What Mathematics Is Most Pertinent For Describing Nature? by Felix M Lev

Dear Florin,

Thank you for your note. You say that you are skeptical about the p-adic approach. Does this imply that you are skeptical about my approach as well? One of the main reasons why I prefer a finite field approach is that it does not contain infinities at all. In my approach there is no problem with standard QM since, as explained in the essay (see also my cited papers and/or my papers in the arXiv), in the formal limit p->\infty my approach recovers the results of standard quantum theory. I am not the author of [4], so you would better ask them about problems bothering you. Let me only note that the authors of the p-adic approach also state that they have a correspondence principle when p->\infty.

Dear Florin,

Thank you for your opinion. If you look at the beginning of the discussion section in [11] (in the arXiv this is the paper http://xxx.lanl.gov/abs/1011.1076 ) you will see that I am aware of this opinion. This is a typical opinion and of course, everyone has a right to have his or her own preferences ("De gustibus non disputandum est!"). I will try to comment this opinion.

You think that "p-adic and Galois approaches to QM seem to be an unnecessarily radical departure from the standard approach". So you think that the existing problems can be solved without radical approaches and you believe that they can be solved by the string theory, noncommutative geometry or something like that. Of course, only the future can be the judge. Nevertheless, some of your remarks seem strange to me.

You say that standard problems look strange when they are considered from the point of view of the p-adic approach. In your opinion, p is only a cutoff parameter. It seems to me, this has something common with the following hypothetical situation. Suppose that Heisenberg or Schroedinger wrote a paper on QM and a referee says: "Since you do not want to take the limit \hbar->0, I have big technical problems describing the motion of the Moon by the Schroedinger equation." And this referee is right! Of course, there is no need to describe the motion of the Moon by the Schroedinger equation. Since this motion is quasiclassical, we can take the limit \hbar->0 immediately and get the Hamilton-Jacobi equation. But now we know that there are other problems where we cannot take the limit \hbar->0. Moreover, the fact that \hbar is not zero leads to a dramatic change of our understanding of nature.

As I explain in my papers, p is not only a cutoff parameter since in Galois fields the rules of arithmetic are essentially different. As a result, some quantities (e.g. the Dirac vacuum energy) which in standard theory are infinite become not of order p (as one might think if p is treated only as a cutoff parameter) but exactly zero. Another new features are that in GFQT the notion of particle-antiparticle is only approximate, there can be no neutral elementary particles and such quantum numbers as the electric, baryon and lepton charges are only approximately conserved. In particular, this completely changes the status of the problem called "Baryon asymmetry of the Universe".

My final remarks are as follows. The absolute majority of physicists treat mathematics only as a tool for solving physics problems. Those physicists don't care much whether or not mathematics is beautiful, rigorous etc. A well known example is that when we subtract one infinity from the other and get correct 8 digits for the electron and muon magnetic moments, physicists typically treat this as a great success (what is true) and believe that there is no need to be bothered by the lack of mathematical rigor. However, several famous physicists did not think so. For example, Dirac wrote:

"The agreement with observation is presumably by coincidence, just like the original calculation of the hydrogen spectrum with Bohr orbits. Such coincidences are no reason for turning a blind eye to the faults of the theory. Quantum electrodynamics is rather like Klein-Gordon equation. It was built up from physical ideas that were not correctly incorporated into the theory and it has no sound mathematical foundation."

Dirac's advice is:

"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."

For me the main reason why GFQT is much more appealing than standard theory is that it is based on extremely simple and beautiful mathematics. I also believe that this mathematics is more physical than standard one. I believe that for quantum physicists it should be obvious that in nature there are no infinitely small or infinitely large quantities and the notions of infinitely small and infinitely large are only idealizations. We have to use these notions because standard quantum physics is based on mathematics developed mainly in the 19th century when people did not know about the existence of elementary particles and believed that any object can be divided by any number of parts. My hope was that physicists should be excited to realize that quantum physics can be based on mathematics involving only finite sets without such unphysical notions as infinitely small and infinitely large. My observation is that, although basics of Galois fields can be taught even at elementary schools, physicists typically are not aware even of those basics as a result of the fact that Galois fields are not taught at Physics Departments. In addition, as I noted above, physicists typically treat mathematics only as a tool for solving physics problems. As a consequence, when a typical physicist encounters some unknown mathematics, he or she has a temptation not to try to understand whether or not this mathematics is more elegant, pertinent etc. but to think that this is an unnecessarily radical departure from the existing approach.

