Curved Momentum Space, Locality, and Generalized Space-Time

Round 1

Reviewer 1 Report

The paper discusses some aspects of relativistic deformed kinematics (RDK) in the context of k-deformed DSR formalism which is based on the Hopf-algebraic scheme. The first part reviews an idea of RDK  its relationship with the geometry of curved momentum space. Aspects of non-locality and the recipes for how to cure them are also discussed. Finally, the comparison between RDK and DSR schemes are presented and some analogies established. The paper is clearly written and deserves publication.

Author Response

We thank the referee for his/her positive remarks about our manuscript. Although the report says that "the paper is clearly written", it also indicates that the presentation of the results "can be improved". We have made a few changes in the text to improve its readibility (see accompanying PDF file). 

Author Response File: Author Response.pdf

Reviewer 2 Report

Unfortunately, I cannot recommend the paper for publication.

 The authors deal with the relation between the geometry of the momentum space in relativistic deformed kinematics and the loss of absolute locality (a.k.a. relative locality). My impression is that the paper, while containing many formulae (which quite often are the straightforward consequences of the assumptions adopted) does not provide sufficient arguments in favour of possible physical relevance. Moreover, the formal consequences of basic assumptions do not seem to be fully developed. Let me give a few examples:

1. the Lorentz transformations in momentum space, eq.(2), are assumed to preserve the origin (this is explicitly stated before eq.(17)). Were the Lorentz group compact we could safely assume that they linearize in appropriate momentum parametrization. For noncompact group this cannot be taken for granted. However, the classification of such Lorentz actions up to a redefinition of momentum variables, is for sure possible;
2. I would expect that the second function (4) obey some kind of cocycle condition resulting from the composition of Lorentz group elements. Such conditions are usually useful in restricting the form of functions we are attempting to describe;
3. It is assumed that the composition law for momenta is associative; I am wondering if this assumption, together with eqs. (4) and (5), implies some further relations;
4.  Eq.(9), already in the global form, follows immediately from (5) by puting p=0;
5. Sec. 3 seems to contain a piece of standard geometry of symmetric spaces. In fact, the example considered there is related to kappa-Poincare group. The relevant Hopf subalgebra generated by the momenta is commutative but not cocommutative. Therefore, it can be viewed as classical (nonabelian but solvable) Lie group and the relevant differential geometry on it may be easily described;
6. The space-time coordinates are introduced as if in passing (before eq.(28)) and their transformation properties are hardly described. The meaning of space-time remains obscure (at least for me).                                                                                                                                        The additional disadvantage seems to be that the only example the author consider is related, as far as I can see, to the kappa-Poincare.                                                                                                                         Concluding, I do not consider the paper to be suitable for publication in Universe. Sorry!

Author Response

We thank the referee for his/her report about our work. His/her comments and criticisms, which we try to answer below, has helped us to improve sections 2 and 3 of our manuscript, which were really a systematic review of previously published results in two Physical Review D articles. Since no objections were made to sections 4 and 5, which contain the main results of the present work, we hope the paper can be published, once the referee's comments have been taken into consideration.

0) The referee has the impression "that the paper, while containing many formulae (which quite often are the straightforward consequences of the assumptions adopted) does not provide sufficient arguments in favour of possible physical relevance."

Answer: The main result of our work is the identification of a relation between the geometry of a curved momentum space, the loss of absolute locality, and a relativistic deformed kinematics, through the identification of a common ingredient, a deformed composition law of momenta. We agree that the physical relevance of this result is an open question. While the relationship between elements that are usually considered in different approaches to quantum gravity is intriguing, the search of a physical application of this result goes beyond the present paper, which just offers a first step in this direction. 

1) "the Lorentz transformations in momentum space, eq.(2), are assumed to preserve the origin (this is explicitly stated before eq.(17)). Were the Lorentz group compact we could safely assume that they linearize in appropriate momentum parametrization. For noncompact group this cannot be taken for granted. However, the classification of such Lorentz actions up to a redefinition of momentum variables, is for sure possible;"

Answer: We have checked explicitely that the Lorentz transformations obtained as the isometries  of the metric in a maximally symetric momentum space preserving the origin can be reduced to a linear transformation by a nonlinear change of momentum variables. 

2) "I would expect that the second function (4) obey some kind of cocycle condition resulting from the composition of Lorentz group elements. Such conditions are usually useful in restricting the form of functions we are attempting to describe;"

Answer: We have checked that the explicit form of the second function in (4) derived from the geometry of momentum space (see (31)) defines a representation of the Lorentz group through (3)-(4). As the referee points out, it  obeys a cocycle condition; however, the geometric procedure gives a way of finding this representation without the use of that algebraic condition.

