Sem II Week 5 — The TFD as a Type III₁ KMS State

Sem II Week 5 — The TFD as a Type III${}_1$ KMS State

Block 2 turns to the standard representation appropriate to a two-sided eternal black hole. The TFD vector is cyclic and separating for each one-sided algebra. Its modular operator is generated by the two-sided Liouvillean ${\beta }_H(H_R-H_L)$ in the finite-dimensional regulator, while the induced automorphism on the right algebra is right-boundary time translation. At large $N$ the one-sided factors act on one TFD GNS Hilbert space, which need not factorize as ${\mathcal{H}}_R\otimes {\mathcal{H}}_L$. This distinction is essential: the TFD is not cyclic and separating for the algebra generated by both mutually commuting factors when that joint algebra is $\mathcal{B}({\mathcal{H}}_{\mathrm{TFD}})$.

Route through this master dossier. Classroom core (two-hour seminar): §§1–4 and the modular-invariance check in §7. Full derivation or self-study: the regulated TFD calculation, the Rindler-mode analogue in §6, and Problems 1–6. Research extension or office hour: §5, the promised BRST guest-lecture slot, and the starred/project problems. The classroom path establishes the standard-pair structure before any geometric interpretation is discussed.

0. Reading

Primary:

• Chandrasekaran, Penington, Witten (CPW), “Large $N$ algebras and generalized entropy,” arXiv:2209.10454, §§2–3 (the two-sided eternal BH and the boundary single-trace algebras).
• Sem I Wk 15 (TFD bridge to Sem II).

Secondary:

• Maldacena, “Eternal black holes in anti-de Sitter,” arXiv:hep-th/0106112 — the foundational TFD/eternal-BH identification.
• Liu, arXiv:2510.07017, especially §§VIII.2–VIII.3 for emergent two-sided causal structure and the conditional algebraic ER=EPR proposal; consult §IX only for the later gravitational-algebra constructions.

Optional research reading:

• Leutheusser & Liu, arXiv:2110.05497 (causal connectability between the two sides).
• Israel, “Thermo-field dynamics of black holes,” Phys. Lett. A 57 (1976) 107 (the original TFD construction in black-hole physics).
• Maldacena & Susskind, “Cool horizons for entangled black holes,” arXiv:1306.0533 (ER=EPR).

Guest-lecture slot promised in the syllabus (30 minutes, after §5): BRST and gravitational observables. The preferred guest topic is the BRST treatment of constraints: why gauge-variant local fields are not yet physical observables, how BRST cohomology organizes the reduction, and where gravitational dressing enters before one assigns a boundary algebra. The guest should distinguish the classical constraint surface, the quantum BRST complex, and the resulting observable algebra; a full BRST construction of the CPW algebra is not claimed here. If no BRST specialist is available, CPW’s microcanonical large-$N$ construction or Liu §VIII.3 may be used as instructor-led alternatives. In either alternative, separate: (i) the one-sided standard-pair theorem; (ii) the commutant/other-boundary identification; and (iii) the geometric bridge interpretation. This slot replaces the research-extension discussion in class; it does not displace the derivation of §§3–4.

1. The two-sided eternal black hole

1.1 The geometry

The maximally extended AdS-Schwarzschild black hole in $(d+1)$ dimensions has the metric

$$d{s}^2=-f(r)d{t}^2+\frac{d{r}^2}{f(r)}+r^{2}d{\Omega }_{d-1}^2,f(r)=1-\frac{\mu }{r^{d-2}}+\frac{r^2}{{\ell }^2},$$

where $\mu =r_h^{d-2}(1+r_h^2/{\ell }^2)$ so that $f(r_h)=0$. Extending through $r=r_h$ gives the full Penrose diagram, with four regions:

• Right exterior (region I): an asymptotically AdS region with a CFT${}_d$ on its boundary at $r\to \infty$.
• Left exterior (region III): a mirror copy of the right exterior, with a second CFT${}_d$ on its boundary.
• Future interior (region II): the part of the geometry behind the horizon, inside the future light cone of the bifurcation surface.
• Past interior (region IV): the mirror of region II.

The two exteriors are causally disconnected — no timelike curve connects them — but they are bulk-geometrically connected through the Einstein–Rosen bridge.

