Remarks on chaos in classical and quantum mechanics

Abstract: 

It is easy to show that 

                                                ||δψ(t)|| = ||δψ(0)||

 where ||( )|| denotes the L2 norm. Therefore it is tempting to conclude chaos cannot exist in quantum world, and this seems to be a consequence of the linear structure of the equation ruling the evolution law. Let us stress that although classical mechanics is typically described by nonlinear equations, it is formally analogous to quantum mechanics in many respects. Indeed, the Liouville equation of classical mechanics affords a linear theory for the evolution of probabilities, at the cost of switching from a finite dimensional phase space to an infinitely dimensional function space, analogously to the quantum mechanics based on the Schrodinger equation. I discuss how the presence of classical chaos has nontrivial impact of the behavior of quantum systems; in particular for: 

a) the classical limit as emergent property, 

b) the relevance of the coarse-graining description 

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