Sem II Week 12 — Ahmad–Jefferson II: Source Audit, Exact Model, and Computation Protocol

This week has three layers, and the layers must remain visible. First, we read Ahmad–Jefferson’s actual formulas: five additional terms at linear order in $h$ and fifteen at $h^2$, through $O(1/N^2)$. Second, we solve a $2\times 2$ Gibbs perturbation exactly through quadratic order. That model teaches the entropy/relative-entropy bookkeeping, but its linear term vanishes only because we chose an off-diagonal perturbation. Third, we formulate a finite-regulator free-field protocol that another student can execute. We do not advertise an unspecified kernel or a cross-wedge symplectic form as a completed calculation: locality makes that cross symplectic form vanish, and the continuum entropy needs a regulator plus a convergence analysis.

Route through this master dossier. Classroom core (two-hour seminar): §1, the exact $2\times 2$ calculation in §3, and the term-counting logic in §4. Full derivation or self-study: §§2–5 in sequence, followed by the nonstarred problems. Research extension or office hour: the regulator-removal questions, the source-audit tasks, and the starred/project problems. The exact matrix model is a checkable lesson in bookkeeping; the free-field part is deliberately presented as a reproducible protocol.

0. Reading and exact source map

Primary. Ahmad and Jefferson, arXiv:2501.01487v2:

• §3.1: covariance of modular data under a unitary transformation;
• §3.2: deformation of the spectral weighting and Jacobian;
• §3.3: entropy expansion through second order;
• §4.1: review of the GJW traversable-wormhole deformation;
• §4.2: deformed exterior algebras and modular charges;
• §4.3: application of the entropy formulas, especially Eqs. (74)–(78);
• §5: discussion. There is no §6 in v2.

Equation numbers here follow the arXiv v2 HTML. The corresponding v2 PDF numbers are listed in Week 11 §0.

Comparison source. GJW, arXiv:1608.05687, is a linear-order treatment of the double-trace deformation.

Course model. Sections 2–3 below concern the fixed-algebra Gibbs family ${\rho }_v\propto e^{-(K+v{\sigma }_x)}$. This is not AAJ’s changing-algebra calculation.

1. What is linear, what is quadratic?

1.1 The source-level answer

AAJ’s own summary is unambiguous. Working through $O(1/N^2)$, they find

$$\boxed{5\text{additional terms at }O(h)+15\text{terms at }O(h^2).}$$

Thus there are twenty new terms through quadratic order, not twenty terms all at quadratic order. The word “additional” matters at linear order: AAJ recover the GJW first-law contribution and display five further linear contributions.

1.2 Where the counts appear

AAJ Eq. (77) groups the linear entropy by powers of $1/N$:

• two displayed structures at $O(N^0)$;
• two at $O(1/N)$, one of which is the GJW term;
• two at $O(1/N^2)$.

There are six displayed linear structures in total: one GJW contribution plus five new ones.

AAJ Eq. (78) groups the quadratic entropy as

• two structures at $O(N^0)$;
• six at $O(1/N)$;
• seven at $O(1/N^2)$.

The sum is $2+6+7=15$.

This is the count students should reproduce. It does not arise from an invented “matter/clock/cross” combinatorics, and it does not come from expanding a Connes cocycle. It comes from AAJ’s deformation-dependent weight and Jacobian after the BCH expansion of the modular charge and the perturbative expansion of $\beta$.

1.3 The two commutators that carry the deformation

In AAJ’s GJW application, the interaction $\delta H_0$ enters through

$$C_1=[\delta H_0,{\tilde{H}}_{L,0}],C_2=[\delta H_0,[\delta H_0,{\tilde{H}}_{L,0}]].$$

Their Eq. (74) rewrites $C_1$ using a Heisenberg time derivative of the left operator, and Eq. (76) displays $C_2$. The paper does not evaluate every resulting correlator for a specified CFT Hamiltonian; it gives the perturbative algebraic structures into which those model-dependent data must be inserted.

