Популярно о конечной математике и ее интересных применениях в квантовой теории - Феликс Лев

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 Gödel’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 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