Towards a Quantum Erlangen Program

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The author revisits a less-explored subject in quantum gravity (QG) known as quantum reference frame (QRF), suggesting an alternative to the most popular view of this notion, whereby quantum geometries may be defined via QRF transformations, at least for the Planckian space-time (toy) model the manuscript considered, pointing to a QG analog of the classical Erlangen programme. It is proposed that observers, embodied as physical (quantum) systems, should contribute to the total degrees of freedom (DOFs) of the system and frame transformations involve partial traces over observers' DOFs, in contrast to mainstream unitary approaches, such that in the proposed framework, there exist ``detectable'' yet unobservable DOFs. This approach offers interesting phenomenological opportunities, as it leads to some kind of generalized uncertainty relations that can be tested (or are already falsified(?)) by experiment. The paper's key contributions are a review of the existing literature on this topic and a new formulation of QRFs as advocated, which is an investigative window connected to the problem of QG.

The rigor of the analysis, careful evaluation of existing research all highlight its quality as a review article and it deserves, therefore, a strong consideration of publication in Universe. I do not have enough expertise to comment on the technical details of the intriguing redefinition of QRFs proposed in this work. For that aspect, I believe that the Editors have contacted another referee. While the paper addresses in a relatively clear manner an technically challenging theme, in my view, there are a few points—from both theoretical and phenomenological sides—that it is better to come with additional explanations.

• On page 4, lines 162–163, the author writes: ``By contrast, there is no way in which the position or momentum of a single quantum particle can be measured, so these are not observables.'' However, I would like to point out that in the well-established framework of nonrelativistic quantum mechanics (QM)—the most famous and fundamental application of the canonical QM discussed in the text—the position x and momentum p operators for a single (quantum) particle are both Hermitian, and thus certainly qualify as observables. Their eigenvalues can, in principle, be obtained in ideal measurements (for instance, position via a spot on a screen, or momentum operationally defined through diffraction on a crystal lattice). That said, this observability relies on the presence of an external reference frame—typically assuming a fixed background space, as in nonrelativistic QM.

  Of course, I fully understand the author's intent here: such quantities are just not observable within the system itself, while the relative displacement and relative momentum between two particles are better examples of observables—particularly in the contexts of QG or relational QM. Nevertheless, for the sake of readers who may not appreciate these subtleties, I still believe the wording in this passage (i.e., in lines 157–163) should be rewritten to avoid confusion or to be better comprehensible.

• In section 3 the manuscript shows that this approach also captures Planck-scale fuzziness in space-time, thereby establishing connections with (extended) generalized uncertainty principles (GUPs) that align with existing QG-inspired phenomenology. However, I am concerned that it is not clear whether the quantum geometry model proposed by the author can be tested or constrained using currently observed GUP-related or fuzzy space-time phenomenologies. A brief discussion of this point is essential in my opinion, especially because GUPs of the type re-derived in equation (9) within this naïve Planck-scale quantized space-time model are expected to yield testable QG corrections to various quantum phenomena, which is crucial for the experimental falsification of the underlying QRF approach.

• The present work argues, in section 4, that classical space-time symmetries can be ``mathematically preserved but operationally broken''—or ``smeared''—in quantum superpositions of geometries, resulting in the so-called smeared frame transformations (cf. page 10, lines 397–401). But what is the fate of Lorentz invariance in this framework? Within my area of expertise on this topic, this symmetry can, among other possibilities, be either explicitly broken (i.e., Lorentz violation) or deformed (Lorentz deformation) by effects of QG. Since GUP alters phase-space structure, naturally connecting with frameworks like deformed or doubly special relativity (DSR, in which Lorentz symmetry is preserved but in a deformed manner), the smeared symmetries discussed here appear to bear some similarities with such scenarios. Could the author briefly comment on the differences between the modified Lorentz transformations proposed here and those in DSR?

