Holographic representation of local operators in de Sitter space
Assuming the existence of the dS/CFT correspondence, we construct local scalar fields with $m^2>\left( \frac{d}{2} \right)^2$ in de Sitter space by smearing over conformal field theory operators on the future/past boundary. To maintain bulk micro-causality and recover the bulk Wightman function in the Euclidean vacuum, the smearing prescription must involve two sets of single--trace operators with dimensions $\Delta$ and $d-\Delta$. Thus the local operator prescription in de Sitter space differs from the analytic continuation from the prescription in anti--de Sitter space. Pushing a local operator in the global patch to future/past infinity is shown to lead to an operator relation between single--trace operators in conformal field theories at $\mathcal{I}^\pm$, which can be interpreted as a basis transformation, also identified as the relation between an operator in CFT and its shadow operator. Construction of spin$-s$ gauge field operators is discussed, it is shown that the construction of higher spin gauge fields in de Sitter space is equivalent to constructing scalar fields with specific values of mass parameter $m^2<\left( \frac{d}{2} \right)^2$. An acausal higher spin bulk operator which matches onto boundary higher spin current is constructed. Implementation of the scalar operator constructions in AdS and dS with embedding formalism is briefly described.