The velocity operator in quantum mechanics in noncommutative space

Abstract

We tested in the framework of quantum mechanics the consequences of a noncommutative (NC from now on) coordinates. We restricted ourselves to 3D rotationally invariant NC configuration spaces with dynamics specified by the Hamiltonian H = H(kin) + U, H(kin) is an analogue of kinetic energy and U = U(r) denotes an arbitrary rotationally invariant potential. We introduced the velocity operator by Heisenberg relation using the commutator of the coordinate and the Hamiltonian operators. We found that the NC velocity operator possesses various general, independent of potential, properties: 1) uncertainty relations indicate an existence of a natural kinetic energy cut-off, 2) vanishing commutator relations for velocity components, which is non-trivial in the NC case, 3) modified relation between the velocity operator and H(kin) that indicates the existence of maximal velocity and confirms the kinetic energy cut-off, 4) All these results sum up in canonical (general, not depending on a particular form of the central potential) commutation relations of the Euclidean group E(4), 5) NC Heisenberg equation for the velocity operator, relating acceleration to derivatives of the potential.Comment: 19 pages, no figure

Magnetic monopoles and symmetries in noncommutative space

“…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].…”

Magnetic Monopoles in (Noncommutative) Quantum Mechanics

“…Even though we find the succession presented in this report more logical, the historical order is that we analyzed NC QM first, see [17,18,19,20], then found out the generalization introducing the monopoles, see [21,22], and considered the commutative theory only afterward, see [23]. The outlook, for now, is to utilize the found symmetries and consider the relativistic generalization of the theory.…”

“…Preserving the rotational symmetry has many advantages, for example, the Hydrogen atom problem remains exactly solvable, see [18]. Study of the velocity operators [17] revealed how the UV cut-off manifest itself in terms of higher symmetry (SO(4) instead of the expected SO (3)).…”

Non-commutativity in Unified Theories and Gravity

“…[86]). The above space has an SO(4) symmetry [87], which one would gauge. In this procedure, the need of including additional generators emerges, typical in non-Abelian non-commutative gauge theories, for the anticommutators to close.…”