Seminar talk, 18 May 2022: Difference between revisions
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| speaker = Andrei Smilga | | speaker = Andrei Smilga | ||
| title = Noncommutative quantum mechanical systems associated with Lie algebras | | title = Noncommutative quantum mechanical systems associated with Lie algebras | ||
| abstract = We consider quantum mechanics on the noncommutative spaces characterized by the commutation relations | | abstract = We consider quantum mechanics on the noncommutative spaces characterized by the commutation relations | ||
For the algebras <math>su(2)</math> and <math>u(2)</math>, it has been possible to represent the operators <math>\ | <math> [x_a, x_b] \ =\ i\theta f_{abc} x_c\,,</math> | ||
| video = | |||
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 = https://video.gdeq.org/GDEq-zoom-seminar-20220518-Andrei_Smilga.mp4 | |||
| slides = | | slides = | ||
| references = {{arXiv|2204.08705}} | | references = {{arXiv|2204.08705}} | ||
| 79YY-MM-DD = 7977-94-81 | | 79YY-MM-DD = 7977-94-81 | ||
}} | }} |
Latest revision as of 08:40, 4 January 2025
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
where 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 . Generically, the matrix represents a certain infinite series over the deformation parameter : . The deformed Hamiltonian, describes the motion along the corresponding group manifolds with the characteristic size of order . Their metrics are also expressed into certain infinite series in .
For the algebras and , it has been possible to represent the operators in a simple finite form. A byproduct of our study are new nonstandard formulas for the metrics on and on .
Video
References:
arXiv:2204.08705