Seminar talk, 18 May 2022: Difference between revisions

Latest revision as of 08:40, 4 January 2025

Title: Noncommutative quantum mechanical systems associated with Lie algebras

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

$[x_{a}, x_{b}] = i θ f_{a b c} x_{c},$

where $f_{a b c}$ 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 ${\hat{x}}_{a} = - i {\hat{D}}_{a} = - i R_{a b} (p) \partial / \partial p_{b}$ . Generically, the matrix $R_{a b} (p)$ represents a certain infinite series over the deformation parameter $θ$ : $R_{a b} = δ_{a b} + \dots$ . The deformed Hamiltonian, $\hat{H} = - \frac{1}{2} {\hat{D}}_{a}^{2},$ describes the motion along the corresponding group manifolds with the characteristic size of order $θ^{- 1}$ . Their metrics are also expressed into certain infinite series in $θ$ .

For the algebras $s u (2)$ and $u (2)$ , it has been possible to represent the operators ${\hat{x}}_{a}$ in a simple finite form. A byproduct of our study are new nonstandard formulas for the metrics on $S U (2) \equiv S^{3}$ and on $S O (3)$ .

Video
References:
arXiv:2204.08705

@@ Line 2: / Line 2: @@
 | 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 <math> [x_a, x_b] \ =\ i\theta f_{abc} x_c\,,</math> where <math>f_{abc}</math> are the structure constants of a Lie algebra. We note that this problem can be reformulated as an ordinary quantum problem in a commuting {\it momentum} space. The coordinates are then represented as linear differential operators <math>\hat x_a = -i \hat D_a = -iR_{ab} (p)\, \partial /\partial p_b </math>. Generically, the matrix <math>R_{ab}(p)</math> represents a certain infinite series over the deformation parameter <math>\theta</math>: <math>R_{ab} = \delta_{ab} + \ldots</math>. The deformed Hamiltonian, <math>\hat H \ =\ - \frac 12  \hat D_a^2\,, </math> describes the motion along the corresponding group manifolds with the characteristic size of order <math>\theta^{-1}</math>. Their metrics are also expressed into  certain infinite series in <math>\theta</math>.
+| abstract = We consider quantum mechanics on the noncommutative spaces characterized by the commutation relations
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<math> [x_a, x_b] \ =\ i\theta f_{abc} x_c\,,</math>
+where <math>f_{abc}</math> 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 <math>\hat x_a = -i \hat D_a = -iR_{ab} (p)\, \partial /\partial p_b </math>. Generically, the matrix <math>R_{ab}(p)</math> represents a certain infinite series over the deformation parameter <math>\theta</math>: <math>R_{ab} = \delta_{ab} + \ldots</math>. The deformed Hamiltonian, <math>\hat H \ =\ - \frac 12  \hat D_a^2\,, </math> describes the motion along the corresponding group manifolds with the characteristic size of order <math>\theta^{-1}</math>. Their metrics are also expressed into  certain infinite series in <math>\theta</math>.
 For the algebras <math>su(2)</math> and <math>u(2)</math>, it has been possible to represent the operators <math>\hat x_a</math> in a simple finite form. A byproduct of our study are new nonstandard formulas for the metrics on <math>SU(2) \equiv S^3</math> and on <math>SO(3)</math>.
-| video =
+| video = https://video.gdeq.org/GDEq-zoom-seminar-20220518-Andrei_Smilga.mp4
 | slides =
 | references = {{arXiv|2204.08705}}
 | 79YY-MM-DD = 7977-94-81
 }}

Seminar talk, 18 May 2022: Difference between revisions

Latest revision as of 08:40, 4 January 2025

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