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.
Highlights
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”
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
The reader not familiar with the formal theory of systems of PD equations may find difficult to deal with the following definitions of the Spencer bundles Cr ⊂ Cr ( E ) and Janet bundles Fr for an involutive system Rq ⊂ Jq ( E ) of order q over E: ( ) Cr = ∧r T * ⊗ Rq δ ∧r−1T * ⊗ gq+1 ( ) Cr ( E ) = ∧r T * ⊗ Jq ( E ) δ ∧r−1T * ⊗ Sq+1T * ⊗ E
Summary
Connected problems: Given a differential operator ξ →η , how can we find compatibility conditions (CC), that is how can we construct a sequence. It is only in 2016 (see [9] and [15] for more details) that we have been able to recover all these operators and confirm with computer algebra that the orders of the operators involved highly depend on the dimension as follows: These results are bringing the need to revisit entirely the mathematical foundations of conformal geometry, in particular when n = 3 because the Weyl type operator is of third order and when n = 4 because the Bianchi type operator is second order in this case contrary to the situation met when n = 5. After presenting two motivating examples, such a procedure will be achieved in Section 3 in such a way that the Spencer sequences involved, being isomorphic to tensor products of the Poincaré sequence for the exterior derivative by finite dimensional Lie algebras, will have vanishing zero, first and second extension modules when n ≥ 3 ([4] [11]). [21] [22] [23])
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