3) "It is assumed that the composition law for momenta is associative; I am wondering if this assumption, together with eqs. (4) and (5), implies some further relations;"

Answer: Eqs. (4) and (5) only involve two momenta. The construction reviewed in section 3.1 allows one to solve eqs. (4)-(5) for all composition laws. However, the referee is right that associativity implies a consequence: the obtained relativistic kinematics is kappa-Poincaré, as is explicitely shown in section 3.2.

4) " Eq.(9), already in the global form, follows immediately from (5) by puting p=0;"

Answer: This is true. Putting p=0 (and then p'=0) in (5), one has $\bar{q}=q'$, which is just what Eq. (9) means. In the manuscript we had mentioned that it is also a consequence of Eq. (8), as it should be since Eq. (8) is derived from (5) by considering an infinitesimal transformation.

From the comment of the referee, we see it was not completely clear, so we have added a new sentence after Eq. (9) to say it explicitely.

5) "Sec. 3 seems to contain a piece of standard geometry of symmetric spaces. In fact, the example considered there is related to kappa-Poincare group. The relevant Hopf subalgebra generated by the momenta is commutative but not cocommutative. Therefore, it can be viewed as classical (nonabelian but solvable) Lie group and the relevant differential geometry on it may be easily described;"

Answer: One of the aims of this paper is to show that, assuming a relation between some of the ingredients in the geometry of symmetric spaces and the ingredients in a deformation of special relativistic kinematics compatible with the relativity principle, one can derive the explicit form of the deformed Lorentz transformations. A description of the differential geometry is beyond the scope of the present paper.

On the other hand, while the first part of section 3 is standard geometry, as the referee indicates, the crucial point comes in section 3.1, where it is shown how to use geometric properties to solve the problem of finding a relativistic deformed kinematics from a modified composition law, in the terms given in section 2.

6) "The space-time coordinates are introduced as if in passing (before eq.(28)) and their transformation properties are hardly described. The meaning of space-time remains obscure (at least for me)."

Answer: The space-time coordinates in (28) are just the generators of transformations with parameters $a_\mu$ in momentum space $p_\mu \to p_\mu + a_\mu$ as in the case of special relativity. A consequence of the deformation of SR is that these coordinates do not have the standard linear transformation of SR but their transformation depends on the momentum. But the loss of locality of interactions in this coordinates leads to look for other alternatives for the physical spacetime. This is the reason why we consider the possibility to find another choice of space-time coordinates where the interactions are local as a better candidate for the physical spacetime.

Motivated by the comment of the referee, we have added a comment on the introduction of the coordinates $x^\mu$ as generators of transformations in momentum space before Eq. (28), and then a comment just before section 3.1 on the transformations properties of these coordinates and their relation with the physical spacetime.

7) "The additional disadvantage seems to be that the only example the author consider is related, as far as I can see, to the kappa-Poincare."

Answer: The complementarity between the geometric, locality and algebraic approaches, leads indeed to identify kappa-Poincaré as a very special example. This is not necessarily a disadvantage, since one could turn this argument around and take the simplicity of the relation between the geometry of a symmetric space and kappa-Poincaré kinematics as an indication that this deformation could play an special role in physics. 

We include the new version of the manuscript with the changes highlighted.

Author Response File: Author Response.pdf

Reviewer 3 Report

The present paper deals with the issue of deformed Poincaré algebra and its related kinematics from the algebraic and curved momentum space points of view. The issue of locality of interactions from the Relative Locality approach is discussed leading to the conclusion that if the deformed composition of momenta is associative, then it is possible to define spacetime coordinates in which the scattering between two particles is local.

The paper succeeds in demonstrating links between these different features of the general problem of deformed kinematics in quantum gravity phenomenology, therefore the paper deserves to be published.

 

Nevertheless, I would like to point out some issues that could improve the presentation of ideas. 

 

1. In the Introduction (Line 32), the authors refer to Deformed Lorentz Transformations (DLT) as an ingredient necessary to preserve the relativity principle. As a matter of fact, I suggest that the authors should highlight that what one should have in fact is a deformed Poincaré symmetry, since as we see along the paper, deformed translation transformations also play a significantly important role in the construction of a relativistic scenario.

 

2. I think Eq.(58) should be calculated at parameter $\tau=0$, since it is derived from the locality condition. If this is indeed the case, this information should be specified in the manuscript. Or if $\tau=0$ is an arbitrary point in the parameter space, which implies that one can write condition (58) for any $\tau$, it should be specified. In any case, I think this issue should be further discussed.

