Symmetry applied by Wigner

The concept of symmetry took a great leap when Eugene Wigner showed how mathematical group theory could be used to create the relatively simple structure for what is now called the standard model of physics.

On Wigner’s use of symmetry:

“[Eugene P.] Wigner’s great contribution to science, for which he won the Nobel Prize in Physics in 1963, was his insight into the fundamental mathematics and physics of quantum mechanics. He applied and extended the mathematical theory of groups to the quantum world of the atom; specifically, he used group theory to organize the quantum energy levels of electrons in atoms in a way that is now standard. With that mathematical approach to the atom, Wigner became one of the first to apprehend the deep implications of symmetry, which has since emerged as one, if not the, key principle of 20th-century theoretical physics. …” Eugene Wigner and Fundamental Symmetry Principles

Here is Wigner’s original paper on applying the mathematical notion of a symmetry group to transformations on vectors in abstract Hilbert space.

Wigner : Symmetry principles in old and new physics

Some other reading on Wigner and the application of symmetry to physics:

Wigner’s theorem – Wikipedia, the free encyclopedia
Symmetry in Physics: Wigner’s Legacy

Genesis – evolving structure of the universe

Some background for my science notes:

About eight years ago I wrote a note outlining the basic steps of the evolution of the universe, cosmogenesis it has been called. My main interest is understanding why the ongoing production of variety over the 13 some billion years going from energy to particles to material to life, and whatever other forms of variety that we can know.

        symmetry -> variety

I have surveyed several academic disciplines trying to sort out a grand, metaphysical but scientific view that makes sense across a range of perspectives from physics to biology to computers to psychology and spirituality. This is a summary of my understanding at that time, and it’s still consistent with what I know now.

Symmetry according to Feynman

from Feynman’s “Six not so easy pieces”, p. 24

Each of the basic species of physical particles and forces are exactly the same, no discernible differences between one electron and another, one photon and another.

Playing catch with a ball in a moving vehicle behaves the same as playing catch standing still, unless the vehicle accelerates. Momentum always acts the same while moving or standing still.

physical laws work the same going forward or backward in time.

Physical law symmetries:
-Interchange of identical atoms or identical particles
-Uniform velocity in a straight line (Lorentz transformations)
-Reversal of time
-Quantum-mechanical phase (of the wave function involving amplitude and probability)
-Matter-antimatter (charge conjugation)

Space/time geometric symmetries:
-Translation in space (move to new location)
-Translation in time (happen at a different time)
-Rotation through a fixed angle (turn to a different direction)
-Reflection of space (mirror image)

Not symmetric – change of scale (ratio of wavelength to length fails)

For each of the rules of symmetry there is a corresponding conservation law.
by adding classical mechanics and principles of quantum mechanics you get:

-translation in space <=> momentum is conserved (similarly, rotation – angular momentum)
-translation in time <=> energy is conserved
-quantum mechanical phase <=> electrical charge is conserved

Symmetry according to Weyl

“As far as I see, all a priori statements in physics have their origin in symmetry.”

geometric concept of symmetry:
translation (change position in space), Rotation (rotate around a point), Reflection (inverted)
ornamental (tiling of plane, packing)
crystallographic

general idea: symmetry = invariance of a configuration of elements under a group of automorphic transformations

[automorphic = element-wise transformations which leave all structural relations undisturbed]

“decorate surfaces with flat ornaments”
“arrangements of atoms in a crystal are [solid ornaments]”

“perhaps one should … substitute…” average distribution density for the atom’s exact position when in equilibrium

“The image of an object traced by light of a certain wave length will be fairly accurate only with respect to details of considerably greater dimensions than that wave length, whereas details of smaller dimensions are leveled down.”

wave length of ordinary light is about a thousand time as big as the atomic distances

“If conditions which uniquely determine their effect possess certain symmetries, then the effect will exhibit the same symmetry.” pg. 125. e.g. the pattern of a stretched sheet will reflect the pattern of the applied stress. the shape symmetries follow the action symmetries.

“morphological laws of crystals are today understood in terms of atomic dynamics”

if there is a state of equilibrium then atoms arrange themselves into a regular system of points.

“The visible characteristics of physical objects usually are the results of constitution and environment.”

duality
genotype/phenotype
nature/nurture
discrete/continuous

space has high symmetry because of indiscernible points and directions.
Leibniz: similarity is individual indiscernibility

Helmholtz: congruence is the only objective relation in space

“What we learn from our whole discussion … is this lesson: Whenever you have to do with a structure-endowed entity, sigma, try to determine its group of automorphisms, the group of those element-wise transformations which leave all structural relations undisturbed.”

element
entity with structural relations among elements

Pattern Language in Art

The idea of a pattern language can be used to describe certain regularities in artistic design. As the term has come to be used, a pattern language is about ways to achieve a good design, based on a history of design elements that serve people well, or based on any other way of sorting out effective and valued solutions that have been or need to be invented.