Local Poincaré Algebra from Quantum Chaos

TL;DR

This work identifies a universal mechanism by which a local 2D Poincaré algebra emerges near bifurcate Killing horizons, connecting bulk geometry to boundary modular dynamics. By leveraging half-sided modular inclusions and modular flow, the authors show that modular future/past subalgebras in quantum systems yield an emergent local Poincaré group in a near-horizon scaling limit, with operators $G_\pm$ generating exponentially growing/decaying modular scrambling modes. In holographic GFF above the Hawking-Page transition, these structures imply maximal modular chaos, a modular second law, and exponential decay of correlators, tying quantum ergodic properties to horizon locality. The results generalize beyond equilibrium to arbitrary state-preserving flows with positive generators and frame a broad ergodic hierarchy: modular K-systems are maximally chaotic and exhibit strong mixing, second law behavior, and information loss-friendly dynamics. Overall, the paper reveals that horizon-locality and chaotic dynamics are not unique to gravity but arise in a wide class of quantum systems through modular algebraic structure, with implications for holography and quantum chaos.

Abstract

The local two-dimensional Poincaré algebra near the horizon of an eternal AdS black hole, or in proximity to any bifurcate Killing horizon, is generated by the Killing flow and outward null translations on the horizon. In holography, this local Poincaré algebra is reflected as a pair of unitary flows in the boundary Hilbert space whose generators under modular flow grow and decay exponentially with a maximal Lyapunov exponent. This is a universal feature of many geometric vacua of quantum gravity. To explain this universality, we show that a two-dimensional Poincaré algebra emerges in any quantum system that has von Neumann subalgebras associated with half-infinite modular time intervals (modular future and past subalgebras) in a limit analogous to the near-horizon limit. In ergodic theory, quantum dynamical systems with future or past algebras are called quantum K-systems. The surprising statement is that modular K-systems are always maximally chaotic. Interacting quantum systems in the thermodynamic limit and large $N$ theories above the Hawking-Page phase transition are examples of physical theories with future/past subalgebras. We prove that the existence of (modular) future/past von Neumann subalgebras also implies a second law of (modular) thermodynamics and the exponential decay of (modular) correlators. We generalize our results from the modular flow to any dynamical flow with a positive generator and interpret the positivity condition as quantum detailed balance.

Figures (13)

• Figure 1: The Killing vector $B$ of a spacetime with a bifurcate Killing horizon splits it into four regions, similar to boosts in Minkowski space. $\sigma$ is the bifurcation surface and the null translations $P_\pm$ on the future and past horizons $\mathfrak{H}^\pm$ are isometries of these surfaces. They can be extended to $G_\pm$ in the near horizon regions.
• Figure 2: The wedge algebras in a spacetime with bifurcate Killing horizons. (a) $W(z^+)$ is the von Neumann subalgebra of all observables the observer (dashed line) has access to from a particular moment in time (black dot) until eternity forms, whereas $W(z^-)$ is the von Neumann subalgebra of all observables the observer had access to since past infinity until now. (b) Given a point $(z^{+},z^{-})$, we define a right wedge $W(z^{+},z^{-}) = (U>z^{+},V>z^{-})$ and a left wedge $W'(z^{+},z^{-}) = (U<z^{+},V<z^{-})$.
• Figure 3: The two-dimensional Poincaré algebra can be viewed as a modular Anosov system with maximal exponents $\lambda=2\pi$ (see section \ref{['sec:ergodicity']}). The blue lines denote the flow generated by the Killing vector field and the red lines denote commuting exponentially growing (decaying) modes $G_{+}$ ($G_{-}$).
• Figure 4: There are enhanced isometry groups on future and past horizons $\mathfrak{H}^\pm$ generated by $P_\pm$ and the Killing flow $B$ that act as dilation and translations: $U\to e^{2\pi t}U+U_0$ and $V\to e^{-2\pi t}+V_0$.
• Figure 5: (a) The time interval algebras of GFF above the Hawking-Page phase transition. (b) The algebra $\mathcal{A}_{(t_1,\infty)}$ is a future subalgebra and $\mathcal{A}_{(-\infty,t_2)}$ is a past subalgebra.

Theorems & Definitions (60)

• Theorem 1: Bisognano-Wichmann CRT Theorem
• proof
• Theorem 2: Borchers CRT Theorem
• proof
• Corollary 3: Poincaré to Future/Past Subalgebras
• Corollary 4: Poincaré to Modular Future/Past Subalgebras
• proof
• Definition 2.1: Spacetime with bifurcate Killing horizon
• Theorem 5
• proof