Abstract
The extended BMS algebra contains a conformal subgroup that acts on the celestial sphere as SO(1, 3). It is of interest to perform mode expansions of free fields in Minkowski spacetime that realize this symmetry in a simple way. In the present work we perform such a mode expansion for massive scalar fields using the unitary principal series representations of SO(1, 3) with a view to developing a holographic approach to gravity in asymptotically flat spacetime. These mode expansions are also of use in studying holography in three-dimensional de Sitter spacetime.
Highlights
Three-dimensional de Sitter spacetime dS3 and the four-dimensional Minkowski spacetime M4, in section 3 and 4, respectively
It is of interest to perform mode expansions of free fields in Minkowski spacetime that realize this symmetry in a simple way
For p2 > 1 we have constructed mode functions of the 4d Klein-Gordon equation corresponding to the Minkowski vacuum, which when restricted to the de Sitter slice ρ = 1 correspond to the Euclidean vacuum of the 3d de Sitter spacetime
Summary
Before we begin our discussion of the representation theory of SO(3, 1) in de Sitter and Minkowski spacetimes, we would like to present the coordinate systems we employ and fix notation. We start with the 4d flat Minkowski spacetime labeled by coordinates (x0, x1, x2, x3) with the following metric ds2 = −(dx0)2 + (dx1)2 + (dx2)2 + (dx3). Ρ and the metric takes the following simpler form ds2 = −ρ2dη2 + dρ2 + ρ2 cosh η dθ2 + sin θ dφ. Note a constant η surface forms a Cauchy slice This will be important in the following, so that free fields states on such spacelike slices provide a complete set. On the 3d de Sitter spacetime the induced metric is ds2 = −dt2 + cosh t dθ2 + sin θ dφ. The first step in building a unitary representation of SO(3, 1) on 3d de Sitter is to solve the scalar field equation with mass μ (∆ − μ2)φ(t, z, z) = 0 where the d’Alembertian is defined in general as. Other combinations will generate modes around an α-vacuum [13,14,15]
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