Appendix F — Notation and Conventions

Notation and Conventions

This appendix is the quick lookup table for symbols and conventions used across the course. All conventions here apply uniformly from Week 4 onward.

Dimension warning. The free-field chapters write ${\mathbb{R}}^{1,d}$ with $d$ spatial dimensions. The holography chapters follow the usual CFT${}_d$/AdS${}_{d+1}$ notation, where $d$ is the boundary spacetime dimension. At a bridge between the two, the note spells out “$3+1$ dimensions” rather than relying on an unqualified value of $d$.

F.1 Algebras and Hilbert Spaces

Symbol	Meaning
$\mathcal{A}$	C*-algebra, often abstract
$\mathcal{M},\mathcal{N}$	von Neumann algebras
$\mathcal{B}(\mathcal{H})$	bounded operators on Hilbert space $\mathcal{H}$
${\mathcal{M}}^{\prime }$	commutant of $\mathcal{M}$
${\mathcal{M}}^{\prime \prime }$	bicommutant of $\mathcal{M}$
$\mathcal{Z}(\mathcal{M})$	center of $\mathcal{M}$
${\mathcal{M}}_{\ast }$	predual of $\mathcal{M}$
$\mathcal{T}(\mathcal{H})$	trace-class operators on $\mathcal{H}$
$\mathcal{A}(\mathcal{O})$	local algebra associated with region $\mathcal{O}$
$W_R,W_L$	right and left Rindler wedges
$\mathcal{F}$	Fock space
$|0_M⟩,|0_R⟩$	Minkowski and Rindler vacua

F.2 States, Weights, and Traces

Symbol	Meaning
$\omega ,\varphi$	states or normal positive linear functionals
${\omega }_{\rho }(a)=\mathrm{Tr}(\rho a)$	density-matrix state (type I or trace-class)
$\tau$	trace, usually finite or semifinite
$\hat{\tau }$	trace on a crossed product / continuous core
$\phi$	weight (possibly non-finite)
$\rho ,\sigma$	density matrices in type I, or trace densities in semifinite settings
$\hat{\rho }$	density of a dressed state relative to $\hat{\tau }$
$\Omega ,{\Omega }_{\omega }$	cyclic-separating GNS vector for state $\omega$

Course rule on “density matrix” language:

• $\rho$ is allowed when the algebra is type I, or when a specified trace is in scope.
• In type III, say “normal state” rather than “density matrix” unless using a regulator or split inclusion.

F.3 Modular Objects

Symbol	Meaning
$S$	Tomita operator, $S(a\Omega )=a^{\ast }\Omega$ on $\mathcal{M}\Omega$
$J$	modular conjugation (antiunitary)
$\Delta$	modular operator (positive self-adjoint)
${\sigma }_t^{\omega }$	modular automorphism group of $\omega$
$K_{\rho }=-\log \rho$	finite-dimensional modular Hamiltonian
$K_{\mathrm{GNS},\omega }=-\log {\Delta }_{\omega }$	modular generator on the GNS/standard-form Hilbert space
${\Delta }_{\omega ,\varphi }$	relative modular operator
$(D\omega /D\varphi {)}_t$	Connes cocycle of $\omega$ relative to $\varphi$

Course modular-flow convention (upper-strip $\beta >0$):

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

Finite-dimensional relation. For $\omega (a)=\mathrm{Tr}(\rho a)$:

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

Gibbs case. If $\rho =e^{-\beta H}/Z$ and ${\alpha }_s(a)=e^{isH}a{e}^{-isH}$ is the physical Heisenberg flow,

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

Modular time runs in the same direction as physical time, at $\beta$-rescaled rate.

Opposite convention (used, for example, in Bratteli—Robinson Vol. II): ${\sigma }_t^{\omega }(a)={\Delta }_{\omega }^{it}a{\Delta }_{\omega }^{-it}$, giving ${\sigma }_t^{\rho }={\alpha }_{-\beta t}$. The two are related by $t\mapsto -t$ and describe the same automorphism group with reversed parametrization. Our convention matches Witten 1803.04993 §3: $K=-\log \rho$ generates ${\sigma }_t=\mathrm{Ad}(e^{itK})=\mathrm{Ad}({\Delta }^{-it})$.

