For English speaking readers
Last update: 22 November 2008. Introduction to a quantized field theory including gravity.
This page is specially written for non French speaking people and try to make the progression of the theory "understandable" for everyone. But for a complete comprehension please visit the French part of the blog.
0) Axiomatic of the theory:
Téléchargement QGD-Principles.pdf [21/11/08; 5 pages; secured]
This document makes a "résumé" of the pillars on which the theory has been built. It also explains the articulations between the different axioms and so tries to give a kind of overview on the whole construction. I hope it can help the readers to get a complete representation of the intern logic of the Gravito-Hydrodynamic-Quantum theory (GHQ). Short said: the theory can be seen as being a roman explaining all about the Lorentz Einstein Law (LEL) and the new mathematical concept of extended products splitting in E4(K) x M4(K).
Where K is either the set of real, complexe, quaternionic or octonionic numbers.
1) Exterior products closed by a cube in a 4D space.
Splits. Associated conics. Associated scalar. Analogy with solutions of GTR. Conditions for that. Comparison with the method of the associated scalar. Explaination of the results in Noemie04.pdf (3D version of the problematic). The role played by antisymmetric cubes. Intern derivations. Bigebroids. Application to the ligth. Interaction between two rays of ligth (in fact between two splits). Necessary introduction of the extended kinematic momentum. Confrontation with the problem : differential operator of second order and Lorentz Einstein Law.
All is in the French document Noemie2008v1106.pdf [1 MB; 10 pages; 06/11/08]
Commentaries in english language:
Téléchargement Noemiegb2008.pdf [1 page; 08/11/08]
2) Non commutative geometry, spinors, extended products:
An evolutive document (46 pages; about 0.5 MB; in French; need Acrobate Reader 9; HGQv20081005.pdf; secured) explains the way I am building the mathematical part of my theory and what is the contribution of the non commutative geometry (Pr. Connes and all.) in it. A link between extended products built with antisymmetric cubes, theirs trivial splits, spinors (à la E. Cartan) and GTR (Generalized Theory of relativity) is explained.
3) The concept of extended product (a modernized introduction):
See Noemiev2.pdf (18 pages; in French; 04/10/08; secured) and Noemiev3.pdf (22 pages; in French; 1.3 MB; 08/10/08; secured).The document Noemiev3.pdf (introduction in English and in French language) has at the beginning only one purpose: to find the solution(s) of the problem connecting a “closed exterior product” with any split of an extended product in a 4D space... but ...
4) The position of my work:
and what I am thinking about it (as independant researcher):
Téléchargement extended_products.pdf [0.5 MB; in English]
5) The answers to the theoretical questions I asked in Noemiev2 and v3.pdf:
Téléchargement gravitation_and_extended_products.pdf [0.7 MB; 13/10/08; 3 pages; in English]
6) Intern derivations:
To introduce an intern derivation there is no necessity for an extended product to be an explicit Lie bracket. It can only be an implicit one and for that purpose the extended product only needs to be anticommutative and the Jacobi's identity must hold. This is the pertinent remark that allows us to open the discussion that seemed to be achieved at the end of the last document. You can read:
PseudoDerint.pdf
[31/10/08; 2 MB; 24 pages; French language; secured]
Re exploring wellknown sectors of fundamental mathematics we try to built Lie brackets with the minimal mathematical structure. The purpose is to reach the discussion about bigedroids... This is the first path we newly discovered to introduce rigourously the concept of split for extended products.
7) The Baker-Hausdorf formula:
We can now write a special version of the Baker Hausdorf formula involving the intern derivation built with extended products; see preliminaries in Baker-Hausdorf.pdf [16/10/08; 0.7 MB; 5 pages; french language]. There is no "scoop" in that document. It is only a first try to apply general considereations made before to the well-known formula.
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August - September 2008
The Fibrev2008xyzt.pdf document will be updated permanently; but extensions will also depend on the evolution of the mathematics needed for the development. So, it is not a bad idea to read the mathematical part of my investigations. Amazingly and fortunately both parts (mathematics and physics) are intimely connected together.
