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

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