There are two related operators here, and the distinction matters. In a finite-dimensional standard representation,

$$-\log {\Delta }_{\rho }=(-\log \rho )\otimes 1-1\otimes (-\log \rho {)}^{\mathsf{T}}.$$

Thus $K_{\rho }=-\log \rho$ generates the automorphism of the observable algebra, whereas $K_{\mathrm{GNS},\rho }=-\log {\Delta }_{\rho }$ acts on the doubled standard-form Hilbert space.

F.4 KMS Convention

Physical Heisenberg KMS for a Gibbs state at inverse temperature $\beta >0$ is upper-strip:

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

The modular flow in our convention satisfies the same upper-strip KMS at $\beta =+1$:

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

Both physical and modular flows live in the same upper strip — that is the structural advantage of our convention over the Bratteli–Robinson one (which puts physical Heisenberg in the upper strip and modular flow in the lower strip).

F.5 Type Classification

Type	Structural marker	Trace status
I${}_n$ ($n<\infty$)	$\cong M_n(\mathbb{C})$	finite trace
I${}_{\infty }$	$\cong \mathcal{B}(\mathcal{H})$, $\dim \mathcal{H}=\infty$	semifinite trace
II${}_1$	no minimal projection, $1$ finite	unique normalized trace
II${}_{\infty }$	no minimal projection, $1$ infinite	semifinite trace
III	every non-zero projection infinite	no faithful normal trace

Connes type-III subclasses (Connes 1973):

$${\mathrm{III}}_0,{\mathrm{III}}_{\lambda }(0<\lambda <1),{\mathrm{III}}_1.$$

Important — common pitfall.

• III${}_1$ is not the $\lambda =1$ endpoint of the Powers III${}_{\lambda }$ family.
• The Powers III${}_{\lambda }$ family has $\lambda \in (0,1)$ strictly. Its $\lambda =1$ tracial endpoint is the hyperfinite II${}_1$ factor.
• III${}_1$ is a separate Connes class with full positive S-invariant $[0,\infty )$. Under the additional hypotheses reviewed in Week 12, local QFT algebras are hyperfinite type III${}_1$; neither nuclearity nor the split property alone is being advertised here as a classification theorem.

F.6 Crossed-Product Notation

Symbol	Meaning
${\alpha }_t$	one-parameter automorphism group used in a crossed product
$t$	parameter of ${\alpha }_t$; for a modular core, modular time
$q$	coordinate on the auxiliary $L^2(\mathbb{R},dq)$ factor
$Q$	multiplication operator, $(Q\xi )(q)=q\xi (q)$
$P$	conjugate momentum $-i{\partial }_q$
$p$	Fourier-dual coordinate in which $P$ is multiplication
${\pi }_{\alpha }(a)$	covariant representation, $({\pi }_{\alpha }(a)\xi )(q)={\alpha }_{-q}(a)\xi (q)$
$\lambda (t)$	translation unitary, $(\lambda (t)\xi )(q)=\xi (q-t)$, hence $\lambda (t)=e^{-itP}$
$\mathcal{M}{⋊}_{\alpha }\mathbb{R}$	crossed product
$\hat{\mathcal{M}}$ or ${\hat{\mathcal{M}}}_{\omega }$	modular crossed product $\mathcal{M}{⋊}_{{\sigma }^{\omega }}\mathbb{R}$
$r$	parameter of the dual action
${\theta }_r$	dual action, implemented by $e^{irQ}$ in the $q$-representation
$\hat{\tau }$	faithful normal semifinite trace on the crossed product

Covariance relation:

$$\lambda (t){\pi }_{\alpha }(a)\lambda (t{)}^{\ast }={\pi }_{\alpha }({\alpha }_t(a)).$$

Dual action:

$${\theta }_r({\pi }_{\alpha }(a))={\pi }_{\alpha }(a),{\theta }_r(\lambda (t))=e^{irt}\lambda (t).$$

It acts by a phase in the $q$-representation and by translation in the $p$-representation. In particular, $g(P)$ is multiplication by $g(p)$ only after Fourier transformation; it is not multiplication by $g(q)$.