Extending the whole discussion to a set E such that dim E = 4, built on K = H (quaternions) and equiped with and anticommutative extended product will give us the possibility to write a collection of equivalent expressions of the Lorentz Einstein Law. A part of these expressions owns a formalism in 4D very similar to the expression (4.2) page 24 of reference 12 in the most recent version of the document (see French part of the blog).
This remark is suggesting the necessity to study Chern-Simons Theories. This is also suggesting the necessity to develop our own conception of "intern derivations" (see etf90.pdf, etf95.pdf... on the French part) when (E4, H, extended product) is a Lie algebra. We explore the "integration" associated with such derivations. Here, like for the Belezin integration, the word "integration" is referring to a functional more than to a Lebesgue function. The introduction of proposition 06 is essential. We now own enough informations to propose a new family of solutions to the Feynman's integrals problem.
The blog is reorganized: diverse pages "archives 200x (FR + GB)" have been opened. Since I want to preserve the historic progression and the initial theoretical spirit of my work, you can continue to read the main chronologic steps below.
May 2008
Téléchargement etgb100v3.pdf [0.725 MB; 16 pages; 16 May 2008]
Update of etgb100.pdf (May 2006; see the "arXiv for english readers" to discover the original investigation).
We re-motivate the proposed decompositions for F and generalize the context of validity. We abandon the discussion about the anticommutative product defined or not for the set K on which the theory is built. That is we restrict our self to the very classical case where K = R, the set of all real numbers.
Because we do leave the restriction concerning extended products built on the Levi-Civita connection, we have no more to discuss the systematic compatibility with the metric and we are now able to propose a new orientation to the discussion.
We propose to consider the decompositions of the EM field tensor F when the extended products are defined via anti-symmetric cubes (e.g: the cubes that can be extracted from the Riemann-Christofel's tensor). This gives us the possibility to introduce the notion of intern derivation into the debate. Starting again from our old investigation (ThP04F.pdf; a GR theory including the variations of the tetrad until the second order) we suggest the existence of circumstances for which deformations of the "underground" geometry generate EM flows. Logical and qualitative considerations lead to the proposition that at most 8 such currents exist. The demonstration of the coherence of the proposition must still be done and is a work for later but qualitative arguments are introduced to justify our intellectual position.
We calculate the Lagrangian density of the proposed model in a very classical manner. It is a bilinear expression. In that sense it is naturally offering a link with Yang-Mills terms. We then propose a computational tool to calculate that quantity in an easier way and explain the advantage of such procedure for investigations in vacuo.
At the end of the document I propose now a field theory treatment of the decompositions. Some new precisions are given for the validity of the proposition. A lot of work has to be done. Very first intuitive solutions are proposed for a special configuration of the Christoffel's cube.
1) errata in etgb100v3.pdf. page 5: read (D-3) in the last sentence; page 9: the definition ofd the "slide product" is (D-6); page 12: it makes no sense if the scalar function lamda is invariant; page 14: ... of which the first order variations satisfy ... = 0; page 15 (5.14) add and if M = F.
2) To be published: some progress has been made in the counting action concerning the number of the components of the Christoffel's cube (when there is no torsion and when condition (5.8) page 13 holds. The solution of this amazing problem seems to be 20 when N (the dimension of the space) = 4. Because of that result, I am now analyzing the existence of an isomorphisme between the set of such Christoffel's cubes, the set of all Riemann-Christoffel's curvature tensors (also when N = 4) and a representation in M5(R).
One must recall here that mathematical properties of this set of Christoffel's cubes have been analyzed in etf31v4.pdf. A link with the (2,2) curvature tensor of Cartan exists.
April 2008
chargement ER.pdf [2 pages; 0,243 MB; 14 April 2007]
This is a fundamental discussion concerning an extension of the notion of "equivalence relation"
2007 - CONCERNING THE NAME OF THE THEORY.
