This booklet is predicated at the Colloquium Lectures awarded via Shlomo Sternberg in 1990. The authors delve into the mysterious position that teams, in particular Lie teams, play in revealing the legislation of nature through concentrating on the typical instance of Kepler movement: the movement of a planet less than the allure of the sunlight in keeping with 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 solar. Kepler's moment legislations is absolutely the statement of the conservation of angular momentum, reflecting the rotational symmetry of the method in regards to the starting place of the strength. In cutting-edge language, we'd say that the gang $O(3)$ (the orthogonal staff in 3 dimensions) is accountable for Kepler's moment legislations. by way of the top of the 19th century, the inverse sq. legislations of appeal was once noticeable to have $O(4)$ symmetry (where $O(4)$ acts on a section of the six-dimensional part area of the planet). Even higher teams have for the reason that been chanced on to be interested by 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 e-book was once written with the final mathematical reader in brain. the rest of the booklet is aimed toward experts. It starts off with an illustration that the Kepler challenge and the hydrogen atom show $O(4)$ symmetry and that the shape of this symmetry determines the inverse sq. legislations in classical mechanics and the spectrum of the hydrogen atom in quantum mechanics. the distance of regularized elliptical motions of the Kepler challenge (also referred to as the Kepler manifold) performs a critical position during this publication. The final part of the ebook experiences some of the cosmological versions during this comparable conformal category (and having various isometry teams) from the perspective of projective geometry. The computation of the hydrogen spectrum offers an indication of the main that enlarging the section area can simplify the equations of movement within the classical surroundings and reduction within the quantization challenge within the quantum atmosphere. The authors supply a brief precis of the homological quantization of constraints and a listing of contemporary functions to many fascinating finite-dimensional settings. The e-book closes with an overview of Kostant's thought, within which a unitary illustration is linked to the minimum nilpotent orbit of $SO(4,4)$ and within which electromagnetism and gravitation are unified in a Kaluza-Klein-type idea in six dimensions.
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