System of classical nonlinear oscillators as a coarse-grained quantum system

Abstract

Constrained Hamiltonian dynamics of a quantum system of nonlinear oscillators is used to provide the mathematical formulation of a coarse-grained description of the quantum system. It is seen that the evolution of the coarse-grained system preserves constant and minimal quantum fluctuations of the fundamental observables. This leads to the emergence of the corresponding classical system on a sufficiently large scale.

Constrained quantum dynamics and coarse-grained description of a quantum system of nonlinear oscillators

“…We now compare the total Hamiltonian (4) on the constrained manifold of the coherent states with h cl = p 2 /(2m) + V (q) representing the Hamilton function of a classical nonlinear oscillator with potential V (q). It can be proven [22] that the total Hamiltonian at a point α ≡ (q, p) on the constrained manifold is…”

Positive-operator-valued measures in the Hamiltonian formulation of quantum mechanics

“…The Hamiltonian formulation of quantum mechanics (HQM) [1][2][3][4] provides an alternative mathematical formulation that is equivalent to the more standard one based on Hilbert spaces and has proven to be useful in discussing such issues as nonlinear constraints [5,6] the geometry of entanglement [2], the classical limit [7,8], hybrid quantumclassical systems [9][10][11], and nonlinear and stochastic generalizations of quantum mechanics (QM) [1,2,12]. In the Hamiltonian formulation quantum pure states are represented by points of an appropriate smooth manifold M and the quantum dynamics is represented by a Hamiltonian flow on M. In order to formulate probabilistic aspects of QM and in particular describe ideal measurements in the sense of von Neumann, the manifold M is equipped with a Riemannian metric.…”