Half-Sided Modular Inclusions and Spacetime Reconstruction

A half-sided modular inclusion is not merely a pair of algebras with modular groups. It is an inclusion $\mathcal N\subset\mathcal M$ with a common cyclic separating vector for which one modular group compresses $\mathcal N$ for an entire time half-line. That semigroup relation forces a positive translation group obeying the affine-group commutation law. With extra standardness and intersection hypotheses, translated copies reconstruct half-line and interval algebras. The chiral current example is returned to affine current algebras and WZW models.

Required background. Standard von Neumann algebras and Tomita–Takesaki theory supplies the modular groups; modular automorphisms, conjugations, and standard forms fixes their state dependence; and the Bisognano–Wichmann theorem and geometric modular action supplies the geometric half-line example.

Helpful background. Half-sided modular inclusions and emergent translations develops the affine-group interpretation, while modular intersections and spacetime organization treats higher configurations.

The half-sided modular hypothesis

Let $\mathcal N\subset\mathcal M$ act on $\mathcal H$, and suppose $\Omega$ is cyclic and separating for both. Denote the modular operators by $\Delta_{\mathcal N}$ and $\Delta_{\mathcal M}$. We choose the convention that the inclusion is positive half-sided modular when

$$\Delta_{\mathcal M}^{-it}\, \mathcal N\, \Delta_{\mathcal M}^{it} \subset\mathcal N, \qquad t\geq0.$$

Other sources reverse the sign or the word “positive”; the displayed semigroup relation is the invariant content used here. It is much stronger than $\Delta_{\mathcal M}^{it}\mathcal M\Delta_{\mathcal M}^{-it}=\mathcal M$, which holds for every standard algebra.

The Wiesbrock theorem constructs a strongly continuous unitary group

$$U(a)=e^{iaP}, \qquad P\geq0, \qquad U(a)\Omega=\Omega,$$

normalized so that

$$\begin{aligned} \Delta_{\mathcal M}^{it}U(a)\Delta_{\mathcal M}^{-it} &=U(e^{-2\pi t}a),\\ \mathcal N&=U(1)\mathcal M U(-1). \end{aligned}$$

Thus the two modular structures determine a representation of the translation-dilation group. The existence, positivity, and modular relations are proved in Wiesbrock 1993, pp. 83–92.

The logarithms of the two modular operators are unbounded and need not have a common domain on which their difference is self-adjoint. The theorem should not be replaced by the formal expression $P\propto\log\Delta_{\mathcal N}-\log\Delta_{\mathcal M}$. It constructs the unitary group through modular relations and closure arguments.

The proof mechanism exploits the half-line, not merely infinitesimal generators. Modular compression gives analytic continuation of suitable matrix elements into a strip; boundary identities and positivity then yield Borchers-type commutation relations. Those relations assemble products of the two modular groups into $U(a)$ and force the spectrum of its generator into $[0,\infty)$. Only after strong continuity and positivity are established may Stone’s theorem be used to write $U(a)=e^{iaP}$. This is why subtracting two unbounded modular Hamiltonians is not an alternative construction.

The normalization $\mathcal N=U(1)\mathcal M U(-1)$ fixes the translation unit, not an absolute length before a geometric model is supplied. Rescaling $a$ rescales $P$ inversely while preserving the affine relation. Reversing the chosen half-sided convention also reverses the translation orientation, so sign claims must be checked against the displayed containment rather than against terminology alone.

From half-lines to a chiral net

Interpret $\mathcal M$ as the algebra of the positive half-line and define

$$\mathcal A(a,\infty) =U(a)\mathcal M U(-a).$$

For $a<b$, candidate interval algebras are intersections such as

$$\mathcal A(a,b) =\mathcal A(a,\infty)\cap \mathcal A(b,\infty)'.$$

To obtain a nondegenerate local net, one needs more than the half-sided relation. A standard half-sided inclusion additionally requires the vacuum to be cyclic for the relative commutant $\mathcal N'\cap\mathcal M$; additivity, duality, and suitable modular-intersection relations may also be needed for the desired line or circle reconstruction. Wiesbrock derives the conformal-net correspondence under the stated extra hypotheses in Wiesbrock 1993, pp. 537–543.

The reconstruction is strongest in one-dimensional chiral geometry, where translations and dilations generate the affine group. Several modular inclusions or intersections can generate larger Möbius or Poincaré symmetry. A lone arbitrary inclusion does not reconstruct a unique higher-dimensional spacetime.

Chiral U(1) current half-lines

In the vacuum representation of the chiral $U(1)$ current net, take

$$\mathcal M=\mathcal A(0,\infty), \qquad \mathcal N=\mathcal A(1,\infty).$$

The vacuum is cyclic and separating for the half-line algebras. Geometric modular action identifies $\Delta_{\mathcal M}^{it}$ with the dilation $x\mapsto e^{-2\pi t}x$. Therefore

$$\Delta_{\mathcal M}^{-it} \mathcal A(1,\infty) \Delta_{\mathcal M}^{it} =\mathcal A(e^{2\pi t},\infty) \subset\mathcal A(1,\infty)$$

for $t\geq0$. The inclusion is half-sided modular. The reconstructed $U(a)$ has positive generator and sends $\mathcal A(0,\infty)$ to $\mathcal A(a,\infty)$; by uniqueness it agrees with the physical translation group after the unit displacement is fixed.

An independent algebraic check differentiates the affine relation. If $L=\log\Delta_{\mathcal M}$, then on a common invariant core

$$\Delta_{\mathcal M}^{it}P\Delta_{\mathcal M}^{-it} =e^{-2\pi t}P \quad\Longrightarrow\quad [L,P]=2\pi iP.$$

This is the Lie algebra of dilations and translations. Since conjugation rescales $P$ by a positive number, positivity of the translation spectrum is consistent for every modular time.

Adversarial test: two unrelated standard factors

Take standard factors $(\mathcal N,\Omega)$ and $(\mathcal M,\Omega)$ with $\mathcal N\subset\mathcal M$, but assume no half-sided containment under $\Delta_{\mathcal M}^{it}$. Each algebra has perfectly good modular data. Products of their modular conjugations are unitaries, but no theorem makes those unitaries a one-parameter translation group with positive generator.

The missing input is the semigroup inclusion for every $t$ in one half-line. Checking it at one value of $t$, or merely observing that both modular groups exist, is insufficient. Even after translations are obtained, interval nontriviality still needs standardness of the relative commutant or an equivalent condition.

Exercises

Show that the half-sided relation holds for nested half-lines under geometric dilations.

Solution

With $\mathcal M=\mathcal A(0,\infty)$ and $\mathcal N=\mathcal A(1,\infty)$, the chosen modular convention gives $\operatorname{Ad}\Delta_{\mathcal M}^{-it}$ as $x\mapsto e^{2\pi t}x$. Hence

$$\operatorname{Ad}\Delta_{\mathcal M}^{-it}(\mathcal N) =\mathcal A(e^{2\pi t},\infty).$$

For $t\geq0$, $e^{2\pi t}\geq1$, so isotony gives $\mathcal A(e^{2\pi t},\infty)\subset\mathcal A(1,\infty)=\mathcal N$.

References

• Wiesbrock, Hans-Werner. 1993a. “Half-Sided Modular Inclusions of von Neumann Algebras.” Communications in Mathematical Physics 157: 83–92. DOI.
• Wiesbrock, Hans-Werner. 1993b. “Conformal Quantum Field Theory and Half-Sided Modular Inclusions of von Neumann Algebras.” Communications in Mathematical Physics 158: 537–543. DOI.