This ebook relies at the Colloquium Lectures provided by means of Shlomo Sternberg in 1990. The authors delve into the mysterious function that teams, specially Lie teams, play in revealing the legislation of nature by way of targeting the standard instance of Kepler movement: the movement of a planet below the appeal of the sunlight in response to Kepler's legislation. Newton discovered that Kepler's moment law--that equivalent parts are swept out in equivalent times--has to do with the truth that the strength is directed radially to the sunlight. Kepler's moment legislation is de facto the statement of the conservation of angular momentum, reflecting the rotational symmetry of the procedure in regards to the foundation of the strength. In ultra-modern language, we'd say that the crowd $O(3)$ (the orthogonal staff in 3 dimensions) is answerable for Kepler's moment legislations. by way of the top of the 19th century, the inverse sq. legislations of appeal used to be noticeable to have $O(4)$ symmetry (where $O(4)$ acts on a part of the six-dimensional part house of the planet). Even better teams have for the reason that been stumbled on to be inquisitive about Kepler movement. In quantum mechanics, the instance of Kepler movement manifests itself because the hydrogen atom. Exploring this circle of rules, the 1st a part of the publication was once written with the final mathematical reader in brain. the rest of the ebook is geared toward experts. It starts off with an indication that the Kepler challenge and the hydrogen atom express $O(4)$ symmetry and that the shape of this symmetry determines the inverse sq. legislation in classical mechanics and the spectrum of the hydrogen atom in quantum mechanics. the gap of regularized elliptical motions of the Kepler challenge (also referred to as the Kepler manifold) performs a primary position during this publication. The final component to the e-book reports some of the cosmological types during this similar conformal type (and having various isometry teams) from the point of view of projective geometry. The computation of the hydrogen spectrum offers an indication of the main that enlarging the section house can simplify the equations of movement within the classical environment and relief within the quantization challenge within the quantum surroundings. The authors supply a brief precis of the homological quantization of constraints and an inventory of modern purposes to many fascinating finite-dimensional settings. The booklet closes with an summary of Kostant's conception, during which a unitary illustration is linked to the minimum nilpotent orbit of $SO(4,4)$ and during which electromagnetism and gravitation are unified in a Kaluza-Klein-type concept in six dimensions.
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