Appendix C — Modular Theory Reference Sheet

Modular Theory Reference Sheet

Compact reference for Tomita–Takesaki objects used throughout Block B and beyond. All conventions match Week 4 §6.1 and the note-quality template.

C.1 Standard Setup

Let $\mathcal{M}\subset \mathcal{B}(\mathcal{H})$ be a von Neumann algebra and $\Omega \in \mathcal{H}$ a vector that is cyclic and separating for $\mathcal{M}$ (Week 5).

Cyclic: $\overline{\mathcal{M}\Omega }=\mathcal{H}$. Separating: $a\Omega =0$ with $a\in \mathcal{M}$ implies $a=0$.

The vector state $\omega (a):=⟨\Omega ,a\Omega ⟩$ is then a faithful normal state (Week 5 Lemma 1.3).

C.2 Tomita Operator

On the dense subspace $\mathcal{M}\Omega \subset \mathcal{H}$, define the conjugate-linear operator

$$S_0(a\Omega ):=a^{\ast }\Omega .$$

$S_0$ is closable (Week 5 Prop 2.2); its closure is denoted $S$.

Polar decomposition. $S=J{\Delta }^{1/2}$, where:

• $J$ is antiunitary with $J^{\ast }=J$ and $J^2=1$ (modular conjugation).
• $\Delta :=S^{\ast }S$ is positive self-adjoint (modular operator).

C.3 Tomita–Takesaki Theorem

[Stated only — refs: B-R Vol. I §2.5; Takesaki Vol. II Ch. VI.] Under the standard setup,

$$J\mathcal{M}J={\mathcal{M}}^{\prime },{\Delta }^{-it}\mathcal{M}{\Delta }^{it}=\mathcal{M}\text{for all }t\in \mathbb{R}.$$

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

$$\boxed{{\sigma }_t^{\omega }(a):={\Delta }^{-it}a{\Delta }^{it}.}$$

This is the course convention (upper-strip $\beta >0$, matching Witten 1803.04993). The opposite-sign convention ${\sigma }_t=\mathrm{Ad}({\Delta }^{+it})$ also appears in the literature (Bratteli–Robinson Vol. II) and gives the same automorphism group running backwards.

Proof status in the course:

• Finite-dimensional/type-I model: computed explicitly (Week 5 §5).
• Full theorem: statement only, used as a standard theorem.

C.4 KMS Condition for the Modular Flow

With the course convention, $\omega$ is KMS at $\beta =1$ for ${\sigma }^{\omega }$ in the upper strip:

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

with $F_{a,b}$ holomorphic on $\{0\le \mathrm{Im}z\le 1\}$.

Both physical Heisenberg + Gibbs (Week 4 §1.1) and modular flow at $\beta =+1$ live in the same upper strip in our convention.

C.5 Finite-Dimensional Computation

For $\mathcal{M}=M_n(\mathbb{C})$ with state ${\omega }_{\rho }(a)=\mathrm{Tr}(\rho a)$, $\rho >0$:

• GNS Hilbert space: ${\mathcal{H}}_{\mathrm{HS}}=M_n(\mathbb{C})$ with $⟨X,Y⟩=\mathrm{Tr}(X^{\ast }Y)$.
• Cyclic-separating vector: ${\Omega }_{\rho }={\rho }^{1/2}$.
• Tomita operator: $S(X)={\rho }^{-1/2}{X}^{\ast }{\rho }^{1/2}$. (Check: extends $S(a{\rho }^{1/2})=a^{\ast }{\rho }^{1/2}$, since $X=a{\rho }^{1/2}\Rightarrow a=X{\rho }^{-1/2}$; and $S^2=1$. For the tracial state $\rho =1/n$ this reduces to $S(X)=X^{\ast }$.)
• Modular operator: ${\Delta }_{\rho }(X)=\rho X{\rho }^{-1}$ on ${\mathcal{H}}_{\mathrm{HS}}$. (Check: $J{\Delta }^{1/2}(X)=({\rho }^{1/2}X{\rho }^{-1/2}{)}^{\ast }={\rho }^{-1/2}{X}^{\ast }{\rho }^{1/2}=S(X)$. ✓)
• Modular conjugation: $J_{\rho }(X)=X^{\ast }$ (Hilbert–Schmidt complex-conjugate-transpose).

Modular automorphism on $\mathcal{M}$:

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

For a Gibbs state $\rho =e^{-\beta H}/Z$:

$${\sigma }_t^{\rho }(a)=e^{i\beta tH}a{e}^{-i\beta tH}={\alpha }_{+\beta t}(a),$$

modular time is the inverse-temperature rescaling of physical Heisenberg time.

