Chaos, Solitons and Fractals 19 (2004) 1323–1334 www.elsevier.com/locate/chaos €dinger equation Quantum derivatives and the Schro Faycßal Ben Adda a a,* , Jacky Cresson b Mathematical Sciences Department, Hail Community College, King Fahd University of Petroleum and Minerals, P.O. Box 2440, 44 Avenue Bartholdi, 72000 le Mans, Hail, Saudi Arabia b Equipe de Math
ematiques de Besancßon, Universit
e de Franche-Comt
e, CNRS-UMR 6623, 16, route de Gray, 25030 Besancßon Cedex, France Accepted 7 July 2003 Abstract We define a scale derivative for non-differentiable functions. It is constructed via quantum derivatives which take into account non-differentiability and the existence of a minimal resolution for mean representation. This justify heuristic computations made by Nottale in scale-relativity. In particular, the Schr€ odinger equation is derived via the scale-relativity principle and NewtonÕs fundamental equation of dynamics.
2003 Elsevier Ltd. All rights reserved. 1. Introduction The aim of this article is to give a complete proof that the Schr€ odinger equation can be obtained from NewtonÕs fundamental equation of dynamics in the scale-relativity setting developed by Nottale [1,11,14]. The articles [1,11,14] contain only a sketch of proof and are based on informal arguments. The main ‘‘ingredient’’ of NottaleÕs work is a new ‘‘derivative’’, 1 that he calls the scale derivative, which applies to non-differentiable functions. Despite its importance, there exists no rigorous definition of this operator. In this article, we give a precise definition of the scale derivative, as well as its geometric interpretation. As a consequence, we can justify completely the computations made by Nottale in the articles [1,11,14]. The main problem is to define an extension of the classical differential calculus which has a clear physical meaning. The starting point of our work is the following informal idea: Let c be a given curve. From the physical view-point, we do not have access to c, but to a ‘‘representation’’ of it, denoted cs , which is always differentiable (up to a finite number of points) at a given scale of observation s, and such that cs converge to c in C 0 topology. Of course, cs is not always sufficient in order to describe the underlying physical process. In particular, if c is non-differentiable, the fluctuations of cs when s goes to zero become non-negligible 2 contrary to what happens in the differentiable case. This transition from a differentiable behaviour to a non-differentiable one when we follow cs must be quantified. In this article, we introduce several concepts R tþs in the special case of graphs of functions f ðtÞ and for a representation given by the s-mean function fs ðtÞ ¼ ð1=2sÞ t
