Summary of the article ''Vacuum Energy Levels''
The application of the fourth steps to verify a theory is necessary to get a consistent theory. Those steps are as the following:
1st step: the respect of the principle of least action: this principle is general and for every physical system its action should be minimum.
2nd step: the respect of the principle of locality: this principle signifies that a physical phenomenon which happen in a region of space & time affect directly only its nearest environment.
3rd step: the respect of the principle of relativity: the equation of motion should be invariant in inertial referential.
4th step: the respect of gauge invariance: when there is a symmetry, there is something which is conserved. A symmetry is a change in the system parameters which doesn't affect the equation of motion.
By applying those steps for a Newton more generalised equation of motion which is the second derivative by time of the product of the inertia and the position of a system, we had concluded that:
1-The field of gravitation and the electromagnetic field can be unified in one field and treated as it is in microscopic scale or great scale;
2-Vacuum had a density non equal to zero and a current equal to zero;
3-The notion of Dark energy & Dark matter are attributed to the action of vacuum in different scales;
4-It is very easy to get the second law of Newton (gravitation law) and the law of Coulomb for electrostatics.
5-There is an electric charge which is an universal constant;
6-The vacuum had a mechanical impedance which is an universal constant;
7-The second pair of Maxwell equations remains inhomogeneous and it call infinitely for new theories to determine new universal constants, to do new experiences & to rise new technologies. Science will never have an end.
The same conclusions can be obtained by doing meshing of space-time as combining the gravitational constant, the speed of light and the Planck constant to get an absolute system of unities called Planck system. Since we had doing this meshing, extension in space-time signify that there is energy: vacuum in its conventional definition doesn't exist. Vacuum becomes like a fluid with a density not equal to zero and a current of this fluid equal to zero. This fluid is like a ''foam'' of space-time which act on the motion of corpuscles and waves. For corpuscles the action of vacuum is in the first order a friction force due to the viscosity of the fluid and its action on waves is absorption so there is dispersion of waves. For the duality of wave-corpuscle correspond another duality of the space-time which viscosity-dispersion duality. Planck constant translate a third duality which is unity-multiplicity duality. Unity signify that a corpuscle is considered like one subject moving with a certain speed and at the same time this subject is a multitude of plane waves which are reinforced in a limit region of space-time and destruct each either away. The group speed of the packet of waves is equal to the speed of the corpuscle with some conditions of that the incertitude of the position of the corpuscle multiplied by the incertitude of its momentum is superior or equal to a certain constant not equal to zero, and the incertitude of the energy of the corpuscle multiplied by the incertitude of time to measure it, is superior or equal to the same constant not equal to zero: this is which we call the principle of incertitude.
Introducing the notion of universal friction, we can associate to a corpuscle an inertial time and an inertial length. Considering the Hamilton equations of motion, we deduce that a corpuscle is like that it is under a friction force along its time component and had a speed equal to light speed along the same time component. We consider that the corpuscle is moving in fourth dimensions which are the three dimensions of space and the dimension of time. Working in a fourth dimensions with a pseudo algebra of space-time which is Minkowski algebra we conclude that the motion of a massive corpuscle can't be in a straight line: the action of vacuum curve the motion of the corpuscle. Only light can move in a straight line. This result can be deduced also from the Restraint Relativity by applying the principle of least action: the minimum of the action is not in the motion in a straight line but in a curved line, we can't deny the action of vacuum.
By determining the motion of a charged corpuscle in an electromagnetic field we had do similitude with the motion of non-charged corpuscle by doing equivalence between the position of the corpuscle and the vector-potential of an electromagnetic field introducing conversions factors. The result is that we can apply the principle of least action on a non-charged corpuscle moving in a viscous medium: the action of the corpuscle is determined by going back with equations of the determining action of a charged corpuscle. The equation of motion of a non-charged corpuscle by using forces and accelerations is valid only locally. Many conclusions had been obtained from this equivalence between the position of corpuscle and the vector-potential:
1-We deduce the second law of gravitation of Newton;
2-We deduce the law of Coulomb for electrostatic interaction
3-We explain what does it mean Dark energy and Dark matter: it is only a problem of scale invariance gauge.
4-We define new system of unities which is the transition between microscopic scale and great scale: there is many unities defined as Maxwell-thing (Maxwell charge, Maxwell mass, Maxwell force...etc.).
5-We show that those considerations can be obtained by putting the Newton equation of motion in a more generalised form where the inertia of the corpuscle is a function of time: this equation is equal the twice derivative by time of the inertia multiplied by the position of the corpuscle.
6-The same conclusions can be obtained referring to quantum mechanics (Klein-Gordon equation or the harmonic oscillator)
After a certain mathematical definitions of algebra tensors, we had determined the transformations of the electromagnetic field between inertial referential and also the same thing for the vector-potentials.
Considering the invariants of the electromagnetic field we had determine the second pair of Maxwell equations with sources as a triumph of the mathematics.
With some considerations in thermodynamics and cosmology we had fix an interval for the mechanical impedance of vacuum. We had also show that vacuum can had an infinite levels of energy and also we can found negative masses in nature.
Note that also for Planck constant we can determine an interval for it before doing any experiment.
As the conversion coefficients between potentials and positions can be a product of two coefficients the second pair of Maxwell equations remains inhomogeneous and this call for new theories to determine new universals constants which appeal for new experiments to do and new technologies to rise: science will never have an end and the humanity will enter for the first time to the City of Science.
Of course which had been done can be generalised for a Riemann space-time and the reader is invited to do this labour.