Chaos, Solitons and Fractals 14 (2002) 553–562
www.elsevier.com/locate/chaos
Scale relativity theory for one-dimensional
non-differentiable manifolds
Jacky Cresson
Equipe de Math
ematiques de Besancßon, CNRS-UMR 6623, Universit
e de Franche-Comt
e,
16 route de Gray, 25030 Besancßon Cedex, France
Accepted 10 October 2001
Abstract
We discuss a rigorous foundation of the pure scale relativity theory for a one-dimensional space variable. We define
several notions as ‘‘representation’’ of a continuous function, scale law and minimal resolution. We define precisely the
meaning of a scale reference system and space reference system for non-differentiable one-dimensional manifolds. 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction
The aim of this paper is to discuss a rigorous foundation of the scale relativity theory developed by Nottale [14,15].
Nottale’s fundamental idea is to give up the hypothesis of differentiability of the space-time continuum. Previous
works in this direction have been done by Ord 1 [17]. As a consequence, one must consider continuous but non-differentiable objects, i.e., a non-differentiable manifold. 2 This leads naturally to the notion of fractal functions and
manifolds.
In order to justify Nottale’s framework, we must develop an analysis on non-differentiable manifolds. This analysis
does not exist already. However, Nottale’s approach to fractal space-time can be used to begin such a work.
In the following, we define a scale and space reference system for graphs of non-differentiable real-valued functions.
The difficulty in defining an intrinsic coordinates system on the graph C of a non-differentiable function, f , comes from
the fact that one cannot define classical curvilinear coordinates. Indeed, a consequence of Lebesgue’s theorem is that the
length of every part of C is infinite. One can overcome this difficulty using representation theory, i.e., associating to f a
one-parameter family of differentiable functions F ðt; Þ, > 0, such that F ðt; Þ ! f ðtÞ when ! 0. For all > 0, we
then define curvilinear coordinates. The study of C is then reduced to the study of the family F ðt; Þ when varies. This
leads to several new concepts like scale laws and minimal resolution.
Using all these notions, we can justify a part of Nottale’s work on scale relativity.
A general programme to study non-differentiable manifolds is discussed in [6] which leads to interesting connections 3 with the non-commutative geometry developed by Connes [8].
2. About non-differentiable functions: definitions and notations
In the following, we consider continuous real-valued functions xðtÞ, defined on a compact set I of R.
We denote by C0 ðIÞ the set of continuous functions (denoted by C 0 ðIÞ) which are nowhere differentiable on I.
1
2
3
E-mail address: cresson@math.univ-fcomte.fr (J. Cresson).
For more details about the history of fractal space-time, we refer to [10, Section 2], and to the book of Sidharth [18].
El Naschie [9] tries to go even further, and gives up the continuity hypothesis in his concept of Cantorian fractal space-time.
For connections of scale relativity with string theory and El Naschie’s Cantorian Eð1Þ space-time, we refer to [7,11].
0960-0779/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved.
PII: S 0 9 6 0 - 0 7 7 9 ( 0 1 ) 0 0 2 2 1 - 1
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J. Cresson / Chaos, Solitons and Fractals 14 (2002) 553–562
Definition 1. Let 0 < a < 1 and xðtÞ 2 C 0 ðIÞ. The a-right and a-left local fractional derivatives of x at point t 2 I are
defined by
daþ x
xðt þ hÞ xðtÞ
;
¼ lim
h!0þ
ha
dt
da x
xðtÞ xðt hÞ
¼ lim
:
h!0þ
dt
ha
ð1Þ
We denote by Ca ðIÞ the set of functions xðtÞ 2 C0 ðIÞ such that daþ x=dt and da x=dt exist for all t 2 I. We refer to [2,3]
for more details.
We refer to the book of Tricot [19, p. 152] for a definition of the fractal dimension of a graph. An important property
of Ca ðIÞ is the following:
Lemma 2. Let 0 < a < 1. For all functions xðtÞ 2 Ca ðIÞ, the fractal dimension of the graph of xðtÞ, t 2 I, is constant, and
equal to 2 a.
This set is very special. Indeed, the order of left–right derivation does not change when t 2 I varies. We introduce a
new functional space in order to allow a changing order of derivation.
