Star products, duality and double Lie algebras

Abstract

Quantization of classical systems using the star-product of symbols of observables is discussed. In the star-product scheme an analysis of dual structures is performed and a physical interpretation is proposed. At the Lie algebra level duality is shown to be connected to double Lie algebras. The analysis is specified to quantum tomography. The classical tomographic Poisson bracket is found.

22 pages, no figures

References in corpus (15)

• f-Oscillators and Nonlinear Coherent States
• Symplectic tomography as classical approach to quantum systems
• Alternative commutation relations, star products and tomography
• Quantum Bi-Hamiltonian Systems
• Spin tomography
• Lyapunov exponent in quantum mechanics. A phase-space approach
• Tomograms and other transforms. A unified view
• Infinitely many star products to play with
• Generating All Wigner Functions
• Geometrical Evaluation of Star Products
• The Pauli Equation for Probability Distributions
• Duality symmetry for star products
• Poisson structures on double Lie groups
• A generalization of the Jordan-Schwinger map: classical version and its q--deformation
• Poisson Lie Group Symmetries for the Isotropic Rotator

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