 

3. What’s the impact of the conclusion: “any associative composition law is compatible with locality” on the different bases of k-Poincaré, like the bicrossproduct and Snyder ones?

Author Response

We thank the referee for his/her positive remarks about our manuscript. We answer here to the suggestions to improve the presentation.

1) "In the Introduction (Line 32), the authors refer to Deformed Lorentz Transformations (DLT) as an ingredient necessary to preserve the relativity principle. As a matter of fact, I suggest that the authors should highlight that what one should have in fact is a deformed Poincaré symmetry, since as we see along the paper, deformed translation transformations also play a significantly important role in the construction of a relativistic scenario."

Answer: When we refer to DLT we mean that the linear Lorentz transformations of SR in momentum space become nonlinear transformations depending on a new energy scale $\Lambda$ such that in the limit $(p/\Lambda)\to 0$ reduce to the linear LT in SR. In this sense one can identify the scale $\Lambda$ as a deformation parameter in the LT. One can also associate the nonlinear infinitesimal transformations  of momenta with a deformed Poincaré algebra by identifying momenta as generators of translations in spacetime. Finally, one can see the transformations generated by momenta, on the new spacetime coordinates in which the interactions are local, as deformed translations transformations. 

We have made explicit in line 32 that we have DLT "of the momenta" to clarify this point. 

2) "I think Eq.(58) should be calculated at parameter $\tau=0$, since it is derived from the locality condition. If this is indeed the case, this information should be specified in the manuscript. Or if $\tau=0$ is an arbitrary point in the parameter space, which implies that one can write condition (58) for any $\tau$, it should be specified. In any case, I think this issue should be further discussed."

Answer: Eq. (58) involves only momenta. These momenta refer to the momenta of the two particles either before the interaction or after the interaction. But those momenta do not depend on $\tau$, as one can see from the equations of motion derived from the variational principle applied to the action (54). All the dependence on $\tau$ is in the space-time coordinates.  We have clarified this in a comment before Eq. (55), and also eliminated the "(0)" which was present in Eq. (55).

3) "What’s the impact of the conclusion: “any associative composition law is compatible with locality” on the different bases of k-Poincaré, like the bicrossproduct and Snyder ones?"

Answer: The composition law of k-Poincaré is associative in any basis, including the bicrossproduct basis. In the case of a relativistic kinematics obtained from a noncommutative Snyder spacetime, the composition is not associative and then the simple relations between the geometric, algebraic and locality perpsectives do not apply. 

We include the new version of the manuscript with the changes highlighted.

 

Author Response File: Author Response.pdf

Reviewer 4 Report

This paper addresses an interesting questions  - the application of RDK to derive a relation between the geometry of momentum space and generalised spacetime - the treatment is mathematically naive and therefore so vague that it does not add to the existing literature. For the kind of general approach taken in this paper to lead to new insights, the mathematical assumptions need to be very clear. This is not the case here,  on several counts:

• Momentum space is only assumed to be a manifold with a `maximally symmetric' metric, but much of the discussion uses coordinates for points on that manifold. Are they assumed to exist globally? The authors refer to an `origin' of momentum space, but manifolds do not have origins. What structures are assumed here?
• The authors always refer to `representations' of Lorentz transformations but the spaces they act on are not linear - so presumably they mean actions on manifolds. Is the orbit structure under these actions the basis for the above `origin' (the unique orbit consisting of one point only?)
• At the top of page 6 the authors suddenly introduce conjugate coordinates and assume Poisson brackets between them and their moment coordinates. Where are these position coordinates defined? On the momentum manifold or a different manifold? What is the origin of the symplectic or Poisson structure which is suddenly assumed here? The geometry of these `conjugate variables' and its relation to the momentum space metric is a key point of the project, and simply assuming it undermines the entire enterprise. 

Author Response

We thank the referee for his/her report, which considers that our work addresses interesting questions, but also raises some criticisms concerning our mathematical assumptions. We try to answer here these criticisms:

1) " the treatment is mathematically naive"

Answer: Indeed it is. The proposal of the work is to assume some simple relations between ingredients in the (partial) description of a maximally symmetric space and those defining a deformation of SR kinematics compatible with the relativity principle. These relations allow one to use simple properties of the geometric perspective to solve a complicated algebraic problem in the kinematic perspective.

2)  "the mathematical assumptions need to be very clear.

Answer: The matematical assumptions are: 1) The states of a particle defined by its energy and momenta can be identified with a subset of points of a manifold. The four coordinates of each point in this subset are just the energy and the momentum of the particle. 2) The manifold is a maximally symmetric space. 3) The six parameter group of isometries of the maximally symmetric space (leaving one point invariant) acting on the coordinates of the points of the subset in correspondence with the states of a particle define the action of the Lorentz group on the energy and momentum of a particle. 4) The remaining four-parameter dependent isometries define the the total energy and momentum of a system of two particles identifying the energy and momentum  of one of the particles with the coordinates of the point on which the isometry acts and the energy and momentum of the second particle with the parameters of the isometry.

