Популярно о конечной математике и ее интересных применениях в квантовой теории

Ref: DARK_2019_25_R1

Title: Cosmological Acceleration as a Consequence of Quantum de Sitter Symmetry, by F. Lev

Author’s appeal on editorial decision

The decision is based on reports of Reviewer 1 and Reviewer 3. Reviewer 1 did not say that the paper should not be published. He/she said that it could not be published in the present form because in his/her opinion the paper contained nothing essentially new in comparison with my previous papers. In view of this remark, I considerably revised the paper and now I explicitly explain why the new paper is fundamentally new. So in fact the decision is based only on the report of Reviewer 3.

At the beginning of the report, Reviewer 3 says the same words as Reviewer 1 without any substantiation. Regardless whether or not Reviewer 1 is the same person as Reviewer 3, for the current version of the paper there were two reviewers with fully opposite recommendations. In such cases in my practice the paper was usually sent to adjudicator. However, in the given case the preference was given to one of the reviewers. Reviewer 3 says that “The related notions of physical dimensions and units of measurement are systematically confused throughout the paper (I recommend https://arxiv.org/abs/physics/0110060 for a clear discussion of the subject).” However, nothing specific is said on what is “systematically confused” and so it is fully unclear whether Reviewer 3 understands what is written about physical dimensions. He/she says nothing on whether or not my paper contradicts this reference. This reference is known and I discuss it in my monograph project https://arxiv.org/abs/1104.4647. The three authors propose considerably different opinions on the problem, and Reviewer 3 says nothing on what opinion (if any) he/she prefers. One of the authors (M. J. Duff) states that the most fundamental physical theory should not contain arbitrary constants at all, and in Sec. 2 I also argue in favor of this statement.

The next part of the report also shows no sign that Reviewer 3 understands my results. First he/she says that "Section 2 is a naive, incomplete and unnecessarily lengthy…" but nothing specific is said on what is naïve, incomplete etc. Reviewer 3 writes: "The Author claims that deSitter and Anti-deSitter spacetimes are more “fundamental” than Minkowski spacetime…" but there is no such a claim in the paper and the comparison of those spacetimes is not discussed at all. I don't know whether Reviewer 3 understands basic facts of quantum theory, whether he/she works in the framework of this theory or he/she works only in the framework of classical theory. As I noted in my previous emails, many physicists do not understand that spacetime is only a classical notion, and spacetime description is only a consequence of quantum theory in semiclassical approximation.

On quantum level symmetry is defined by the commutation relations in the symmetry algebra as explicitly explained in Sec. 2, and in the formulation of this symmetry nothing is said about spacetime. In the theory of Lie groups and algebras a well-established fact is that if symmetry B is obtained from symmetry A by contraction then symmetry A is higher than symmetry B. In Sec. 2 I refer to famous Dyson’s paper [7] where this fact is explained for groups and I explain this fact for algebras. Since Poincare algebra can be obtained from dS or AdS algebra by contraction, this automatically implies that dS and AdS symmetries are more fundamental than Poincare symmetry, and this has nothing to do with the relation between de Sitter and Minkowski spaces. The notion of contraction is a fundamental notion of the theory of Lie groups and algebras, and the report shows no sign that Reviewer 3 has a basic knowledge in this theory.

Reviewer 3 says ”It is simply incorrect to speak of a given spacetime geometry as being more fundamental than another; rather, the attribute “fundamental” should be used with reference to a dynamical theory having a broader regime of applicability compared to a particular limit.”. Again, as noted above, I do not discuss spacetime at all because this is only a classical notion. In Sec. 2 I give Definition when theory A is more general than theory B, and this definition explicitly says that a more general theory has a broader regime of applicability compared to a particular limit. So a question arises whether Reviewer 3 read my Definition and whether he/she tried to understand it.

Reviewer 3 says: “Moreover, neither the value nor the sign of the cosmological constant can be fixed following the arguments in the paper.” As I explain in detail, the problem of the value of Λ does not arise for the same reasons as the problems of the values of c and ћ do not arise. Indeed this statement contradicts the usual dogma that Λ should be somehow fixed. However, Reviewer 3 says nothing specific on why in his/her opinion my explanation is incorrect or unacceptable, and so his/her objection cannot be treated as a scientific argument. It is known that relativistic quantum theory itself does not need the values of c and ћ, and in all textbooks on this theory the presentation is given in units c=ћ=1. The numerical values of c and ћ are needed only if one wants to express some quantities in (kg,m,s). The notion of the system of units was proposed many years ago when quantum theory and relativity did not exist. The notion of (kg,m,s) is pure classical and physical quantities are expressed in these units only for convenience. The problem why the values of c and ћ in units (kg,m,s) are as are does not exist since the answer is: because people want to measure c and ћ in these units.

