The Quantum-Classical Transition: The Fate of the Complex Structure

Details The Quantum-Classical Transition: The Fate of the Complex Structure The Quantum-Classical Transition: The Fate of the Complex Structure

2005-01-13

arxiv.org/abs/hep-th/0501094v1

Int.J.Geom.Meth.Mod.Phys. 2 (2005) 127-145

Physics

High Energy Physics

High Energy Physics - Theory

19 pages

Scientific paper

According to Dirac, fundamental laws of Classical Mechanics should be recovered by means of an "appropriate limit" of Quantum Mechanics. In the same spirit it is reasonable to enquire about the fundamental geometric structures of Classical Mechanics which will survive the appropriate limit of Quantum Mechanics. This is the case for the symplectic structure. On the contrary, such geometric structures as the metric tensor and the complex structure, which are necessary for the formulation of the Quantum theory, may not survive the Classical limit, being not relevant in the Classical theory. Here we discuss the Classical limit of those geometric structures mainly in the Ehrenfest and Heisenberg pictures, i.e. at the level of observables rather than at the level of states. A brief discussion of the fate of the complex structure in the Quantum-Classical transition in the Schroedinger picture is also mentioned.

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