Download Scales are fishy!

Scales are fishy!
Stephane LeBohec!
April 21st 2011!
The relativity principle is one of the founding basis of the !
laws of physics. In its Galilean, special and general
implementations, it accounts for the fact observations and
measurements depend on the relative state of motion of the
observers. In many situations, observations and
measurements also depend on the relative scales/resolutions
with which they are done. Differences in scale and resolution
are however generally not included explicitly in the
fundamental equations of physics. We will explore these ideas
with a few simple examples.!
Fractal fish by Thomas Schönenbach!
This coming summer, I will …!
…bring my Mac to the beach?!
&!
…wear a pink tie?!
Well, why not?!
Relativity!
Galileo transformation!
y’!
y!
⎧ t' = t
⎪
⎨ x' = x − uK ' / K ⋅ t
⎪
⎩ y' = y
Combination of velocities:!
v' = v − uK ' / K
K!
€
K’!
€
u!
x!
x’!
Relativity!
y’!
y!
The fundamental relation of dynamics!
In K:!
∂ϕ(t, x, y)
d 2 x(t)
−
= m⋅
2
∂x
dt
In K’:!
2
∂ϕ(t', x', y')
d x'(t')
−
= m⋅
∂x'
dt'2
€
K!
K’!
u!
x!
€ Is covariant under Galileo transform!
x’!
Lagrange formalism!
S=
∫
t2
t1
L(q, q˙,t)dt
δS = 0
d ∂L ∂L
−
=0
dt ∂q˙ ∂q
1 2
L = mv − U(r)
2
∂U
mv˙ = −
∂r
Relativity!
Maxwell’s equations are !
" " " " "not Galileo covariant:!
y’!
y!
1
while Michelson and Morley did
c=
ε 0 µ0 not find any evidence for an
aether.
€
K!
D!
K’!
Linear mass density μ!
Linear charge density λ!
u!
x’!
x!
 Lorentz transform of special relativity!
Euclidean geometry!
From: Addison Wesley!
Non-Euclidean geometries!
From: Addison Wesley!
General relativity!
From: Addison Wesley!
λ
λ
Dv
dv
λ µ ν
Geodesic equation:!
=
+ Γµν
v v =0
Dt
dt
Quantum mechanics!
∗ System state fully specified by state vector ϕ
∗ Physical quantities A associated to operators A acting on state vectors
∗ Results of measurements are operator eigenvalues ai
* ϕ measurement outcome probabilities P(ai ) = ui ϕ
(A ui = a i ui )
2
 Born postulate!
∗ Right after the measurement of A yielded ai , the system is in state ui
 von Neumann postulate!
∗ The time evolution of a state is given by Schrodinger equation :
d
i ϕ(t) = H ϕ(t)  Ehrenfest theorem!
dt
where H is the operator associated with the system energy
Different things …!
http://htwins.net/scale/!
… appear at different scales!!
Brownian motion model!
At each time step δt0, probability p=1/2 of !
taking a δx0 step to the left or to the right!
<K+>=Np=N/2 !
<(K+-<K+>)2>=<K+2>-<K+>2=Np(1-p)=N/4!
so <K+2>=N/4+N2/4!
Measuring the length of a Brownian path!
x n = k+ ⋅ δx 0 − (n − k + )δx 0
= (2k+ − n)δx 0
< x n >= 0
< x n 2 >=< 4k +2 − 4nk+ + n 2 > δx 0 2
 ΔT
= (n + n 2 − 2n 2 + n 2 )δx 0 2
ΔT
= nδx 0 2 =
δx 0 2
€
δt 0
T ΔT
T
2
=
δx =
δx 0
ΔT δt 0 0
ΔT⋅ δt 0
= 0
δt 0
ΔT
ΔT
δx 0
δt 0
 Δx δx 0
we obtain
=
0
Δx
with Δx = < x n 2 > =
T
 0 = N⋅ δx 0 =
δx 0
δt 0
The Hausdorff-Besicovitch dimension!
