Популярно о конечной математике и ее интересных применениях в квантовой теории

Dear Sergei,

Thank you for your response to my detailed letter. However, you probably will not be surprised if I say that I am disappointed with your response. You asked whether my approach “is capable of obtaining new results or of consistently explaining known results in a new way”. I was very glad that you asked this question and hoped that you will read my response. But now I am not sure that you were interested in my response at all and probably you decided from the beginning that the answer is negative.

I tried to answer your question in such a way that (in my understanding) the answer should be appreciated and understood by any mathematician, even by students of mathematical departments. For example, I give a popular explanation why in modular mathematics I have one irreducible representation (IR) which splits into two IRs in the formal limit p→∞. This (mathematically beautiful!) example immediately shows that, even from a pure mathematical point of view, modular theory is more general than standard one. Since you said nothing about this example then either you even did not read it at all or were unable to understand it.

I also give other simple MATHEMATICAL examples which show that there are cases when modular theory can solve problems which standard theory cannot. However, again, no specific comments on those examples are given, and so you either did not read those examples or were unable to understand them.

I understand that everybody has his/her own problems, and nobody can insist on what other people should or should not read. But it is beyond any logic that you asked a question and said nothing explicit about my response. You say: “My criticism of your article – and in it I agree with the referee – is that it essentially just a declaration that one can build quantum theory based on the rings Z/pZ, but it doesn't provide examples or any details.”

In my paper and the last letter, I give many simple MATHEMATICAL arguments but neither you nor the referee give any comments on these arguments. And so again, you either did not read the arguments or were unable to understand them. In the literature, criticism is defined as “the practice of judging the merits and faults of something”. But since there is no sign that you and the referee tried to understand my arguments, the word “criticism” in your letter is fully inappropriate. If it were only a discussion between two people, then everybody has a full right to read or not to read what he/she wants. But, in the given content, your opinion is understood not only as your personal opinion but as the opinion of the readers of your journal. I am not sure that your understanding of this opinion is realistic. For example, several physicists and mathematicians told me that they would be interested in reading a popular discussion of my approach. You say, “It well could be that your subject is too technical to be explained in an expository article in a magazine for general mathematical audience.” But I just tried to explain my results in an extremely popular (expository) level, and, in my understanding, this is fully what the editorial policy requires.

You explain to me that “From the mathematical point of view, one needs to obtain a consistent and, preferably, elegant theory capable of explaining the relevant phenomena in the framework of the model at hand”. According to the present knowledge, quantum theory is the most general model of nature which mankind has developed. So, in the content of your letter, your words can be understood only such that you do not think that my theory is elegant and capable of explaining the relevant phenomena. But in your letter, as I already noted, I do not see any sign that you tried and/or were able to understand what my theory is capable of.

Now let me comment on the following extract of your letter: “You also seem to claim that mathematics could be rebuilt starting with the rings Z/pZ, instead of Z (Peano arithmetic). This may be the case, but such an undertaking would take an enormous amount of work and, in my opinion, even if successful it will have little bearing on modeling nature. As I said, no mathematics is off limit if it's relevant in description of nature, and there is no need to rebuild the foundations for this purpose.”

In my paper and the last letter, I note that (during the last 80 years) there is a great problem that, by using the existing mathematics, physicists cannot construct a quantum theory which is mathematically consistent and can explain many existing experimental data. This is acknowledged by famous scientists and even Nobel Prize laureates (as I noted, even one of them wrote a paper titled “Living with Infinities”). Also, many authors and even some Nobel Prize laureates wrote papers conjecting that the ultimate quantum theory will be based on finite mathematics. Of course, constructing such a theory would take an enormous amount of work. However, in your opinion “, even if successful it will have little bearing on modeling nature” and “there is no need to rebuild the foundations”.

So, you do not know about existing fundamental problems, do not have ideas how to solve them, the opinion of famous scientists is not important to you, but your opinion is that “there is no need to rebuild the foundations”. So, your remarks are like those from the known Chekhov’s story “Letter to a learned neighbor” when a man writes to his neighbor: “You say that there are spots on the Sun; this cannot be because this can never be”.

You note that there are even similarities between our approaches because both start from a discrete approach. You will probably be indignant if someone, without any attempt to figure it out, says that your results are only declarations. You will probably say that you already have recognized works on this topic and this topic is related to generally recognized problems. But I can also say that I have papers in so-called prestigious journals, there are many other papers on this topic, and even Nobel Prize laureates wrote about this.

In summary, you asked me a question, I tried to answer this question in detail, there are no sign that you and the referee read my arguments and/or were able to understand them, but you say that my paper “is essentially just a declaration”. Giving negative statements about my arguments without any explicit mentioning them, contradicts scientific ethics and is simply indecent. I understand your last letter such that you do not want to spend any time on our discussion, and I also think that your attitude is such that any further correspondence is meaningless.

I wish all the best to you and your journal. Felix.