Music of the Spheres:

Symmetry Lost, Found and Hidden

David W. Siegrist, PhD

(Example use of the Schrödinger Equation)

© David W. Siegrist 2011. Material may not be reproduced except with

proper attribution for personal/educational but non-commercial purposes

Ancients thought Numbers were Deeply Important

but were Disappointed

• 20th-century physics began around 600 B.C. when

Pythagoras discovered that the perception of harmony is

connected to numerical ratios.

– Notes sound harmonious when the ratio of the lengths

of string can be expressed in small whole numbers

• 2:1 sounds a musical octave, 3:2 a musical fifth,

and 4:3 a musical fourth.

• “All is Number”- Pythagoras

– Pythagoras' followers were grossly disappointed when the

ratio of radius to circumference of a circle (pi) proved to be

an Irrational number

Following Frank Wilczek, “Numerical Recipe of the Universe” © David W. Siegrist 2011

• Platonic solids are “ideal shapes” showing symmetries.

– The faces of a Platonic solid are congruent regular polygons, with

the same number of faces meeting at each vertex; thus, all their

edges, vertices and angles are congruent.

• There are five Platonic solids (shown above):

• The name of each figure is derived from its number of

faces: respectively 4, 6, 8, 12, and 20.

• The aesthetic beauty and symmetry of the Platonic solids

have made them a favorite subject of geometers for

thousands of years.

– They are named for the ancient Greek philosopher Plato who

theorized that the classical elements (fire, earth, air, water and

quintessence) were constructed from the regular solids.

Source: Wikipedia: Platonic Solids

Tetrahedron Cube Octahedron Dodecahedron Icosahedron

(Animation)

(Animation)

(Animation)

(Animation)

(Animation)

© David W. Siegrist 2011

Galileo Notes Mathematical and Geometric

Nature of the World

• The book of nature lies continuously

open before our eyes (I speak of the

Universe) but it can't be understood

without first learning to understand the

language and characters in which it is

written. It is written in mathematical

language, and its characters are

geometrical figures.

- Galileo Galilei

© David W. Siegrist 2011

Newton and the Rhythm

• Isaac Newton founded modern physics

• He was into dynamics (as opposed to the

Ancients and statics).

– He was able to predict future positions of matter from

the current position and momentum of the object

– Newton (co)invented calculus and featured the

Inverse Square relationship in gravity, for instance

– However, his calculations did not support a

conceptual framework that would explain and predict

size or qualities of matter

– His was not deeply into symmetries

Following Wilczek, “Numerical Recipe of the Universe”

© David W. Siegrist 2011

Johannes Kepler, Early Astronomer.

• Analyzed the motion of planets and

(prematurely) heralded that they represent the

Music of the Spheres

– He thought planets were in concentric circular orbits

at a fixed ratio from the sun

– He later found that orbits are not (completely) circular,

but elliptical and not regularly spaced, which

detracted from his contention

Following Wilczek, “Numerical Recipe of the Universe”

© David W. Siegrist 2011

Dirac and Quantum Mechanics Find Rhythm

• 20th Century: Dirac found harmony in the subatomic space

• He thought electrons circled around nuclei in a regular harmonic fashion

• Einstein responded with great empathy and enthusiasm, referring to quantum mechanics as “the highest form of musicality in the sphere of thought.”

(We now think that electrons are “wavicles” that set up a standing wave around the nucleus, rather than being separate particles in orbits.)

• The mathematics describing the vibratory patterns that define the states of atoms in quantum mechanics is identical to that which describes the resonance of musical instruments. (HT: Pythagoras!)

• The stable states of atoms correspond to pure tones.

Following Wilczek, “Numerical Recipe of the Universe”

© David W. Siegrist 2011

Symmetries and (Numerical) Group Theory

• Physics relies on equations (Schrödinger's, etc)

– It‟s all about matter and motion, following Conservation laws.

• Some equations yield infinities, making them useless

• Equations that feature symmetries (invariances) sometimes

provide results in which infinities “miraculously” cancel out

other infinities, leading to finite (useful) answers

– Such equations are termed “renormalizable.”

