Magnetized Kepler manifolds and their quantization

Abstract

Abstract Let k ≥ 1 be an integer and μ be the half of a nonzero integer. It is reported here that the phase space of the ( 2 k + 1 ) -dimensional magnetized Kepler problem with magnetic charge μ can be identified with the elliptic co-adjoint orbit of the real Lie algebra so ( 2 , 2 k + 2 ) that corresponds to the dominant weight ( | μ | , … , | μ | ︸ k + 1 , μ ) . As a corollary, one has the following result: the aforementioned elliptic co-adjoint orbit admits a natural quantization in which the Hilbert space of quantum states is the space of square integrable sections of a Hermitian vector bundle over the punctured Euclidean space in dimension 2 k + 1 ; moreover, this Hilbert space provides a geometric realization for the unitary highest weight so ( 2 , 2 k + 2 ) -module with highest weight ( − k − | μ | , | μ | , … , | μ | ︸ k , μ ) .