Download Limits in imaging - The Brijuni Conference

Limits of imaging
S. Danko Bosanac
Brijuni 2006
Imaging
Primitive
Direct inversion
Object
optical media
Identification
Advanced
Modeling
Model
media
mathematical
interpretation
Limitations of direct inversion
• Mimicry
Not sufficient information on the object
• Illusion
Optical media is at fault
• No identification (lack of reference object or limitations
by physical laws)
Advanced imaging
• Human invention
• Roots in organized societies (prophets,
soothsayers, priests)
– Basic want: to predict events
– Basic information: model of Nature
– Basic intermediary: supreme being
• Birth of science (scientists)
– Supreme being is replaced by mathematics
Mimicry
The nature of gravity
Ptolemy model
xj
xz
r1 cos
1t
r2 cos
2
t
r3 cos
3t
yj
yz
r1 sin
1t
r2 sin
2
t
r3 sin
3t
xJupiter-xEarth
xJupiter-xEarth
t
Spectrum of xJupiter-xEarth
w
Epicycles for Jupiter
w1
0; w2
0.2247; w3
r1
0.6089; r2
0.4492; w4
2.6828; r3
1
0.2; r4
1
Mimicry
The nature of electromagnetic field
Newton conjecture
Light is a stream of particles
• Supporting argument:
Particles go in straight lines and
obey simple laws of reflections.
Beam of light behaves likewise.
Light is a stream of particles
Wave nature of light
• Experiment: beam of light
cannot have a sharp edge
(Fresnel diffraction).
• Theory: the wave that has
the wavelength much shorter
than the width of the beam
obeys the rules of geometric
optics.
The argument for the particle
nature of light is
invalidated.
Particle nature resurrected
• Lewis argument
– Charge-EM wave energy/momentum transfer is the
same as for particles
• Argument against
– Interference, diffraction
• Proof against particle nature
– Frequency dependent energy/momentum transfer is
derived from quantum dynamics
Compromise
• Instead of rejecting particle nature of the
electromagnetic field a new idea in
science is introduced
dualism
Essence of particle manifestation of
the electromagnetic field
i
A
f
t
1
f
2m
f
i k.r i t
A0 e
f0
Vf
e
i k.r i t
A0 e
i k.r i t
A. f
f0
Photon counter
• Based on the effect of ejecting electrons
from atoms by the electromagnetic plane
wave.
• Probability of ejecting is proportional to:
amplitude squared of the field - modulus of
the Poyting vector - number of photons
Weaknesses of the photon model
• Compton effect must assume a bound
charge
• Rabi oscillations not explainable
• Drift velocity not explainable
Illusion
Resonances
• Typical quantum phenomenon in dynamics
of particles:
– Decay
• Nuclei, elementary particles, dissociation of
molecules
– Collisions
• Elementary particles, long lived transition states,
quantum gas
Time delay
• Solution for delocalized
probability amplitude
• Solution for localized
probability amplitude
• Stationary phase of
integrand
• Error: stationary phase
method is not valid if the
modulus of the integrand is
sensitive to integration variable
inc
sc
b
sc
fsc
e
A
C
e
e
er
e
t
e
er
ikr iet
i
b
e
e
er
e
ikr iet
e
i
ikr iet i ArcTan
e
e er
e
i Arg A e
0
mr
k
1
e
Arg A
er
e
Time delay and resonance
scattering
Time delay
Long
Short
Illusion
Klein paradox
• Great impact on merging relativity and
quantum mechanics
• Based on its predictions it is accepted that
merger is not possible without quantizing
field of particles
• Its predictions are:
– Creation of particle-antiparticle pairs
– Particles cannot be bound
Scattering on infinite potential wall
non relativistic dynamics
Standard derivation
f
ipz
e
a p e
ipz
iP p z
b p e
e
V
P p
e
v
2
; z
; z
0
1
z
p 1
a p
b p
a
2
e
v
P 1
e
1 e P p 1 e
2 p 1 e v
v
1
v
e P
p 1
P 1
e
1 e
v p
e
b
p
v
1
v
2
1
p
e
e
0
Errors in derivation
there are many
• Plane wave in z>0 does
not represent outgoing
wave
• Equation for wave
function has singularity
eiPz
z,z f
ei v z
v
1
e v 1
• Exact solution is
v
iet
z
f
e v
iet
A f e
Im(p)
A f e
z
i pz
• where integration path
should be defined in
Riemann p-plane
i
-p
2
-p
p
1
-i
1
p
2
Re(p)
2
1 f
p
Scattering on hard wall
standard boundary condition
Scattering on hard wall
“physical” boundary condition
No reference object
spin-statistics connection
• Fundamental empirical postulate
– Spin dependent symmetry of the probability
amplitude for a system of identical particles
• Problem
– Symmetry has origin in dynamics, what is it
for this postulate?
Example
super fluidity
• He4 and He3 have different super fluid
properties.
• Dynamic parameters that determine
fundamental difference
– Mass and magnetic dipole moment
• Mass determines the smallest stable cluster
• Potential energy of two magnetic dipole
moments for protons at r=10-10 m
– V~10-10 eV
Claim
• Fundamental difference of two liquids is
the consequence of the nuclear spins of
the two He isotopes, resulting in:
different symmetry of the probability
amplitude for the two systems of He atoms
• Question:
What is the interaction that causes the
difference in symmetry?
Role of symmetry
I
Particle is localized in
II
B
B
well I
f
fs
fa
well II
f
fs
fa
For 2 particles one defines
fa
fs
f1
fs 1 fs 2 ; f2
f3
fs 1 fa 2
fs 2 fa 1 ;
f4
fs 1 fa 2
fs 2 fa 1
Particle 1 in well I and particle 2 in well II
f
f1
f2
f4
fa 1 fa 2
Two identical particles in harmonic field
Model for connecting interaction with permutation
symmetry
1
1
1
2
1
2
2
1
r
r
r2
4
r1
2
r1
2
1
R
2
2
4
2
2
r2
2
2
2
r
1
R
1
R
2
2
r2
V
r2
r1
V r
r1
Solution
r1, r2
f r g
For repulsive V r
R
boundary condition is f r
In terms of unperturbed solutions
r1, r2
n
r1
m
r2
n
r2
n
m
r
r1
0
No reference object
• What is a reference object?
– Built from classical experience
Force plays the central role
• Limits of classical experience
– Quantum effects
– Relativistic effects
– Field reaction
Imaging without reference object
Mathematical modeling
• Examples