Week 6 — Modular Flow and the KMS Condition

Last week we built $\Delta$ and $J$. The theorem said that ${\Delta }^{-it}$ does something highly non-obvious: it sends the algebra back to itself. This week we give that fact its physical name. The automorphisms ${\sigma }_t(a)={\Delta }^{-it}a{\Delta }^{it}$ are the modular flow, and the state that produced them is KMS at inverse temperature $1$. In finite dimensions this reduces to Gibbs calculus. In type III it supplies the standard equilibrium language that survives without an intrinsic Gibbs density matrix. Takesaki’s uniqueness theorem then says that, at the fixed normalization $\beta =1$, this is the unique $\sigma$-weakly continuous flow for which the faithful normal state is KMS.

How to use this chapter

• In class: begin with ${\rho }^{it}A{\rho }^{-it}$ in matrix standard form, translate it into the course’s ${\Delta }^{-it}$ convention, and then prove the KMS boundary relation before discussing uniqueness and outerness.
• For self-study: make a one-page sign dictionary linking $\Delta$, the modular Hamiltonian, physical time, and inverse temperature. Recheck every later example against the matrix calculation.
• Instructor checkpoint: separate point-norm continuity from $\sigma$-weak continuity, uniqueness of a KMS dynamics from a rescaling of its time parameter, and a state’s centralizer from the centre of the algebra.

0. Reading

Primary:

• Bratteli & Robinson, Vol. I, §2.5 (modular automorphisms; basic structure).
• Bratteli & Robinson, Vol. II, §5.3 (KMS states and the modular/KMS link; equilibrium statistical mechanics in operator algebras).

Secondary:

• Takesaki, Theory of Operator Algebras II, ch. VIII (Takesaki uniqueness theorem and consequences).
• Haag, Local Quantum Physics, ch. V (the QFT interpretation and the thermal-time hypothesis).

Optional research reading:

• Connes & Rovelli, “Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories,” CQG 11 (1994) 2899 — the thermal-time hypothesis.
• Borchers, “On revolutionizing quantum field theory with Tomita’s modular theory,” J. Math. Phys. 41 (2000) 3604 — AQFT viewpoint.
• Witten, “Notes on some entanglement properties of QFT,” arXiv:1803.04993, §§3–4 — physics applications of modular theory.

1. Modular automorphisms

1.1 Construction

Let $\mathcal{M}\subset \mathcal{B}(\mathcal{H})$ be a von Neumann algebra and let $\Omega$ be cyclic and separating. Week 5 produced the closed Tomita operator

$$S=J{\Delta }^{1/2}$$

and Tomita-Takesaki gave (in our sign convention)

$${\Delta }^{-it}\mathcal{M}{\Delta }^{it}=\mathcal{M},t\in \mathbb{R}.$$

This says: the one-parameter family of unitaries ${\Delta }^{-it}$ implements automorphisms of $\mathcal{M}$ by conjugation.

Definition 1.1. The modular automorphism group of $(\mathcal{M},\Omega )$ is

$${\sigma }_t^{\Omega }(a):={\Delta }^{-it}a{\Delta }^{it},a\in \mathcal{M},t\in \mathbb{R}.$$

Equivalently, if $\omega (a)=⟨\Omega ,a\Omega ⟩$, we write ${\sigma }_t^{\omega }$ — the modular automorphism group of the state $\omega$. We use the two notations interchangeably; the state is the algebraic object, the vector is its Hilbert-space realization.

1.2 Group properties

Proposition 1.2. [Stated only — refs: B-R Vol. I §2.5.1.] The maps ${\sigma }_t^{\omega }$ form a $\sigma$-weakly continuous one-parameter group of $\ast$-automorphisms of $\mathcal{M}$:

$${\sigma }_{t+s}^{\omega }={\sigma }_t^{\omega }\circ {\sigma }_s^{\omega },{\sigma }_0^{\omega }=\mathrm{id},{\sigma }_t^{\omega }(ab)={\sigma }_t^{\omega }(a){\sigma }_t^{\omega }(b),{\sigma }_t^{\omega }(a^{\ast })={\sigma }_t^{\omega }(a{)}^{\ast }.$$

The group property follows formally from ${\Delta }^{-it}{\Delta }^{-is}={\Delta }^{-i(t+s)}$ (functional calculus of the positive self-adjoint operator $\Delta$). The structural content — that conjugation by ${\Delta }^{-it}$ preserves $\mathcal{M}$ — is Tomita-Takesaki.

$\sigma$-weak continuity is the natural regularity condition: the map $t\mapsto {\omega }^{\prime }({\sigma }_t^{\omega }(a))$ is continuous for every $a\in \mathcal{M}$ and every normal functional ${\omega }^{\prime }$. This is weaker than point-norm continuity. Tomita-Takesaki guarantees the former; it does not guarantee the latter. Norm-continuous and entire analytic elements form useful dense subalgebras on which stronger manipulations are legitimate.

