Scale relativistic formulation of non-differentiable mechanics I: Application to the harmonic oscillator

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by  Mei-Hui Teh

Noname manuscript No.
(will be inserted by the editor)
Scale relativistic formulation of non-differentiable
mechanics I
Application to the harmonic oscillator
Mei-Hui Teh · Laurent Nottale ·
Stephan LeBohec
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Received: date/Accepted: date
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Abstract This article is the first in a series of two presenting the scale rela-

0 tivisticapproachtonon-differentiabilityinmechanicsanditsrelationtoquan-
3 tum mechanics. In this first paper, we present the definition of a complex

scale-covariant time-differential operator and show that mechanics of non-
]
h differentiable paths is implemented in the same way as classical mechanics
p but with the replacement of the time derivative and velocity with the time-
- differential operator and associated complex velocity. With this, the general-
n
e ized form of Newton’s fundamental relation of dynamics is shown to take the
g form of a Langevin equation in the case of stationary motion characterized
.
s by a null averageclassicalvelocity.The numericalintegrationofthe Langevin
c equationin the case of a harmonic oscillatorrevealsthe same statistics as the
i
s stationary solutions of the Schr¨odinger equation for the same problem. This
y
motivates the second paper which makes the relation to quantum mechanics
h
explicitbydiscussingtheaxiomsofquantummechanicsintheScaleRelativity
p
[ framework.

2 Keywords ScaleRelativity FoundationofQuantumMechanics Stochastic
v Mechanics · ·
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7 1 Introduction
0
.
1 The relativity principle prescribes the laws of physics to have the same ex-
0
pression in all reference systems. It is implicitly understood that the relation
6
between two reference systems is entirely specified by their relative position,
1
:
v M.-H.Teh S.LeBohec
i Dept.ofPh·ysicsandAstronomy,UniversityofUtah,SaltLakeCity,UT84112-0830,USA.
X
E-mail:[email protected]
r
a L.Nottale
LUTH,ObservatoiredeParis,5,placeJulesJanssen,92195Meudon,France.
2 Mei-HuiTehetal.
orientation,andmotion.The Galileanimplementationofthe relativityprinci-
ple with the least action principle establish the whole classical mechanics [9].
The special relativistic implementation with the identification of the speed of
light as an invariant under the Lorentz transformations of the coordinates al-
lows for a simple and natural theory of electrodynamics. Galilean and special
relativity apply to the class of inertial reference systems, all in uniform rela-
tive motionwith respectto eachother.The transformationofthe coordinates
from one reference system to another is then linear. The next step consists
in extending the relativity principle to non inertial reference systems. Coor-
dinates transformations are then not necessarily linear anymore but they are
at least twice differentiable. The least action principle takes the form of the
geodesicprincipleand,withtheequivalenceprinciple,thisleadstoconsidering
gravitation as a manifestation of the curvature of space-time[15,6].
From a purely geometrical point of view, the next step toward greater
generality would be to consider transformations that are still continuous but
not necessarily differentiable. The task of implementing such a generalization
of the relativity principle is absolutely formidable. However, there are two
foreseeable direct implications of this idea. First, any two points would be
connectedby aninfinite number ofgeodesics.Consequently,determinism,im-
plemented by the principle of least action, has to give way to a probabilistic
and statistical description. Second, geodesics are still continuous but they are
non-differentiable. This implies their scale dependent and divergent nature,
that is fractal in a general meaning[10]. This identifies scaling laws as the
essential tool for the implementation of a non-differentiable relativity. This
proposal, known as Scale Relativity, was originally formulated by one of the
authors [13,14].
The consideration of non-differentiability and the associated scale depen-
dence may provide new insights in two major aspects of the physical world.
First, in standard quantum mechanics, the scale dependance is explicit in the
Heisenberguncertaintyrelationsandseveralauthors[5,?,2]commentedonthe
fractal nature of the quantum path. Second, complex and chaotic systems of-
ten involve the coupling between phenomena occurring at different scales and
alwaysdemonstratestructuresoverbroadrangesofscales.Thereisnogeneral
framework for modeling these systems while chaos is a universal phenomenon
investigated in many domains of science [7].
