The Conformal Group Revisited

TL;DR

This paper revisits the conformal group in arbitrary dimensions using modern mathematical tools like Spencer cohomology to provide a unified and precise definition applicable to any dimension n.

Contribution

It introduces a new, mathematically rigorous definition of the conformal group for any dimension n using Spencer operator and cohomology, clarifying its structure beyond classical cases.

Findings

Provides a unified definition of the conformal group for all n

Uses Spencer cohomology to analyze systems of differential equations

Clarifies the structure of conformal transformations in higher dimensions

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 > 2, 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 " , 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 as the biggest group of invariance of the constitutive law of electromagnetism (EM) in vacuum, even though the two sets of field and induction Maxwell equations are respectively invariant by any local invertible transformation. 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 -cohomology, in order to provide a unique striking definition that could be valid for any n. The concept of a " finite type " system is crucial for such a new definition.