Solving Particle–Antiparticle and Cosmological Constant Problems

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The author during this chapter concentrates in some open issues in Cosmology, such as particle-antiparticle problem, the cosmological constant problem, etc. In the text, there is a "transition" between quantum theory and classical approximation. I think that the author describes satisfactory the current situation. This work could be accepted by the journal. 

Author Response

I am grateful to Reviewer1 for his/her report. 

Reviewer 2 Report

Comments and Suggestions for Authors

 

Dear Editor,

I recommend that this paper will be reconsidered for publication after some revisions would be considered by the author.

The author proposes that our current quantum field theories of the fundamental building blocks of Nature and their interactions, built in terms of irreducible representations of the Poincare group defined by eq.(1.1), should be rewritten in terms of irreducible representations of the deSitter or anti-deSitter groups defined by eq.(1.2), from which the Poincare group can be obtained as a limit case when a certain parameter R goes to infinity. The author argues that this development might help to solve some outstanding issues in fundamental high energy physics and cosmology.

I find his proposal interesting, but there is one technical issue that maybe needs further clarification. If I understand his arguments correctly, the author explains that the actual value of the parameter R, being dimensional, should not play any role in the actual predictions of the theory. Hence, it seems that unless there is some other fundamental scale R_0 with respect to which the limit R/R_0 >> 1 could be properly defined, the limit in which the dS/AdS groups approach the Poincare group is singular in the sense described by M.Berry in "Asymtotics, singularities and the reduction of theories", Studies in Logic and the Foundations of Mathematics, 134 (1995) 597-607.

I think that only after identifying/justifying such a fundamental scale R_0, one may have an actual alternative physical theory that could make experimentally testable predictions.   

Besides that, I generally agree with many of the arguments and motivations raised by the author regarding the current status of our understanding of fundamental physics. In particular, I strongly agree with his observation that " ... when a new phenomenon is discovered, physicists should try to first explain it proceeding from the existing science. Only if all such efforts fail, something exotic can be involved.", which has certainly has not been the case regarding the introduction of now commonly used concepts like dark matter, dark energy or even inflation.

Author Response

I am very grateful to Reviewer 2 for his/her report.

In particular, I am grateful for the remark that “I strongly agree with his observation that " ... when a new phenomenon is discovered, physicists should try to first explain it proceeding from the existing science. Only if all such efforts fail, something exotic can be involved.", which has certainly has not been the case regarding the introduction of now commonly used concepts like dark matter, dark energy or even inflation.”

For example, I think that the situation with dark energy is contrary to basic scientific principles. Many publications note that the physical nature of dark energy is a mystery and that there are an almost endless number of explanations for dark energy. However, many papers on dark energy are published in so-called prestigious journals and some of these publications are awarded prestigious awards. At the same time, my experience shows that attempts to explain cosmological acceleration within the framework of the existing theory are rejected by these journals, even without explanation and without scientific reviews. I am grateful to the editors of Axioms for the fact that my paper was considered based on the principles of scientific ethics.

The meaning of the Reviewer’s sentence “If I understand his arguments correctly, the author explains that the actual value of the parameter R, being dimensional, should not play any role in the actual predictions of the theory.” is not quite clear to me. Indeed, I note that fundamental dS and AdS quantum theories do not involve dimensional parameters. However, these theories are quite complex and in them it is often very difficult to carry out calculations that can be compared with experiments. Therefore, in many cases it is desirable to find good approximations in which calculations can be made. Now we live in a world in which Poincare symmetry is satisfied with great accuracy because the parameter R is very large. Therefore, often the effects specific to dS symmetry are small and they can be calculated in the approximation when R is large. Therefore, the value of R is important because, if, for example, R is small, then the Poincare approximation does not work and only an exact calculation in dS theory is necessary.

Then Reviewer 2 writes: “Hence, it seems that unless there is some other fundamental scale R_0 with respect to which the limit R/R_0 >> 1 could be properly defined, the limit in which the dS/AdS groups approach the Poincare group is singular in the sense described by M.Berry…”

In his paper, Berry considers 6 cases, in particular transition from relativistic to nonrelativistic theory and transition from quantum to classical theory. He describes the transitions in terms of the parameter δ which is the series in v/c in the first case and ћ/S in the second one. However, the first case is considered only in non-quantum theory and nothing specific is said about the implementation of the second case.

