The Theory of the (E) Question.
Looking for unification is a leitmotiv in physics since the end of the 19th century. It is rooted in the success of A. Einstein’s theories: special relativity and theory of gravitation.
The appearance of the quantum theory at the beginning of the 20th century, of which the logic, the tools and the results cannot easily be put in the same pot than the ones of Einstein’s approach is challenging the actors of that quest for unification since more than hundred years.
They have a new goal: the search for rationalistic links between the gravitation and the standard model for particles. As related in diverse books and articles, attempts are numerous but as far it is known, none of them was able to give a satisfactory answer in a four-dimensional space formalism (nota bene: there is a recent thesis working only in a two-dimensional context). One of the followed paths lies on the fact both, Maxwell’s, and Dirac’s theories, can be formulated in the spinor language; unfortunately -for technical reasons- that formal similarity is neither a strong enough nor a sufficient argument to get the dreamed unification.
Hence, the quest must go on into another direction and that is exactly what I try to do with a theory studying the deformation and the decomposition of tensor products. A trivial and first motivation explaining my choice is the fact that the covariant formulation of the Lorentz law (eventually mentioned in the literature as Lorentz-Einstein law) is both, (i) at the frontier between electromagnetism since it is built on the classical formulation of Lorentz law and (ii) with a foot in Einstein’s theory of gravitation since it contains a so-called gravitational term which is nothing, but a tensor product deformed by a cube of Christoffel’s symbols of the second kind.
Within that context and with that viewpoint, I interpret the Lorentz-Einstein law as a concrete and natural illustration of the mathematical spirit guiding my quest. Since gravitational sources are universally present, there must be non-exactly zero Christoffel’s cube all around us and, as consequence of that law, every motion should be accompanied with the appearance of an EM field and of an acceleration.
This “a priori” interpretation justifies the study the decomposition of tensor products, in general. This study is done through the development of mathematical methods (the intrinsic, the extrinsic, etc.…) and their confrontation. The intrinsic method works only in a three-dimensional environment whilst the extrinsic one can be involved in any-dimensional context but, is only an approximative one. A calibration can be realized in a three-dimensional space. The calibration is made with the help of the Holtz decomposition of vectors, and it introduces a “topological” vector related to the deforming cube which, in that context, is reduced to an element in M(3, C). That means -as already known through the Lorentz-Einstein law- that the deforming background strongly influence the result of any decomposition.
Furthermore, the first argument in the pair ([P], z) characterizing any decomposition can be associated with its transposed [P]t and it can be proved that each pair ([P], [P]t) is a suitable tool for the construction of canonical anticommutative variables (see: the algebraic dynamics). Since any such decomposition is also associated with a polynomial form of degree two, the variations of these forms can be translated into the matrix language.
Further explorations in that direction can be done in involving the Klein-Gordon equation describing the propagation of massive waves because the matrices [P] can be corelated to the Euler-Rodrigues parametrizations. They exhibit a link with the deformation of elastic strings (see the analysis of the Klein-Gordon equation in a 3 and in a 4-D space; and the document vacuums and strings as well) and with the standard model for particles. These matrices can also be incorporated into (4-4) matrices representing the action of Hodge’s operator acting on 2-forms in a 4-D space. Short: the matrices [P] can become the heart of matrices symbolizing the electromagnetic dualism, hence perfectly suited for the description of particles behaving like waves and propagating along strings; that means: in taking care of external interaction, and the gravitation can be one of them.
© Thierry PERIAT, Waldkirch, 17 February 2022.