The generators of radial conformal symmetries in Minkowski space-time can be put in correspondence with the generators of time evolution in conformal quantum mechanics. Within this correspondence it is shown that in conformal quantum mechanics the state corresponding to the inertial vacuum for a conformally invariant field in Minkowski spacetime has the structure of a thermofield double. The latter is built from a bipartite "vacuum state” corresponding to the ground state of the generators of hyperbolic time evolution, which cover only a portion of the time domain. When such generators are the ones of conformal Killing vectors mapping a causal diamond in itself and of dilations, the temperature of the thermofield double reproduces, respectively, the diamond temperature and the Milne temperature. This result indicates that, for conformally invariant fields, the fundamental ingredient at the basis vacuum thermal effects in flat-space time is the non-eternal nature of the lifetime of observers rather than their acceleration.