Colloquium Publications 1991; 88 pp; softcover Volume: 42 ISBN10: 082184184X ISBN13: 9780821841846 List Price: US$30 Member Price: US$24 Order Code: COLL/42.S
 This book is based on the Colloquium Lectures presented by Shlomo Sternberg in 1990. The authors delve into the mysterious role that groups, especially Lie groups, play in revealing the laws of nature by focusing on the familiar example of Kepler motion: the motion of a planet under the attraction of the sun according to Kepler's laws. Newton realized that Kepler's second lawthat equal areas are swept out in equal timeshas to do with the fact that the force is directed radially to the sun. Kepler's second law is really the assertion of the conservation of angular momentum, reflecting the rotational symmetry of the system about the origin of the force. In today's language, we would say that the group \(O(3)\) (the orthogonal group in three dimensions) is responsible for Kepler's second law. By the end of the nineteenth century, the inverse square law of attraction was seen to have \(O(4)\) symmetry (where \(O(4)\) acts on a portion of the sixdimensional phase space of the planet). Even larger groups have since been found to be involved in Kepler motion. In quantum mechanics, the example of Kepler motion manifests itself as the hydrogen atom. Exploring this circle of ideas, the first part of the book was written with the general mathematical reader in mind. The remainder of the book is aimed at specialists. It begins with a demonstration that the Kepler problem and the hydrogen atom exhibit \(O(4)\) symmetry and that the form of this symmetry determines the inverse square law in classical mechanics and the spectrum of the hydrogen atom in quantum mechanics. The space of regularized elliptical motions of the Kepler problem (also known as the Kepler manifold) plays a central role in this book. The last portion of the book studies the various cosmological models in this same conformal class (and having varying isometry groups) from the viewpoint of projective geometry. The computation of the hydrogen spectrum provides an illustration of the principle that enlarging the phase space can simplify the equations of motion in the classical setting and aid in the quantization problem in the quantum setting. The authors provide a short summary of the homological quantization of constraints and a list of recent applications to many interesting finitedimensional settings. The book closes with an outline of Kostant's theory, in which a unitary representation is associated to the minimal nilpotent orbit of \(SO(4,4)\) and in which electromagnetism and gravitation are unified in a KaluzaKleintype theory in six dimensions. Table of Contents  Classical brackets and configurational symmetries
 Quantum mechanics and dynamical symmetries
 The conformal group and hidden symmetries
 The conformal completion of Minkowski space
 Homogeneous models in general relativity
 Appendices
 References
 Index
