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Probabilistic deconstruction of a theory of gravity, Part I: flat space
by S. Josephine Suh
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Submission summary
Authors (as registered SciPost users):  Josephine Suh 
Submission information  

Preprint Link:  https://arxiv.org/abs/2108.10916v3 (pdf) 
Date accepted:  20230921 
Date submitted:  20230817 04:08 
Submitted by:  Suh, Josephine 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We define and analyze a stochastic process in antide Sitter JackiwTeitelboim gravity, induced by the quantum dynamics of the boundary and whose random variable takes values in $AdS_2$. With the boundary in a thermal state and for appropriate parameters, we take the asymptotic limit of the quantum process at short time scales and flat space, and show associated classical joint distributions have the Markov property. We find that Einstein's equations of the theory, sans the cosmological constant term, arise in the semiclassical limit of the quantum evolution of probability under the asymptotic process. In particular, in flat JackiwTeitelboim gravity, the area of compactified space solved for by Einstein's equations can be identified as a probability density evolving under the Markovian process.
Author comments upon resubmission
I have composed replies to both of the referee's reports, answering their questions, and have resubmitted a revised version of the manuscript that includes additional exposition as appropriate.
Best regards,
Josephine Suh
List of changes
The new Conclusion and Discussion section includes an exposition of
1. the role that the microscopic SYK model plays in our discussion
2. how the bulk spacetime AdS_2 emerges and our new proposal regarding the emergent spacetime
2. the role of Dirichlet b.c. and negative cosmological constant in the JT action
3. relation to previous research deriving Einstein's equations from entanglement entropy.
Published as SciPost Phys. 15, 174 (2023)
Josephine Suh on 20230826 [id 3932]
We thank the referee for helpful questions in the previous round of refereeing, and hope our answers below can be satisfactory. (We are posting our answers here as they were not vetted in time to appear on the page for our previous submission.)
Dilatonic theories on an $AdS_2$ background often arise by dimensional reduction of an $AdS_2 \times X$ geometry. (See e.g. reference 1402.6334.) For example, $X$ can be $S^2$ and after dimensional reduction, the the value of the dilaton defined as in our manuscript corresponds to the area of the compact twosphere.
Our arguments work to all orders in a perturbative expansion in $N$ of SYK, or $\gamma$ in our setup. To this accuracy, one has a background spacetime in Lorentzian signature which is $AdS_2$ with two boundaries. AdS spacetimes with nontrivial topologies enter Euclidean path integrals to subleading order in an expansion in $e^N$. These are relevant in situations where one is calculating quantities that are sensitive to such corrections. In our setup we are not asking such questions.
We do not expect to have a black hole when we take the flatspace limit in our setup. In particular, our limit of flat space involves taking the temperature to be zero, see eq. (63).
$n$ is normal to the level curve of the dilaton.
Dirichlet boundary conditions ensure that the JT action reduces to the Schwarzian action, which captures the lowenergy dynamics of the SYK model.
The stress tensor can exist as an arbitrary source for the dilaton field in the bulk spacetime.
There is no gauge chosen for the metric. However, to derive the formulas, it is convenient to work in null coordinates. We did not include the derivation in an Appendix because (11)(12) are not important to the main content of the paper. It merely serves as motivation.
We thank the referee's suggestion. We will include a description of the $T_i$'s as the times of the projections $P_i$.
In (2), we are considering the most general equation of motion resulting from (1). In Eq. (6), we are working in JT gravity without matter. We may include a clarification on this point.
As we describe in the secondtolast paragraph in the introduction, we conjecture that the general identification of spacetime as target space and volume measure as probability measure, and Einstein's equations as a generator equation, will hold quite generally in general relativity, and not just in two dimensions.
However, the technical derivation of the above results in JT gravity certainly do depend on the details and solvable nature of the theory. In ongoing and future work, we hope to give proposals and theoretical machinery for calculating the relevant quantities we have identified in this paper (i.e. joint quantum distributions) in higherdimensional gravitational theories.
Josephine Suh on 20230826 [id 3931]
We thank the referee for helpful questions in the previous round of refereeing, and hope our answers below can be satisfactory. (We are posting our answers here as they were not vetted in time to appear on the page for our previous submission.) We have included more quantitative expositions as appropriate in the current revision of our manuscript.
