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In [a], cross products have been deformed and then non-trivially decomposed. In [b], I gave an illustration of the results via a renewed analysis of the Klein-Gordon equation+.

 

In that new exploration, I scrutinize the consequences of the approach developed in [a] further and I apply them to an analysis of the line element+. The limit case of a Euclidean geometry+ is envisaged. It forces me to think about the eventual links between deformed scalar and deformed cross products.

 

The mathematical coherence is only obtained when (i) the second argument involved in the cross product is an isotropic vector in a three-dimensional Euclidean space related to a spinor+ “à la E. Cartan” and (ii) that isotropic vector is orthogonal to the non-vanishing residual part of the non-trivial decomposition. These conditions, although they are very technical, will be the starting point for the next step of that exploration.

 

Hence, the discussion must be developed in E(3, C) and, if a non-trivial decomposition is also needed for the transposed cross product, both arguments must be isotropic vectors. As a direct consequence: the theory which has been initiated in [a] is meaningful if it manages bi-spinors+.

 

Although mathematically uncomfortable, the appearance of bi-spinors is a good point when one is constructing a theory. Il allows a natural connection with the quantum mechanics+ or with considerations on the supersymmetry+.

 

In a next step, the main parts of the non-trivial decompositions are analyzed with the help of Euler parametrizations+. There exists a simple correspondence. The ADM slicing+ and the Klein-Gordon equation, again, are two examples illustrating the dissertation.

 

© Thierry PERIAT, 26 March and 14 April 2019.

 

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