Dual-action trace scaling:

$$\hat{\tau }\circ {\theta }_r=e^{-r}\hat{\tau }.$$

F.7 Rindler and TFD Symbols

Symbol	Meaning
${\Lambda }^{\mathrm{boost}}(s)$	Lorentz boost in $(x^0,x^1)$ plane by rapidity $s$
$K$	boost generator, $U({\Lambda }^{\mathrm{boost}}(u))=e^{iuK}$
$u$	boost rapidity; for the right wedge, $u=2\pi t$
$\eta$	Rindler time coordinate
$\xi$	Rindler radial coordinate (lapse ${\xi }_0$ for an observer with acceleration $a=1/{\xi }_0$)
$|{\mathrm{TFD}}_{\beta }⟩$	thermofield double state at inverse temperature $\beta$
$H_R,H_L$	right and left boundary Hamiltonians

Course Rindler conventions (Bisognano–Wichmann; our sign):

$${\Delta }_{W_R}=e^{-2\pi K},{\sigma }_t^{W_R}=\mathrm{Ad}(U({\Lambda }^{\mathrm{boost}}(2\pi t))).$$

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

F.8 Entropy Notation

Symbol	Meaning
$S(\rho )=-\mathrm{Tr}(\rho \log \rho )$	von Neumann entropy (finite-dim or type I with trace)
$S(\rho \Vert \sigma )$	Umegaki relative entropy (finite-dim or type I)
$S(\omega \Vert \varphi )$	Araki–Uhlmann relative entropy (any vN algebra)
$S_{\hat{\tau }}(h)=-\hat{\tau }(h\log h)$	entropy of a normal state with density $h$ relative to the fixed semifinite trace $\hat{\tau }$
$S_{\mathrm{gen}}$	generalized entropy in gravity (only in holographic settings)
$\mathcal{B}(\omega ,\varphi )$	modular boundary term

Finite-dimensional identity (derived in Week 14 §3.1):

$$S(\rho )-S(\sigma )=-S(\rho \Vert \sigma )+\mathrm{Tr}((\rho -\sigma )K_{\sigma }),K_{\sigma }=-\log \sigma .$$

Thus, in the finite-dimensional model,

$$S({\rho }_{\omega })-S({\rho }_{\varphi })=-S({\rho }_{\omega }\Vert {\rho }_{\varphi })+\mathcal{B}(\omega ,\varphi ),\mathcal{B}(\omega ,\varphi )=\omega (K_{\varphi })-\varphi (K_{\varphi }).$$

The boundary term uses the modular Hamiltonian of the reference state $\varphi$, not that of $\omega$. A corresponding core statement requires a fixed trace, a specified lift of both states to that core, and the appropriate density-domain hypotheses; it is not an identity for arbitrary type-III state pairs.

Trace rescaling. If ${\hat{\tau }}^{\prime }=c\hat{\tau }$ and the same normalized state has density $h^{\prime }=h/c$, then

$$S_{{\hat{\tau }}^{\prime }}(h^{\prime })=S_{\hat{\tau }}(h)+\log c.$$

Type-I weighted-trace check. For

$$\hat{\tau }(a\otimes f)=\mathrm{Tr}(a){\int }_{\mathbb{R}}f(p)e^{-p}dp$$

and a product probability density $\rho \mu (p)$ relative to the flat product measure, the density relative to $\hat{\tau }$ is $h(p)=\rho \mu (p)e^p$. Hence

$$S_{\hat{\tau }}(h)=S(\rho )-\int \mu \log \mu dp-\int p\mu (p)dp.$$

The last term is fixed by the non-flat trace; it cannot be omitted or absorbed into the differential entropy without changing conventions.

Safe distinctions (Week 12 §4):

• $S(\omega \Vert \varphi )$: exact type-III-safe relative entropy.
• $S_{\hat{\tau }}(h)$: well-defined for a normal state on a specified semifinite algebra once $\hat{\tau }$ is fixed; rescaling the trace shifts it by a state-independent constant.
• $S_{\mathrm{gen}}$: requires the holographic dictionary as additional input.

All three are distinct controlled objects, not the same thing.