Популярно о конечной математике и ее интересных применениях в квантовой теории

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policy of the journal for which they write a report and they believe that they know better what should or should not be published.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.