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Weyl correspondence for a charged particle in the field of a magnetic monopole. (English. Russian original) Zbl 1346.81055

Theor. Math. Phys. 187, No. 2, 782-795 (2016); translation from Teor. Mat. Fiz. 187, No. 2, 383-398 (2016).
Summary: We construct a generalized Weyl correspondence for an electrically charged particle in the field of the Dirac magnetic monopole. Our starting points are a global Lagrangian description of this system as a constrained system with \(U(1)\) gauge symmetry given in terms of the fiber bundle theory and a reduction of the presymplectic structure arising on the constraint surface. In contrast to the recently proposed quantization scheme based on using a quaternionic Hilbert module, the quantum operators corresponding to classical observables in our construction act in the complex Hilbert space of \(U(1)\)-equivariant functions introduced by Greub and Petry. These functions are defined on the total space of a fiber bundle that is topologically equivalent to the Hopf fibration. [Dedicated to Igor Viktorovich Tyutin on the occasion of his 75th birthday]

MSC:

81S10 Geometry and quantization, symplectic methods
53D55 Deformation quantization, star products
78A97 Mathematically heuristic optics and electromagnetic theory (must also be assigned at least one other classification number in Section 78-XX)
78A35 Motion of charged particles
70S05 Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems
53D12 Lagrangian submanifolds; Maslov index
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)

Biographic References:

Tyutin, Igor Viktorovich
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