I beg my pardon: readers of this blog have to understand French and English languages. it certainly doesn't simplify the comprehension of my work. Despite of this imperfection, I am sure you are able to understand the progression. Another "mea culpa" because of the name of my theory: "The (E) Theory". Its first "official" publication started in September 2003 when I was invited at the IVth Vigier's Symposium in PARIS. At this time I didn't know that K. Thomsen, Pr Connes, for example, and many others were working on an extension of the C*-algebras called the E-Theory. Quite more: I had no idea what the C*-algebras were looking like! So, there is sometimes in the history of sciences, some strange coincidences. My approach was and is always not motivated by the mathematics but by the dream to understand the "birth" of particles. The coincidence is very disturbing. If we consider, as said by Camenzind in one of his lessons, that we did paid our mathematical dues during the last months in paying attention to the (mathematical) part of every modern theory; we must also recognized that we are now prepared to begin with the concept of asymtotic homomorphism. With the document "Group.pdf" (see archives 2008 FR + GB) we did begin the discovery of a group to work with. We have to push this investigation forward. We have also to understand the "intellectual" similarity between "our" splits and the asymptotic homomorphisms. Indeed, we can consider the trivial splits inside the (E) Theory as being an illustration of exact morphisms if extended products under consideration are associative; and we have to ask when the non trivial splits can be understood as asymptotic morphisms. We shall develop this point of view during the next weeks; at least I hope it.
The former point which was asking how far the (HGQ) Theory could be understood as an extrapolation of the GTR received the begining of a positive answer in the French document: "Fondamentaux.pdf" (See archives 2008 FR + GB). Indeed, the introduction of some modern operators and their interpretation inside the (HGQ) Theory could give us the interesting opportunity to define a Laplacian (not a Lagrangian) for that theory. Amazingly and surprizingly we are able to connect this intuitive work with a part of the Einstein's equations for the field. This is encouraging us to study more about operators, Dirac's operators, and so on...
CAN THE HGQ-Theory BE UNDERSTOOD AS AN EXTRAPOLATION OF THE GTR?
Any matrix in which an (ad hoc) extended product splits in minimizing the equivalent scalar that can be associated to that split, can be considered as a good potential candidate challenging against the matrix representation of the Ricci tensor in M4(R).
Indeed, last days I did re-read attentively the first informal argument that has been used to start the construction of the generalized theory of relativity; namely the wish to extend the Poisson equation. This purpose is obtained within the GTR via the Ricci tensor; in fact its temporal (0,0) component et the Newtonian limit.
For the (HGQ) Theory, since 1) a Poisson equation involves a Laplacian, 2) any Laplacian is built on a Hessian and 3) any Hessian is the limit formalism of a matrix in which an extended product splits in minimizing the equivalent scalar associated to this split (Coming : the French document "topologie.pdf"; see "Nouveautés"), we have good reasons to get the intuition that such matrices can play an equivalent theoretical role than those of the matrix representing the Ricci tensor in M4(R).
This is the subject of the next coming parution and certainly a way to test the difference (if any one exists at this step) between the GTR and the (HGQ) approach.
SOME IMPORTANT PRECISIONS CONCERNING PHYSICS CONTAINED WITHIN THE HGQ-Theory:
The document "Group.pdf" (see archives 2008 - FR + GB) introduces a representation of (E4, H) in M4(R). This representation does not always come into coincidence with the representation of quaternions in M4(R). But it sometimes do. In that case, we have the strange situation that four quaternions [the components of a vector in (E4,H)] can be represented by only one quaternion. This is the first important presicion made inside this provisory short work.
It gives me, by side, the opportunity to re-introduce a set of M(n)(K) matrices characterized by the fact that one can "reduce" them to a set containing a scalar k of K, a vector k of [E(n - 1), K] and a sub-matrix of M(n-1)(K). Allthough I gave my name to this special feature, the perian matrices (I did call them so) represent a similar situation than M. Jourdain making "prose" without to know it (see Molière: French writer). The link inside the document, for example, shows that M. A. Nüchter did make use of similar matrices inside his thesis in 2002 (At the Fraunhofer Institute). This family of matrices is very important and it should be paid attention to its mathematical structure. One interesting property is that it can be associated with a "conoide".