1.3 $\omega$-invariance

The modular flow leaves the defining state invariant:

$$\omega ({\sigma }_t^{\omega }(a))=\omega (a)$$

for all $a\in \mathcal{M}$ and $t\in \mathbb{R}$. This is because $\Delta \Omega =\Omega$ (the vacuum is in the kernel of $\log \Delta$, which is the abstract content of ”$\Omega$ is invariant under the modular flow”). Concretely:

$$\omega ({\sigma }_t^{\omega }(a))=⟨\Omega ,{\Delta }^{-it}a{\Delta }^{it}\Omega ⟩=⟨{\Delta }^{it}\Omega ,a{\Delta }^{it}\Omega ⟩=⟨\Omega ,a\Omega ⟩=\omega (a).$$

So ${\sigma }_t^{\omega }$ is an automorphism of $\mathcal{M}$ that fixes the state $\omega$. Every modular flow is a one-parameter family of “$\omega$-preserving automorphisms.”

2. The modular KMS theorem

Week 4 defined the KMS condition using the upper strip:

$$0\le \mathrm{Im}z\le \beta ,$$

with boundary values

$$F_{a,b}(t)=\omega (a{\sigma }_t(b)),F_{a,b}(t+i\beta )=\omega ({\sigma }_t(b)a).$$

This is a property of a state-flow pair $(\omega ,\sigma )$ at a specific $\beta >0$. The modular theorem sets $\beta =1$ and identifies the unique candidate flow.

2.1 The theorem

Theorem 2.1 (Modular KMS theorem). [Stated only — refs: B-R Vol. II §5.3.1; Takesaki Vol. II Ch. VIII §1.] Let $\mathcal{M}$ be a von Neumann algebra and let $\omega$ be a faithful normal state. Then $\omega$ is KMS at $\beta =1$ for its modular automorphism group ${\sigma }_t^{\omega }$.

Explicitly, for $a,b\in \mathcal{M}$ there is a bounded continuous function $F_{a,b}$ on the closed strip $\{0\le \mathrm{Im}z\le 1\}$, holomorphic in the interior, such that

$$F_{a,b}(t)=\omega (a{\sigma }_t^{\omega }(b)),F_{a,b}(t+i)=\omega ({\sigma }_t^{\omega }(b)a).$$

2.2 The uniqueness theorem

The KMS-modular link goes both ways:

Theorem 2.2 (Takesaki uniqueness). [Stated only — refs: Takesaki Vol. II, ch. VIII, modular/KMS section.] Conversely, let $\omega$ be faithful and normal on $\mathcal{M}$. If a $\sigma$-weakly continuous one-parameter automorphism group ${\alpha }_t$ of $\mathcal{M}$ makes $\omega$ a KMS state at $\beta =1$, then

$${\alpha }_t={\sigma }_t^{\omega }.$$

This is why the modular flow is not merely one possible equilibrium dynamics. For a faithful normal state, it is the unique $\sigma$-weakly continuous one-parameter automorphism group for which that state is KMS at $\beta =1$. The pair $(\mathcal{M},\omega )$ canonically determines a flow; no external Hamiltonian or time-translation is needed.

Physical picture. The uniqueness theorem inverts the usual logic of statistical mechanics. Ordinarily one starts with a clock (a Hamiltonian), and asks which states are in equilibrium with respect to it; many states qualify (one per temperature, per phase). Takesaki uniqueness says the reverse question has a unique answer: given the equilibrium state, there is exactly one clock with respect to which it is thermal at $\beta =1$. A sufficiently complete thermometer determines the clock. The physical mechanism is the detailed-balance identity of Week 4 §2.2: the state’s correlation functions $\omega (a{\sigma }_t(b))$ know all the Boltzmann ratios between all pairs of “levels,” and a full set of Boltzmann ratios at known temperature reconstructs the energy differences — hence the generator, hence the flow. What is remarkable is that this reconstruction survives in type III, where there are no levels, no $\rho$, and no trace: the strip analyticity alone carries the full dynamical information.

2.3 Why this matters

The combination of Theorems 2.1 and 2.2 is a rigidity result. Two consequences:

• The temperature fixes the normalization of time. Once we have a faithful normal state, there is exactly one $\sigma$-weakly continuous flow that makes it KMS at modular $\beta =1$. More generally, if $\omega$ is KMS at inverse temperature $\beta >0$ for ${\alpha }_t$, then ${\alpha }_{\beta t}={\sigma }_t^{\omega }$. The same nontrivial flow cannot normally make the same faithful state KMS at two different inverse temperatures; the tracial state with trivial flow is the important degenerate exception.

