Magnetic Monopoles in (Noncommutative) Quantum Mechanics

We utilize the close relation between the complex space $\textbf{C}^2$ and the real space $\textbf{R}^3$ to reformulate quantum mechanics in a manner which allows to, either or both, describe magnetic monopoles and quantize the underlying space, obtaining (noncommutative) quantum mechanics (with magnetic monopoles).Comment: 8 page

Magnetic monopoles and symmetries in noncommutative space

In this paper, we review the progress in the analysis of magnetic monopoles as generalized states in quantum mechanics. We show that the considered model contains rich algebraic structure that generates symmetries which have been utilized in different physical contexts. Even though are we focused on quantum mechanics in noncommutative space $\textbf{R}_\lambda^3$, the results can be reconstructed in ordinary quantum mechanics in $\textbf{R}^3$ as well.Comment: 7 page

COULOMB SCATTERING IN NON-COMMUTATIVE QUANTUM MECHANICS

Recently we formulated the Coulomb problem in a rotationally invariant NC configuration space specified by NC coordinates xi, i = 1, 2, 3, satisfying commutation relations [xi, xj ] = 2iλεijkxk (λ being our NC parameter). We found that the problem is exactly solvable: first we gave an exact simple formula for the energies of the negative bound states Eλn < 0 (n being the principal quantum number), and later we found the full solution of the NC Coulomb problem. In this paper we present an exact calculation of the NC Coulomb scattering matrix Sλj (E) in the j-th partial wave. As the calculations are exact, we can recognize remarkable non-perturbative aspects of the model: 1) energy cut-off — the scattering is restricted to the energy interval 0 < E < Ecrit = 2/λ2; 2) the presence of two sets of poles of the S-matrix in the complex energy plane — as expected, the poles at negative energy EIλn = Eλn for the Coulomb attractive potential, and the poles at ultra-high energies EIIλn = Ecrit − Eλn for the Coulomb repulsive potential. The poles at ultra-high energies disappear in the commutative limit λ→0

Magnetic monopoles in noncommutative quantum mechanics 2

In this paper we extend the analysis of magnetic monopoles in quantum mechanics in three dimensional rotationally invariant noncommutative space R^3_λ . We construct the model step-by-step and observe that physical objects known from previous studies appear in a very natural way. Nonassociativity became a topic of great interest lately, often in connection with magnetic monopoles. We show that this model does not possess this property

Magnetic Monopoles in (Noncommutative) Quantum Mechanics

We utilize the close relation between the complex space C^2 and the real space R^3 to reformulate quantum mechanics in a manner which allows to, either or both, describe magnetic monopoles and quantize the underlying space, obtaining (noncommutative) quantum mechanics (with magnetic monopoles)

Alternative Description of Magnetic Monopoles in Quantum Mechanics

We present an alternative description of magnetic monopoles by lifting quantum mechanics from 3-dimensional space into a one with 2 complex dimensions. Magnetic monopoles are realized as a generalization of the considered states. Usual algebraic relations and magnetic fields describing monopoles are reproduced, with the Dirac quantization condition satisfied naturally.Comment: 6 page