Course Conventions

This is the canonical convention reference for the algebraic-QFT course. Every formula involving modular flow, boosts, crossed products, or dressed entropy must be read against this page. When a source uses the opposite sign, we translate the source into the course convention before continuing.

For a comprehensive symbol table, see Appendix F.

Spacetime and Free-Field Signs

We use the mostly-plus Minkowski metric

$$\eta =\mathrm{diag}(-1,+1,\dots ,+1),x^2=-(x^0{)}^2+|\mathbf{x}{|}^2.$$

The Pauli—Jordan distribution is real and is defined by $[\varphi (x),\varphi (y)]=i\Delta (x-y)$. Wightman functions are boundary values with the time prescription $x^0\mapsto x^0-iϵ$. These three choices are a single sign package; a source written with the opposite metric must be translated before its formulas are imported.

Dimension notation. In the free-field chapters, ${\mathbb{R}}^{1,d}$ means one time and $d$ spatial dimensions, so physical Minkowski spacetime has $d=3$. In the holography chapters, the standard notation CFT${}_d$/AdS${}_{d+1}$ instead counts the boundary spacetime dimension. We state which convention is active whenever the two parts of the course meet; a bare phrase such as “for $d=4$” is not sufficient there.

Modular Flow

The course uses the Witten 1803.04993 §3 convention:

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

For $\mathcal{M}=M_n(\mathbb{C})$ and ${\omega }_{\rho }(a)=\mathrm{Tr}(\rho a)$:

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

For a Gibbs state $\rho =e^{-\beta H}/Z$ with physical Heisenberg flow ${\alpha }_s(a)=e^{isH}a{e}^{-isH}$:

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

Modular time runs in the same direction as physical time, at $\beta$-rescaled rate. This is the structural reason our convention is “natural” for thermal physics.

Opposite convention. Some references, including Bratteli–Robinson Vol. II, use ${\sigma }_t^{\omega }=\mathrm{Ad}({\Delta }^{+it})$, giving ${\sigma }_t^{\rho }={\alpha }_{-\beta t}$. The two conventions describe the same automorphism group running in opposite directions; they are related by $t\mapsto -t$. Our convention is fixed; do not mix formulas before making this translation.

KMS Convention

We use the upper-strip $\beta >0$ KMS condition:

A state $\omega$ is KMS at $\beta >0$ for a one-parameter automorphism group ${\sigma }_t$ if for every $a,b$ there is a function $F_{a,b}$ on the closed strip $\{0\le \mathrm{Im}z\le \beta \}$, holomorphic in the interior, bounded and continuous, with

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

Two facts under this convention:

1. The Gibbs state $\rho =e^{-\beta H}/Z$ with physical Heisenberg flow is KMS at $\beta >0$ upper-strip (Week 4 §1.1, derived).
2. A faithful normal state $\omega$ with its modular flow (in our ${\Delta }^{-it}$ convention) is KMS at $\beta =+1$ upper-strip (Week 6 §3, model proof).

Both physical and modular flows live in the same upper strip in our convention — that is the pedagogical advantage.

Rindler / Bisognano–Wichmann

For the right Rindler wedge $W_R$ in any Wightman QFT (Week 10):

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

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

We define the boost generator by

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

With this definition, ${\Delta }_{W_R}^{-it}=e^{i2\pi tK}$ and the two formulas above agree without an additional sign. A reference using $U(\Lambda (u))=e^{-iu{K}_{\mathrm{ref}}}$ is using $K_{\mathrm{ref}}=-K$.

Crossed-Product Variables

Several real variables occur in a modular crossed product. They are related, but they are not interchangeable.

