  The question how to quantize a classical system where an angle phi is one of
the basic canonical variables has been controversial since the early days of
quantum mechanics. The problem is that the angle is a multivalued or
discontinuous variable on the corresponding phase space. The remedy is to
replace phi by the smooth periodic functions cos phi and sin phi. In the case
of the canonical pair (phi,l),l: orbital angular momentum (OAM), the phase
space S_(phi,l) ={phi in R mod 2pi, l in R} has the global structure S^1 x R of
a cylinder on which the Poisson brackets of the 3 functions cos phi, sin phi
and l obey the Lie algebra of the euclidean group E(2) in the plane. This
property provides the basis for the quantization of the system in terms of
irreducible unitary representations of the group E(2) or of its covering
groups. A crucial point is that - due to the fact that the subgroup SO(2) = S^1
is multiply connected - these representations allow for fractional OAM l = n +
c, c in [0,1). Such c not 0 have already been observed in cases like the
Aharonov-Bohm and the fractional quantum Hall effects and they correspond to
the quasi-momenta of Bloch waves in ideal crystals. The proposal of the present
paper is to look for fractional OAM in connection with the quantum optics of
Laguerre-Gaussian laser modes in external magnetic fields. The quantum theory
of the phase space S_(phi,l) in terms of unitary representations of E(2) allows
for two types of "coherent" states the properties of which are discussed in
detail: Non-holomorphic minimal uncertainty states and holomorphic ones
associated with Bargmann-Segal Hilbert spaces.
