# Christoffel meets Heisenberg

Christoffel meets Heisenberg.

1. Historical context of the discussion.

The search for unity inside theoretical physics, especially between the theory of relativity (A. Einstein) and the quantic world is an approximately hundred years old fundamental quest. It has occupied many brilliant brains and it does yet occupy them today.

The first formulation of Einstein’ theory is mainly rooted in a world-known experiment of interferometry (due to Morley and Michelson in 1887) and, in its analysis. The speed of light does not depend on the observer, if this one is working at the center of an inertial frame (definition: the sum of all forces acting on him/her vanishes).

This interpretation is the starting point for the mathematical construction of Lorentz transformations linking the observations which can be made by two separate observers (each of them being at the center of an inertial frame) moving with a relative constant speed.

2. A new approach.

Although the affirmation may sound a little bit pretentious, my toy-theory proposes a new path connecting the two pillars of modern physics. Let me describe the main idea.

In 1887, the quantum theories were not yet really born, even if some mathematical premises were already present since a long time [Abel]. For example, the Heisenberg’ uncertainty principle had not been proposed since its publication only occurred in 1926.

In exposing his uncertainty principle, W. Heisenberg has pointed out the fact that positions and speeds form two separate and independent sets of observables.

He also suggested that infinitesimal variations of energies and lapses of times were following a similar logic. Unfortunately, there is up to now no operator representing the time and, at least at a technical level, this absence is blocking the progression of any development for the second pair.

Whatever the interpretation of that principle is, a fact is that it introduces a finite limit for the relative variations of these pairs. The situations realizing the minimum value of a product between both elements of a given pair are labeled as “quantum limit”.

And this is the information on which I build my theoretical proposition. Following a well-known and accepted principle affirming that the laws of physics must be the same everywhere, I look for technical circumstances allowing to go further on that road. Concretely, that quantum limit must be the same for two observers, wherever they stay in the world.

I consider the simplest transposition of Heisenberg’ suggested interdependence between the elements of the pair (delta energy, delta lapse of time) in a simple one-dimensional world. And I ask me where, in the arsenal of formulas and laws of physics, I can find a plausible manner to introduce these ingredients softly.

Analyzing the physical units appearing in a product dE.dt, I realize that it is possible to involve the Lorentz-Einstein force density via a technical tool that I have called the “associated scalar” within my extrinsic method.

That special scalar is a polynomial of degree two. It can be reduced to a simple quadratic form when the ad hoc constraints are imposed. In certain conditions, these constraints are compatible with the preservation of the elements of lengths appearing within Einstein’s theory. Furthermore, they allow the use of E. B. Christoffel’s historical work on the preservation of quadratic forms.

This logical path is yielding the conditions allowing the preservation of the quantum limit and a (4-4) matrix playing in some extent a similar role than the one which is devoted to the metric tensor. That matrix contains three essential ingredients offering a new insight on physics.

3. Conclusion.

This rough description is the central skeleton of my toy-approach. It must certainly be fine-tuned and generalized. If it is the right manner to connect the two main pillars of modern theoretical physics, this is a scoop and an essential stone for future progressions.

I could parody my own approach in saying with an ounce of irony and sadness that, if Heisenberg would have open his eyes on the horrific reality he was supporting through his administrative position and if he would have accepted to confront his scientific thoughts with the calculations of E. B. Christoffel’s and Einstein’s ideas, he would certainly have suggested a similar logical path than mine. But for all those who know the dramatic history of the period (1933-1945), it is also evident that this confrontation was highly not probable.

Therefore, I apologize for my impertinence in presenting the fruits of that politically indelicate essay.

© Thierry PERIAT, 12 August 2020.

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event Date de dernière mise à jour : 09/09/2020

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