• Sets of numbers describing symmetries are “groups”

• A Group can apply to both mathematics and geometry

– One may perform mathematical functions such as the continuous

transformation of a circle by small amounts (Lie groups)

– Mathematically, protons in equations morph (“rotate”) into neutrons

• Groups/Symmetries are Key in modern particle physics

– Group theory is the mathematical approach to symmetry

© David W. Siegrist 2011

Symmetry “Chalk Talk”

• Squares can be rotated by 90; 180 270 and 360 and

look the same. These are „discrete‟ transformations.

• Circles can be rotated an arbitrary amount and look the

same. These are „continuous‟ transformations.

• Spheres can be rotated continuously in 3 dimensions and

remain the same.

• These physical transformations all represent symmetries

– They are invariant under specified mathematical operations also

X

Non-symmetric polygon

© David W. Siegrist 2011

Mathematical Symmetry (from Wolfram MathWorld)

• A property of a mathematical object which causes it to remain invariant under certain classes of transformations (such as rotation, reflection, inversion, or more abstract operations).

• The mathematical study of symmetry is systematized and formalized in the extremely powerful and beautiful area of mathematics called group theory.

• Symmetry can be present in the form of applied mathematical functions and constants, as well as in the physical arrangement of objects.

• In physics, the extremely powerful Noether's symmetry theorem states that each symmetry of a system leads to a physically conserved quantity. (Which leads to forces).

© David W. Siegrist 2011

Mathematical Group Theory

• Any set of objects with an associated rule (operation)

that combines pairs of objects in the set

• Must obey four rules.

– Closed. Product of the group must be member of group

• Whole numbers closed under multiplication, but not division

• Rotating a circle maintains the circle.

– Associative. Numbers could be combined in many ways.

• Not necessarily Commutative. (1+2) +3= 1+(2+3)

– Identity Elements. (I) One process gives the original #.

– Inversion. There must be one reciprocal that yields I.

• Rotate a circle backwards it goes back to the beginning.

• Mathematicians originally pursued Groups as ways to

factor complex physics equations.

– Now they are key to them and to progress in theoretical physics

© David W. Siegrist 2011

Symmetry and Group Theory Glossary

• Rotating a line of fixed length around a fixed point (like a

second hand on a watch) is an Orthogonal transformation

– An Orthogonal transformation in 2 dimensions like this forms a

continuous group called O(2)

– If the radius can‟t jump out from the plane, it is described as a

„Special‟ group, SO(2) in this example.

• Lie Groups describe continuous transformations of circles

– If the example were a sphere, its symmetry would be SO(3)

• It has 3 possible rotational angles.

• Groups that can be transformed by operations in any

particular order are called Abelian (after Niels Abel)

– Groups that must be transformed in order are Non-Abelian.

– Such groups are designated „Unitary” groups U(x)

* *

* *

* * * * * *

* *

Dice are Non-Abelian * *

Lie Group © David W. Siegrist 2011

Spirograph by Kenner

via Wikipedia

1. There is a link between symmetries and conserved quantities. It is

true even in classical mechanics, and also true in quantum

mechanics and field theories.

2. No evidence for violation of energy, momentum, or angular

momentum conservation is seen.

Symmetry Space translation Time translation Rotation

Conserved

quantity

Linear

momentum Energy

Angular

momentum

There are also discrete symmetries associated with reversing the

direction of some quantity. These include:

1. Charge conjugation – changing particles into anti-particles.

2. Parity inversion – reversing the direction of each of the three

spatial coordinates. (Mirror Image). [Broken in Weak Force]

A Conserved Quantity is Associated with Each

Continuous Symmetry Operation: from Molzon

hep.ps.uci.edu/quarknet/lectures/LectureMolzon.ppt © David W. Siegrist 2011

U(1), SU(2) and SU(3) Symmetry Groups

Describe Physical Forces

• A circle, called U(1) by physicists, is the simplest example

of a Lie group. It has a single symmetry: if we rotate a circle,

it remains the same. It applies to electric charge.

– A small rotation like this is called a “generator” of the Lie group.

Following a generator, just like drawing with a compass, takes us

around a circle.

