Appendix D — Crossed-Product Reference Sheet

Crossed-Product Reference Sheet

This sheet fixes the notation used in Weeks 13–15 and throughout Semester II. The point of the notation is not cosmetic: the regular-representation coordinate, its conjugate momentum, modular time, physical time, and the parameter of the dual action play different roles.

D.1 Seven symbols that must not be conflated

Symbol	Meaning	Where it appears
$t$	dimensionless parameter of the automorphism ${\alpha }_t$; for a modular crossed product, ${\alpha }_t={\sigma }_t^{\omega }$	$\lambda (t)$ and the covariance relation
$u$	physical time or rapidity	$u=\beta t$; for a Rindler wedge, $u=2\pi t$
$q$	scalar coordinate in the regular representation	argument of $\xi (q)$ in $L^2({\mathbb{R}}_q)$
$Q$	multiplication by $q$: $(Q\xi )(q)=q\xi (q)$	implementation of the dual action in the ambient algebra
$P$	momentum conjugate to $Q$: $P=-i{\partial }_q$, $[Q,P]=i$	generator of $\lambda (t)=e^{-itP}$
$p$	Fourier spectral variable of $P$	multiplication variable after Fourier transform; the trace model uses $e^{-p}dp$
$r$	parameter of the dual action ${\theta }_r$	${\theta }_r(\lambda (t))=e^{itr}\lambda (t)$

Thus $q$ is a scalar coordinate and $Q$ is the operator that multiplies by it, while $p$ is the Fourier-dual scalar variable and $P$ is the operator whose spectral value is $p$. A function $g(Q)$ is multiplication by $g(q)$; a function $f(P)$ becomes multiplication by $f(p)$ only after Fourier transform. They are not interchangeable.

D.2 Setup and regular representation

Let $\mathcal{M}\subset \mathcal{B}(\mathcal{H})$ be a von Neumann algebra and let $\alpha :\mathbb{R}\to \mathrm{Aut}(\mathcal{M})$ be a $\sigma$-weakly continuous action. On $\mathcal{H}\otimes L^2({\mathbb{R}}_q)$ define

$$({\pi }_{\alpha }(a)\xi )(q)={\alpha }_{-q}(a)\xi (q),(\lambda (t)\xi )(q)=\xi (q-t).$$

Since $P=-i{\partial }_q$, the second formula is $\lambda (t)=e^{-itP}$. The crossed product is

$$\mathcal{M}{⋊}_{\alpha }\mathbb{R}:=\{{\pi }_{\alpha }(\mathcal{M}),\lambda (t):t\in \mathbb{R}{\}}^{\prime \prime }.$$

The defining covariance relation is

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

This relation says precisely that the action becomes inner after passing to the crossed product. Calling $q$ a physical clock reading is an additional interpretation, appropriate only when the physical construction supplies such a clock or relational degree of freedom.

D.3 The dual action

The dual action $\theta :\mathbb{R}\to \mathrm{Aut}(\mathcal{M}{⋊}_{\alpha }\mathbb{R})$ is fixed by

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

In the regular representation it is implemented in the ambient algebra by $1\otimes e^{irQ}$. Since

$$e^{irQ}P{e}^{-irQ}=P-r,$$

the same action is a translation of the Fourier variable:

$${\theta }_r(f(P))=f(P-r).$$

The fixed-point algebra is

$$(\mathcal{M}{⋊}_{\alpha }\mathbb{R}{)}^{\theta }={\pi }_{\alpha }(\mathcal{M}).$$

D.4 Modular crossed product and continuous core

For a faithful normal state or weight $\omega$, the course convention is

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

The modular crossed product

$$c_{\omega }(\mathcal{M}):=\mathcal{M}{⋊}_{{\sigma }^{\omega }}\mathbb{R}$$

is called the continuous core. Cores obtained from two faithful normal weights are canonically isomorphic up to the Connes-cocycle identification. This state independence concerns the algebra up to canonical isomorphism; a particular realization and a particular lifted state still depend on choices.

