The role of conformal symmetry and instability in determining the temperature of a causal diamond

A finite-lifetime observer detects thermal particles in the Minkowski vacuum. Such an observer is constrained to a diamond shaped region in the flat spacetime -- known as a causal diamond. In the literature, it has been shown that the generator of the time evolution of a diamond observer is a non-compact SO(2,1) hyperbolic transformation generator $S$ and is intimately connected to conformal quantum mechanics (CQM). In this paper, using an explicit representation of this $S$ operator as described in the de Alfaro-Fubini-Furlan model involving the inverse square potential and the inverted harmonic oscillator potential, we explore the connection between CQM and thermality in causal diamonds via a semiclassical approach. In doing so, we probe the role of instability in determining the diamond temperature by using the Gutzwiller Trace Formula. We further make some comments about the quantum chaotic nature of the $S$ operator.