Canonical quantization of a massive particle on AdS3

The classical theory for a massive free particle moving on the group manifold AdS₃congSL(2,Bbb R) is analysed in detail. In particular a symplectic structure and two different sets of canonical coordinates are explicitly found, corresponding to the Cartan and Iwasawa decomposition of the group. Canonical quantization is performed in two different ways; either by imposing the future-directed constraint before or after quantization. It is found that this leads to different quantum theories. The Hilbert space of either theory decomposes into the sum of certain irreducible representations of sl(2,Bbb R)oplussl(2,Bbb R); however, depending on how the constraint is imposed we get different representations. Quantization of the mass occurs, although a continuum exists in the unconstrained theory corresponding to particles that can reverse their direction in time. A representation on function space is worked out in detail, and from this the wavefunctions of tilde-{AdS₃} and the BTZ black hole are deduced. A quantization in terms of the ``chiral'' variables of the theory is also carried out giving the same results. Comparisons are made between QFT in AdS₃ and the quantum mechanics derived.