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 2. Figure 2. CERN collision ring The basic Einstein-Hilbert action principle given by the action Z. 1 . SE [ g ] = r g d4 x G M. which holds in empty space with the possible addition of a cosmological term, is re- placed in the presence of matter by the combination (1) S = SE + SSM.
NONCOMMUTATIVE GEOMETRY AND PHYSICS 5 v Figure 4. Hydrodynamics its inertial mass m by the renormalized mass m ! m + 1 2 M where M is the mass of the °uid corresponding to the volume of the ball (as in Archimedes law).
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