Definition 3. Let a : R ! R be a continuous function such that 0 < aðtÞ < 1 for all t 2 I. We denote by CaðtÞ ðIÞ the set of
functions xðtÞ 2 C0 ðIÞ such that, for all t 2 I, the aðtÞ-right and -left derivatives exist.
This functional space will play a crucial role in special scale relativity. In particular, using Lemma 2, we can see that
locally, the fractal dimension of a function belonging to CaðtÞ ðIÞ for a given continuous function aðtÞ is more or less
constant, but it can strongly vary along the path.
3. Galilean scale relativity
3.1. About the scale reference system
Let 0 > 0 be a real number which, in the following, is a resolution variable. A given absolute resolution > 0 can be
described with respect to a given origin of resolution, 0 , by the new variable
s0 ðÞ ¼
;
0
ð2Þ
which is now a scale.
In order to obtain a ‘‘classical’’ reference system for scale, we introduce, for each 0 > 0 fixed, the function
E0 ðÞ ¼ lnð=0 Þ:
ð3Þ
In this scale reference system, we have 0 which is sent by E0 to 0, and for resolutions such that > 0 (resp. < 0 ), we
have E0 ðÞ > 0 (resp. E0 ðÞ < 0).
Definition 4. We denote by RE0 the scale reference system, related to resolution via the function E0 defined by
E0 ðÞ ¼ lnð=0 Þ.
The scale reference system is less natural than the resolution reference system. However, in order to easily write the
analogy between the relativity principle of Einstein and the scale relativity principle of Nottale, the scale reference
system is more appropriate.
3.1.1. Change of origin in the scale reference system
We now study the effect of changing the origin of a scale reference system RE0 from 0 to 1 . We have
E1 ðÞ ¼ lnð=1 Þ ¼ lnðð=0 Þð0 =1 ÞÞ ¼ E0 ðÞ þ lnð0 =1 Þ:
ð4Þ
The basic effect of changing origin of resolution is then a translation in the scale reference system.
The quantity lnð0 =1 Þ is the scale speed of the scale reference system RE1 with respect to RE0 . This terminology is
justified by the following ‘‘Galilean’’ composition rule of scale speed:
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J. Cresson / Chaos, Solitons and Fractals 14 (2002) 553–562
1
3
Lemma 5. Let 1 , 2 and > 0 be three resolutions. The scale speed of R with respect to R , denoted S3=1 , satisfies
S3=1 ¼ S3=2 þ S2=1 ;
ð5Þ
where S3=2 (resp. S2=1 ) is the scale speed of R3 (resp. R2 ) with respect to R2 (resp. R1 ).
Proof. We have
S3=1 ¼ lnð1 =3 Þ ¼ ln
1 2
2 3
¼ lnð1 =2 Þ þ lnð2 =3 Þ ¼ S2=1 þ S3=2 :
3.2. Construction of a reference system for non-differentiable one-dimensional manifolds
In this section, we discuss the construction of an intrinsic coordinates system for one-dimensional non-differentiable
manifolds. The basic example is the graph of an everywhere non-differentiable continuous function.
3.2.1. Curvilinear coordinates and Lebesgue’s theorem
Let xðtÞ be a continuous differentiable function, defined on a compact set I of R. The basic way to construct an
intrinsic coordinates system on the graph C of xðtÞ is to introduce the so-called curvilinear coordinate, which is defined,
an origin t0 2 I being given, by the length Lðx; t; t0 Þ of the graph of xðtÞ between the points xðt0 Þ and xðtÞ.
If xðtÞ is nowhere differentiable, one cannot use this construction. Indeed, we have the converse of Lebesgue’s
theorem:
Theorem 6. If xðtÞ is almost everywhere non-differentiable then the length of xðtÞ is infinite.
As a consequence, we cannot define the analogue of curvilinear coordinates on a nowhere differentiable curve.
Following Nottale, we introduce a new point of view on this problem: in general, we can have access not to the nowhere
differentiable function xðtÞ, but to a ‘‘representation’’ of it, controlled by the resolution constraint, and to the behaviour
of this representation when the resolution changes. 4 In the following we study the one-parameter family of mean
representation of xðtÞ and study its properties.
