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

An observer with a finite lifetime $\mathcal{T}$ perceives the Minkowski vacuum as a thermal state at temperature $T_D=2\mathrm{\hslash }\mathrm{/}(\pi \mathcal{T})$, 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 ${\lambda }_L=\pi T_D\mathrm{/}\mathrm{\hslash }$, 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.