Sem II Week 11 — Ahmad–Jefferson I: Unitary Perturbations, Araki Comparison, and GJW

Ahmad and Jefferson ask how the crossed-product entropy changes when the background-defining algebra is unitarily deformed. Their paper does not derive the answer by expanding a Connes cocycle. It transports the modular data covariantly, computes the changed weight in the type-II trace, and organizes the deformation by BCH nested commutators. Because the course has already developed Connes cocycles, we place a separate Araki perturbation interlude beside their calculation. The comparison is valuable precisely when the two constructions are not identified: one changes a state by a Gibbs-type modular perturbation; the other transports an algebra/state system by a unitary and tracks the crossed-product weight.

Route through this master dossier. Classroom core (two-hour seminar): §§1–2, §3.1, and §4, with the AAJ/Araki distinction and the GJW sign convention kept on the board. Full derivation or self-study: the finite-dimensional reconstruction in §3, the computation protocol in §5, and the core exercises. Research extension or office hour: the AAJ term ledger, the starred problems, and the project/source-comparison tasks. The comparison interlude is pedagogical scaffolding and is never substituted for AAJ’s actual calculation.

0. Reading and source map

Primary.

• S. Ali Ahmad and R. Jefferson, “Algebraic perturbation theory: traversable wormholes and generalized entropy beyond subleading order,” arXiv:2501.01487v2.
  • §2: crossed products in AdS–Schwarzschild;
  • §3: weighting of expectation values under a general unitary perturbation;
  • §4: black-hole evaporation and the traversable-wormhole application;
  • §5: discussion.
• P. Gao, D. L. Jafferis, and A. C. Wall, “Traversable Wormholes via a Double Trace Deformation,” arXiv:1608.05687. GJW work to linear order in their deformation strength.

Equation-number convention. This dossier follows the arXiv v2 HTML numbering. In the v2 PDF, HTML Eqs. (52)—(53) are Eqs. (3.41)—(3.42), while HTML Eqs. (74), (76)—(78) are Eqs. (4.17), (4.19)—(4.21).

Course interlude. Sem I Week 7 supplies the Connes cocycle. Sections 2–3 below reconstruct a bounded finite-dimensional Araki perturbation as a comparison model. It is not attributed to AAJ.

Optional. T. Faulkner, M. Li, and H. Wang, “A modular toolkit for bulk reconstruction,” arXiv:1806.10560.

1. The perturbation that AAJ actually study

1.1 Unitarily related algebra-state systems

AAJ begin with a type-III${}_1$ algebra ${\mathfrak{A}}_0$, a cyclic-separating vector $|{\Psi }_0⟩$, and the corresponding crossed product ${\hat{\mathfrak{A}}}_0$. A unitary $U$ defines

$$\mathfrak{A}=U{\mathfrak{A}}_0{U}^{†},|\Psi ⟩=U|{\Psi }_0⟩.$$

The adjoined $L^2(\mathbb{R})$ factor is not acted on by $U$, so the full crossed product transforms covariantly. The modular operator and modular Hamiltonian transform by conjugation as well. This covariance is exact; the perturbative work enters when AAJ compare the trace weight and entropy of the deformed construction with the undeformed one.

In the two-sided application, $U$ mixes left and right observables. It is therefore unsafe to say that each one-sided deformed algebra is simply $U{\mathfrak{A}}_{R,0}{U}^{†}$ or $U{\mathfrak{A}}_{L,0}{U}^{†}$ in isolation. AAJ emphasize the full exterior algebra and then analyze its one-sided structure.

1.2 Where the series comes from

AAJ choose

$$U=e^{-i\tau H},H=H_0+\delta H_0,$$

where $H_0=H_{L,0}+H_{R,0}$ is the decoupled part and $\delta H_0$ couples the two sides. AAJ impose, by construction,

$$[H_0,{\tilde{H}}_{L,0}]=0,$$

and in their Eqs. (52)–(53) replace $H$ by $\delta H_0$ in the BCH series. For a checkable model, this step should be read as using the interaction-picture evolution generated by the interaction (with a time-independent $\delta H_0$ over the interval), or as adding the stronger commutator assumptions needed to make the replacement exact. The single condition $[H_0,{\tilde{H}}_{L,0}]=0$ does not, by itself, remove a second-order term $[H_0,[\delta H_0,{\tilde{H}}_{L,0}]]$ from a Schrödinger-picture BCH expansion of $e^{-i\tau (H_0+\delta H_0)}$. Within AAJ’s stated replacement, if ${\tilde{H}}_{L,0}$ is the relevant undeformed left modular charge, then

