Magnetic Monopoles in (Noncommutative) Quantum Mechanics

Abstract

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).

Magnetic monopoles and symmetries in noncommutative space

“…It was shown in the same paper that the four-vector Va transforms under the SO(4) rotations generated by (8) and squares to a constant V 2 a = λ 2 , which demonstrates the UV regularization of the theory. This result holds only for states in H 0 , but it has been generalized to H κ (which is defined in the next section) in [11].…”

“…Let us now move our attention to a generalized class of states which was investigated in [11]. It contains functions containing an unequal number of creation and annihilation operators, with their difference held fixed and equal to κ…”

“…However, the identification is perfect since κ describes the difference between the number of creation and annihilation operators it follows that κ 2 ∈ Z/2, which is the same condition as has to be satisfied by µ. It was therefore concluded in [11] that H κ is the Hilbert space of monopoles states of strength − κ 2 . Monopole states can in some scenarios cause the non-associativity of the algebra [14,15].…”

“…where the +1 was added to ensure that r 2 − x 2 i = O(λ 2 ) (so there is no term linear in λ). This construction of 3D rotational invariant space R 3 λ was developed in [4] and explored by the present authors and their collaborator in [5,6,7,8,9,10,11].…”

Quaternionic electrodynamics

“…We notice that magnetic monopole descriptions in non-commutative terms are speculated in various models, as in non-abelian quantum field theories [33,34], in the quaternionic description of hyperbolic monopoles in Yang-Mills theories [35,36], in non-commutative quantum mechanics [37][38][39], and in connections between non-commutative quantum mechanics and anti-hermitian HQM as well [40][41][42][43].…”