The connection between the concept of a 'Lagrangian manifold' and Liouville’s theorem in classical mechanics and the concept of a global information field in quantum physics is discussed.
In this article, it is argued that the Gibbs-Liouville theorem is a mathematical representation of the statement that closed classical systems evolve deterministically. From the perspective of an observer of the system, whose knowledge about the degrees of freedom of the system is complete, the statement of deterministic evolution is equivalent to the notion that the physical distinctions between the possible states of the system, or, in other words, the information possessed by the observer about the system, is never lost. Thus, it is proposed that the Gibbs-Liouville theorem is a statement about the dynamical evolution of a closed classical system valid in such situations where information about the system is conserved in time. Furthermore, in this article it is shown that the Hamilton equations and the Hamilton principle on phase space follow directly from the differential representation of the Gibbs-Liouville theorem, i.e. that the divergence of the Hamiltonian phase flow velocity vanish. Thus, considering that the Lagrangian and Hamiltonian formulations of classical mechanics are related via the Legendre transformation, it is obtained that these two standard formulations are both logical consequences of the statement of deterministic evolution, or, equivalently, information conservation.
In terms of the coherent state evolution in phase space, we present a quantum mechanical version of the classical Liouville theorem. The evolution of the coherent state from | z 〉 to | sz − rz *〉 corresponds to the motion from a point z ( q , p ) to another point sz − rz * with | s | 2 − | r | 2 = 1. The evolution is governed by the so-called Fresnel operator U ( s , r ) that was recently proposed in quantum optics theory, which classically corresponds to the matrix optics law and the optical Fresnel transformation, and obeys group product rules. In other words, we can recapitulate the Liouville theorem in the context of quantum mechanics by virtue of coherent state evolution in phase space, which seems to be a combination of quantum statistics and quantum optics.
We clarify certain confusions in the literature of the density operator in quantum mechanics, and demonstrate that the quantum Liouville theorem has the same form in both the Schrodinger and the Heisenberg pictures.Our starting point is to treat the density operator as an observable which has its specific time dependence in each of the two pictures.It is further shown that such a formulation will provide the exact correspondence between classical and quantum statistical mechanics with the Liouville theorem being interpreted as a conservation law, which is derivable from the equation of motion only in the quantum case.
Despite its status as a core part of contemporary physics, there is no consensus among physicists on the question of what, if anything, the success of quantum theory is telling us about the physical world. This gives rise to the collection of debates and discussions known as "the interpretation of quantum mechanics". Describing quantum mechanics is complicated by there being so many interpretations of it. In text books, classical mechanics is an approximation of the quantum mechanics when ℏ → 0. In a first part, we recall the fundamental differences between the classical mechanics in different formalisms: Newtonian, Lagrangian and Hamiltonian (Jacobi). A second part is dedicated to the deep similarities and differences between classical and quantum mechanics. The third part deals with the misunderstanding between scientific communities about the meaning ℏ → 0. The last part deals with the porous frontier between classical and quantum physics to the light of the algebraic topology illustrated by two major theorems: Borsuk Ulam and Non squeezing Gromov's theorem.