3.2.2. Representation theory of real-valued functions
We introduce the general idea of representation of a given real-valued function. This notion comes from Nottale’s
original work on fractal functions [16].
Let xðtÞ be a real-valued function, defined on a compact set I of R (or defined on R).
Definition 7. A representation of xðtÞ is a one-parameter family of real-valued functions, denoted X ðt; Þ, 2 Rþ , such
that
1. for all 2 Rþ , the function X ðt; Þ is differentiable;
!0
2. we have simple convergence toward xðtÞ when goes to zero, i.e., X ðt; Þ ! xðtÞ.
R tþ
A basic example is to take the -mean function as a representation of x, i.e., X ðt; Þ ¼ ð1=2Þ t xðsÞ ds. This is the
representation that we use in the following. A general study of representation of real-valued functions is done in [6].
Following Nottale [16, p. 75], we define the converse notation of fractal functions:
Definition 8. A fractal function is a real-valued function F ðt; Þ, depending on a parameter > 0, such that
1. F ðt; Þ is differentiable (except at a finite number of points) for all > 0;
2. there exists a non-differentiable continuous function f ðtÞ such that lim!0 F ðt; Þ ¼ f ðtÞ.
The main point in this definition is that, contrary to the representation of the continuous function, we only know
that the limit f ðtÞ exists. This does not imply that f ðtÞ is explicit.
Representations correspond to fractal functions for which the limiting function is explicit. We denote by F the set of
fractal functions. An interesting example of fractal functions is introduced by Nottale [16]:
4
This can be considered as the beginning of the renormalization group approach.
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J. Cresson / Chaos, Solitons and Fractals 14 (2002) 553–562
Definition 9. For all > 0 and l > 0, we denote by U;l ðx; yÞ a continuous function such that
Z 1
U;l ðx; yÞ dy ¼ 1 8x 2 R:
ð6Þ
1
Such a function is called a smoothing function.
We call Nottale’s functions fractal functions satisfying for all > 0, 80 < l < ,
Z
xðt; Þ ¼ U;l ðt; yÞxðy; lÞ dy:
ð7Þ
As a natural example, we can take mean functions by using the Dirac window U ðx; yÞ ¼ ð1=2Þ1½x;xþ ðyÞ, where the
function 1 is defined for the whole interval by: 1 ðyÞ ¼ 1 if y 2 and 1 ðyÞ ¼ 0 otherwise.
We denote by NðU;l Þ the set of Nottale functions satisfying (7) with a smoothing function equal to U;l .
Remark 10. It is important to fix the smoothing function. If not, we have no interesting equivalence relation on this set
(see Section 3.2.3).
3.2.3. About equivalence relations on fractal functions
We have a natural equivalence relation on F, given by:
Definition 11. We say that two fractal functions, denoted F1 ðt; Þ and F2 ðt; Þ, are equivalent, and we denote F1
lim!0 F1 ðt; Þ ¼ lim!0 F2 ðt; Þ.
We easily verify that is an equivalence relation. The basic idea behind the equivalence relation
non-differentiable function admits an infinite number of fractal functions as representation.
F2 if
is that a given
Problem 12. Can we define a canonical representation of a non-differentiable function?
The mean representation used in Section 3.2.4 seems to be a good candidate.
The main problem of this relation is that it is based on the limiting function, which is not always accessible. To solve
this problem, Nottale introduces the following binary relation:
Definition 13. Let F1 and F2 be in F. We say that F1 and F2 are equivalent, and we denote F1 RF2 if 8 ; 8t, we have
jF1 ðt; Þ F2 ðt; Þj < .
The problem is that R is not an equivalence relation on F because it does not respect the transitivity property, as
proved by the following counter-example.
Let F1 ðt; Þ be given. We define F2 ðt; Þ ¼ F1 ðt; Þ þ ð2=3Þ and F3 ðt; Þ ¼ F1 ðt; Þ þ ð4=3Þ. We have
jF1 ðt; Þ F2 ðt; Þj ¼ ð2=3Þ < for all > 0 and t. Moreover, we have jF2 ðt; Þ F3 ðt; Þj ¼ ð2=3Þ < for all > 0 and t.
