Pour accéder à tous les contenus il faut activer ce widget
Since we now have at least one way to deform vector products, generalizing the work contained in the booklet "J.C. Maxwell's Vacuum" requires knowing how to decompose these deformed vector products and their dual images in a non-trivial way.
The approach requires clarifying two concepts: (i) that of dual image decomposition and (ii) that of the triviality of a decomposition.
There exists, in a sense, a double isomorphism of vector spaces between E(3, R) and its dual E*(3, R), and then between the latter and R^3 = R x R x R (Cartesian product).
The approach to spatial positioning in three dimensions, as developed by Descartes, partially explains this fact.
Since the affine space is referenced to a point origin O, any pair of points (O, P) defines a vector OP to which a triplet of real numbers (x, y, z) corresponds.
The development of quantum physics introduced a symbolic matrix notation to represent vectors of E(3, R) in R^3. These representations are either elements of M(1, 3) or elements of M(3, 1); that is, respectively: row representations denoted . This is the "row-column" convention (or Bra-ket notation in Anglo-Saxon literature). In what follows, I use column representations and the standard matrix multiplication rule.
The lessons on rotations are sufficient to understand that any column representation of a given vector can be decomposed in many ways according to the following scheme:
|OP> = [R].|OQ> + |Q'P>
This can be translated literally by saying that a vector OQ can undergo a rotation in place (synonym: the endpoint O is fixed) whose matrix [R] of M(3, R) contains all the information. As a result, the vector OQ becomes a vector OQ'. By applying a translation Q'P to it, the final result is obviously a vector OP.
La théorie qui me préoccupe et que j’ai baptisée : « Théorie de la question (E) » cherche des réponses à une série de questionnements.
I assume that the characteristics of a twist are known. It amounts to the same thing to say that the pair ([R], |Q'P>) is given at the start. The first consists of asking the following:
"Can the vector OP obtained by the twist acting on OQ be the result of the product of an unknown vector OX with the vector OQ:
|OP> = [R].|OQ> + |Q'P> = |OX x OQ> ?"
In this statement, everything is known, except for the vector OX which needs to be discovered. The question is whether a twist can be equivalent to a classical cross product?
A second question, the inverse of the previous one, is as follows. The pair of vectors (OX, OQ) is known and its cross product is a priori the result of a twist. The question becomes: "Can I discover the characteristics of this twist?" In this formulation, the problem consists of discovering the pair ([R], |Q'P>).
A third question extends the first and asks: "Can the vector OP obtained by the torsion acting on OQ be the result of a product of the unknown vector OX with the vector OQ deformed by a matrix [A]:
|OP> = [R].|OQ> + |Q'P> = |[OX x OQ][A]> ?"
The goal is then to discover the pair ([A], OX).
The fourth question reverses the logic of the third. The pair of vectors (OX, OQ) and the deforming matrix [A] are known. I calculate the deformed vector product [OX, OQ][A].
The question becomes: "Can I discover decompositions ([P], z) of the type:
|[OX, OQ][A]> = [P].|OQ> + |z> ?"
The concept of decomposition of deformed vector products is thus clearly established.
By analogy with the semantics characterizing division, the matrix [P] will be called the principal part of the decomposition and the vector z will be its remainder or residue.
Intuitively, a decomposition will be qualified as "trivial" when its remainder is zero.
A detailed analysis of this intuition will show that a given deformed vector product does not necessarily have only one trivial decomposition.
These are the few questions I would like to answer in order to extend the conclusions of the work presented in the first chapter of the booklet "J.C. Maxwell's Void".
© Thierry PERIAT, 4 August 2026.