These assumptions are made to give a geometric interpretation of a relativistic deformed kinematics, which is defined by a deformed composition law.  Assumptions 2-4 are summarized in the text around Eq. (88). We have made some stylistic changes in section 5 to appreciate better the different approaches that are trying to be related. We have also included assumption 1 in a paragraph at the beginning of section 3.

3) "Momentum space is only assumed to be a manifold with a 'maximally symmetric' metric, but much of the discussion uses coordinates for points on that manifold. Are they assumed to exist globally?"

Answer: We use the expression of the tetrad in some coordinates. Obviously these coordinates can not exist globally. They apply only on the subset of the manifold which is in correspondence with the states of a particle. 

4) "The authors refer to an 'origin' of momentum space, but manifolds do not have origins. What structures are assumed here?"

Answer: Coordinates of points in a subset of a manifold do have origins. According to assumption 1) the origin is in correspondence with the state of a massless particle when its momentum tends to zero. Obviously, the specific point of the manifold which is coordinated as the origin is arbitrary.

We have included some sentences trying to clarify this point raised by the referee at the beginning of secton 3.

5) "The authors always refer to `representations' of Lorentz transformations but the spaces they act on are not linear - so presumably they mean actions on manifolds. Is the orbit structure under these actions the basis for the above `origin' (the unique orbit consisting of one point only?)"

Answer: Yes, it is, see assumption 3) above.

6) "At the top of page 6 the authors suddenly introduce conjugate coordinates and assume Poisson brackets between them and their moment coordinates. Where are these position coordinates defined? On the momentum manifold or a different manifold? What is the origin of the symplectic or Poisson structure which is suddenly assumed here? The geometry of these `conjugate variables' and its relation to the momentum space metric is a key point of the project, and simply assuming it undermines the entire enterprise."

Answer: The Poisson brackets are a formal way to define a continuous group of transformations in the manifold, by identifying the infinitesimal transformations of the coordinates of a point in the manifold with the parameter of the transformation multiplied by the Poisson bracket of the "conjugate coordinates" and the momenta. These "position coordinates" are the generators of these transformations.

In the present work, the geometric perspective is restricted to momentum space. An extension of the geometric perspective to a phase space have been considered by one of the authors of this paper in Phys. Rev. D 101, 064062 (2020).

To clarify the introduction of these (auxiliary) space-time coordinates, we have added a comment of their role as generators of transformations in momentum space before Eq. (28), and then a comment just before section 3.1 on their relation with the physical spacetime.

We include the new version of the manuscript with the changes highlighted.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Unfortunately, in spite of some improvements I don't consider the paper to be suitable for publication in Universe. For example, is the case of kappa-Poincare a very special example or the only possibility? Also the notion of coordinates remains obscure.  The explanations the authors have added are hardly entightening.

Author Response

We thank the referee for his/her report. It helped us to introduce new corrections and add-ons that we think clarify the content of our paper.  

1-"Is the case of kappa-Poincare a very special example or the only possibility?" 

Answer: In this work we try to combine two different perspectives related with a deformed relativistic kinematics. On one hand, there are several papers in the literature showing a connection between a curved momentum space and a deformed relativistic kinematics. In our specific proposal, presented in Sec. 3,  we are able to reproduce several deformed kinematics from our geometrical perspective: kappa-Poincaré, Snyder, and hybrid models. 

On the other hand, as we summarize at the beginning of Sec. 4, there is also a clear connection between a deformed composition law and a noncommutativity of spacetime (well known fact in the Hopf algebra scheme). In our proposal of implementing locality of interactions, we ask the noncommutative coordinates to be defined as a sum of two terms, each one having only the phase-space coordinates of one of the particles (see Eq.(56)). The condition Eq.(69) related to the associativity properties of the composition law makes kappa-Poincaré indeed the only possible kinematics within those obtained from our geometrical approach that is compatible with this particular implementation of locality.

2-" Also the notion of coordinates remains obscure. The explanations the authors have added are hardly entightening ...The space-time coordinates are introduced as if in passing (before eq.(28)) and their transformation properties are hardly described. The meaning of space-time remains obscure (at least for me)."

