Transcription of Introduction - Alain Connes
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Introduction - Alain Connes
NONCOMMUTATIVE GEOMETRY AND PHYSICS. Alain Connes . 1. Introduction The two existing theories which successfully encode our knowledge of space-time are : General Relativity The Standard Model General relativity describes space-time as far as large scales are concerned (cf. Figure 1. and [2] for many more suggestive thoughts and pictures) and is based on the geometric paradigm discovered by Riemann. It replaces the flat (pseudo) metric of Poincar e, Einstein, and Minkowski, ds2 = dt2 + dx2 + dy 2 + dz 2. by a curved space-time metric whose components form the gravitational potential g . ds2 = g dx dx . Figure 1. Suspected black hole in center of galaxy 1.
NONCOMMUTATIVE GEOMETRY AND PHYSICS 4 y x Figure 3. Feynman Graph 2. Lessons from renormalization In QFT the recipe of Dirac and Feynman gives the probability amplitude of a classical fleld conflguration A as ei S(A) where the classical action is the integral of the Lagrangian density
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