vol I chap 1 sect 2

Previous: 1.1. Mathematics and science in ancient Greece.

1.2. Conceptualizations of spaces and quantum statistics.¶

Conceptual developments in ancient Greece.¶

The first part of this section is based on the conference Géométrie et astronomie sphérique dans la première cosmologie grecque by Vernant (1963). He considered how the evolution of ancient Greek society generated two kinds of conceptual developments, all of them before the Christian era:

(1) the building of circular spaces in family life and community (the reference is made to the philosopher Pherecydes, the poet Homer, the historian and geographer Herodotus and the architect Hippodamus), and

(2) the concept of circularity in the definition of astronomical spaces (reference is made to the poet and philosopher Hesiod and to the philosopher, geographer and physicist Anaximander).

Building circular spaces according to political organizations.¶

Before political institutions were established in Greece, the highest level in the social organization was occupied by the king-priest. The entire population was always under conditions of domination and submission. There were no spaces for communications. Public affairs were not discussed, they were imposed. However, laws and rules were promulgated and made public in written.

Pherecydes of Syros (580 BC – 520 BC).¶

When Greek cities were built (the polis) writing was used as a transforming instrument that made public private issues. Probably it was Pherecydes of Syros (580 BC – 520 BC) the first writer to publish a philosophical work in prose. He wrote books for transforming private knowledges into public possibilities for discussion.

Herodotus of Halicarnassus (484 BC- 425 BC).¶

Herodotus of Halicarnassus (484 BC- 425 BC) mentioned that when the dictator Polycrates of Samos died, his successor asked all the citizens to meet in an assembly and informed them that he will behave in a democratic way in total disagreement with his predecessor.

Homer (VIII century).¶

Homer (VIII century) wrote in Odyssey (chant II) that Telemachus, son of Ulysses and Penelope, was worried because many pretenders to the throne were harassing his mother. He asked soldiers to protect her by forming an agora (Greek word for an assembly). It is attributed to the same writer the use of the word ageirien laon, that means to congregate the army.

Hippodamus of Miletus (498 BC- 408 BC).¶

One of the first urban architects Hippodamus of Miletus (498 BC- 408 BC) rebuild Miletus around 479 BC. The city was sacked and destroyed by Persians in 490 BC. The map made by Hippodamus shows an ordered reticular structure having an agora as its center with broader and straight streets (Figure 1.4a). For comparison, plans of other old sites are shown in Figure 1.4, b, c and d.

The agora.

As a part of a demystification and rationalization process of the social life, the agora was a center for political actions and decisions with social, urban and cultural implications for the community. In that place democracy and law application were expressions of equality, equilibrium, symmetry and reciprocity. This was an indication of the appearance of new politically centralized institutions. See Figure 1.4.


(a)	(b)

(c)	(d)

(Images credit: CC Wikimedia Commons)

Figure 1.4. Plans of different archeological sites: (a) Mileto in Asia Minor, nearby 490 BC.; (b) Sun pyramid in Teotihuacan, Mexico, nearby 450 AC; © residential palace in Machu Pichu, Peru, nearby 1500 AC, and (d) Massada in Israel, nearby 20 BC.

Astronomical and geometrical spaces.¶

According to Vernant, the new image of the democratic Greek society went in parallel with a new image of the astronomical and geometrical spaces. This conceptual change implied overcoming astronomical Babylonian ideas as well as eliminating myths about the form of the Earth.

Astronomy in Babylon characterized a religion believing that stars were divinities whose intentions might be perceived if their positions in the sky were carefully observed and registered. Scribes serving the king were in charge for registering the economic activity of the kingdom and for accounting all celestial and terrestrial events. Their knowledge was arithmetic without connections to any geometrical system of spatial representations of positions and movements.

However, Greek ancient astronomy was looking for explanations about the structure of the world without appeal to divinities nor ritual ceremonies, although it was full of myths.

Hesiod of Askra, Thebes (nearby 800 BC).¶

Hesiod of Askra, Thebes (nearby 800 BC) wrote a Theogony, a mythological treatise describing the origin of gods. According to him the Earth was a vessel whose interior contained a structured world enclosed by Zeus to avoid the perception of light in the disordered world: at the upper level lived Zeus and immortal gods, then the human beings and downward death and underground gods.