In my essay and papers I argue that only a finite mathematics can describe reality. In particular, the notion of probability can be only approximate. Indeed, this notion implies that we should carry out an infinite number of experiments within an infinite time interval. In reality this can be never done and when one says about probabilities, he or she believes that a finite number of experiments gives a value close to a hypothetical limit when the number of experiments is infinite. For example, we can never guarantee that the probability is exactly zero since if some event has not been found even in a very large number of experiments, there is no guarantee that it will not be found in a greater number of experiments.

Dear Florin,

My impression is that our discussion becomes rather strange. I submitted an essay entitled "What Mathematics is Most Pertinent for Describing Nature?" and I argue that only a finite mathematics can be fundamental. I believe that my essay is fully in the spirit of this essay contest entitled "Is Reality Digital or Analog?" So I discuss mathematics. However, my impression is (maybe I am wrong?) that you are not interested in discussing mathematics but instead you try to convince me that my approach, which I call GFQT is unphysical or unnecessary or what? (I would prefer if you say explicitly what you want to prove). If this discussion has nothing to do with the essay contest then it's probably not correct to continue this discussion here; for example, we can communicate via email if you are interested.

As I noted in my previous response, our philosophies are almost orthogonal and, as a consequence, we have even a communication problems. For example, you think that my goal is to fix problems, which do not exist since they are well understood. For me your phrase "field theory is already on a solid foundation and by now we understand the infinities from both the mathematical and physical point of view" seems strange since if a theory has a solid foundation, it will not contain infinities at all. The arguments that QFT does not have a solid foundation are well known (do you know them?). They have been given not only by Dirac (whose remarks you treat as obsolete) but by Heisenberg, Wigner and other famous physicists. Even Weinberg, who contributed much to QFT, acknowledges in his textbook that infinities are still a big problem. Nobody has shown that those arguments are not correct but since QFT has achieved several impressive successes, the majority of physicists believe that there is no need to worry about foundations.

I have proposed a new approach and I believe that in view of the present situation in physics different approaches should be considered. I am not saying that you should like my approach. You believe that the existing problems can be better solved by the string theory or similar approaches and of course you have a right to think so. In that case could you, please say explicitly what you are going to prove in our discussion? If you think that my approach is incorrect, I am very eager to know why. If I understand you correctly, so far you try to convince me that the approaches you like will solve the existing problems more successfully. If this is your only goal then I do not see any further point for discussion since, as I noted, "De gustibus non disputandum est".

In my essay and papers I note that GFQT fully changes the status of such problems as particle-antiparticle, elementary neutral particles, conservation laws etc. You do not mention those problems at all. Meanwhile if you are going to prove that my approach is unacceptable then probably it is better to explicitly say why. Let me try to help you.

My original goal was indeed to get 8 correct digits with a solid mathematics (see [11] for a discussion). However, this naïve expectation fails for the following reason. In GFQT there are no independent irreducible representations (IRs) for a particle and its antiparticle but one IR describes an object such that a particles and its antiparticle are different states of this object. As a consequence, there are no neutral elementary particles, the electric, baryon and lepton charges can be only approximately conserved and even the notion of particle-antiparticle is only approximate. I believe that this is an extremely interesting result but you, probably have another opinion. For example, you can say that if even the photon cannot be elementary then GFQT is unphysical. Could you, please, tell me explicitly what you think about this situation? Let me also repeat that if I understand you correctly and you are interested only in discussing physics then the present forum is not an appropriate place for this discussion.

In probability theory, probability is defined as a measure of sets belonging to a sigma-algebra. Those sets can have only a positive nonzero measure. In physics, a definition of a physical quantity is a description how this quantity should be measured. So a question arises whether in physics it is possible to define probability in accordance with mathematics. Mathematics prompts us that we cannot define such a quantity as "the probability to find 5.0724" since the set containing only the point 5.0724 has measure zero and does not belong to the sigma-algebra. But we can try to define the probability to find a number in some interval. As I noted in the previous note, the only known way of defining probability in physics is that we should carry out an infinite number of experiments within and infinite time interval and this is problematic. Also, in quantum physics probability can be zero if there are superselection rules.

Dear Rick P,

Thank you for this reference. If you read my essay you could see that it is in the spirit of these remarks.

Dear Cristi,

Thank you for encouraging words about my works.

I am not saying that standard mathematics is wrong. The question is whether we

i) accept a principle that only those statements have a physical significance, which can be experimentally verified (at least in principle) or ii) we agree that some statements (axioms) can be accepted without proof (for some reasons). Since you pose two questions [?] in your note, you probably think that we should accept i), right? But then we should acknowledge that standard axioms cannot be verified. For example, how can we verify that a+b=b+a for any natural numbers a and b?