Conformal quantum mechanics of causal diamonds: Time evolution, thermality, and instability via path integral functionals

• URL: http://arxiv.org/abs/2407.18177v2
• Date: Wed, 01 Jan 2025 15:54:43 GMT
• Title: Conformal quantum mechanics of causal diamonds: Time evolution, thermality, and instability via path integral functionals
• Authors: H. E. Camblong, A. Chakraborty, P. Lopez-Duque, C. R. Ordóñez,
• Abstract summary: An observer with a finite lifetime perceives the Minkowski vacuum as a thermal state at temperature $T_D = 2 hbar/(pi mathcalT)$. In this paper, we explore the emergence of thermality in causal diamonds due to the role played by the symmetries of conformal quantum mechanics.
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• Abstract: An observer with a finite lifetime $\mathcal{T}$ perceives the Minkowski vacuum as a thermal state at temperature $T_D = 2 \hbar/(\pi \mathcal{T})$, as a result of being constrained to a double-coned-shaped region known as a causal diamond. In this paper, we explore the emergence of thermality in causal diamonds due to the role played by the symmetries of conformal quantum mechanics (CQM) as a (0+1)-dimensional conformal field theory, within the de Alfaro-Fubini-Furlan model and generalizations. In this context, the hyperbolic operator $S$ of the SO(2,1) symmetry of CQM: (i) is the generator of the time evolution of a diamond observer; (ii) its dynamical behavior leads to the predicted thermal nature; and (iii) its associated quantum instability has a Lyapunov exponent $\lambda_L = \pi T_D/\hbar$, which is half the upper saturation bound of the information scrambling rate. Our approach is based on a comprehensive framework of path-integral representations of the CQM generators in canonical and microcanonical forms, supplemented by semiclassical arguments. The properties of the operator $S$ are studied with emphasis on an operator duality with the corresponding elliptic operator $R$, using a representation in terms of an effective scale-invariant inverse square potential combined with inverted and ordinary harmonic oscillator potentials.

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