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

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 $T_D=2\mathrm{\hslash }\mathrm{/}(\pi \mathcal{T})$ is associated with a Lyapunov exponent ${\lambda }_L=\pi T_D\mathrm{/}\mathrm{\hslash }$, which is half the upper saturation bound of the information scrambling rate.