On a quantum-like structure of classical mechanics

Andréa Oldofredi, Gabriele Carcassi, C. Aidala

Giuseppe Marmo, G. Scolarici, Alberto Simoni +1 more

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 Schrödinger picture is also mentioned.

Enrico G. Beltrametti, Slawomir Bugajski

Nuno Costa Dias, João Nuno Prata

We investigate the possibility that the semiclassical limit of quantum mechanics might be correctly described by a classical dynamical theory, other than standard classical mechanics. Using a set of classicality criteria proposed in a related paper, we show that the time evolution of a set of quantum initial data satisfying these criteria is fully consistent with the predictions of a new theory of classical dynamics. The dynamical structure of the new theory is given by the Moyal bracket. This is a Lie bracket that was first derived as the dynamical structure of the Moyal-Weyl-Wigner formulation of quantum mechanics. We present a new derivation of the Moyal bracket, this time in the context of the semiclassical limit of quantum mechanics and thus prove that both classical and quantum dynamics can be formulated in terms of the same canonical structure.

Ghenadie N. Mardari

A. Horzela