ARTICLE IN PRESS
Progress in Biophysics and Molecular Biology 97 (2008) 115–157
www.elsevier.com/locate/pbiomolbio
Review
Scale relativity theory and integrative systems biology:
2 Macroscopic quantum-type mechanics
Laurent Nottalea,b, Charles Auffrayb,Ã
aLUTH, CNRS, Observatoire de Paris and Paris Diderot University—Paris VII, 5 Place Jules Janssen, 92190 Meudon, France
bFunctional Genomics and Systems Biology for Health, UMR 7091-LGN, CNRS/Pierre & Marie Curie University—Paris VI,
7 rue Guy Moquet, BP 8, 94801 Villejuif Cedex, France
Available online 2 October 2007
Abstract
In these two companion papers, we provide an overview and a brief history of the multiple roots, current developments and
recent advances of integrative systems biology and identify multiscale integration as its grand challenge. Then we introduce the
fundamental principles and the successive steps that have been followed in the construction of the scale relativity theory, which
aims at describing the effects of a non-differentiable and fractal (i.e., explicitly scale dependent) geometry of space–time. The first
paper of this series was devoted, in this new framework, to the construction from first principles of scale laws of increasing
complexity, and to the discussion of some tentative applications of these laws to biological systems. In this second review and
perspective paper, we describe the effects induced by the internal fractal structures of trajectories on motion in standard space.
Their main consequence is the transformation of classical dynamics into a generalized, quantum-like self-organized dynamics.
A Schro¨dinger-type equation is derived as an integral of the geodesic equation in a fractal space. We then indicate how gauge fields
can be constructed from a geometric re-interpretation of gauge transformations as scale transformations in fractal space–time.
Finally, we introduce a new tentative development of the theory, in which quantum laws would hold also in scale space,
introducing complexergy as a measure of organizational complexity. Initial possible applications of this extended framework to the
processes of morphogenesis and the emergence of prokaryotic and eukaryotic cellular structures are discussed. Having founded
elements of the evolutionary, developmental, biochemical and cellular theories on the first principles of scale relativity theory, we
introduce proposals for the construction of an integrative theory of life and for the design and implementation of novel
macroscopic quantum-type experiments and devices, and discuss their potential applications for the analysis, engineering and
management of physical and biological systems and properties, and the consequences for the organization of transdisciplinary
research and the scientific curriculum in the context of the SYSTEMOSCOPE Consortium research and development agenda.
r 2007 Elsevier Ltd. All rights reserved.
Keywords: Scale relativity; Systems biology; Scale covariance; Macroscopic quantum mechanics; Self-organization
Contents
1.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
2.
New macroscopic quantum-type mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
ÃCorresponding author.
E-mail addresses: laurent.nottale@obspm.fr (L. Nottale), charles.auffray@vjf.cnrs.fr (C. Auffray).
0079-6107/$ - see front matter r 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.pbiomolbio.2007.09.001

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2.1.
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
2.2.
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
2.2.1.
Fractality of the paths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
2.2.2.
Infinite number of geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
2.2.3.
Discrete symmetry breaking from irreversibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
2.2.4.
Covariant total derivative operator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
2.3.
Covariant mechanics in scale relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
2.3.1.
Geodesic form of the motion equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
2.3.2.
Emergence of the quantum tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
2.3.3.
Schro¨dinger form of the motion equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
2.3.4.
Fundamental wavelengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
2.3.5.
Fluid mechanics form of the motion equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
2.3.6.
Relation to diffusion processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
2.3.7.
Generalization of quantum laws. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
2.4.
Application to biological processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
2.4.1.
Conditions of application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
2.4.2.
Morphogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
2.4.3.
Formation, duplication and bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
2.4.4.
Prediction of biological sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
2.4.5.
Prospects: other quantum-type properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
2.5.
Gauge field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
3.
Quantum-type mechanics in scale space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
3.1.
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
3.2.
Schro¨dinger equation in scale space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
3.3.
Complexergy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
3.3.1.
Third quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
3.3.2.
A new conservative quantity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
3.4.
Application to biological systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
4.
Discussion, conclusions and perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
4.1.
Validated predictions of scale relativity theory in astrophysics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
4.2.
Founding elements of an integrative theory of life on first principles . . . . . . . . . . . . . . . . . . . . . . . . . 145
4.3.
Defining biological space–time, biological fields and charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
4.4.
Extending the classical framework for multiscale integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
4.5.
Design of macroscopic quantum-type experiments and devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
4.6.
Applications in integrative systems biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
4.7.
Consequences for transdisciplinary research, development and training in integrative systems biology . 149
Aknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
Appendix A Proof of fundamental remarkable identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
Appendix B From Schro¨dinger equation to Euler and continuity equations . . . . . . . . . . . . . . . . . . . . . . . . 151
Appendix C Schro¨dinger process as anti-diffusion process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
C.1. Fluid representation of diffusion processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
C.2. Continuity equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
C.3. Euler equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
C.3.1.
Case of vanishing mean velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
C.3.2.
Basic remarkable identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
C.3.3.
Diffusion potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
1. Introduction
Caminante, no hay camino, se hace camino al andar—Antonio Machado
In the first paper, we provided a first indication that scale invariant laws and generalized scale laws with
variable fractal dimensions have the potential to found elements of the evolutionary, developmental and
cellular biology theories on the common first principles of scale relativity theory. In this second paper we
explore the effects of fractal structures on the laws of dynamics in standard space, then in scale space, before