C.6 Type-I Bipartite Computation

For $\mathcal{M}=\mathcal{B}({\mathcal{H}}_A)\otimes 1$ acting on ${\mathcal{H}}_A\otimes {\mathcal{H}}_B$, and $\Omega ={\sum }_i\sqrt{p_i}{e}_i\otimes f_i$ in Schmidt form (all $p_i>0$):

• ${\rho }_A={\sum }_i{p}_i|e_i⟩⟨e_i|$ (reduced density on ${\mathcal{H}}_A$).
• $\Delta ={\rho }_A\otimes {\rho }_A^{-1}$ (the inverse acts on ${\mathcal{H}}_B$ via the Schmidt isomorphism).
• $J(e_i\otimes f_j)=e_j\otimes f_i$ with complex conjugation in the Schmidt bases.
• ${\sigma }_t^{\Omega }(a\otimes 1)=({\rho }_A^{-it}a{\rho }_A^{it})\otimes 1$.

Modular Hamiltonian: $K_A:=-\log {\rho }_A$ (positive operator on ${\mathcal{H}}_A$).

C.7 Relative Modular Operator and Connes Cocycle (Week 7)

Given two faithful normal states $\omega ,\varphi$ with GNS vectors ${\Omega }_{\omega },{\Omega }_{\varphi }$:

• Relative Tomita operator: $S_{\omega ,\varphi }(a{\Omega }_{\varphi }):=a^{\ast }{\Omega }_{\omega }$.
• Relative modular operator: ${\Delta }_{\omega ,\varphi }:=S_{\omega ,\varphi }^{\ast }{S}_{\omega ,\varphi }$.

In finite dim, ${\Delta }_{\rho ,\sigma }(X)=\rho X{\sigma }^{-1}$ on ${\mathcal{H}}_{\mathrm{HS}}$.

Connes cocycle (our convention):

$$u_t=(D\omega /D\varphi {)}_t:={\Delta }_{\omega ,\varphi }^{-it}{\Delta }_{\varphi }^{it}.$$

In finite dim: $u_t={\rho }^{-it}{\sigma }^{it}$. Lies in $\mathcal{M}$.

Cocycle identity: $u_{t+s}=u_t{\sigma }_t^{\varphi }(u_s)$.

Intertwining: ${\sigma }_t^{\omega }(a)=u_t{\sigma }_t^{\varphi }(a)u_t^{\ast }$.

C.8 Araki–Uhlmann Relative Entropy (Week 7)

$$S(\omega \Vert \varphi ):=-⟨{\Omega }_{\omega },\log {\Delta }_{\varphi ,\omega }{\Omega }_{\omega }⟩.$$

Note the index order: the modular operator inside $\log$ has $\varphi$ first (reference state) and $\omega$ second.

In finite dim, reduces to Umegaki:

$$S({\omega }_{\rho }\Vert {\omega }_{\sigma })=\mathrm{Tr}(\rho (\log \rho -\log \sigma )).$$

Properties.

• $S(\omega \Vert \varphi )\ge 0$, with equality iff $\omega =\varphi$ (Klein’s inequality + spectral theorem).
• Monotone under restriction to subalgebras: $S(\omega {|}_{\mathcal{N}}\Vert \varphi {|}_{\mathcal{N}})\le S(\omega {|}_{\mathcal{M}}\Vert \varphi {|}_{\mathcal{M}})$ for $\mathcal{N}\subset \mathcal{M}$.

C.9 Bisognano–Wichmann (Week 10)

For the right Rindler wedge $W_R=\{x:x^1>|x^0|\}$ in a Wightman QFT, with vacuum $|0_M⟩$:

$${\Delta }_{W_R}=e^{-2\pi K},J_{W_R}=\Theta \cdot R_{\perp }(\pi ),$$

where $K$ is the boost generator and the displayed expression for $J$ is the neutral scalar form; fields with spin, charge, or braid statistics require the usual extra factors. We define

$$U({\Lambda }^{\mathrm{boost}}(u))=e^{iuK}.$$

The modular flow is therefore

$${\sigma }_t^{W_R}(a)=U({\Lambda }^{\mathrm{boost}}(2\pi t))aU({\Lambda }^{\mathrm{boost}}(2\pi t){)}^{\ast }.$$

Modular time $t$ corresponds to boost rapidity $+2\pi t$. Unruh temperature $T_U=a/(2\pi )$.

