Mysticism

Quantum physics, magic and the unconscious

Symmetry as the language of physics, number as the archetype of order, and a 30-gon at the heart of the Lie group E8. Where matter, symbol and soul touch.

Magic, mysticism and modern physics seem to have nothing to do with one another. And yet all three work with the same raw material: symmetry. The magician draws a circle around a centre, the Kabbalist joins 22 letters into a circle of 231 gates, and the physicist describes the fundamental forces of nature with symmetry groups. The question of this chapter: is it the same order, once in the image, once in the soul and once in matter?

Symmetry is physics

In 1918 the mathematician Emmy Noether proved a theorem that became a foundation of modern physics: every continuous symmetry of a physical system has a conserved quantity. That the laws of nature are the same tomorrow as today means conservation of energy. That they are the same here as there means conservation of momentum. That they hold in every direction means conservation of angular momentum. Symmetry is not nature’s ornament but its law.

Particle physics today is at its core a theory of symmetry groups. The forces between elementary particles are described by Lie groups, continuous symmetries that can be pictured as rotations in abstract spaces. Their structure can be represented by root systems: sets of vectors arranged symmetrically around a centre. Drawn in the plane, they produce figures that look like simplexes, stars and mandalas.

E8 and the 30-gon

The largest of the five so-called exceptional groups, and the most enigmatic, is E8. It has 248 dimensions. Its root system consists of 240 vectors in eight-dimensional space. Projected onto the right plane, these vectors arrange themselves in eight rings of 30 points each: 8 × 30 = 240. Each ring is a regular 30-gon.

240 roots · 8 rings of 30 · 6720 edges

The 240 roots of E8, projected into the Coxeter plane and computed here from the coordinates. The colours distinguish the eight rings, the lines join neighbouring roots.

The 30-gon is no accident. 30 is the so-called Coxeter number of E8, and it follows necessarily from the structure of the group. And there is a second, less well-known fact: E8 has eight numbers called its exponents, and they are 1, 7, 11, 13, 17, 19, 23 and 29. They are exactly the numbers below 30 that share no divisor with 30. That 30 = 2 × 3 × 5 is at the same time the grid that orders the primes next after the rhythm of six (see The order of the primes) is an appealing parallel, for in both cases the numbers coprime to 30 are at stake. A deeper connection, however, is not proven by this. For comparison, the simplex of the 30-gon, every point joined to every other:

Points

The 30-gon simplex: 30 points, 435 connections. Its figures show all divisors of 30 and all numbers coprime to it.

The figures of the E8 projection and of the 30-gon simplex look very alike. Both have the symmetry of the 30-gon, and in both the numbers coprime to 30 play a part. This is a real kinship, but one should not stretch it. The simplex is a figure from number theory, E8 a structure of algebra in eight dimensions, and that both look alike in the plane is due to their common symmetry, not to one being contained in the other. In 2007 the physicist Garrett Lisi even proposed fitting all particles and forces into E8. His theory has not held up in this form: in 2010 Jacques Distler and Skip Garibaldi showed that it cannot produce the three families of particles. Independently of this, in 2010 excitations were measured in a magnetic crystal of the kind Alexander Zamolodchikov had predicted in 1989 for a chain of magnetic atoms with the symmetry of E8: the energies of the two lowest stand almost in the ratio of the golden section.

All is frequency

Quantum physics began with a discovery by Max Planck in 1900: energy is emitted in packets, and the energy of a packet is proportional to its frequency, E = h · f. From this Einstein for light and de Broglie for matter later concluded that every particle has a wave and every wave a particle. With this the Pythagorean idea that the world is ordered by ratios of vibration arrived in physics (see All is number, all is frequency). In a lecture in Florence in 1944 Planck himself expressed the view that behind matter stands a mind that holds it together (see Great thinkers on the structure of the world). That was his personal conviction, not a consequence of the formula E = h · f.

