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
law—that equal areas are swept out in equal times—has 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
six-dimensional 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 finite-dimensional
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 Kaluza–Klein-type theory in six
dimensions.