As discussed in my paper, symmetry at the quantum level is defined by commutation relations in the symmetry algebra. So, one has to investigate transitions from the more general algebra to the less general one. For example, Poincare invariance is defined by the Poincare algebra and Galilei invariance – by the Galilei algebra. As I note in the revised version of the paper, in agreement with Berry, the transition from Poincare invariance to Galilei invariance at c→∞ involves singularities and in general, Poincare algebra does not become the Galilei algebra if the actions of those algebras are considered on all elements of the Hilbert space. The very meaning of the nonrelativistic approximation is that, in the process under consideration, only those elements of the Hilbert space are important for which singularities can be resolved and the transition from the Poincare to the Galilei can be implemented. The fact that the Galilei algebra is simpler than the Poincare algebra follows from the fact that the former contains a greater number of zero commutators than the latter.

I think that this is the most general description of the transition from Poincare symmetry to Galilei symmetry and in each concrete case it should be investigated whether the transition can be implemented. Analogous remarks can be made about other transitions from more general algebras to less general ones.

Reviewer 3 Report

Comments and Suggestions for Authors

Please, find my report in the file attached.

Comments for author File: Comments.pdf

Author Response

I am very grateful to Reviewer 3 for his/her report.

In particular, I am grateful for the Reviewer’s remark that “The presence of antigravity without considering a background geometry from the dS algebra in quantum theory in the semi-classical approximation impressed me particularly.”

When I discussed my papers with physicists, I was surprised by the following. In the abstract and in the main text, I note that I do not use dS space, but proceed from representations of the dS algebra because, as noted by Dyson, this algebra is more general than the Poincare algebra. However, even very qualified physicists treated this such that implicitly I still use dS space with some curvature Λ and therefore I already have the cosmological repulsion in advance. So, those physicists think that representations of the dS algebra imply that dS space is used. This is the first time in my practice when a reviewer explicitly noticed that the dS algebra does not necessarily imply that a background geometry is used.

Following the item 2 in the report, I added in section 3.2 the description of Lundmark's and Lemaître’s results.

The item 1 in the report contains a criticism of my statement “Therefore, our approach gives a clear explanation why Λ is as is.” Reviewer 3 writes: “However, the issue with the cosmological constant is not this. The real issue is why is Λ ∼ 10^-122 ћG/c³ with G the Newton constant of gravity? The author has not solved this problem and I demand that this is recognized along the manuscript.

Apparently, here there is a typo in the relationship between Λ and G. For example, in units c= ћ=1, Λ has the dimension m^-2, and G has the dimension m², so Λ should be inversely proportional to G, and not directly proportional, right? In this units, Λ≈10^-122/G. For now, this equation only says that the experimental values of Λ and G are such that Λ≈10^-122/G and does not say that this relation follows from some theory. In my paper I explain why Λ is as is, and, if one wants to compare Λ and G then Λ≈10^-122/G. So, I have solved the problem why Λ≈10^-122/G. But my understanding of this Reviewer’s statement is that he/she does not accept my explanation and believes that some theory should derive analytically that Λ≈10^-122/G, right? However, Reviewer 3 does not explain why, in his/her opinion, my conclusion is not correct.

Below I will note that Λ does not depend on G, but first let me note the following. Originally, the relations (1.2) do not contain any dimensional quantities, i.e., one can say that those relations are written in units c= ћ=R=1. This is reasonable because the relations (1.2) are written in quantum theory while (kg,m,s) are taken from macroscopic physics. However, if physicists want to describe c in m/s, ћ in  kg·m²/s and R in meters then the relations (1.2) begin to depend on (c,ћ,R).

As I note in my paper, physicists usually understand that physics cannot (and should not) derive that c≈3·10⁸m/s and ћ≈1.054·10^-34 kg·m²/s. Those values are purely kinematical (i.e., they do not depend on gravity and other interactions) and are as are simply because people want to describe velocities in m/s and angular momenta in kg·m²/s. I explain that, at the quantum level, (c,ћ,R) are the contraction parameters for transitions from more general theories to less general ones. So, as explained in detail, at the level of the contraction parameters, R is fundamental to the same extent as c and ћ. At this level R has nothing to do with the radius of the background space. It is only the coefficient of proportionality between M^4μ and P^μ. It is purely kinematical (i.e., it does not depend on gravity and other interactions) and is as is simply because people want to describe distances in meters. 