Anaximander of Miletus, Ionia, (619 – 546).¶

Anaximander of Miletus, Ionia, (619 – 546) had a spherical notion of the universe whose center contained the Earth as a cylindrical column at equilibrium, stagnant and without falling. This was a democratic conceptualization of space were all positions and distances were mathematically defined.

Cognitive spaces in quantum statistics: fermions or bosons.¶

The notion of quantum statistics is now analyzed by looking at two Physics Nobel Prizes: the 1945 Prize to Wolfgang Pauli “for the discovery of the Exclusion Principle, also called the Pauli Principle”, and the 1954 Prize to Max Born “for his fundamental research in quantum mechanics, especially for his statistical interpretation of the wavefunction”. The 1945 Prize was also awarded to Walther Bothe "for the coincidence method and his discoveries made therewith".

Next, for each Nobel Laureate we include a brief description of his work as presented in the document called Works published in the Nobel wave page and indicate the titles of their Nobel Lectures. References for both documents are mentioned in MLA format (MLA means Modern Language Association).

Furthermore, each Nobel Lecture is described in two boxes: one for Accepted knowledge or questions under discussion in Laureate´s time and another for Laureat´s contributions or explanations. Quotation marks indicate complete quoted paragraphs.

1945 Physics Nobel Prize awarded to Pauli.

WORK: "In Niels Bohr’s model of the atom, electrons move in fixed orbits around a nucleus. As this model developed, electrons were assigned certain quantum numbers corresponding to distinct states of energy and movement. In 1925, Wolfgang Pauli (1900-1958) introduced two new numbers and formulated the Pauli principle, which proposed that no two electrons in an atom could have identical sets of quantum numbers. It was later discovered that protons and neutrons in nuclei could also be assigned quantum numbers and that Pauli’s principle applied here too."

MLA style: Wolfgang Pauli – Facts. NobelPrize.org. Nobel Prize Outreach AB 2023. Sat. 25 Feb 2023. https://www.nobelprize.org/prizes/physics/1945/pauli/facts/

NOBEL LECTURE: Exclusion Principle and Quantum Mechanic.

MLA style: Wolfgang Pauli – Nobel Lecture. NobelPrize.org. Nobel Prize Outreach AB 2023. Sat. 25 Feb 2023. https://www.nobelprize.org/prizes/physics/1945/pauli/lecture/

BOX 1.1: Accepted knowledge or questions under discussion in Pauli´s times.

Einstein relativity theories were not well understood.
The complete description of the quantum state of an electron in an atom required four quantum numbers: the principal quantum number \(n\), the azimuthal quantum number \(l\), the magnetic quantum number \(m\), and the spin quantum number \(m_s\).
Uhlenbeck and Goudsmit introduced the idea of electron spin (intrinsic angular momentum).
Three crucial contributions to quantum mechanics have been made:
1. Matter waves by De Broglie: To each quantum particle a wave function is associated in such a way that the square of the amplitude of the function describes the probability of occupation in space.
2. Wave mechanics equation by Schrödinger: This equation corresponds to the quantum counterpart of Newton´s second law in classical mechanics.
3. Matrix formulation by Heisenberg: The formulation of quantum mechanics serves to interpret the physical properties of particles as matrices that evolve in time.

BOX 1.2: Pauli´s contributions or explanations.

