Achilles, the Tortoise and Quantum Mechanics

CSR: Culture, Science and Religion Achilles, the Tortoise and Quantum Mechanics pagina 1 5-1-2018

Content
Introduction
Playing with lines and trains
Aristotle and
Quantum Mechanics
Quantum Contextuality
Summary

Achilles and the Tortoise
figure taken from: https://ibmathsresources.com/2014/08/27/zenos-paradox-achilles-and-the-tortoise/
paradox of
Zeno of Elea, 490-430 BC
figure taken from
https://www.youtube.com/watch?v=EfqVnj-sgcc

preliminary solution: via mathematics
By Jochen Burghardt - Own work, CC BY-SA 3.0,
https://commons.wikimedia.org/w/index.php?curid=29475780
Distance vs. time,
assuming the tortoise to
run at Achilles' half speed
time:
20 × 𝐾 = 20 [𝑠]
distance:
200 × 𝐾 = 200 [𝑚]
𝐾 = lim
𝑘→∞
෍
𝑛=1
𝑘
1
2𝑛
= 1

levels of abstraction, according to Aristotle
1st level of abstraction
considers things which can neither exist nor be
understood without matter
example:
bird: many forms, colors, seize,
first level of abstraction:
bird as such: no specific color, form,...
is an animal with feathers, two legs, with wings
result: the concept: bird
by abstraction from things which enter in the mind by
the aid of senses
concrete bird:
is something changing, each time different
concept of a bird:
does not change, is stable in time
with concepts: science becomes possible

2nd level of abstraction
it allows to study things which can be
understood without any references to sense
qualities, but which exist only implemented
in matter
example: circle:
may by abstracted completely
from the material
arriving at the level of geometry, more
general mathematics
example of increasing abstraction:
2 apples plus 3 apples are 5 apples
2+3=5
a+b=c
a+b=b+a (addition is commutative)

3rd level of abstraction
considers things which can be thought
of -and exist- apart from matter
example:
being (ens), form, substances,
essence, love
level of the being as such
enter realm of metaphysics
studies the general properties and
structure of beings,
and the relations between beings;
among others causality
Raphael, School of Athens
with Plato and Aristotle

Content
Introduction
Aristotle and
Quantum Mechanics
Summary

playing with lines and trains
the static train-map of the
Netherlands
detail of line Haarlem-Amsterdam

the three levels of abstraction of a line
starting point: the railway line between Amsterdam and Haarlem
1. level: the railway connection with certain physical properties
example: rail connection between two cities
2. level, mathematical representation
example: a line between point A and point B
3. level, metaphysical representation
example: the thing with a certain extension, the continuum:
“ens extensum”
Leibnitz:
1672-1682
the Labyrinth
of the
continuum

What are the parts of a line?
step 1: take the line between A and B at cut at point C1,
result: two lines
step 2: step: take the line between A and C1 at cut at point C2,
result: two + 1 lines
step n: step: take the line between A and Cn-1 at cut at point Cn,
result: two + n lines
repeat step n ad infinitum
result: -any line consists of infinite lines, each of these parts has on its
own infinite line-parts, which on its own have infinite line-parts, which on its
own………
-points (zero dimension) can not be obtained by cutting a line
C1
A B

playing with lines and trains
the dynamical train-map of
Haarlem-Amsterdam
t=0
t=20
t=18
t=16
t=14
t=12
t=10
t=8
t=6
t=4
t=2
train
is moving from Haarlem, t=0 min,
to Amsterdam at t=20 min
Haarlem (A) Amsterdam (B)

what are the parts of a dynamical line?
the dynamical train-map of
Haarlem-Amsterdam
t=0
t=20
t=18
t=16
t=14
t=12
t=10
t=8
t=6
t=4
t=2
train
is moving from Haarlem, t=0 min,
to Amsterdam at t=20 m
A (s=0 km, t=0 min)
B (s=22 km, t=20 min)
reduced to 1 space-
and 1 time-
coordinate
(line A-B)
Haarlem (A) Amsterdam (B)