• State-dependent vs. external Hamiltonians. In ordinary quantum mechanics, the time evolution ${\alpha }_t=\mathrm{Ad}(e^{itH})$ is determined by an externally chosen Hamiltonian $H$; states are then “thermal” for ${\alpha }_t$ at appropriate $\beta$. In modular theory, the flow is determined by the state — flipping the relationship. This is the algebraic root of the Connes–Rovelli “thermal time hypothesis” (§7).

3. Model proof: finite-dimensional KMS

Let $\mathcal{M}=M_n(\mathbb{C})$, let $\rho >0$ with $\mathrm{Tr}\rho =1$, and set

$${\omega }_{\rho }(a)=\mathrm{Tr}(\rho a).$$

The state ${\omega }_{\rho }$ is faithful (since $\rho >0$) and normal. The GNS representation can be carried out on the Hilbert–Schmidt space ${\mathcal{H}}_{\mathrm{HS}}=M_n(\mathbb{C})$ with cyclic-separating vector ${\Omega }_{\rho }={\rho }^{1/2}$, and the modular operator computed in Week 5 is ${\Delta }_{\rho }=L_{\rho }{R}_{{\rho }^{-1}}$. The modular automorphism on $\mathcal{M}$ acts by

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

(Week 5 §5.3, in our convention).

3.1 Verifying KMS at $\beta =1$

Proposition 3.1. [Model proof.] ${\omega }_{\rho }$ is KMS at $\beta =1$ for ${\sigma }^{\rho }$ in the upper-strip convention.

Proof. For $a,b\in M_n(\mathbb{C})$, define

$$F_{a,b}(z)=\mathrm{Tr}\left(\rho a{\rho }^{-iz}b{\rho }^{iz}\right).$$

Since $\rho >0$ has finite spectrum, ${\rho }^{-iz}=e^{-iz\log \rho }$ is an entire matrix-valued function. Therefore $F_{a,b}$ is entire, hence holomorphic on the strip and continuous on its boundary.

Boundary value at $z=t$.

$$F_{a,b}(t)=\mathrm{Tr}(\rho a{\rho }^{-it}b{\rho }^{it})=\mathrm{Tr}(\rho a{\sigma }_t^{\rho }(b))={\omega }_{\rho }(a{\sigma }_t^{\rho }(b)).$$

Boundary value at $z=t+i$. Compute ${\rho }^{-i(t+i)}={\rho }^{-it+1}=\rho {\rho }^{-it}$ and ${\rho }^{i(t+i)}={\rho }^{it-1}={\rho }^{-1}{\rho }^{it}$. So

$$\begin{array}{rl}{F}_{a,b}(t+i)& =\mathrm{Tr}\left(\rho a\rho {\rho }^{-it}b{\rho }^{-1}{\rho }^{it}\right)\\ & =\mathrm{Tr}\left(a\rho {\rho }^{-it}b{\rho }^{-1}{\rho }^{it}\rho \right)\text{(cyclicity, one rotation)}\\ & =\mathrm{Tr}\left(a\rho {\rho }^{-it}b{\rho }^{it}\right)\text{(}{\rho }^{-1}{\rho }^{it}\rho ={\rho }^{it}\text{, functions of }\rho \text{)}\\ & =\mathrm{Tr}\left(\rho {\rho }^{-it}b{\rho }^{it}a\right)\text{(cyclicity)}\\ & =\mathrm{Tr}\left(\rho {\sigma }_t^{\rho }(b)a\right)\\ & ={\omega }_{\rho }({\sigma }_t^{\rho }(b)a).\end{array}$$

So $F_{a,b}(t)={\omega }_{\rho }(a{\sigma }_t^{\rho }(b))$ and $F_{a,b}(t+i)={\omega }_{\rho }({\sigma }_t^{\rho }(b)a)$ — exactly the KMS at $\beta =1$ identity in the upper strip. $\square$

3.2 Discussion of the proof

The proof is a sequence of trace cyclicities, exploiting that ${\rho }^{-iz}$ and ${\rho }^{iz}$ are both functions of $\rho$ and commute with $\rho$ itself. The crucial moment is when we use ${\rho }^{-1}{\rho }^{it}\rho ={\rho }^{it}$ — this is the place where the analytic continuation in $z$ to the upper strip gives a clean boundary value.

The classroom version is finite-dim; the full theorem (2.1) says the same strip identity persists when the trace expression no longer exists — i.e., when $\mathcal{M}$ is type II or III. In those cases, the proof uses the analytic structure of ${\Delta }^{-iz}$ on a dense subspace of “analytic-for-$\Delta$” vectors, but the structural identity is the same.