1.2 The bifurcation surface

At the center of the Penrose diagram, the Killing vector ${\partial }_t$ vanishes. This codimension-2 locus is the bifurcation surface $\Sigma$, topologically $S^{d-1}$ (the angular sphere at $r=r_h,t=0$).

The bifurcation surface plays two roles:

• It is the intersection of the future and past horizons; the null horizons themselves emanate from it.
• It is the edge of the entanglement wedge of either boundary: the surface anchoring the Ryu–Takayanagi formula.

1.3 The Hawking temperature

The surface gravity at the horizon gives the Hawking temperature:

$$T_H=\frac{f^{\prime }(r_h)}{4\pi }=\frac{(d-2)+d{r}_h^2/{\ell }^2}{4\pi r_h}.$$

For a large black hole, $r_h\gg \ell$, this becomes $T_H\sim d{r}_h/(4\pi {\ell }^2)$.

The inverse Hawking temperature ${\beta }_H=1/T_H$ enters as the periodicity of imaginary time in the Euclidean continuation of the geometry. It is the universal scale in the two-sided story.

1.4 Maldacena’s duality

Holographic identification (Maldacena 2001). [Stated only.] The two-sided eternal AdS-Schwarzschild black hole is dual, in the semiclassical regime, to two copies of the boundary CFT prepared in the thermofield-double state

$$|{\mathrm{TFD}}_{{\beta }_H}⟩=\frac{1}{\sqrt{Z({\beta }_H)}}\sum _n{e}^{-{\beta }_H{E}_n/2}|n{⟩}_R\otimes |n{⟩}_L,$$

where $|n{⟩}_R,|n{⟩}_L$ are energy eigenstates of the right and left CFTs (with the same Hamiltonian spectrum).

This is the foundational identification of the entire Sem II program. The two-sided geometry has a single state-level dual: the boundary CFTs in the TFD.

2. The two boundary algebras

2.1 Setup

CPW do not begin with an algebra of operators “at one fixed time.” They take the large-$N$ thermal correlation functions of subtracted single-trace operators, build the corresponding GNS Hilbert space, and then take the von Neumann closure of bounded functions of finite products. Equivalently, one may use an appropriate boundary time band/timelike tube whose causal completion captures the exterior algebra. In this week we suppress CPW’s subscript $0$ until the Hamiltonian/clock is adjoined in Week 6, and denote the two GNS algebras by:

• ${\mathcal{A}}_R$: the right thermal GNS von Neumann algebra generated by the large-$N$ single-trace data;
• ${\mathcal{A}}_L$: its left counterpart/commutant in the two-sided representation, once the CPW duality identification is imposed.

These are the two-sided boundary single-trace algebras. In the strict microcanonical large-$N$ GNS regime analyzed by CPW, before the Hamiltonian/clock extension is formed, each is type III${}_1$ on the appropriate Hilbert space (Week 1 §4.2). This is CPW’s limiting construction, not Witten’s separate perturbative $1/N$ algebra.

2.2 Causal complementarity

The right and left boundaries are causally disconnected (§1.1). At the operator-algebra level, this means:

$$[{\mathcal{A}}_R,{\mathcal{A}}_L]=0.$$

Any right-boundary operator commutes with any left-boundary operator. This is the algebraic content of “the two CFTs are non-interacting.”

In holographic terms: causal complementarity on the boundary corresponds to bulk causal separation between the two asymptotic regions through the wormhole.

2.3 The joint algebra is not the modular pair

The algebra generated by both sides is

$${\mathcal{A}}_R\vee {\mathcal{A}}_L:=({\mathcal{A}}_R\cup {\mathcal{A}}_L{)}^{\prime \prime },$$

acting on ${\mathcal{H}}_{\mathrm{TFD}}$. In the standard factorial situation, wedge duality identifies ${\mathcal{A}}_L={\mathcal{A}}_R^{\prime }$, and then ${\mathcal{A}}_R\vee {\mathcal{A}}_L=\mathcal{B}({\mathcal{H}}_{\mathrm{TFD}})$. The TFD vector is not separating for this joint algebra. Tomita–Takesaki theory is applied to the pair $({\mathcal{A}}_R,{\Omega }_{\mathrm{TFD}})$, or symmetrically to $({\mathcal{A}}_L,{\Omega }_{\mathrm{TFD}})$, not to the joint algebra.

