Infinity

Abstract: The Road to Reality, some 1000 pages long, aims to provide a comprehensive account of our present understanding of the physical universe, and the essentials of its underlying mathematical theory. No particular mathematical knowledge on the part of the reader is assumed - the early chapters providing the essential mathematical background for the physical theories described in the remainder of the book. The aim is to convey something of an overall understanding - a feeling for the deep beauty and philosophical connotations of the subject, as well as of its intricate logical interconnections. Clearly, a work of this nature is challenging, but there is enough descriptive material to carry the less mathematically inclined reader through, as well as some 450-500, mostly hand-drawn, figures. The book provides a feeling for all the key issues and deep current controversies, and counters the common complaint that cutting-edge science is fundamentally inaccessible. The topics covered in this book include: the roles of different kinds of numbers and of geometry in physics; the ideas - and magic - of calculus and of modern geometry; notions of infinity; the physics and mathematics of relativity theory; the foundations and controversies of quantum mechanics; the standard model of particle physics; cosmology; the big bang; black holes; the profound challenge of the second law of thermodynamics; string and M theory; loop quantum gravity; twistors; fashions in science; and new directions.

Abstract: The time-dependent density-functional theory of Runge and Gross [Phys. Rev. Lett. 52, 997 (1984)] is reexamined on the basis of its limitations, and the criticisms raised by Xu and Rajagopal [Phys. Rev. A 31, 2682 (1985)] are addressed, within the imposition of natural boundary conditions of vanishing density and potential at infinity. Also, for a single-particle system characterized by an arbitrary time-dependent potential, the uniqueness of the density-to-potential mapping is established explicitly for both bound and scattering states.

Abstract: The Uncertainty Principle (up) as understood in this lecture is the following informal assertion: a non-zero “object” (a function, distribution, hyperfunction) and its Fourier image cannot be too small simultaneously. “The smallness” is understood in a very broad sense meaning fast decay (at infinity or at a point, bilateral or unilateral), perforated (or bounded, or semibounded) support etc. The UP becomes a theorem for many “smallnesses” and has a multitude of quite concrete quantitative forms. It plays a fundamental role as one of the major themes of classical Fourier analysis (and neighboring parts of analysis), but also in applications to physics and engineering. The lecture is a review of facts and techniques related to the UP; connections with local and non-local shift invariant operators are discussed at the end of the lecture (including some topical problems of potential theory). The lecture is intended for the general audience acquainted with basic facts of Fourier analysis on the line and circle, and rudiments of complex analysis.