$$\delta {\tilde{H}}_{L,0}:=U{\tilde{H}}_{L,0}{U}^{†}-{\tilde{H}}_{L,0}$$

has the BCH expansion

$$\delta {\tilde{H}}_{L,0}=-i\tau [\delta H_0,{\tilde{H}}_{L,0}]-\frac{{\tau }^2}{2}[\delta H_0,[\delta H_0,{\tilde{H}}_{L,0}]]+O(h^3),$$

when the interaction is of order $h$. Outside that setup one must expand with the full $H$ or use the time-ordered interaction-picture Hamiltonian $\delta H_I(t)$; mixed commutators may then survive. AAJ insert their displayed $\delta H_0$ series, together with

$$\beta ={\beta }_0+{\beta }_1+{\beta }_2+\cdots ,{\beta }_n=O(h^n),$$

into the spectral weight and Jacobian that appear in the type-II trace. The resulting entropy formulas are organized simultaneously in $h$ and $1/N$.

This is the method to remember: unitary covariance $\to$ changed modular charge $\to$ changed trace weight and Jacobian $\to$ entropy expansion.

1.3 What not to say

AAJ do not present §2 as a Connes-cocycle construction, and §2 is not their perturbation section. Nor do they claim twenty quadratic terms. Their current v2 result, through $O(1/N^2)$, is:

• five additional contributions at linear order in $h$;
• fifteen contributions at order $h^2$;
• twenty new contributions through quadratic order in total.

Week 12 audits that count against the displayed AAJ equations.

2. Course interlude: two different notions of perturbation

The comparison with Connes theory is illuminating if we first keep two operations distinct.

2.1 Unitary transport

Given $({\mathfrak{A}}_0,{\Psi }_0)$ and a unitary $U$, transport both:

$$({\mathfrak{A}}_0,{\Psi }_0)⟼(U{\mathfrak{A}}_0{U}^{†},U{\Psi }_0).$$

All modular data transform covariantly. If one transports an observable and the state together, its ordinary expectation value is unchanged. AAJ nevertheless obtain a nontrivial entropy comparison because the crossed-product trace is expressed with a deformation-dependent spectral weighting and a changed subsystem/algebraic assignment.

2.2 Araki/Gibbs perturbation of a state

Fix one algebra and, in a finite-dimensional model, take

$${\rho }_0=\frac{e^{-K}}{Z_0},{\rho }_V=\frac{e^{-(K+V)}}{Z_V},V=V^{\ast }.$$

This changes the state on the same algebra. Abstract Araki perturbation theory reconstructs an analogous state from modular data under suitable hypotheses. The Connes cocycle relates the two states. This is not the same operation as transporting both the algebra and vector by $U$.

The distinction answers a common student question: “If both are called perturbations, why do their series look different?” Because they compare different objects.

3. Correct cocycle convention and a checkable finite model

3.1 The sign convention

Throughout the course,

$${\sigma }_t^{\rho }(a)={\rho }^{-it}a{\rho }^{it}=e^{itK}a{e}^{-itK},$$

and therefore

$$\boxed{{\left(\frac{D{\omega }_{\rho }}{D{\omega }_{\sigma }}\right)}_t={\rho }^{-it}{\sigma }^{it}.}$$

It follows that the normalized-state cocycle for the Gibbs perturbation is

$$u_t={\left(\frac{Z_V}{Z_0}\right)}^{it}{e}^{it(K+V)}{e}^{-itK}.$$

The scalar partition-function phase does not cancel. It drops out of the inner automorphism $\mathrm{Ad}{u}_t$, but it belongs to the Connes cocycle of the normalized states.

It is useful to isolate the implementing cocycle

$$v_t:=e^{it(K+V)}{e}^{-itK}.$$

It obeys

$$\frac{d{v}_t}{dt}=i{v}_t{\sigma }_t^0(V),v_0=1,$$

and hence

$$v_t=\overline{\mathcal{T}}\exp \left(i{\int }_0^t{\sigma }_s^0(V)ds\right),$$

with earlier modular times to the left. This is a bounded finite-dimensional reconstruction of an Araki implementing cocycle. Multiplying by $(Z_V/Z_0{)}^{it}$ gives the normalized-state Connes cocycle.

3.2 Why this is not AAJ’s series

Both calculations involve modular objects, but their expansion variables are arranged differently:

Course Araki model	AAJ v2
same algebra, Gibbs-perturbed state	unitarily deformed algebra-state system
expand modular-time-ordered ${\sigma }_s(V)$	expand conjugated modular charge with BCH
cocycle relates two states	trace weight/Jacobian compares crossed products
partition-function phase required for normalized states	inverse-temperature and spectral-variable changes enter explicitly

The course uses the left column to consolidate Sem I. The source audit and entropy calculation use the right column.

