Analytic vectors, anomalies and star representations

Alcalde, Carlos Alberto; Sternheimer, Daniel

doi:10.1007/bf00402326

Cited by 7 publications

(1 citation statement)

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“…The euclidean group E(2) also plays a role in several papers on Moyal ⋆-product ("deformation") quantization of the phase space (13) [104][105][106][107][108][109].…”

Section: Appendices a The "Fault" Of The Anglementioning

confidence: 99%

Quantization of the canonically conjugate pair angle and orbital angular momentum

Kastrup¹

2006

Phys. Rev. A

162

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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, p ϕ ) , p ϕ : orbital angular momentum (OAM), the phase space S ϕ,pϕ = {ϕ ∈ R mod 2π, p ϕ ∈ R} has the global topological structure S 1 × R of a cylinder on which the Poisson brackets of the three functions cos ϕ, sin ϕ and p ϕ 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 + δ), n ∈ Z, δ ∈ [0, 1). Such δ = 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 ϕ,pϕ 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.PACS number(s): 03.65. Fd, 78.20.Ls

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“…The euclidean group E(2) also plays a role in several papers on Moyal ⋆-product ("deformation") quantization of the phase space (13) [104][105][106][107][108][109].…”

Section: Appendices a The "Fault" Of The Anglementioning

confidence: 99%