A Quantum-Classical Brackets from p-Mechanics

Abstract

We provide an answer to the long standing problem of mixing quantum and classical dynamics within a single formalism. The construction is based on p-mechanical derivation (quant-ph/0212101, quant-ph/0304023) of quantum and classical dynamics from the representation theory of the Heisenberg group. To achieve a quantum-classical mixing we take the product of two copies of the Heisenberg group which represent two different Planck's constants. In comparison with earlier guesses our answer contains an extra term of analytical nature, which was not obtained before in purely algebraic setup. Keywords: Moyal brackets, Poisson brackets, commutator, Heisenberg group, orbit method, representation theory, Planck's constant, quantum-classical mixing

LaTeX, 7 pages (EPL style), no figures; v2: example of dynamics with two different Planck's constants is added, minor corrections; v3: major revion, a complete example of quantum-classic dynamics is given; v4: few grammatic corrections

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• Classical-quantum limits
• Spin dynamics with non-abelian Berry gauge fields as a semiclassical constrained hamiltonian system
• Comment on `Do we have a consistent non-adiabatic quantum-classical mechanics?'
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• Erlangen Program at Large: Outline
• Classical signal viewpoint to bunching and anti-bunching