Quantization of the canonically conjugate pair angle and orbital angular momentum
The question how to quantize a classical system where an angle φ 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 φ by the smooth periodic functions cos φ and sin φ. In the case of the canonical pair (φ,pφ), where pφ is the orbital angular momentum (OAM), the phase space Sφ,pφ=φ∊R mod 2π,pφ∊R has the global topological structure S1×R of a cylinder on which the Poisson brackets of the three functions cos φ,sin φ, and pφ 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)≅S1 is multiply connected, these representations allow for fractional OAM l=ℏ(n+δ),n∊Z,δ∊[0,1). Such δ≠0 have already been observed in cases like the Aharonov-Bohm and fractional quantum Hall effects, and they correspond to the quasimomenta 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 Sφ,pφ in terms of unitary representations of E(2) allows for two types of “coherent” states, the properties of which are discussed in detail: nonholomorphic minimal-uncertainty states and holomorphic ones associated with Bargmann-Segal Hilbert spaces.