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117
discussing the possible consequences of this extended framework for multiscale integration in systems biology
and the development of novel experiments and devices with macroscopic quantum-type behaviours.
2. New macroscopic quantum-type mechanics
2.1. Motivation
Let us now consider an essential part of the theory of scale relativity, namely, the description of the effects in
standard space–time that are induced by the internal fractal structures of its geodesics. The companion paper
introduced pure scale laws describing the dependence on scale of fractal paths at a given point of space–time.
The next step consists of considering a displacement of such a structured point, i.e., the consequences on
motion of the non-differentiability. As we shall see, these consequences are radical since they amount to a
transformation of Newton’s equation of dynamics into a generalized Schro¨dinger equation.
Note that, in the perspective of potential applications to biological systems, we consider here only the non-
relativistic case (i.e., velocities small with respect to the velocity of light c). One can show (Nottale, 1993) that
this case corresponds to a fractal space, while time keeps its regular behaviour.
As a first step, we shall mainly consider only the simplest case of fractal internal structures, namely, the self-
similar ones that are characterized by a constant fractal dimension, and more precisely fractal dimension
DF ¼ 2 that plays a critical role in the theory (Nottale, 1996a). As we have seen in the companion paper, this
behaviour can be derived from a simple, scale-inertial differential equation of first order. We shall see that the
laws of mechanics constructed from such internal structures of the geodesics of a fractal space become a
quantum-type mechanics. Therefore the various generalizations of internal scale laws that have been
considered in the companion paper naturally lead to generalized quantum laws, as we shall briefly see in
Section 2.3.7.
Actually, the discovery that typical quantum mechanical paths (those that contribute mainly to the path
integral) are non-differentiable and of fractal dimension 2 is due to Feynman (Feynman and Hibbs, 1965),
even though the word ‘‘fractal’’ was coined by Mandelbrot only in 1975. But the various properties of
quantum paths described by Feynman in his approach (which is not a return to determinism, since his paths
are in infinite number) correspond very closely to the later definition of fractals (Abbott and Wise, 1981; Ord,
1983; Nottale and Schneider, 1984). Now, Feynman derives the fractal and non-differentiable properties of
quantum paths from quantum mechanics and its sets of axioms, while the scale relativity approach attempts to
do the reverse, namely, found quantum mechanics on the non-differentiable and fractal geometry of
space–time.
2.2. Method
The method is as follows. We start from a generic description of the displacements in a non-differentiable
and continuous space, which is fractal as a consequence, following the fundamental founding theorem of the
theory (Nottale, 1993, 1996a; Cresson, 2001, 2003). As we shall see, the paths in a fractal space are
characterized by three minimal properties: fractality, infinite number and time irreversibility.
These three conditions are mathematically expressed at the level of the elementary displacements, then their
effect on a physical quantity is described in terms of a ‘‘quantum-covariant’’ derivative. This means that, since
the dynamical effects of a space geometry are internal (instead of being external as in the case of an externally
added force or field), they are included in the differential calculus itself (Einstein, 1916). But in addition to
the general (motion) relativity case, in the scale relativity case we have to deal not only with the effects of the
geometry, but also with those of the non-differentiability (which does not mean that we cannot define
differential elements, but that their ratios, i.e., the derivatives, are sometimes undefined).
Finally, the principle of relativity–equivalence–covariance allows one to write the equation of geodesics as a
2
free form motion equation, which expresses the acceleration b
d X =dt2 ¼ 0 in terms of the new covariant
derivative b
d (see its construction in what follows). This means that one writes that, locally, there is rectilinear
uniform motion, so that all the final complexity comes from the change of reference system itself. The final