The general mechanism, in one paragraph. Where the trace computation above uses cyclicity, the general proof uses the two relations from Week 5: ${\Delta }^{1/2}a\Omega =J{a}^{\ast }\Omega$ (from $S=J{\Delta }^{1/2}$) and $\Delta \Omega =\Omega$. For $a,b\in \mathcal{M}$ define $F_{a,b}(z):=⟨a^{\ast }\Omega ,{\Delta }^{-iz}b\Omega ⟩$. On vectors analytic for $\Delta$ this is holomorphic on the strip; at the real boundary, $F(t)=⟨a^{\ast }\Omega ,{\Delta }^{-it}b{\Delta }^{it}\Omega ⟩=\omega (a{\sigma }_t(b))$. At the top boundary, insert ${\Delta }^{-i(t+i)}=\Delta {\Delta }^{-it}$ and use $\Delta =S^{\ast }S=(J{\Delta }^{1/2}{)}^{\ast }(J{\Delta }^{1/2})$ to convert the extra ${\Delta }^{1/2}$ factors into Tomita flips: $⟨a^{\ast }\Omega ,\Delta {\sigma }_t(b)\Omega ⟩=⟨{\Delta }^{1/2}{a}^{\ast }\Omega ,{\Delta }^{1/2}{\sigma }_t(b)\Omega ⟩=⟨Ja\Omega ,J{\sigma }_t(b{)}^{\ast }\Omega ⟩=⟨{\sigma }_t(b{)}^{\ast }\Omega ,a\Omega ⟩=\omega ({\sigma }_t(b)a)$, using anti-unitarity of $J$ in the second-to-last step. The KMS identity is thus a two-line consequence of the polar decomposition — all the analytic effort of the full proof goes into controlling domains, i.e. into showing enough vectors are analytic for $\Delta$ to make the formal manipulation legitimate. [Sketched.]

4. Relation to physical Gibbs evolution

The finite-dim modular flow has a clean physical interpretation when $\rho$ is a Gibbs density matrix.

4.1 Identification

Suppose $\rho$ is a Gibbs density matrix at physical inverse temperature $\beta$:

$${\rho }_{\beta }=\frac{e^{-\beta H}}{Z},Z=\mathrm{Tr}(e^{-\beta H}).$$

Then ${\rho }_{\beta }^{-it}=Z^{it}{e}^{i\beta tH}$, and the modular flow is

$${\sigma }_t^{\rho }(a)={\rho }_{\beta }^{-it}a{\rho }_{\beta }^{it}=Z^{it}{e}^{i\beta tH}a{e}^{-i\beta tH}{Z}^{-it}=e^{i\beta tH}a{e}^{-i\beta tH}.$$

The scalar phases cancel.

The physical Heisenberg evolution at inverse temperature $\beta$ is

$${\alpha }_s(a)=e^{isH}a{e}^{-isH}.$$

Comparing:

$${\sigma }_t^{\rho }={\alpha }_{\beta t}.$$

4.2 Modular time vs. physical time

So in our upper-strip convention, modular time is the inverse-temperature rescaling of physical time, in the same direction. The Gibbs state at physical inverse temperature $\beta$ is KMS at the physical $\beta$ for $\alpha$ (Week 4 §1.1 derivation) and equivalently KMS at ${\beta }_{\mathrm{mod}}=1$ for ${\sigma }^{\rho }$ (the modular flow obtained by rescaling).

The translation:

$$\text{Gibbs state at inverse temperature }\beta ⟺\text{KMS state at }{\beta }_{\mathrm{mod}}=1\text{ for modular time}.$$

4.3 Why the rescaling?

This is the algebraic content of the “thermal time hypothesis” (§7): there is a canonical modular time scale, intrinsic to the state, which differs from physical time by the temperature factor. Equivalently: in a thermal system, “modular time” is “physical time, measured in units of the thermal coherence time $1/(k_{B}T)$.”

In QFT (Bisognano–Wichmann, Week 10), the same idea gives: modular time on a Rindler wedge is the boost-parameter time, related to the proper time of an accelerated observer by the Unruh-temperature factor.

5. Three benchmark examples

5.1 Non-tracial type-I state

Let

$$\rho =\mathrm{diag}(p_1,\dots ,p_n),p_i>0,\sum _i{p}_i=1.$$

For the matrix unit $E_{ij}=|e_i⟩⟨e_j|$,

$${\sigma }_t^{\rho }(E_{ij})={\rho }^{-it}{E}_{ij}{\rho }^{it}=p_i^{-it}{E}_{ij}{p}_j^{it}={\left(\frac{p_j}{p_i}\right)}^{it}{E}_{ij}.$$

The diagonal matrix units $E_{ii}$ are fixed by the flow; the off-diagonal matrix units $E_{ij}$ for $i\ne j$ rotate with frequencies $\log p_j-\log p_i$.

Periodicity. If all log-ratios $\log p_j-\log p_i$ are commensurate (rational multiples of a common number), the flow is periodic. Generically (for irrational ratios), the flow is not periodic — it is an almost-periodic flow on the matrix algebra.

