General relativity and quantum mechanics
The scale covariance and (or) complementarity ?
"Je vous ai dit plus d'une fois que je suis un partisan acharné non pas des équations différentielles, mais bien du principe de relativité générale (i.e. du principe de covariance),
dont la force heuristique nous est indispensable. Or en dépit de bien des recherches, je n'ai pas réussi à satisfaire le principe de relativité générale autrement que grâce à des
équations différentielles; peut-être quelqu'un découvrira-t-il une autre possibilité, s'il cherche avec assez de persévérance."
A. Einstein in the conclusion of its letter to Pauli of May 2, 1948.
Implicit aspects in the axiomatization of
general relativity:
Here are three:
In the axioms of general relativity,
nothing is known as on the behavior of metric by change of
system of units.
In fact it is implicitly allowed an invariance
by scaling, under pretext of homogeneity of
dimensions. Under this same pretext, an invariance
not being to confuse with a covariance, let us pose:
There exists a scale covariance.
A second axiom is significant, it is that which stipulates that
the free falling bodies follow trajectories which are the geodesic ones.
The usual translation of the axiom of the trajectories followed
by the free falling bodies consists in saying that they are solutions
of the geodesic equations. It
is implicitly supposed that the geodesics are twice
differentiable.
A third axiom is that
translating the fact that in any point one can wipe out the gravitation and
which states the existence of a locally inertial reference frame
(the gravitation is arased in such a reference frame), in other words
tangent space at this point is the Minkowski space
of the special relativity. One thus admits locally the duality
(space, time)-(momentum, energy), via the Fourier transform
which uses explicitly a constant usually
identified with the Planck's constant. This third axiom
thus supposes the Heisenberg inequalities (called improperly
" uncertainty relations ").
If one wants to make compatible these two last axioms, while
keeping the physics of laboratory (i.e. axiom 3), one is obliged
to weaken the usual translation of the second axiom (by
weakening the implicit assumption of
differentiability).
And taking
account of the recent result of Abbott and Wise (1981), let us pose:
The trajectories of free falling bodies
are fractals, half differentiable solutions of the geodesic
equations.
It is the base of the scale relativity, cf.
Laurent Nottale.
See also (in French) théorie des quanta .
Within this framework the gravitation is not any more to quantify, IT IS " QUANTUM "
scale relativity and quantum theory can
go well together.
There is many work to do!
