Популярно о конечной математике и ее интересных применениях в квантовой теории

"Proposal for a letter to the Editor

The title of the letter is “Discussion of cosmological acceleration and dark energy”. The current version of manuscript contains 8 printed pages. It be found in the HAL archive: https://hal.archives-ouvertes.fr/hal-03581039.

The problem of cosmological acceleration (CA) is one of the hot topics of modern physics and cosmology. In the vast majority of works on this topic, the cosmological expansion is explained as a manifestation of dark energy, quintessence or similar mechanisms. For example, explaining the Nobel Prize for Peebles, some members of the Nobel committee said that he opened our eyes that we know only 5 % of the universe because almost 70 % is dark energy and 25 % is dark matter.

The generally accepted approach in theoretical physics is such that when new experimental data appear, then, first of all, they should be explained on the basis of the available proven theory. Only if this fails, then some new exotic explanations must be invoked. However, in the case of CA, the opposite approach was taken: there were practically no works in which this phenomenon is explained on the basis of the available results, and in most works the effect is explained on the basis of dark energy and other exotics.

Probably, one of the historical reasons was that Einstein said that introducing Λ was the greatest blinder of his life. Even in textbooks written before 1998 a point of view was advocated that"…there are no convincing reasons, observational and theoretical, for introducing a nonzero value of and that"… introducing Λ to the density of the Lagrange function a constant term which does not depend on the field state would mean attributing to space-time a principally ineradicable curvature which is related neither to matter nor to gravitational waves".

However, several authors (see e.g., Refs. [1,2]) give clear arguments that the explanation of CA by dark energy is not physical. In my publications [2–6] I show that the problem of CA has a clear solution based on well-established results of quantum theory, and the explanation does not need dark energy or other exotic mechanisms the validity of which has not been proved. More details on my publications can be found in my ORCID: https://orcid.org/0000–0002–4476–3080

The generally accepted opinion is that since the problem of CA deals with large macroscopic bodies located at large distances from each other, there is no need to involve quantum theory to study this problem, and the problem must be considered within the framework of General Relativity and other classical theories. However, ideally, every result of classical theory should be obtained from quantum theory in semiclassical approximation.

Consideration of the CA problem from the point of view of quantum theory sheds essentially new light on this problem. For example, in classical theory the case Λ=0 corresponds to the flat Minkowski space while the case Λ ≠ 0 corresponds to the de Sitter (dS) space. As noted above, the usual philosophy is that empty space should be flat and therefore the case Λ=0 is preferable than Λ ≠ 0. However, the concepts of background space-time and Λ are pure classical. On quantum level the problem is what symmetry group or algebra is preferable. As shown by Dyson in his famous paper “Missed Opportunities”, the dS group is more general (fundamental) than the Poincare one because it is more symmetric, and the latter can be obtained from the former by contraction. In addition, since the dS group is semisimple, it has a maximum symmetry and cannot be obtained from other groups by contraction. This Dyson’s result has nothing to do with the relation between dS and Minkowski spaces and with the value of Λ. Consequently, quantum theory based on dS symmetry is more general (fundamental) than quantum theory based on Poincare symmetry.

It is difficult to imagine standard quantum theory without irreducible representations (IRs) of the Poincare algebra. Therefore, quantum theory based on dS symmetry should involve IRs of the dS algebra. However, my observation is that even physicists working on dS quantum theory are not familiar with such IRs. Some of them give a strange argument that such IRs are not needed because fields are more important than particles.

My results in [2–6] and other publications are based on large calculations. To understand them, the readers must be experts not only in quantum theory, but also in the theory of representations of Lie algebras in Hilbert spaces. Therefore, understanding my results can be a challenge for many physicists. Since the problem of CA and dark energy is very important, I decided to write a short note, which outlines only the ideas of my approach without calculations. I hope that after reading this note, many readers will have an interest in studying my approach because it gives a clear solution of the problem of cosmological acceleration and considerably differs from approaches of other authors.

References

[1] Bianchi, E., Rovelli, C.: Why all These Prejudices Against a Constant? arXiv:1002.3966v3 (2010).

[2] Lev, F. M.: Finite Mathematics as the Foundation of Classical Mathematics and Quantum Theory. With Application to Gravity and Particle theory. ISBN 978–3–030–61101–9. Springer, https://www.springer.com/us/book/9783030611002 (2020).

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

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

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

[6] Lev, F. M.: Cosmological Acceleration as a Consequence of Quantum de Sitter Symmetry. Physics of Particles and Nuclei Letters 17, 126–135 (2020).