Variations on a Theme by KeplerThis 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. |
Contents
Quantum mechanics and dynamical symmetries | 17 |
The conformal group and hidden symmetries | 29 |
The conformal completion of Minkowski space | 53 |
Homogeneous models in general relativity | 65 |
| 83 | |
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action of G acts transitively antisymmetric canonical classical mechanics coadjoint orbit conformal class conformal structure consisting corresponding curve decomposition defined definition denote determined dimensional dimensions double cover element fiber find finite finite-dimensional first fixed function geodesic flow geometry given gives group G Hamiltonian Hamiltonian action hence Hilbert space Hom(Z homological hydrogen atom hypersurface identified induced infinitesimal invariant irreducible representations isomorphic Kepler problem Lenz vector Lie algebra Lie bracket logic Marsden-Weinstein reduction Math metric Minkowski space moment map multiplication nondegenerate nonzero null cone null geodesics null lines pair phase space Poisson algebra Poisson bracket polynomials preceding equation projection quadratic form quantization quantum mechanics reduced space restriction rotations satisfies scalar product selfadjoint selfadjoint operator Sitter six-dimensional skew-adjoint operator sp(U subalgebra subgroup submanifold subset subspace super Poisson algebra symmetry symplectic form symplectic manifold symplectic vector space tangent theorem theory unitary vector field