There is an important convention behind these two commutators. AAJ decompose $H=H_0+\delta H_0$, impose $[H_0,{\tilde{H}}_{L,0}]=0$, and then use the interaction-only replacement displayed in their Eqs. (52)–(53). We will reproduce that ledger only in an interaction-picture model generated by $\delta H_0$, or under extra assumptions that eliminate all mixed nested commutators. If a regulator instead uses the literal Schrödinger-picture unitary $e^{-i\tau (H_0+\delta H_0)}$, the full Hamiltonian belongs in every BCH commutator.

1.4 Why a first-order term is expected

For two nearby faithful states on one fixed finite-dimensional algebra,

$$S({\rho }_h\Vert {\rho }_0)=O(h^2),$$

because relative entropy is minimized at $h=0$. But the entropy itself obeys the first-law relation

$$\Delta S^{(1)}=\Delta ⟨K_0{⟩}^{(1)},K_0=-\log {\rho }_0,$$

and need not vanish. AAJ’s situation is more elaborate because the algebra and crossed-product weighting also change, but the same basic warning applies: positivity of relative entropy never proves that the entropy has no linear term.

The proposed argument “$JVJ=V$ and $JKJ=-K$, therefore the GJW linear term vanishes” mixes different modular objects. The $J$-odd standard Liouvillean of the doubled representation is not the one-sided reference modular Hamiltonian that appears in the entropy first law, and AAJ explicitly display linear terms. We therefore discard that cancellation argument.

2. An exact identity on a fixed finite algebra

Before reading the more elaborate crossed-product formula, it helps to have one identity we can prove without approximation. For faithful density matrices $\rho$ and ${\rho }_0$, let $K_0=-\log {\rho }_0$. Then

$$D(\rho \Vert {\rho }_0)=\mathrm{Tr}\rho (\log \rho -\log {\rho }_0)=-S(\rho )+\mathrm{Tr}(\rho K_0).$$

Subtract the same expression at ${\rho }_0$ to obtain

$$\boxed{S(\rho )-S({\rho }_0)=-D(\rho \Vert {\rho }_0)+\left[\mathrm{Tr}(\rho K_0)-\mathrm{Tr}({\rho }_0{K}_0)\right].}$$

This identity separates distinguishability from modular energy. In a gravitational crossed product, one may sometimes interpret parts of the type-II entropy as matter and area contributions, but AAJ caution that the fundamental quantity is the total von Neumann entropy and that the area/matter split is normalization- and regime-sensitive. The boxed finite-dimensional identity should therefore be used as a pedagogical comparison, not declared to be AAJ’s derivation.

3. Exact mini-calculation: the off-diagonal $M_2$ model

3.1 Setup and scope

Take

$$K=\left(\begin{array}{cc}0& 0\\ 0& \epsilon \end{array}\right),{\rho }_0=\frac{e^{-K}}{Z_0},Z_0=1+e^{-\epsilon },$$

and perturb the Gibbs exponent by

$$V=v{\sigma }_x,{\rho }_v=\frac{e^{-(K+V)}}{Z_v}.$$

Because $V$ is off-diagonal in the eigenbasis of $K$, both $\mathrm{Tr}({\rho }_{0}V)$ and the first-order change of $⟨K_0⟩$ vanish. This is why this particular model starts at $v^2$. It is not a theorem about GJW.

3.2 Exact spectrum and entropy

The two eigenvalues of $K+V$ are

$$E_{\pm }=\frac{\epsilon \pm \sqrt{{\epsilon }^2+4v^2}}{2}.$$

Only the gap

$$d(v)=\sqrt{{\epsilon }^2+4v^2}=\epsilon +\frac{2v^2}{\epsilon }+O(v^4)$$

affects the entropy. A two-level Gibbs state with gap $d$ has

$$S(d)=\log (1+e^{-d})+\frac{d}{1+e^d}.$$

Since

$$\frac{dS}{dd}=-\frac{d}{4{\cosh }^2(d/2)},$$

we find

$$\boxed{S({\rho }_v)-S({\rho }_0)=-\frac{v^2}{2{\cosh }^2(\epsilon /2)}+O(v^4).}$$

The negative sign has a simple interpretation: the off-diagonal perturbation increases the energy gap by level repulsion and makes the Gibbs probabilities less balanced.