First, some background regarding the relationship between microscopic SYK and the JT action:
The bulk $AdS_2$ spacetime emerges from the lowenergy dynamics of the microscopic SYK model, with the coordinates of the bulk spacetime being directly related to form of the fermionic Green's function in SYK. In short, the form of the Green's function at lowenergies ("soft mode") and the effective action governing the emergent mode ("Schwarzian") implies a description of the system in terms of a free matter field in $AdS_2$ interacting with a specific mode propagating between bulk points.
(The new point of view we are contributing in this paper with regards to this "emergent" spacetime is that we can understand it operationally and quantummechanically as the target space of a quantum observable, $i.e.$ the one resulting from quantizing the soft mode ("boundary" d.o.f.) . We then show the utility of this viewpoint by deriving the JT equivalent of Einstein's equations ("bulk" equations) using only the quantum mechanics resulting from quantizing the boundary mode.)
Now, the emergent Schwarzian action can be derived from the SYK model when one works in the $N \gg \beta J \gg 1$ limit ("holographic limit") and importantly, at long time scales $\tau_1\tau_2 J \gg 1$. The JT action in eq. 1 can be viewed as a completion to short timescales of the Schwarzian action. The JT action, with Dirichlet boundary conditions and negative cosmological constant, reduces to the Schwarzian action at long timescales. (Intuitively, the boundary particle resulting from quantizing the JT action in eq. 1 lives in exact $AdS_2$, whereas in the Schwarzian limit it lives in an asymptotic geometry near the boundary of $AdS_2$.)
Now, to address the referee's individual questions:
For the JT action to reduce to the Schwarzian action which describes an emergent mode in the SYK model, it is both true that i) the Dirichlet boundary condition should be imposed, and ii) that the cosmological constant should be set to be negative. (The negative cosmological constant enters because an explicit expression for extrinsic curvature $K$ depends on the curvature of the spacetime.)
Our arguments will not generalize. Without Dirichlet boundary conditions, we cannot view the boundary action of JT as simply describing a particle propagating in background spacetime, and proceed with quantization, etc.
See preliminary exposition. For the action in eq. 15 to describe the shorttime completion of the emergent dynamics of SYK, it must be put on an AdS background.
What we are attempting to do is identify the direct operational meaning of the emergent spacetime and its volume measure relative to the quantum mechanics describing the boundary d.o.f. We argue that one should view "emergent spacetime" as the target space of a quantum observable, and the "volume measure" at a point of the emergent spacetime, as a probability measure constrained by the dynamics of the said observable.
Perhaps another way to put it is this: the bulk doesn't mysteriously "emerge", it exists as the target space of an (emergent) observable of a quantum mechanical system.
The $AdS_2$ geometry is a consequence of the lowenergy dynamics of the SYK model. What we do in this paper is identify the direct operational meaning of this geometry relative to the quantum mechanics resulting from quantizing said dynamics.
As we take the shorttime limit in joint quantum distributions in eq. (4) to go to flatspace, the quantities retain knowledge of the density matrix that we started out with, i.e. the spectral density of the JT gravity in AdS. In short, AdS/CFT correspondence gave us a quantum system producing certain expectation values which one could one then deform to reach the flatspace limit.
Also, it is only in the limit eq. (16) or equivalently, eq. (29), the analogue of large $N$, large $\lambda$ in higherdimensional AdS/CFT , that we can renormalize the action (15). Furthermore, all of our calculations of the quantum theory are done in the same limit, and our derivation of the bulk Einstein's equations from boundary quantum theory would not occur outside the limit.
We found that such a double expansion does not exist in closed form, $i.e.$ the coefficients do not come in closedform functions.
The large $N$, $E \ll N$ limit in eq. (29) (analogue of large $N$, large $\lambda$ in higherdimensional AdS/CFT) as well as the semiclassical limit $E \gg 1$ in eq. (31) are all crucial to the derivation of the Markovian property. A final ingredient is the shortdistance or flatspace limit. Without the latter, we suspect that the Markovian property only holds in a ``local" sense.
I believe the Lifshitz asymmetry will show up when going from the LHS to the RHS. The coordinates l, t on the LHS are relative coordinates between two point measuring geodesic distance and direction.
In technical terms, the work in eg. 1312.7856 crucially relied on an expansion around the CFT vacuum and also had the limitation of obtaining only the linearized Einstein's equations. Conceptually, entanglement entropy only characterizes the quantum state, and it is natural to expect that the dynamics of the boundary quantum system should be accounted for in order to fully reproduce Einstein's equations. In our work we are proposing to quantify the necessary quantum dynamics using joint quantum distributions in eq.(4).