D.5 Type and centre of the core

The continuous core is semifinite. The factor statement requires care:

• If $\mathcal{M}$ is a type III${}_1$ factor, $c_{\omega }(\mathcal{M})$ is a type II${}_{\infty }$ factor.
• If $\mathcal{M}$ is type III${}_{\lambda }$ with $0\le \lambda <1$, its core is a semifinite type-II algebra with a nontrivial centre. The induced action on that centre is the flow of weights. It is therefore incorrect to call the whole core a factor in these cases.
• If $\mathcal{M}$ is semifinite, every modular automorphism group is inner. Crossing by that inner action gives

$$\mathcal{M}{⋊}_{{\sigma }^{\omega }}\mathbb{R}\cong \mathcal{M}\bar{\otimes }{L}^{\infty }(\mathbb{R}),$$

not $\mathcal{M}\bar{\otimes }\mathcal{B}(L^2(\mathbb{R}))$. The latter algebra appears after crossing once more by the dual action.

For a hyperfinite type III${}_1$ source, the core is the hyperfinite type II${}_{\infty }$ factor.

D.6 Canonical trace and its scaling

The continuous core has a faithful normal semifinite trace $\hat{\tau }$ satisfying the course normalization

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

For a type II${}_{\infty }$ factor this trace is unique up to a positive multiplicative constant. The formula is an abstract theorem; in a general type III representation the trace is not a vector functional of a non-normalizable vector such as $\Omega \otimes e^{-q/2}$.

D.7 Exact inner-action Fourier model

Let $\mathcal{M}=\mathcal{B}(\mathcal{H})$ and $\omega (a)=\mathrm{Tr}(\rho a)$ with $\rho >0$. Since

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

is inner, an untwisting unitary followed by Fourier transform gives

$$\mathcal{M}{⋊}_{{\sigma }^{\omega }}\mathbb{R}\cong \mathcal{B}(\mathcal{H})\bar{\otimes }{L}^{\infty }({\mathbb{R}}_p).$$

Here $p$ is the spectral variable of $P$, not the original coordinate $q$. On positive simple tensors in its natural trace ideal,

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

Because ${\theta }_r(f(P))=f(P-r)$,

$$\hat{\tau }({\theta }_r(a\otimes f(P)))=\mathrm{Tr}(a)\int f(p-r)e^{-p}dp=e^{-r}\hat{\tau }(a\otimes f(P)).$$

Compactly supported positive $f$ and finite-rank positive $a$ give a concrete $\hat{\tau }$-finite core. Notice that a generic $g(Q)$ is not an element of the abelian factor $L^{\infty }(P)$ in this model.

For a normalization comparison one may define $E:=p/\beta$, after which the same weight is $e^{-\beta E}dE$ up to an overall constant. Calling $E$ a physical energy requires a further model-dependent identification. For Rindler flow $u=2\pi t$, the comparison weight is $e^{-2\pi E}dE$. The factor $2\pi$ comes from rescaling the spectral variable, not from replacing the coordinate $Q$ by modular time or from proving that $E$ is the matter boost generator.

D.8 Product-state density and entropy in the exact model

Let $\rho$ be a density matrix on $\mathcal{H}$ and let $\mu (p)dp$ be a probability distribution with the required integrability. The product state

$$\hat{\omega }(a\otimes f)=\mathrm{Tr}(\rho a)\int \mu (p)f(p)dp$$

has density, relative to $\hat{\tau }$,

$$D_{\hat{\omega }}(p)=\rho \mu (p)e^p.$$

Consequently,

$$\boxed{S_{\hat{\tau }}(\hat{\omega })=S(\rho )+H(\mu )-{\mathbb{E}}_{\mu }[p],}$$

where

$$H(\mu )=-\int \mu (p)\log \mu (p)dp.$$

The last term is part of the entropy. It cannot be discarded or renamed as a normalization constant when the mean of $p$ changes from state to state.