3.3 Relative entropy

The first-order density variation is off-diagonal:

$$(\delta \rho {)}_{12}=(\delta \rho {)}_{21}=-\frac{v(1-e^{-\epsilon })}{\epsilon (1+e^{-\epsilon })}=:w.$$

The quadratic Kubo–Mori formula is

$$D({\rho }_0+\delta \rho \Vert {\rho }_0)=\frac{1}{2}{\int }_0^{\infty }ds\mathrm{Tr}\left[\delta \rho ({\rho }_0+s{)}^{-1}\delta \rho ({\rho }_0+s{)}^{-1}\right]+O(\delta {\rho }^3).$$

Writing ${\rho }_0=\mathrm{diag}(p_1,p_2)$, the integral reduces to

$$\begin{array}{rl}D({\rho }_v\Vert {\rho }_0)& =w^2{\int }_0^{\infty }\frac{ds}{(p_1+s)(p_2+s)}+O(v^4)\\ & =w^2\frac{\log (p_1/p_2)}{p_1-p_2}+O(v^4)\\ & =\boxed{\frac{v^2}{\epsilon }\tanh \left(\frac{\epsilon }{2}\right)}+O(v^4).\end{array}$$

This is positive, as it must be.

3.4 Modular-energy term and checks

The exact identity of §2 now gives

$$\Delta ⟨K_0⟩=\frac{v^2}{\epsilon }\tanh \left(\frac{\epsilon }{2}\right)-\frac{v^2}{2{\cosh }^2(\epsilon /2)}+O(v^4).$$

Three checks are worth assigning:

1. High-temperature point. At $\epsilon =0$, $K_0$ is scalar and the modular-energy difference vanishes. Direct diagonalization of $V$ gives $$S({\rho }_v)=\log (2\cosh v)-v\tanh v=\log 2-\frac{v^2}{2}+O(v^4).$$
2. Fixed positive $\epsilon$. Exact numerical diagonalization agrees with the $v^2$ coefficient and the Kubo–Mori expression.
3. Large-$\epsilon$ asymptotics. Taking the coefficient at finite $\epsilon$ and then sending $\epsilon \to \infty$ gives $\Delta S\to 0$ and an approximate cancellation between $D$ and $\Delta ⟨K_0⟩$. At strictly zero temperature ${\rho }_0$ is nonfaithful, so relative entropy requires a separate support analysis; do not infer it by naive substitution.

3.5 What the model does not contain

The model has neither a changing algebra, a spectral Jacobian, an independent clock coordinate, nor a $1/N$ hierarchy. It therefore cannot “reproduce AAJ’s twenty terms.” What it does provide is a fully checkable example of noncommuting Gibbs perturbation, entropy, relative entropy, and modular energy.

4. AAJ term map: follow the source organization

The following map keeps the paper’s actual organizational variables visible.

Order	AAJ location	Source of structures	Count at $N^0,1/N,1/N^2$	Status
$h$	Eq. (77)	$C_1$, ${\beta }_1$, weight/Jacobian	$2+2+2$ displayed	one is GJW; five are additional
$h^2$	Eq. (78)	$C_2$, $C_1^2$, ${\beta }_2$, ${\beta }_1^2$, mixed terms	$2+6+7$	fifteen quadratic terms

Terms involving ${\beta }_1$ or ${\beta }_2$ are candidates for an area interpretation because the inverse-temperature change is tied to the shifted horizon radius in the chosen gravitational setup. Terms containing modular-charge commutators can contribute to matter and quantum-gravity pieces. AAJ themselves caution that the type-II von Neumann entropy is primary and that the area/matter split should be made with care. The course therefore does not impose an independent “matter/clock/cross” label on every term unless the paper or a subsequent calculation supplies it.

Instructor comparison procedure

For each displayed term in Eqs. (77)–(78), students record:

1. its perturbative order in $h$;
2. its stated order in $1/N$;
3. whether it contains $C_1$, $C_2$, ${\beta }_1$, or ${\beta }_2$;
4. whether AAJ explicitly interpret it as area, matter, or quantum correction;
5. which Hamiltonian/correlator data would be needed to evaluate it.

This creates a source-faithful table and makes the missing dynamical inputs visible.

