The Timelike Tube Theorem in Curved Spacetime

TL;DR

The paper extends the timelike tube theorem to quantum fields on real analytic curved spacetimes by leveraging microlocal analysis, specifically the analytic wavefront set ${\mathrm{WF}}_a$, to show that local observables in ${\mathcal{U}}$ constrain those in the timelike envelope ${\mathcal{E}}({\mathcal{U}})$. The core method reduces the theorem to commutation relations of local operators via a dense set of analytic states, and then propagates vanishing matrix elements from ${\mathcal{U}}$ to ${\mathcal{E}}({\mathcal{U}})$ using the causal structure of ${\mathrm{WF}}_a$. The Reeh-Schlieder theorem is generalized to curved spacetime under analytic-state assumptions, with WF$_a$-based arguments ensuring density and entanglement properties. The article also discusses explicit constructions and prospects for analytic states, including tempered analytic states and Euclidean (Hartle–Hawking-type) constructions, highlighting avenues to realize the analytic-state framework in both free and interacting theories.

Abstract

The timelike tube theorem asserts that in quantum field theory without gravity, the algebra of observables in an open set U is the same as the corresponding algebra of observables in its ``timelike envelope'' E(U), which is an open set that is in general larger. The theorem was originally proved in the 1960's by Borchers and Araki for quantum fields in Minkowski space. Here we sketch the proof of a version of the theorem for quantum fields in a general real analytic spacetime. Details have appeared elsewhere.

Figures (4)

• Figure 1: In these figures, time runs vertically. Illustrated in (a) is the extension in a timelike direction from ${\mathcal{U}}$ to $D({\mathcal{U}})$ and in (b) the extension in a spacelike direction from ${\mathcal{U}}$ to ${\mathcal{E}}({\mathcal{U}})$.
• Figure 2: A distribution $\phi$ vanishes to the left of the hypersurface $G=0$ and not to the right. $\phi$ does not vanish in any neighborhood of the point $x_0$.
• Figure 3: The hypersurface ${\mathcal{U}}$ can "grow" outward to $\widetilde{{\mathcal{U}}}$ in such a way that its normal vector is always everywhere spacelike. This can continue until ${\mathcal{U}}$ expands to fill out the timelike envelope ${\mathcal{E}}({\mathcal{U}}),$ which has null boundaries.
• Figure 4: The lightly shaded region is the $t>0$ portion of a real analytic spacetime of Lorentz signature. From $t=0$, we continue the spacetime in the direction of imaginary time, truncating it at $t=-{\mathrm i}\epsilon$. The Euclidean part of the picture is more darkly shaded.