At finite $N$, before taking the GNS/large-$N$ limit, one may use the regulator ${\mathcal{H}}_R\otimes {\mathcal{H}}_L$ and $\mathcal{B}({\mathcal{H}}_R)\otimes 1$. The tensor-product notation below belongs to this regulator.

3. The TFD as cyclic-separating

3.1 Statement

TFD standardness. [Finite-dimensional proof; large-$N$ statement imported from CPW.] In the finite-dimensional regulator, a full-Schmidt-rank TFD vector is cyclic and separating for $\mathcal{B}({\mathcal{H}}_R)\otimes 1$. In the large-$N$ GNS representation, the TFD vector is cyclic and separating for the one-sided factor ${\mathcal{A}}_R$.

By symmetry, the same is true for the left one-sided algebra. It is not true for the joint algebra ${\mathcal{A}}_R\vee {\mathcal{A}}_L$ when that algebra equals $\mathcal{B}({\mathcal{H}}_{\mathrm{TFD}})$.

3.2 The Tomita–Takesaki data

By Sem I Wk 5 (Tomita–Takesaki theorem) and cyclic-separating from §3.1, we obtain:

• A modular operator ${\Delta }_{\mathrm{TFD}}$ on ${\mathcal{H}}_R\otimes {\mathcal{H}}_L$, positive self-adjoint.
• A modular conjugation $J_{\mathrm{TFD}}$, antiunitary.
• A modular flow ${\sigma }_t^{\mathrm{TFD}}(a):={\Delta }_{\mathrm{TFD}}^{-it}a{\Delta }_{\mathrm{TFD}}^{it}$ on ${\mathcal{A}}_R$.

The TFD vector is KMS at $\beta =+1$ for ${\sigma }^{\mathrm{TFD}}$, in our upper-strip convention (Sem I Wk 6).

3.3 Modular Hamiltonian = ${\beta }_H(H_R-H_L)$

In finite dimensions (Sem I Wk 15 §1.2), the modular operator of the TFD is

$${\Delta }_{\mathrm{TFD}}={\rho }_R\otimes {\rho }_L^{-1}=e^{-{\beta }_H(H_R-H_L)},$$

and the modular Hamiltonian is

$$K_{\mathrm{TFD}}=-\log {\Delta }_{\mathrm{TFD}}={\beta }_H(H_R-H_L).$$

The partition functions cancel between ${\rho }_R$ and ${\rho }_L^{-1}$. Thus there is no $Z^{-2}$ factor in $\Delta$ and no $2\log Z$ term in the standard Liouvillean.

In the large-$N$ holographic setting, the same identification holds (CPW §2, hypothesis-explicit):

$$\boxed{K_{\mathrm{TFD}}={\beta }_H(H_R-H_L).}$$

The crucial structural feature. The operator implementing the modular automorphism on the standard Hilbert space is the two-sided Liouvillean ${\beta }_H(H_R-H_L)$. The formal one-sided expression $-\log {\rho }_R={\beta }_H{H}_R+\log Z$ generates the same automorphism on right observables in the type-I regulator, but it is not the standard modular operator on the doubled space.

Physical picture: the standard generator versus its formal halves. The modular Hamiltonian $K=-\log \Delta$ of a standard pair is a genuine self-adjoint operator on the GNS Hilbert space. What generally fails in the type-III setting is a reduced density matrix ${\rho }_R$ and a decomposition of $K$ into separately defined self-adjoint operators ${\beta }_H{H}_R$ and $-{\beta }_H{H}_L$ on that same Hilbert space. Stress-tensor expressions for such one-sided pieces may first exist only as quadratic forms and can have divergent fluctuations at the bifurcation surface. The two-sided Liouvillean ${\beta }_H(H_R-H_L)$ is the well-defined standard-space implementer; $H_R$ survives safely as shorthand for the restricted right automorphism, not necessarily as an element or separate operator of the limiting algebra. The finite-dimensional TFD formulas, where both halves exist, are a regulator that makes this distinction visible rather than erasing it.

3.4 The two-sided boost interpretation

In the bulk, the Killing vector ${\partial }_t$ that generates time translation on the right boundary also generates backward time translation on the left boundary (by the time-reversal symmetry of the maximally extended geometry). This is the two-sided boost: $H_R$ on the right minus $H_L$ on the left.