s f ðsÞ ds. In Section 2 we define the notion of s-differentiability, which leads to a natural transition quantity for the differentiable–non-differentiable behaviour of fs ðtÞ with respect to f ðtÞ, called minimal resolution and denoted sðf Þ. * Corresponding author. Address: Laboratoire dÕanalyse Numerique, tour 55-65, 5e etage, Universite Pierre et Marie-Curie, 4 Place Jussieu, 75252 Paris Cedex, France. E-mail addresses: fbenadda@kfupm.edu.sa, benadda@ann.jussieu.fr (F. Ben Adda), cresson@math.univ-fcomte.fr (J. Cresson). 1 We will see that this terminology is not appropriate. 2 Following Greene [8, p. 149] this is the basic reason for which differentiable (Riemanian) manifolds of EinsteinÕs relativity theory cannot be used to describe the structure of space–time in quantum mechanics. 0960-0779/$ - see front matter
2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0960-0779(03)00339-4 1324 F. Ben Adda, J. Cresson / Chaos, Solitons and Fractals 19 (2004) 1323–1334 Heuristically, we can say that for s > sðf Þ, the approximation of f ðtÞ by fs ðtÞ is sufficient to describe the behaviour of f up to s small perturbations, which is not the case for s < sðf Þ. When s < sðf Þ, we must take into account the non-differentiability of the limiting function f . We then define left and right quantum derivatives, denoted þ f =t and 
f =t respectively, which are nothing else than the derivatives of the R tþrsðf Þ right and left sðf Þ-mean function of f , i.e. fr ðtÞ ¼ ðr=sðf ÞÞ t f ðsÞ ds, r ¼ . The fact to consider separately fþ ðtÞ and f
ðtÞ is due to the non-differentiability of f . The scale derivative, defined in Section 3 and denoted by f =t, is a complex operator which takes into account the two quantity r f =t, r ¼ , in such a way that when f is differentiable f =t ¼ f 0 ðtÞ. This ‘‘gluing’’ property of =t to d=dt on the set of differentiable functions is a necessary constraint to be satisfied by all extended ‘‘differential’’ calculus. Paragraph 4 gives some properties of the scale derivative. In Section 5, we generalize, following NottaleÕs scale relativity principle, NewtonÕs fundamental equation of dynamics, by replacing the classical derivative by our scale derivative. We prove that the new equation leads to a generalized Schr€ odinger equation. This justify heuristic computations made by Nottale [1,11,14]. 2. Non-differentiable functions and minimal resolution In the following, f is a continuous function, defined on an open set I of R. For all s 2 Rþ , we denote by fs the smean function defined by Z tþs fs ðtÞ ¼ ð1=2sÞ f ðsÞ ds: ð1Þ t
s We call s-oscillation of f the quantity oscs f ðtÞ ¼ supff ðt0 Þ
f ðt00 Þ; t0 ; t00 2 ½t
s; t þ sg: ð2Þ We say that the graph of f is fractal according to Tricot [12] if the quantity oscs f ðtÞ ! þ1 when s ! 0; s ð3Þ uniformly with respect to t, contrary to the differentiable case. For 0 < a 6 1, we denote jf ðtÞja;s ¼ sup s;s0 2½t
s;tþs; s6¼s0 jf ðsÞ
f ðs0 Þj : js
s0 ja ð4Þ Remark 1. We have jf ja 6 jf ðtÞja;s 6 jf ja ; ð5Þ where jf ja ¼ inf t6¼t0 2I jf ðtÞ
f ðt0 Þj ; jt
t0 j jf ja ¼ sup t6¼t0 2I jf ðtÞ
f ðt0 Þj ; jt