The projectof abandoning the hypothesis of differentiability seems daunt-
ing as differential calculus is the main mathematical tool of physics. We may
however proceed without abandoning differential analysis tools by smooth-
ing out any non-differentiable structures smaller than some parametric scale.
In this paper we demonstrate this approach in the case of non-differentiable
spacecoordinateswhilemaintainingtimeasanexternalandabsoluteparame-
ter. Furthermore,we arerestricting ourselvesto the type of non-differentiable
pathswiththescaledependencecorrespondingtoquantummechanicalpathor
BrownianmotionasmotivatedbySection2.Withinthisframework,inSection
3, we define a time-differential operator.In Section 4, applying the stationary
action principle, we show that the transition from differentiable to the con-
Scalerelativisticformulationofnon-differentiablemechanics I 3
sidered non-differentiable paths is simply implemented by replacing the usual
time derivative with this new time-differential operatorwhile keeping trackof
changes in the Leibniz product rule, which result from a higher order differ-
ential term. This extends the principle of covariance to scaling laws with the
new time-differential operator playing the role of a scale-covariant derivative.
In Section 5, we show that, under the restriction to stationary solutions, the
fundamentalrelationofdynamicsgeneralizedtonon-differentiablepathstakes
theformofaLangevinequation.Wethenproceedtothenumericalintegration
ofthe Langevinequationin the caseof a simple harmonic oscillatorandshow
we recover the statistic of quantum mechanics for the same system. Finally,
Section 6 summarizes the results andprovides a discussionand anoutlook on
their implications.
2 Fractal dimension
We may approach the concept of fractality from the point of view of physi-
cal measurements. The measurement of any quantity Q amounts to counting
the number of times the unit quantity δQ fits in Q. The result of the
δQ
M
measurement is then noted Q= δQ. The measurement unit δQ is gen-
δQ
M ·
erally chosen in a way that closely relates to the precision with which the
measurement is carried out. For this reason and by simplification, we do not
distinguish the resolution from the measuring unit.
In practice, it is usually implicitly assumed that two measurements of the
same quantity performedwith different resolutionsδQ andδQ are related by
′
δQ δQ δQ′ δQ′ up to the experimental errors.The presumption then
M · ≈M ·
isthatthequantityQdoesnotdependonthescaleofinspection.Thisleadsto
theideathatthemeasurementaccuracyisimprovedbytheuseofameasuring
device with an improved resolution.
This logic however breaks down when the structures contributing to the
measurement outcome themselves depend on the scale of inspection. This is
better seen when the quantity Q is of geometrical nature, such as a length,
an area or a volume. One may then write the resolution as δQ = (δx)DT
where δx is the length scale with which the object is inspected (for sim-
plification, we assume this scale to be the same in all directions, which is
not necessarily the case) and D is the topological dimension of the quantity
T
being measured. Comparing measurements carried out with different resolu-
tions, one may write ( δx/ δx′) = (δx/δx′)−DF with the fractal dimen-
M M
sion D = D +δD, where δD, an anomalous exponent, is introduced to
F T
account for the resolution or scale dependance of the measurement. With
this, Qδx/Qδx′ = (δx/δx′)−DF (δx/δx′)DT = (δx/δx′)DT−DF = (δx/δx′)−δD.
This can be inverted to establish the fractal dimention of an object as D =
F
D logQδx/Qδx′.
T − logδx/δx′
We may apply this to the case of the measurement of the length, D =
T
1, of the von Koch curve (See Figure 1). Take δx/δx = 3p, with p some
′
integer. Then by the construction of the von Koch curve, δx/ δx′ = 4−p
M M
4 Mei-HuiTehetal.
Fig. 1 Left panel: Five iterations in the construction of the von Koch curve. Each iter-
ation consists in replacing each segment of the previous iteration by 4 segments of length
3 times smaller, ultimately resulting in a curve of length diverging with resolution at a
ratecharacterized bythe fractal dimensionDF = llnn34. Alternatively, the successive curves
can be regarded as representations of a same object inspected with different resolutions.
