  The classical theory for a massive free particle moving on the group manifold
$AdS_3 \cong SL(2, \mathbb{R})$ 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; by imposing
the future-directed constraint before and 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
$sl(2,\mathbb{R}) \oplus sl(2,\mathbb{R})$; 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
quantization in terms of the ``chiral'' variables of the theory is also carried
out giving the same results. Comparisons are made between QFT in $AdS_3$ and
the quantum mechanics derived, and it is found that one of the quantum theories
is consistent with the Breitenlohner-Freedman bound.
