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

TL;DR

The paper addresses how a finite-lifetime observer in Minkowski spacetime experiences thermality due to causal horizons, by formulating a $(0+1)$-D conformal quantum mechanics (CQM) framework with SO$(2,1)$ symmetry. It develops a path-integral description of the CQM generators, identifies a duality between the hyperbolic $S$ and elliptic $R$ operators, and uses analytic continuation to derive the diamond temperature $T_D = {\hbar}/(\pi \alpha)$ and associated thermodynamic structure; it also links the instability of $S$-driven dynamics to a Lyapunov exponent $\lambda_L = 1/\alpha$, connecting thermality with quantum chaos and information scrambling. The work provides a fully quantum-mechanical account of causal-diamond thermality via canonical and microcanonical path integrals, and shows that the same framework reproduces results consistent with modular and Unruh-type arguments, while highlighting a finite-lifetime observer’s dynamics as a controllable laboratory for spacetime thermodynamics. By tying diamond thermality to an inverted-harmonic-oscillator-like instability and to a dual $R$-sector with closed periodic orbits, the paper suggests deep connections to black-hole physics, holography, and SYK-like chaos, with potential experimental realizations in time-dependent quantum systems.

Abstract

An observer with a finite lifetime $\mathcal{T}$ perceives the Minkowski vacuum as a thermal state at temperature $T_D = 2 \hbar/(π\mathcal{T})$, 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 $λ_L = πT_D/\hbar$, 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.

Figures (5)

• Figure 1: The green region denotes the causal diamond of size $2\alpha$ ("radius" $\alpha$), which is the spacetime region accessible to an observer with a finite lifetime $\tau = 2 \alpha$. The diamond's boundaries are causal horizons.
• Figure 2: Graph $V_{\!_{S}}(q)$ vs $q$ of the classical effective potential associated with the operator $S$. The chosen parameters are $g=1$, $\alpha=1$. The potential is monotonic with no lower or upper bounds, and has two competing behaviors: an effective centrifugal-like barrier for small $q$ and an inverted harmonic oscillator behavior as $q \rightarrow \infty$.
• Figure 3: The integral curves of the RCKF operator $S_K$ (with $\alpha=2$) are shown. The diamond-shaped region is the causal diamond. In this diagram, the variable $r$ is continued to both positive and negative values to cover the whole real line. In particular, the integral curves within the diamond stay within the diamond.
• Figure 4: Phase-space direction field for the trajectories for the $S$ operator (with $g=1, \alpha =2$). The red line $p=q/\alpha$ shows the asymptotes of the trajectories.
• Figure 5: Example of a closed trajectory for $R$ (with $g=1,\alpha=2$, and $E_{\gamma}=2$).