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From Koopmanvon Neumann theory to quantum theory
AuthorKlein, U.
Published in
Quantum Studies: Mathematics and Foundations, 2018, Vol. 5, Issue 2, page 219-227
PublishedSpringer International Publishing, 2018
Document typeJournal Article
Keywords (EN)Quantum theory / Liouville equation / Koopmanvon Neumann theory / Derivation of Schrödinger equation / Quantumclassical relation
URNurn:nbn:at:at-ubl:3-1333 Persistent Identifier (URN)
 The work is publicly available
From Koopmanvon Neumann theory to quantum theory [0.41 mb]
Abstract (English)

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 Schrödingers 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 derivation of quantum operators and evolution equations from corresponding classical entities. It is performed by replacing p by 0 and p by ıq , thus providing an explanation for the common quantization rules.

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