Physical picture. This computation is spectroscopy without an externally supplied Hamiltonian. Off-diagonal matrix units precess at frequencies $\log p_j-\log p_i$, read from the state populations. The fixed-point algebra

$${\mathcal{M}}^{\omega }=\{a:{\sigma }_t^{\omega }(a)=a\text{for all }t\}$$

is the centralizer of the state. Its size is state-dependent—even on a type III${}_1$ factor it need not be small—so one should not infer the factor subtype from one centralizer alone. The crossed product in Block D has a canonical semifinite trace for a different structural reason: adjoining the modular translation turns the outer modular action into an inner one on the enlarged algebra.

Powers state. For the Powers state ${\rho }_{\lambda }=\mathrm{diag}(1,\lambda )/(1+\lambda )$ with $\lambda \in (0,1)$, the ratio is $\lambda$, and the modular flow has period $T_{\lambda }=2\pi /|\log \lambda |$ (Week 4 §7.4). On the infinite tensor product, this period survives and becomes the Connes T-invariant of the type III${}_{\lambda }$ factor (Week 12).

5.2 Tracial state

For the normalized trace on $M_n(\mathbb{C})$,

$$\rho =\frac{1}{n}1.$$

Then ${\rho }^{-it}=n^{it}1$, and

$${\sigma }_t^{\rho }(a)={\rho }^{-it}a{\rho }^{it}=(n^{it}1)a(n^{-it}1)=a.$$

The trace is KMS for the trivial flow at every inverse temperature (Week 4 §5), and the modular flow agrees: ${\sigma }_t^{\tau }=\mathrm{id}$. In modular language, ${\Delta }_{\tau }=1$.

The same statement holds in the tracial GNS representation of a finite factor $\mathcal{M}$ (e.g., the hyperfinite II${}_1$ factor). On $L^2(\mathcal{M},\tau )$,

$$S[a]=[a^{\ast }],\Delta =1,{\sigma }_t^{\tau }=\mathrm{id}.$$

This is the type II${}_1$ situation: the modular dynamics is degenerate. Tracial states have no preferred thermal time — the modular clock is at rest.

5.3 Powers factor

Return to the Powers state from Week 4:

$${\rho }_{\lambda }=\frac{1}{1+\lambda }\left(\begin{array}{cc}1& 0\\ 0& \lambda \end{array}\right),0<\lambda <1.$$

On a single copy of $M_2(\mathbb{C})$,

$${\sigma }_t^{\lambda }(E_{12})={\lambda }^{it}{E}_{12},{\sigma }_t^{\lambda }(E_{21})={\lambda }^{-it}{E}_{21}.$$

(Here $E_{12}=|e_1⟩⟨e_2|$, with $\rho$-eigenvalues $p_1=1/(1+\lambda )$, $p_2=\lambda /(1+\lambda )$, so $p_2/p_1=\lambda$.)

The local modular flow has period

$$T_{\lambda }=\frac{2\pi }{|\log \lambda |}.$$

On the infinite tensor product. The modular flow on ${\mathcal{R}}_{\lambda }$ (the type III${}_{\lambda }$ factor of Week 4 §7) is the product of single-site flows. The period $T_{\lambda }$ survives to the infinite-product GNS representation — every single site returns to identity at the same modular time, so the global flow is periodic at $T_{\lambda }$.

Theorem 5.1 (Powers factor plus later Connes invariant). [Stated only — refs as Week 4 §7.3.] In the infinite-tensor-product GNS representation of ${\omega }_{\lambda }^{\otimes \infty }$, the von Neumann closure is the hyperfinite type III${}_{\lambda }$ factor. Connes’ later classification expresses its algebra-level invariant as $S({\mathcal{R}}_{\lambda })=\{0\}\cup \{{\lambda }^n:n\in \mathbb{Z}\}$.

We use the explicit product-state flow as a guide, not as a proof of type. Its periodicity survives the infinite limit and is consistent with the later invariant, but passing from one state’s modular data to an algebra-level classification requires the quoted theorem.

6. Intrinsic character of modular flow

The finite-dimensional formula

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

looks representation-dependent because it uses a matrix $\rho$. The general theorem says the opposite.

6.1 Representation independence

Proposition 6.1. [Stated only — refs: Takesaki Vol. II Ch. VIII §1.] Let $\omega$ be a faithful normal state on an abstract von Neumann algebra $\mathcal{M}$. The modular automorphism group ${\sigma }_t^{\omega }$ depends only on the pair $(\mathcal{M},\omega )$, not on the particular faithful representation used to construct it.

More concretely: if ${\pi }_i:\mathcal{M}\to \mathcal{B}({\mathcal{H}}_i)$ are two faithful normal representations carrying cyclic-separating vectors for the same state $\omega$, the resulting modular flows on $\mathcal{M}$ agree (after identifying the two representations via the abstract algebra).