Reviewer 3 writes: "Moreover, it is quite challenging to build a theory of gravity where the cosmological constant (or, equivalently, the deSitter radius) matches the observed value without introducing new tunable parameters…". As explained in Sec. 2, quantum dS or AdS theories themselves do not need the numerical value of Λ for the same reasons as relativistic quantum theory does not need the numerical values of c and ћ. I also explain the known fact that even for classical dS and AdS theories themselves the numerical value of R is not needed. Since Reviewer 3 again raises this question, I will try to explain this obvious point again.

Suppose for simplicity that our world is a surface of two-dimensional sphere. Then the coordinates on the sphere can be described by two dimensionless polar angles (φ,θ). For the description of geometry we do not need the radius of the sphere R and we can assume that R=1. The quantity R in meters has the meaning of the radius of the sphere seen from the three-dimensional space where the sphere is embedded in. But we know nothing and do not need to know about this space and its coordinates. Those coordinates are of interest only when we want to attribute to R some value and consider a formal limit R→∞. In this limit a vicinity of the Northern pole of the sphere becomes the flat two-dimensional space.

Analogously, for dS or AdS theories themselves the value of R is not important; we can assume that R=1 and describe geometry on dS or AdS space by using only dimensionless polar and hyperbolic angles. The value of R becomes important only when we consider transition from dS or AdS space to Minkowski one. So the desire to describe R in meters does not have a fundamental physical meaning. The question why R is as is does not arise since the answer is: because people want to measure R in meters.

The only problem which is indeed important is whether dS quantum theory is more fundamental than AdS one or vice versa. I discuss this problem in my paper in J. Phys. A [9] and in my papers published in J.Math. Phys., Finite Fields and Applications, Phys. Rev. D and other journals where I argue that a quantum theory based on a finite ring or field is more fundamental than standard quantum theory based on complex numbers.

The cosmological constant problem is purely artificial. One first tries to build quantum gravity from Poincare invariance because it is associated with Minkowski background. Then he/she realizes that the expression for the vacuum energy-momentum tensor strongly diverges, and after the cutoff which is called reasonable he/she obtains that Λ is of the order of 1/G as expected. However, as noted above, on quantum level Poincare symmetry is a special degenerate case of dS or AdS symmetry not because Minkowski space is less symmetric than dS or AdS space but because Poincare algebra can be obtained from dS or AdS algebra by contraction. With the same success one can discuss the speed of light problem or the Planck constant problem.

Finally, let me note the following. Reviewer 3 claims that my paper is of no interest for the readers of Physics of Dark Universe and for this reason he/she does not want the readers to know about my results. I believe, however, that the readers are interested in knowing different approaches to the problems of their interest. My paper shows that a known problem can be tackled from a fully different approach. I believe that for the readers it would be extremely interesting to know that the result of General Relativity on cosmological acceleration obtained from dS space can be obtained from semiclassical approximation of dS quantum mechanics without using dS space at all (i.e. its metric, connection etc.). This result is obviously more general than the result of General Relativity because any classical result should be a consequence of quantum theory in semiclassical approximation. As I note in my explanations, while in [9] this result has been obtained after lengthy mathematical calculations, in the present paper I give a short description on three pages such that the reader will understand the necessary steps.

Let me also note that my paper is fully in the scope of Physics of the Dark Universe because the editorial policy contains "cosmic acceleration and its alternative explanations". At the same time, Reviewer 3 does not allow alternative explanations and accepts only those approaches which are in the spirit of his/her mentality.

The report cannot be treated as a scientific recommendation because: 1) it contains no sign that Reviewer 3 understands what is done in the paper; 2) scientific ethics implies that all negative statements in the report should be substantiated but all of them are made without any substantiation; 3) the report contains no specific statement on why anything in my paper is incorrect or unacceptable, my only “fault” is that my statements contradict known dogmas which have no physical justification. For those reasons I would appreciate if the editorial decision is reconsidered. I am also grateful to Reviewer 2 for the recommendation to publish the paper and for important remarks which will be taken into account in the next version of the paper.