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step amounts to make changes of variables (without any change of the number of degrees of freedom) which
transform the classical type of physico-mathematical tool into a quantum-type tool.
2.2.1. Fractality of the paths
Strictly, the non-differentiability of the coordinates means that the standard velocity
dX
X ðt þ dtÞ À X ðtÞ
V ðtÞ ¼
¼ lim
(1)
dt
dt!0
dt
is undefined. Namely, when dt tends to zero, either the ratio dX =dt tends to infinity, or it fluctuates without
reaching any limit.
However, as recalled above, continuity and non-differentiability imply an explicit dependence on scale
(and even a divergence) of the various physical quantities. As a consequence, the coordinate X ðtÞ and the
velocity, V ðtÞ are themselves re-defined as explicitly scale-dependent functions X ðt; dtÞ and V ðt; dtÞ. We can
therefore use again all the arguments developed in the companion paper, and conclude that, in the simplest
case, we expect it to be solution of a first order scale differential equation, i.e.,
(
 
)
T 1À1=DF
V ðt; dtÞ ¼ vðtÞ þ wðt; dtÞ ¼ vðtÞ 1 þ ZðtÞ
.
(2)
dt
Here the position on the curve is now located by using the time t itself as parameter s, while the resolution is a
time resolution. The scale T must be introduced for dimensional reasons, and appears as a constant of
integration. Its presence manifests once again the fact that only scale ratios do have a physical meaning,
not the scales themselves.
This result means that the velocity is now the sum of two independent terms of different orders of
differentiation, since their ratio v=w is, from the standard viewpoint, infinitesimal (see Fig. 1). The v
component is what we have called the ‘‘classical part’’ (or differentiable part) of the velocity (Ce´le´rier and
Nottale, 2004), and w is its ‘‘fractal part’’ (or non-differentiable part). The new component w is an explicitly
scale-dependent fractal fluctuation (see Fig. 4 of companion paper) which is described in terms of a
dimensionless normalized stochastic variable ZðtÞ such that hZi ¼ 0 and hZ2i ¼ 1. As we shall see later on, the
final result is totally independent of the probability distribution of this variable.
Eq. (2) multiplied by dt gives the elementary displacement, dX , of the system as a sum of two infinitesimal
terms of different orders
dX ¼ dx þ dx,
(3)
which are such that
dx ¼ v dt,
(4)
dx ¼ Zð2DÞ1À1=DF dt1=DF ,
(5)
where the parameter D is a reformulation of the previous scale T of Eq. (2).
Only the critical case of fractal dimension DF ¼ 2 will be now considered (see Nottale, 1995, 1996a, b and
Section 2.3.7 for generalization to a different fractal dimension). The fluctuation becomes
pffiffiffiffiffiffiffi
dx ¼ Z 2D dt1=2.
(6)
The fundamental parameter D is bound to play a very important role in what follows. It can be considered to
be defined by the above relation, namely,
1 hdx2i
D ¼
.
(7)
2 dt
It looks like a coefficient of diffusion, but here its meaning is of geometric essence, namely, it manifests the
intrinsic diffusive property of a fractal space, but no external agent or particle is the cause of this ‘‘diffusion’’.
This coefficient intervenes in the determination of the fundamental transition from scale dependence
(fractality) to scale independence (see Fig. 1). But, as we shall see, it may also be identified, modulo