3.3 $M_2$ warmup through second order

Let

$$K=\left(\begin{array}{cc}0& 0\\ 0& \epsilon \end{array}\right),V=v{\sigma }_x=v(E_{12}+E_{21}).$$

The unperturbed modular flow gives

$${\sigma }_s^0(V)=v\left(e^{-is\epsilon }{E}_{12}+e^{is\epsilon }{E}_{21}\right).$$

For the implementing cocycle $v_t=1+v_t^{(1)}+v_t^{(2)}+O(v^3)$,

$$v_t^{(1)}=\frac{v}{\epsilon }\left[(1-e^{-it\epsilon })E_{12}+(e^{it\epsilon }-1)E_{21}\right].$$

At second order,

$$v_t^{(2)}=-v^2\left[I(t)E_{11}+\overline{I(t)}{E}_{22}\right],$$

where

$$I(t)=\frac{it}{\epsilon }-\frac{e^{it\epsilon }-1}{{\epsilon }^2}.$$

The calculation is short: $E_{12}^2=E_{21}^2=0$, while $E_{12}{E}_{21}=E_{11}$ and $E_{21}{E}_{12}=E_{22}$. A decisive check is

$$v_t^{(2)}+v_t^{(2)\ast }+v_t^{(1)\ast }{v}_t^{(1)}=0,$$

which verifies unitarity through second order. The full normalized-state cocycle is obtained only after multiplying by the exact phase $(Z_V/Z_0{)}^{it}$.

3.4 Temperature limits, carefully stated

If $K\to 0$ while $V$ is held fixed, the implementing cocycle tends to $e^{itV}$. The normalized cocycle retains its scalar partition-function phase. At the opposite limit, a finite-dimensional Gibbs state tends toward a nonfaithful ground-state density matrix. The faithful-state Connes cocycle used above therefore has no automatic $\beta \to \infty$ limit. One may formulate vacuum perturbation theory with support projections or a different standard form, but it is not simply Rayleigh–Schrödinger theory obtained by taking the limit in these formulas.

4. GJW and AAJ’s application

4.1 The original deformation

Gao, Jafferis, and Wall add a time-dependent double-trace coupling between the two boundaries. In their Hamiltonian convention,

$$\boxed{\delta H(t)=-\int d^{d-1}xh(t,x){\mathcal{O}}_R(t,x){\mathcal{O}}_L(-t,x).}$$

For the positive $h$ profile used in their controlled scalar example, this convention produces negative horizon-averaged null energy. If one instead writes $\delta H=+g{\mathcal{O}}_L{\mathcal{O}}_R$, the traversable choice is correspondingly $g<0$ after matching the left-time argument and smearing. Invariantly, the required sign is the one for which the computed horizon integral is negative. GJW work to linear order in the deformation. The resulting gravitational backreaction shifts a null trajectory and makes the wormhole traversable for an appropriate window of signals.

The qualifier “appropriate” matters. An arbitrary sign, profile, or operator does not automatically open the wormhole.

4.2 AAJ’s use of GJW

AAJ use the GJW interaction as a concrete $\delta H_0$ in their general unitary framework. The nested commutators

$$[\delta H_0,{\tilde{H}}_{L,0}],[\delta H_0,[\delta H_0,{\tilde{H}}_{L,0}]]$$

feed into the changed crossed-product weight. Their calculation extends the linear-order GJW entropy discussion and produces the five-plus-fifteen term ledger quoted above. It does not follow by identifying a Connes cocycle with a bulk shockwave.

5. A genuine free-field comparison protocol

The free-field analog is useful if its scope is kept modest. It checks unitary conjugation, Weyl commutators, and Gaussian expectation values. It does not reproduce gravitational backreaction or a Shapiro time advance.

5.1 Regulated joint algebra

Let $f_L$ and $f_R$ be real test functions supported in opposite Rindler wedges. Since the supports are spacelike separated,

$$\sigma (f_L,f_R)=0.$$

Define

$$Q=W(f_L)W(f_R)=W(f_L+f_R),V=\frac{1}{2}(Q+Q^{\ast }).$$

Then $V$ is a bounded self-adjoint element of the joint algebra $\mathcal{A}(W_L)\vee \mathcal{A}(W_R)$, not of either one-sided algebra alone. Work first with a finite-volume or finite-mode regulator if entropy or density matrices will be used.

Take

$$U_g=e^{-igV},A_g=U_{g}A{U}_g^{\ast }.$$

For bounded $A$,

$${\frac{d{A}_g}{dg}|}_{g=0}=-i[V,A],{\frac{d^2{A}_g}{d{g}^2}|}_{g=0}=-[V,[V,A]].$$

These identities are the exact finite-regulator counterpart of the BCH structure used by AAJ.

5.2 A checkable Weyl commutator

For a right-wedge Weyl observable $W(h_R)$,

$$[Q,W(h_R)]=W(f_L)[W(f_R),W(h_R)].$$

The Weyl relations give

$$[Q,W(h_R)]=-2i\sin \left(\frac{\sigma (f_R,h_R)}{2}\right)W(f_L+f_R+h_R).$$

The left-right symplectic form does not generate this response—it vanishes by locality. The response is controlled by the same-side form $\sigma (f_R,h_R)$, while the left Weyl factor remains in the resulting joint operator.

