  We study non-interacting automorphic quantum scalar fields with positive mass
in two-dimensional de Sitter space. We find that there are no Hadamard states
which are de Sitter invariant except in the periodic case, extending the result
of Epstein and Moschella for the anti-periodic case. We construct the two-point
Wightman functions for the non-Hadamard de Sitter-invariant states by
exploiting the fact that they are functions of the geodesic distance between
the two points satisfying an ordinary differential equation. We then examine a
certain Hadamard state, which is not de Sitter invariant, and show that it is
approximately a thermal state with the Gibbons-Hawking temperature when
restricted to a static region of the spacetime.
