A hidden non-Abelian monopole in a 16-dimensional isotropic harmonic oscillator

Abstract

We suggest one variant of generalization of the Hurwitz transformation by adding seven extra variables that allow an inverse transformation to be obtained. Using this generalized transformation we establish the connection between the Schrödinger equation of a 16-dimensional isotropic harmonic oscillator and that of a nine-dimensional hydrogen-like atom in the field of a monopole described by a septet of potential vectors in a non-Abelian model of 28 operators. The explicit form of the potential vectors and all the commutation relations of the algebra are given.

Generalized five-dimensional Kepler system, Yang-Coulomb monopole, and Hurwitz transformation

“…Let us mention that these results can have a wider applications, it was shown that systems with monopole can be extended in N dimensions [80] and a generalized Hurwitz transformation were obtained for a 9-dimensional hydrogenlike system with so(8) non-Abelian monopole [81]. Let us mention also recent works on monopoles [82][83][84][85][86].…”

Normed Division Algebras Application to the Monopole Physics

“…Hurwitz transformations is a class of bilinear transformations [11,[22][23][24]30] connecting two Euclidean spaces, R n+1 and R 2n , that satisfy the Euler identity…”

“…Although these transformations were discovered independently later, they were similar to the maps given by Hurwitz while proving the composition theorem. For this reason, in several references, these were named the Hurwitz transformations [11,[22][23][24]30]. The first (2 − 2) Hurwitz transformation was constructed by Levi-Civita in 1904 [21].…”

“…This gauge field also appeared in the work of Zhang et al in 2001 on the 8D Hall effect [9,10]. Then, Le et al constructed the SO(8) monopole's explicit form in 2009 using the generalized Hurwitz transformation [11,12].…”

“…One of the fascinating problems in mathematical physics is the duality between the two most basic systems: a harmonic oscillator and a hydrogen atom [16]. This duality led to the birth of MICZ-Kepler problems in three-, five-, and nine-dimensional spaces [11,[17][18][19]. Indeed, the harmonic oscillator is equivalent to the Kepler problem adding the monopole field in three cases of dimension.…”