The classical limit of quantum theory

TL;DR

This paper formalizes the classical limit of quantum observables, showing how quantum mechanics converges to classical mechanics in phase space as Planck's constant approaches zero, with implications for dynamics and algebraic structures.

Contribution

It introduces a rigorous notion of convergence for quantum observables to classical functions, unifying various approaches and analyzing the behavior of quantum dynamics in the classical limit.

Findings

Quantum observables converge to phase space functions in norm

Quantum commutators scaled by $ o 0$ approach Poisson brackets

Quantum dynamics converge to classical Hamiltonian flow

Abstract

For a quantum observable $A_\hbar$ depending on a parameter $\hbar$ we define the notion ``$A_\hbar$ converges in the classical limit''. The limit is a function on phase space. Convergence is in norm in the sense that $A_\hbar\to0$ is equivalent with $\Vert A_\hbar\Vert\to0$. The $\hbar$-wise product of convergent observables converges to the product of the limiting phase space functions. $\hbar^{-1}$ times the commutator of suitable observables converges to the Poisson bracket of the limits. For a large class of convergent Hamiltonians the $\hbar$-wise action of the corresponding dynamics converges to the classical Hamiltonian dynamics. The connections with earlier approaches, based on the WKB method, or on Wigner distribution functions, or on the limits of coherent states are reviewed.