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

• URL: http://arxiv.org/abs/2407.18191v1
• Date: Thu, 25 Jul 2024 16:57:35 GMT
• Title: Conformal quantum mechanics of causal diamonds: Quantum instability and semiclassical approximation
• Authors: H. E. Camblong, A. Chakraborty, P. Lopez-Duque, C. Ordóñez,
• Abstract summary: 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. We show that the unbounded nature of $S$ endows it with a quantum instability.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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