Why Does Quantum Field Theory In Curved Spacetime Make Sense? And What Happens To The Algebra of Observables In The Thermodynamic Limit?

TL;DR

The paper argues that quantum field theory in curved spacetime, especially for open universes, naturally demands an algebraic, density-matrix-centric framework due to the absence of a unique vacuum and the proliferation of entanglement leading to Type III von Neumann algebras. It introduces the thermofield double as a robust method to describe infinite-volume thermodynamics in a separable Hilbert space and demonstrates how different algebraic types (I, II, III) arise in various limits, including open universes, thermodynamic limits, and large-N gauge theories. The work connects these ideas to familiar concepts like the Unruh effect, Hagedorn temperature, Hawking-Page transitions, and AdS/CFT, illustrating a unifying algebraic perspective on quantum gravity, horizon thermodynamics, and gauge theory limits. It also clarifies how modular theory underpins time evolution in these open-system contexts and highlights the profound role of entanglement in governing the structure of observables across spacetime regions. Overall, the article provides a conceptual roadmap linking QFT in curved spacetime, thermodynamic limits, and large-$N$ gauge theories through the lens of operator algebras and the thermofield double.

Abstract

This article aims to explain some of the basic facts about the questions raised in the title, without the technical details that are available in the literature. We provide a gentle introduction to some rather classical results about quantum field theory in curved spacetime and about the thermodynamic limit of quantum statistical mechanics. We also briefly explain that these results have an analog in the large N limit of gauge theory.

Figures (3)

• Figure 1: (a) A Cauchy hypersurface $S$, drawn as a two-sphere, is divided by the equator $B$ into an upper hemisphere $V$ and a lower hemisphere $V'$. (b) In this view, $S$ is drawn as a one-dimensional curve and $B$ is depicted as a point that divides $S$ into the two open sets $V$ and $V'$. ${\mathcal{U}}$ and ${\mathcal{U}}'$ are open sets in spacetime that are the domains of dependence of $V$ and $V'$. It is apparent from this picture that near $B$, ${\mathcal{U}}$ and ${\mathcal{U}}'$ can be modeled by opposite Rindler wedges in Minkowski space.
• Figure 2: (a) A thermal density matrix $\rho=\frac{1}{Z}e^{-\beta H}$ can be computed by a path integral on the strip $0\leq \tau\leq \beta$. The horizontal direction in the picture represents the "spatial" manifold (or lattice) $W$. (b) The thermofield double state $\Psi_{\mathrm{TFD}}=\rho^{1/2}$ can be computed by a path integral on a strip of half the width, $0\leq \tau\leq \beta/2$. Operators of the "right" or "left" copy are inserted, respectively, on the boundaries at $\tau=0$ and $\tau=\beta/2$. .
• Figure 3: (a) In a two-dimensional spacetime, we consider a Cauchy hypersurface consisting of a circle $S$. The entangling surface $B$ is a pair of points $p,p'$, whose complement in $S$ consists of the two intervals $V$ and $V'$. (b) A path integral on the disc $W$, whose boundary is $S$, computes the ground state of a conformal field theory. (c) The disc with the points $p$ and $p'$ removed can be conformally mapped to ${\mathbb{R}}\times I$, where $I$ is a unit interval. $V$ and $V'$ become the two boundary components of ${\mathbb{R}}\times I$, and the points $p,p'$ are projected to infinity.