de Sitter isometry

Four-dimensional de Sitter spacetime has ten independent isometries. In the expanding flat slicing they act on the spatial boundary as three translations, three rotations, one dilation, and three special conformal transformations.

A spatial translation in de Sitter flat slicing sends $\mathbf{x}\mapsto \mathbf{x}+\mathbf{a}$ at fixed conformal time. Translation invariance of a momentum-space correlator produces its momentum-conserving Dirac delta distribution.

A spatial rotation acts by $\mathbf{x}\mapsto R\mathbf{x}$ with $R$ orthogonal. A rotationally invariant scalar two-point function depends only on the magnitude of its momentum.

In flat conformal coordinates, a de Sitter dilation acts as $(\tau ,\mathbf{x})\mapsto (s\tau ,s\mathbf{x})$. It is an isometry because the common coordinate rescaling cancels the inverse-square scaling of the conformal factor.

The three special conformal transformations complete translations, rotations, and dilations to the ten-dimensional de Sitter isometry group in flat slicing. A propagation speed different from the metric light speed generally breaks these transformations.