We deduce that F1 RF2 and F2 RF3 . However, we have jF1 ðt; Þ F3 ðt; Þj ¼ ð4=3Þ > for all > 0 and t, such that
F1 F3 .
However, we have the following lemma:
Lemma 14. For all smoothing functions U;l satisfying (6), the binary relation R is an equivalence relation on Nottale’s set
NðU;l Þ.
Proof. The only non-trivial part is the transitivity property. Let f ðx; Þ, gðx; Þ and hðx; Þ be three Nottale functions of
NðU;l Þ such that f Rg and gRh. We have
Z
jf ðx; Þ hðx; Þj ¼ U;l ðt; yÞðf ðy; lÞ hðy; lÞÞ dyj:
ð8Þ
As f Rg and gRh, we have, for all x 2 R and for all l > 0,
jf ðx; lÞ hðx; lÞj < jf ðx; lÞ gðx; lÞj þ jgðx; lÞ hðx; lÞj < 2l:
ð9Þ
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J. Cresson / Chaos, Solitons and Fractals 14 (2002) 553–562
Hence, we obtain for all x 2 R and 8 > 0 and 0 < l < ,
Z
jf ðx; Þ hðx; Þj < 2l U;l ðx; yÞ dy ¼ 2l:
By choosing l ¼ =2, we obtain f Rh. This concludes the proof.
ð10Þ
Remark 15. A notion of the scale equivalence relation is defined in Section 3.2.4.
3.2.4. One-parameter family of mean functions and reference system
The idea, which is taken from Nottale’s work, is to associate to xðtÞ the one-parameter family of mean functions x ðtÞ
defined for all > 0 by
Z tþ
1
x ðtÞ ¼
xðtÞ dt:
ð11Þ
2 t
For all > 0, the mean function x ðtÞ is differentiable, with a derivative equal to ðdx =dtÞðtÞ ¼ ðxðt þ Þ xðt ÞÞ=2.
As a consequence, for all > 0, a t0 2 I being fixed, we can define an intrinsic coordinates system, which is simply the
curvilinear coordinate associated to x ðtÞ, denoted X ðtÞ and defined by
X ðtÞ ¼ Lðx ; t; t0 Þ:
ð12Þ
We are led to the following natural reference system associated to xðtÞ:
ðE;t0 Þ
Definition 16. Let t0 2 I be fixed. For all > 0, the ðEðÞ; t0 Þ-space reference system, denoted by RX
X ðtÞ.
, is defined by
As can be recovered by E0 ðÞ, we will now denote X by XE . We remark that, by definition of XE , we have
dXE xðt þ Þ xðt Þ
¼
ð13Þ
;
2
dt
so that we recover the classical mean velocity.
The approach of non-differentiable functions via the one-parameter family of mean functions induces a natural
equivalence relation in scale for continuous functions.
Definition 17. Let f ðtÞ and gðtÞ be two continuous real-valued functions defined on a compact set I of R. For all > 0,
we denote by f ðtÞ and g ðtÞ the -mean function of f and g, respectively. We say that f and g are -scale equivalent, and
we denote f g if f ðtÞ ¼ g ðtÞ for all t 2 I.
One can easily verify that is an equivalence relation.
For a differentiable function f , this equivalence relation is not interesting, because the asymptotic object, f , can be
studied via ordinary differential calculus. If f is a non-differentiable function, a new phenomenon appears. Indeed, there
exists a non-zero minimal resolution (see Section 3.5), denoted ðf Þ, under which one must take into account the nondifferentiability of f . In that case, for a given and a given function f , we have an infinite-dimensional equivalence class
of functions, g, which cannot be distinguished by f at resolution ðf Þ. This is why one must deal with an infinite number
of representations of f when a minimal resolution exists 5 (which is the case in scale relativity).
3.2.5. Relation between the space reference system and scale: the scale law
The relation between the space reference system and scales can be described by a scale law, which is an ordinary
differential equation controlling the behaviour of X ðtÞ when changes.
Definition 18. We say that xðtÞ satisfy a scale law if there exists a function A : R ! R such that for all > 0 we have
dXE
¼ AðXE Þ;
dE
where E is the scale variable, an origin of resolution 0 > 0 being given.
