Quantization of the nilpotent orbits in so(1,2)* and massless particles on (anti-)de Sitter space–time

Abstract

The nontrivial nilpotent orbits in so(1,2)*≂su(1,1)* are the phase spaces of the zero mass particles on the two-dimensional (anti-)de Sitter space–time. As is well known, the lack of global hyperbolicity (respectively, stationarity) for the anti-de Sitter (respectively, de Sitter) space–time implies that the canonical field quantization of its free massless field is not uniquely defined. One might nevertheless hope to get the one-particle quantum theory directly from an appropriate ‘‘first’’ quantization of the classical phase space. Unfortunately, geometric quantization (the orbit method) does not apply to the above orbits and a naive canonical quantization does not yield the correct result. To resolve these difficulties, we present a simple geometric construction that associates to them an indecomposable representation of SO0(1,2) on a positive semidefinite inner product space. It is shown that quotienting out its one-dimensional invariant subspace yields the first term of the holomorphic discrete series of representations of SO0(1,2)≂SU(1,1)/Z2. We interpret these results physically by showing that the above positive semidefinite inner product space is naturally isomorphic to a space of solutions of the conformally invariant zero-mass Klein–Gordon equation on the (anti)-de Sitter space–time, equipped with the usual Klein–Gordon inner product, obtained by integrating over a suitable spacelike hypersurface. As such, our version of geometric quantization selects in a natural way a quantization of the free massless particle. We show it is conformally [i.e., SO0(2,2)] invariant and behaves correctly in the classical limit.