t0 j ð6Þ and I is the closure of I. For all t 2 I, we define aðtÞ such that aðt; f ; sÞ ¼ supfa > 0 j jf ðtÞja;s 6¼ 0g: ð7Þ In the following, we do not write the explicit dependence of aðt; f ; sÞ with respect to f and s. We denote as f ðtÞ ¼ oscs f ðtÞ : 2sjf ðtÞjaðtÞ;s ð8Þ We have jfs ðtÞ
f ðtÞj 6 2sjf ðtÞjaðtÞ;s as f ðtÞ; so that as f ðtÞ is a measure of the approximation of f by the differentiable function fs . ð9Þ F. Ben Adda, J. Cresson / Chaos, Solitons and Fractals 19 (2004) 1323–1334 1325 Remark 2. For a differentiable function, we have aðtÞ ¼ 1 for all t 2 I and as f ðtÞ 6 1 for all t and all s. We then have the following notion of s-differentiability at a point t: Definition 1. Let f be a continuous function, defined on a compact set I of R, such that for all t 2 I, ð10Þ 1 P aðtÞ > 0: Let s > 0 be given. We say that f is s-differentiable at point t 2 I if as f ðtÞ 6 1: ð11Þ A large class of continuous functions satisfy condition (10). At least, H€ olderian functions of order 0 < a < 1. Moreover, there exists explicit construction of continuous functions with prescribed H€ older regularity, as discussed in [16]. The simplest case is that of continuous function such that aðtÞ ¼ a for all t 2 I. As an example, we can consider the Weierstrass function which possesses a uniform H€ older regularity [12]. Remark 3. The concept of s-differentiability can be understood as a way to characterize when the fluctuations of f are ‘‘small’’ with respect to the mean function fs . Of course, the notion of ‘‘smallness’’ depends on a normalization, which is fixed. In our case, this normalization is contained in the choice of the number 1 for the upperbound of as;a f in (11), by comparison with the differentiable case. Different kinds of normalization can be taken, leading to different notions of non-differentiability. This is just a matter of choice. We denote by sðf ÞðtÞ the minimal order of s-differentiability at point t: sðf ÞðtÞ ¼ inffs P 0 j f is s-differentiable at point tg: ð12Þ Definition 2. Let f be a continuous real valued function defined on an open interval I  R. We call minimal resolution the quantity sðf Þ ¼ inf sðf ÞðtÞ: ð13Þ t2I We remark that for all k 2 R, we have sðkf Þ ¼ sðf Þ and sðf þ kÞ ¼ sðf Þ: ð14Þ If f is a fractal function (see (3)), then sðf Þ > 0. In fact, we have a more general result: Lemma 1. If a continuous function f is differentiable then sðf Þ ¼ 0. Proof. This follows from Remark 2.  As a consequence, if sðf Þ > 0, then f is an everywhere non-differentiable function. Of course, if f is differentiable on a given subset J of I, then sðf Þ ¼ 0, despite the fact that sðf ÞðtÞ > 0 for all t 2 I n J . In the following, we consider only continuous everywhere non-differentiable functions. 3. Quantum derivatives and the scale derivative We define left and right quantum derivatives. Definition 3. Let f : R ! R be a continuous function such that its minimal resolution sðf Þ satisfies sðf Þ > 0. We call right and left quantum derivative the quantities 1326 F. Ben Adda, J. Cresson / Chaos, Solitons and Fractals 19 (2004) 1323–1334 þ f f ðt þ sÞ