In the free-field Rindler analog (Sem I Wk 15 §2), the analog is the Lorentz boost in the $(x^0,x^1)$-plane: it generates positive time translation in $W_R$ and negative time translation in $W_L$, with the bifurcation surface $\{x^0=x^1=0\}$ fixed.

This identification is a holographic analogue of Bisognano–Wichmann in the stationary eternal-black-hole setting. We use it as part of the holographic dictionary; it is not a general theorem for arbitrary states or boundary regions.

4. KMS at the Hawking temperature

4.1 The KMS condition

The modular flow ${\sigma }_t^{\mathrm{TFD}}=\mathrm{Ad}({\Delta }_{\mathrm{TFD}}^{-it})=\mathrm{Ad}(e^{i{\beta }_{H}t(H_R-H_L)})$ acts on $a\in {\mathcal{A}}_R$ as time evolution by ${\beta }_{H}t$ on the right, with the left-side rotation acting trivially because $a\in {\mathcal{A}}_R$ commutes with $H_L$.

So for $a\in {\mathcal{A}}_R$:

$${\sigma }_t^{\mathrm{TFD}}(a)=e^{i{\beta }_{H}t{H}_R}a{e}^{-i{\beta }_{H}t{H}_R}={\alpha }_{{\beta }_{H}t}^{H_R}(a).$$

Identifying $u={\beta }_{H}t$, the modular flow ${\sigma }^{\mathrm{TFD}}$ on ${\mathcal{A}}_R$ is the ADM time translation with rapidity $u$. The modular time $t$ is the dimensionless time, related to ADM time by $u={\beta }_{H}t$.

4.2 KMS at ${\beta }_H$

The TFD is KMS at modular $\beta =1$ for ${\sigma }^{\mathrm{TFD}}$ (Sem I Wk 6 Theorem 2.1). Translating to ADM time $u={\beta }_{H}t$, this is KMS at inverse temperature ${\beta }_H$ for the ADM time flow:

$${\omega }_{\mathrm{TFD}}(a{\alpha }_u^{H_R}(b))\text{ extends to }0\le \mathrm{Im}u\le {\beta }_H,\text{ with }{\omega }_{\mathrm{TFD}}(a{\alpha }_{u+i{\beta }_H}^{H_R}(b))={\omega }_{\mathrm{TFD}}({\alpha }_u^{H_R}(b)a).$$

The TFD vacuum is KMS at the Hawking temperature for ADM time translation on the right boundary. This is the holographic Unruh effect.

4.3 One automorphism, two representations

The formulas ${\beta }_H{H}_R+\log Z$ and ${\beta }_H(H_R-H_L)$ should not be called two different modular Hamiltonians of the same Tomita pair:

• Density-matrix language: $-\log {\rho }_R={\beta }_H{H}_R+\log Z$ is an operator on ${\mathcal{H}}_R$ that generates the right Gibbs automorphism.
• Standard representation: $-\log {\Delta }_{\mathrm{TFD}}={\beta }_H(H_R-H_L)$ acts on ${\mathcal{H}}_R\otimes {\mathcal{H}}_L$ and annihilates the TFD vector.

They induce the same automorphism on ${\mathcal{A}}_R$ because the left term lies in the commutant. At large $N$, the automorphism survives even though neither one-sided density matrix nor a tensor-product factorization need exist.

In the finite-dimensional regulator, the standard modular Hamiltonian ${\beta }_H(H_R-H_L)$ acts on the doubled Hilbert space. In the large-$N$ algebraic limit, the robust statement is instead that the standard Liouvillean implements the modular automorphism on the GNS Hilbert space; a literal tensor product need not survive.

5. Modular conjugation: ER=EPR

5.1 The modular conjugation $J_{\mathrm{TFD}}$

By Sem I Wk 5, the Tomita–Takesaki theorem gives an antiunitary modular conjugation $J_{\mathrm{TFD}}$ on ${\mathcal{H}}_R\otimes {\mathcal{H}}_L$.

Theorem 5.1 (Tomita commutant theorem). [Stated only — ref: Sem I Wk 5.] For the standard pair $({\mathcal{A}}_R,{\Omega }_{\mathrm{TFD}})$,

$$J_{\mathrm{TFD}}{\mathcal{A}}_R{J}_{\mathrm{TFD}}={\mathcal{A}}_R^{\prime }.$$

This theorem identifies the right algebra with its commutant. The further identification ${\mathcal{A}}_R^{\prime }={\mathcal{A}}_L$ is a duality assumption about the two-sided representation—automatic in the finite-dimensional TFD model, and supplied by wedge duality in the Rindler example. It is not a consequence of cyclicity and separatingness alone. With that extra input, $J_{\mathrm{TFD}}$ exchanges the two represented sides.

