Article Dans Une Revue Quantum Studies: Mathematics and Foundations Année : 2018

From Koopman–von Neumann theory to quantum theory

Résumé

Koopman and von Neumann (KvN) extended the Liouville equation by introducing a phase space function $S^{(K)}(q,p,t)$ whose physical meaning is unknown. We show that a different $S(q,p,t)$, with well-defined physical meaning, may be introduced without destroying the attractive "quantum-like" mathematical features of the KvN theory. This new $S(q,p,t)$ is the classical action expressed in phase space coordinates. It defines a mapping between observables and operators which preserves the Lie bracket structure. The new evolution equation reduces to Schr\"odinger's equation if functions on phase space are reduced to functions on configuration space. This new kind of "quantization" does not only establish a correspondence between observables and operators, but provides in addition a \emph{derivation} of quantum operators and evolution equations from corresponding classical entities. It is performed by replacing $\frac{\partial}{\partial p}$ by $0$ and $p$ by $\frac{\hbar}{\imath} \frac{\partial}{\partial q}$, thus providing an explanation for the common quantization rules.

Fichier principal
Vignette du fichier
koopman.pdf (570.2 Ko) Télécharger le fichier
Origine Fichiers éditeurs autorisés sur une archive ouverte
Licence

Dates et versions

hal-04078500 , version 1 (23-04-2023)

Licence

Identifiants

Citer

U. Klein. From Koopman–von Neumann theory to quantum theory. Quantum Studies: Mathematics and Foundations, 2018, 5 (2), pp.219-227. ⟨10.1007/s40509-017-0113-2⟩. ⟨hal-04078500⟩
93 Consultations
132 Téléchargements

Altmetric

Partager

  • More