Right panel: Cavalier projection of five iterations in the construction of a curve in three
dimensions[13].Each segment of the previous iteration is replacedby 9segments of length
3timessmaller,resultinginafractaldimensionDF = llnn93 =2.
so that Qδx/Qδx′ = (3/4)p and, DF = log4/log3 1.261859 . The fact
≈ ···
thatD >D indicatesthe divergenceofthe lengthasthe curveis inspected
F T
atsmallerscales.The rightpanelofFigure1presentsanotherexamplewith a
fractal dimension D =2. In self-similar objects such as the curves in Figure
F
1, the fractal dimension remains constant under successive changes of scale.
This,however,is notnecessarilythe case,andcurveswithafractaldimension
that depends on scale are easy to imagine. We can even imagine changes of
scale resulting in transitions between fractal and non fractal regimes. A very
good example of this is the Brownian motion of a particle. The inspection
of the trajectory with a fine enough resolution reveals the kinks resulting
from collisions with individual molecules of the surrounding fluid. Between
collisions,theparticleisobservedintheballisticregime,withapathoffractal
Scalerelativisticformulationofnon-differentiablemechanics I 5
dimension1.0,while atpoorerresolution,inthe diffusive regime,we willsoon
see that the fractal dimension of the path is 2.0.
It is useful to further concentrate on the specific case of the measurement
of the length of a curve. From what precedes, when a path is inspected
with two differLent resolutions δx and δx′, we have δx/ δx′ =(δx/δx′)1−DF.
L L
Instead of measuring the length of the curve by inspection at a given reso-
lution δx or δx, we may consider it being traveled at a constant rate and
′
inspectedatregulartimeintervalsδtorδt.Forthis,wecanconsiderasection
′
of the path to be traveled in a given time T = δt( δx/δx) = δt′( δx′/δx′),
L L
in such a way δt and δt are the times required to travel distances δx and
′
δx′ respectively. This implies δt/δt′ = (δx/δx′)( δx′/ δx) = (δx/δx′)DF and
1 1 L L
δt/ δt′ =(δt/δt′)DF− .
L L
Withthis, wecanconsiderthe caseofBrownianmotionobservedatscales
corresponding to the diffusion regime. The distance traveled during a given
time interval δt scales with √δt and the total distance traveled over the full
durationT of the observationis proportionalto √δtT/δt=T(δt)−1/2, corre-
sponding to D =2, which implies a divergence of the velocity at small scale
F
until the ballistic regime is reached.
We can also consider the case of the path of a quantum particle. In 1965,
FeynmanandHibbswrote”Itappears that quantum-mechanicalpaths arevery
irregular” and ”( ) the ’mean’ square value of a velocity averaged over a
···
shorttimeintervalisfinite,butitsvaluebecomeslargerastheintervalbecomes
shorter”insuchawaythat” althoughameansquarevelocitycanbedefined,
···
no mean square velocity exists at any point. In other words, the paths are
non-differentiable”[5]. It should be noted that this preceded the word fractal
coined by BenoˆıtMandelbrotin 1975.In order to see this quantitatively from
a simple argument[2] basedonthe Heisenberg uncertainty relations,consider
aparticleofmassmwhosepositionismeasuredatregulartimeintervals∆t.If
theparticleisatrestonaverage,themeasuredpathittravelsresultsfromthe
quantumfluctuationsδpofthe momentumsothatafterN measurements,the
length of path that will have been recorded on average will be =Nδp∆t.
hLi m
UsingtheHeisenberguncertaintyrelationδp δx ¯h/2,thismaybe rewritten
· ≈
hLi = N2h¯m∆δxt and identifying with hLi ∝ (δx)1−DF we see that DF = 2.
From one measurement to the next, the smallest significant change in path
length ∆ is the position measurement resolution itself ∆ δx. With
this, wehobLtiain δx2 h¯ ∆t. Comparing this with δt δxDhF Lwie≈identify the
≈ 2m ∝
effective diffusion coefficient = h¯ .
