Algebraic perturbation theory: traversable wormholes and generalized entropy beyond subleading order

The authors study von Neumann algebras in the context of AdS/CFT and the double trace deformation of Gao, Jafferis, and Wall. It is an interesting topic, deserving of study. However, the paper is extremely difficult to read because there are several deep conceptual issues and incorrect assertions. I have included some (but potentially not all) of them below, which I recommend the authors to address. The paper requires a major revision if it is to be published.

On page 2, it is said that the reason why the area term in equation (1.2) is not well-defined is because the surface approaches the asymptotic boundary, i.e., \( A \) diverges. This is not the actual issue with the formula, as can be seen in cases where \( A \) is finite, such as for black holes. Rather, it is the denominator that is problematic in the semi-classical limit, and the renormalization of Newton's constant is what is related to the divergence in the second term, \( S_{\text{matter}} \).

Later on page 2, it is said that the crossed product only contains "trivial excitations which do not backreact on the spacetime." This is incorrect, as the matter excitations crucially backreact on the spacetime in that story, changing areas at order \( G \).

On page 4, it says that \( X = h_l \), which is incorrect. \( X \) is the (perturbation to) the left ADM mass, not the bulk modular Hamiltonian. As such, footnote 5 is incorrect because it does not generate singular states of the left algebra.

Minor: The term "density state" is used, which is nonstandard. The authors should use the usual terminology to avoid confusion unless there is a reason not to — e.g., "density matrix" or "density operator."

It is crucial to the crossed product story that the trace (equation (2.5)) is only defined up to rescalings that cannot be canonically fixed, which leads to an overall additive ambiguity in the definition of von Neumann entropy. This must be acknowledged.

Equation (2.9) is imprecise and it is not explained clearly in the text what \( \epsilon \) is. The exact density matrix was computed for the states in (2.4) in reference [30]. The "semi-classical" expansion in (2.9) has corrections, and this was justified in {arXiv:2312.07646}, which has not been cited.

Section 3.1 is a very convoluted way of stating the commonly known fact that letting a system interact with its environment can change the entropy. This section only causes confusion.


Equation (3.11) is referred to as the "separating property," but this is not the statement of the vacuum being separating. "Separating" means that the vacuum is not annihilated by any nonzero element of the algebra.

Minor: Equations (3.17)--(3.20) are overly pedantic. A wavefunction on \( L^2(\mathbb{R}) \) is self-explanatory.

In section 4.1, the ``modular charge'' is the one-sided modular hamiltonian which is not well-defined. This is distinct from the full modular Hamiltonian, discussed earlier, which is well-defined. These are being conflated in this paper and they are very different.

I do not think that footnote 15 makes sense. The ``vacuum contribution'' is $O(1/G)$ and so it must be subtracted to have well-defined operators. I looked at the previous reference in footnote 15 and the same mistake is made there. This was explained in the author's references [20] and [25].

There is a significant issue in the discussion of the traversable wormhole in that it requires $O(1/G)$ stress energy to make the wormhole traversable, but this is out of the regime of validity of the original algebraic construction. This makes it not meaningful to compare entropies between states with the double trace deformation and without because there is no way to sync the ambiguous additive constant within the algebraic framework discussed.

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The authors consider the calculation of entropies for traversable wormhole spacetimes which occur due to the injection of negative energy. They claim to compute the corrections to the generalized entropy using recent methods of semiclassical gravitational algebras. While the paper is clearly written, I have some fundamental reservations about the set-up that the authors should clearly address.

The basic set up of gravitational algebras are to consider a fixed background spacetime and a region "R" that is defined in a diffeomorphism invariant way. Given this background spacetime one considers the algebra of O(\sqrt{G_{N}) graviton fluctuations in R which corresponds to what the authors call "\mathfrak{A}{0}". The crossed product arises from considering the backreaction of these gravitons on the region "R" which occurs at O(G) in perturbation theory (see [1-5] below for a more thorough explanation). At this order in perturbation theory the backreaction perturbs the black hole mass at O(G_N) and yields a constraint on the gravitational algebra. The algebra that satisfies this constraint is a Type II algebra and entropy is defined up to an additive constant. In this sense the entropy is really only defined as an entropy difference and the O(1/G_N) part of the entropy is not accessible in this framework. The approach can be applied to different backgrounds such as in references [1-5] where the charge associated to the boost symmetry of horizon will be different depending on the background spacetime.

At this stage the authors particular setting raises some questions:

1. The traversable wormhole is created by injecting negative energy into the black hole. The stress-energy of such matter must scale like O(1/G_N) in order to have a non-trivial backreaction in G_N->0 limit. The above references work on the crossed product utilizes the time translation symmetry of the spacetime to relate the gravitational constraints to the modular crossed product. However, the spacetime here is necessarily time-dependent. The authors should clearly explain why the gravitational constraints are related to the modular crossed product in a non-stationary spacetime.