DT=1!
Unit length!
Unit area!
DT=2!
The Hausdorff-Besicovitch dimension!
2 x 1!
DT=1!
3 x 1!
DT=2!
The Hausdorff-Besicovitch dimension!
4 x 1/2!
12 x (½)2!
DT=1!
DT=2!
€
€
The Hausdorff-Besicovitch dimension!
9 x 1/4!
€
DH
2 = 2 ⇒ DH = 1
DH
L1 L0 ⎛ λ0 ⎞
= ⎜ ⎟
λ1 λ0 ⎝ λ1 ⎠
D H −1
⎛
⎞
€ λ0
L1 = L0 ⎜ ⎟
= L0
⎝ λ1 ⎠
DH
2 = 4 ⇒ DH = 2
⎛ λ0 ⎞ D H −DT
A1 = A0 ⎜ ⎟
= A0
⎝ λ1 ⎠ 41 x (¼)2!
D =2!
H
DT=1!
DH=1!
DT=2!
Scale divergence and fractal dimension!
⎛ 4 ⎞
 n =  0 ⎜ ⎟
⎝ 3 ⎠
n
ln 4
3 = 4 ⇒ DH =
= 1.2618...
ln 3
D H −1
 n ⎛ λ0 ⎞
= ⎜ ⎟
 0 ⎝ λn ⎠
n=0!
DT=1!
n=1!
DH
⎛ 1 ⎞ n
For example with λ0 = 1 & λn = ⎜ ⎟
⎝ 3 ⎠
n
n×(D H −1)
n×0.2618...
=3
=3
0
n=2!
n=3!
n=4!
DH=1.2618…!
n=5!
Hausdorff dimension of a Brownian path!
D H −1
⎛
⎞
 δx 0
δx 0
=
= ⎜
⎟
 0 δx ⎝ δx ⎠
so we find DH = 2
Hausdorff dimension of a quantum path!
∗ N measurements of the position of a particle of mass m at time intervals Δt
∗ Average path length <  >= N < Δ >= N < v > Δt
t
<p>

T⋅ 
∗ <  >= N
Δt ≈ N Δt =
m
m⋅ δx m⋅ δx
∗ Measuring the path length we can set Δt so that δx ≈< Δ >
1 ⎛ 1 ⎞
∗ Then <  >∝
= ⎜
⎟
< Δ > ⎝ < Δ > ⎠
Δl!
δx!
D H −1
so that DH = 2
Δt!
x!
Angular momentum of a fractal path!
2
•
M z = mr ϕ
q=4 & p=16!
DH=ln(p)/ln(q)=2!
•
M z n = mr 2 ϕ
€
€
€
n
p
M z n +1 = 2 M z n
q
p
lim M z n defined ⇒ 2 ≤ 1
n→∞
q
p
lim M z n non vanishing ⇒ 2 ≥1
n→∞
q
p
⇒ 2 = 1 & DH = 2
q
L.Nottale, Fractal space time and microphysics!
Galilean relativity of scales!
Selfsimilar fractal : τ constant , τ1 = τ 2 = τ = DH − DT
−τ
−τ
−τ
⎛
⎞
⎛
⎞
⎛
⎞
1
λ1
2
λ2
 1  2  1 λ2
= ⎜ ⎟ &
= ⎜ ⎟ =
= ⎜ ⎟
 0 ⎝ λ0 ⎠
 0 ⎝ λ0 ⎠
 0  1  0 ⎝ λ1 ⎠
⎛  2 ⎞
⎛  1 ⎞
⎛ λ2 ⎞
ln⎜ ⎟ = ln⎜ ⎟ − τ ln⎜ ⎟
⎝  0 ⎠
⎝  0 ⎠
⎝ λ1 ⎠
⎛ λ2 ⎞
⎛ λ1 ⎞
⎛ λ2 ⎞
ln⎜ ⎟ = ln⎜ ⎟ + ln⎜ ⎟
⎝ λ0 ⎠
⎝ λ0 ⎠
⎝ λ1 ⎠
Galilean relativity of scales!