• The Weak Force is described by a 3D Lie group called SU(2).

– Its shape has three symmetry generators, corresponding to the

three weak-force boson particles: W+, W− and Z.

• SU(3) is an eight-dimensional internal space composed of

eight sets of circles twisting around one another in an intricate

pattern, producing interactions among eight kinds of photon-

like particles called gluons. It describes the Strong Force.

(Source: A Geometric Theory of Everything A. Garrett Lisi and James Owen Weatherall Scientific American

303, 54 - 61 (2010) Published online: 17 November 2010 doi:10.1038/scientificamerican1210-54

© David W. Siegrist 2011

SU (3) Symmetry Group and the Strong Force From Wikipedia and James Bottomley and John Baez, 1996.

• Example: “SU(3)” is a symmetry group

that describes the strong force

• The SU(3) mathematical group has 8

parameters in a matrix

• The strong force features 8 particles

• SU(3) is the group of 3 x 3 unitary

matrices with determinant 1.

• Gluons transform in the adjoint

representation of SU(3), which is 8-

dimensional. What this means is that,

the state of a particle is given by a

vector in some vector space on which

elements of SU(3) act as linear (in fact

unitary) operators.

• We say the particle "transforms under

some representation of SU(3)".

SU(3) Matrix by Gell-Man

The relative tendencies of all

flavor transformations are

described by a mathematical

table, called the Cabibbo–

Kobayashi–Maskawa matrix. © David W. Siegrist 2011

Su(3) Patterns of Particle Descriptions from Wikipedia

SU(3) weight diagram A combination of three u, d or s-quarks with a total spin

of 3/2 form the so-called baryon decuplet. The lower

six are hyperons. S = strangeness, Q = electric charge

Mesons of spin 0 form a nonet,

K = kaon, π = pion, η = eta meson

The octet of spin-1/2

baryons. n = neutron,

p = proton, Λ = Lambda

baryon, Σ = Sigma

baryon, Ξ = Xi baryon

• SU(3) allows groups of 8, 10,

• Quarks form Baryons,

including Protons,

Neutrons and Mesons.

• Symmetry group is a little

like Periodic Table

• Gell Mann predicted the Ω

particle and its properties

based on symmetry group

Predictions of Particles in the Standard Model

Based on SU(3)×SU(2)×U(1) Symmetries

• Using the symmetry groups to plug into their equations, physicists were

able to predict several SM particles & their most important properties

– A little like the periodic table of elements

• See Wilczek, Lightness of Being, Chapter 9, pp. 122-127 for detail

• Symmetries, Conservation, and Forces are deeply linked.

• Symmetries may be providing the long-sought „music of the spheres”

© David W. Siegrist 2011

So

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Th

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niv

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e 2

002 G

am

e

Particles, Charges, and Spins Depicted

Geometrically: Electroweak Theory

(Source: A Geometric Theory of Everything A. Garrett Lisi and James Owen Weatherall Scientific American

303, 54 - 61 (2010) Published online: 17 November 2010 doi:10.1038/scientificamerican1210-54

© David W. Siegrist 2011

E8-based

Theory of

Everything:

Ready

for its

Close-up?

(Source: A Geometric Theory of Everything A. Garrett Lisi and James Owen Weatherall Scientific American

303, 54 - 61 (2010) Published online: 17 November 2010 doi:10.1038/scientificamerican1210-54

Animation of E8

from New Scientist

Symmetry: Lost, Found and Hidden

• There is profound evidence of symmetry in the creation

• However, it is complex and not total

– It is not the “Onion-like” symmetry of the Middle Ages, with the earth (and people) in the center, with heaven above and hell below.

– Broken symmetry creates structure, including us.

• Nevertheless, the complex symmetries seem to work together for good to create the world we live in

• Symmetries imply order and beauty, things one might associate with a Creator, as opposed to a strictly random process

– What are the odds that something demonstrating complex mathematical symmetries happened just by accident?

• Symmetries are hidden in greater symmetries. SU(3) in E8?

• Beauty connected to truth? "The Inconceivable Nature of Nature" -

Richard Feynman © David W. Siegrist 2011