Quantization of the nilpotent orbits in so(1,2) and massless particles on (anti-)de Sitter space-time
199419 pages
Published in:
- J.Math.Phys. 35 (1994) 3775-3793
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Abstract: (AIP)
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)/Z 2. 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.- particle: massless
- massless: particle
- phase space
- space-time: de Sitter
- dimension: 2
- quantization: orbit
- symmetry: SO(1,2)
- Klein-Gordon equation: solution
- invariance: conformal
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