My approach to the problem of cosmological acceleration is basically new because I describe this problem from the point of view of quantum theory. In view of Dyson’s observation that the dS algebra is more general than the Poincare one, I consider representations of the dS algebra. To the best of my knowledge, such an approach has not been considered in literature. As noted in Sec. 3.3, the results for cosmological acceleration in my approach and in General Relativity (GR) are given by the same expression (3.3) but the crucial difference between my approach and GR is as follows. While in GR, R is the radius of dS space and can be arbitrary, in my approach, R is defined uniquely because it is a parameter of contraction from the dS algebra to the Poincare one. Therefore, my approach indeed explains why Λ is as is.

However, a typical approach in the literature is that physics should analytically derive the relation between Λ and G and that the solution to the dark energy problem depends on the value of Λ. Physicists also believe that QFT of gravity should analytically confirm the experimental result that, in units c=ћ=1, Λ≈10^-122/G. This theory proceeds from Poincare symmetry and Minkowski space, i.e., here physicists do not consider Poincare symmetry as a special degenerate case of dS symmetry. QFT of gravity contains only one phenomenological parameter - the gravitational constant G, and Λ is defined by the vacuum expectation value of the energy-momentum tensor. The theory contains strong divergencies which cannot be eliminated because the theory is not renormalizable. The results can be made finite only with a choice of the cutoff parameter. Since G is the only parameter in the theory, the usual choice of the cutoff parameter in momentum space is ћ/l_P where l_P is the Plank length. Then, if c=ћ=1, G has the dimension length² and Λ is of the order of 1/G. This value is approximately 122 orders of magnitude greater than the experimental one, and this situation is called vacuum catastrophe.

In spite of the fact that, as explained in the literature (and in my paper), standard QFT of gravity contains fundamental inconsistencies, physicists usually believe that the inconsistences will be resolved and, instead of Λ≈1/G the theory will derive the correct result Λ≈10^-122/G.

However, as follows from the very problem statement about the cosmological acceleration, Λ  should not depend on G. Indeed, as noted in Sec. 3.1, in this problem it is assumed that the bodies are located at large (cosmological) distances from each other, and sizes of the bodies are much less than distances between them. Therefore, all interactions between the bodies (including gravitational ones) can be neglected and, from the formal point of view, the description of our system is the same as the description of N free spinless elementary particles.

Reviewer 3's comments about Λ≈10^-122/G were helpful in encouraging me to describe this issue in more detail in Sec. 3.4.

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

Dear Editor,

I cannot recommend that this paper will be accepted for publication in its present form. The author proposes a nice idea that in my opinion deserves being further explored and published once the author will make a good case to support it.  But this is not the case with the current version of the manuscript. 

Current fundamental physical theories are based on the principle that the laws of physics are covariant under Poincare transformations. Hence, such theories are naturally built in terms of linear representations of this group on a (locally) Minkowskian space-time. These representations naturally include particle/antiparticle pairs and, indeed, Lorentz covariance guarantees CPT invariance. 

The author notices that the Poincare algebra is a limit case of the dS and AdS algebras when certain dimensionless parameter R is taken to infinity, and he nicely suggests that the fundamental building blocks of Nature might actually be representations of these algebras. Unfortunately, the case that he builds to support this idea is very weak and even flawed.

In particular, the argumentations that he makes in the Introduction to his paper in order to motivate his idea are flawed. There are certainly many objections that can be made to the original Dirac equation, but these objections were already solved almost a hundred years ago through the formulation of the equation in terms of Grassmann algebras. Once this is done one can get a well-defined Lorentz covariant quantum field theory for which many of the author's argumentations in the Introduction to his paper are not relevant. 