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4
2
ln V
0
-2
-4
-10
-5
0
5
10
ln (dt / T)
Fig. 1. Dependence on the time-scale, lnðdt=T Þ, of the logarithm of the velocity lnðV =V 0Þ on a fractal geodesic, including a classical
(differentiable) part, which is dominant at large scale (it tends to a scale-independent velocity toward the right in the figure), and a fractal
(non-differentiable) fluctuating part, which is dominant at small scales (to the left). The fluctuation has been taken here to be Gaussian.
The fluctuating fractal part dx is of order dt1=2 for fractal dimension 2, so that the velocity diverges toward small scales as dtÀ1=2, which
expresses the non-differentiability of the fractal coordinate.
fundamental constants, to a generalization of the Compton scale, _=mc, that is the fundamental wavelength
which has phenomenologically appeared in quantum mechanics (without having, up to now, been
theoretically understood from first principles).
2.2.2. Infinite number of geodesics
One of the main geometric consequences of the non-differentiability is that there is an infinity of fractal
geodesics relating to any couple of points of a fractal space (Nottale, 1993; Cresson, 2001). This can be easily
understood already at the level of fractal surfaces, which can be described in terms of a fractal distribution of
conic points of positive and negative infinite curvature (see Nottale, 1993, Sections 3.6 and 3.10). As a
consequence, we are led to replace the velocity V ðt; dtÞ on a particular geodesic by the fractal velocity field
V ½xðt; dtÞ; t; dtŠ ¼ v½xðtÞ; tŠ þ w½xðt; dtÞ; t; dtŠ of the whole infinite ensemble of geodesics. This representation is
similar to that of fluid mechanics (Landau and Lifchitz, 1959), in which the motion of a fluid is described in
terms of its velocity field vðxðtÞ; tÞ, its density rðxðtÞ; tÞ and possibly its pressure. We shall indeed recover the
fundamental equations of fluid mechanics (Euler and continuity equations), but written in terms of a density
of probability (as defined by the set of geodesics) instead of a density of matter, and with an additional term of
quantum pressure which occurs as a manifestation of the underlying fractal geometry (see below).
2.2.3. Discrete symmetry breaking from irreversibility
A last fundamental consequence of the non-differentiability is the breaking of a discrete symmetry, namely,
of the reflection invariance on the differential element of time (it is said to be discrete since it is not a
continuous symmetry, such as e.g., translation or rotation, but a discontinuous one such as a mirrored
symmetry). It implies a two-valuedness of velocity which can be subsequently shown to be the origin of the
fundamental use of complex numbers in quantum mechanics (Ce´le´rier and Nottale, 2004). This use determines
a large part of the particularities of quantum mechanics with respect to classical mechanics.
The derivative with respect to the time t of a differentiable function f can be written twofold
df
f ðt þ dtÞ À f ðtÞ
f ðtÞ À f ðt À dtÞ
¼ lim
¼ lim
.
(8)
dt
dt!0
dt
dt!0
dt
The two definitions are equivalent in the differentiable case. In the non-differentiable situation, both
definitions fail, since the limits are no longer defined. In the new framework of scale relativity, the physics is
related to the behaviour of the function during the zoom operation on the time resolution dt, identified with
the differential element dt. The non-differentiable function f ðtÞ is replaced by an explicitly scale-dependent

