  The generators of radial conformal symmetries in Minkowski space-time can be
mapped to the generators of time evolution in conformal quantum mechanics.
Within this correspondence we show that in conformal quantum mechanics the
state associated to the inertial vacuum in Minkowski space-time has the
structure of a thermofield double. Such state is built from a bipartite "vacuum
state", the ground state of the generators of hyperbolic time evolution, which
cover only part of the time domain. When time evolution is restricted to a
finite time domain one obtains the temperature perceived by static diamond
observers in the Minkowski vacuum. When time evolution is determined by
dilations, covering only half of the time line, the temperature of the
thermofield double corresponds to the non-vanishing temperature perceived by
Milne observers whose proper time evolution is confined to the future cone
(Milne universe) of Minkowski space-time. The two pictures are related by a
conformal transformation on the real line. Our result provides a purely group
theoretical derivation of the Milne and diamond temperature and shows that the
fundamental ingredient for vacuum thermal effects is the presence of a horizon
rather than acceleration.