Another point of concern, which I already noticed to the author in my previous report, and I still find unclear, refers to the role of the dimensionless parameter R.  Obviously, I agree with the author that there is no role for physics in trying to "explain" the value of the dimensional Nature's constants, like c or h, which can be set to 1 so that they define a fundamental set of units. However, the same cannot be said regarding the dimensionless constants of Nature (for example, the parameter alpha that describes the electromagnetic interaction) and, in particular, the dimensionless parameter R discussed by the author. Indeed, it is only through the value of this dimensionless parameter that the author's proposal might one day be falsified. Hence, I got completely confused by the author's argumentations about this issue: I do not understand what he argues for.

Along the same line, I would also criticize the author's claim that his proposal solves the problem of the origin of the cosmological baryon asymmetry. All existing experimental data is consistent with the statement that known particles and their antiparticles have exactly the same mass to a great degree of precision. The author's arguments prompt me to ask him:

a) Which mass difference between particles and their antiparticles would be needed to quantitatively explain the observed baryonic asymmetry of the Universe, n_B ~ 10^(-10) x n_s? 

b) Are these required mass differences consistent with the existing experimental data? 

c) If they are, could such differences be actually measured in present of future colliders?

Hence, I think this paper should be presented as a proposal for future research, rather than as an actual solution to current big questions because it does not contain any such answer. As a research proposal the paper might be interesting.   

Comments on the Quality of English Language

English is fine in most parts of the text, but some sentences and even paragraphs are confusing, maybe due to the English style. 

Author Response

The first Reviewer 2’s report seems, in general, favorable but they wrote that my paper had problems in view of the paper by Berry. My understanding of their recommendation was that when the problems are solved, my paper can be published. In response to this recommendation, I revised the paper and explained in detail that my results did not contradict the remarks by Berry. It was natural to expect that Reviewer 2 would somehow express their opinion about the changes I made in response to their report. Do they think my changes are correct, incorrect, partially correct, and so on? However, in the new round of reviews, Reviewer 2 did not express any opinion about this. But they raised three new problems stating that my paper has problems in the interpretation of the following issues: 1) the Dirac equation, 2) the interpretation of the value of R, 3) the difference in the masses of a particle and its antiparticle. These issues have already been discussed in the original version of the paper, and no changes have been made to the original discussion of 1)-3) in the new version. Therefore, if Reviewer 2 had objections to my discussions of 1)-3), then it is unclear why they did not make them in Round 1 but did them only now.

 

Now I will make this remark. The journal is called Axioms, and the paper was sent to the section Mathematical Physics. Therefore, it is probably natural to assume that the reviewer is familiar with at least the most basic concepts of the theory of representations of Lie algebras. However, as explained below, Reviewer 2 is not familiar with these concepts. But the main thing is not even this, but the fact that they make their comments in a categorical tone and, apparently, does not even allow for the possibility that they may not understand something. In addition, as explained below, Reviewer 2’s objections are formulated very unclearly and typically it is difficult to understand what the meaning of the objections are.

 

Revier 2 writes: “Unfortunately, the case that he builds to support this idea is very weak and even flawed.” These comments were made only in Round 2 but not in Round 1.  And Reviewer 2 does not recommend publications of my paper in the present form. But they write: “Hence, I think this paper should be presented as a proposal for future research, rather than as an actual solution to current big questions because it does not contain any such answer. As a research proposal the paper might be interesting.” Should I understand these words to mean that Reviewer 2 allows the publication of my paper only if I admit that the problems that I declared solved are in fact not solved? But below I explain that he/she does not understand some basic mathematical concepts, and in my paper all the stated problems have been solved.

 

Now I will discuss Reviewer 2’s objections on 1)-3).

• Dirac equation

Reviewer2 writes: “Hence, such theories are naturally built in terms of linear representations of this group on a (locally) Minkowskian space-time. These representations naturally include particle/antiparticle pairs and, indeed, Lorentz covariance guarantees CPT invariance.”
However, as explained in the paper, if we proceed only from representations of the Poincare algebra (Approach A), then there is no guaranty that the mass of a particle is equal to the mass of the corresponding antiparticle. CPT invariance arises only if we additionally assume that elementary particles are described by local covariant equations (Approach B) and in this assumption there is a problem that the solutions of such equations are not described in terms of well-defined self-adjoint operators.