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fractal function f ðt; dtÞ, which therefore is a function of two variables, t (in space–time) and dt (in scale space).
The two functions f 0 and f 0 are therefore defined as explicit functions of the two variables t and dt
þ
À
f ðt þ dt; dtÞ À f ðt; dtÞ
f 0 ðt; dtÞ ¼
,
(9)
þ
dt
f ðt; dtÞ À f ðt À dt; dtÞ
f 0 ðt; dtÞ ¼
.
(10)
À
dt
Here we have assumed that dt40. By taking dt algebraic, these two functions would correspond, respectively,
to the positive and negative parts of a same unique function. One passes from one definition to the other by
the transformation dt2 À dt (differential time reflection invariance), which actually was an implicit discrete
symmetry of differentiable physics. When applied to fractal space coordinates xðt; dtÞ, these definitions yield,
in the non-differentiable domain, two velocity fields instead of one, that are fractal functions of the resolution,
V þ½xðtÞ; t; dtŠ and V À½xðtÞ; t; dtŠ. Each of these fractal velocity field can in turn be decomposed in terms of a
classical part and a fractal part, namely, V þ½xðt; dtÞ; t; dtŠ ¼ vþ½xðtÞ; tŠ þ wþ½xðt; dtÞ; t; dtŠ and V À½xðt; dtÞ;
t; dtŠ ¼ vÀ½xðtÞ; tŠ þ wÀ½xðt; dtÞ; t; dtŠ.
The important fact appearing here is that there is no a priori reason for the two classical parts to be the same.
In several works Ord (Ord and Deakin, 1996; Ord and Galtieri, 2002) also insists on the importance of
introducing entwined paths for understanding quantum mechanics (but without giving a cause for this
fundamental two-valuedness), while Jumarie (2006) supports the scale-relativistic view that the use of
complex-valued variables appears as a direct consequence of the irreversibility of time.
A simple and natural way to account for this doubling consists of using complex numbers a þ ib and the
complex product, according to which ða þ ibÞðc þ idÞ ¼ ðac À bdÞ þ iðad þ bcÞ. This is the origin of the
complex nature of the wave function of quantum mechanics. Actually, the choice of complex numbers to
represent the two-valuedness of the velocity can be proven to be a simplifying and covariant choice (Ce´le´rier
and Nottale, 2004; Nottale, 2008), in the sense of the principle of covariance, according to which the form of
the equations of physics should be conserved under all transformations of coordinates. Indeed, the choice of
the complex product allows one to suppress what would be additional infinite terms in the final equations of
motion.
Another consequence of the combination of the two velocity fields into a single complex velocity field is
that, in terms of this physical tool, one recovers a global reversibility of physical laws, as we are going to see
from the derivation of the Schro¨dinger equation.
2.2.4. Covariant total derivative operator
We are now led to describe the elementary displacements for both processes, dX Æ, as the sum of a classical
part, dxÆ ¼ vÆdt, and of a fractal fluctuation dxÆ, i.e.,
dX þðtÞ ¼ vþdt þ dxþðtÞ,
dX ÀðtÞ ¼ vÀdt þ dxÀðtÞ,
ð11Þ
and similar relations for the other variables. One passes from one process to the other by the transformation
dt2 À dt. More generally we define two classical derivatives, dþ=dt and dÀ=dt, such that
dþ
d
x
À
ðtÞ ¼ v
xðtÞ ¼ v
dt
þ;
dt
À.
(12)
These expressions are also valid for three space variables by considering that x and v represent vectors.
The two derivatives can now be combined to construct a complex derivative operator that allows
recovering local differential time reversibility in terms of the new complex process (Nottale, 1993). We
define it as
b




d
1 dþ
dÀ
i
dþ
dÀ
¼
þ
À
À
.
(13)
dt
2
dt
dt
2
dt
dt

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121
This choice is motivated by the need to recover, at the classical limit (where the two velocities are equal), the
classical real velocity as real part of this complex velocity and a vanishing imaginary part. This is the main tool
of the theory.
Applying this operator to the classical part of the position vector yields a complex velocity
b
d
vþ þ vÀ
vþ À vÀ
V ¼
xðtÞ ¼
À i
.
(14)
dt
2
2
We call V the real part of this complex velocity and U its imaginary part, i.e. V ¼ V À iU, with V ¼
vþ þ vÀ=2 and U ¼ vþ À vÀ=2.
After having defined the covariant derivative, we now need to find its expression. This will be achieved by
explicitly calculating its effect on a given physical quantity.
For this purpose, let us first calculate the derivative of a scalar function f. Since the fractal dimension is 2,
we need to go to second order of expansion (this is reminiscent of Einstein’s argument about Brownian
motion). For one variable it reads
df
qf
qf dX
1 q2f dX 2
¼
þ
þ
.
(15)
dt
qt
qX dt
2 qX 2 dt
Once again the generalization of this writing to three dimensions is straightforward (Nottale, 1993).
Let us now take the stochastic mean of this expression (i.e., we take the mean of the stochastic variable Z
which appears in the definition of the fractal fluctuation dx). By definition, since dX ¼ dx þ dx and hdxi ¼ 0,
we have hdX i ¼ dx, so that the second term is reduced (in three dimensions) to v:rf . (Recall that this
expression denotes the scalar product of the velocity v ¼ ðvx; vy; vzÞ by the gradient rf ¼ ðqf =qx; qf =qy; qf =qzÞ
of the function f, i.e., in decompacted form, v:rf ¼ vx qf =qx þ vy qf =qy þ vz qf =qz).
Now concerning the term dX 2=dt, it is infinitesimal and therefore not taken into account in the standard
differentiable case. But in the non-differentiable case considered here, the mean squared fluctuation is non-
vanishing and of order dt, namely, hdx2i ¼ 2D dt, so that the last term of Eq. (15) amounts in three dimensions
to a Laplacian (defined as D ¼ q2=qx2 þ q2=qy2 þ q2=qz2). We obtain, respectively, for the ðþÞ and ðÀÞ
processes,