f ðtÞ ðtÞ ¼ ; t s 
f f ðtÞ
f ðt
sÞ ðtÞ ¼ ; t s ð15Þ respectively (where s ¼ sðf Þ). If s ¼ 0, then For a differentiable function, we have r f t þ f t ðtÞ ¼ lims!0 f ðtþsÞ
f ðtÞ . s ðtÞ ¼ t
f ðtÞ ¼ f 0 ðtÞ. Remark 4. We refer to [17] for a careful study of such kind of difference operators in the context of what they call time scales calculus. We define quantum functions fþ ðtÞ and f
ðtÞ as Z 1 tþs f ðsÞ ds; fþ ðtÞ ¼ s t Z t 1 f
ðtÞ ¼ f ðsÞ ds; s t
s ð16Þ respectively. We have r f ðtÞ ¼ fr0 ðtÞ; t Remark 5. As þ f t r ¼ : ðtÞ and 
f t ð17Þ ðtÞ are continuous, the functions fþ ðtÞ and f
ðtÞ are well defined. Using the quantum functions fþ ðtÞ and f
ðtÞ, we can write f as f ðtÞ ¼ fþ ðtÞ þ nfþ ðtÞ; f ðtÞ ¼ f
ðtÞ þ nf
ðtÞ; ð18Þ where nfþ and nf
are non-differentiable functions representing fluctuations with respect to the right (resp. left) mean function. In the following, we will denote nfþ and nf
by nþ and n
if no confusion is possible. Lemma 2. Let f be a continuous function such that sðf Þ > 0. Then for h sufficiently small, we have f ðt þ rhÞ ¼ f ðtÞ þ r where r f t r f ðtÞh þ ðnr ðt þ hÞ
nr ðtÞÞ þ oðhÞ; t ð19Þ ÞÞ
f ðtÞ , r ¼ . ðtÞ ¼ f ðtþrsðf rsðf Þ Proof. We have f ðt þ rhÞ ¼ fr ðt þ rhÞ þ nr ðt þ rhÞ þ oðhÞ for r ¼ . A first order TaylorÕs expansion of fr ðt þ rhÞ gives fr ðt þ rhÞ ¼ fr ðtÞ þ rfr0 ðtÞh þ oðhÞ: ð20Þ Using (17), we obtain for h sufficiently small, þ f ðtÞh þ nþ ðt þ hÞ þ oðhÞ; t 
f f ðt
hÞ ¼ f
ðtÞ þ ðtÞh þ n
ðt
hÞ þ oðhÞ: t f ðt þ hÞ ¼ fþ ðtÞ þ Replacing fr ðtÞ by f ðtÞ
nr ðtÞ, we obtain (19).  In the following, we denote rrf ðt; hÞ ¼ nfr ðt þ rhÞ
nfr ðtÞ for r ¼ , or rr ðt; hÞ if no confusion is possible. ð21Þ F. Ben Adda, J. Cresson / Chaos, Solitons and Fractals 19 (2004) 1323–1334 1327 Definition 4. Let f : R ! R be a continuous function and sðf Þ its minimal resolution. We call scale derivative of f at point t the quantity     f 1 þ f 
f 1 þ f 
f ð22Þ ðtÞ ¼ ðtÞ þ ðtÞ
i ðtÞ þ ðtÞ ; i2 ¼
1: t 2 t t 2 t t  only to be coherent with NottaleÕs terminology Remark 6. We use the terminology of scale derivative for the operator t [1,11]. However, as quantum derivatives, the scale derivative is not a derivation on the set of continuous functions, i.e. it does not satisfy the Liebniz rule. 3 When f is differentiable, we obtain the classical derivative. The real part of the scale derivative is the derivative of the s-mean function of f , fs . We have fþ ðtÞ þ f
ðtÞ ¼ fs ðtÞ: 2 In our computations about the Schr€ odinger equation, we will need the following definition of the scale derivative for complex valued functions: Let f : R ! C, t 7! f ðtÞ, be a continuous function. We have f ðtÞ ¼ Reðf ðtÞÞ þ i Imðf ðtÞÞ where Re and Im are the real and imaginary part of f ðtÞ respectively. The functions Reðf ðtÞÞ and Imðf ðtÞÞ are continuous real valued functions. We define the scale derivative of f as f Reðf Þ Imðf ÞðtÞ ðtÞ ¼ ðtÞ þ i : t t t ð23Þ 4. Consequences of non-differentiability We keep the notations of Section 3. 