Otherwise, we come back to our previous investigations showing that the Faraday-Maxwell tensor accepts a representation in M4(R) involving the local metric tensor G and a local trivial matrix relatively to the split of the extended product between the EM potential 4-vector A and the position vector x. This fact has been investigated, for example, in etgbYMPv2.pdf (see the link below on the same page of this website). We suggest here that the existence of such representation is an argument to believe that the Lorentz transformations only are a special restricted feature of more general expressions. We specially think that Lorentz transformations are concerning circumstances with no force; that means in the absence of gravitation (G is then reduced to its Minkowskian version) and in the absence of EM field (F = 0). We insist on the fact that we don't want to reject the well established transformations and only suggest a limitation to their validity. Indeed, Lorentz transformations have been (historically) built to connect two "inertial" observers (that means observers each of them being situated at the origin of a 4D frame where no force was to be found) measuring the same speed, c, for the light propagating in vacuum.
RELATIVE INTENSITY OF GRAVITATIONAL AND EM FIELDS: an explanation?
One of the mysteries of physics (and unsolved problem) is the scale problem between the intensity of the EM fields and the intensity of the gravitational fields. One road to surround this problematic (and it is the usual one today) is to believe that the “unity” is recovered for the very high energies (where one expects to obtain circumstances very similar to the initial big-bang).
But can we not imagine another road? Let us abandon for a while our dream to build a cosmology and instead of that let us only start from the evidence: gravitational fields are very tiny but omnipresent. What would it be if the disproportion were not only the result of the time (temporal distance in the actual cosmology starting from the big-bang) but would in some way accelerated by the fact that variations of the second order of the geometry are also producing fields isomorphic to an EM field while the variations of the first order are the origin for the gravitational field (GR approach)? Quantitatively, because of the successive interferences and accumulations, one could expect that the intensity of the EM field would be increasing with the time and would always surpass the intensity of the gravitational field. This is in some way the story contained in ThP04F.pdf (French language; actually no more available). See the new link on the right: "Archives de la theorie (HGQ)".
A SHORT DISCUSSION ON THE DIRAC'S gamma MATRICES:
See a short note concerning Dirac's relation (in french language: Dirac.pdf; actually no more available). The background behind the alpha matrices. Ideas to extend the relation to a generalized theory of relativity (including a non vanishing Christoffel's cube). Unfortunately it does not work because of some technical problems of which we do not have the solutions.
TRIVIAL SPLIT AND NOISELESS CHANNELS?
Téléchargement NC.pdf [03 November 2007; 10 pages; 1.08MB]
Explaining an intuition: "Is the trivial split of an extended product equivalent to a noiseless channel?".
JUMPING FROM A SPLIT TO THE NEXT ONE ?
The splits of a any extended product can be assimilated to a mathematical transformation. It introduces the concept of extended momentum and an original scenario to describe the evolution of any particle under the influence of a force. This is in fact the english version of the mce.pdf document in french language (actually no more available). The former is actually a little bit more developed and gives two important informations: 1) a relation between the metric tensor and the EM field tensor in some peculiar circumstances; 2) a suggestion to built a BF-type action in a N = 4 dimensions space.
(This point of view has still been roughly studied a few months ago (2005) in the very naive and partially false document: Téléchargement etgb28.pdf; old naive document with very important ideas)
[0.5 MB; 10 pages; 2005; protected]
NEUTRAL STREAMS IN VACUUM (v2)?
After four long years of work, I present the v2 of my fundamental work (proposed for the IVth Vigier Symposium in September 2003 in Paris; but I did never present it). This is an evolutive English version (some typo must be revised). Nevermind, the spirit is preserved and appears to be more and more actual.
