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

TL;DR

This paper investigates how conformal quantum mechanics explains the thermal perception and instability of causal diamonds in Minkowski space, revealing a connection between symmetries, time evolution, and information scrambling.

Contribution

It demonstrates that the SO(2,1) symmetry generator in conformal quantum mechanics governs the time evolution and thermal nature of causal diamonds, introducing a path integral framework and operator duality analysis.

Findings

The generator S induces thermal behavior in causal diamonds.

Quantum instability of S exhibits a Lyapunov exponent related to temperature.

Path integral methods elucidate the duality between operators S and R.

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.