Symbol	Role
$t$	modular-group parameter in ${\sigma }_t$
$u$	physical rapidity or physical time; for the wedge, $u=2\pi t$
$q$	coordinate on the auxiliary $L^2(\mathbb{R},dq)$ factor
$Q$	multiplication by $q$: $(Q\xi )(q)=q\xi (q)$
$P$	momentum conjugate to $Q$: $P=-i{\partial }_q$
$r$	parameter of the dual action ${\theta }_r$

In the covariant representation,

$$({\pi }_{\sigma }(a)\xi )(q)={\sigma }_{-q}(a)\xi (q),(\lambda (t)\xi )(q)=\xi (q-t),\lambda (t)=e^{-itP}.$$

The covariance relation is

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

The dual action is implemented by $e^{irQ}$:

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

Thus the dual action is a phase in the $q$-representation and a translation in the Fourier-dual $p$-representation. A function of $P$ is not a multiplication function of $q$. Whenever we Fourier transform, we name the new variable $p$ and state the transformation explicitly.

The canonical trace on the continuous core is normalized so that

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

In a Fourier realization this scaling may be represented by a weight $e^{-p}dp$. If one instead uses a rapidity-normalized variable $p=2\pi q_{\mathrm{rap}}$, the same weight is proportional to $e^{-2\pi q_{\mathrm{rap}}}d{q}_{\mathrm{rap}}$. The factor $2\pi$ is therefore a change of variable, not a second trace convention.

Modular Boundary Term

In a type-I representation, for two states $\omega ,\varphi$ with density matrices and with modular Hamiltonian $K_{\varphi }=-\log {\rho }_{\varphi }$ of the reference state $\varphi$, the modular boundary term is

$$\mathcal{B}(\omega ,\varphi ):=\omega (K_{\varphi })-\varphi (K_{\varphi })=\mathrm{Tr}(({\rho }_{\omega }-{\rho }_{\varphi })K_{\varphi }).$$

Note: uses $K_{\varphi }$ (the second argument’s modular Hamiltonian), not $K_{\omega }$. This is the relative modular energy of $\omega$ measured by the reference flow ${\sigma }^{\varphi }$.

Vanishes when $\omega =\varphi$.

Entropy and Trace Normalization

Let $\hat{\omega }(x)=\hat{\tau }(hx)$ be a normalized state on a semifinite algebra. If the trace is rescaled to ${\hat{\tau }}^{\prime }=c\hat{\tau }$, the density becomes $h^{\prime }=h/c$, and hence

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

Only differences computed with the same trace normalization are invariant.

For the type-I model

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

and a product state with system density $\rho$ and clock probability density $\mu (p)$, the density relative to $\hat{\tau }$ is $h(p)=\rho \mu (p)e^p$. Its entropy is

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

The final term is the clock-energy term induced by the non-flat trace. It must not be dropped or renamed as a Shannon entropy.

Entropy-Difference Identity

Let $D_{\omega }$ and $D_{\varphi }$ be normalized trace-densities in the same semifinite algebra with respect to the same faithful normal semifinite trace $\hat{\tau }$, and set $K_{\varphi }=-\log D_{\varphi }$. Whenever the displayed terms are well defined (or after a stated common regularization), the exact same-algebra identity is

$$\boxed{S_{\hat{\tau }}(D_{\omega })-S_{\hat{\tau }}(D_{\varphi })=-D_{\hat{\tau }}(D_{\omega }\Vert D_{\varphi })+\hat{\tau }\left[(D_{\omega }-D_{\varphi })K_{\varphi }\right],}$$

where $D_{\hat{\tau }}(D_{\omega }\Vert D_{\varphi })=\hat{\tau }[D_{\omega }(\log D_{\omega }-\log D_{\varphi })]$.

In finite dimensions this becomes

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

with $\mathcal{B}$ as above. Week 14 proves the algebraic identity directly and checks it in the type-I Fourier model.

Relating $D_{\hat{\tau }}(D_{\omega }\Vert D_{\varphi })$ on a continuous core to Araki relative entropy of states on the original type-III algebra requires more data: a fixed core, a fixed trace normalization, and a specified lift of both states. Witten and CPW construct such data in their respective large-$N$ settings. We do not promote that lift to a theorem for arbitrary pairs of type-III states without these hypotheses.

