I will discuss peculiarities of quantum fields in de Sitter space on the example of the self-interacting massive real scalar. Non-conformal quantum field theories in de Sitter space show very special infrared behavior, which is not shared by quantum fields neither in flat nor in anti-de-Sitter space: in de Sitter space loops are not suppressed in comparison with tree level contributions and dominate at late times. That is true even for massive fields. I will discuss how the invariance of the quantum field theory under the de Sitter isometry gets broken due to the loop corrections and such symmetry could be respected only by one state. Also I will discuss how physics in even dimensional de Sitter spaces is different from odd dimensional ones.
Based on recent works with A. Polyakov, E. Akhmedov and U. Moschella