5. Mini-Calc 4: a reproducible finite-regulator protocol

The continuum calculation previously promised by these notes was not specified enough to be checked. We replace it with a protocol whose output can be audited.

5.1 Required input

Choose and record:

• a finite-dimensional regulator ${\mathcal{H}}_L^{(D)}\otimes {\mathcal{H}}_R^{(D)}$;
• matrices $H_{L,0}$ and $H_{R,0}$ and the reference TFD vector/density matrix;
• a bounded self-adjoint interaction $\delta H_0$;
• a duration $\tau$ and one of two explicitly labeled evolutions: either the full Schrödinger-picture unitary $U=e^{-i\tau (H_0+\delta H_0)}$, or the time-independent interaction-picture model $U_I=e^{-i\tau \delta H_0}$ used to reproduce AAJ’s simplified commutator ledger;
• a finite clock regulator only if a crossed-product trace is being modeled;
• numerical values or a declared model assumption for ${\beta }_1$ and ${\beta }_2$;
• the precise entropy/subalgebra being compared.

Setting ${\beta }_1={\beta }_2=0$ is allowed as a toy-model choice, but it must not be described as the GJW or AAJ result.

5.2 Algebraic checks before entropy

First decide which evolution is being checked.

Full Schrödinger-picture check. Put $H=H_0+\delta H_0$ and compute

$$D_1=[H,H_{L,0}],D_2=[H,[H,H_{L,0}]].$$

Then verify

$$U{H}_{L,0}{U}^{\ast }=H_{L,0}-i\tau D_1-\frac{{\tau }^2}{2}{D}_2+O({\tau }^3).$$

Expanding the second commutator gives

$$\begin{array}{rl}{D}_2=& [H_0,[H_0,H_{L,0}]]+[H_0,[\delta H_0,H_{L,0}]]\\ & +[\delta H_0,[H_0,H_{L,0}]]+[\delta H_0,[\delta H_0,H_{L,0}]].\end{array}$$

Thus $[H_0,H_{L,0}]=0$ removes the first and third displayed pieces, but it generally leaves both $[H_0,[\delta H_0,H_{L,0}]]$ and the interaction-only term $[\delta H_0,[\delta H_0,H_{L,0}]]$. It is not legitimate to drop the mixed term silently.

AAJ interaction-only check. If the regulator is explicitly placed in the time-independent interaction-picture model $U_I=e^{-i\tau \delta H_0}$, compute

$$C_1=[\delta H_0,H_{L,0}],C_2=[\delta H_0,[\delta H_0,H_{L,0}]].$$

and verify

$$U_I{H}_{L,0}{U}_I^{\ast }=H_{L,0}-i\tau C_1-\frac{{\tau }^2}{2}{C}_2+O({\tau }^3)$$

against exact matrix conjugation. Check Hermiticity: $C_1^{\ast }=-C_1$, $C_2^{\ast }=C_2$, so the displayed correction is self-adjoint through second order. For a time-dependent interaction-picture generator, replace the simple exponential by a time-ordered exponential and evaluate the corresponding Dyson integrals; the constant-$\delta H_0$ formula is then only a special case.

If the model uses a right-local observable $A_R$, also compare

$$U{A}_R{U}^{\ast }=A_R-i\tau [H,A_R]-\frac{{\tau }^2}{2}[H,[H,A_R]]+O({\tau }^3)$$

with exact conjugation.

5.3 Entropy computation

There are two legitimate routes.

Route A: direct finite matrix. Construct the specified reduced or type-II-regulator density matrix after the deformation, normalize it, diagonalize it, and compute $-\mathrm{Tr}\rho \log \rho$. Fit the difference to $a_{1}h+a_2{h}^2$ over a shrinking symmetric interval and report convergence of $a_1,a_2$ with the regulator.

Route B: AAJ ledger. Evaluate the matrix analogs of the terms in Eqs. (77)–(78), including the weight and Jacobian chosen for the regulator. Sum the linear and quadratic pieces separately and compare them with Route A. Any mismatch is diagnostic: a normalization, Jacobian, or BCH term has been omitted.