Reviewer 2 also writes: “There are certainly many objections that can be made to the original Dirac equation, but these objections were already solved almost a hundred years ago through the formulation of the equation in terms of Grassmann algebras. Once this is done one can get a well-defined Lorentz covariant quantum field theory for which many of the author's argumentations in the Introduction to his paper are not relevant.”

Those phrases are pronounced without any specific explanations and references. It is not clear whether, in Reviewer 2’s opinion, all objections were already resolved or only part of them. For example, Pauli has pointed out to the problem that in the case of fields with an integer spin it is not possible to define a positive-definite charge operator while in the case of fields with a half-integer spin it is not possible to define a positive-definite energy operator, and this problem has not been solved. The phrase: “…through the formulation of the equation in terms of Grassmann algebras” is given without any references and so it is not clear what Reviewer 2 means. For example, Grassmann variables are used in the path integral formulation of the Dirac equation, but standard QFT does not involve path integrals. The phrase: “… many of the author's argumentations in the Introduction to his paper are not relevant” is given without explanation which argumentations are meant. It is contrary to scientific ethics when negative statements about the author's results are given without any explanation.

 

• Interpretation of R

 

Reviewer 2 writes: “The author notices that the Poincare algebra is a limit case of the dS and AdS algebras when certain dimensionless parameter R is taken to infinity.” This phrase shows that Reviewer 2 does not understand the meaning of the transition from the dS and AdS algebras  to the Poincaré algebra since such a transition can only be carried out if R is dimensional. I define R as a dimensional parameter, note several times that R is dimensional and note that at the present stage of the universe R is of the order of 10²⁶m. So, a question arises whether Reviewer 2 carefully read the paper. He/she writes that the meaning of dimensionless R differs from the meaning of dimensional quantities (c,ћ). However, I explain that R is fundamental to the same extent as (c,ћ) because (c,ћ,R) are the contraction parameters for contraction from the dS and AdS algebras to less general algebras (including the Poincare algebra). I note that Eq. (1.2) does not contain (c,ћ,R). This can be treated as a choice of units c=ћ=R=1 but in that case, there is no transition from dS and AdS algebras to less general algebras. Reviewer2 writes: “Hence, I got completely confused by the author's argumentations about this issue: I do not understand what he argues for.” I do not know what is not clear in my explanations because Reviewer 2 does not explicitly indicate what my arguments are formulated unclearly. If they are not clear to Reviewer 2, then the question arises whether he/she understands the meaning of contraction from one algebra to another.

• Barion asymmetry of the universe

Reviewer 2 writes: “The author's arguments prompt me to ask him:

1. a) Which mass difference between particles and their antiparticles would be needed to quantitatively explain the observed baryonic asymmetry of the Universe, n_B ~ 10^(-10) x n_s? 
2. b) Are these required mass differences consistent with the existing experimental data? 
3. c) If they are, could such differences be actually measured in present of future colliders?”

The meaning of the formula n_B ~ 10^(-10) x n_s is not explained. When Reviewer 2 talks about a particle and an antiparticle, then there must be two irreducible representations (IRs), and I explain that I am considering only one IR which splits into two only in the limit R→∞. Even when R is very large but finite, there is only one IR and the concept of particle-antiparticle is only approximate. In that case, all the elements of this IR have the same mass. Therefore, the question of the difference between the masses of a particle and its antiparticle does not arise.

 

Let me now make the following remark. I understand that the reviewers and most of the potential readers did not work on the problems discussed in my paper. Therefore, it is quite natural that many of them may have problems understanding my results. I carefully consider any comments from reviewers, including incorrect ones, because they may help me to understand why potential readers may have problems. In response to any comments, even incorrect ones, I try to improve the text to make it easier for readers to understand the paper.

 

In response to Reviewer 2's comments, I have revised the paper as follows. At the very beginning of the paper, I note that I solve the particle-antiparticle and cosmological constant problems proceeding from the conditions (H,O,S) while in the literature, when considering these problems, not all those conditions are met. Since the journal is called Axioms, and the paper was sent to the section Mathematical Physics, I ask Reviewer 2 to formulate objections not just in words, but in mathematical terms. Therefore, I propose the following. If Reviewer 2 still does not agree with my explanations, then I ask him/her to construct his/her answer as follows. First, I ask him/her to say whether he/she accepts the conditions (H,O,S). If he/she doesn’t, then I ask to explain why. And if he/she accepts, then I ask to formulate objections only proceeding from (H,O,S).

