# A Bound on Chaos from Stability

## TL;DR

This paper derives an upper bound on quantum chaos, specifically the Lyapunov exponent, from the stability conditions of coadjoint orbit actions, connecting semi-classical analysis with chaos bounds.

## Contribution

It establishes a direct link between the stability of coadjoint orbit actions and the chaos bound, showing the bound is saturated by certain orbits and not by others.

## Key findings

- The stability condition leads to an upper bound on the Lyapunov exponent.
- The bound matches the chaos bound proven in prior work.
- Different coadjoint orbits exhibit maximal or non-maximal Lyapunov exponents.

## Abstract

We explore the quantum chaos of the coadjoint orbit action. We study quantum fluctuation around a saddle point to evaluate the soft mode contribution to the out-of-time-ordered correlator. We show that the stability condition of the semi-classical analysis of the coadjoint orbit found by Witten (1988) leads to the upper bound on the Lyapunov exponent which is identical to the bound on chaos proven in arXiv:1503.01409. The bound is saturated by the coadjoint orbit $\text{Diff}(S^1)/SL(2)$ while the other stable orbit $\text{Diff}(S^1)/U(1)$ where the $SL(2,\mathbb{R})$ is broken to $U(1)$ has non-maximal Lyapunov exponent.