4.1. Main lemma In order to derive the Schr€ odinger equation from NewtonÕs fundamental equation of dynamics in Section 5, we need to compute the scale derivative of a composed function of the form f ðxðtÞ; tÞ where f ðx; tÞ is a differentiable function, and xðtÞ is not. The following lemma gives the formula. Lemma 3. Let f ðx; tÞ be a C 3 function. Let xðtÞ be a continuous function such that sðxÞ > 0 and xðt þ rhÞ ¼ xðtÞ þ r r x ðtÞh þ rrx ðtÞ t ð24Þ for h > 0 sufficiently small, r ¼ . We consider the function gðtÞ ¼ f ðxðtÞ; tÞ. We have (i) sðgÞ > 0. For all h sufficiently small, we denote gðt þ rhÞ ¼ gðtÞ þ r r g ðtÞh þ rrg ðt; hÞ; t r ¼ : ð25Þ We assume that for r ¼ , we have 2 x (*) the function admits  g ðrr ðt; hÞÞ a right derivative at point h ¼ 0 for all t, of x (**) we have rr ðt; hÞ
ox rr ðt; hÞ h ! 0 when h ! 0r . 3 An operator D on an algebra A is a derivation if 8x; y 2 A we have Dðx  yÞ ¼ Dx  y þ x  Dy which is usually called Liebniz identity (see [10]). 1328 F. Ben Adda, J. Cresson / Chaos, Solitons and Fractals 19 (2004) 1323–1334 Then, we have ðiiÞ r g of r x of 1 o2 f ¼ ðxðtÞ; tÞ ðtÞ þ ðxðtÞ; tÞ þ ðxðtÞ; tÞar ðtÞ; t ox t ot 2 ox2 ð26Þ where aþ ðtÞ (resp. a
ðtÞ) is the right derivative at point h ¼ 0 of rþ2 ðt; hÞ (resp. r
2 ðt; hÞ). Remark 7. If f is a flat function at point ðxðtÞ; tÞ then ðof =oxÞðxðtÞ; tÞ ¼ 0 and assumption (**) is equivalent to the differentiability of rr ðt; hÞ, r ¼  at h ¼ 0, which is not possible as sðxÞ > 0. Proof. For (i) this follows easily from Lemma 1. For (ii), we have   þ x ðtÞh þ oðhÞ þ rþx ðt; hÞ; t þ h ; gðt þ hÞ ¼ f ðxðt þ hÞ; t þ hÞ ¼ f xðtÞ þ t using (19). As rþx ðt; hÞ ! 0 when h ! 0, we have, doing a TaylorÕs expansion of f in the neighborhood of ðxðtÞ; tÞ,   of þ x of ðtÞh þ oðhÞ þ rþx ðt; hÞ þ ðxðtÞ; tÞh gðt þ hÞ ¼ gðtÞ þ ðxðtÞ; tÞ ox t ot  2 2 1 of þ x þ ðtÞh þ oðhÞ þ rþx ðt; hÞ ðxðtÞ; tÞ 2 ox2 t !   o2 f þ x o2 f x 2 ðxðtÞ; tÞ ðtÞh þ oðhÞ þ rþ ðt; hÞ h þ 2 ðxðtÞ; tÞh þ    þ2 oxot t ot ð27Þ ð28Þ As ðrþx Þ2 ðt; hÞ is differentiable at point h, we have ðrþx Þ2 ðt; hÞ ¼ aþ ðtÞh þ oðhÞ. By factorizing terms of order 1 in h, we obtain   of þ x of 1 o2 f ðxðtÞ; tÞ ðtÞ þ ðxðtÞ; tÞ þ ðxðtÞ; tÞa ðtÞ h þ oðhÞ þ Rþ ð29Þ gðt þ hÞ ¼ gðtÞ þ þ g ðt; hÞ; ox t ot 2 ox2 where Rþ g ðt; hÞ ¼ of ðxðtÞ; tÞrþx ðt; hÞ ox ð30Þ is non-differentiable and rþg ðt; 0Þ ¼ 0. We deduce   þ of þ x of 1 o2 f ðxðtÞ; tÞ ðtÞ þ ðxðtÞ; tÞ þ ðxðtÞ; tÞa ðtÞ f ðxðtÞ; tÞ
þ ox t ot 2 ox2 t 1 þ 8h > 0: ¼ ðrg ðt; hÞ þ oðhÞ
Rþ g ðt; hÞÞ h ð31Þ By replacing Rg ðt; hÞ by (30) and taking the limit in (31), we obtain, using (**), þ of þ x of 1 o2 f ðxðtÞ; tÞ ðtÞ þ ðxðtÞ; tÞ þ f ðxðtÞ; tÞ ¼ ðxðtÞ; tÞaþ ðtÞ; ox t ot 2 ox2 t ð32Þ where aþ ðtÞ is the derivative of ðrþx Þ2 ðt; hÞ at point h ¼ 0. Similar computations allow us to prove that 
of 
x of 1 o2 f ðxðtÞ; tÞa
ðtÞ; f ðxðtÞ; tÞ ¼ ðxðtÞ; tÞ ðtÞ þ ðxðtÞ; tÞ þ t ox t ot 2 ox2 where a
ðtÞ is the derivative of ðr