5.2 Algebraic ER=EPR

The structural picture:

• ${\mathcal{A}}_R,{\mathcal{A}}_L$ are causally disconnected (cannot communicate).
• $|\mathrm{TFD}⟩$ is cyclic-separating for both.
• $J_{\mathrm{TFD}}$ realizes a canonical isomorphism between them.

These facts are a useful algebraic shadow of the eternal-black-hole construction: they organize the two exteriors and their correlations without presupposing a tensor factorization. They do not give an if-and-only-if criterion for a bulk bridge. Standard pairs with the same commutant relation occur already for opposite Rindler wedges in ordinary QFT, where no dynamical Einstein–Rosen bridge is being inferred.

Maldacena–Susskind’s heuristic (ER=EPR): the TFD/eternal-black-hole example motivates a relation between entanglement and bulk connectivity. Here $J{\mathcal{A}}_{R}J={\mathcal{A}}_L$ is a necessary structural consistency check in the chosen two-sided representation, not a sufficient reconstruction theorem for a wormhole.

5.3 What ER=EPR is not

This is not the claim that any entanglement generates a bulk wormhole geometry. In the eternal-black-hole setting, the modular data are compatible with the known geometric dual; recovering that geometry additionally requires the holographic dictionary, a suitable semiclassical state, and bulk reconstruction.

The Sem II Block 3 (Liu lectures) discusses the algebraic ER=EPR proposal in more detail.

6. Worked example: free-field Rindler-Rindler

The two-sided analog of the Sem I Wk 15 §2 free-field Rindler model. Same machinery, two sides.

6.1 Setup

Take the 4D massless free scalar field on Minkowski space. The commuting wedge algebras generate

$$\mathcal{A}(W_R)\vee \mathcal{A}(W_L)\subset \mathcal{B}(\mathcal{F})$$

on the Fock space $\mathcal{F}$. Reeh–Schlieder makes $|0_M⟩$ cyclic and separating for each one-sided wedge algebra. If wedge duality gives $\mathcal{A}(W_L)=\mathcal{A}(W_R{)}^{\prime }$, their join is $\mathcal{B}(\mathcal{F})$; then the vacuum cannot be separating for that joint algebra.

6.2 Minkowski vacuum is the TFD

Theorem 6.1 (Bisognano–Wichmann; one-sided standard pair). [Stated only — ref: Sem I Wk 10.] For $(\mathcal{A}(W_R),|0_M⟩)$, the modular group is the Lorentz-boost group with the conventional $2\pi$ normalization, and modular conjugation maps the right wedge to its commutant.

In a Rindler-mode regulator one often writes the mnemonic

$$|0_M⟩=\prod _{\omega }\frac{1}{\sqrt{Z_{\omega }}}\sum _{n_{\omega }}{e}^{-\pi n_{\omega }\omega }|n_{\omega }{⟩}_R\otimes |n_{\omega }{⟩}_L.$$

Mode by mode, this is a TFD at inverse boost temperature $2\pi$. In the continuum the displayed infinite product is formal: the Minkowski and Rindler Fock representations are not related by a literal Hilbert-space tensor factorization with a normalizable product vector. The intrinsic statement is the KMS/modular statement. In a standard representation, the corresponding modular Liouvillean is conventionally written

$$K_{0_M}=2\pi (K_R-K_L),$$

where the notation records the opposite boost orientations on the two wedges.

6.3 KMS at $\beta =2\pi$

The Minkowski vacuum is KMS at modular $\beta =1$ for ${\sigma }_t^{0_M}=\mathrm{Ad}(e^{2\pi it(K_R-K_L)})$, which after rescaling $s=2\pi t$ is KMS at $\beta =2\pi$ for the boost flow with rapidity $s$.

This is Bisognano–Wichmann plus the Unruh interpretation. Each one-sided restriction is KMS at inverse boost temperature $2\pi$; the formal TFD expansion is a useful mode-by-mode picture of those correlations.

