×

Diffusion, quantum theory, and radically elementary mathematics. (English) Zbl 1104.81003

Mathematical Notes (Princeton) 47. Princeton, NJ: Princeton University Press (ISBN 0-691-12545-7/pbk). x, 243 p. (2006).
[The articles of this volume will not be indexed individually.]
This book came out of the conference, held at the University of British Columbia on June 17 and 18, 2004. It consists of ten Chapters written by different ten authors.
Except for the music theory on the generalized interval system (GIS) in Chap. 9, the editor William G. Faris and other seven authors aim to treat the theme relating to the works of the last author Edward Nelson (Princeton Univ.). For example, Chap. 2 treats the Theorem: A logarithmic Sobolev inequality holds for the measure \(\mu\) if and only if \(\|\exp(-tA_\mu)\|_{{\mathcal L}^p(\mu)\to{\mathcal L}^q(\mu)}= 1\) for all \(t\geq t_N(p,q)= (1/2)\log((q- 1)/(p-1))\) and for \(1<p \leq q<\infty\). \(A_\mu\): Dirichlet form operator for \(\mu\) on \({\mathcal L}^2(\mathbb{R}^n,\mu)\) (cf. p. 47).
Introduction: “Diffusive motion and where it leads” is written by the editor, in Chap. 1. Diffusion process satisfying \(\Delta x^2= \sigma^2\Delta t\) and quantum field theory as one of its applications are the common themes in this book. One major theme treated in Chaps. 2–6 is Markovian diffusion (which is defined in Chap. 4, p. 100), where a particle wanders randomly but also feels the influence of systematic drift. Another important theme is the need for a closer look at the irregular path arising from the diffusion. Chap. 7 makes this motion precise by using a syntactic approach to nonstandard analysis. A finite sequence of random variables discribing a diffusion process becomes an unlimited one in this sense. This leads to a related topic, a syntactic description of natural number, which is explored in Chap. 8. The author of Chap. 9 explores the mathematical structure of musical composition (transformation group in GIS), which turns out to parallel the structure of space-time. At last, Edward Nelson writes “afterword” in Chap. 10. We can say that these works describe the different profiles of the diffusion process.

MSC:

81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
60J60 Diffusion processes
03H10 Other applications of nonstandard models (economics, physics, etc.)
81P05 General and philosophical questions in quantum theory
81P20 Stochastic mechanics (including stochastic electrodynamics)
PDFBibTeX XMLCite