Crédit: European Southern Observatory (ESO) – 10 April 2019.

Nota bene: The purpose of presenting this video is to illustrate a recent event and the concept of invisibility about which I am speaking in the text below. Of course, it doesn’t mean that the ESO organization endorses in any way the text below. Thierry PERIAT

Back to the page: “Physics”.

 

The quest for the interpretation of the fundamental equation in the theory of deformed Lie products

What has been reached

The first steps of that exploration have been made and can be read on this website. From a mathematical point of view, the theory works perfectly in a three-dimensional E(3, C) space; inclusively at the Euclidean limit… if one accepts to work with isotropic vectors and their associated two complex components spinors.

 

Complex numbers in physics

In physics, complex numbers are no aliens and, today, nobody can imagine be working with electromagnetic fields without these numbers. It is a mystery of the nature, but it is so. Everything is going “as if” our universe would have an invisible complex part catching a part of the reality. Strange: “Isn’t it?”

 

There is a branch in modern physics developing this fact and proposing a renewed vision of the theory of relativity [e.g.: bi-metric gravity; extern link Wikipedia-GB = +]. I must say (I may be wrong) that I am not convinced by these approaches.

 

The concept of invisibility as alternative for the probabilistic interpretation?

Nevertheless, the concept of invisibility is a fascinating one, and not only because of its military applications. It is enough to focus the attention on cosmology+ to understand that that concept may eventually be involved in some explanation justifying the dark energy+.

 

There is something there, that we don’t directly see, but that is existing. Or do we have made a collective mistake in the construction of fundamental physics? Let us dream for a while and try to formulate rationalistic explanations.

 

Quantum dynamics denies the concept of point particle and privileges the statistic interpretation, introducing the one of “probability of presence”; well. But what would happen if, instead of that interpretation, we would promote the idea that a point particle can oscillate between two different states: (i) a visible one and (ii) an invisible one (in both cases: for our instruments)? Could not we substitute the probability of presence with a ratio of visibility? The existence of any physical phenomenon would be related to an optic problem.

 

The first contra-argument would certainly be that the frontier of sensibility of our instruments are century-dependant; hence: the frontier between visibility and invisibility would also not be the same during Galileo’s life and today, especially after the first picture of a black hole (Galaxy M 87).

 

But, anyway, let us think that -whatever the future technological progresses will be- there is an effective natural limit impeaching the perception of signals, and not destroying the signal. Remark that that though is very close to the one concerning the event horizon surrounding a black hole. Note also that, if what happens in the micro-cosmos is like what is existing in the cosmos, point particles would be an ideal incarnation for these crazy ideas! Particles as event horizons? Who knows? There are modern researches of which the skeleton is not so far from this suggestion.

 

The theory of spinors to the rescue for the understanding of decompositions?

Coming back to my modest theoretical toy-model, I now concentrate my (visible) energy on a search for an interpretation concerning the concept of decomposition itself. Once again, the opinion of the mathematician is clear: this is an extrapolation of the division. But what about the opinion of the physicist? Why should the manner to calculate an angular momentum be deformed by the geometry? Is it effectively what happens? And if yes, why must this angular momentum decompose non trivially as soon as the geometry is no more Euclidean?

 

I must say that I have no pertinent answers to these questions. But, inspired by some general considerations inherited from the theory of spinors, I can imagine further justifications for the mathematical existence of such decomposition.

 

In E*(3, C): |[projectile, target][Deforming matrix] > = [Main part]. |target> + |residual part>

 

Let suppose that that fundamental equation can be transposed in a one-to-one way into a four-dimensional world; this would be yielding:

 

In E*(4, C): |[projectile, target][Deforming matrix] > = [Main part]. |target> + |residual part>

 

As a matter of observation, the target appears on both sides of the equivalence. This fact suggests that that relation should be (i) interpreted independently on the target and (ii) understood as the signature of the existence of the following application acting in M(4, C) x E(4, C):

 

Y: ([A], projectile) ® Y([A], projectile) = ([P], residual part)

 

The decomposition is the result of that application forcing the deforming matrix [A] to act on the projectile.

 

At this stage, let recall that, in that specific four-dimensional world, a pair of any orthogonal 4-vectors is equivalent to a pair of non-orthogonal isotropic 4-vectors.

 

This fact is offering several possible interpretations for the relation I am studying here. One of them is to suppose that (i) the main part [P] is representing any 4-vector, (ii) that any 4-vector is also orthogonal to the residual part, (iii) the deforming matrix [A] is representing an isotropic 4-vector and (iv) the projectile too.

 

Noon, as we know, a 4 x 4 x 4 antisymmetric cube A is not equivalent to a 4 x 4 [A] matrix. But fortunately, such antisymmetric cube can be anti-reduced. It then directly becomes a 4-vector. The realization of point (iii) of that first proposition would impose a supplementary constraint on it: it must be an isotropic 4-vector.

 

© Thierry PERIAT, 14 April 2019.