D 2m
In the following section, we establish a general method of approach to
these non-differentiable and scale dependent paths and we apply this method
specifically to the case D =2 because of it’s importance in physics.
F
3 Scale covariant time-differential operator
The usual derivative f (t) of a function f(t) is defined by considering the
′
changeofthevalueofthefunctioninthelimitofinfinitesimallysmallincrease
6 Mei-HuiTehetal.
Fig. 2 The black curve represents a non-differentiable path x(t). It was obtained by di-
chotomy iterations.Theredlinerepresents thefirstiteration,the bluelinethesecond, the
green line the third and the black line the ninth. When going from one iteration to the
next, the position x of the midpoint of each segment is shifted randomly in such a way
that, onaverage, the combined displacements correspondingto the tworesultingsegments
is √2 larger than the displacement represented by the original one. This corresponds to a
fractaldimensionof2.0.Asthecurveisconsidered withfiner resolution,theleftandright
derivativesint=0forexamplekeepfluctuating withanamplitudethatdiverges.
(+) or decrease ( ) of the argument:
−
f(t+δt) f(t) f(t) f(t δt)
f+′ (t)=δtlim0+ δt− and f−′ (t)=δtlim0+ −δt −
→ →
with f (t)=f (t)=f (t) as long as the function is differentiable in t. If, on
′ +′ ′
the contrary, the funct−ion is continuous but non-differentiable in t, then f (t)
′
is no longer uniquely defined as f (t)=f (t). Furthermore,if the function is
non-differentiable in a dense set o+′f poi6nts−′(a set is said dense if any neighbor-
hood of any point not in the set contains at least one point in the set) , then,
taking the limit δt 0+ becomes impossible in practice as the outcome fluc-
→
tuates indefinitely with a diverging amplitude as the limit is approached (see
Figure 2). We then have to refrain from taking the limit and define a double-
Scalerelativisticformulationofnon-differentiablemechanics I 7
valued and explicitly scale dependent differential, which can be applied to a
scale dependent function f(t,δt):
f(t+δt,δt) f(t,δt)
f+′ (t,δt)= δt− δt>0;
f(t+δt,δt) f(t,δt)
f′ (t,δt)= − δt<0.
− δt
With this in mind, we can consider the displacement of a point of position
x(t) along a non-differentiable path with the following representation with
two terms:
dx =v dt+db dt>0
+ + +
dx =v dt+db dt<0 (1)
− − −
The first term proceeds from a usual velocity v , which depends on the
±
scale of inspection (see Figure 2). By usual velocity we mean a finite displace-
ment δx divided by the time δt it takes to complete it. The displacement δx
or the time δt correspondto the scale or resolutionwith which the path is in-
spected to yield a specific representation. On the contrary, b represents the
±
residual, possibly a stochastic process, to be revealed by finer observations.
There is an infinite number of paths whose inspection at a finite resolutionδt
yield the same representation. They differ from each other only by their re-
spective db . Consequently, the expectation value of db over this ensemble
must cancel± b = 0. Indeed, the non cancelation of th±is expectation value
h ±i
wouldbetraysomeknowledgeaboutthepathinspectedwitharesolutionfiner
than actually considered. We will shortly come back to the statistical prop-
erties of b as they determine how the path representation is affected by a
±
changein resolution.Indeed, the usual velocity ata givenresolutionδt canbe
seen as deriving from the residual b in the representation of the same path
±
inspected atacoarserresolution.It is worthstressingthat this representation
of the path with two terms is a formalization of the usual way we deal with
trajectories: the details smaller than some resolution scale are disregarded or
smoothedout.Thiscanberegardedasthepromotionofresolutiontobeoneof
the relative characteristics of reference frames, at the same level as position,
orientation and motion in a way which constitutes the essence of the Scale
Relativity approach[13,14].
