2020/11/30 by Rodrigo Andrade e Silva, Ted Jacobson · 1 citation
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Casimir effect #Hilbert space #Line bundle #Magnetic monopole #Noncommutative and Quantum Gravity Theories #Phase space #Quantization (signal processing) #Quantum Electrodynamics and Casimir Effect #Symmetry (geometry) #Symmetry group #Symplectic geometry #Unitary group #gr-qc #hep-th #math-ph #math.MP #quant-ph
paper · pdf · doi:10.1088/1751-8121/abf961
v2: Minor editing, references added, 47 pages, version published in J. Phys. A
openalex created_date 2020/11/23 · openalex publication_date 2021/04/22 · arxiv created 2021/06/20 · arxiv updated 2021/06/22 · openalex updated_date 2026/08/06
Abstract The problem of quantizing a particle on a two-sphere has been treated by numerous approaches, including Isham’s global method based on unitary representations of a symplectic symmetry group that acts transitively on the phase space. Here we reconsider this simple model using Isham’s scheme, enriched by a magnetic flux through the sphere via a modification of the symplectic form. To maintain complete generality we construct the Hilbert space directly from the symmetry algebra, which is manifestly gauge-invariant, using ladder operators. In this way, we recover algebraically the complete classification of quantizations, and the corresponding energy spectra for the particle. The famous Dirac quantization condition for the monopole charge follows from the requirement that the classical and quantum Casimir invariants match. In an appendix we explain the relation between this approach and the more common one that assumes from the outset a Hilbert space of wave functions that are sections of a nontrivial line bundle over the sphere, and show how the Casimir invariants of the algebra determine the bundle topology.