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 $\mathbf{x}\mapsto R\mathbf{x}$, where $R\in SO(3)$ satisfies $R^{T}R=I$ and $\det R=+1$. Orthogonality preserves distances and angles; the positive determinant excludes reflections. In de Sitter flat slicing it leaves conformal time fixed. A rotationally invariant scalar two-point function in a translation-invariant state 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.

A special conformal transformation is a conformal transformation whose finite Euclidean form in one sign convention is $x^{\prime i}=(x^i-b^i{x}^2)/(1-2b\cdot x+b^2{x}^2)$ wherever the denominator is nonzero. Its position-dependent scale change preserves local angles.

In three boundary dimensions, the three parameters $b^i$ supplement translations, rotations and dilation to form the ten-dimensional de Sitter isometry algebra. The full transformation of the flat-slicing spacetime also acts on conformal time; the boundary formula alone is not that bulk transformation. A matter equation with a propagation speed different from the metric speed of light generally need not preserve these isometries.