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JT Gravity on a Finite Lorentzian Strip: Time dependent Quantum Gravity Amplitudes
by Jose Alejandro Rosabal Rodriguez
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Jose Alejandro Rosabal Rodriguez |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2302.11863v2 (pdf) |
| Date submitted: | 2023-03-02 09:57 |
| Submitted by: | Rosabal Rodriguez, Jose Alejandro |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We formulate JT quantum gravity on a finite Lorentzian strip. Due to the spatial boundaries of the strip, it is possible to define left and right proper times. With respect to these times we compute non-perturbatively the quantum gravity (QG) time dependent transition amplitude. Lagrangian and Hamiltonian formulations are presented. Special attention is paid to the four corner terms (Hayward terms) in the action that are needed in order to have a well defined variational problem. From a detailed analysis of the gravity boundary condition on the spatial boundary, we find that while the lapse and the shift functions are independent Lagrange multipliers on the bulk, on the spatial boundary, these two are related. This fact leads to an algebraic equation of motion for a particular degree of freedom that is conveniently introduced on the spatial boundaries whose solution can be plugged back into the action allowing to fully determine the time dependent transition amplitude. The final result suggests that the time evolution could be non-unitary. Other wave functions for other topologies obtained from strip are also presented, remarkably these do not exhibit any non-unitary evolution behavior.
Current status:
Reports on this Submission
Report #2 by Chitraang Murdia (Referee 2) on 2023-9-24 (Contributed Report)
Strengths
1. This paper aims to further the understanding of Lorentzian quantum gravity. The computation of the transition amplitude is very interesting.
2. The paper is self-contained and detailed.
Weaknesses
1. Although the paper tries to make contact with known results about the double trumpet wave function, it is not clear if this relationship exists.
Report
Understanding Lorezntian quantum gravity is a very important goal in the field. This paper attempts to further this goal by studying Lorentzian JT on a finite strip.
The results are novel and interesting. I would recommend that this paper be published after minor revision.
Requested changes
1. The function χ(C) is an important object in the analysis. It would be great to have some comments made on what this function means physically. Also, the connection to unitarity can be explored further.
2. It would be nice to include some comments on how to incorporate topology change (e.g., splitting/merging of baby universes) in this framework.
Strengths
1- Considers JT gravity in the presence of spacelike plus timeline boundaries plus corners, which is more general than most existing analyses
2- Discusses quantum aspects, including arbitrary topologies
Weaknesses
1- Pertinent work missing from the references has substantial overlap with the first half of the paper (see report)
Report
The paper deals with (Lorentzian) JT gravity on a finite strip, i.e., on a spacetime region bounded by four corners, which in turn are connected by spacelike and timeline boundaries. This is nicely captured in Fig. 1.
Note, however, that Fig. 1 is essentially identical to Fig. 1 in https://arxiv.org/abs/gr-qc/9612021, which is not cited. Moreover, that reference has substantial overlap with the first half of the paper (albeit in somewhat different notation). In particular, the setup on a finite Lorentzian strip, the relevance of boundary and corner terms, the introduction of the rapidity variable, the imposition of suitable boundary conditions, and related issues were all done already in that paper. At least, I was not able to find anything new in the first three sections as compared to the paper from '96.
I am a bit unsure whether or not section 4 contains new content, though it might. Section 5 seems to be genuinely new, combining the results of the '96 paper with more recent discussions along the lines of Ref. [6].
Overall, the results in sections 4-5 (together with the ingredients from sections 2-4) seem sufficiently interesting so that eventually the paper could be published in SciPost Physics.
However, some changes are required (see requested changes).
Requested changes
1- My main recommendation to the author is to carefully read the '96 paper quoted above and to check (and state clearly) which of the statements until the end of section 4 are already covered by that work. While I do not necessarily think the paper has to be shortened (since there is value in it being self-contained), it should be pointed out clearly which of the results are novel and which are contained already in the '96 paper.
2- Given that the '96 paper applies to models more general than JT it seems natural to pose the question whether or not the results of sections 4-5 might be generalized beyond JT as well. Perhaps the author can comment on such generalizations in the conclusions.
3- Sticking with JT gravity, it is natural to ask about the holographic interpretation of the results. Is there some clear statement of what the authors' construction means in the context of the SYK model? If so, this should be mentioned in the conclusions. If not, it might still be worthwhile to point toward such considerations in the conclusions.
Author: Jose Alejandro Rosabal Rodriguez on 2023-06-16 [id 3736]
(in reply to Report 1 on 2023-06-09)Dear referee
Thank you so much for this fair report. I agree with you in all points and these issues will be addressed in the new version of the manuscript.
1- The paper you mention will be properly cited. I would like to make some comments on this point because it is true that this 96 paper has some overlap with the first part of my paper and I was not aware of this work. It is worth mentioning that I got these results independently in a more general context reported in ref. 11. Also that the notation in this 96 paper is quite involved and it discusses topics that at some point overlap with mine but they are orthogonal to my work. Nonetheless, in general, this work will be a good complement for mine.
2-Regarding the other results I used, first, they are properly cited in my work. Second, I would like to add that unfortunately nowadays a huge amount of calculations has been performed in different contexts, so it is quite likely that some calculations today use similar methods to those used in the past. What we should highlight here, as you already mentioned, is that overall, combining all these old methods and results with my new ideas I get something completely new.
3- Although some results in section 1-3 are presented in the 96 paper, in my work they are presented clearly and combined in a novel way that is adapted for the quantum calculations in section 4 and 5. Making the paper self contained.
The subsequent results in section 4 and 5 are new. Moreover, the canonical transformation in my work is motivated by the classical solution of the constraints. In this way we establish a connection between the early work in ref. 8 and ref. 13 (or what would be the 2d version of it). This is also a straightforward method to find any generating function associated with any possible solution of the constraints equ's. (4.11) - (4.14) (possible with matter). This procedure could potentially be exploited to solve similar problems in higher dimensions. For instance the calculation of the time dependent amplitude in a spherically symmetric quantum gravity.
4- The requested changes will appear in the new version.