6.2 Why this matters for QFT

This is the main reason modular theory belongs in an AQFT course. Local type-III algebras do not have intrinsic density matrices relative to a trace on the local algebra: there is no faithful normal semifinite trace $\tau$ on $\mathcal{M}$ with respect to which one writes $\omega (a)=\tau (\rho a)$. In a concrete embedding $\mathcal{M}\subset \mathcal{B}(\mathcal{H})$, a vector state can of course be written using the ambient rank-one operator $|\Omega ⟩⟨\Omega |$ and the trace on $\mathcal{B}(\mathcal{H})$; that ambient formula does not supply a trace or density matrix internal to $\mathcal{M}$. But:

• Faithful normal states exist (the vacuum, by Reeh–Schlieder).
• Faithful normal states have canonical modular flows (Tomita–Takesaki + Proposition 6.1).
• These flows are intrinsic to the algebra-state pair; they do not depend on any representation choice.

So even when the finite-dim formula ${\sigma }_t^{\rho }(a)={\rho }^{-it}a{\rho }^{it}$ doesn’t literally apply (because $\rho$ doesn’t exist), the modular flow does. This is the technical content of “modular theory works in any type.”

6.3 Computing modular flow in practice

In practice, one often computes the modular flow as follows:

1. Find a Hilbert-space realization with $\Omega$ cyclic-separating (e.g., the GNS representation, or the standard form of $\mathcal{M}$, or a physical representation like Fock space for QFT).
2. Compute $\Delta$ explicitly in that representation (via the Tomita operator + polar decomposition, Week 5).
3. Define ${\sigma }_t^{\omega }(a)={\Delta }^{-it}a{\Delta }^{it}$ and check that the result lies in $\mathcal{M}$.

For a Wightman QFT on the Rindler wedge, step 2 is solved by Bisognano–Wichmann: $\Delta =e^{-2\pi K}$ with $K$ the boost generator (using $U(\Lambda (s))=e^{isK}$). This is the principal model-independent geometric example. Other explicit modular flows exist—for instance for vacuum balls in conformal field theory and in certain free-field or half-sided-modular settings—but they require additional symmetry or structure.

7. Type III and thermal time

For type I algebras, modular flow is density-matrix dynamics in disguise. For finite type II algebras with the trace, it is trivial. For type III algebras, there is no faithful normal trace, and modular flow cannot be dismissed as a trace artifact.

7.1 Careful wording

A few precision points worth dwelling on:

• A type III algebra can have many one-parameter automorphism groups. Modular flow is not “the only possible time.” For example, the Bisognano–Wichmann modular flow on $\mathcal{A}(W_R)$ is the boost; one could also consider other Poincaré-subgroup actions on the same algebra. Each gives a different one-parameter dynamics.

• For each faithful normal state $\omega$, ${\sigma }_t^{\omega }$ is the canonical time flow attached to $(\mathcal{M},\omega )$. Two different states give two different modular flows.

• The two flows are related by the Connes cocycle. If one changes the state, the modular flow changes, but Connes’ Radon–Nikodym theorem (Week 7) says different faithful normal states have modular flows related by an inner cocycle in $\mathcal{M}$. The cocycle is the noncommutative “Radon–Nikodym derivative.”

7.2 The thermal-time hypothesis

This is the mathematical core behind the Connes-Rovelli (1994) thermal-time hypothesis:

In a generally covariant setting, where no external Hamiltonian is preferred — for example, in canonical quantum gravity, where there is no preferred time coordinate — the state itself can supply a distinguished flow. Modular time is the algebraic substitute for external clock time.

The hypothesis: in any physical situation where one has an algebra of observables and a state (vacuum, thermal, ground state, …), the modular flow of that state is the natural time evolution. There is no need to specify a Hamiltonian; the state does it for you.

This is philosophical but mathematically precise. Whether it gives the right physical time in any given situation is an open question. In QFT, Bisognano–Wichmann says that for a wedge in Wightman QFT with the Minkowski vacuum, the modular flow is the boost — which matches the natural “time” for accelerated observers. In curved spacetime / gravity, the picture is more delicate; this is one of the motivations for the Sem II Witten/CPW/AAJ program.

7.3 The AQFT slogan

For AQFT, the slogan becomes practical:

“In the Rindler wedge, Bisognano–Wichmann identifies the vacuum modular flow with Lorentz boosts. In that special case, the abstract modular clock coincides with a geometric flow.”

This is the structural anchor of all the Sem II calculations. Without Bisognano–Wichmann, modular flow would be an abstract object that one cannot compute. With it, modular flow becomes a geometric symmetry of spacetime, and the whole crossed-product machinery (Block D) becomes computable in specific physical models.