6.4 Modular conjugation: PCT

By Bisognano–Wichmann (Sem I Wk 10 §6), the modular conjugation is PCT:

$$J_{0_M}=\Theta \cdot R_{\perp }(\pi ),$$

where $\Theta$ is the antiunitary CPT operator and $R_{\perp }(\pi )$ is a $\pi$-rotation in the perpendicular directions $(x^2,x^3)$ in $3+1$ spacetime dimensions—that is, $d=3$ spatial dimensions in the course convention. This swaps $W_R$ with $W_L$; identifying the image algebra with the full commutant also uses wedge duality.

In 2D ($d=1$), $J_{0_M}=\Theta$ (no perpendicular rotation needed), and $\Theta$ acts as $(x^0,x^1)\to (-x^0,-x^1)$, swapping $W_R\leftrightarrow W_L$.

6.5 The free-field analog mirror

The dictionary between the gravitational and free-field analog settings:

CPW (gravitational)	Free-field Rindler-Rindler
Right CFT boundary	Right Rindler wedge $W_R$
Left CFT boundary	Left Rindler wedge $W_L$
Bifurcation surface (BH horizon)	Bifurcation surface $\{x^0=x^1=0\}$
TFD at ${\beta }_H$	Minkowski vacuum: KMS at boost inverse temperature $2\pi$; formal mode TFD
$H_R-H_L$ (two-sided ADM)	$K_R-K_L$ (two-sided boost)
Modular Hamiltonian ${\beta }_H(H_R-H_L)$	Modular Hamiltonian $2\pi (K_R-K_L)$
Modular flow = ADM time difference	Modular flow = Lorentz boost
Modular data used in the ER=EPR discussion	Lorentz-boost modular flow and wedge duality

The right column supplies a controlled analog for the modular and KMS statements. It does not verify the gravitational interpretation: replacing the boost generator by an ADM charge is a holographic input, and the existence of Rindler modular data alone does not imply a wormhole.

7. Why $-H_L$? An invariance argument

A pedagogically helpful aside: why does the difference $H_R-H_L$ appear in the modular Hamiltonian, rather than just $H_R$ on the right?

7.1 TFD invariance

The TFD is invariant under the combined right-minus-left time translation:

$$(H_R-H_L)|\mathrm{TFD}⟩=0.$$

Proof. $H_R$ acts as ${\sum }_n{E}_n|n{⟩}_R⟨n{|}_R$, similarly $H_L$. On the TFD,

$$H_R|\mathrm{TFD}⟩=\sum _n\frac{E_n{e}^{-{\beta }_H{E}_n/2}}{\sqrt{Z}}|n{⟩}_R\otimes |n{⟩}_L=H_L|\mathrm{TFD}⟩,$$

since the sum is symmetric in the index $n$ and $H_L$ acts on the left with the same eigenvalue. Hence $(H_R-H_L)|\mathrm{TFD}⟩=0$. $\square$

So $H_R-H_L$ annihilates the TFD, while $H_R+H_L$ does not. The difference is the boost-like generator that fixes the TFD (and the bifurcation surface in the bulk picture).

7.2 Modular Hamiltonian fixes the modular state

By Tomita–Takesaki, ${\Delta }_{\Omega }\Omega =\Omega$, and hence $K_{\Omega }\Omega =-\log {\Delta }_{\Omega }\Omega =0$. So whatever implements the modular flow of the one-sided TFD standard pair must annihilate the TFD vector. The natural candidate is ${\beta }_H(H_R-H_L)$, consistent with the explicit finite-dimensional calculation in §3.3.

This is a consistency check on the identification $K_{\mathrm{TFD}}={\beta }_H(H_R-H_L)$, not a derivation — but it’s a useful sanity check.

7.3 Bulk interpretation

In the bulk, the Killing vector field $\xi ={\partial }_t$ that generates ADM time on the right boundary continues across the horizon (with sign flip) to generate ADM time backwards on the left boundary. This is the boost Killing vector of the maximally extended geometry, which fixes the bifurcation surface.

On the boundary CFT, this Killing vector projects to $H_R$ on the right and $-H_L$ on the left, i.e., the bulk Killing vector is dual to $H_R-H_L$. This is the boost-symmetry of the eternal BH.

So the standard-space modular Hamiltonian ${\beta }_H(H_R-H_L)$ has a clean bulk interpretation: it implements the boundary action of the bulk boost Killing field. Calling this a “holographic Bisognano–Wichmann” relation is useful shorthand, but it is an identification in the eternal-black-hole code subspace, not a general theorem for arbitrary holographic regions or states.