Other than that, while their presence did not seriously obstruct my understanding of the text, a fair number of typos should be rectified:

1. In line 50 on page 2, there is an incorrect cross-reference: in ``the statement (3) no longer holds'', where the statement ``(3)'' the author is referring to ought to be the ``(1)'', first appearing on the title page.
2. The second to the last word in line 92 on page 3, ``access it'' should be corrected as ``access to''. And in the following paragraph, in line 105 there is a missing period.
3. The third word ``the'' in line 132 on page 4 should be deleted, while in the next line the preposition ``to'' is missing for the phrase ``corresponding to''.
4. In line 277 on page 7, `` and it well known that'' should be `` and it is well known that''.
5. There is also a notational issue regarding the operator form of ``(k’ – k)'' used in section 4. On page 9, the generalized or, ``perturbed'', momentum operator, P, in line 358, appears to be identical with the one shown in line 355, hence, two redundant operator notation ^'s above ``k’'' and ``k'' in the second term ``(k’ – k)'' of the former expression should be removed. The same problem also occurs in line 315 on page 8.

For clearer illustrations, the author should consider redrawing or adapting the hand-drawn Figure 1 (which is directly taken from reference [21]) using TikZ or other tools, although this is optional.

Finally, a small suggestion on abbreviations: I would recommend against using abbreviations without expanding them in full the first time they appear, even for those seemingly well-known terms like ``quantum mechanics''. It is best to provide the full form on first use (e.g., ``QM'' in footnote 2), as not all readers will immediately know QM means that.

With that said, my recommendation is for Minor Revision. Once the author responds to the points being raised here, the paper can be considered for publication in Universe.

Author Response

Please see attached PDF.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

In this manuscript the author's aim is to study Quantum Reference Frame (QRF) as an extension to what is called Erlangen Program (EP) for classification of metric spacetimes. The goal is to see if it can be used in the context of a quantum model for gravity. The text is in a large extent a highlight of previous works of the authors.  

Sections 1 to 3.3 introduce the QRF, the EP and its extension to quantum system. In 
particular, they discuss the issue of observers as physical quantum systems, and how this fact may affect measurements and their interpretations. These sections are descriptive and globally well written and interesting. However, beginning from section 3.4 where the text becomes more technical, it also becomes more vague and misleading. Moreover, it includes 
misunderstandings, which eventually lead to wrong conclusions. In particular, relations in eq.(3), which are crucial for the formalism and conclusions of the manuscript are 
incorrectly used and interpreted. 

Specifically, equalities in eq.(3) are simplistic error propagation relations for random variables $x_A \pm x_B$ and $p_A \pm p_B$. They are general relations, and even in this respect not optimized, because the optimal uncertainty estimation for the above variables - from any statistic textbook - is: 

$(\Delta x_A)^2 (\Delta x_B)^2 / ((\Delta x_A)^2 + (\Delta x_B)^2)$ 

As for the application of these relations, which along with the fundamental Heisenberg uncertainty relation presented in line 204 lead to eq.(4) and other relations and conclusions presented in the manuscript, there is a clear misunderstanding. The line 195 indicates that only two subsystems are considered in this section. Thus, according to discussions of Sec.1-3, only one position measurable exists: $x_B-x_A$ (assuming A as reference). This means that without loss of generality the value of $x_A$ can be fixed, for instance $x_A = 0$. This is equivalent to assuming A in an eigen state of position operator. However, definition of the total wave function $\Psi_{AB}$ of two subsystems in line 195 depends on both $x_A$ and $x_B$. Thus, it implicitly assumes a third subsystem - the reference with respect to which $x_A$ and $X_B$ are measured. Therefore, the claim in lines 
202-203 that unobservable quantities can affect "detectable physical effects" and the content of footnote 4 are wrong. In addition, the decomposition of $\Psi_{AB}$ as shown in line 196 is correct only if subsystems A and B are not entangled. However, because $x_A$ and $X_B$ are measured with respect to an unmentioned third subsystem, and $\psi_B$ is expressed as a function of $x_B-x_A$, there is necessarily a correlation - an entanglement- between states of A and B. This is a direct consequence of perspective dependence mentioned in Sec. 1-3. These misunderstandings and errors make the rest of formulation and conclusions invalid.

