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An Inelastic Bound on Chaos

Gustavo J. Turiaci

TL;DR

This work extends the quantum chaos bound to OTOCs involving four different operators, including non-Hermitian cases, under the MSS framework. It proves an inelastic bound λ_ABCD ≤ λ_diag ≤ 2π/β and derives amplitude constraints that relate off-diagonal growth to diagonal OTOC data, with extensions to linear combinations and non-Hermitian operators. In holography, inelastic OTOCs probe non-gravitational bulk interactions (Pomeron exchange) while elastic OTOCs are gravity-driven, and the analysis suggests gravity is the highest spin force. In 2d CFTs at large c, elastic OTOCs saturate the bound via the vacuum block, whereas inelastic OTOCs are governed by non-vacuum blocks with small amplitudes and a growth-decay pattern determined by effective spin and quasinormal modes. Overall, the paper connects bulk scattering data and OPE coefficients to universal chaos bounds, offering a framework to constrain non-elastic bulk interactions and OPE data in holographic theories and beyond.

Abstract

We study a generalization of the chaos bound that applies to out-of-time-ordered correlators between four different operators. We prove this bound under the same assumptions that apply for the usual chaos bound and extend it to non-hermitian operators. In a holographic theory, these correlators are controlled by inelastic scattering in the bulk and we comment on implications. In particular, for holographic theories the bound together with the equivalence principle suggests that gravity is the highest spin force, and the strongest one with that spin.

An Inelastic Bound on Chaos

TL;DR

This work extends the quantum chaos bound to OTOCs involving four different operators, including non-Hermitian cases, under the MSS framework. It proves an inelastic bound λ_ABCD ≤ λ_diag ≤ 2π/β and derives amplitude constraints that relate off-diagonal growth to diagonal OTOC data, with extensions to linear combinations and non-Hermitian operators. In holography, inelastic OTOCs probe non-gravitational bulk interactions (Pomeron exchange) while elastic OTOCs are gravity-driven, and the analysis suggests gravity is the highest spin force. In 2d CFTs at large c, elastic OTOCs saturate the bound via the vacuum block, whereas inelastic OTOCs are governed by non-vacuum blocks with small amplitudes and a growth-decay pattern determined by effective spin and quasinormal modes. Overall, the paper connects bulk scattering data and OPE coefficients to universal chaos bounds, offering a framework to constrain non-elastic bulk interactions and OPE data in holographic theories and beyond.

Abstract

We study a generalization of the chaos bound that applies to out-of-time-ordered correlators between four different operators. We prove this bound under the same assumptions that apply for the usual chaos bound and extend it to non-hermitian operators. In a holographic theory, these correlators are controlled by inelastic scattering in the bulk and we comment on implications. In particular, for holographic theories the bound together with the equivalence principle suggests that gravity is the highest spin force, and the strongest one with that spin.

Paper Structure

This paper contains 10 sections, 41 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Constraints on generic OTOC
  3. An inelastic chaos bound
  4. Non-Hermitian Operators
  5. An Example: 2d CFTs
  6. Kinematics
  7. Elastic case
  8. Inelastic case
  9. Open Questions
  10. Semiclassical Virasoro Blocks

Figures (1)

  • Figure 1: Sketch of a typical behavior of inelastic OTOC $F(t)$ in 2d CFT (black curve) as a function of time, assuming its approximated by a non-vacuum block with effective dimension $\Delta_{\rm eff}$ and spin $s_{\rm eff}$. Initially the OTOC grows exponentially with rate $\lambda = \frac{2\pi}{\beta} (s_{\rm eff}-1)$. For late times the fast decay is controlled by the quasi-normal modes. In blue we show a typical elastic OTOC.