Conformal quantum mechanics of causal diamonds: Quantum instability and semiclassical approximation

TL;DR

This paper explores the quantum instability of conformal quantum mechanics in causal diamonds, linking it to thermality and information scrambling, with a semiclassical analysis revealing a temperature related to observer lifetime.

Contribution

It demonstrates the quantum instability of the hyperbolic operator in conformal quantum mechanics and connects it to thermality and scrambling rates in a semiclassical framework.

Findings

Quantum instability of the hyperbolic operator S.

Detected temperature inversely proportional to observer lifetime.

Lyapunov exponent is half the maximal scrambling rate.

Abstract

Causal diamonds are known to have thermal behavior that can be probed by finite-lifetime observers equipped with energy-scaled detectors. This thermality can be attributed to the time evolution of observers within the causal diamond, governed by one of the conformal quantum mechanics (CQM) symmetry generators: the noncompact hyperbolic operator $S$. In this paper, we show that the unbounded nature of $S$ endows it with a quantum instability, which is a generalization of a similar property exhibited by the inverted harmonic oscillator potential. Our analysis is semiclassical, including a detailed phase-space study of the classical dynamics of $S$ and its dual operator $R$, and a general semiclassical framework yielding basic instability and thermality properties that play a crucial role in the quantum behavior of the theory. For an observer with a finite lifetime $\mathcal{T}$, the detected temperature $T_D = 2 \hbar/(\pi \mathcal{T})$ is associated with a Lyapunov exponent $\lambda_L = \pi T_D/\hbar$, which is half the upper saturation bound of the information scrambling rate.