8. Discussion: two-sided modular structure

8.1 What the TFD knows

The TFD vector defines correlated restrictions to ${\mathcal{A}}_R$ and ${\mathcal{A}}_L$. The standard pair $({\mathcal{A}}_R,{\Omega }_{\mathrm{TFD}})$ determines:

1. The right CFT’s reduced state: a thermal Gibbs state at ${\beta }_H$ (the bulk Hawking effect from the perspective of an observer outside the BH).
2. The modular Hamiltonian ${\beta }_H(H_R-H_L)$, which equals the bulk Killing flow at the bifurcation surface.
3. The commutant $J{\mathcal{A}}_{R}J={\mathcal{A}}_R^{\prime }$; identifying it with ${\mathcal{A}}_L$ requires the two-sided duality input.

In the eternal-black-hole example these data match Hawking thermality and boost symmetry. The bridge interpretation comes from the holographic state/geometry dictionary, not from Tomita–Takesaki theory by itself.

8.2 What’s the same as Block 1

The Block 2 setup is the two-sided extension of Block 1:

Block 1 (single-sided)	Block 2 (two-sided)
Algebra ${\mathcal{A}}_R$ alone	Two commuting represented sides ${\mathcal{A}}_R,{\mathcal{A}}_L$
Cyclic-separating: one-sided thermal state	TFD vector standard for each one-sided algebra
Right modular automorphism: formally generated by ${\beta }_H{H}_R$	Standard-space implementer: ${\beta }_H(H_R-H_L)$
Modular flow on ${\mathcal{A}}_R$: ADM time on right	Same (when restricted to ${\mathcal{A}}_R$)
Type III${}_1$: yes	Yes (each side)
KMS at ${\beta }_H$: yes	Yes (each side)

For the single-side restriction to ${\mathcal{A}}_R$ alone, the two stories are equivalent — the modular flow on ${\mathcal{A}}_R$ generated by ${\beta }_H(H_R-H_L)$ equals the one generated by ${\beta }_H{H}_R$ (since $H_L$ commutes with ${\mathcal{A}}_R$).

The new structural content in Block 2 is the two-sided representation: how the one-sided algebra and its commutant are related, when the commutant can be identified with the other boundary algebra, and how dressing works in that setting.

8.3 What’s new in Block 2

Beyond Block 1, Block 2 will address:

1. The CPW dressing (Week 6): both algebras dressed in a way that respects the two-sided structure.
2. Entropy of the relevant right/dressed algebra (Week 7): in CPW’s construction this entropy, evaluated on the stated semiclassical states, matches the generalized entropy of the black-hole bifurcation surface up to the trace-normalization constant. It is not an entropy of the full joint algebra.
3. Bell–CHSH between two sides (Week 8): the algebraic content of “the TFD is maximally entangled across the bifurcation horizon.”

These are the new things the two-sided structure unlocks.

9. What to take away

• Two-sided eternal BH (Maldacena 2001): dual to the boundary CFTs in the TFD state at the Hawking temperature.
• TFD as cyclic-separating (stated only, hypothesis-explicit): for each of ${\mathcal{A}}_R$ and ${\mathcal{A}}_L$ at large $N$. This licenses Tomita–Takesaki for either one-sided standard pair, not for their join.
• Modular Hamiltonian = ${\beta }_H(H_R-H_L)$: the two-sided modular Hamiltonian, projecting to ${\beta }_H{H}_R$ when restricted to ${\mathcal{A}}_R$ alone.
• KMS at ${\beta }_H$ for ADM time translation on either side. Holographic Unruh effect.
• Modular conjugation = PCT/boost-reflection in the free-field analog; its identification with the other boundary algebra in gravity is conditional on the two-sided duality input.
• Free-field Rindler analog: an explicit check of modular flow, KMS thermality, and wedge duality—not a derivation of bulk connectivity.
• Why $-H_L$: the TFD is annihilated by $H_R-H_L$, not by $H_R$ alone. This is the algebraic content of “the TFD is boost-invariant.”

