BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Andrei Smilga DTSTART:20220518T162000Z DTEND:20220518T180000Z DTSTAMP:20260423T024831Z UID:GDEq/66 DESCRIPTION:Title: No ncommutative quantum mechanical systems associated with Lie algebras\n by Andrei Smilga as part of Geometry of differential equations seminar\n\n \nAbstract\nWe consider quantum mechanics on the noncommutative spaces cha racterized by the commutation relations\n$$ [x_a\, x_b] \\ =\\ i\\theta f_ {abc} x_c\\\,\, $$\nwhere $f_{abc}$ are the structure constants of a Lie a lgebra. We note that this problem can be reformulated as an ordinary quant um problem in a commuting {\\it momentum} space. The coordinates are then represented as linear differential operators $\\hat x_a = -i \\hat D_a = - iR_{ab} (p)\\\, \\partial /\\partial p_b $. Generically\, the matrix $R_{a b}(p)$ represents a certain infinite series over the deformation parameter $\\theta$: $R_{ab} = \\delta_{ab} + \\ldots$. The deformed Hamiltonian\, $\\hat H \\ =\\ - \\frac 12 \\hat D_a^2\\\,\, $ describes the motion alon g the corresponding group manifolds with the characteristic size of order $\\theta^{-1}$. Their metrics are also expressed into certain infinite se ries in $\\theta$.\n\nFor the algebras $su(2)$ and $u(2)$\, it has been po ssible to represent the operators $\\hat x_a$ in a simple finite form. A b yproduct of our study are new nonstandard formulas for the metrics on $SU( 2) \\equiv S^3$ and on $SO(3)$.\n LOCATION:https://researchseminars.org/talk/GDEq/66/ END:VEVENT END:VCALENDAR