What are the parts of a dynamical line?
step 1: take the dynamical line between A and B at cut at point C1,
result: two dynamical lines
step 2: take the dynamical line between A and C1 at cut at point C2,
result: two + 1 dynamical lines
step n: take the dynamical line between A and Cn-1 at cut at point Cn,
result: two + n dynamical lines
repeat step n ad infinitum
result: -any dynamical line consists of infinite dynamical lines, each of
these parts has on its own infinite dynamical line-parts, which on its own
have infinite dynamical line-parts, which on its own………
-motionless points (zero dimension) can not be obtained by
cutting a dynamical line
A (s=0 km, t=0 min)
B (s=22 km, t=20 min)
C1

What are the parts of a line?
Leibnitz: -Labyrinth of continuum,
ens extensum
dilemma: -how to build-up something
from infinite, unequal parts?
physical level: -there are minima for any physical line:
e.g. rail of iron: minimum the Fe atom:
mathematical -there are no minima, but even the smallest of the infinite
level number of parts are lines;
-it is not always possible to construct the whole from the
parts;
metaphysical: -a line has potentially infinite parts, but as long as it is not
level actually devided, it has no parts
-

To be, or not to be?
great insight of Aristotle:
there is a third between being and not being:
virtual or potential being
applied to lines or continuum:
the line (continuum) is a whole,
but there are potentially parts
smallest seize of the potential parts:
so called minima naturalia
is determined by the physical level
what about dynamical lines (continua)?
are there minima naturalia?
already discussed by Suarez Disp. Met, 1597

Content
Introduction
Aristotle and
Quantum Mechanics
Summary

Achilles and the Tortoise
Solution of Aristotle for Zeno’s Antinomy
1) movement is a dynamical continuum,
has to be considered as a whole.
2) Being a whole,
movement is largely determined by the
initial and final point (telos).
3) Can a certain movement potentially be divided in parts?
This is determined by the physical properties.
4) In Quantum Mechanics (QM):
- movement (changes) are quantified;
- there are minima of movement;
- the initial and final state enter symmetrically in the
mathematical description.

Movement as a whole
-- If movement (change) have to be considered as a whole then the final
state and initial state have causal influence on the whole.
-- During the -not interrupted- movement, information about the
condition of the movement is contained in the initial and final state, and
the physical conditions of the possible trajectories.
-- Initial state and final state may be (or not be) separated in time.
-- In Quantum Mechanics the movement of (small)
objects is quantified.
-- In Aristotelian terms: movement has to be
considered as a whole: there are quanta of
movement.
-- In the metaphysical and mathematical level:
there are potential parts
-- In the physical level there are no parts: the
quantum of movement can not be divided.
Aristotle by Rembrandt

Fermi’s Golden rule
Sommerfeld 1930 (Scientia 1930 II, p 85; translation by AD)
When on occasions I spoke about a new, conditioned causality, it was
mathematically founded. It appears namely that we have to calculate
emission by a formula, in which the initial and final condition of the atom
enters equally and symmetrically.
(...) By the way, this is not completely new. Aristotle considered besides
the efficient cause also the final cause.
taken from: Snoeks et al.
PRL, 74, 13, 1995, p 2459

Content
Introduction
Aristotle and
Quantum Mechanics
Summary

quantum contextuality
Quantum contextuality:
-- All what we may know is the initial and final state; there is nothing in-
between: no information, no physical properties of moving particles....
-- Final state: is often called the observer.
-- The outcome of an experiment with a given initial state depends
critically on the observer, or better said, the final state set-up by the
experimentalist or by other causes involved.
Weirdness of Quantum contextuality
expressed by Ian Durham:
When you go to your garage, you get a car;
you don’t get a llama.
Perhaps if you live on a ranch in the Andes
Mountains.
quoted in plus-magazine by Brendan Foster