Selfsimilar fractal : τ constant , τ1 = τ 2 = τ = DH − DT
t1=t2!
−τ
−τ
−τ
⎛
⎞
⎛
⎞
⎛
⎞
1
λ1
2
λ2
 1  2  1 λ2
= ⎜ ⎟ &
= ⎜ ⎟ =
= ⎜ ⎟
 0 ⎝ λ0 ⎠
 0 ⎝ λ0 ⎠
 0  1  0 ⎝ λ1 ⎠
⎛  2 ⎞
⎛  1 ⎞
⎛ λ2 ⎞
ln⎜ ⎟ = ln⎜ ⎟ − τ ln⎜ ⎟
⎝  0 ⎠
⎝  0 ⎠
⎝ λ1 ⎠
⎛ λ2 ⎞
⎛ λ1 ⎞
⎛ λ2 ⎞
ln⎜ ⎟ = ln⎜ ⎟ + ln⎜ ⎟
⎝ λ0 ⎠
⎝ λ0 ⎠
⎝ λ1 ⎠
X2=x1-t.v2/1!
V2/0=v1/0+v2/1!
Quantum paths as geodesics of a fractal space !
A!
B!
Quantum paths as geodesics of a fractal space !
A!
B!
Quantum paths as geodesics of a fractal space !
A!
B!
Quantum paths as geodesics of a fractal space !
A!
B!
Quantum paths as geodesics of a fractal space !
A!
B!
Quantum paths as geodesics of a fractal space !
A!
B!
* Born postulate!
* Ehrenfest theorem!
* von Neumann
postulate!
Back to Zeno’s paradox?!
“That which is in locomotion must arrive at the halfway stage before it arrives at the goal.”!
S=1.0!
S=0.0!
S=0.!
Back to Zeno’s paradox?!
“That which is in locomotion must arrive at the halfway stage before it arrives at the goal.”!
S=0.2!
S=0.0!
S=0.1!
S=0.3!
S=0.0!
S=0.2!
S=1.0!
Back to Zeno’s paradox?!
“That which is in locomotion must arrive at the halfway stage before it arrives at the goal.”!
S=0.20!
S=0.21!
S=0.0!
S=0.1!
S=0.3!
S=0.0!
S=0.21!
S=1.0!
Back to Zeno’s paradox?!
“That which is in locomotion must arrive at the halfway stage before it arrives at the goal.”!
S=0.20!
S=0.213!
S=0.210!
S=0.0!
S=0.1!
S=0.3!
S=0.0!
S=0.213!
S=1.0!
Back to Zeno’s paradox?!
“That which is in locomotion must arrive at the halfway stage before it arrives at the goal.”!
S=0.20!
S=0.2132!
S=0.210!
S=0.0!
S=0.1!
S=0.0!
S=0.3!
S=1.0!
S=0.2132…!
Finite curvilinear coordinate (proper time)
on an infinite length curve…!
Implications of non-differentiability!
* Infinite number of geodesic connecting any two points!
* Geodesics are fractal, with DH=2 playing a special role!
f (x,t) → f (x,t,δt)
* Time reflection symmetry breaking dt
€
dˆ 1 ⎛ d+ d− ⎞ i ⎛ d+ d− ⎞
= ⎜ + ⎟ − ⎜ − ⎟
dt 2 ⎝ dt dt ⎠ 2 ⎝ dt dt ⎠

-dt!
* Complex lagrange function and action S!
* Equations of motion from δS = 0 in Terms of ψ = e iS / S is !
0
∂
U

D Δψ + iD ψ −
ψ = 0 with possibly D =
∂t
2m
2m
2
€
€
€
…, no, in fact, this coming summer, !
I will be at the!
Observatoire de Paris-Meudon,!
trying to understand what is all this about!!
Have a good final week
.!
and a good summer afterward!