Laurent NottaleCNRS

LUTH, Paris-Meudon Observatory

http://www.luth.obspm.fr/~luthier/nottale/

Scales in naturePlanck scale10 cm-33

10 cm-28

10 cm-16

3 10 cm-13

4 10 cm-11

1 Angstrom

40 microns

1 m

6000 km700000 km1 millard km

1 parsec

10 10

10 20

10 30

10 40

10 50

10 60

1

Grand Unification

accelerators: today's limitelectroweak unification

electron Compton lengthBohr radius

quarks

virus bacterieshuman scale

Earth radiusSun radiusSolar Systemdistances to StarsMilky Way radius10 kpc

1 Mpc100 Mpc

Clusters of galaxiesvery large structuresCosmological scale10 cm28

atoms

Scales of living systems

Relationsbetween

length-scalesand mass-

scales

l/lP = m/mP

l/lP = mP / m

(GR)

(QM)

RELATIVITY

COVARIANCE EQUIVALENCE

weak / strong

Action Geodesical

CONSERVATIONNoether

FIRST PRINCIPLES

Giving up the hypothesis of differentiability of

space-time

Explicit dependence ofcoordinates in terms of

scale variables+ divergence

Generalize relativity of motion ?

Transformations of non-differentiable coordinates ? ….

Theorem

FRACTAL SPACE-TIME

Complete laws of physics by fundamental scale laws

Continuity +SCALE RELATIVITY

Principle of scale relativity

Scale covariance Generalized principleof equivalence

Linear scale-laws:  “Galilean”self-similarity,

constant fractal dimension,scale invariance

Linear scale-laws :  “Lorentzian”varying fractal dimension,

scale covariance,invariant limiting scales

Non-linear scale-laws:  general scale-relativity,

scale dynamics,gauge fields

Constrain the new scale laws…

A

A

0

1X

t0 10.1

1. Continuity + nondifferentiability Scale dependence

0.01 0.11

Continuity + Non-differentiability implies Fractality

when

Continuity + Non-differentiability impliesFractality

0 0.2 0.4 0.6 0.8 1

0.25

0.5

0.75

1

1.25

1.5

0 0.2 0.4 0.6 0.8 1

0.25

0.5

0.75

1

1.25

1.5

0 0.2 0.4 0.6 0.8 1

0.25

0.5

0.75

1

1.25

1.5

0 0.2 0.4 0.6 0.8 1

0.25

0.5

0.75

1

1.25

1.5

0 0.2 0.4 0.6 0.8 1

0.25

0.5

0.75

1

1.25

1.5

0 0.2 0.4 0.6 0.8 1

0.25

0.5

0.75

1

1.25

1.5

0 0.2 0.4 0.6 0.8 1

0.25

0.5

0.75

1

1.25

1.5

0 0.2 0.4 0.6 0.8 1

0.25

0.5

0.75

1

1.25

1.5

0 0.2 0.4 0.6 0.8 1

0.25

0.5

0.75

1

1.25

1.5

0 0.2 0.4 0.6 0.8 1

0.25

0.5

0.75

1

1.25

1.5

0

1

2

3

4

5

6

7

8

9

Construction by

successivebisections

Continuity + Non-differentiability impliesFractality

Continuity + Non-differentiability implies Fractality

divergence

Lebesgue theorem (1903):«  a curve of finite length is almost everywhere differentiable »

Since F is continuous and no where or almost no where differentiable

i.e., F is a fractal curve

2. Continuity + nondifferentiability

when

Fractal function: construction by bisection

DF = 1.5

Fractal function: construction by bisection. 2

DF = 2

Feynman (1948), Feynman & Hibbs (1965):

« It appears that quantum-mechanical paths are very irregular.However these irregularities average out over a reasonablelength of time to produce a reasonable drift, or « average »velocity, although for short intervals of time the « average »value of the velocity is very high. »

« Typical paths of a quantum-mechanical particle are highlyirregular on a fine scale. Thus, although a mean velocity can bedefined, no mean-square velocity exists at any point. In otherwords, the paths are nondifferentiable. »

Typical paths of a quantum mechanical particle

x

t

Fractals:

Abbott & Wise 1981

Typical paths of a quantum mechanical particle

Feynman & Hibbs (1965) p. 177:

Typical paths of a quantum mechanical particle

«[…] this complete description could not be content with thefundamental concepts used in point mechanics. I have toldyou more than once that I am an inveterate supporter, not ofdifferential equations, but quite of the principle of generalrelativity whose heuristic force is indispensable to us.However, despite much research, I have not succeededsatisfying the principle of general relativity in another waythan thanks to differential equations; maybe someone willfind out another possibility, provided he searches withenough perseverance.»

Cf also G. Bachelard 1927, 1940, A. Buhl 1934…: non-analycity, see Alunni 2001

Einstein, letter to Pauli (1948):

*Re-definition of space-time resolution intervals ascharacterizing the state of scale of the coordinate system

*Relative character of the « resolutions » (scale-variables):onlyscale ratios do have a physical meaning, never an absolute scale

*Principle of scale relativity: « the fundamental laws of nature arevalid in any coordinate system, whatever its state of scale  »

*Principle of scale covariance: the equations of physics keeptheir form (the simplest possible)* in the scale transformations

of the coordinate system

Weak: same form under generalized transformationsStrong: Galilean form (vacuum, inertial motion)

Principle of relativity of scales

Origin

Orientation

Motion

VelocityAcceleration

Scale

Resolution

Coordinate system

x

t

δ x

δ t

Scale Relativity : structure of the theory

Laws of scaletransformations (in

scale space)

« Galilean » DT=2

Special scale relativity

Generalized scale relativity

Quantum laws in scalespace

Induced dynamics(motion laws) in

standard space-(time)

Standard quantummechanics

Scale-motioncoupling

Generalized quantummechanics

Abelian and non-Abelian

gauge fields