5
This phenomenon is responsible for the existence of an infinity of geodesics in scale relativity.
ð14Þ
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J. Cresson / Chaos, Solitons and Fractals 14 (2002) 553–562
In [4], we prove the following lemma:
Lemma 19. Let 0 < a < 1 and x 2 Ca ðIÞ. For all P 0 sufficiently small, we denote by rþ ðt; Þ (resp. r ðt; Þ) the remainder
of the generalized Taylor expansion of f ðt þ Þ (resp. f ðt Þ), i.e.,
xðt þ rÞ ¼ xðtÞ þ ra
dar x
þ rr ðt; Þ;
dt
r ¼ ;
ð15Þ
and rðt; Þ ¼ rþ ðt; Þ r ðt; Þ.
If rðt; Þ is differentiable on an open neighbourhood of 0 with respect to , then xðtÞ satisfies the scale law
AðxÞ ¼ ða 1Þx þ Oðx2 Þ:
ð16Þ
We refer to [4] for a proof.
This result motivates the introduction of a new functional space:
Definition 20. We denote by CaL ðIÞ the subset of Ca ðIÞ whose functions satisfy the linear scale law
dXE
¼ ða 1ÞXE :
dE
ð17Þ
Galilean relativity is based on functions belonging to CaL ðIÞ.
3.3. Djinn variable
For 0 > 0 fixed, and for all > 0 being given, we have defined an intrinsic coordinate on the graph of a non-differentiable function xðtÞ 2 CaL ðIÞ by taking X , an origin t0 2 I being given.
We introduce a new variable XE which is defined by
XE ¼ ln XE
if XE > 0:
ð18Þ
Remark 21. We refer to [16, p. 218] for a possible justification of the logarithmic variable form.
In this new variable, the scale law (17) reduces to
dXE
¼ a 1:
dE
ð19Þ
We stress that the assumption XE > 0 is nothing else than saying that only the behaviour for t > t0 can be described,
which means that the phenomenon is strongly non-reversible.
We denote by d the parameter
d¼1a
ð20Þ
in the following. The parameter d is called the djinn variable by Nottale.
3.4. The two basic effects: translation in scale and space
We now investigate the two basic effects of translating origin of scale and origin of space.
3.4.1. Translation in scale
As we have see in Section 3.1.1, a change in the origin of resolution between 0 and 1 translates for scale reference
system in a translation given by
E1 ¼ E0 þ Sð1 ; 0 Þ;
ð21Þ
where the scale state Sð1 ; 0 Þ is given by
Sð1 ; 0 Þ ¼ lnð0 =1 Þ:
ð22Þ
J. Cresson / Chaos, Solitons and Fractals 14 (2002) 553–562
559
This translation in scale is viewed in the space reference system using the scale law (17) by integrating the ordinary
differential equation between E0 and E1 . We obtain
XE1 ¼ XE0 expðdSð1 ; 0 ÞÞ:
ð23Þ
For the variable XE , translation in scale gives
XE1 ¼ XE0 dSð1 ; 0 Þ;
ð24Þ
which is, following Nottale, the Galilean version of the scale relativity theory.
We stress that, in this case, the djinn variable is constant under scale translations, which is of course, a very particular
case, i.e.,
dE1 ¼ dE0 ;
ð25Þ
dd
¼ 0:
dE
ð26Þ
or
The djinn variable plays the same role as the time t in classical Galilean relativity theory.
3.4.2. Translation in space
Assume that we make a change of origin in the space reference system, by changing t0 to t1 . We have the following
relation, the origin of resolution 0 begin fixed:
XEt1 ¼ XEt0 þ T ðt0 ; t1 ; EÞ;
where the space state is defined by
Z t0
T ðt0 ; t1 ; EÞ ¼
xE ðsÞ ds:
ð27Þ
ð28Þ
t1
The translation depends on E. If we assume that T is independent of E (which is the case in Nottale’s papers), a minimal
resolution appears (see the remark below). Here, we prove that such an effect does not exist. Indeed, by differentiating
Eq. (27) as
dT
¼ dT ;
dE
ð29Þ
we obtain
dXEt1 dXEt0
¼
dT ¼ dXEt0 dT ¼ dXEt1 þ dT dT ¼ dXEt1 ;
dE
dt
ð30Þ
which is the same equation as for XEt0 .
