least a very basic knowledge of finite mathematics is needed. The reviewer reports show that the reviewers do not have this knowledge. As I note above, this is not a drawback. However, scientific ethics implies that it is not decent to judge an approach without having at least very basic knowledge about the approach.

In particular, finite mathematics does not involve continuity, derivatives or integrals; those notions are approximations which might or might not work in different situations. In finite mathematics finite sums are possible. In some cases such sums can be approximated by integrals. So in this case not discrete is an approximation of continuous but vice versa. In my proposal I also explain that the continuous spectrum is an approximation of the discrete one but not vice versa.

После этого ответа Angela Lahee написала мне, чтобы я прислал ей свои предложения о рецензентах. Я их прислал и думал, что теперь мне надо ждать что напишут рецензенты и что она скажет. Но неожиданно получил такой емайл:

I have now received back some further comments on your manuscript. Although two of the reviews by persons you had suggested were positive about the work you present, I’m afraid that other established researchers in quantum theory remain skeptical. In particular they question the sense of applying finite mathematics to QFT in place of the well established renormalisation theory.

They are nonetheless open to new approaches. But they propose (and I agree) that the better way to disseminate new ideas of this kind is first to publish a series of short(er) self-contained papers demonstrating the power of this approach. If the published results have some impact in the community, this would be the right moment to publish a longer book-length treatment.

So I am sorry, but we will not change our decision about this proposal. I hope you will be successful in publishing your ideas as one or more journal papers.

То есть опять, раз у меня не QFT, то пошел подальше, а слова, что ”They are nonetheless open to new approaches” противоречат предыдущему. Мой ответ был такой:

Dear Angela,

Thank you for this info. To be honest, it looks rather strange for me. You say that “other established researchers in quantum theory remain skeptical. In particular they question the sense of applying finite mathematics to QFT in place of the well established renormalisation theory.” Did they send you their reports or these are only words? Do they have at least very basic understanding of finite mathematics? They propose me to publish new papers. My proposal is based on papers published in known journals: Annals of Physics, J.Math.Phys., J.Phys.A, Phys.Rev. D, Physics of Particles and Nuclei and Theor. Math. Phys. (the last two journals are published by Springer). If this is not sufficient then what are their requirements for publications? You say that “If the published results have some impact in the community…”. Several physicists support my approach. You say that you received two reports from physicists I proposed. But my list contains six names. Will you wait for other reports? Indeed, many physicists do not accept my approach but so far I failed to receive clear explanations of their reasons and to be honest, I suspect that one of the main reasons is that they do not have at least very basic understanding of finite mathematics. For the problems I discuss I do not need QFT and renormalization theory because I consider only systems of free particles in the framework of standard de Sitter symmetry or de Sitter symmetry based on finite math. I show that those symmetries result in effective interactions which have not been discussed in the literature, they change the notion of elementary particles, conservation laws etc.

Let me also note that in 2017 Springer published a monograph by Vourdas where applications of finite math are discussed and this monograph has nothing to do with QFT and renormalization theory. And finally my MOST fundamental result is: standard continuous math with infinitely small, continuity etc. (which was started by Newton and Leibniz approx.. 370 years ago) is a degenerated special case of finite mathematics in the formal limit when the characteristic of the field or ring in the latter goes to infinity. Moreover, in view of existence of elementary particles it is obvious that in nature there are no infinitely small quantities and no continuity but fundamental quantum theories are based on continuous math and many physicists oppose results where the other math is used. This result fundamentally changes the usual philosophy on what math and what physics are the most fundamental. I have no doubt that sooner or later this result will be acknowledged.

In summary, I would be very grateful if you explain me the following. Will you wait for the reports of other physicists proposed in my list? Could you tell me what are the requirements that my results have an impact in the community? And to be honest, I would be very grateful if you tell me without diplomacy whether I have real chances to be published by Springer. If the clear answer is “no” then no questions will be asked and I will not bother you anymore.

И после этого получил такой ответ:

Dear Felix,

Given the consensus among four long-standing advisors who I have now consulted, I am afraid that your book will not be further considered by Springer. It is a difficult case, as one of the reviewers commented, so perhaps another publisher will come to a different conclusion.

То есть теперь она уже ясно говорит, что шансов опубликовать монографию нет. Я не могу ее осуждать потому что она может принимать только те решения которые ей разрешают. И в данной ситуации у нее не было выбора.

Но через два года ситуация изменилась. Я переработал монографию и послал ей новую просьбу. В ней написал, что моя просьба простая: раз я посылаю proposal в раздел Fundamental theories of physics, то те которые будут рассматривать мой proposal, обязаны рассмотреть его в рамках editorial policy этой серии. Приведу полностью эту policy:

The international monograph series “Fundamental Theories of Physics” aims to stretch the boundaries of mainstream physics by clarifying and developing the theoretical and conceptual framework of physics and by applying it to a wide range of interdisciplinary scientific fields. Original contributions in well-established fields such as Quantum Physics, Relativity Theory, Cosmology, Quantum Field Theory, Statistical Mechanics and Nonlinear Dynamics are welcome. The series also provides a forum for non-conventional approaches to these fields. Publications should present new and promising ideas, with prospects for their further development, and carefully show how they connect to conventional views of the topic. Although the aim of this series is to go beyond established mainstream physics, a high profile and open-minded Editorial Board will evaluate all contributions carefully to ensure a high scientific standard.

То есть, произносится много хороших слов о том что могут быть non-conventional approaches, new and promising ideas и то что члены Editorial Board должны быть open-minded. Как я уже писал, предыдущие рецензии в этой серии не имели ничего общего с этой policy. Angela Lahee ответила, что она согласна, что моя просьба разумная и будет просить членов Editorial Board написать рецензию в соответствии с этой editorial policy. И она просила многих написать рецензию, но никто не захотел. Как она писала, кто-то долго тянул, а потом написал, что он занят своими делами и у него нет времени, но большинство просто не ответили.

Я не могу понять их логику: если они согласились быть в этой Editorial Board, то, казалось бы, они взяли на себя моральное обязательство писать рецензии в рамках editorial policy. А если они не хотят, то зачем они в этой Editorial Board? Я думаю, что основная причина такого отношения в том, что они, как и абсолютное большинство физиков, не знают даже самых основ конечной математики. Может быть, были и другие причины.

После того как получить рецензию не удалось, Angela написала, что, т.к. никто не был против публикации монографии, а рецензии, которая она просила два года тому назад, были в целом положительными, то монография может быть опубликована, но уже не в серии Fundamental theories of physics, а как stand-alone книга. И она была опубликована! Более того, мое название монографии Angela предложила изменить, и ее предложение мне понравилось. Поэтому монография вышла с названием: "Finite Mathematics as the Foundation of Classical Mathematics and Quantum Theory. With Application to Gravity and Particle theory". Более подробную ссылку на книгу см. в [23].

Так что общий вывод от общения с Springer: все как обычно, если предлагаешь что-то новое, что establishment не понимает, то шансов почти нет, а надо только то, что укладывается в их догмы. И, как обычно, все что написано в editorial policy никакого значения не имеет, т. е. Springer не выполняет те правила, которые сам провозгласил.

Но мне очень повезло, что Angela Lahee оказалась очень порядочным человеком. В рамках ее служебных обязанностей она могла поступить по-разному, но она поступила по высшим критериям порядочности. Я очень благодарен ей за помощь в публикации книги.

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