The Conformal Group Revisited

Abstract

Since 100 years or so, it has been usually accepted that the conformal group could be defined in an arbitrary dimension n as the group of transformations preserving a non-degenerate flat metric up to a nonzero invertible point depending factor called “ conformal factor ”. However, when n ≥3 , it is a finite dimensional Lie group of transformations with n translations, n ( n -1)/2 rotations , 1 dilatation and n nonlinear transformations called elations by E. Cartan in 1922, that is a total of ( n +1)( n +2)/2 transformations. Because of the Michelson-Morley experiment, the conformal group of space-time with 15 parameters is well known for the Minkowski metric and is the biggest group of invariance of the Minkowski constitutive law of electromagnetism (EM) in vacuum, even though the two sets of field and induction Maxwell equations are respectively invariant by any local diffeomorphism. As this last generic number is also well defined and becomes equal to 3 for n= 1 or 6 for n =2, the purpose of this paper is to use modern mathematical tools such as the Spencer operator on systems of OD or PD equations, both with its restriction to their symbols leading to the Spencer δ -cohomology , in order to provide a unique definition that could be valid for any n ≥1 . The concept of an “ involutive system ” is crucial for such a new definition.