From the representation of de Sitter group to the quantum field theory

A rigorous mathematical consideration of quantum field theory in de Sitter universe is briefly presented in ambient space formalism. This formalism permit us to solve some physical and mathematical problems in de Sitter space-time. The concept of mass is defined from the light-cone propagation: massless fields or particles are propagated on the de Sitter light-cone and the massive cases are propagated inside the light-cone. In this respect, there appears a big problem of mass definition for the massless minimally coupled scalar field. This field play an important role in quantum gravity and mass generation in quantum inflationary scenario. 
The classical field theory consideration is based on group theory and Lie algebra. From the Casimir operators of the de Sitter group the free field equation is derived. By using the Lie algebra of de Sitter group one-particle Hilbert state and unitary irreducible representation can be constructed and then the solution of the free field equations can be extracted.
In QFT level, the field operator is defined as a map on the Fock space and then is constructed by the annihilation and creation operators on the Fock space. The free massive field operator must be transform by the unitary irreducible representation of the dS group. The act of annihilation operator on the one-particle state results to the vacuum state which must be fixed by the auxiliary condition. For the free massless field with s≥1 and massless minimally coupled scalar field for obtaining a covariant quantization the field operator must be transform by an indecomposable representation of dS group. The Hilbert space of the interaction fields and the time evolution operator are the most challenging problems in de Sitter universe.