Random renormalization in the semiclassical long-time limit of a precessing spin
type: Item
instance of: scholarly article
title: Random renormalization in the semiclassical long-time limit of a precessing spin
zbMATH Open document ID: 0667.10026
DOI: 10.1016/S0167-2789(98)90006-9
author: M. V. Berry
published in: Physica D
publication date: 1988-01-01T00:00:00Z
review text: Several authors have studied the (complicated) behavior of the sum \(S_ j(\tau)=\sum^{j}_{m=1}\exp (i\pi \tau^ 2_ m)\) [see for instance \textit{F. M. Dekking} and \textit{M. Mendès France}, J. Reine Angew. Math. 329, 143-153 (1981; Zbl 0459.10025); for related results see also \textit{J. M. Deshouillers}, Elementary and analytic theory of numbers, Banach Cent. Publ. 17, 75-82 (1985; Zbl 0616.10031)]. In the paper under review the author gives a physical interpretation for this sum: the trace of the propagator for a particle of spin \(J^ 2=\hslash^ 2j(j+1)\) with Hamiltonian \(J^ 2_ z\times c^ r\) can be expressed in terms of \(S_ j(t/j)\) where t is a dimensionless propagation time. The long-limit corresponds to \(t\to \infty\) (j fixed), and the semiclassical limit is \(j\to \infty\) (t fixed). Taking the combined limit \(j\to \infty\), \(t\to \infty\) \((\tau =t/j\) fixed), one obtains complex patterns for \(S_ j(\tau)\) which consist of an infinite hierarchy of scaling spirals: these pattern strongly depend on the arithmetic properties of \(\tau\).
Mathematics Subject Classification ID: 11L40
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 81V99
zbMATH DE Number: 4091606
zbMATH Keywords: renormalization
zbMATH Keywords: semiclassical long-time limit
zbMATH Keywords: curlicues
zbMATH Keywords: exponential sums
reviewed by: Jean-Paul Allouche
MaRDI profile type: Publication
cites work: Renormalisation of curlicues
cites work: Disorder, renormalizability, theta functions and Cornu spirals
cites work: Uniform distribution modulo one: a geometrical viewpoint.
cites work: Q3963837
cites work: Q3317878
full work available at URL: s0167-2789(98)90006-9
OpenAlex ID: W2029738311
description: scientific article; zbMATH DE number 4091606