Remark 22. If we assume for simplicity that the space state is a function independent of E, we have
XEt1 ¼ XEt0 þ T ðt0 ; t1 Þ:
ð31Þ
We obtain
dXEt1
¼ dXEt1 þ dT ðt0 ; t1 Þ;
dE
ð32Þ
which gives
dXtE1
¼ d þ d expðXtE1 Þ:
dE
ð33Þ
As a consequence, we have
XEt1 ¼ T ½1 þ Sðk; 1 Þd ;
1
ð34Þ
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J. Cresson / Chaos, Solitons and Fractals 14 (2002) 553–562
where k is a resolution defined by
!1=d
XEt0
1
0
:
k¼
0
T
ð35Þ
As explained by Nottale [14], the resolution k has a particular status. Indeed, for k, we have XEt1
T , which is a
1
typical differentiable behaviour. In contrast, for k, we must take into account Sðk; 1 Þd , which comes from the nondifferentiable character of xðtÞ. By definition k is a relative resolution (it depends on 0 ) and is not at all scale invariant,
being dependent on x via XEt0 .
0
We define in the next section a natural notion of minimal resolution, which is compatible with the result of this
section.
3.5. Minimal resolution
The domain of validity of scale relativity is mainly beyond classical mechanics, in particular, particle physics and
quantum mechanics. A common idea, even at the basis of superstring theory (see [12]), is that there exists a scale at
which we must take into account the non-differentiable character of space-time. In the following, we define a natural
notion of minimal resolution for a given non-differentiable function xðtÞ, denoted ðxÞ, such that for > ðxÞ, the nondifferentiable character of x is not dominant, and for < ðxÞ, we must take into account non-differentiable effects.
3.5.1. -differentiability
Let xðtÞ be a continuous real-valued function defined on an open set I of R. We call -oscillation of x the quantity
osc xðtÞ ¼ supfxðt0 Þ xðt00 Þ; t0 ; t00 2 ½t ; t þ g:
We denote
Ka xðtÞ ¼
X
s;s0 2½t;tþ
jxðxÞ xðs0 Þj
:
js s0 ja
Let xðtÞ be a continuous function on I such that Ka xðtÞ 6¼ 0 for all t 2 I. We denote
a;a xðtÞ ¼
osc xðtÞ
:
2Ka xðtÞ
ð36Þ
For a differentiable function, we have a;1 xðtÞ 6 1 for all t 2 I and all . We introduce the following notion of -differentiability:
Definition 23. Let x be a continuous real-valued function defined on I. We assume that there exists a > 0 such that
Ka xðtÞ 6¼ 0 for all t 2 I. Let > 0 be given. We say that x is -differentiable if a;a xðtÞ 6 1 for all t 2 I.
We denote by ðxÞ the minimal order of -differentiability:
ðxÞ ¼ inff P 0; x is -differentiableg:
ð37Þ
We remark that for all k 2 R we have the following stability results:
ðkxÞ ¼ ðxÞ;
ðx þ kÞ ¼ ðxÞ:
3.5.2. Minimal resolution
The basic idea is that oscillation of non-differentiable functions increases toward infinity. In particular, fractal
functions, according to Tricot [19], are precisely functions such that lim!0 osc xðtÞ= ¼ 1 uniformly in t. We easily
deduce that fractal functions possess a non-zero minimal order of -differentiability. This result is in fact general for
non-differentiable functions:
Lemma 24. Let xðtÞ be a continuous, non-differentiable real-valued function. Then, its minimal order of -differentiability is
non-zero.
We refer to [4] for a proof.
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J. Cresson / Chaos, Solitons and Fractals 14 (2002) 553–562
In the following, we call ðxÞ the minimal resolution of x. The minimal resolution is a pure geometric constant
associated to the regularity of x.
Remark 25. Minimal resolution allows us to define the notion of quantum derivatives and scale derivative in [5].