dÆf
q
¼
þ v
f .
(16)
dt
qt
Æ:r Æ DD
The last step consists of recombining the two derivatives into the complex covariant derivative. Substituting
Eqs. (16) into Eq. (13), we finally obtain the expression for the covariant time derivative operator (Nottale,
1993)
b
d
q
¼
þ V:r À iDD.
(17)
dt
qt
This is one of the main tools of the theory of scale relativity. Indeed, the passage from standard classical (i.e.,
almost everywhere differentiable) mechanics to the new non-differentiable theory can now be implemented by
replacing the standard time derivative d=dt by the new complex operator b
d=dt (Nottale, 1993).
Note that in this replacement, one should remain aware of the fact that this derivative operator is a linear
combination of first and second order derivatives, in particular when dealing with the Leibniz rule about
derivatives of a product and of composed functions. Now it is possible to build more efficient, fully covariant,
tools (Pissondes, 1999). This can be made by introducing a velocity operator c
V ¼ V À iDr (Nottale, 2004),
in terms of which the first order Leibniz rule still applies, since the covariant derivative now reads
b
d
q
¼
þ c
V:r.
(18)
dt
qt
In other words, this means that b
d=dt plays the role of a ‘‘covariant derivative operator’’, namely, we are
able, by using it, to write the fundamental equations of physics under the same form they had in the
differentiable case.

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2.3. Covariant mechanics in scale relativity
Let us now summarize the main steps by which one may generalize the standard classical mechanics using
this covariance. We consider below only the classical parts of the variables, which are differentiable and
independent of resolutions. But the same final equation can be proved to be valid also for the full velocity field
including its non-differentiable part, after redefinition of the wave functions in terms of fractal functions,
see Nottale (2008).
We define a Lagrange function Lðx; V; tÞ that keeps the usual form but now in terms of the complex
velocity, then a complex action S which is still defined as
Z t2
S ¼
Lðx; V; tÞ dt.
(19)
t1
One finds that generalized Euler–Lagrange equations that keep their standard form (see companion paper) in
terms of the new complex variables can be derived from this action (Ce´le´rier and Nottale, 2004), namely
b
d qL
qL
À
¼ 0.
(20)
dt qV
qx
Since we now consider only the classical parts of the variables (while the effects on them of the fractal parts are
included in the covariant derivative) the basic symmetries of classical physics hold. From the homogeneity of
standard space, one defines a generalized complex momentum given by the same form as in standard
mechanics, namely,
qL
P ¼
.
(21)
qV
If we now consider the action as a function of the upper limit of integration in Eq. (19), the variation of the
action from a trajectory to another nearby trajectory yields a generalization of another well-known relation of
standard mechanics,
P ¼ rS.
(22)
2.3.1. Geodesic form of the motion equations
Let us now consider the special case of Newtonian mechanics, in the general case when the structuring
external scalar field is described by a potential energy F. The Lagrange function of a closed system,
L ¼ 1 mv2 À F, is generalized as Lðx; V; tÞ ¼ 1 mV2 À F. The Euler–Lagrange equations then keep the form
2
2
of Newton’s fundamental equation of dynamics F ¼ m dv=dt, namely, for a force that derives from a
potential,
b
d
ÀrF ¼ m
V,
(23)
dt
which is now written in terms of complex variables and complex operators.
In the case when there is no external field (F ¼ 0), the covariance is explicit, since Eq. (23) takes the free
form of the equation of inertial motion, i.e., of a geodesic equation,
b
d V ¼ 0.
(24)
dt
This is analog to Einstein’s general relativity, where the equivalence principle of gravitation and inertia leads
to a strong covariance principle, expressed by the fact that one can always find a coordinate system in which
the metric is locally Minkowskian. This means that, in this coordinate system, the covariant equation of
motion of a free particle is that of inertial motion Dum=ds ¼ 0 in terms of the general-relativistic covariant
derivative D, four-vector um and proper time differential ds. The expansion of the covariant derivative
subsequently transforms this free-motion equation in a local geodesic equation in a gravitational field.
The covariance induced by scale effects leads to an analogous transformation of the equation of motions,
which, as we show below, become after integration the Schro¨dinger equation, which we can therefore consider