D.9 Trace rescaling

If ${\hat{\tau }}^{\prime }=c\hat{\tau }$, the density of a fixed normalized state becomes $D^{\prime }=D/c$. Therefore

$$S_{{\hat{\tau }}^{\prime }}(D^{\prime })=-c\hat{\tau }\left(\frac{D}{c}\log \frac{D}{c}\right)=S_{\hat{\tau }}(D)+\log c.$$

Thus trace rescaling shifts every normalized-state entropy by $+\log c$. Entropy differences computed with the same trace normalization are unchanged.

D.10 Entropy-difference identity: exact scope

For two faithful densities $D_{\omega },D_{\varphi }$ in the same semifinite algebra with the same trace, whenever the displayed terms are well defined (or after a stated common regularization),

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

Writing $K_{\varphi }=-\log D_{\varphi }$ gives the exact identity

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

This is the precise semifinite version of “minus relative entropy plus modular-energy difference.” To replace the relative entropy on the right by the Araki relative entropy of two states on the original type III algebra, one must specify a lift to the core that preserves that relative entropy. It is not a statement about arbitrary pairs of extensions.

D.11 Inner-unitary check

If $D_{\omega }=U{D}_{\varphi }{U}^{\ast }$ for a unitary $U$ in the same semifinite algebra, then traciality implies

$$S_{\hat{\tau }}(D_{\omega })=S_{\hat{\tau }}(D_{\varphi }).$$

The entropy-difference identity therefore yields an exact cancellation,

$$S(D_{\omega }\Vert D_{\varphi })=\hat{\tau }\left[(D_{\omega }-D_{\varphi })K_{\varphi }\right].$$

This is the correct consistency check for a Weyl coherent excitation implemented by a unitary belonging to the algebra and dressed with the same clock. A nonzero entropy difference requires a different comparison: for example, a non-inner excitation relative to the chosen algebra, a changed clock distribution, or a state-dependent identification of two cores.

D.12 Takesaki duality

The full duality statement is

$$(\mathcal{M}{⋊}_{\alpha }\mathbb{R}){⋊}_{\theta }\mathbb{R}\cong \mathcal{M}\bar{\otimes }\mathcal{B}(L^2(\mathbb{R})).$$

The second tensor factor is the stabilization produced by the second crossing. It should not be inserted into the first crossed product of an inner action.

D.13 Common pitfalls

1. $Q$ and $P$ are conjugate operators, not two names for the same clock observable.
2. The dual action is a phase on $\lambda (t)$, or equivalently a translation of the $P$-spectrum. It is not a translation of $Q$ in the regular representation.
3. The weighted integral $e^{-p}dp$ belongs to the Fourier variable in the exact inner-action model. A formal ${\delta }_q$ diagonal is not a proof of the trace formula for a type III core.
4. For a type III factor, the continuous core is a factor precisely in the type III${}_1$ case relevant here; III${}_0$ and III${}_{\lambda }$, $0<\lambda <1$, retain a nontrivial centre.
5. The crossed product is an algebraic construction. Its identification with a gravitationally dressed algebra requires the physical hypotheses of Witten, CPW, CLPW, or the relevant model.
6. Clock width and the matter UV cutoff are independent. In $1+1$ dimensions matter entanglement is logarithmic; in $3+1$ dimensions its leading divergence is proportional to $A/{ϵ}^2$.
7. Under $\hat{\tau }\to c\hat{\tau }$, entropy shifts by $+\log c$.

D.14 Use in the course

Week	Main use
13	Definition, dual action, core, exact inner-action Fourier model
14	Semifinite entropy, product-state formula, entropy-difference identity
15	One-sided modular standard form and the TFD analogy
Sem II 2–4	Witten construction, with physical variables matched to the abstract dictionary
Sem II 6–7	CPW right algebra and its commutant; generalized entropy
Sem II 10–13	Structural synthesis and perturbations

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Reference sheet for the algebraic-QFT course. Last revised 2026-08-24.