  We study bosonic Quantum Field Theory on the double covering
$\widetilde{dS}_{2}$ of the 2-dimensional de Sitter universe, identified to a
coset space of the group $SL(2,{\Bbb R})$. The latter acts effectively on
$\widetilde{dS}_{2}$ and can be interpreted as it relativity group. The
manifold is locally identical to the standard the Sitter spacetime ${dS}_2$; it
is globally hyperbolic, geodesically complete and an inertial observer sees
exactly the same bifurcate Killing horizons as in the standard one-sheeted
case. The different global Lorentzian structure causes however drastic
differences between the two models. We classify all the $SL(2,{\Bbb
R})$-inveriant two-point functions and show that: 1) there is no
Hawking-Gibbons temperature; 2) there is no covariant field theory solving the
Klein-Gordon equation with mass less than $1/2R$ , i.e. the complementary
fields go away.