5.4 Free-field specialization

For a Gaussian free-field regulator, take

$$V=\frac{1}{2}\left[W(f_L)W(f_R)+W(-f_L)W(-f_R)\right].$$

Opposite-wedge locality gives $\sigma (f_L,f_R)=0$. Expectations of the Weyl monomials generated by nested commutators follow from

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

The student must supply the finite covariance matrix $\mu$, the symplectic matrix $\sigma$, the smearing functions, and the truncation. There is no universal formula proportional to $\sigma (f_L,f_R)$: that quantity is zero. Nontrivial response of a right observable arises from same-side commutators such as $\sigma (f_R,h_R)$, as shown in Week 11.

5.5 Continuum claim only after convergence

A finite-regulator number becomes evidence for a continuum result only after:

1. increasing the mode/energy cutoff;
2. controlling clock-window dependence;
3. checking normalization and positivity;
4. reporting quadrature/matrix precision;
5. demonstrating convergence or giving an error bound.

Without these steps the honest deliverable is “computation protocol plus finite-regulator example,” not “explicit free-field reproduction of AAJ.”

6. Interpretation map

Statement	Status
GJW contains a linear-order double-trace result	Source result
AAJ add five linear and fifteen quadratic terms through $O(1/N^2)$	Source result, v2
Eq. (77) has $2+2+2$ displayed structures	Direct source count
Eq. (78) has $2+6+7$ structures	Direct source count
$M_2$ off-diagonal Gibbs model starts at $v^2$	Exact model calculation
GJW must start at $h^2$ by $J$ symmetry	Incorrect; removed
A specified finite Gaussian regulator evaluates the nested commutators	Checkable protocol
The free scalar protocol reproduces a gravitational area shift	Not established

7. What to take away

• The central correction is factual: five additional linear terms plus fifteen quadratic terms, twenty new terms through $h^2$.
• AAJ’s §3 is the general perturbation framework and §4 is the GJW application; §5 is discussion.
• Relative entropy begins quadratically near an identical faithful state, while entropy and modular energy can begin linearly.
• The missing linear term in the $M_2$ example follows from the chosen off-diagonal perturbation, not from a universal TFD/GJW symmetry.
• The $M_2$ entropy coefficient

$$-\frac{1}{2{\cosh }^2(\epsilon /2)}$$

and the relative-entropy coefficient

$$\frac{1}{\epsilon }\tanh (\epsilon /2)$$

are independently checkable.

• A credible continuum mini-calc specifies a regulator, covariance data, nested commutators, trace normalization, and convergence. Until then it is a protocol.

8. Problem set

Core problems

1. Linear-term logic. Derive the first law $\Delta S^{(1)}=\Delta ⟨K_0{⟩}^{(1)}$ from the exact identity in §2. Explain why it does not imply either side vanishes.

2. Exact $M_2$ calculation. Starting from the exact gap $d(v)$, derive the entropy coefficient in §3.2. Verify it by diagonalizing the density matrix at three decreasing values of $v$.

3. Kubo–Mori calculation. Evaluate the integral in §3.3 and derive $v^2\tanh (\epsilon /2)/\epsilon$.

4. AAJ linear ledger. Read Eq. (77), identify the GJW term singled out by AAJ, and list the five additional structures with their $1/N$ order.

5. AAJ quadratic ledger. Read Eq. (78) and reproduce $2+6+7=15$. Tag every term by $C_2$, $C_1^2$, ${\beta }_2$, ${\beta }_1^2$, or mixed dependence.

Starred problems

6. Break the off-diagonal choice.* Add $a{\sigma }_z$ to $V$. Compute the linear entropy change and show explicitly which covariance with $K_0$ produces it.

7. BCH verification.* Choose a two-qubit interaction, evaluate $C_1$ and $C_2$, and compare the quadratic BCH approximation with exact conjugation over a range of $\tau$.

8. Gaussian Weyl protocol.* Supply four explicit finite-mode smearings and a valid covariance/symplectic matrix. Evaluate one first-order and one second-order nested-commutator expectation. State exactly which AAJ structure the calculation is intended to model.

Project problem

9. Mini-Calc 4 report. Submit a reproducible notebook or symbolic worksheet plus a three-page interpretation. Separate: exact finite-regulator result, convergence evidence, AAJ source comparison, and gravitational interpretation.

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