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Achilles, the Tortoise and Quantum Mechanics

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Achilles, the Tortoise and Quantum Mechanics
Alfred Driessen
prof. emer. University of Twente

In several places of his Physica Aristotle analyzes the famous antimony of Zeno about the competition between Achilles and the Tortoise. He emphasizes that any movement, or more general any change, is actually a continuum, i.e. an unity. It depends on the specific movement or change whether this continuum is potentially divisible in parts. In fact, there could be certain minima of the division. In line with this approach, Quantum Mechanics states that there are minima or quanta of movement (or change), with other words, there are no gradual changes in the world of micro- and nano-structures. This behavior is completely unexpected when starting with the mechanistic approach of classical physics.
Taking another finding of Aristotle, the four aspects of causality including final cause, one gets another ingredient of Quantum Mechanics. Movements and changes are not only influenced by the initial state -describing the present situation- but also by the final state which takes account of the future situation. As an example one may mention Fermi’s golden rule, where the initial and final state symmetrically determine the transition probability.
Bringing these two philosophical concepts of Aristotle together namely quanta of movement and final cause, a new light is shed on fundamental issues in Quantum Mechanics. One may mention the experimental evidence for contextuality, which is considered one of the weird phenomena in Quantum Mechanics. As illustration, some of the examples of experiments with optical microresonators are given.

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

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Achilles, the Tortoise and Quantum Mechanics

  1. 1. CSR: Culture, Science and Religion Achilles, the Tortoise and Quantum Mechanics pagina 1 5-1-2018
  2. 2. CSR: Culture, Science and Religion Achilles, the Tortoise and Quantum Mechanics pagina 2 5-1-2018 Content Introduction Playing with lines and trains Aristotle and Quantum Mechanics Quantum Contextuality Summary
  3. 3. CSR: Culture, Science and Religion Achilles, the Tortoise and Quantum Mechanics pagina 3 5-1-2018 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
  4. 4. CSR: Culture, Science and Religion Achilles, the Tortoise and Quantum Mechanics pagina 4 5-1-2018 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
  5. 5. CSR: Culture, Science and Religion Achilles, the Tortoise and Quantum Mechanics pagina 5 5-1-2018 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
  6. 6. CSR: Culture, Science and Religion Achilles, the Tortoise and Quantum Mechanics pagina 6 5-1-2018 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) levels of abstraction, according to Aristotle
  7. 7. CSR: Culture, Science and Religion Achilles, the Tortoise and Quantum Mechanics pagina 7 5-1-2018 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 levels of abstraction, according to Aristotle
  8. 8. CSR: Culture, Science and Religion Achilles, the Tortoise and Quantum Mechanics pagina 8 5-1-2018 Content Introduction Playing with lines and trains Aristotle and Quantum Mechanics Quantum Contextuality Summary
  9. 9. CSR: Culture, Science and Religion Achilles, the Tortoise and Quantum Mechanics pagina 9 5-1-2018 playing with lines and trains the static train-map of the Netherlands detail of line Haarlem-Amsterdam
  10. 10. CSR: Culture, Science and Religion Achilles, the Tortoise and Quantum Mechanics pagina 10 5-1-2018 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
  11. 11. CSR: Culture, Science and Religion Achilles, the Tortoise and Quantum Mechanics pagina 11 5-1-2018 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
  12. 12. CSR: Culture, Science and Religion Achilles, the Tortoise and Quantum Mechanics pagina 12 5-1-2018 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)
  13. 13. CSR: Culture, Science and Religion Achilles, the Tortoise and Quantum Mechanics pagina 13 5-1-2018 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)
  14. 14. CSR: Culture, Science and Religion Achilles, the Tortoise and Quantum Mechanics pagina 14 5-1-2018 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
  15. 15. CSR: Culture, Science and Religion Achilles, the Tortoise and Quantum Mechanics pagina 15 5-1-2018 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 -
  16. 16. CSR: Culture, Science and Religion Achilles, the Tortoise and Quantum Mechanics pagina 16 5-1-2018 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
  17. 17. CSR: Culture, Science and Religion Achilles, the Tortoise and Quantum Mechanics pagina 17 5-1-2018 Content Introduction Playing with lines and trains Aristotle and Quantum Mechanics Quantum Contextuality Summary
  18. 18. CSR: Culture, Science and Religion Achilles, the Tortoise and Quantum Mechanics pagina 18 5-1-2018 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.
  19. 19. CSR: Culture, Science and Religion Achilles, the Tortoise and Quantum Mechanics pagina 19 5-1-2018 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
  20. 20. CSR: Culture, Science and Religion Achilles, the Tortoise and Quantum Mechanics pagina 20 5-1-2018 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
  21. 21. CSR: Culture, Science and Religion Achilles, the Tortoise and Quantum Mechanics pagina 21 5-1-2018 Content Introduction Playing with lines and trains Aristotle and Quantum Mechanics Quantum Contextuality Summary
  22. 22. CSR: Culture, Science and Religion Achilles, the Tortoise and Quantum Mechanics pagina 22 5-1-2018 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
  23. 23. CSR: Culture, Science and Religion Achilles, the Tortoise and Quantum Mechanics pagina 23 5-1-2018 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
  24. 24. CSR: Culture, Science and Religion Achilles, the Tortoise and Quantum Mechanics pagina 24 5-1-2018 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)
  25. 25. CSR: Culture, Science and Religion Achilles, the Tortoise and Quantum Mechanics pagina 25 5-1-2018 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
  26. 26. CSR: Culture, Science and Religion Achilles, the Tortoise and Quantum Mechanics pagina 26 5-1-2018 evolution of the numerical solution of the Helmholtz Equation (Stoffer et al.) In Through Drop Basic principle of a MR
  27. 27. CSR: Culture, Science and Religion Achilles, the Tortoise and Quantum Mechanics pagina 27 5-1-2018 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 Gedanken experiments with MRs
  28. 28. CSR: Culture, Science and Religion Achilles, the Tortoise and Quantum Mechanics pagina 28 5-1-2018 Results of experiments With single “monochromatic” photons? a) detected with slow detector 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 II: probability to detect photon at the output ------------------------------------------------------------------------ b) detected with high time-resolution detector (resolution << τr) in out 1 out 2 out 3 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 Gedanken experiments with MRs input output 1 output 3 input output 1 output 2 output 3 input output 1 output 2
  29. 29. CSR: Culture, Science and Religion Achilles, the Tortoise and Quantum Mechanics pagina 29 5-1-2018 Content Introduction Playing with lines and trains Aristotle and Quantum Mechanics Quantum Contextuality Summary
  30. 30. CSR: Culture, Science and Religion Achilles, the Tortoise and Quantum Mechanics pagina 30 5-1-2018 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
  31. 31. CSR: Culture, Science and Religion Achilles, the Tortoise and Quantum Mechanics pagina 31 5-1-2018 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.
  32. 32. CSR: Culture, Science and Religion Achilles, the Tortoise and Quantum Mechanics pagina 32 5-1-2018 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
  33. 33. CSR: Culture, Science and Religion Achilles, the Tortoise and Quantum Mechanics pagina 33 5-1-2018 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

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