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

TL;DR

This paper investigates which geometric structures of Quantum Mechanics, like the complex structure, persist or vanish in the Classical limit, focusing on the Ehrenfest and Heisenberg pictures and their implications for the quantum-classical transition.

Contribution

It analyzes the fate of non-symplectic geometric structures, such as the complex structure and metric tensor, during the quantum-to-classical transition at the observable level.

Findings

Symplectic structure survives the classical limit.

Complex structure and metric tensor may not survive the classical limit.

Focus on observables in Ehrenfest and Heisenberg pictures.

Abstract

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.