3.6. Summary about Galilean scale relativity
We summarize the previous results in the following:
• Let 0 > 0 and 1 > 0 be two given resolutions. We denote by E and E0 the scale variables with respect to 0 and 1 ,
respectively.
• For all > 0, we denote by X the associated space coordinate on C .
• Let T be a given translation of origin in the space reference system. We denote by Y ¼ X T the normalized coordinate on C .
• Let Y ¼ ln Y , Y > 0, and S be a translation of origin in the scale reference system. Then we have
oY
¼ d;
oE
oY
¼ S:
od
ð38Þ
Note that we have taken d as a fundamental variable by obtaining S via a differentiation of Y with respect to d, just as
the classical speed is obtained as a derivative of the space variable with respect to time.
• If S1=0 and S2=1 are the scale speeds of the scale reference system 1 with respect to 0 and 2 with respect to 1, then the
scale speed of the scale reference system 2 with respect to 0, denoted S2=0 , is given by the Galilean composition rule
S2=0 ¼ S2=1 þ S1=0 :
ð39Þ
4. Special pure scale relativity
The existence of a minimal resolution ðxÞ allows us to fix a specific origin of resolution by setting 0 ¼ ðxÞ. We keep
notations from the previous section.
4.1. The Planck length as a scale invariant
The basic idea of Nottale is to generalize the previous Galilean point of view by allowing more general transformation laws of coordinates, respecting the relativity principle. In order to simplify the discussion, we restrict ourselves
to a pure scale relativity theory, i.e., we only pay attention to the set of variables ðY ; dÞ. We will discuss the general form
of the special scale relativity (integrating the time variable) in a forthcoming paper.
If d is not taken as an absolute variable, we know that the most general transformation rules of the form
Y0 ¼ AðSÞY þ BðSÞd;
d0 ¼ CðSÞY þ DðSÞd;
ð40Þ
which respect the relativity principle are given by Lorentz transformations [13], i.e.,
Y Sd
Y0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;
1 ðS 2 =L2 Þ
where L is a constant.
d SðY=L2 Þ
d0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;
1 ðS 2 =L2 Þ
ð41Þ
Remark 26. The fact that d is variable implies that the fractal dimension of quantum mechanical paths is fluctuating. As
a consequence, by Lemma 2, we must consider continuous functions with a variable H€
older exponent, as, for example,
the Riemann function
RðtÞ
1
X
cosðn2 tÞ
:
n2
n¼1
In order to understand L, we separate the geometric contribution of x and the ‘‘universal’’ one, by posing
ðxÞ
;
L ¼ ln
K
where K is a constant.
ð42Þ
ð43Þ
562
J. Cresson / Chaos, Solitons and Fractals 14 (2002) 553–562
We have the following relation:
ðxÞ
logððxÞ=Þ þ log S
¼
log
:
0
1 þ logððxÞ=Þ log S= log2 ððxÞ=KÞ
ð44Þ
We denote by TS the mapping giving 0 as a function of following Eq. (44), i.e., 0 ¼ TS ðÞ. The following lemma
makes precise the status of this constant:
Lemma 27. The constant K is scale invariant, i.e., TS ðKÞ ¼ ðKÞ for all S 2 R.
Proof. We put ¼ K in Eq. (44). We obtain
logððxÞ=KÞ þ log S
¼ logððxÞ=KÞ;
1 þ logððxÞ=KÞ log S= log2 ððxÞ=KÞ
which concludes the proof.
The minimal resolution is a geometrical constant, depending on the regularity of x (i.e., on the regularity of the
space-time). The constant K is a universal constant, not depending on the geometry of the space-time manifold.
Remark 28. In [16, p. 94–5], the minimal resolution is identified with the De Broglie length, using the result of Abbott
and Wise [1] on thep
Hausdorff
ffiffiffiffiffiffiffiffiffiffiffiffiffi dimension of a quantum mechanical path. The universal constant K is identified with the
Planck length K ¼ hG=c3 , where h is the reduced Planck constant, G is the gravitational constant, and c the speed of
light (see [16, p. 235]).
Acknowledgements
It is a pleasure to thank Dr. L. Nottale for very useful remarks about this paper, and Dr. M.S. El Naschie for his
interest and references about fractal space-time.
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Jacky Cresson