[Paper Review] Addendum to Fast Scramblers

Leonard Susskind|arXiv (Cornell University)|Jan 31, 2011

Noncommutative and Quantum Gravity Theories7 references54 citations

TL;DR

This addendum establishes that de Sitter space and Rindler space are fast scramblers, meaning they scramble quantum information in time logarithmic in entropy. It proposes that their holographic duals may be finite-temperature matrix quantum mechanics, with scrambling governed by non-Abelian matrix interactions, supporting the idea that such systems achieve the fastest possible information scrambling consistent with quantum gravity bounds.

ABSTRACT

This paper is an addendum to [arXiv:0808.2096] in which I point out that both de Sitter space and Rindler space are fast scramblers. This fact naturally suggests that the holographic description of a causal patch of de Sitter space may be a matrix quantum mechanics at finite temperature. The same can be said of Rindler space. Some qualitative features of these spaces can be understood from the matrix description.

Motivation & Objective

• To demonstrate that de Sitter space and Rindler space are fast scramblers, meaning they scramble information in time logarithmic in entropy.
• To argue that the holographic description of a causal patch in these spacetimes may be a matrix quantum mechanics at finite temperature.
• To explore how the fast-scrambling behavior arises from non-Abelian matrix interactions in the dual theory.
• To address the challenge of realizing global symmetries like observer complementarity in finite-entropy matrix models.
• To discuss the limitations of holographic duality in de Sitter space, particularly regarding precision and long-time behavior.

Proposed method

• Analyzes the scrambling time $ t^* $ in terms of diffusion time $ t_D $, showing $ t^* T \approx \hbar \log S $ for black holes, indicating fast scrambling.
• Applies the same logic to de Sitter and Rindler spacetimes, showing their causal patches exhibit the same logarithmic scrambling time.
• Uses the Rindler horizon's local geometry to derive the diffusion time for charge spreading, yielding $ t^* \sim MG \log(L/l_s) $, consistent with fast scrambling.
• Connects the matrix model's quartic couplings $ -\text{Tr}[X^i,X^j]^2 $ to maximal coupling between all matrix elements, enabling rapid information spreading.
• Applies the thermofield double formalism to describe the global state of de Sitter space, allowing non-compact $ O(4,1) $ symmetry to emerge in the large $ N $ limit.
• Considers the large $ N $ limit to evade a no-go theorem on realizing $ O(4,1) $ symmetry in finite-entropy models, analogous to Matrix theory.

Experimental results

Research questions

• RQ1 Can de Sitter space and Rindler space be classified as fast scramblers, given their causal patch structure?
• RQ2 What is the nature of the holographic dual for a causal patch of de Sitter space, and does it involve matrix quantum mechanics at finite temperature?
• RQ3 How do non-Abelian matrix interactions in the dual theory account for the logarithmic scrambling time observed in black holes and horizons?
• RQ4 Can the global symmetries of de Sitter space, such as observer complementarity, be realized in a finite-entropy matrix model?
• RQ5 What are the limits of precision in holographic descriptions of de Sitter space, particularly on long time-scales?

Key findings

• De Sitter space and Rindler space are fast scramblers, with scrambling time $ t^* \sim \hbar \log S / T $, matching the black hole bound.
• The causal patch of de Sitter space may be holographically described by finite-temperature matrix quantum mechanics, with $ N \sim \mathcal{R}/l_s $.
• The fast scrambling arises from the maximal coupling in matrix models, where every matrix element is directly coupled to every other via quartic interactions.
• The $ O(4,1) $ symmetry of de Sitter space can only be realized in the large $ N $ limit, where entropy diverges, evading a no-go theorem for finite entropy.
• Holographic duals of de Sitter space are expected to be approximate, with precision limited by the maximum observable area $ \mathcal{R}^2 $, and not exact for arbitrary time-scales.
• The time to fall to the Rindler horizon scales as $ t \sim \mathcal{R} \log(\mathcal{R}/l_s) $, consistent with the fast-scrambling behavior.