2. More importantly the authors claim to relate the unbackreacted geometry (i.e. the TFD with no incoming negative energy) to the backreacted geometry by a unitary on the algebra constructed above. However the geometries differ at O(1) (i.e., the areas differ at O(1/G_N)) and the semiclassical algebras correspond to O(\sqrt{G_N}) fluctuations off their backgrounds. On these grounds alone it is clear that there cannot be a unitary in the algebra that relates these two different spacetime backgrounds. This point is explained in the above references (see also section 5.1 of [6]).

Thus, while the authors claims are interesting and worthy of investigation I do not see how their approach circumvents this basic issue. Indeed, as explained in [6], such a construction would require a background independent algebra in order to relate the semiclassical algebras in different backgrounds. The authors should clearly state how their approach relates or differs from approach outlined [1-5] and [6] which states the necessary steps required to generalize approach to the case considered in this paper.

1. "Large N algebras and generalized entropy" [2209.10454]
2. "An Algebra of Observables for de Sitter Space" [2206.10780]
3. "Generalized black hole entropy is von Neumann entropy" [2309.15897]
4. "A clock is just a way to tell the time: gravitational algebras in cosmological spacetimes" [2406.02116]
5. "Algebraic Observational Cosmology" [2406.01669]
6. "A Background Independent Algebra in Quantum Gravity" [2406.01669]

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The use of operator algebras to study thermodynamic aspects of gravity has a long history. Recently, this topic has been revitalized by Witten's observation that the general structure of the von Neumann algebra of observables is modified in certain cases. For example, in AdS/CFT, when accounting for certain gravitational corrections, the so-called crossed product construction applies. This construction amounts to appropriately adjoining the generator of modular evolution. The advantage is that if the original algebra is of type III, the new algebra becomes type II, allowing one to define entropies and density matrices in a rigorous way. The crossed product construction is now understood to apply more generally in various cases.

The authors of this paper study how this construction is modified in perturbation theory when introducing deformations. They consider perturbations which can be written as the action of a unitary operator on the whole Hilbert space, but which do not act unitarily on the algebra or on its commutant. In this case the von Neumann entropy will generically change due to the perturbation.

In particular, they focus on a deformation introduced by Gao, Jafferis, and Wall (GJW) in the context of AdS/CFT, which involves terms from both the algebra and its commutant, effectively coupling the two boundaries of the eternal AdS black hole. The authors determine the general structure of the deformation of the algebras for an arbitrary perturbation of this sort, including the change in von Neumann entropy. They then apply their formalism to the GJW perturbation and discuss its physical implications.

This topic is certainly worth studying, and the authors make an important contribution. Therefore, I recommend this paper for publication. However, I have a few comments and corrections that I would like the authors to address briefly.

1) When formula (2.5) for the trace is used, at least in the context of Witten's result for the canonical ensemble, the factor of (X) in the exponential should depend on (N), the rank of the dual gauge theory. This dependence appears to be generic in the canonical ensemble, and formulas must be interpreted only formally as functions of (N) (see, for example, Appendix A of the follow-up paper 2209.10454). Is this (N)-dependence present here as well? If so, does it appear in the (e^{\lambda (\lambda_0)}) factor in equation (3.24)? How does it impact the analysis of the perturbative expansion, for instance, in Section 4.3?

I believe a comment from the authors could clarify this situation. Note that these formulas for the trace appear elsewhere in the literature, and it is commonly accepted that they can be interpreted as formal results without invalidating other arguments. That would be fine if the same applies here.

2) There is a typo in formula (3.28) on page 11: there is an extra ")" before the ket ( | \Psi \rangle ).

3) In footnote 12, as well as in Section 4.2, the authors remark that their definition of the left Hamiltonian does not involve subtracting the vacuum expectation value. They justify this by referring to their previous work. I believe the readability of the paper would improve if they expanded on this remark.

4) In the first line of page 12, the authors write: "the state of the (L^2) factor above." To what does "above" refer? Could they be more explicit?

5) In formula (3.33), the authors consider the possibility that the temperature changes after the perturbation. This is further discussed on page 22, where they consider the GJW deformation. In the deformed KMS state, the (inverse) temperature is now (\beta = \beta_0 + \delta \beta), where (\beta_0) is the temperature of the unperturbed state. But isn't (\delta \beta) computable from the perturbation? In ordinary statistical mechanics, one expects that, in linear perturbation theory, if the Hamiltonian is perturbed as (H + V), then the change in temperature at first order is controlled by (\langle V \rangle_{\beta_0}). Could something similar hold in this case? I believe it would be helpful for the authors to add a comment on this point.

6) The authors consider deformations which act unitarily on the full Hilbert space but not unitarily on the algebra of observables on one of the boundaries. Is it possible to show directly that 4.1 is of this form? What I mean is are there particular conditions that the coupling $h$ or the operators have to satisfy, or is this true in general due to the coupling of the two boundaries?

7) There is a typo above equation (4.7): "the (U) dependence between operators and states allows us to to take the difference..." One "to" should be removed.

8)There is a typo in the statement before equation (4.16) on page 22: should (\delta H_0 = - \delta S) be (\delta H_0 = - \delta I)?

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