Proof-Status Labels

Every theorem-level claim should carry one of these labels in the prose:

• Proved. Full proof is included.
• Model proof. Proved in finite dimensions, type I, or another controlled case.
• Sketched. Main ideas only; missing analytic details are named.
• Stated only — refs. Used as a theorem from the literature; reference given.
• Hypothesis-explicit. Stated with the technical assumptions required, even if the proof is in references.
• Heuristic. Physical or conceptual motivation, not a proof.
• Exact calculation. An exact computation inside the explicitly defined model.
• Controlled perturbative. Derived through the stated order in a named expansion parameter.
• Formal analogy. A structural comparison, not an equality or derivation.
• Example-only. Demonstrated in one model; not claimed generally.

Scope Discipline

The course must not blur:

• Type-I density-matrix calculations with type-III local-algebra facts.
• Wedge-specific Bisognano–Wichmann geometry with generic bounded-region modular flow (which is not geometric).
• Structural Bell-correlation theorems (Summers–Werner, hypothesis-explicit) with explicit Weyl-test-function computations (which approach but don’t attain $2\sqrt{2}$).
• Proved operator-algebra statements (Takesaki’s structure theorem) with holographic or gravitational interpretation (Witten 2022 identification).
• The clock coordinate, its conjugate momentum, the modular parameter, and the Fourier-dual variable.
• The clock wavefunction width with a matter UV cutoff. The crossed product gives a semifinite trace and can support finite entropy differences under explicit integrability hypotheses; it does not derive a QFT area-law coefficient from clock localization.

Common Errors to Avoid

1. Type III${}_1$ ≠ Powers III${}_{\lambda }$ at $\lambda =1$. The tracial endpoint of the Powers family is type II${}_1$, not type III${}_1$. III${}_1$ is a separate Connes class with full positive S-invariant $[0,\infty )$. Week 12 gives the hypothesis-sensitive AQFT route to the hyperfinite III${}_1$ result; no single condition such as split or nuclearity is treated as sufficient by itself.

2. Type III${}_1$ universality is NOT due to Haag–Hugenholtz–Winnink. HHW 1967 is about KMS/equilibrium states, not local-algebra classification. The correct citation chain is Driessler 1975/77 / Fredenhagen 1985 / Buchholz–Wichmann 1986 / Buchholz–D’Antoni–Fredenhagen 1987, plus Connes 1976 + Haagerup 1987 for operator-algebra uniqueness.

3. Tomita–Takesaki does NOT produce a state. Given a faithful normal state $\omega$ on $\mathcal{M}$ (equivalently, a cyclic-separating vector in a standard representation), Tomita–Takesaki produces the state’s canonical modular flow ${\sigma }_t^{\omega }$, and $\omega$ is KMS for that flow at $\beta =1$. The state is input, not output.

4. Murray–vN equivalence of projections is INTERNAL to the algebra. A non-tracial vector state cannot certify or rule out Murray–vN equivalence. Use a partial isometry $u\in \mathcal{M}$, not a state expectation.

5. The crossed product is NOT by itself the gravitational dressing. The crossed product is an operator-algebraic construction (Block D, Week 13). Witten and CPW identify a crossed-product algebra with a gravitationally dressed observable algebra in their stated large-$N$ regimes, after adding the appropriate energy variable and gravitational constraints. That identification is a source-specific physical result with perturbative and holographic hypotheses, not a consequence of the abstract continuous-core theorem.

6. Tensor-product factorization does NOT hold for AQFT bounded local algebras. Type III${}_1$ algebras do not split $\mathcal{H}=\mathcal{H}(\mathcal{O})\otimes \mathcal{H}({\mathcal{O}}^{\prime })$. Tensor reasoning is valid only under the split property (Doplicher–Longo) or in type-I regulators.

7. A shared clock unitary is not central. In a crossed product, $\lambda (t)$ implements a nontrivial automorphism and therefore does not commute with the represented original algebra. Two commuting dressed algebras cannot both be generated by the same noncentral $\lambda (t)$ in the naive way; the commutant construction must be used.

8. Type II${}_1$ is not finite-dimensional. A II${}_1$ factor has a finite normalized trace but is generally infinite-dimensional. In the de Sitter construction, the finite trace follows from the observer-clock spectral restriction, not from replacing QFT by a finite-dimensional Hilbert space.