C.10 Three Distinguishing Properties (Week 12)

• Type II${}_1$: tracial state has $\Delta =1$, ${\sigma }_t=\mathrm{id}$, $S(\mathcal{M})=\{1\}$.
• Type III${}_{\lambda }$ ($\lambda \in (0,1)$): modular flow on Powers state is periodic, $T_{\lambda }=2\pi /|\log \lambda |$, $S(\mathcal{M})=\{0\}\cup \{{\lambda }^n:n\in \mathbb{Z}\}$.
• Type III${}_1$: $S(\mathcal{M})=[0,\infty )$. Since $S(\mathcal{M})={\bigcap }_{\phi }\mathrm{Sp}{\Delta }_{\phi }$ over the faithful normal states (or weights) of a factor, this says that every such state has $\mathrm{Sp}{\Delta }_{\phi }=[0,\infty )$, equivalently a logarithmic modular generator with full real spectrum. One state with this spectrum proves nothing: on $\mathcal{B}(\mathcal{H})$ a faithful state whose eigenvalue ratios $p_i/p_j$ are dense in $(0,\infty )$ also has $\mathrm{Sp}\Delta =[0,\infty )$.

C.11 Use in the Course

Week	Modular object used
5	$S,J,\Delta$ (Tomita operator + polar decomp)
6	${\sigma }_t^{\omega }$ (modular automorphism + KMS)
7	${\Delta }_{\omega ,\varphi }$, $(D\omega /D\varphi {)}_t$, $S(\omega \Vert \varphi )$
9	Reeh–Schlieder licenses Tomita–Takesaki for local algebras
10	Bisognano–Wichmann: $\Delta =e^{-2\pi K}$ on wedges
11	Modular tools in the hypothesis-sensitive Summers—Werner analysis; explicit Weyl lower bounds
12	S-invariant classifies type III subtypes
13	Crossed product by ${\sigma }^{\omega }$
14	Core entropy with a fixed trace + the finite-dimensional boundary-term identity
15	One-sided TFD modular operator ${\Delta }_R=e^{-\beta (H_R-H_L)}$ in the type-I purification model

C.12 Math ↔ Physics Dictionary

Modular object	Physical meaning
Cyclic vector $\Omega$	The orbit $\mathcal{M}\Omega$ is dense. In a finite bipartite model this corresponds to full Schmidt support; in AQFT it expresses the Reeh—Schlieder reach of local operations, not a tensor-factor claim.
Separating vector	No nonzero $a\in \mathcal{M}$ annihilates $\Omega$; equivalently, the induced vector state is faithful on $\mathcal{M}$.
Tomita operator $S$	State-flip $a\to a^{\ast }$; its non-isometry encodes the state’s thermal tilt
$J$ (modular conjugation)	Mirror into the commutant; CPT × rotation for wedges; system ↔ environment duality
$\Delta$ (modular operator)	Boltzmann-weight ratios of the state; $e^{-2\pi K}$ for wedges
${\sigma }_t^{\omega }$ (modular flow)	Canonical state-dependent automorphism group; a boost for wedges and identified with an ADM flow only in the specified holographic construction. Calling it “thermal time” is an interpretation.
$K_{\mathrm{GNS}}=-\log \Delta$	Modular generator on standard form; it equals $2\pi K_{\mathrm{boost}}$ for the right wedge and $\beta (H_R-H_L)$ in the regulated TFD model. The one-sided type-I generator $-\log {\rho }_R$ is related but acts on a different space.
KMS at $\beta =1$	Equilibrium; imaginary-time periodicity; detailed balance ${\hat{G}}_{-}=e^{-\beta \nu }{\hat{G}}_{+}$
Spectrum of $\Delta$	Thermal fingerprint; its state-independent core is the Connes invariant (type)
Connes cocycle $(D\omega /D\varphi {)}_t$	Noncommutative Radon–Nikodym derivative; dressing converting one equilibrium dynamics into another
$S(\omega \Vert \varphi )$	Araki relative entropy; an operational distinguishability rate under the hypotheses of an appropriate quantum Stein theorem; intrinsic in type III but possibly $+\infty$
Crossed product $\mathcal{M}{⋊}_{{\sigma }^{\omega }}\mathbb{R}$	The continuous core in the abstract theory. It becomes an algebra of relational or gravitationally dressed observables only after a model-specific physical identification.
Canonical trace $\hat{\tau }$	Faithful normal semifinite trace on the continuous core, characterized up to scale and obeying $\hat{\tau }\circ {\theta }_r=e^{-r}\hat{\tau }$.
Core entropy $S_{\hat{\tau }}(h)$	Semifinite entropy relative to a fixed trace. Its identification with $A/(4G_N)+S_{\mathrm{out}}$ requires the Witten/CPW holographic state construction.