Light can be considered as a particle, called photon,
with energy and momentum
that propagates with the speed of light c
h: Planck constant; ħ = h/2 π; k and ω: spatial and (time) angular frequency
The photon is the number one quantum particle
In the following, gedachten-experiments with photons in optical micro resonators
(MC) will be presented.
They show that the outcome of the experiments is critically dependent on the
detection set-up (the observer).
Interference (wave-picture) or quantum superposition (particle picture) lead to
pronounced quantum effects: quantum contextuality
for more details, see:
photons and quantum contextuality
E kP 
https://www.researchgate.net/project/The-weird-properties-of-a-photon

top view spectral respons
The integrated optical MR
wavelength [nm]
1550 1560
FSR
-3dB
P/P[dB]throughinP/P[dB]dropin
Lorentzian lineshape Free Spectral Range: FSR
linewidth: Δλ-3dB
Finesse: FSR/ Δλ-3dB
roundtrip time τr
literature: A. Driessen et al., Optics Com. 270, 217-224 (2007)

input output 1
output 3
input output 1
output 2
output 3
input output 1
output 2
Experiments with straight (green)
And bent waveguides (yellow)
Experiment I:
1 straight and 1 curved
waveguides coupled with loss-less
10% intensity coupler
Experiment II:
2 straight and 1 curved
waveguides coupled with loss-less 10%
couplers
Experiment III:
2 straight waveguides and
1 ring-resonator in resonance coupled
with loss-less 10% intensity couplers
Gedanken experiments with MRs

evolution of the numerical solution of the
Helmholtz Equation (Stoffer et al.)
In Through
Drop
Basic principle of a MR

Results of experiments
With monochromatic cw laser
in out 1 out 2 out 3
experiment I 1 0.9 --- 0.1
experiment II 1 0.9 0.01 0.09
experiment III 1 0 1 ---
Table I: normalized intensities at input and output
With single “monochromatic” photons?
a) detected with slow detector
b) detected with high time-resolution detector
(resolution << τr , the roundtrip time in the MR)
input output 1
output 3
input output 1
output 2
output 3
input output 1
output 2

Results of experiments
With single “monochromatic” photons?
a) detected with slow detector
experiment I 1 0.9 --- 0.1
experiment II 1 0.9 0.01 0.09
experiment III 1 0 1 ---
Table II: probability to detect photon at the output
------------------------------------------------------------------------
b) detected with high time-resolution detector
(resolution << τr)
exper. I 1 0.9 --- 0.1
exper. II 1 0.9 0.01 0.09
exper. III, 1 0.95 0.05 ---
Table III: probability to detect photon at the output
input output 1
output 3
input output 1
output 2
output 3
input output 1
output 2

Content
Introduction
Aristotle and
Quantum Mechanics
Summary

The antinomy of Zeno leads Aristotle to a deeper understanding of
movement (change), and the relation between the whole and the parts.
By distinguishing three main levels of abstraction (physics, mathematics
and metaphysics) he is able to find a balance between empirical
(physics) and a-priori (mathematics and metaphysics) sciences.
He introduces a third alternative between being and not being: potentially
being.
His emphasize on the wholeness of a movement opens a road to what
later is called quantum mechanics.
His analysis of causality including final cause appears explicitly in the
mathematical formalism of quantum mechanics.
It is hopefully shown that it is worthwhile to involve metaphysics for a
deeper understanding of the main theories of modern physics, relativity
and QM.
Summary

Summary
Can Science and Technology Shape a New Humanity?
As science progresses, it is constrained to
introduce into its theories concepts of a
metaphysical nature – like those of time, space,
objectivity, causality, individuality.
De Broglie, 1892-1987
Revue de metaphysique et de morale, 1947, 3, p 278.
In the current situation of relativism and postmodern thinking science
could contribute that old and new truth gathered in the best philosophical
traditions will show unexpected actuality.
It is the dialogue between scientists and philosophers that could lead to
satisfying answers for the many big questions that challenge our
understanding.

Acknowledgment
Many of the ideas of this presentation
have been derived from a book of
Prof. P. Hoenen, S.J. (1880-1961)
Pontifical Gregorian University
Philosophie der Anorganische Natuur
Standaard Boekhandel, Antwerpen ,
3rd edition 1947
(philosophy of inorganic nature)
Latin edition:
Cosmologia, Rome, 1945

This talk has been presented at the
20th International Interdisciplinary Seminar
Can Science and Technology Shape a New Humanity?
Netherhall House, London, 5-1-2018

Achilles, the Tortoise and Quantum Mechanics

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