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Physical Review

Quantum Recurrence Theorem

P. Bocchieri, A. Loinger

Published Jul 15, 1957• 287 citations
Publisher

Abstract

A recurrence theorem is proved, which is the quantum analog of the recurrence theorem of Poincaré. Some statistical consequences of the theorem are stressed.Received 9 October 1956DOI:https://doi.org/10.1103/PhysRev.107.337©1957 American Physical Society

DOI
10.1103/physrev.107.337

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A recurrence theorem is proved, which is the quantum analog of the recurrence theorem of Poincaré. Some statistical consequences of the theorem are stressed.Received 9 October 1956DOI:https://doi.org/10.1103/PhysRev.107.337©1957 American Physical Society
Poincaré's recurrence theorem, which states that every Hamiltonian dynamics enclosed in a finite volume returns to its initial position as close as one wishes, is a mathematical basis of statistical mechanics. It is Liouville's theorem that guarantees that the dynamics preserves the volume on the state space. A quantum version of Poincaré's theorem was obtained in the middle of the 20th century without any volume structures of the state space (Hilbert space). One of our aims in this paper is to establish such properties of quantum dynamics from an analog of Liouville's theorem, namely, we will construct a natural probability measure on the Hilbert space from a Hamiltonian defined on the space. Then we will show that the measure is invariant under the corresponding Schrödinger flow. Moreover, we show that the dynamics naturally causes an infinite-dimensional Weyl transformation. It also enables us to discuss the ergodic properties of such dynamics.