  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 is the generator of the time
evolution of a diamond observer, and its dynamical behavior leads to the
predicted thermal nature. 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.
