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

details or even a formulation of the theory that you purport to exist that allows for baryon asymmetry. In fact, it is not clear what you mean by baryon symmetry since you do not specify a theory of particle physics on which this symmetry would act”.

However, the problem statement is given in the very first paragraph of the manuscript: “The problem of the baryon asymmetry of the universe (BAU) is a long standing problem of modern physics described in a vast literature (see e.g. Ref. [1] and references therein). According to modern quantum theories, the baryon number is a conserved quantum number, and, according to modern cosmological theories, the universe has been created with equal numbers of baryons and antibaryons. Then a problem arises why there is an imbalance in baryonic matter and antibaryonic matter in the observable universe.”

What is unacceptable in this paragraph? The conservation of the baryon number in all modern particle theories is a well-known fact. The fact that modern cosmological theories state that the universe has been created with equal numbers of baryons and antibaryons is known to all quantum cosmologists. Ref. [1] contains several references where the BAU problem is discussed, and the title of Ref. [1] contains the words “Baryon Asymmetry of the Universe”. My manuscript is not a review of particle and cosmological theories, and the only purpose of the first paragraph is to mention facts known to all quantum cosmologists (and even the abbreviation BAU is well-known to them). Those facts are described even in Wikipedia in an article titled “Baryon Asymmetry”. So, any physicist interested in the BAU problem can easily find a vast literature on this problem.

The second Dr. Fayyazuddin’s objection is: “It is also not clear whether the (unspecified) purported theory satisfies the extensive tests that the Standard Model has passed over the last several decades”.

Standard Model is a successful model, but it is only a model based on Poincare symmetry. However, quantum theories describing early stages of the universe cannot be based on Poincare symmetry. As I note in the introduction, in his famous paper “Missed Opportunities”, Dyson explains that de Sitter symmetry is more general (fundamental) than Poincare one. As shown in my publications (e.g., in paper [3] in Physical Review D), the latter is a special degenerate case of the former in the formal limit R→∞ where R is the parameter of contraction from the de Sitter algebra to the Poincare one, and (as shown e.g., in section 2 of Ref. [3]) in semiclassical approximation R coincides with the radius of de Sitter space in General Relativity. Since now this radius is very large, Poincare symmetry works with a high accuracy. However, at early stages of the universe this parameter cannot be large, and Poincare symmetry cannot work at those stages.

So, Dr. Fayyazuddin’s argument with Standard Model is inadequate in the context of my work. Moreover, in view of this argument, all physicists working on de Sitter quantum theories should justify their results by investigating their agreement with Standard Model, but this is not consistent.

Let me also comment Dr. Fayyazuddin’s terminology where he talks about my theory only with an adjective “purported” and says: “you provide no details or even a formulation of the theory that you purport to exist”.

For explaining the BAU problem I do not need any theory describing specific interactions (e.g., QED, QCD and electroweak theory). I need only properties of irreducible representations (IRs) of the de Sitter algebra. Those properties are described in detail in sections 2 and 3. Dr. Fayyazuddin does not explicitly mention those sections, and so a question arises whether he carefully read them.

Let me now briefly describe why I think that the results of the manuscript are fundamental.

The notions of particle-antiparticle, baryon number and its conservation arise because the energy in IRs describing particles in Poincare invariant theories can be either strictly positive or strictly negative. The corresponding IRs are associated either with particles or with antiparticles. However, this is not the case for more general (fundamental) IRs of the de Sitter algebra. I note that one IR of the de Sitter algebra contains both, positive and negative energies. When symmetry is broken such that de Sitter symmetry becomes Poincare one then one IR for the former splits into two IRs for the latter with positive and negative energies. So, the very notions of particle-antiparticle, baryon number and its conservation arise as a result of symmetry breaking from a more general symmetry to a less general one. Since now the value of R is very large, Poincare symmetry works with a high accuracy, and those notions have a physical meaning with a high accuracy. However, they do not have a physical meaning at early stages of the universe. So, standard statements that the universe has been created with equal numbers of baryons and antibaryons do not have a physical meaning.

As I note in the manuscript, the Dyson paper appeared in 1972, and, in view of Dyson’s results, a question arises why modern particle theories (e.g., QED, QCD and the electroweak theory) are still based on Poincare symmetry and not de Sitter symmetry. I think that the problem of constructing particle theory based on de Sitter symmetry is one the most fundamental problems of quantum theory. Probably, many particle physicists think that since now R is much greater than sizes of elementary particles, then there is no need to construct such a theory. This argument is not consistent because usually more general theories shed a new light on standard concepts. As noted above, the very notions of particle-antiparticle, baryon number and its conservation arise as a result of symmetry breaking from de Sitter symmetry to Poincare one. So, in de Sitter quantum theory those notions will be replaced by fundamentally new ones. The fact that such a theory does not yet exist does not mean that investigation of de