Table Of ContentThe Mathematical Principles
of Scale Relativity Physics
The Concept of Interpretation
Nicolae Mazilu
Maricel Agop
Ioan Mercheş
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Table of Contents
Chapter 1. Introduction.......................................4
Chapter 2. Madelung Fluid Dynamics ...................... 18
The Madelung Fluid.........................................19
Classical and Quantum Conservation Laws...................25
Hydrodynamics of Free Point Particles: Universality
of the Schr¨odinger Equation...............................29
A Definition of the Interpretation............................32
Chapter 3. De Broglie’s Interpretation of Wave Function.44
The Appropriate Geometry of de Broglie’s Idea .............. 52
Lessons and Mandatory Developments.......................55
Chapter 4. The Planetary Model as a Dynamical
Kepler Problem ........................................ 60
A Newtonian Brief on Density...............................69
The Concept of Confinement.................................71
A Clasic Example of Affine Reference Frame: Maxwell
Stress Tensor..............................................74
Chapter 5. The Light in a Schr¨odinger Apprenticeship....76
A Special Contribution of Helmholtz.........................79
Enters Erwin Schr¨odinger....................................83
Chapter 6. The Wave Theory of Geometric Phase.........86
Enters Sir Michael Berry.....................................86
A Kepler Motion Analysis: the Geometrical Condition
of Yang-Mills Fields.......................................91
The Berry Moment of Human Knowledge....................96
1
A Classical Implementation of the Idea of Interpretation .... 101
A Characterization of the Hertz’s Material Point............105
The General Meaning of Berry’s Curvature ................. 107
Chapter 7. The Physical Point of View in the
Theory of Surfaces....................................110
A Few Mathematical Prerequisites..........................110
The Differential Theory of Surfaces.........................114
Rainich’s Description of Surrounding Space.................118
A Physical Parametrization of Surface......................121
The Three-Dimensional Space of Accelerations..............124
Force at an Outward Distance..............................128
Chapter 8. Nonconstant Curvature ........................ 132
The Infinitesimal Deformation..............................134
Summing up the Differential Geometry of Curvature
Parameters...............................................138
A Definition of Surface Tension.............................139
The Statistics of Fluxes on a Material Point ................ 142
The Stress by a Statistic....................................145
The Tensions: Conclusions and Outlook .................... 147
Chapter 9. The Nonstationary Description of Matter....150
The Louis de Broglie Moment .............................. 150
Airy Moment of Berry and Balazs .......................... 154
Cosmological Moment of Berry and Klein...................161
Chapter 10. The Idea of Continuity in Fluid Dynamics..166
The Mass Transport in a Volume Element .................. 166
The Transport Theorem in Finite Volume...................170
Some Classic Physical Examples............................173
The Hamiltonian Transport in Finite Volume...............176
Transcendence between Volume Element and a Control
Volume .................................................. 178
Chapter 11. A Hertz-type Labelling in a Madelung Fluid184
Torsion Induced by Space Variations of Density.............196
The Reference Frame and the Torsion.......................198
The Torsion and the Waves.................................201
Chapter 12. Theory of Nikolai Alexandrovich Chernikov 208
Enters Chernikov...........................................209
Chernikov’s Theory in the Three-Dimensional Case ......... 216
2
Conclusions: Concept of Interpretation and Necessary Fur-
ther Elaborations.............................................221
References .................................................... 231
Subject index.................................................249
3
Chapter 1. Introduction
Among the newest theories of physics, the Laurent Nottale’s scale the-
ory of relativity deserves, in our opinion, a special attention. The scale
relativity theory (SRT in what follows) really means business, and big
business at that, and we are set here on demonstrating this fact: SRT
targets in fact the very foundations of our positive knowledge. The
proof will be effectively done by showing that SRT follows a line of
essential achievements of the physical knowledge of the world, and fol-
lows it properly. As a matter of fact the bottom line of our conclusion
here is that, once the principle of scale invariance is adopted, there is
no other way to follow but the right way, which is the line of thought
marked by those essential achievements of knowledge. All of the works
to date of Laurent Nottale, regarding the problems raised by scale rel-
ativity testify of a well guided thinking, and such a guidance cannot
come but from an inherent fundamental principle of knowledge. If
there is an ambition from our part here, that would therefore be none
otherthantomakethisprincipleasobviousaspossible, maybeevenby
giving it an explicit verbalization. In doing this, we make use both of
common and own results upon the fractal theory of space, expounded
though along a special line indicated by Laurent Nottale himself, in
an evaluation of thirty years of development of the theory. We quote
the final words from a relatively recent book of Nottale:
Giving up the differentiability hypothesis, i.e. general-
izing the geometric description to general continuous mani-
folds, differentiable or not, involves an extremely large num-
ber of new possible structures to be investigated and de-
scribed. In view of the immensity of the task, we have chosen
toproceedbysteps, using presently-known physics as a guide.
Such an approach is rendered possible by the result accord-
ing to which the small scale structures, which manifest the
nondifferentiability, are smoothed out beyond some relative
transitions toward the large scales. One therefore recovers
4
the standard classical differentiable theory as a large scale
approximation of this generalized approach. But one also
obtains a new geometric theory, which allows one to under-
standquantummechanicsasamanifestationofanunderlying
nondifferentiable and fractal geometry and finally to suggest
generalizations of it and new domains of application for these
generalizations.
Nowthedifficultythatalsomakestheirinterestwiththe-
ories of relativity is that they are meta-theories rather than
theories of some particular systems. Hence, after the con-
struction of special relativity of motion at the beginning of
the 20 th century, the whole of physics needed to be rendered
relativistic (from the viewpoint of motion), a task that is not
yet fully achieved.
The same is true regarding the program of construct-
ing a fully scale- relativistic science. Whatever the already-
obtained successes, the task remains huge, in particular when
onerealizesthatit is no longer only physics that is concerned,
but also many other sciences. Its ability to go beyond the
frontiers between sciences may be one of the main interests
of the scale relativity theory, opening the hope of a refounda-
tion on mathematical principles and on predictive differential
equations of a philosophy of nature in which physics would
no longer be separated from other sciences. [(Nottale, 2011),
p. 712; our Italics]
The Italics in this excerpt roughly mark our points of intervention with the
present work, ‘using presently-known... SRT as a guide’. In broad strokes, we
aim here to clarify the idea of “general continuous manifolds” and of the general
“transition between scales”. We also construct a “new geometric theory”, with
the task of “understanding the quantum mechanics”, with a slight change in
emphasis: the quantum mechanics in its wave mechanical form.
One of the ideas that occurred to us, regarding the fact that theories of
relativity are indeed meta-theories rather than theories of some particular sys-
tems, is that these theories should in fact not be axiomatically forced upon such
systems. It is sufficient that the description of a particular one truly significant
for the whole our knowledge be accomplished properly in order to reveal the
theory of relativity in it, in all its fundamental features. This shows that the
relativity is indeed a meta-theory, by the manner in which it acts as such a
meta-theory. Following the usual concept of meta-theory is perhaps the reason
why that task of rendering the physics relativistic mentioned by Nottale “is
not yet fully achieved”. The physical system we have in mind as significant,
is significant for the whole modern knowledge indeed: the classical planetary
5