x Þ2 ðt; hÞ at point h ¼ 0. ð33Þ  About Ito’s stochastic calculus. As a consequence, the non-differentiability of xðtÞ introduces additional spatial terms in the derivative of f ðxðtÞ; tÞ with respect to the differentiable case. This formula is similar to ItoÕs formula in stochastic calculus [15]. However, ItoÕs formula is obtained under probabilistic assumptions. In Lemma 4, this follows from the F. Ben Adda, J. Cresson / Chaos, Solitons and Fractals 19 (2004) 1323–1334 1329 geometric assumption that f is non-differentiable. For more details about our scale derivative and ItoÕs formula, we refer to [2,3]. 4.2. The complex case Let RR ! ðx; tÞ 7! C: C : Cðx; tÞ We denote Aðx; tÞ ¼ ReCðx; tÞ and Bðx; tÞ ¼ ImCðx; tÞ, then Cðx; tÞ ¼ Aðx; tÞ þ iBðx; tÞ. Let x: ! 7! R t R xðtÞ be a continuous function such that its minimal resolution sðf Þ satisfies sðf Þ > 0. We define the functions xþ ðtÞ, x
ðtÞ, nþ ðtÞ and n
ðtÞ as in Section 3. We denote by ar ðtÞ the right derivative of ðnr ðt þ rhÞ
nr ðtÞÞ2 at point h ¼ 0. Lemma 4. The scale derivative of the function C: R ! t 7! C CðxðtÞ; tÞ is C oC x oC 1 o2 C ¼ þ þ aðtÞ 2 ; t ot t ox 2 ox ð34Þ where aðtÞ ¼  aþ ðtÞ
a
ðtÞ 2 
i   aþ ðtÞ þ a
ðtÞ : 2 ð35Þ Proof. We denote by AðtÞ and BðtÞ the functions AðxðtÞ; tÞ and BðxðtÞ; tÞ respectively. By Lemma 4, we have r A oA r x oA 1 o2 A ¼ þ þ r ar ðtÞ 2 : t ox t ot 2 ox We deduce A oA 1 ¼ t ox 2  þ x 
x þ t t  þ     oA 1 o2 A 1 oA 1 þ x 
x 1 o2 A 1 þ ða ða
þ ðtÞ
a ðtÞÞ
i ðtÞ þ a ðtÞÞ ; þ
þ
ot 2 ox2 2 ox 2 t t 2 ox2 2 that is A oA x oA 1 o2 A ¼ þ þ aðtÞ 2 t ox t ot 2 ox     aþ ðtÞ
a
ðtÞ aþ ðtÞþa
ðtÞ with aðtÞ ¼
i . 2 2 We obtain a similar formula for BðtÞ. Hence, by definition, we have  2      C x oA oB oA oB 1 oA o2 B þ i : ¼ þi þ þi þ aðtÞ t t ox ox ot ot 2 ox2 ox2 We then obtain C x oC 1 o2 C oC ¼ þ aðtÞ 2 þ ; t t ox 2 ox ot which concludes the proof.  1330 F. Ben Adda, J. Cresson / Chaos, Solitons and Fractals 19 (2004) 1323–1334 4.3. About the regularity assumption (*) Feynman and Hibbs [7] have proved that generic trajectories of quantum particles are continuous non-differentiable curves. However, there exists a quadratic velocity, that is, the quantity lim0 x!x ðf ðxÞ
f ðx0 ÞÞ2 exists: x
x0 ð36Þ Hence, the following quantities ðf ðt þ hÞ
f ðtÞÞ2 h and ðf ðtÞ
f ðt
hÞÞ2 h ð37Þ keep sense when h > 0þ , and are equals. For all h > 0, we have f ðt þ hÞ
f ðtÞ
rþ2 ðt; hÞ ¼ h h þ f t h þ oðhÞ

2 : ð38Þ When h ! 0þ , we obtain limþ h!0 rþ2 ðt; hÞ ðf ðt þ hÞ
f ðtÞÞ2 : ¼ limþ h!0 h h ð39Þ Similar computations prove that lim
h!0 r
2 ðt; hÞ ðf ðtÞ
f ðt
hÞÞ2 : ¼ lim
h!0 h h We denote by aþ ðtÞ (resp. a
ðtÞ) the right derivative of rþ2 ðt; hÞ (resp. r
2 ðt; hÞ). As the quadratic velocity is well defined, we must have aþ ðtÞ ¼ a
ðtÞ: ð40Þ Assumption (*) is then satisfied by functions describing quantum trajectories. Remark 8 • For the Browninan motion, Einstein [9] has proved that f ðt þ hÞ