"As soon as the symmetry classes for electrons were cleared, the question arose which are the symmetry classes for other particles with symmetrical wave functions like photons. We note that the symmetrical class for photons occurs together with the integer value 1 for their spin, while the antisymmetrical class for the electron occurs together with the half-integer value ½ for the spin."
"In order to prepare for the discussion of more fundamental questions, we want to stress here a law of Nature which is generally valid, namely, the connection between spin and symmetry class. A half-integer value of the spin quantum number is always connected with antisymmetric states (exclusion principle), an integer spin with symmetrical states. This law holds not only for protons and neutrons but also for protons and electrons. Moreover, it can easily be seen that it holds for compound systems, if it holds for all of its constituents. If we search for a theoretical explanation of this law, we must pass to the discussion of relativistic wave mechanics, since we saw that it can certainly not be explained by non-relativistic wave mechanics."
"In his lecture delivered here in Stockholm (*) he himself explained his proposal of a new interpretation of his theory, according to which in the actual vacuum all the states of negative energy should be occupied and only deviations of this state of smallest energy, namely holes in the sea of these occupied states are assumed to be observable. It is the exclusion principle which guarantees the stability of the vacuum, in which all states of negative energy are occupied. Furthermore, the holes have all properties of particles with positive energy and positive electric charge, which in external electromagnetic fields can be produced and annihilated in pairs. These predicted positrons, the exact mirror images of the electrons, have been actually discovered experimentally."

(*) Pauli refers to a lecture delivered by Dirac.

1954 Physics Nobel Prize awarded to Born.

WORK: "In Niels Bohr’s theory of the atom, electrons absorb and emit radiation of fixed wavelengths when jumping between orbits around a nucleus. The theory provided a good description of the spectrum created by the hydrogen atom, but needed to be developed to suit more complicated atoms and molecules. Following Werner Heisenberg’s initial work around 1925, Max Born (1882-1970) contributed to the further development of quantum mechanics. He also proved that Schrödinger’s wave equation could be interpreted as giving statistical (rather than exact) predictions of variables."

MLA style: Max Born – Facts. NobelPrize.org. Nobel Prize Outreach AB 2023. Sat. 25 Feb 2023.
https://www.nobelprize.org/prizes/physics/1954/born/facts/

NOBEL LECTURE: The Statistical Interpretations of Quantum Mechanics.

MLA style: Max Born – Nobel Lecture. NobelPrize.org. Nobel Prize Outreach AB 2023. Sat. 25 Feb 2023. https://www.nobelprize.org/prizes/physics/1954/born/lecture/

BOX 1.3: Accepted knowledge or questions under discussion in Born´s times.

Niels Bohr introduced the idea that electronic energy levels were discrete stationary states. This assumption explained the stability of the atoms, the structure of their electronic shells, and the Periodic System of the elements. Transition between these discrete levels corresponded to quantized emissions or absorptions.
Bohr also formulated his principle of correspondence which requires that for transitions in between energy levels with very high quantum numbers the changes in energy are so small that practically coincide with the continuum of values predicted by classical mechanics.
Einstein proposed that the concept of intensity of radiation must be replaced by the statistical concept of transition probability. For each transition there must be a corresponding intensity. However, the idea of an amplitude of oscillation associated with each transition was indispensable.
Heisenberg disregarded the picture of electron orbits with definite radii and periods of rotation because these quantities were not observable. Instead of describing the motion by giving a coordinate as a function of time, \(x(t)\), an array of transition amplitudes \(x_{nm}\) should be determined.
By solving his equation, Schrödinger obtained the stationary states of the hydrogen atom and demonstrated the complete equivalence of the two systems of explanation: matrix mechanics and wave mechanics. Instead of speaking of electrons as particles, he considered them as continuous density distributions \(‖ψ‖^2\).

BOX 1.4: Born´s contributions or explanations.

By following Einstein’s idea that for photons the square of the optical wave amplitude represents the probability density for their occurrence, \(‖ψ‖^2\) could represent the probability density for electrons (or other particles).
Introduction of the square arrays of quantities called matrices and applied to Heisenberg’s quantum condition requiring that coordinates \(q\) and momenta \(p\) cannot be represented by figure values but by symbols, and that their product depends upon the order of multiplication - they are “non-commuting”.
The statistical interpretation of quantum mechanics indicates that transition probabilities can be calculated if the wave function is time and space dependent as well as a function of the discrete index \(n\) which enumerates the electronic stationary states.
Discussion of the essentially indeterministic statistical interpretation of quantum mechanics and the questions of reality, indeterminacy, and complementarity.

Next: 1.3. Contexts learning for conceptualizing cognitive spaces.