Holographic Representation of Local Operators In De Sitter Space

Understand

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.