f ðtÞ  h1=2 for h > 0, which is in accordance with (*). • The existence of a quadratic velocity is equivalent to 1/2-right differentiability of f following [4]. 5. Scale relativity principle and Schrödinger equation In [1,11,14], Nottale announce that the Schr€ odinger equation can be obtained from the classical NewtonÕs equation of dynamics using a quantization procedure which comes from the scale relativity theory. The scale relativity theory is developed by Nottale since 1980. Its aim is to generalize EinsteinÕs relativity principle in order to derive quantum mechanics from a first principle. We refer to his work for more details [1]. The quantization procedure is based on a generalized Euler–Lagrange equation coming from NottaleÕs theory and the use of the scale derivative instead of the classical derivative. The computations made by Nottale in [1,11,14] are informal and based on heuristic arguments. Using the scale derivative defined in the previous paragraph, we give a complete and detailed proof of his approach. 5.1. Action functional and wave function Let x : R ! R, t 7! xðtÞ, be a continuous, non-differentiable function, describing the trajectory of a quantum particle of mass m. Let v: R ! t 7! C x vðtÞ ¼ t 1331 F. Ben Adda, J. Cresson / Chaos, Solitons and Fractals 19 (2004) 1323–1334 be its velocity. Let U: RR ðx; tÞ ! 7! C Uðx; tÞ ð41Þ be a differentiable function, called scalar potential. The action functional is then defined by L: RCR ! ðx; v; tÞ 7! C 1 2 : mv
Uðx; tÞ 2 ð42Þ We note that the map Lðx; v; tÞ is differentiable with respect to x and v. Scale assumption. We assume, following Nottale [1], that equation of motion for particles is given by the following Euler–Lagrange generalized equation:    oL oL : ð43Þ ¼ t ov ox Nottale deduce this equation informally via his scale relativity principle. We refer to his work [1,5,6] for more details. We then have m v oU ¼
: t ox ð44Þ We call this equation, fundamental equation of dynamics, by analogy with NewtonÕs classical equation. The momentum is defined by p ¼ oL , which gives p ¼ mv. We introduce an action A as ov A: RR ðx; tÞ ! 7! C ; Aðx; tÞ ð45Þ which is a differentiable function, related to the momentum via the relation p ¼ oAðx;tÞ . We then obtain v ¼ m1 ox We can introduce a function w: RR ðx; tÞ ! 7! C ; wðx; tÞ oA . ox ð46Þ differentiable, such that iAðx;tÞ wðx; tÞ ¼ e 2mc ; ð47Þ where c 2 R is a normalization constant to be determined. This function is of course the wave function of a particle. We note that Aðx; tÞ ¼
2mci lnwðx; tÞ and v ¼ lnw , where ln is the complex logarithm. vðx; tÞ ¼
i2c o ox Remark 9. We obtain the classical correspondence principle of quantum mechanics for momentum and energy, that is p ¼
2imc ow 1 ; ox w E ¼ 2imc ow 1 : ot w ð48Þ 5.2. Schr€odinger equation Using the wave function, the fundamental equation of dynamics looks like    o oU 2icm ðln wÞ ¼ : t ox ox Lemma 5. The fundamental equation of dynamics is equivalent to  2  aðtÞ ow 1 o lnw o2 w 1 þ i2cm þ icaðtÞ 2 ¼ U þ aðxÞ:
i2cm ic þ 2 2 ox ot ox w w ð49Þ ð50Þ 1332 F. Ben Adda, J. Cresson / Chaos, Solitons and Fractals 19 (2004) 1323–1334 Proof. The fundamental equation of dynamics is    o lnw oU 2icm ¼ : t ox ox lnw ðxðtÞ; tÞ. We have, using Lemma 5, We denote NðtÞ ¼ nðxðtÞ; tÞ ¼ o ox    o N on x on 1 o2 n ðln wÞ ¼ ¼ þ þ aðtÞ 2 : t ox t ox t ot 2 ox A simple computation gives on ot ¼ oxo o lnw ot
on x on ¼
2ic n; ox t ox and as function of w. We then have by definition of v ¼ x t  2 ! on x on2 o 1 ow ¼
ic ¼
ic : ox t ox w2 ox ox Moreover, we have o2 n o ¼ ox2 ox o2 w 1
ox2 w  ow ox 2 1 w2 ! : We deduce, by gathering these terms !   2 N o aðtÞ ow 1 aðtÞ o2 w 1 o lnw ¼ þ
ic þ : þ t ox 2 ox 2 ox2 w ot w2 By replacing in the fundamental equation of dynamics, we obtain !   2 o aðtÞ ow 1 o2 w 1 o lnw oU :
2icm ic þ ¼ þ icmaðtÞ 2 þ 2icm ox 2 ox ox w ot ox w2 We conclude the proof by integration. h As a particular case, when the non-differentiability of xðtÞ is uniform, we obtain the classical Schr€ odinger equation. Corollary 1. Let xðtÞ be a continuous, non-differentiable function such that aðtÞ ¼
i2c: ð51Þ Then the fundamental equation of dynamics takes the form i2cm ow o2 w þ 2c2 m 2 ¼ ðU þ aðxÞÞw: ot ox ð52Þ We can always choose a solution of (52) such that aðxÞ ¼ 0. In this case, when c¼ h ; 2m ð53Þ where h is the Planck constant, we obtain the classical Schr€ odinger equation ih ow h2 o2 w ¼ Uw: þ ot 2m ox2 ð54Þ Þ2 w12 : In this case, by replacing aðtÞ and remarking that Proof. The choice of aðtÞ allows us to cancel the term ðow ox o lnw ow 1 , we obtain Eq. (52). ¼ ot ot w F. Ben Adda, J. Cresson / Chaos, Solitons and Fractals 19 (2004) 1323–1334 1333 Let w be a solution of (52). We search for a function w~ solution of the equation i2cm ow~ o2 w~ þ 2c2 m 2 ¼ Uw~ ot ox ð55Þ of the form Aðx;tÞ w~ ¼ ei 2mc þhðxÞ ¼ wðx; tÞHðxÞ; ð56Þ where HðxÞ ¼ ehðxÞ : That is, we modify the phase of the wave function w. We then have ow~ ow ¼ H þ wH0 ; ox ox o2 w~ o2 w ow ¼ 2 H þ 2 H0 þ wH00 ; ox2 ox ox ow ow ¼ H; ot ot where H0 ðxÞ and H00 ðxÞ are the first and second derivative of HðxÞ. By replacing in (55), we obtain  2  ow ow ow 0 00 2 H þ 2c m i2cm H þ 2 H þ wH ¼ UwH: ot ox2 ox We deduce an ordinary differential equation in H of the form   ow o2 w ow 0 H þ 2c2 mwH00 ¼ 0: þ 2c2 m 2
Uw þ 4c2 m H i2cm ot ox ox ð57Þ ð58Þ ð59Þ As w is a solution of (52), we have HaðxÞw þ 4c2 m ow 0 H þ 2c2 mwH00 ¼ 0: ox ð60Þ This differential equation has always a solution. Hence, we can always choose a solution of (52) such that aðxÞ ¼ 0. The choice of c in order to obtain Eq. (54) is then done by identification. h Remark 10. Our derivation of the Schr€ odinger equation is done under the scale assumption, which follows from NottaleÕs physical concept of scale relativity principle. We refer to [13, pp. 254–257] for a completely different proof. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] Nottale L. Scale-relativity and quantization of the universe I. Theoritical framework. Astronom Astrophys 1997;327:867–89. Ben Adda F. Diff
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