Un site pour les amateurs de physique mathématique
© Thierry PERIAT
As I explained in the first chapter of the booklet "J.C. Maxwell's Void", there are forces in empty regions located far from sources (masses, electric charges) as soon as the laws of electromagnetism apply there; see the previous page.
The demonstration takes place in a geometric and physical context entirely faithful to what the contemporaries of this Scottish physicist could have imagined.
The demonstration technique uses (i) classical vector products, (ii) their projection into the dual space of the vector space E(3, R) referred to a direct orthonormal basis of dimension three, and (iii) the trivial decomposition of these projections.
This protocol can therefore potentially be generalized at several levels: (i) by deforming the vector products, (ii) by projecting the deformed vector products into the dual space in a more sophisticated way, and (iii) by decomposing these projections in non-trivial ways.
Certainly, only a few people do know the existence of the so-called “synergetic theory (1971)” (see, e.g. [01]) even if they are, like me, French citizen. This is entirely because, despite of some polemic at the time of its publication and later [02], that forgotten theory has been disproved by a rigorous experiment [03].
At this time (1974-1976), I was a young man studying mathematics and physics and this gave me the opportunity to follow the whole story: the enthusiasm, the disputes, and finally the disaster.
Although that episode has been chronologically in coincidence (but without any relation) with a breakdown in my personal life, that sad story did not destroy my passion for mathematical physics and, a little bit later, following a rationalistic and scientific attitude, I got an exemplar of the demonstration.
The lecture has achieved to convince myself of some inacceptable approximations. But, to be honest, I have been very impressed by two ideas: the first one (a) was the search for unity in physics and the second one (b) was a sort of non-said claim; in extenso, the author seemed to have an obsession (there was in fact no explanation for that intuition in the text); namely: it should be possible to recover the equivalence “mass-energy” just in starting with the J. C. Maxwell’s laws for electromagnetism [04].
For relatively obscure reasons rooted in my adolescence, I decided to rework the synergetic theory and to confront it with my own visions. The battle has yielded the demonstration exposed in the booklet.
It doesn’t exactly realize the dream contained in (b) but it proves that, in introducing a very modest mathematical tool in a so-called “Maxwell’s vacuum” where Einstein’s mass-energy equivalence holds true a priori, neutral energetic flows exist.
That tool is nothing but the cross product. I shall deform and decompose it in a way which will be described in detail on this website. This approach will prove that this modest tool was only the visible top of a gigantic iceberg.
Modern readers or actors involved in the initial theory may be shocked by that introduction and decide to leave the lecture at this stage. I insist on the fact that I am only an amateur and that my purpose is not to interfere with the professional research. I ask for your patience and hope that you give me a chance. If the ideas exposed here can be fruitful and help some teams, I shall be the happiest man in the world. Feel free to send me your feedback too, if it is allowed for you to do that.
[01] Vallée, R. L. : La théorie synergétique, exposé d’ensemble présenté à la mutualité le 04 février 1976.
[02] Wikipedia contributors, "Synergetic theory," Wikipedia, The Free Encyclopedia, [This page was last edited on 17 August 2026, at 09:04 (UTC)]; extern link, online, accessed on the 27th of September 2026.
[03] Jean-Marc Lévy-Leblond, « La « théorie synergétique » de monsieur Vallée », La Recherche, vol. 7, no 662, juillet-août 1976, p. 661-662.
[04] A Dynamical Theory of the Electromagnetic Field: J. Clerk Maxwell; Philosophical Transactions of the Royal Society of London, 1865, 155: 459-512.
You may also want discover my work on vixra.
As its name suggests it, a deformed tensor product is a classical tensor product which is deformed by the intervening of some cube; concretely if E(D, K) is a vector space with canonical basis Ω = (e0, e1, ..., eχ, ..., eD - 1), then:
A ∈ ⊞(D, K) : ⊗ → ⊗A :
(a, b) ∈ E2(D, K)
↓
⊗A(a, b)
=
∑χ {∑α ∑β Aχαβ. aα. bβ}.eχ ∈ E(D, K)
As its name may suggests it, an alternated and deformed tensor product is such that:
A ∈ ⊞(D, K) : ⊗A → ∧A :
(a, b) ∈ E2(D, K)
↓
∧A(a, b)
=
⊗A(a, b) - ⊗A(b, a)
=
∑χ {∑α ∑β Aχαβ. (aα. bβ - bα. aβ}}.eχ ∈ E(D, K)
The definition of an alternated and deformed tensor product is obviously inspired by the one of the wedge product, but there is a fundamental difference between both definitions; precisely: the alternated and deformed tensor product is an intern operation acting on elements in E(D, K) of which the result is again an element in E(D, K). This is not the case for the wedge product.
One can easily be stated that:
∀ a, b ∈ E(D, K):
∧A(a, a) = 0
∧A(a, b) + ∧A(b, a) = 0
∧A(a, b)
=
∑χ {∑α < β ∑β (Aχαβ - Aχβα).(aα. bβ - bα. aβ}}.eχ ∈ E(D, K)
Per definition in this theory, a deformed cross product is the half of an alternated and deformed tensor product when the cube is antisymmetric:
∀ a, b ∈ E(D, K):
A ∈ ⊞-(D, K)
[a, b][A] = (1/2). ∧A(a, b)
1) In opposition with the classical construction, the definition of a deformed cross product remains true for any D ≥ 2.
2) The deformed cross product inherits the basic properties of the alternated and deformed product on which it is built.
3) Its generic mathematical definition is:
[a, b][A]
=
∑χ {∑α < β ∑β Aχαβ.(aα. bβ - bα. aβ}}.eχ ∈ E(D, K)
4) When D = 3, it can be demonstrated that a deformed cross product really is a deformation of some classical cross product.
5) Neither the alternated and deformed tensor product, nor the deformed cross product is systematically a deformed Lie product. This is only the case when supplementary conditions are added. This affirmation is developped on the next page.
© Thierry PERIAT
This page has been updated on the 18th of October 2026