[11 pages; 08 October; 0.592 MB; protected]
A NEW ANALYZE OF THE FARADAY-MAXWELL TENSOR:
I present here a new version of the document etgbYMP and two complementary works. I come back on the reduction propose intuitively for the Faraday-Maxwell tensor and analyze it again in taking care of what I did learn during the last year. Relationships with the possibility for this reduction to be a) the adhoc formalism to transform an ordinary differential operator into an expression of the Lorentz Einstein Law or b) a veiled formalism for the Lorentz Einstein Law itself are examined.
[0,938 MB; 19 pages in English; September 2007; protected]
For the better understanding of these documents you will also need the following introductory works:
Téléchargement etgb93.pdf [0,305 MB; 27 december 2006]
Téléchargement etgb95.pdf [O,438 MB; 27 March 2007]
SOME IDEAS TO EXTEND THE LORENTZ TRANSFORMATIONS:
Téléchargement etgb2007.pdf [0.35 MB; free; 20 August 2007]
THE FIRST WORK (first offical publication in 2003)
English speaking readers can find here some of the documents that have been published on the personal homepage vacuum-world-net.eu; this is the continuation of the work presented in September 2003 for the IV Vigier's symposium in Paris. This work demonstrates the existence of a dynamic equation in vacuum based on the Maxwell's laws and suggests the possibility, via the Newton's second law, to discover the masses of the particles in a totally new approach.
(there is a CD version of this work with a ISBN number)
This first and naive work (in 3D) gave rise to a systematic mathematical exploration of a family of "extended products" in a 3D space [Noemie04.pdf, a French document; actually no more available]. Because the Einstein-Lorentz law contains such a product (the gravitational part of the force) in a 4D space formulation, we are encourage to push our investigations further. Extended products have been extensively studied in a 3D space and main results in English language can be read here:
A CONSEQUENCE: GRAVITO-MAGNETISM.
A first important consequence and prediction of this 3D approach is a document concerning the Gravito-magnetism:
VIRTUAL PARTICLES
The Lorentz Einstein law can be understood as a particular split of a particular extended product in 4D: namely, the extended squared product of the proper speed of a particle moving through a changing topology of which the continuity is given by the Levi Civita connection. I analyze and examine this fact attentively. This work is a partial translation of the very important French document etf31v4.pdf. The fundamental idea which is developped here is the following: a quantum approach allows virtual and short life particles. We accept this idea and suppose that such particles also follow trajectories governed by the Lorentz Einstein Law (LEL). The only difference is that, according to the uncertainty principle, such particles can get credits by the Universal Bank of Energy. This kind of analysis is shelding a little bit ligh on the relationship between the masses of such particles and the deformations of the geometry. The latter is resulting from the amount of energy they are carrying for a short while and then giving very quickly back. The whole only is a toy model but also a new vision in physics; I think.
Téléchargement etgb98.pdf [0.335 MB; 5 pages; free; June 2007]
GAMMA MATRICES, INVARIANT CONNECTION AND OSCILLATING METRIC
Perhaps it is more acceptable to present this old investigation today. I did investigate the idea of an invariant connection with an oscillating metric:
Téléchargement etgb102.pdf [0.289 MB; 30 August 2006]
OLD INVESTIGATIONS
You will find here some old documents trying to translate my first and naives intuitions concerning the structure of vacuum. These sections must be analyzed more carefuly but they give the essence of my thougths. Namely, we should consider vacuum like a super-fluid at 3°K behaving relatively to the photons like a crystal would do for electrons at this low temperature. One section demonstrates that if gravitation acts on a tube of this super-fluid like it does on a material string with cross-section zero, then one recovers one solution of Einstein's fields equations for vacuum; thus naturally establishing a relationship between the classical string theory and the generalized theory of relativity.
Téléchargement Foundations.pdf
This document will appear very esoteric to you but it tries to develop a new method of calculation based on the extended products and it yields a path to recover the terms of second order in the formulation of EM forces (Yang-Mills terms).
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