Популярно о конечной математике и ее интересных применениях в квантовой теории

The monograph will be based on my results published in:

[1] F. M. Lev, Some Group-theoretical Aspects of SO(1,4)-Invariant Theory. J. Phys., A21, 599–615 (1988).

[2] F. Lev, Representations of the de Sitter Algebra Over a Finite Field and Their Possible Physical Interpretation. Yad. Fiz., 48, 903–912 (1988).

[3] F. Lev, Modular Representations as a Possible Basis of Finite Physics. J. Math. Phys., 30, 1985–1998 (1989).

[4] F. Lev, Finiteness of Physics and its Possible Consequences. J. Math. Phys., 34, 490–527 (1993).

[5] F. Lev, Exact Construction of the Electromagnetic Current Operator in Relativistic Quantum Mechanics. Ann. Phys. 237, 355–419 (1995).

[6] F. M. Lev, The Problem of Interactions in de Sitter Invariant Theories. J. Phys., A32, 1225–1239 (1999).

[7] F. Lev, Massless Elementary Particles in a Quantum Theory over a Galois Field. Theor. Math. Phys., 138, 208–225 (2004). The journal is published by Springer.

[8] F. M. Lev, Could Only Fermions Be Elementary? J. Phys., A37, 3287–3304 (2004).

[9] F. Lev, Why is Quantum Theory Based on Complex Numbers? Finite Fields and Their Applications, 12, 336–356 (2006).

[10] F. M. Lev, Quantum Theory and Galois Fields, International J. Mod. Phys. B20, 1761–1777 (2006).

[11] F. M. Lev, Positive Cosmological Constant and Quantum Theory. Symmetry 2(4), 1401–1436 (2010).

[12] F. M. Lev, Introduction to a Quantum Theory over a Galois Field. Symmetry 2(4), 1810–1845 (2010).

[13] F. M. Lev, Is Gravity an Interaction? Physics Essays, 23, 355–362 (2010).

[14] F. Lev, Do We Need Dark Energy to Explain the Cosmological Acceleration? J. Mod. Phys. 9A, 1185–1189 (2012).

[15] F. Lev, de Sitter Symmetry and Quantum Theory. Phys. Rev. D85, 065003 (2012).

[16] F. M. Lev, A New Look at the Position Operator in Quantum Theory. Physics of Particles and Nuclei, 46, 24–59 (2015). The journal is published by Springer.

[17] F. M. Lev, Why Finite Mathematics Is The Most Fundamental and Ultimate Quantum Theory Will Be Based on Finite Mathematics. Physics of Elementary Particles and Atomic Nuclei Letters, 14, 77–82 (2017). The journal is published by Springer.

[18] F. M. Lev, Fundamental Quantal Paradox and its Resolution. Physics of Elementary Particles and Atomic Nuclei Letters, 14, 444–452 (2017). The journal is published by Springer.

and possibly in other journals.

I graduated from the Moscow Institute for Physics and Technology, got a PhD from the Institute of Theoretical and Experimental Physics in Moscow and a Dr. Sci. degree from the Institute for High Energy Physics (also known as the Serpukhov Accelerator). In Russia there are two doctoral degrees; Dr. Sci. degree is probably an analog of Habilitationsschrift in Germany. In Russia I worked at the Joint Institute for Nuclear Research (Dubna, Moscow region) and now I work at a software company in Los Angeles, USA.

I have many papers published in known journals (Ann. Phys., Few Body Systems, J. Math. Phys., J. Phys. A, Nucl. Phys. C, Phys. Rev. C and D, Phys. Rev. Letters and others). The majority of those papers are done in the framework of more or less mainstream approaches. On the other hand, the proposed monograph will be done in the fully new approach which I am working on for many years. In this approach quantum theory is based on finite mathematics.

I think that the main problems in convincing physicists that ultimate quantum theory will be based on finite mathematics are not scientific but subjective. First of all, 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.

Probably this situation has happened in view of several reasons. For example, the successes of QED at the end of the 40th were very impressive and it is of course impressive that the theory gives correct eight digits for the electron and muon magnetic moments and five digits for the Lamb shift. From mathematical point of view QED has several inconsistencies the reasons of which are clear. The above famous results are obtained by subtracting infinities from each other. However, in view of these and other results the mentality of the majority of physicists is that agreement with the data is much more important than mathematical consistency and many of those physicists believe that all fundamental problems of quantum theory can be solved in the framework of QFT or string theory (which has similar mathematical inconsistencies).

The meaning of «quantum» is discrete and historically the name «quantum theory» has arisen because it was realized that some physical quantities have discrete spectrum. 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. Several famous physicists (e.g. Schwinger, Wigner, Nambu, Gross and others) discussed a possibility that ultimate quantum theory will be based on finite mathematics. One of the reasons is that in this case infinities cannot exist in principle. However, standard quantum theory is based on continuous mathematics. Efforts of many physicists to resolve fundamental difficulties of this theory (e.g. existence of infinities) have not been successful so far. Continuous mathematics describes many data with high accuracy but this does not necessarily imply that ultimate quantum theory will be based on continuous mathematics. For example, classical mechanics describes many data with high accuracy but fails when v/c is not small. Continuous mathematics is not natural in quantum theory. For example, the notions of infinitely small and infinitely large have arisen when people did not know about atoms and elementary particles and believed that any object can be divided by any number of parts. Ultimate quantum theory cannot be based on continuous mathematics because the latter has its own foundational problems (as follows, for example, from G?del’s incompleteness theorems).

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 G?del’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 h?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 M

we introduce standard Poincare four-momentum P such that P= M

/R where R is a formal parameter which can be called the radius of the world. The spectrum of the operators M

is discrete, the distances between the spectrum eigenvalues are of the order of h 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 h/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 M

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.