Vacuum expectations of all products can then be evaluated from

$${\omega }_0\left(\prod _{j=1}^{n}W(f_j)\right)=\exp \left[-\frac{i}{2}\sum _{j<k}\sigma (f_j,f_k)-\frac{1}{2}\mu \left(\sum _j{f}_j,\sum _j{f}_j\right)\right].$$

This is a real mini-calculation: specify the functions and covariance, evaluate the nested commutators, and compare their Taylor series with exact finite-matrix conjugation.

5.3 Modular boosts do not push support to the horizon

Bisognano–Wichmann transports $W(f)$ by a Lorentz boost. In Rindler coordinates that boost translates Rindler time and preserves the radial coordinate. A common boost of the full vacuum configuration therefore does not “slide both test functions toward the bifurcation surface.” A horizon-localization family must be defined by changing the support itself, not by relabeling a common boost orbit.

5.4 What this protocol cannot claim

The free scalar has no dynamical gravity. Its Weyl commutator is not a negative-energy shockwave, and there is no bulk Shapiro shift to match. The calculation provides an algebraic perturbation laboratory. Any equality between its commutator coefficient and a gravitational time advance would be a new, separately derived result.

6. What to take away

• AAJ v2 use unitary covariance, spectral reweighting, a Jacobian, and BCH nested commutators; they do not compute their entropy series from a Connes cocycle.
• The course’s Araki interlude is complementary and explicitly labeled as such.
• With ${\sigma }_t=\mathrm{Ad}{\rho }^{-it}$,

$${\left(D{\omega }_V/D{\omega }_0\right)}_t={\left(Z_V/Z_0\right)}^{it}{e}^{it(K+V)}{e}^{-itK}.$$

• The partition-function phase is part of the normalized-state cocycle, although it does not affect the implemented automorphism.
• GJW is a linear-order result for a suitable double-trace profile/sign. AAJ extend the entropy analysis with five additional linear terms and fifteen quadratic terms through $O(1/N^2)$.
• The free-field protocol checks BCH and Gaussian algebra. It does not derive traversability.

7. Looking ahead

Week 12 audits AAJ’s actual $O(h)$ and $O(h^2)$ ledgers, preserves the exact $M_2$ calculation as a model, and turns the continuum “mini-calc” into a reproducible protocol. The deliverable will distinguish what AAJ calculate, what the finite regulator verifies, and what remains a gravity interpretation.

8. Problem set

Core problems

1. Cocycle sign audit. Starting from ${\sigma }_t^{\rho }(a)={\rho }^{-it}a{\rho }^{it}$, derive $(D{\omega }_{\rho }/D{\omega }_{\sigma }{)}_t={\rho }^{-it}{\sigma }^{it}$. Insert the two Gibbs densities and recover the partition-function phase.

2. Implementer versus normalized cocycle. Verify that $v_t=e^{it(K+V)}{e}^{-itK}$ and $u_t=(Z_V/Z_0{)}^{it}{v}_t$ implement the same automorphism. Which one is the cocycle of the normalized states?

3. Second-order check. Derive $I(t)$ in §3.3 and verify the second-order unitarity identity. Compare the truncated series with direct exponentiation for numerical values of $t,\epsilon ,v$.

4. Two perturbations. Give one sentence explaining why ${\rho }_0\mapsto e^{-(K+V)}/Z_V$ is not the same as $({\mathfrak{A}}_0,{\Psi }_0)\mapsto (U{\mathfrak{A}}_0{U}^{\ast },U{\Psi }_0)$.

5. AAJ source map. Locate unitary covariance, the general weight, the entropy expansion, and the GJW application in v2. Record the section and equation number for each.

Starred problems

6. Weyl response.* Derive the exact commutator in §5.2, including its sign. Repeat for $Q^{\ast }$ and assemble $[V,W(h_R)]$.

7. Gaussian verification.* Choose an explicit finite covariance matrix satisfying the Weyl uncertainty condition. Evaluate the first two derivatives of ${\omega }_0(A_g)$ and compare them with exact matrix data in a truncated oscillator representation.

8. Faithfulness at zero temperature.* Explain precisely why the $\beta \to \infty$ density matrix ceases to be faithful and which support projection enters a relative modular construction.

Project problem

9. GJW/AAJ ledger. Make a two-column memo: “proved/computed in GJW at linear order” and “added in AAJ v2.” Cite exact equations and keep bulk negative-energy statements separate from crossed-product entropy statements.

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Notes prepared for the algebraic-QFT course, Semester II Block 4. Last revised 2026-08-24.