10. Looking ahead

Week 6 builds the CPW dressed algebra in its microcanonical large-$N$ setting and distinguishes the right modular charge from the two-sided standard Liouvillean. Week 7 studies the entropy of the relevant right/dressed algebra and CPW’s source-level match to generalized entropy. The free-field two-sided Rindler model checks the algebra/clock formulas and inner-unitary cancellations; it does not derive a gravitational area coefficient or verify an area law. Week 8 closes the block with Bell–CHSH between the two sides.

11. Problem set

Core problems.

1. TFD reduced state. Verify directly that ${\mathrm{Tr}}_L(|{\mathrm{TFD}}_{{\beta }_H}⟩⟨{\mathrm{TFD}}_{{\beta }_H}|)=e^{-{\beta }_H{H}_R}/Z({\beta }_H)$ in finite dimensions. Conclude that the right CFT’s reduced state is the Gibbs state at ${\beta }_H$.

2. Modular operator of the TFD. In finite dimensions, verify ${\Delta }_{\mathrm{TFD}}={\rho }_R\otimes {\rho }_L^{-1}$ with ${\rho }_R={\rho }_L=e^{-{\beta }_{H}H}/Z$. Compute $K_{\mathrm{TFD}}=-\log {\Delta }_{\mathrm{TFD}}$ and show that the two normalization constants cancel, leaving ${\beta }_H(H_R-H_L)$.

3. The TFD is annihilated by $H_R-H_L$. Verify $(H_R-H_L)|{\mathrm{TFD}}_{{\beta }_H}⟩=0$ from the explicit form. Why does this not hold for $H_R+H_L$?

4. Two-sided modular flow. Show that for $a\in {\mathcal{A}}_R\otimes 1$, the modular flow ${\sigma }_t^{\mathrm{TFD}}(a)=e^{i{\beta }_{H}t(H_R-H_L)}a{e}^{-i{\beta }_{H}t(H_R-H_L)}$ reduces to $e^{i{\beta }_{H}t{H}_R}a{e}^{-i{\beta }_{H}t{H}_R}$ (one-sided ADM time on the right). Where does $H_L$ go?

5. Modular conjugation in a wedge. Using the Bisognano–Wichmann statement supplied in the notes, show that in two dimensions the geometric action of $J$ sends $W_R$ to $W_L$. State clearly which additional assumption turns this geometric inclusion into the equality $\mathcal{A}(W_L)=\mathcal{A}(W_R{)}^{\prime }$.

Starred problems.

6*. KMS at ${\beta }_H$. Take a thermal two-point function in a 2D CFT on the nonrotating BTZ boundary and verify its KMS continuation with inverse temperature ${\beta }_H=2\pi {\ell }^2/r_h$. Distinguish KMS exchange of operator order from simple periodicity. (Hint: use the conformal map from the plane to the thermal cylinder.)

7*. Read CPW §2. Identify the precise hypotheses CPW use for the two-sided large-$N$ structure. Compare with our statement in §3.1.

8*. Read Maldacena 2001 §§1–3. Identify the precise argument for the eternal BH ↔ TFD duality. Where does the holographic dictionary enter, and where is it taken as input?

9*. Boundary Killing vector. Sketch why the restriction of the standard modular flow to ${\mathcal{A}}_R$ is the right ADM-time automorphism with parameter $u={\beta }_{H}t$. Use ${\beta }_H(H_R-H_L)$ in the finite regulator, and explain why writing ${\beta }_H{H}_R$ for the restricted action does not assert that a separate $H_R$ exists in the limiting GNS algebra. (Hint: the boundary projection of ${\partial }_t$ generates ADM time on the right; ${\beta }_H$ comes from the surface gravity.)

Project problems.

10. Connection to Liu §VIII.3. Liu (arXiv:2510.07017) §VIII.3 discusses a conditional algebraic ER=EPR proposal. Write a five-page exposition comparing CPW’s two-sided modular structure with Liu’s distinction between classical and quantum-volatile bulk descriptions. Record the semiclassical and ${\alpha }^{\prime }$ qualifications rather than replacing the proposal by the generic identity $J\mathcal{A}J={\mathcal{A}}^{\prime }$.

11. Higher-genus generalization. Sketch what changes in the two-sided story if we replace the eternal AdS-Schwarzschild by an eternal AdS black hole at higher genus (e.g., a torus boundary). What’s the analog of the TFD? How does the modular Hamiltonian change?

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Notes prepared for the algebraic-QFT course, Semester II Block 2. Last revised 2026-08-24.