With this, it is convenient to define the classical time-differentials as the
expectation values of the time-differentials after and before the considered
point:
d db d db
+x=v++ + =v+ and −x=v + − =v
dt h dt i dt − h dt i −
Furthermore, rather than continuing to manipulate the two classical differ-
ential operators separately, we combine them linearly into a single complex
time-differential operator[13]:
8 Mei-HuiTehetal.
dˆ 1 d d i d d
+ +
= + − − . (2)
dt 2 dt dt − 2 dt − dt
(cid:18) (cid:19) (cid:18) (cid:19)
The real part corresponds to the average of the after and before differentials.
Itcanbethoughtofastheclassicaldifferential,whichispreservedforadiffer-
entiable path in the limit dt 0. The imaginary part is the halved difference
→
between the after and before differentials. It can be thought of a the kink
differential which vanishes for a differentiable path in the limit dt 0.
→
When acting on x with the complex time-differential operator, we can
define the complex velocity:
dˆ v +v v v
= x= + − i +− − =V iU
V dt 2 − 2 −
whereVcanberegardedastheclassicalvelocityandUisanadditionalterm,
thekinkvelocitywhichpersistsundertheinspectionofnon-differentiablepaths
with ever finer resolutions.
Equipped with these definitions, we can consider the time-differential of
a regular field f(x(t,dt),t), along a possibly non differentiable path x(t,dt),
describedintermsofafiniteresolutionrepresentation(Equation1).Thetotal
derivativeoff(x,t)iswrittenas(therepetitionofanindexmeanstheimplicit
summation over that index):
df ∂f ∂f dx 1 ∂2f dx dx 1 ∂3f dx dx dx
i i j i j k
= + + + +
dt ∂t ∂x dt 2∂x ∂x dt 6∂x ∂x ∂x dt ···
i i j i j k
At this point we need to make a choice as to the statistical nature of
the residual process b . As we would like to concentrate on the case D =
F
±
2, specific to Brownian motion and quantum mechanical paths, we write it
as a Wiener process with db = 0 as already discussed, db db =
i+ i
h ±i h · −i
0, and db db = db db = 2 δ dt with akin to a diffusion
i+ j+ i j i,j
h · i h − · −i D D
coefficient.Itshouldbestressedthatthischoiceisveryspecific.Itcorresponds
to describing Markovian or random walks. The underlying statistical process
can naturally be thought as Gaussian by virtue of the central limit theorem.
However any other statistical distribution would be equally valid and would
make no difference for the restof the development.We now consider both the
afterandthebeforetime-differentialswhilekeepingonlythetermsthatdonot
vanish with dt. Note that in taking the limit here, we do not change the scale
aswearenowconsideringaspecificrepresentationofthepathwithaclassical
component linear in dt and a stochastic component accounted for through an
expectation value with a fractional power of dt:
d ∂f ∂f ∂f db 1 ∂2f db db
i i j
±f = + vi + ± + ± ±
dt ∂t ∂xi ± ∂xih dt i 2∂xi∂xjh dt i
Scalerelativisticformulationofnon-differentiablemechanics I 9
Since b is of order √dt, the third term would diverge with dt 0 if it were
not for±the expectation value, which makes it cancel as db =→0. The after
h ±i
and before time-differentials become:
d ∂f
±f = +v f 2f
dt ∂t ±·∇ ±D∇
Combining the after and before differential operators in a single complex dif-
ferential operator as before:
dˆ ∂
= + i ∆ (3)
dt ∂t V ·∇− D
Itcanbeanticipatedthatgoingfromdifferentiablegeometrytonon-differentiable
geometryshouldbeimplementedbyreplacingtheusualtimederivative d with
dt
this complex time-differential operator dˆ provided proper attention is given
dt
to the changes in the Leibniz product rule implied by the second derivative
appearing in the last term of Equation 3. In the next section, we verify that
this is indeed the case for the Lagrange formulation of mechanics.
4 Mechanics of non-differentiable paths
Consideringnon-differentiablepathswiththecorrespondingdouble-valuedve-
locities with after and before components x˙ , we assume here that the me-
chanicalsystemwiththe configurationcoord±inatexcanbe characterizedbya
nowcomplex Lagrangefunction (x, ,t). We canthen expressthe actionfor
the evolution of the system betwLeenVtimes t and t as = t2 (x, ,t)dt.