The magician and the unconscious

A very speculative but intriguing field belongs here too: seeing symmetries in altered states of consciousness, through meditation or psychedelic substances. Many reports describe patterns of stars, lattices and mandalas, ordered around a centre. This can only stand here as a prompt for one’s own inquiry.

Magic, as handed down, works not with physics but with images. The magic circle (see The throne of God and the number 24) is a symbol meant to act on the magician’s unconscious: he stands at its centre, invokes the four elements and enters a trance. The Kabbalistic ritual of the 231 gates also works with visualisation: each letter is imagined, held and joined with all the others until the whole 22-gon stands before the inner eye.

The magic circle: an order around a centre, built of square, hexagrams and pentagrams.

Why should these figures in particular act on the soul? One answer was given by the psychologist Carl Gustav Jung. He observed that people in crisis spontaneously draw mandalas: circles around a centre, usually divided into four. He read them as images of the Self, the wholeness of the psyche, and fourness, the quaternity, as its basic pattern. Jung’s psychology knows four basic functions of consciousness, thinking, feeling, sensation and intuition, which recall the four elements and the four temperaments (see The tetractys in mysticism and religion).

For Jung, number was an archetype, a primal form of order that belongs to the soul as much as to the world. It is just as possible, in his view, that the natural numbers were not invented but found. They have properties nobody knew of beforehand. This matches Philolaus’ statement that the order of number exists by nature and not by convention.

Pauli and Kepler

The bridge between physics and soul was built by none other than one of the founders of quantum physics. The Nobel laureate Wolfgang Pauli worked with Jung for years. In 1952 they published a book together. Pauli’s contribution was titled The Influence of Archetypal Ideas on the Scientific Theories of Kepler. In it Pauli showed how Kepler’s planetary laws grew out of an inner image: the sphere with its centre as an image of the Trinity, and the conviction that God created according to geometric archetypes. Pauli’s thought can be summed up thus: exact science too draws on inner images, from the same sources as mysticism and religion. But its ideas are tested against experience, and Kepler even gave up the perfect circular orbits when observation disagreed, although he held on to his image of the five solids all his life (see Astronomy).

Three worlds, one order?

Set the three realms side by side and a picture emerges:

In all three columns stands the geometry of circle and centre, structured by polygons and star polygons. What follows from this? Here it pays to distinguish carefully, as the four questions from Chance and pattern demand.

Proven are Noether’s theorem, the Coxeter number 30 of E8 and the fact that its exponents are the numbers coprime to 30. Observed is that people in many cultures and states of mind draw and see circular figures around a centre. Interpreted is everything that links the three columns. The table shows similarities of form, not a common mechanism. That physicist, magician and soul reach for the circle may simply be because the circle is the simplest form in which everything is equal towards the centre.

On this view the tetractys is a key with which the three worlds can be read side by side, and a spur to ask for their common ground. That this common ground really exists is a hope shared by Pythagoras, Kepler and Pauli. It is not proven.

Sources and further reading

Sources

• Emmy Noether, Invariante Variationsprobleme, Nachrichten der Gesellschaft der Wissenschaften zu Göttingen, 1918, pp. 235 to 257
• James E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge 1990 (Coxeter number and exponents of E8)
• A. Garrett Lisi, An Exceptionally Simple Theory of Everything, 2007, arXiv:0711.0770
• Radu Coldea et al., Quantum Criticality in an Ising Chain, Science 327, 2010, pp. 177 to 180
• Carl Gustav Jung and Wolfgang Pauli, The Interpretation of Nature and the Psyche, German original Zurich 1952
• Max Planck, Das Wesen der Materie (The nature of matter), lecture in Florence 1944, Archive for the History of the Max Planck Society

Further reading

• Wolfgang Pauli and Carl Gustav Jung, Atom and Archetype. The Pauli/Jung Letters 1932 to 1958, ed. C. A. Meier, Princeton 2001
• Carl Gustav Jung, Mandala Symbolism, Princeton 1972