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 T, the detected temperature TD = 2 /(π T) is associated with a Lyapunov exponent λL = π TD/, which is half the upper saturation bound of the information scrambling rate.