In conclusion, the author should correct these errors and revise their consequences. This needs a full revision of the manuscript, which should be reassessed to see if it presents any new, correct, and interesting result.

Author Response

Please see attached PDF.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

My report is attached.

Comments for author File: Comments.pdf

Author Response

I thank the referee for their report and for taking the time to critically analyse my manuscript. I have referred to the Editorial Guidlines, regarding the revision. All changes to the original draft are highlighted in red. 

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The author has successfully addressed my concerns and added text. I believe the paper is now suitable for publication.

Author Response

I thank the referee for their time and attention to detail in reviewing my manuscript.

I am glad that my reply, and the amendments made to the text, were satisfactory. 

Reviewer 2 Report

Comments and Suggestions for Authors

The response of the author confirms the confusion that I had spotted in the first version of the manuscript. Specifically, in their response the author uses coordinates of objects - quantum or classical - without considering the fact that they are meaningful only with respect to a reference - this is what I called a 3rd particle / (sub)system in my first comments. Otherwise, it is meaningless to even mention e.g. x_A. This reference may be virtual, meaning not a physical system - the author call such references "classical". However, there is extended literature about inconsistencies of 
mixed quantum and classical systems. Therefore, it is preferable to consider such reference as virtual. Below it is shown that it can be a real quantum system.

As the author correctly states, a physical reference is quantum and in general may have a non-sharp position with respect to the virtual reference. Nonetheless, in both classical and quantum case physical processes and measurables depend only on the relative distance between particles. Therefore, if there are N particles / (sub)systems are available, their dynamics depends on (N-1) D-dimensional 
vector variables, where D is the dimension of space(time). For instance, consider the simpler case of randomly moving classical particles - the classical Brownian motion. Position of particle i with respect to a virtual fixed reference is assumed to have probability distribution P_i(x_i). But, its interaction with particle j depends only on the vector x_ij = x_j - x_i. It is clear that this quantity is independent of the assumed virtual reference, because x_ij = (x_j - x_ref) - (x_i - x_ref).  In particular, coordinates x_i in the definition R6 of the center of mass can be replaced by (x_j - x_ref). In this case, the left hand side of R6 in the response document becomes x_CM - x_ref. Thus, there are only (N-1) independent random vectors x_ij.

The probability distribution of x_ij can be obtained from P(X_i) and P(X_j) (Here random variables are indicated in upper case and their outcomes in lower case): 

P__ij (X_ij = x_ij) = \int dx_i P_i(x_i) P_j(x_j | X_i = x_i). 

Assuming independent position distribution for particles - equivalent to untangled particles in quantum case - P(x_ij) = \int dx_i P_i(x_i) P_j(x_i + x_ij). Assuming one particle, e.g. i=0 as reference P_0(x_0) can be included in P_j for j \neq 0. Alternatively, without loss of generality, one can assume x_0 = 0 i.e. P_i(x_i) = 1 for x_i =0, and P_i(x_i) = 0, otherwise. Indeed, as particles are assumed untangled, their positions can be simultaneously measured, either with respect to each others or with respect to a 
virtual fixed reference. The above choice corresponds to a projective measurement of the position of particle 0 and measurement of others with respect to it. Irrespective of how the reference is chosen, only N-1 random positions determine physical processes and measurements. Interestingly, the author uses an analogous formulation using wavefunction instead of probability, that is R3-R5 in the  response file. However, the author continues their wrong claim about measurability and influence of the quantum reference on 
measurements and processes. 

For these reasons I do not recommend the publication of this manuscript.

Author Response

Please see attached PDF.

Author Response File: Author Response.pdf