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

I would like to thank the author for his detailed reply to my previous referee report. I agree with the author that according to his point of view what he explains is correct. I recommend the publication of this manuscript in its present form in Axioms.

Author Response

I am grateful to Reviewer 3 for his/her recommendation to publish my paper.

Round 3

Reviewer 2 Report

Comments and Suggestions for Authors

Dear Editor,

I cannot recommend yet that this paper will be accepted for publication since the author has not addressed any of the comments that I raised in my previous report.

According to the experimental data collected by the particle data group (Prog.Theor.Exp.Phys.2022, 083C01) the mass difference between positrons and electrons is smaller than 10^(-8) of their average masses. Is this tiny mass difference consistent with the present value of the parameter as estimated by the author, R ~ 10^26 meter ?

Can this tiny mass difference quantitatively reproduce the observed baryonic asymmetry of the Universe, n_B ~ 10^(-10) n_gamma. If so, at which stage in the evolution of the Universe did the asymmetry develop according to the scenario proposed by the author?

If the author cannot answer these question then he cannot claim that he is solving either one of the problems that he claims to have solved, and he should instead write the paper as a proposal for others to take into consideration.

I further suggest that the author will revise the arguments that he brings in the Introduction to the paper. They are misleading, and unnecessary.  In particular, the discussion about the superposition of states is wrong. For example, the physical properties of positronium - the bound state of an electron and a positron - can be obtained within the framework of QED. The same can be said about the discussion on the spatial localization of physical observables: the author may wish to read  Mukhanov, Feldman and Branderberger, Physics Reports 215, 203 (1992), where a model to describe the expected spatial correlations of the primordial cosmological density inhomogeneities.  Also the discussion about the predictability of the actual value of the parameter R is also quite confusing.

 

Comments on the Quality of English Language

Some improvements in the quality of the English language might help the author to convey his idea.

Author Response

In the first part of his/her Round 3 report, Reviewer 2 writes about the electron-positron mass difference and asks whether “this tiny mass difference quantitatively reproduces the observed baryonic asymmetry of the Universe”. However, in the last section of my response to Round 2 of Reviewer 2’s report, I note: “Even when R is very large but finite, there is only one IR and the concept of particle-antiparticle is only approximate. In that case, all the elements of this IR have the same mass. Therefore, the question of the difference between the masses of a particle and its antiparticle does not arise.” To make this point even clearer, in the revised version of the paper, on page 20 in the paragraph which begins with: “Since the number of states…” I note that: “according to the Schur lemma, the operator W has only one eigenvalue in this IR and all states have the same mass μ.”

I explain that the baryon asymmetry of the universe arises not because a particle and its antiparticle have different masses but because the very concept of the baryon number is only approximate when R is finite. At the present stage of the universe, this concept works with a high accuracy, but it was not the case when R was much less than now. The fact that now R is of the order of 10²⁶m is not my estimation as Reviewer 2 writes but, as noted in the paper, is the experimental result described in [35].

Reviewer 2 writes: “If the author cannot answer these question…” (what does he/she mean “these questions” or “this question"?). But I answer all the required questions and Reviewer 2 does not explain why my statement about the zero-mass difference is not correct.

In the last paragraph of his/her Round 3 report, Reviewer 2 writes that the properties of positronium can be obtained within the framework of QED. But this is reasonable because at the present stage of the universe R is very large and so QED should work with a very high accuracy. As noted in the literature, the problem of primordial cosmological density inhomogeneities is essentially model dependent.

Finally, I note the following. Reviewer 2 writes that “the discussion about the superposition of states is wrong” and that “the discussion about the predictability of the actual value of the parameter R is also quite confusing”. I don’t know how it is possible to discuss scientific results if statements about their incorrectness are made without any explanation. Those problems are the main ones in my paper, and it is not clear to me why Reviewer 2 makes these remarks only in Round 3 of his/her report, why those remarks have not been made from the very beginning in Round 1.