• DOI:10.1088/0264-9381/24/3/002
• Corpus ID: 119644807

On obtaining classical mechanics from quantum mechanics

@article{Date2006OnOC,
  title={On obtaining classical mechanics from quantum mechanics},
  author={Ghanashyam Date},
  journal={Classical and Quantum Gravity},
  year={2006},
  volume={24},
  pages={535 - 550}
}

• G. Date
• Published 18 June 2006
• Physics
• Classical and Quantum Gravity

Constructing a classical mechanical system associated with a given quantum-mechanical one entails construction of a classical phase space and a corresponding Hamiltonian function from the available quantum structures and a notion of coarser observations. The Hilbert space of any quantum-mechanical system naturally has the structure of an infinite-dimensional symplectic manifold (‘quantum phase space’). There is also a systematic, quotienting procedure which imparts a bundle structure to the quantum phase space and extracts a classical phase space as the base space. This works straightforwardly when the Hilbert space carries weakly continuous representation of the Heisenberg group and one recovers the linear classical phase space . We report on how the procedure also allows extraction of nonlinear classical phase spaces and illustrate it for Hilbert spaces being finite dimensional (spin-j systems), infinite dimensional but separable (particle on a circle) and infinite dimensional but non-separable (polymer quantization). To construct a corresponding classical dynamics, one needs to choose a suitable section and identify an effective Hamiltonian. The effective dynamics mirrors the quantum dynamics provided the section satisfies conditions of semiclassicality and tangentiality.