1 2 S t1 L V
In orderto lightenthe notationwithout loosingany generality,we proceedby
R
considering a one-dimensional problem with = 1(x˙ +x˙ ) i (x˙ x˙ )
and (x, ,t) = (x, 1 ix˙ + 1+ix˙ ,t). TheVstati2ona+ry act−ion−p2rinc+ip−le c−an
L V L −2 + 2 −
be written as
t2 ∂ ∂ ∂
δ = Lδx+ L δx˙ + L δx˙ dt=0.
+
S Zt1 (cid:18)∂x ∂x˙+ ∂x˙− −(cid:19)
Provided the complex Lagrange function is an analytic function of and
making use of dˆ (See equation 2), it becomes: V
dt
t2 ∂ ∂ 1 i ∂ 1+i t2 ∂ ∂ dˆ
δ = Lδx+ L − δx˙ + L δx˙ dt= Lδx+ L δx dt=0
+
S Zt1 (cid:18)∂x ∂V 2 ∂V 2 −(cid:19) Zt1 ∂x ∂V dt !
We recover the usual form of the expression of action stationarity with the
velocityreplacedwithourcomplexvelocity andthetimederivativereplaced
with our complex time-differential dˆ. FromVthis point, we need to integrate
dt
by parts. This requires care as dˆ includes a second derivative which affects
dt
10 Mei-HuiTehetal.
the productrule.One canverifythat dˆ (f g)= dˆfg+fdˆg+2i f g and
dt · dt dt D∇ ·∇
we obtain
t2 ∂ dˆ ∂ dˆ ∂
δ = Lδx+ L δx L δx dt=0,
S Zt1 ∂x dt(cid:18)∂V · (cid:19)− dt(cid:18)∂V(cid:19) !
where the term in dδx was discardedas δx is not a function of x. Considering
dx
thatδx(t )=δx(t )=0 andrequiringthis equationto holdfor anyinfinitesi-
1 2
malδx(t), weobtaintheusualEuler-Lagrangeequationbut withthe complex
time-differential operator and velocity:
∂ dˆ ∂
L L =0
∂x − dt ∂
(cid:18) V(cid:19)
Using the usual form of the kinetic energy and including a purely real
potential energy term Φ associated with a conservative force acting on the
particle: = 1m 2 Φ,the Euler-Lagrangeequationresults ina generalized
L 2 V −
form of Newton’s relation of dynamics
dˆ
m = Φ. (4)
dtV −∇
The recovery of the velocity and time derivative replaced with their com-
plexconterparts and dˆ indicatesthatthisreplacementimplementsthetran-
V dt
sitionfromtheusuallyassumeddifferentiablegeometrytoanon-differentiable
geometry with dˆ playing the role of a scale-covariant derivative. In the next
dt
section, we explore the implications of this transitionin the case of one of the
simplest mechanical systems: the harmonic oscillator.
5 Application to the one-dimensional harmonic oscillator
Thefundamentalrelationofdynamicsobtainedabovehasbothrealandimag-
inaryparts,whichwecanwriteseparately,replacing dˆ and bytheirexpres-
dt V
sions.
∂ 1
V ∆U+(V )V (U )U = Φ
∂t −D ·∇ − ·∇ −m∇
∂
U+ ∆V+(V )U+(U )V = 0
∂t D ·∇ ·∇
This system of differential equations is the same as Equations (34) in E.
Nelson’s article entitled ”Derivation of the Schr¨odinger equation from New-
tonian Mechanics” [12]. However, they originate quite differently. In Nelson’s
stochastic quantization, these equations result from the additional hypothe-
sis of some underlying Brownian motion, which, in the quantum mechanical
context, is characterized by a diffusion coefficient = h¯ . In the Scale Rela-
D 2m
tivityapproachfollowedhere,theseequationsresultfromtheabandonmentof