8. What to take away

• Stated only: Tomita-Takesaki makes ${\sigma }_t^{\omega }(a)={\Delta }^{-it}a{\Delta }^{it}$ a $\sigma$-weakly continuous one-parameter automorphism group of $\mathcal{M}$.
• Stated only (Theorem 2.1): $\omega$ is KMS at $\beta =1$ for its own modular flow.
• Stated only (Theorem 2.2): Takesaki uniqueness — the modular flow is the unique one-parameter flow for which $\omega$ is KMS at $\beta =1$.
• Model proof: for $M_n(\mathbb{C})$, KMS at $\beta =1$ upper-strip follows from $F_{a,b}(z)=\mathrm{Tr}(\rho a{\rho }^{-iz}b{\rho }^{iz})$ and trace cyclicity.
• Computed here: non-tracial density matrices rotate matrix units (with frequencies $\log p_j-\log p_i$); tracial states have trivial modular flow; Powers states have periodic modular flow with period $2\pi /|\log \lambda |$.
• Intrinsic character: modular flow depends only on the algebra-state pair, not on the representation.
• Physical interpretation: in type III, modular flow is the algebraic replacement for Gibbs evolution. The Connes–Rovelli thermal-time hypothesis takes this seriously as a substitute for external time in generally covariant settings.

9. Looking ahead

Week 7 asks how two different faithful normal states on the same algebra compare. The answer is the Connes cocycle:

$$(D\omega /D\varphi {)}_t.$$

It is the noncommutative Radon-Nikodym derivative that converts ${\sigma }_t^{\varphi }$ into ${\sigma }_t^{\omega }$ by inner-automorphism dressing. The same relative modular operator also gives Araki-Uhlmann relative entropy — the entropy notion that survives in type III, replacing $S(\rho )=-\mathrm{Tr}(\rho \log \rho )$ where the latter is undefined.

10. Problem set

Core problems.

1. Finite-dimensional KMS, step by step. Reproduce the proof of Proposition 3.1 in full detail. Check every equality at $z=t+i$, identifying each use of trace cyclicity and each combination of $\rho$-powers. Verify that the result is well-defined as $\rho \to 1/n$ (tracial limit) — does anything degenerate?

2. Matrix-unit flow. Let $\rho =\mathrm{diag}(p_1,\dots ,p_n)$. Show that

$${\sigma }_t^{\rho }(E_{ij})={\left(\frac{p_j}{p_i}\right)}^{it}{E}_{ij}.$$

Which elements of $M_n(\mathbb{C})$ are fixed by the flow? Identify the fixed-point subalgebra.

3. Thermal qubit. For $H=\mathrm{diag}(0,E)$ and ${\rho }_{\beta }=e^{-\beta H}/Z$, compute ${\sigma }_t^{\rho }({\sigma }_x)$ and ${\sigma }_t^{\rho }({\sigma }_y)$. Compare with the physical Heisenberg evolution ${\alpha }_s({\sigma }_x),{\alpha }_s({\sigma }_y)$ at $s=\beta t$. Verify the relation ${\sigma }_t^{\rho }={\alpha }_{\beta t}$.

4. Tracial modular flow. In the GNS representation of a finite factor $(\mathcal{M},\tau )$, prove that $\Delta =1$ and ${\sigma }_t^{\tau }=\mathrm{id}$. (Hint: on the tracial GNS space $L^2(\mathcal{M},\tau )$, the Tomita operator is $S[a]=[a^{\ast }]$ with $S^2=1$, so $\Delta =S^{\ast }S=1$.)

5. Powers period. For ${\rho }_{\lambda }=\frac{1}{1+\lambda }\mathrm{diag}(1,\lambda )$, compute the smallest positive period of the modular flow on $E_{12}$. Compare with the period of the modular flow on a tensor product ${\rho }_{\lambda }\otimes {\rho }_{\lambda }$ (acting on $M_2\otimes M_2$).

6. Modular invariance of the state. Verify directly that ${\omega }_{\rho }({\sigma }_t^{\rho }(a))={\omega }_{\rho }(a)$ for all $a\in M_n(\mathbb{C})$ and $t\in \mathbb{R}$, using $\rho \cdot {\rho }^{-it}={\rho }^{1-it}$ and trace cyclicity.

Starred problems.

7*. Periodicity criterion. For $\rho =\mathrm{diag}(p_1,\dots ,p_n)$, find a necessary and sufficient condition on the $p_i$ for ${\sigma }_t^{\rho }$ to have a positive period. Treat the tracial case separately: then the flow is trivial and every $T>0$ is a period. In the nontracial case, show that a period $T$ exists iff every difference $\log p_j-\log p_i$ lies in $(2\pi /T)\mathbb{Z}$; equivalently, all nonzero log-ratios are rational multiples of one chosen nonzero log-ratio.