Answer: The space-time coordinates are introduced before Eq.(28) as canonically conjugated variables to the momenta. This construction was used to describe the generators of translations and Lorentz isometries. This amounts to write these generators as 

T^\alpha=\mathcal{T}^\alpha_\mu\frac{\partial}{\partial k_\mu}, 
J^{\alpha\beta}=\mathcal{J}^{\alpha\beta}_\mu\frac{\partial}{\partial k_\mu}, 

regarding them as the generators of transformations preserving the distance defined by the momentum metric.

Summary of changes:

We have made an extensive revision of the text incorporating the issues pointed out by the referee. In particular, associativity is now not introduced from the starting of the paper, since the discussions of Sec. 2 and 3 (before 3.2) are indeed valid for non-associative kinematics, such as Snyder or hybrid models. In section 3.2 we consider the specific case of an associative kinematics, which leads to kappa-Poincaré. The introduction of the (auxiliary) space-time coordinates is now better described. The conclusions have also been rewritten with a better clarification. The changes with respect to the previous (number 2) version are remarked in the accompanying PDF file.

 

Author Response File: Author Response.pdf

Reviewer 4 Report

I thank the authors for the response to my comments, but those replies and the (minor) changes to the manuscript do not address my main point: the papers uses mathematical terms like `manifold', `Poisson brackets' etc  superficially, and without incorporating their full and rather precise meaning. Examples, already mentioned, is the use of coordinates on the momentum manifold  to define physical states (not coordinate independent, and therefore not well-defined on the manifold) or the assumption that transformations are generated by Poisson brackets (not all flows on a manifold are generated by Poisson brackets, and if Poisson brackets are part of the story the should be defined in a coordinate-independent way). The imprecise use of precise terminology in this paper creates confusion and muddies the  discussion.  I therefore remain of the view that this paper should not be published. 

Author Response

We thank the referee for his/her report. It helped us to introduce new corrections and add-ons that we think clarify the content of our paper.  

1- "The use of coordinates on the momentum manifold  to define physical states (not coordinate independent, and therefore not well-defined on the manifold)..."

We agree that the physical states we define are coordinate dependent. However, it is important to note that this goes in the same spirit of DSR theories: different
bases could represent different physics. In fact, there is a current debate in the literature about this fact. The possible "physical basis" in which physical states should be described, or a possible definition of a coordinate independent physical state, goes beyond the scope of this paper. 

We have added an explanation on this point at the beginning of Section 3 in the revised version of the manuscript.

2- "The assumption that transformations are generated by Poisson brackets (not all flows on a manifold are generated by Poisson brackets, and if Poisson brackets are part of the story the should be defined in a coordinate-independent way)" 

The isometries can be formulated in a coordinate independent way, as the referee points out. This invariance is manifest when one writes the generators of the isometries as 

T^\alpha=\mathcal{T}^\alpha_\mu\frac{\partial}{\partial k_\mu}, 
J^{\alpha\beta}=\mathcal{J}^{\alpha\beta}_\mu\frac{\partial}{\partial k_\mu},

in terms of the coordinates (k_\mu) on each patch of the manifold.  

We have used an alternative (equivalent) notation based on the introduction of canonical Poisson brackets and canonical conjugated (spacetime) coordinates just to make clear the relation between the geometric perspective and the perspective based on the locality of particle interactions.  

This has been explained after Eq. (29) in the new version of the manuscript.

Apart from these changes, we have modified other parts of the text, including a rewriting of the conclusions, to try to give an answer to the need for clarifications expressed by the referees. The changes with respect to the previous (number 2) version are remarked in the accompanying PDF file.

Author Response File: Author Response.pdf

Round 3

Reviewer 2 Report

I am still not very satisfied with the modifcations made by the authors. In the final paragraph of the paper they claimed that their main result is "... a relation between the geometry of a curved momentum space, the loss of absolute locality, and a relativistic deformed kinematics, through the identification of of a common ingredient, a deformed composition law of momenta".

However, to speak about locality one should have a clear notion of space-time coordinates which, in my opinion still remains obscure in the paper. Moreover, as I have mentioned previously, the momentum composition law defines solvable Lie group (in the classical sense) so its geometry can be easily derived from standard theory.

The only point which has been, to some extent, clarified is the relation to kappa-Poincare. Since the latter is still fairly popular I think that the paper could be published. However, I am far from being enthusiastic.

Reviewer 4 Report

The authors' replies confirm my  main points: to work on a manifold and then assign the different coordinate systems `different physics'  is a contradiction. If this really has become standard practice in papers on DSR then this underlines the danger of publishing papers like the present manuscript. On a manifold different coordinates have, by definition, no intrinsic meaning. Similarly to work on a manifold with metric and to invoke `canonical Poisson brackets'  makes no sense. There are no `canonical brackets' in that context.