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123
as the integral of a geodesic equation. Note that one also obtains in the motion-relativistic case the
Klein–Gordon equation (Nottale, 1994a, 1996a) and the Dirac equation on spinors (Ce´le´rier and Nottale,
2004), since spinors arise as a consequence of a new two-valuedness from the breaking of the symmetry
dx2 À dx (see also Ce´le´rier and Nottale, 2006 about the Pauli equation).
In the Newtonian case the complex momentum P reads
P ¼ mV,
(25)
so that, from Eq. (22), the complex velocity V appears as a gradient, namely the gradient of the complex
action
V ¼ rS=m.
(26)
2.3.2. Emergence of the quantum tools
Up to now the various concepts and variables used were of a classical type (space, geodesics, velocity fields),
even when they were generalized to the fractal, explicitly scale-dependent case.
We shall now make essential changes of variable, which transform this classical-like tool (that will finally
reveal not to be classical) into quantum tools (but without any hidden parameter or degree of freedom). We
now introduce a complex wave function c which is nothing but another expression for the complex action S
by making the transformation
c ¼ eiS=S0 .
(27)
The factor S0 has the dimension of an action (i.e., an angular momentum) and must be introduced because S
is dimensioned while the phase should be dimensionless. When this formalism is applied to microphysics, S0 is
nothing but the fundamental constant _ of standard quantum mechanics. As a consequence, since
S ¼ ÀiS0 ln c,
(28)
one finds that the function c is related to the complex velocity appearing in Eq. (26) as follows:
S0
V ¼ Ài
rðln cÞ.
(29)
m
Since we have P ¼ ÀiS0r ln c ¼ ÀiS0ðrcÞ=c, we obtain the equality Pc ¼ Ài_rc (Nottale, 1993) in the
standard quantum mechanical case S0 ¼ _, which establishes a correspondence between the classical
momentum p, which is the real part of the complex momentum in the classical limit, and the operator Ài_r.
Therefore the correspondence principle is no longer an independent axiom as it is in standard quantum
mechanics.
2.3.3. Schro¨dinger form of the motion equation
We have now at our disposal all the mathematical tools needed to write the fundamental equation of
dynamics (23) in terms of the new quantity c. It takes the form
b
d
iS0
ðr ln cÞ ¼ rF.
(30)
dt
This equation can be integrated in a general way under the form of a Schro¨dinger equation.
Such an equation could be integrated provided its left-hand side be a gradient. But one should be aware that
b
d and r do not commute. However, as we shall now see, b
dðr ln cÞ=dt is nevertheless a gradient.
Replacing b
d=dt by its expression, given by Eq. (17), yields


q
rF ¼ iS0
þ V:r À iDD ðr ln cÞ,
(31)
qt
and replacing once again V by its expression in Eq. (29), we obtain

&
'!
q
S0
rF ¼ iS0
r ln c À i
ðr ln c:rÞðr ln cÞ þ DDðr ln cÞ
.
(32)
qt
m

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This expression may be simplified thanks to the remarkable identity (see Nottale, 1993 and its proof in
Appendix A),
 
Dc
r
¼ 2ðr ln c:rÞðr ln cÞ þ Dðr ln cÞ.
(33)
c
We recognize, in the right-hand side of the identity (33), the two terms of Eq. (32), which were, respectively, in
factor of S0=m and D. Therefore, in order to simplify the right-hand side of Eq. (32), the arbitrary parameter
S0 in the definition of the wave function can be taken to be
S0 ¼ 2mD.
(34)
One can prove that the final result is actually independent of this choice, see Nottale (2008).
Note that this relation is more general than the standard quantum mechanical one, in which S0 is restricted
to the only value S0 ¼ _. The function c in Eq. (27) is therefore now defined as
c ¼ eiS=2mD,
(35)
so that the fundamental equation of dynamics now reads

