Neutral-element.

Title: Looking for a neutral element in V4(H, Γ(2)).

Author: © Thierry PERIAT.

Matriculation: ISBN 978-2-36923-004-5, EAN 9782369230045.

Version: 2.

Language : GB/USA.

Number of pages: 19.

Publication: 9 August 2022.

Document: 978 2 36923 004 5 periat v2978 2 36923 004 5 periat v2 (355.7 Ko).

Abstract:

In looking for a neutral element for the space E(4, H) equipped with a tensor product deformed by the (4-4-4) cube containing the Christoffel’s symbols, I discover a link with the Lorentz transformations.

Introduction.

The first version of this exploration has been written between the 9 December 2007 and the 4 January 2008 but never published. This document is a modernized formulation of it.

At this time Lisi's proposal [01] had revived interest for the E8 x E8 model. The latter stayed at the top of the research during the seventies and seemed to be a reasonable candidate for a theory of (fast) everything [02].

This way has been abandoned because experiments could not verify the decay of the proton.

Nevertheless, the E8 group generates a brainstorm and can be proposed as an application in the way of thinking developed by Smolin at the Perimeter Institute [03]. The scheme described by Smolin works for any gauge group [03; page 8]. But interesting groups must contain SO (3, 1) [03; page 2] to give rise to a theory including general relativity.

In 2008, Pr. Connes did explain very clearly why one should make the choice to work on the algebra A = M(2, H) ⊗ M(4, C) [04] as soon as one wants to build something coherent with the standard model. This information gave me a first indication on where I should start.

Motivations.

In 2008 too, theorists were aware of the theoretical predictions that were implicitly induced by A. Einstein's theory [05] (gravitational waves [06], Thirring-Lense effects [07], etc.). These predictions were all exhibiting an obvious property linking them all together: a kind of elasticity able to act on the geometry of the four-dimensional spacetime.

But they did not yet have any experimental fact proving that elasticity. The proofs came later: [08; 2011] for the Thirring-Lense effect and [09; 2016] for the gravitational waves. The existence of the covariant version of the Lorentz law (electromagnetism) was known since a long time. It contains a so-called gravitational term. At a technical level, that term is a vector, more precisely a tensor product which is deformed by a (4-4-4) cube containing the 64 Christoffel's symbols of the second kind. These symbols describe the first order variations of the metric. Therefore, they seemed like the perfect tools for those who wanted to describe that predicted elasticity further, eventually in extending its domain of validity until the smallest scales

The unsaid motivation for this ambition was the secret hope to unify the predictions which were essentially concerning physical phenomenon occurring at a cosmic scale with the physics of particles, which -as it is obvious- concerns what happens at a very, very small scale.

That dream of unification is not new and relatively clearly expressed in the Einstein-Rosen article published in 1935 [10]. You may, by the way, visit on this website my exploration revisiting this article.

Although brilliant spirits like Freeman Dyson [11; 2018] have recently presented arguments explaining why this dream is perhaps an unnecessary quest and a waste of time [12], not all theorists have abandoned the quest and I belong to that category.

Taking into account the previous thoughts, I decided to gradually explore the mathematical properties of V4(H, Γ(2)) = {E(4, H), Γ(2)}. The specific topic which I want to develop here on this page is the search for a neutral element.

Before doing this, I want to make two remarks:

  • For a mathematician, the classical version of the Lorentz law is a decomposition of the null vector.

  • Because the metric is permanently changing, the structure of V4(H, Γ(2)) may vary.

© Thierry PERIAT.

You may eventually want to confront this document with my exploration concerning the "Klein-Gordon-Equation-in-4D".

Go back to the page: "Mathematical structures".

Bibliography (abstract).

[01] Lisi, A. G.: An extraordinary simple theory of everything; arXiv 0711.0770v1 [hep-th], 06 November 2007.

[02] Hawking, S.: Is the end in sight for theoretical physics? Conference at the Lucasian chair of mathematics, April 1980; Cambridge.

[03] Smolin, L.: The Plebanski action extended to a unification of gravity and Yang Mills theory; arXiv: 0712.0977v1 [hep-th], 06 December 2007.

[04] Connes, A. and Chamseddine, A. H.: Why the Standard Model; arXiv: 0706.3688v1 [hep-th] 25 June 2007.

[05] Einstein, A. : Die Grundlage der allgemeinen Relativitaetstheorie; Annalen der Physik, vierte Folge, Band 49, (1916), N 7.

[06] MTW: Gravitation.

[07] On the Gravitational Effects of Rotating masses: The Thirring-Lense Papers; General Relativity and Gravitation, Vol. 16, No 8, 1984, 40 pages.

[08] Standford's Gravity Probe B confirms two Einstein's theory; news.stanford.edu, Stanford Report, May 4, 2011.

[09] Einstein's gravitational waves found at last; nature.com, 11 february 2016.

[10] A. Einstein, N. Rosen: The particle problem in the theory of relativity; pp. 73-77, physical review, vol. 48, July 1, 1935.

[11] Dyson Quantenfeldtheorie, die weltbekannte Einführung von einem der Väter der QED; © Springer Verlag Berlin Heidelberg 2014, 288 Seiten.

[12] Freeman Dyson: Why general relativity and Quantum Mechanics can’t be unified; Nomen Nominendum, 2018.

... and more inside the document.

Date de dernière mise à jour : 12/08/2022