8*. Takesaki uniqueness in finite dimensions. Suppose ${\omega }_{\rho }$ is faithful on $M_n(\mathbb{C})$ and KMS at $\beta =1$ for a one-parameter group ${\alpha }_t=\mathrm{Ad}(e^{itK})$. Show that, up to an additive scalar in $K$,

$$K=-\log \rho .$$

Equivalently, $K=K_{\omega }$ is the modular Hamiltonian and ${\sigma }_t^{\rho }=\mathrm{Ad}(e^{it{K}_{\omega }})$. (Hint: use the KMS condition on matrix units $a=E_{ij},b=E_{ji}$ to derive the energy weights, then conclude $K$ is diagonal in the $\rho$-eigenbasis with eigenvalues $-\log p_i$.)

9*. Inner versus outer. In $M_n(\mathbb{C})$, every automorphism is inner (implemented by conjugation with a unitary). Explain why this makes modular flow less informative than in type III factors. Then state what it means for the image of ${\sigma }_t^{\omega }$ in $\mathrm{Out}(\mathcal{M})=\mathrm{Aut}(\mathcal{M})/\mathrm{Inn}(\mathcal{M})$ to be non-trivial (this is one form of the type III diagnostic).

10*. Modular flow on a tensor product. Let $\rho \in M_n(\mathbb{C})$ and $\sigma \in M_m(\mathbb{C})$ both be faithful states. Compute the modular flow ${\sigma }_t^{\rho \otimes \sigma }$ on $M_n\otimes M_m$ and express it as ${\sigma }_t^{\rho }\otimes {\sigma }_t^{\sigma }$. Verify by direct computation that the KMS condition at $\beta =1$ holds for the joint state.

Project problems.

11. Thermal time memo. Write a two-page note explaining the Connes-Rovelli thermal-time hypothesis using only concepts introduced in Weeks 4–6. Keep separate: theorem, interpretation, and speculative physics. Discuss whether the hypothesis is empirically distinguishable from “ordinary” external-Hamiltonian time evolution.

12. Rindler preview. Read the statement of the Bisognano-Wichmann theorem (Week 10 preview). Identify which part corresponds to ${\Delta }^{-it}$ (the modular operator → boost) and which part corresponds to $J$ (the modular conjugation → PCT × rotation) for the right-wedge algebra.

Self-study answer checkpoints

These checks close the six core calculations. Starred and project problems are deliberately source-led: use the named theorems and later QFT chapters rather than treating the lines below as a replacement.

1. Finite-dimensional KMS. For

   $$F_{a,b}(z)=\mathrm{Tr}\left(\rho a{\rho }^{-iz}b{\rho }^{iz}\right),$$

   the real boundary is ${\omega }_{\rho }(a{\sigma }_t^{\rho }(b))$. At $z=t+i$, move the leading block $\rho a$ cyclically to the end and cancel ${\rho }^{it-1}\rho ={\rho }^{it}$; the result is ${\omega }_{\rho }({\sigma }_t^{\rho }(b)a)$. When $\rho =1/n$, all powers are scalar and the flow simply becomes the identity.

2. Matrix units and the centralizer.

   $${\sigma }_t^{\rho }(E_{ij})={\left(\frac{p_j}{p_i}\right)}^{it}{E}_{ij}.$$

   The fixed algebra consists of matrices preserving every eigenspace of $\rho$: it is the direct sum of full matrix blocks associated with equal eigenvalues, and reduces to the diagonal algebra only when the spectrum is simple.

3. Thermal qubit. If $\theta =\beta Et$, the formulas are

   $${\sigma }_t^{\rho }({\sigma }_x)=\cos \theta {\sigma }_x+\sin \theta {\sigma }_y,{\sigma }_t^{\rho }({\sigma }_y)=\cos \theta {\sigma }_y-\sin \theta {\sigma }_x.$$

   Since ${\rho }_{\beta }^{-it}$ differs from $e^{i\beta tH}$ only by a scalar phase, ${\sigma }_t^{\rho }={\alpha }_{\beta t}$.

4. Trace. Traciality makes $S[a]=[a^{\ast }]$ antiunitary, so $S^{\ast }S=1$, $\Delta =1$, and ${\sigma }_t^{\tau }=\mathrm{id}$.

5. Powers period. On $E_{12}$ the phase is ${\lambda }^{it}=e^{it\log \lambda }$, so the least positive period is

   $$T_{\lambda }=\frac{2\pi }{|\log \lambda |}.$$

   The two-site flow has phases ${\lambda }^{ikt}$ with $k=-2,-1,0,1,2$; because $k=1$ still occurs, its least positive period is the same $T_{\lambda }$.

6. State invariance. Directly,

   $$\mathrm{Tr}\left(\rho {\rho }^{-it}a{\rho }^{it}\right)=\mathrm{Tr}(\rho a),$$

   using cyclicity and commutativity of the powers of $\rho$. Thus the modular state is stationary under its own flow.

---

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