!
q
rF ¼ 2imD
r ln c À if2Dðr ln c:rÞðr ln cÞ þ DDðr ln cÞg .
(36)
qt
Now using the above remarkable identity and the fact that q=qt and r commute, it becomes
&
'
rF
q
Dc
À
¼ À2Dr i
ln c þ D
.
(37)
m
qt
c
Finally, using the fact that d ln c ¼ dc=c, the full equation becomes a gradient,
&

'
F
i qc=qt þ DDc
r
À 2Dr
¼ 0.
(38)
m
c
This equation can now be easily integrated, to finally obtain a generalized Schro¨dinger-like equation
(Nottale, 1993)
q
F
D2Dc þ iD
c À
c ¼ 0,
(39)
qt
2m
up to an arbitrary phase factor which may be set to zero by a suitable choice of the c phase.
Therefore the Schro¨dinger equation is the new form taken by the energy equation in the non-differentiable
context. The standard Schro¨dinger equation of microphysics corresponds to the particular case D ¼ _=2m, but
the important point here is that all the physical–mathematical structure of the description is preserved for any
constant value of the parameter D, which therefore does not need to depend on the universal Planck constant
_. It is therefore possible for some particular systems to be described by such a Schro¨dinger-type equation, in
terms of a parameter D which would be characteristic of this system (e.g., as a self-organization constant).
Arrived at that point, several steps have been already made toward the final identification of the function c
with a wave function. Indeed, it is complex, solution of a Schro¨dinger equation, so that its linearity is also
ensured (namely, if c1 and c2 are solutions, a1c1 þ a2c2 is also a solution). We shall now complete the proof
by showing that it fundamentally describes a wave, defined by a generalized Einstein–deBroglie wavelength,
see below (Nottale, 1993, 2008; Ce´le´rier and Nottale, 2004), and that it is ultimately a wave of proba-
bility, since Born’s postulate, according to which the probability of presence is given by the square of its
modulus, can be derived from first principles in the scale relativity framework (Ce´le´rier and Nottale, 2004;
Nottale, 2008).
2.3.4. Fundamental wavelengths
Let us first briefly recall how quantum mechanics is, from its origin, fundamentally a wave mechanics.
Historically, this led to a very profound unification of matter and radiation by Einstein and de Broglie. In
1905, while the main view about light was that it was an electromagnetic wave, Einstein introduced, after
Planck’s work, the concept of a quantum of light which later became known as the photon. He related the

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Document Outline

• Scale relativity theory and integrative systems biology: 2 Macroscopic quantum-type mechanics
  • Introduction
  • New macroscopic quantum-type mechanics
    • Motivation
    • Method
      • Fractality of the paths
      • Infinite number of geodesics
      • Discrete symmetry breaking from irreversibility
      • Covariant total derivative operator
    • Covariant mechanics in scale relativity
      • Geodesic form of the motion equations
      • Emergence of the quantum tools
      • Schrödinger form of the motion equation
      • Fundamental wavelengths
      • Fluid mechanics form of the motion equations
      • Relation to diffusion processes
        • Motivation
        • Diffusion approach to quantum laws
        • Diffusion potential
      • Generalization of quantum laws
    • Application to biological processes
      • Conditions of application
      • Morphogenesis
      • Formation, duplication and bifurcation
      • Prediction of biological sizes
      • Prospects: other quantum-type properties
    • Gauge field theory
  • Quantum-type mechanics in scale space
    • Motivation
    • Schrödinger equation in scale space
    • Complexergy
      • Third quantization
      • A new conservative quantity
    • Application to biological systems
  • Discussion, conclusions and perspectives
    • Validated predictions of scale relativity theory in astrophysics
    • Founding elements of an integrative theory of life on first principles
    • Defining biological space-time, biological fields and charges
    • Extending the classical framework for multiscale integration
    • Design of macroscopic quantum-type experiments and devices
    • Applications in integrative systems biology
    • Consequences for transdisciplinary research, development and training in integrative systems biology
  • Aknowledgements
  • Proof of fundamental remarkable identity
  • From Schrödinger equation to Euler and continuity equations
  • Schrödinger process as anti-diffusion process
    • Fluid representation of diffusion processes
    • Continuity equation
    • Euler equation
      • Case of vanishing mean velocity
      • Basic remarkable identity
      • Diffusion potential
  • References

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