Seminar talk, 18 May 2022

From Geometry of Differential Equations
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Speaker: Andrei Smilga

Title: Noncommutative quantum mechanical systems associated with Lie algebras

Abstract:
We consider quantum mechanics on the noncommutative spaces characterized by the commutation relations

                         [xa,xb] = iθfabcxc,

where fabc are the structure constants of a Lie algebra. We note that this problem can be reformulated as an ordinary quantum problem in a commuting momentum space. The coordinates are then represented as linear differential operators x^a=iD^a=iRab(p)/pb. Generically, the matrix Rab(p) represents a certain infinite series over the deformation parameter θ: Rab=δab+. The deformed Hamiltonian, H^ = 12D^a2, describes the motion along the corresponding group manifolds with the characteristic size of order θ1. Their metrics are also expressed into certain infinite series in θ.

For the algebras su(2) and u(2), it has been possible to represent the operators x^a in a simple finite form. A byproduct of our study are new nonstandard formulas for the metrics on SU(2)S3 and on SO(3).

Video
References:
arXiv:2204.08705